On the Electric-Field-Induced Responses of Charged ...digitool.library.mcgill.ca/thesisfile21951.pdf · Colloids in Uncharged Hydrogels ... understanding and support during various

On the Electric-Field-InducedResponses of Charged SphericalColloids in Uncharged Hydrogels

and the Anomalous Bulk Viscosity ofPolymer-Nanocomposite Melts

Mu Wang

Master of Engineering

Department of Chemical Engineering

McGill University

Montreal,Quebec

June 2008

A thesis submitted to McGill University in partial fulfilment of therequirements of the degree of Master of Engineering

c© Mu Wang 2008. All rights reserved.

ABSTRACT

Colloidal particles dispersed in complex fluids such as hydrogels and poly-

mer melts are important because nano-scale inclusions often impart unex-

pected and commercially attractive changes in the dispersed phase. Future

development of these colloidal composites, and diagnostics to characterize

their microstructure, demand a sound understanding of micro-scale dynamics.

Accordingly, this thesis addresses (i) the steady and dynamic electric-field-

induced displacements of spherical colloidal particles embedded in hydrogels,

and (ii) the anomalous viscosity reduction of polymer-nanocomposite melts.

The first problem is undertaken by solving a multi-phase electrokinetic model

that quantifies how the viscoelasticity, compressibility, and hydrodynamic per-

meability of the hydrogel skeleton, and physicochemical properties of the in-

clusions, modulate the particle dynamics and electroacoustic responses. For

the second problem, a hydrodynamic model is developed, and its analytical so-

lution and numerical extension are adopted to interpret recent experiments in

the literature where the bulk viscosity decreases anomalously with increasing

particle volume fraction.

ii

ABREGE

Les particules colloıdales dispersees dans les fluides complexes comme les

hydrogels et des fontes de polymeres sont importantes parce que les inclusions

a nano-echelle repandent souvent des changements inattendus et commerciale-

ment interessants dans la phase dispersee. Les developpements futurs de ces

composites colloıdales et des diagnostiques pour caracteriser leur microstruc-

ture, demande une bonne comprehension de la dynamique a micro-echelle. En

consequence, cette these porte sure (i) la progression reguliere et dynamique

des deplacements de particules colloıdales spheriques embarques dans des hy-

drogels induits par le champ electrique, et (ii) la reduction anormale de la

viscosite des fontes en polymeres nanocomposites. Le premier probleme est

entrepris par la resolution d’un modele electrocinetique a multiple phases qui

quantifie de facon ou la viscoelasticite, de compression, la permeabilite hy-

drodynamiques de squelette d’hydrogel et des proprietes physico-chimiques

des inclusions, et de moduler la dynamique des particules et reponses elec-

troacoustiques. Pour le deuxieme probleme, un modele hydrodynamique est

developpe, sa solution analytique et son extension numerique sont adoptees

pour interpreter les experiences recentes en litterature ou la plus grande vis-

cosite diminue anormalement avec l’augmentation du volume fraction des par-

ticules.

iii

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to Professor R. J. Hill for his

guidance, support and patience during this research project. I was very lucky

to work with him on the exciting subjects of colloid and interface science. His

physical insights and critical thinking abilities for solving complex problems

make every discussion with him very fruitful. Professor Hill is not only a

supervisor, but also a good mentor of life. I am especially grateful for his

understanding and support during various phases of the project.

It is a pleasure to work with members of Hill’s group: Jan van Heiningen,

Aliasghar Mohammadi, Huaiying Zhang, and Savnit Raj, as our delightful

conversations over various topics are not only relaxing, but also enlightening.

I would like to thank my families and my friends for their help, and especially

my parents, for their unconditional love and support.

Finally, I would like to thank the Department of Chemical Engineering,

McGill University, for financial support through the William H. Gauvin Fel-

lowship and an Eugenie Ulmer Lamothe Award.

iv

COPYRIGHT CLEARANCES

I, Reghan J. Hill, hereby give copyright clearance of the following papers,

of which I am a co-author, in the master thesis of Mu Wang:

• Chapter 3: Wang, M. & Hill, R. J. 2008 Electric-field-induced dis-

placement of charged spherical colloids in compressible hydrogels. Soft

Matter 4, 1048–1058.

• Chapter 4: Wang, M. & Hill, R. J. Dynamic electric-field-induced

response of charged spherical colloids in uncharged hydrogels. Submitted.

• Chapter 5: Wang, M. & Hill, R. J. Anomalous bulk viscosity of

polymer-nanocomposite melt. Submitted.

Reghan J. Hill, Associate Professor

Department of Chemical Engineering

McGill University

Montreal, Quebec, Canada

v

RE: Permission Request Form: Mu Wang imap://exchange.mcgill.ca:993/fetch%3EUID%3E/INBOX%3E1997...

1 of 2 2008-6-9 0:42

Subject: RE: Permission Request Form: Mu Wang

From: "CONTRACTS-COPYRIGHT (shared)" <[email protected]>

Date: Tue, 3 Jun 2008 10:09:06 +0100

To: <[email protected]>

Dear Mu Wang The Royal Society of Chemistry (RSC) hereby grants permission for the use of your paper(s) specified below in theprinted and microfilm version of your thesis. You may also make available the PDF version of your paper(s) that theRSC sent to the corresponding author(s) of your paper(s) upon publication of the paper(s) in the following ways: in yourthesis via any website that your university may have for the deposition of theses, via your university’s Intranet or via yourown personal website. We are however unable to grant you permission to include the PDF version of the paper(s) onits own in your institutional repository. The Royal Society of Chemistry is a signatory to the STM Guidelines onPermissions (available on request). Please note that if the material specified below or any part of it appears with credit or acknowledgement to a third partythen you must also secure permission from that third party before reproducing that material. Please ensure that the published article states the following: Reproduced by permission of The Royal Society of Chemistry Regards Gill CockheadContracts & Copyright Executive Gill Cockhead (Mrs), Contracts & Copyright ExecutiveRoyal Society of Chemistry, Thomas Graham HouseScience Park, Milton Road, Cambridge CB4 0WF, UKTel +44 (0) 1223 432134, Fax +44 (0) 1223 423623http://www.rsc.org -----Original Message-----From: [email protected] [mailto:[email protected]] Sent: 30 May 2008 07:16To: CONTRACTS-COPYRIGHT (shared)Subject: Permission Request Form: Mu Wang Name : Mu WangAddress : Department of Chemical EngineeringMcGill UniversityRoom 3060, Wong Building, 3610 University StreetMontreal, Quebec H3A 2B2 Tel : 514-991-6000Fax :Email : [email protected] I am preparing the following work for publication: Article/Chapter Title : Electric-field-induced displacement of charged sphericalcolloids in compressible hydrogelsJournal/Book Title : On the Electric-field-induced Responses of Charged SphericalColloids in Uncharged Hydrogels and the Anomalous Bulk Viscosity ofPolymer-nanocomposite Melts Editor/Author(s) : Mu Wang Publisher : Thesis for Master of Engineering I would very much appreciate your permission to use the following material:

vi

RE: Permission Request Form: Mu Wang imap://exchange.mcgill.ca:993/fetch%3EUID%3E/INBOX%3E1997...

2 of 2 2008-6-9 0:43

Journal/Book Title : Soft MatterEditor/Author(s) : Mu Wang and Reghan J. HillVolume Number : 4Year of Publication : 2008 Description of Material : a research paper titled: Electric-field-induced displacementof charged spherical colloids in compressible hydrogels Page(s) : 1048-1058 Any Additional Comments : I am the first author of the above research paper (Soft Matter, 4, 2008, 1048-1058), andI would like to include a clearly duplicated version (not reprints) of the above paperin my thesis for the degree of Master of Engineering in McGill University, Montreal,Canada. Thank you very much in advance for your kind permission.

DISCLAIMER:

This communication (including any attachments) is intended for the use of the addressee only and may contain confidential, privileged or copyright material. It may not be relied upon or disclosed to any other person without the consent of the RSC. If you have received it in error, please contact usimmediately. Any advice given by the RSC has been carefully formulated but is necessarily based on the information available, and the RSC cannot be held responsible for accuracy or completeness. In this respect, the RSC owes no duty of care and shall not be liable for any resulting damage or loss. The RSC acknowledges that a disclaimer cannot restrict liability at law for personal injury or death arising through a finding of negligence. The RSC does not warrant that its emails or attachments are Virus-free: Please rely on your own screening.

vii

CONTRIBUTION OF AUTHORS

Contents of chapters 3–5 of this thesis are reproduced or adapted from the

papers that have been published or submitted for publication in scientific jour-

nals under the supervision of my research supervisor, Professor R. J. Hill, who

is also a co-author. Chapter 3 discussed the steady electric-field-induced dis-

placement of a colloidal particle in compressible hydrogels, chapter 4 extended

the steady displacement to the dynamic responses and connected the single

particle response to the bulk electroacoustic signals, and chapter 5 theoreti-

cally interpreted the recently discovered bulk viscosity reduction in polymer-

nanocomposite melts.

The research project was initiated by Professor Hill. In chapter 3, using a

displacement construction proposed by Professor Hill, I devised the computa-

tional methodology and boundary layer solution for the steady displacement,

and highlighted the importance of compressibility. In chapter 4, I numerically

calculated dynamic particle displacement and derived a boundary layer ap-

proximation. To overcome numerical difficulties, I also analytically solved a

so-called two-fluid model, and performed a far-field asymptotic analysis on the

governing equations. Furthermore, I have theoretically shown that electric mi-

crorheology and electroacoustic diagnostics can be applied to hydrogel-colloid

composites at low and high frequencies, respectively. In both chapters, I devel-

oped robust computer programs in C and FORTRAN to compute the steady

and dynamic particle responses. In chapter 5, I proposed that the bulk vis-

cosity reduction may arise from Rouse dynamics in entangled melts at small

scales. Professor Hill and I derived a hydrodynamic model to verify this idea,

and I have showed that the layer thickness is qualitatively independent of the

continuous layer profile.

viii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ABREGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . iv

COPYRIGHT CLEARANCES . . . . . . . . . . . . . . . . . . . . . . v

CONTRIBUTION OF AUTHORS . . . . . . . . . . . . . . . . . . . . viii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Thesis motivation . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives of the thesis . . . . . . . . . . . . . . . . . . . . 21.3 Thesis organization . . . . . . . . . . . . . . . . . . . . . . 4

2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Theoretical models for hydrogels . . . . . . . . . . . . . . . 52.2 Theoretical development of electroacoustics . . . . . . . . . 62.3 Numerical solutions of electrokinetic models . . . . . . . . 7

3 Electric-field-induced displacement of charged spherical colloids incompressible hydrogels1 . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Theoretical model and solution . . . . . . . . . . . . . . . 15

3.2.1 Coupled electrokinetic transport and elastic deforma-tion model . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 Solution methodology . . . . . . . . . . . . . . . . . 173.2.3 Force evaluation and inclusion displacement . . . . . 20

3.3 Boundary-layer analysis for κa 1, ` a and |ζ| < kT/e 213.3.1 Outer solution . . . . . . . . . . . . . . . . . . . . . 223.3.2 Inner solution and matching . . . . . . . . . . . . . 24

3.4 Numerically exact results . . . . . . . . . . . . . . . . . . . 273.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Dynamic electric-field-induced response of charged spherical col-loids in uncharged hydrogels . . . . . . . . . . . . . . . . . . . . 38

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 39

ix

4.2 Two-fluid model and response for uncharged colloids . . . . 454.2.1 Fluid velocity and polymer displacement fields . . . 464.2.2 Force and response function . . . . . . . . . . . . . 49

4.3 Multi-phase electrokinetic model . . . . . . . . . . . . . . . 504.3.1 Governing equations and boundary conditions . . . 514.3.2 Solution methodology . . . . . . . . . . . . . . . . . 534.3.3 Simplification for incompressible hydrogels . . . . . 564.3.4 Force and dynamic electrokinetic response . . . . . . 57

4.4 Far-field asymptotic analysis . . . . . . . . . . . . . . . . . 604.4.1 Far-field decays of ψ and nj . . . . . . . . . . . . . 614.4.2 Far-field decays of f,r, g1 and g2 . . . . . . . . . . . 634.4.3 Far-field analysis for incompressible hydrogels . . . . 64

4.5 Connection to electroacoustics . . . . . . . . . . . . . . . . 664.6 High-frequency boundary-layer approximation . . . . . . . 704.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.7.1 Response functions for an uncharged particle . . . . 764.7.2 Numerical solution of the multi-phase electrokinetic

model . . . . . . . . . . . . . . . . . . . . . . . . 794.7.3 High-frequency boundary-layer approximation and ap-

plication to electroacoustics . . . . . . . . . . . . 864.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.A Point-force representation of a particle in an uncharged hy-

drogel matrix . . . . . . . . . . . . . . . . . . . . . . . . 944.B Numerical solution of the field equations . . . . . . . . . . 97

5 Anomalous bulk viscosity of polymer-nanocomposite melt . . . . 101

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3 Intrinsic viscosity from the single-layer model . . . . . . . . 1125.4 Theoretical interpretation of experiments . . . . . . . . . . 1155.5 Summary and conclusions . . . . . . . . . . . . . . . . . . 121

6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . 122

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

x

LIST OF TABLESTable page

3–1 Poisson’s ratios of selected hydrogels ascertained from experi-ments under undrained and drained conditions . . . . . . . . 13

4–1 Parameters for the results shown in figure 4–1. . . . . . . . . . 76

5–1 Summary of the parameters that characterize the experimentsof Mackay et al. (2003) and Tuteja et al. (2005) with Rg/h &1 and φ = 0.005, and theoretical interpretations (providingfitted values for δ) based on (5.8) with χ = 0 . . . . . . . . . 118

5–2 Best-fit polymer correlation lengths ξ ascertained from experi-ments of Mackay et al. (2003) and Tuteja et al. (2005) and thetheoretical interpretation based on a continuous-layer-profilemodel with ka = 0 . . . . . . . . . . . . . . . . . . . . . . . . 120

xi

LIST OF FIGURESFigure page

3–1 Streamlines, polymer and particle displacement, and electro-static potential isocontours . . . . . . . . . . . . . . . . . . 29

3–2 The ratio Z/E as a function of Poisson’s ratio . . . . . . . . . 31

3–3 The ratio Z/E as a function of the scaled reciprocal doublelayer-layer thickness . . . . . . . . . . . . . . . . . . . . . . 33

3–4 The scaled displacement as a function of the scaled ζ-potential 35

3–5 The ratio Z/E as a function of scaled Brinkman screening length 36

4–1 Response function α(ω) as a function of angular frequency ω . 77

4–2 Representative frequency spectrum of Z/E . . . . . . . . . . . 81

4–3 Comparison of compressible hydrogel Z/E with the classicalNewtonian response (Z/E)∗ = −µd∗/(iω) . . . . . . . . . . . 82

4–4 Frequency spectra of Z/E for various Poisson ratios . . . . . . 83

4–5 Frequency spectra of Z/E for various Young’s moduli . . . . . 84

4–6 Frequency spectra of Z/E for various scaled ζ-potentials . . . 85

4–7 Frequency spectra of Z/E for various Brinkman screening lengthswith κa = 1 and 10 . . . . . . . . . . . . . . . . . . . . . . . 87

4–8 Frequency spectra of Z/E for various Brinkman screening lengthswith κa = 100 and 1000 . . . . . . . . . . . . . . . . . . . . 88

4–9 Frequency spectra of the dynamic electrophoretic mobility µd =−iωZ/E for various Young’s moduli . . . . . . . . . . . . . 90

4–10 Frequency spectra of the dynamic electrophoretic mobility µd =−iωZ/E for various scaled ζ-potentials . . . . . . . . . . . . 92

5–1 Intrinsic viscosity as a function of the scaled layer thickness δ/afor χ ≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5–2 Intrinsic viscosity as a function of the scaled layer thickness δ/afor χ ≥ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5–3 Intrinsic viscosity as a function of the parameter χ . . . . . . . 115

5–4 Intrinsic viscosity as a function of the scaled reciprocal slippinglength ka . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xii

CHAPTER 1Introduction

1.1 Thesis motivation

Colloidal dispersions are formed by distributing a discrete phase into a

continuous, immiscible phase (Cosgrove 2005). The discrete phase is usually

referred to as the disperse phase or colloid, and the continuous phase as the

dispersing medium (Hunter 2001). The size of the disperse phase, ranging

from several nanometers to a few micrometers, distinguishes colloidal disper-

sions from solutions where the kinetic units of the solute and the solvent are

similar in size. Colloidal dispersions encompass a broad scope of materials,

e.g., smoke (aerosols), milk (emulsions), printing inks, and biological cells

(dispersions) (Hunter 2001; Cosgrove 2005), and they are crucial for many in-

dustrial processes, for example, oil recovery, ceramic processing, and mineral

processing, to name a few (Cosgrove 2005).

Colloid and interface science traditionally focuses on aqueous dispersions

due to their simplicity and practical importance, and even these relatively

“simple” systems exhibit extremely rich mechanical, rheological, optical, and

electrical behaviors (Hunter 2001; Lyklema 1995; Cosgrove 2005). Recently,

dispersing colloidal particles in complex fluids such as hydrogels, polymer solu-

tions, and polymer melts forms novel colloidal dispersions, also termed colloid

composites. In these systems, colloidal particles can serve as probes to char-

acterize the mechanical, rheological, and physicochemical properties of the

base material (Lin et al. 2005; Schnurr et al. 1997; Hunter 1998), and can

also introduce new dynamics and enhancements, such as light-wave-sensitive

swelling (Sershen et al. 2005) and enhanced mechanical and optical proper-

ties (Haraguchi & Takehisa 2002; Haraguchi et al. 2002). Further development

of colloid composites requires the following questions to be answered: (i) How

1

does the continuous phase, e.g., hydrogel, influence the dynamics of colloidal

particles when subjected to external fields, e.g., electrical, optical, and mag-

netic fields? (ii) How do the colloidal particles affect the microstructure, and,

consequently, how do microstructural changes affect bulk properties of the

composite? These questions are extremely challenging to answer, since they

cover a broad range of topics in colloid and interface science.

Motivated by these questions, this thesis seeks answers to the foregoing

questions by focusing on the interactions between the continuous and dispersed

phases, and neglecting the particle-particle interactions. To make the project

more tractable, two representative systems with appealing characteristics are

selected. For question (i), hydrogel-colloid composites, i.e., colloidal particles

dispersed in water-saturated swollen polymer networks, are chosen as a model

system. The single particle and collective bulk responses of the composite

to electric fields are investigated in detail using a multi-phase electrokinetic

model. These investigations not only reveal the influences of the continuous

phase, but also serve as a rigorous theoretical foundation for novel electric-

field-based characterization techniques for hydrogel-colloid composites. For

question (ii), colloidal dispersions in polymer melts are selected (Buscall &

Ettelaie 2006), as changes in polymer configurations can significantly affect

the melt bulk properties (de Gennes 1979). Using a hydrodynamic model,

this project interprets the anomalous bulk viscosity reduction recently discov-

ered in polymer-nanocomposite melts, i.e., nanoparticles dispersed in polymer

melts (Mackay et al. 2003; Tuteja et al. 2005). The theory shows how the

nanoparticle-induced changes in the polymer microstructure affect the bulk

viscosity of the composite.

1.2 Objectives of the thesis

As evident from § 1.1, the overall objective of this project is to understand

how electric-field-induced colloid dynamics are affected by the viscoelastic-

ity of hydrogels, and how nanoparticle-induced microstructural changes influ-

ence the bulk viscosity of the polymer-nanocomposite melts. It is important

2

to note that the electric-field-induced displacement of a spherical colloidal

particle embedded in incompressible hydrogels has been addressed by Hill &

Ostoja-Starzewski (2008) using a special case of the multi-phase electrokinetic

model. Their work serves as an important first step for this and future in-

vestigations. In this project, the multi-phase electrokinetic model of Hill &

Ostoja-Starzewski (2008) is adopted to describe the fluid velocity, polymer

displacement, electrostatic potential and ionic concentrations in hydrogels.

The first objective, as a direct extension of the theory by Hill & Ostoja-

Starzewski (2008), is to investigate the effect of hydrogel compressibility, as

characterized by the Poisson ratio, on the steady particle displacement. The

steady multi-phase electrokinetic model is solved numerically, and an analyt-

ical approximation is also derived. Similarly to the Smoluchowski formula for

electrophoretic mobility (Hunter 2001), the analytical approximation can be

extremely useful for interpreting microrheological experiments that measure

the quasi-steady particle displacement.

Extending the steady electric-field-induced responses to the dynamic re-

sponses in hydrogel-colloid composites is the second objective of this project.

This is also crucial for developing new electric-field-based characterization

techniques—including electrophoretic microrheology (Mizuno et al. 2000, 2001)

and electroacoustics (Hunter 1998)—for these materials. Note that the dy-

namic multi-phase electrokinetic model is an augmentation of a two-fluid

model currently used in microrheology, where only approximate solutions are

available (Levine & Lubensky 2000, 2001). Therefore, the two-fluid model

is first solved exactly, and the results are compared with several approxima-

tions. Spectra of electric-field-induced particle responses can be obtained by

numerically solving the full multi-phase electrokinetic model for compress-

ible and incompressible hydrogels. An analytical approximation, which can

be valuable for designing and interpreting experiments, is also derived. The

connection between the single particle dynamic response and electroacoustic

signals is also established.

3

The last objective of this project is to understand how nanoparticle-

induced microstructural changes affect the bulk viscosity of polymer melts.

Specifically, the contribution of a nanoparticle encapsulated by a layer of dif-

ferent properties to the bulk viscosity is derived and examined thoroughly

using a hydrodynamic model. The model is then compared with the available

experiments of Mackay et al. (2003) and Tuteja et al. (2005) to help elucidate

the physical origin of the bulk viscosity reduction.

1.3 Thesis organization

This thesis is arranged as a collection of manuscripts published and sub-

mitted to meet the objectives presented in § 1.2. Since each manuscript con-

tains an exhaustive literature review in its introduction, the literature reviews

presented in chapter 2 serve as a brief complement to the manuscripts. Chap-

ters 3–5 are the main body of the thesis, organized in the form of manuscripts.

The steady response of a spherical colloidal particle embedded in a compress-

ible hydrogel matrix is presented in chapter 3. Numerically exact solutions and

an analytical approximation of the steady multi-phase electrokinetic model are

presented. Chapter 4 addresses the dynamic response of hydrogel-colloid com-

posites. The two-fluid model of Levine & Lubensky (2001) is first solved ana-

lytically, and it is the basis for the dynamic multi-phase electrokinetic model,

which is solved numerically. A high-frequency analytical approximation is also

derived. In addition, the connection between electroacoustic signals and the

single particle response is established. In chapter 5, a hydrodynamic model

that incorporates nanoparticle-induced microstructural changes is presented

to interpret the experimentally observed bulk viscosity reduction in polymer-

nanocomposite melts. Comparison between the model and the experimental

data reveals the physical origin of the bulk viscosity reduction. Chapter 6

provides a brief summary and conclusions.

4

CHAPTER 2Literature Review

Since the introductions of chapters 3–5 provide exhaustive literature re-

views on their respective subjects, this chapter serves as a complement to these

reviews, and presents additional important information.

2.1 Theoretical models for hydrogels

Hydrogels are an important class of complex fluids that exhibit viscoelas-

ticity depending on the characteristic time scales, i.e., if the time scales are

long, they behave as elastic solids, and if the time scales are short, they behave

as viscous fluids. Their viscoelasticity arises from their composite nature: the

fluid contributes to the viscosity, and the elasticity comes from the polymer

network (Tanaka et al. 1973). Modeling the dynamics of hydrogels is crucial

for understanding the viscoelastic response of inclusions in the hydrogels, as

well as for developing acoustical diagnostic techniques (Snieder & Page 2007).

The viscoelasticity of hydrogels was first modeled as a complex, frequency-

dependent modulus or viscosity. Using this idea, Oestreicher (1951) derived

the stress-strain relation and equations of motion for general viscoelastic me-

dia, including hydrogels. He also calculated the force on an oscillating sphere

in a viscoelastic medium using the method of Lamb (1945), and the results

agreed well with experimental data. However, this model is phenomenological,

and does not account for the fluid-network interactions. In the more recent

two-fluid model of Levine & Lubensky (2001), the viscoelasticity of hydrogels

arises from the hydrodynamic coupling of the viscous and elastic phases. The

exact solution of the two-fluid model, which is the foundation of the multi-

phase electrokinetic model, is not as straighforward as that of Oestreicher

(1951), and Levine & Lubensky (2001) derived only an approximation for the

force on an oscillating sphere.

5

The exact solution of the two-fluid model is largely inspired by the field of

poroelasticity, which studies the dynamics of porous, liquid-containing rocks

in the Earth’s crust (Frenkel 1944; Biot 1941; Coussy 2004). Although rocks

are quite different from hydrogels, their theoretical treatment, as evidenced

from the poroelasticity constitutive equations (Biot 1941, 1956a,b), is mathe-

matically similar to the two-fluid model. Noteworthy is the theory by Markov

(2005), where he analytically solved the wave propagation problem in a fluid-

saturated porous medium with spherical inclusions. In his work, the consti-

tutive equations of poroelasticity are solved by constructing scalar and vector

potential solutions for the fluid and elastic skeleton displacements based on

plane-wave propagations in porous medium (Biot 1956a,b). The same pro-

cedure can be applied to the hydrogel two-fluid model for the fluid velocity

and polymer displacement. Note that the waves propagate differently: in the

two-fluid model, there are two propagating shear waves and one compressional

wave, and the fluid is incompressible; whereas in poroelasticity, there are two

propagating compressional waves and one shear wave, and both the elastic and

viscous media are considered compressible. These differences will modify the

construction of Markov (2005) accordingly, but the underlying ideas remain

the same.

The above models are restricted to linear responses of statistically homo-

geneous media, and do not consider non-linear effects due to large deforma-

tions, or local inhomogeneity (Fung & Tong 2001). Moreover, the assumption

of an incompressible fluid in the hydrogel two-fluid model is valid when the

fluid wavelength is much longer than the characteristic length scale. For a

typical speed of sound in water ∼ 103 m s−1 and colloidal particles of sizes

∼ 10−6 m, the incompressible assumption is valid at frequencies less than

∼ 1 GHz.

2.2 Theoretical development of electroacoustics

In this thesis, electroacoustics, first introduced by Debye (1933), is pro-

posed to characterize hydrogel-colloid composites at ultrasonic frequencies.

6

An electroacoustic response arises from the surface charge of colloidal particles

in the medium, which produces an electric field when applying a sound wave,

i.e., the colloidal vibration potential/current (CVP/CVI), and pressure distur-

bances when subjected to external electric fields, i.e., the electrokinetic sonic

amplitude (ESA). In chapter 4, the modern theory of O’Brien (1988, 1990) for

the electroacoustics of Newtonian suspensions is extended to hydrogel-colloid

composites. The theory of O’Brien (1988, 1990) connects the micro-scale single

particle response to macro-scale electroacoustic signals of the composite.

There is another class of theories of suspension electroacoustics primar-

ily focused on CVP/CVI phenomena. These were first published by Enderby

(1951) and Booth & Enderby (1952). Unlike O’Brien’s approach, these au-

thors directly connected the CVI/CVP signals of suspensions to the surface

physicochemical properties of colloidal particles. However, the theories of En-

derby (1951) and Booth & Enderby (1952) are restricted to low frequencies,

and are not applicable to modern ultrasonic electroacoustic techniques. The

high frequency theory, valid over a wide range of particle concentrations, was

developed by Dukhin et al. (2000, 1999a,b) using a coupled-phase model and

cell boundary conditions for non-conducting colloidal suspensions. Together

with the computation of the electrophoretic mobilities for concentrated sys-

tems (Rider & O’Brien 1993; O’Brien et al. 2003), the theory of O’Brien (1988,

1990) is equivalent to the theory of Dukhin et al. (2000, 1999a,b). Note that

both classes of experiments with complementary theories have been success-

fully commercialized (Hunter 1998; Dukhin & Goetz 2002).

This thesis adopts the methodology of O’Brien (1988, 1990) for the elec-

troacoustic response of dilute hydrogel-colloid composites. The approach of

Dukhin et al. (2000, 1999a,b), although valid for both dilute and concentrated

systems, is complicated and beyond the scope of the present project.

2.3 Numerical solutions of electrokinetic models

The multi-phase electrokinetic model adopted in this thesis is an extension

of the standard electrokinetic model first developed by Overbeek (1943). The

7

standard electrokinetic model describes a wide range of colloidal dynamics,

ranging from electrophoretic mobilities to dielectric responses (Hunter 2001).

O’Brien & White (1978) presented the first numerical solution of the full steady

model over the entire experimental accessible parameter space by linearly per-

turbing an equilibrium base state. They decomposed the non-linear problem

to a non-linear equilibrium base state governed by the Poisson-Boltzmann

equation, and a linearly perturbed state. Using far-field asymptotic analysis,

the model was solved using a technique analogous to the multiple shooting

method (Ascher et al. 1988). Using a very similar approach, the dynamic

model was solved by DeLacey & White (1981) for the dielectric response of

dilute colloidal suspensions. The results are only valid at low frequencies,

since fluid inertia is neglected. Mangelsdorf & White (1992) first solved the

full dynamic problem by removing the numerical difficulties through a careful

reformulation of the differential equations. As a result, they calculated the

dynamic electrophoretic mobility up to several MHz. The same methodology

was later applied to determine the dielectric response of dilute colloidal sus-

pensions (Mangelsdorf & White 1997). A very powerful method developed by

Preston et al. (2005) using a general-purpose boundary value problem soft-

ware package COLSYS (Ascher et al. 1988) can compute the dynamic elec-

trophoretic mobility over a wide range of frequencies, from several Hz to GHz.

Hill et al. (2003a) developed the software package MPEK to calculate the

electrophoretic mobility of polymer-coated colloidal particles using a modified

electrokinetic model. MPEK also calculates the high-frequency polarizabil-

ity of dilute colloidal suspensions up to several GHz (Hill et al. 2003b). Fi-

nally, based on the MPEK package, Hill & Ostoja-Starzewski (2008) solved the

steady multi-phase electrokinetic model for incompressible hydrogels. Their

model is the basis for the present study.

Evidently, numerical solution of electrokinetic models can benefit signif-

icantly from the development of powerful boundary value problem solvers

8

with automatic mesh adjustments, such as COLSYS, COLNEW, and TW-

PBVPL (Ascher et al. 1988; Cash & Mazzia 2006). Also, far-field asymptotic

analysis can greatly improve the accuracy and stability of the numerical ap-

proach. In this thesis, the solution of the Poisson-Boltzmann equation is based

on the methodology of MPEK, and the perturbed multi-phase electrokinetic

model is solved using TWPBVPL, a general-purpose boundary value problem

solver (Cash & Mazzia 2006). To help improve the computational accuracy,

asymptotic analysis of the governing equations is undertaken.

9

CHAPTER 3Electric-field-induced displacement of charged spherical colloids in

compressible hydrogels1

This chapter concerns the electric-field-induced displacement of a charged

spherical colloid embedded in an uncharged compressible hydrogel. Previ-

ous theoretical calculations for incompressible polymer skeletons predict sub-

nanometer particle displacements within the experimentally accessible param-

eter space (e.g., particle surface charge density, polymer shear modulus, and

electric field strength). Accordingly, the prevailing expectation is that an ex-

perimental test of the theory would be extraordinarily difficult. In this work,

however, we solved the electrokinetic model for compressible polymer skele-

tons with arbitrary Poisson’s ratio. The most striking result, obtained from

numerically exact solutions of the full model and an analytical boundary-layer

approximation, is that polymer compressibility admits particle displacements

that increase linearly with particle size when the radius is greater than the

Debye length. This scaling is qualitatively different than previously obtained

for incompressible skeletons, where the ratio of the particle displacement to

the electric field approaches a particle-size-independent constant. The dis-

placement is also much more sensitive to the hydrodynamic permeability of

the polymer skeleton. Therefore, when compressible hydrogels are deformed

at frequencies below their reciprocal draining time, our theory identifies the

parameter space where displacements could be registered using optical mi-

croscopy. In turn, this will help to establish a quantitative connection between

the electric-field-induced particle displacement and physicochemical character-

istics of the particle-polymer interface.

1 Reproduced by permission of The Royal Society of Chemistry

10

3.1 Introduction

Hydrogels are polymer networks that have found widespread use in tis-

sue engineering (Barndl et al. 2007), drug delivery (Qiu & Park 2001), and

molecular separations, e.g., gel-electrophoresis, isoelectric focusing, and iso-

tachophoresis (Westermeier 2005). The networks are often synthesized from

polymers such as poly(methyl methacrylate) (PMMA), poly(vinyl alcohol)

(PVA), and polyacrylamide (PA); as well as from macromolecules of biologi-

cal origin, such as collagen and agar.

Recently, several novel applications of hydrogel nano-composites have

been demonstrated where organic and inorganic nanoparticles are immobi-

lized in otherwise conventional hydrogel matrices. For example, wavelength-

selective light-induced swelling from gold and gold-coated silica nanoparticle

inclusions makes these intriguing materials useful as light-activated microflu-

idic valves (Sershen et al. 2005). Nanoparticles have also been introduced

into soft biological tissues to increase the sensitivity of ultrasound imaging for

early tumor detection (Liu et al. 2006; Dayton & Ferrara 2002), and to adsorb

optical energy for treating certain cancers (Loo et al. 2005). Larger colloidal

inclusions have been used to probe the local and bulk viscoelastic response

of polymer solutions and gels (Schnurr et al. 1997; MacKintosh & Schmidt

1999). Finally, in an effort to control the otherwise diffusion-limited transfer

of uncharged molecules across membranes in biosensing, silica nanoparticles

have been embedded in uncharged hydrogel gels to produce electroosmotic

flow (Matos et al. 2006).

As a first step toward understanding the coupling of electroosmotic flow

and polymer deformation in hydrogel composites, Hill & Ostoja-Starzewski

(2008) calculated the electric-field-induced displacement of particles embed-

ded in incompressible polymer skeletons. Their work demonstrates that a

simple balance between the bare Coulomb force and an elastic restoring force

on the particles prevails only when the particle radius a is smaller than the

11

Debye screening length κ−1. Otherwise, the theory quantifies how electroos-

motic flow—in the diffuse layer of countercharge that envelops each particle—

deforms the polymer skeleton and, therefore, influences the particle displace-

ment. For incompressible skeletons, the ratio of the particle displacement to

the electric field strength bears a striking resemblance to the electrophoretic

mobility (O’Brien & White 1978) at all values of κa. Accordingly, the electric-

field-induced particle displacement reflects the size and charge of the inclu-

sions, the viscosity and concentration of the electrolyte, and the shear modu-

lus and hydrodynamic permeability of the polymer skeleton. In principle, the

electric-field-induced displacement is an appealing diagnostic for probing the

physicochemical characteristics of the particle-polymer interface, in a similar

way that electrophoresis is routinely used to ascertain the surface charge of

colloidal particles dispersed in Newtonian electrolytes. However, in the exper-

imentally accessible parameter space, the particle displacements predicted by

Hill & Ostoja-Starzewski (2008) are extraordinarily small, making it difficult

to envision practical diagnostic applications.

In the absence of electroosmotic flow, the elastic restoring force of the

polymer skeleton varies by up to 25 percent over the experimentally accessible

range of Poisson’s ratios for hydrogels (Schnurr et al. 1997). Therefore, when

κa 1, the electric-field-induced displacement of a particle in a compressible

polymer network increases by only 25 percent over the value for an incom-

pressible skeleton. It is therefore unlikely that finite compressibility would

significantly influence the sensitivity of an experiment to test Hill and Ostoja-

Starzewski’s theory when κa 1. However, the situation when κa & 1, which

is generally achieved for particles larger than about one micron, is not as

straightforward to interpret. In this chapter, we show that the electric-field-

induced particle displacement of sufficiently large particles in compressible

matrices is qualitatively different than in incompressible skeletons. Rather

than tending to a size-independent value, the particle displacement increases

linearly with particle size when κa 1. For particles whose radius is greater

12

Table 3–1: Poisson’s ratios of selected hydrogels ascertained from experimentsunder undrained and drained conditions. Note that HEMA-AA represents2-hydroxyethyl methacrylate (HEMA) acrylic acid (AA) co-monomer gel.

Class I (undrained)Hydrogel νHEMA-AA (Johnson et al. 2004a,b) 0.42 – 0.45polyacrylamide (Boudou et al. 2006) 0.487± 0.013polyacrylamide (Takigawa et al. 1996) 0.457± 0.011polyacrylamide (Engler et al. 2004) 0.4 – 0.45poly(vinyl alcohol) (Urayama et al. 1993) 0.433

Class II (drained)Hydrogel νagarose gel (Freeman et al. 1994) 0.15± 0.09resorcinol-formaldehyde (Gross et al. 1997) 0.124 – 0.233silica gel (Scherer 1992) 0.216 – 0.244polyacrylamide (Li et al. 1993) 0.24 – 0.36polyacrylamide (Geissler & Hecht 1980, 1981) 0 – 0.25

than about one micron, our theory predicts displacements of tens to hundreds

of nanometers with modest electric-field strengths and electrolyte concentra-

tions.

While the thermodynamically admissible range for Poisson’s ratio is from

−1 to 0.5, Geissler & Hecht (1980, 1981) established Poisson’s ratio’s of 0 and

0.25 for polymer skeletons in poor and good solvents, respectively. This range

is corroborated to some extent by experiments, but only after two classes of

experiments are identified. As summarized in table 3–1, class I experiments,

which often involve measurements of strain immediately after the initial de-

formation, or with boundary conditions that prevent draining, yield Poisson’s

ratios greater than about 0.4. These experiments reflect the incompressibil-

ity of the solvent. In contrast, the Poisson’s ratios from class II experiments,

where the polymer is permitted to drain, are often in the range 0–0.25 pre-

dicted by Geissler & Hecht (1980, 1981).

The electrokinetic (multi-phase) model of Hill & Ostoja-Starzewski (2008)

generalizes a bi-phasic model (polymer and solvent) where the solvent is hy-

drodynamically coupled to a linearly elastic polymer skeleton. The bi-phasic

13

model can be traced to early works of Biot (1941) and Frenkel (1944) per-

taining, respectively, to the consolidation and seismoelectric behavior of soils,

as well as the propagation of sound waves in geological exploration (Biot

1956a,b). More recently, the bi-phasic model—also termed a two-fluid model—

has been adopted in the relatively new field of microrheology to interpret the

dynamics of entangled polymer solutions and gels (Brochard & de Gennes

1977; de Gennes 1976a; Milner 1993; Barrire & Leibler 2003; Levine & Luben-

sky 2001; Cicuta & Donald 2007).

Central to microrheology (Valentine et al. 1996; Lin et al. 2005; Schnurr

et al. 1997; Ziemann et al. 1994; Mason & Weitz 1995; MacKintosh & Schmidt

1999) are the static and dynamic susceptibilities of a colloidal sphere embedded

in a fluid-saturated polymer network. Early applications of the two-fluid model

addressed dynamics in which the polymer skeleton and fluid are hydrodynami-

cally coupled to yield divergence-free (incompressible) states of strain (Schnurr

et al. 1997). However, as first identified by Tanaka et al. (1973), the apparent

compressibility of a fluid-saturated network depends, in part, on the deforma-

tion time scale. If this is shorter than the characteristic draining time, which

depends on the elastic modulus, characteristic length, and hydrodynamic per-

meability, then the polymer skeleton and incompressible solvent are coupled

as a single phase. Consequently, at sufficiently high frequencies, the polymer

skeleton adopts non-equilibrium, divergence-free states of strain. On longer

time scales (or at lower frequencies), however, draining permits the displace-

ment field of a compressible skeleton to adopt non-zero divergence. Under

these conditions, the particle dynamics also reflect the compressibility and

hydrodynamic permeability of the polymer, similarly to the class II experi-

ments presented in table 3–1. Levine & Lubensky (2001) derived closed-form

approximations for the dynamic susceptibility and, accordingly, their theory

provides a basis for interpreting particle dynamics in the absence of electroki-

netic influences.

14

Returning to the electric-field-induced particle displacement considered by

Hill & Ostoja-Starzewski (2008), their theory for incompressible skeletons also

applies to compressible polymer networks when the frequency of an oscillatory

electric field is higher than the reciprocal draining time. Therefore, our effort

to solve the model for compressible skeletons (arbitrary Poisson’s ratio) also

makes an important step toward a theory for dynamics at frequencies below the

reciprocal draining time. Such a theory would facilitate an interpretation of

electro-acoustic phenomena in hydrogel nanocomposites, in a similar way that

acoustic spectroscopy has recently been adopted to probe the microstructure

of model food gels (Strubulevych et al. 2007), for example.

This chapter is set out as follows. Section 3.2 presents the model and

methodology for calculating the particle displacement from the polymer dis-

placement and fluid velocity fields. A boundary-layer analysis is undertaken in

§ 3.3 to verify the numerics and obtain a convenient closed-form expression for

the particle displacement when the Debye and Brinkman screening lengths are

both small compared to the particle radius. Numerical and analytical solutions

of the model are compared in § 3.4, where the separate influences of Poisson’s

ratio and several other important parameters on the particle displacement are

investigated; this section also highlights the significant influence of polymer

compressibility and hydrodynamic permeability. This chapter concludes with

a brief summary in § 3.5.

3.2 Theoretical model and solution

A charged spherical colloid with radius a and surface charge density σ is

embedded in an unbounded, uncharged hydrogel matrix with Darcy permeabil-

ity `2 (` is the Brinkman screening length), Young’s modulus E , and Poisson’s

ratio ν. The hydrogel is saturated with an aqueous electrolyte (e.g., NaCl)

whose concentration determines the Debye screening length κ−1. The prime

objective of the following theory is to determine the particle displacement Z

when the particle and gel are placed in a uniform electric field E.

15

First, consider the situation where the electrolyte concentration is so low

that the only forces acting on the colloid are the bare Coulomb force f e,E =

σ4πa2E and the elastic restoring force (Hill & Ostoja-Starzewski 2008)

fm,Z = − 2πaE(1− ν)

(5/6− ν)(1 + ν)Z. (3.1)

Balancing these forces to achieve static equilibrium (fm,Z = −f e,E) gives

Z =2ζεoεs(5/6− ν)(1 + ν)

E(1− ν)E as κa→ 0. (3.2)

Note that the surface charge density has been written in terms of the surface

potential ζ = σa/(εoεs) appropriate when κa → 0. Here, εs and εo are, re-

spectively, the dielectric constant of the solvent (water) and permittivity of

a vacuum. Equation (3.2) demonstrates that a Poisson’s ratio less than 0.5

increases the particle displacement by up to 25 percent in the range appro-

priate for hydrogel skeletons (0 < ν < 0.5). This is due to the vanishing

electroosmotic flow as κa→ 0. To capture the influence of flow on the parti-

cle displacement when κa & 1, the full electrokinetic and elastic deformation

model must be solved. This requires calculating the electroosmotic flow and

the degree to which it distorts the (compressible) polymer skeleton.

3.2.1 Coupled electrokinetic transport and elastic deformation model

The full set of equations involving N electrolyte ions is

εoεs∇2ψ = −N∑j=1

njzje (3.3)

jj = −Dj∇nj − zjeDj

kTnj∇ψ + nju (3.4)

η∇2u−∇p = (η/`2)u+N∑j=1

njzje∇ψ (3.5)

µ∇2v + (λ+ µ)∇(∇ · v) = −(η/`2)u (3.6)

with ion and electrolyte conservation equations

∇ · jj = 0 and ∇ · u = 0. (3.7)

16

Here, ψ, p, u and v denote the electrostatic potential, fluid pressure, fluid

velocity, and polymer displacement; and nj and jj denote the concentration

and flux of the jth electrolyte species. Moreover, η is the shear viscosity of

the fluid; and zj and Dj are the valence and diffusion coefficient of the jth

electrolyte species. The Debye length is κ−1 =√kTεoεs/(2Ie2), where I =

(1/2)∑N

j=1 z2jn∞j is the bulk ionic strength and n∞j is the bulk concentration

of the jth electrolyte species. Finally, the Lame constants µ and λ, written

in terms of Young’s modulus and Poisson’s ratio, are µ = E/[2(1 + ν)] and

λ = Eν/[(1 + ν)(1− 2ν)] (Landau & Lifshitz 1986).

Boundary conditions at the surface of the colloid particle (r = a) ensure:

u = 0 (zero fluid slip), v = Z (zero polymer slip); jj · er = 0 (zero radial ion

flux); and εoεs(er ·∇>)ψ − εoεp(er ·∇<)ψ = −σ (constant surface charge).

Here, er is the radial unit vector, εp is the particle dielectric constant, and

the subscripts attached to the gradient operators distinguish the particle (<)

and the solvent (>) sides of the interface. In the far field (r → ∞), the

fluid velocity and polymer displacement vanish (u → 0, v → 0); the ion

concentrations approach their bulk values (nj → n∞j ); and ψ → −rE · er(uniform undisturbed electric field).

Our analysis considers linearized perturbations to an equilibrium base

state (with E = Z = 0) that is governed by the non-linear Poisson-Boltzmann

equation (Verwey & Overbeek 1948). The continuum model is valid when the

inclusion size a is larger than the hydrogel Brinkman screening length `. To

this level of approximation, the Darcy permeability equals its homogeneous

and isotropic equilibrium value `2 (Hill & Ostoja-Starzewski 2008). Finally,

because the inclusion is assumed to be much stiffer (due to its higher density)

than the polymer skeleton, it remains spherical.

3.2.2 Solution methodology

The equilibrium base state and the linearized perturbations and boundary

conditions are similar to those of Hill & Ostoja-Starzewski (2008). Moreover,

17

we adopt the same methodology in which the solution is obtained by super-

posing the solutions of two simpler problems: one where the particle is fixed at

the origin (Z = 0) and subjected to an electric field E; and another where the

particle is displaced a distance Z in the absence of an electric field (E = 0).

The latter solution can be obtained analytically and provides the force given

by (3.1), so the following addresses the more challenging task of calculating

u, v, ψ and nj for a fixed particle in the presence of an electric field.

For a compressible polymer skeleton, a more general functional form for

the polymer displacement is necessary. Therefore, by considering the linearity

and symmetry of the perturbations to equilibrium, the polymer displacement

must have the form

v = g1(r)E + g2(r)(E · er)er, (3.8)

where g1 and g2 are scalar functions of radial position r. For a spherical

inclusion in an unbounded isotropic continuum, (3.8) is the most general con-

struction of a real, first-order tensor (v) that depends on position r = rer

and is linear in E = Eez. Here, ez is the polar axis of a coordinate sys-

tem (r, θ, φ) that has mutually orthogonal unit basis vectors (er, eθ, eφ); e.g.,

er · ez = cos θ. With axisymmetry, the radial and tangential components

of v are vr = (g1 + g2)E cos θ and vθ = −g1E sin θ. These demonstrate the

equivalence of our construction of v with classical treatments based on scalar

and vector potentials (Temkin & Leung 1976; Oestreicher 1951; Lamb 1945;

Markov 2005).

The fluid velocity takes the usual form (Landau & Lifshitz 1987)

u = ∇×∇× f(r)E, (3.9)

where f is another scalar function of radial position. Substituting (3.8) and

(3.9) into (3.6) provides two independent linear ordinary differential equations

18

for g1 and g2:

(µ+ λ)(r−1g1r + r−1g2r + 2r−2g2) +

µ(g1rr + 2r−1g1r + 2r−2g2) = (η/`2)(frr + r−1fr), (3.10)

(µ+ λ)(g1rr + g2rr − r−1g1r + r−1g2r − 4r−2g2) +

µ(g2rr + 2r−1g2r − 6r−2g2) = (η/`2)(r−1fr − frr) (3.11)

with boundary conditions g1 = g2 = 0 at r = a; and, more importantly,

g1 → ZE1 r−1 and g2 → ZE

2 r−1 as r →∞. (3.12)

Here, the subscripts r indicate differentiation, and the asymptotic coefficients

ZE1 and ZE

2 measure the strength of the far-field decay of v when the particle is

fixed at the origin (Z = 0) and subjected to an electric field E. As identified

by the boundary-layer analysis in § 3.3.1, the r−1 far-field decays of g1 and g2,

which reflect a net force, arise from the symmetry of the biharmonic potentials

of the irrotational and solenoidal contributions to the polymer displacement.

Our numerical solutions of (3.10) and (3.11) are accomplished using an

established finite-difference methodology with an adaptive mesh (Hill et al.

2003a). First, however, f(r) is obtained from a numerical solution of the

electrokinetic transport equations (involving u, ψ and nj) with a perfectly

rigid (v = 0) polymer skeleton (Hill 2006d). Briefly, taking the curl of the

momentum conservation equation with u written in terms of f according to

(3.9) leads to a system of ordinary differential equations that couple f(r), ψ(r)

and nj(r), where nj = n0j + njE cos θ and ψ = ψ0 − rE cos θ + ψE cos θ (Hill

et al. 2003a; Hill 2006d). Here, ψ0(r) and n0j(r) denote spherically symmetric

solutions of the Poisson-Boltzmann equation (with E = Z = 0) (Hill et al.

2003a). Our stepwise approach for obtaining g1 and g2 simply reflects the one-

way coupling of the electrolyte and polymer deformation under the prevailing

steady conditions.

19

3.2.3 Force evaluation and inclusion displacement

Using the superposition methodology adopted by Hill & Ostoja-Starzewski

(2008), the force balance on the inclusion demands

0 = fm,Z + fm,E + f e,E + f d,E, (3.13)

where fm,Z and fm,E are the mechanical-contact forces acting on the inclusion

by the polymer. The superscript Z indicates that the force is calculated with

the particle displaced a distance Z in the absence of an electric field (E = 0);

and the superscript E indicates that the force is calculated with the particle

fixed at the origin (Z = 0) and subjected to an electric field E. Furthermore,

f e,E and f d,E are electrical and hydrodynamic drag forces acting on the inclu-

sion by the electrolyte; both arise from the electric field while the particle is

fixed at the origin. Note that the calculation of fm,Z involves only the poly-

mer equation of static equilibrium and, consequently, the analytical solution

of this problem is given by (3.1).

The asymptotic coefficients ZE1 and ZE

2 determine fm,E as follows. First,

the mechanical-contact force on the inclusion is transformed to the far field

via

fm,E =

∫r=a

Te · erdA =

∫r→∞

Te · erdA+

∫ r→∞

r=a

(η/`2)udV, (3.14)

where the elastic stress tensor is

Te = 2µe + λ(∇ · v)I, (3.15)

where I is the identity tensor. Writing the strain e = (1/2)[∇v + (∇v)T ] in

terms of g1(r) and g2(r) gives

2e = g1r[Eer + (Eer)T ] + 2(g2r − 2r−1g2)(E · er)erer

+r−1g2[2(E · er)I +Eer + (Eer)T ], (3.16)

20

with dilation

∇ · v = (g1r + g2r + 2r−1g2)E · er. (3.17)

Therefore, from the far-field decays of g1 and g2, the surface and volume inte-

grals in (3.14) become∫r→∞

Te · erdA =−4πE

3(1 + ν)

[2ZE

1 +ν

1− 2ν(ZE

1 − ZE2 )

]E (3.18)

and ∫ r→∞

r=a

(η/`2)udV = −(8/3)π(η/`2)CEE, (3.19)

so

fm,E =−4πE

3(1 + ν)

[2ZE

1 +ν(ZE

1 − ZE2 )

(1− 2ν)

]E − 8π

3(η/`2)CEE. (3.20)

Note that f e,E+f d,E can be expressed in terms of the asymptotic coefficient CE

that characterizes the strength of the r−3 far-field decay of u, giving (Hill

2006d)

f e,E + f d,E = 4π(η/`2)CEE. (3.21)

Finally, the force balance in (3.13) becomes

aE(1− ν)

(5/6− ν)(1 + ν)Z =

2

3(η/`2)CEE − 2E

3(1 + ν)

[2ZE

1 +ν(ZE

1 − ZE2 )

(1− 2ν)

]E.

(3.22)

3.3 Boundary-layer analysis for κa 1, ` a and |ζ| < kT/e

The following boundary-layer analysis addresses the limit in which the

Debye length κ−1 and Brinkman screening length ` are both small compared

to the inclusion radius a. In practice, κ−1 is less than ∼ 100 nm at practical

ionic strengths, and ` is less than ∼ 10 nm for hydrogel skeletons. It follows

that the boundary-layer analysis is appropriate for inclusions with radii greater

than∼ 1 µm. Our analysis also adopts the Debye-Huckel approximation where

|ζ| < kT/e; fortunately, this proves to be satisfactory when |ζ| > kT/e if κa

is sufficiently large.

21

Using boundary-layer scaling approximations, we derive analytical solu-

tions of the governing equations in inner and outer regions, and match these to

determine the asymptotic coefficients in (3.22). Hill (2006b) used the method-

ology to calculate the incremental pore velocity for a dilute random array of

inclusions in a rigid Brinkman medium, and Hill & Ostoja-Starzewski (2008)

calculated the particle displacement in an incompressible hydrogel (ν = 0.5),

finding

Z → ζεoεsµ−1E as κa→∞. (3.23)

It is noteworthy that the particle displacement under these conditions is inde-

pendent of the hydrodynamic permeability, fluid viscosity, and particle size.

In striking contrast, our analysis for a compressible skeleton reveals a parti-

cle displacement that grows linearly with the particle radius, and decreases

with increasing hydrodynamic permeability. Similarly to the situation when

κa → 0 in (3.2), the scaling with particle size reflects a balance between an

electrical force that increases with a2—as expected for an unscreened surface

charge with constant surface charge density—and an elastic restoring force as

shown in (3.1), that is proportional to a.

3.3.1 Outer solution

The outer region is distinguished by the electrically neutral space where

κ(r − a) 1. Here, the electrical body force in the fluid momentum con-

servation equation is zero, and the viscous stresses are overwhelmed by the

Darcy drag force where r − a `. Accordingly, the fluid momentum equa-

tion reduces to Darcy’s equation u = −(`2/η)∇p with ∇2p = 0. The general

decaying solution is

p = −(η/`2)CEr−2E · er, (3.24)

where the asymptotic coefficient CE is determined by matching to the inner

solution.

The polymer displacement in the outer region can be written

v = ∇φs + ∇×Ψs, (3.25)

22

where ∇ ·Ψs = 0. Note that φs and Ψs specify the irrotational and solenoidal

(incompressible) contributions to v. Axisymmetry requires φs = φs(r, θ) and

Ψs = Ψs(r, θ)eθ, so, from the polymer equation of static equilibrium in (3.6),

∇[(2µ+ λ)∇2φs − p] = −∇× (µ∇2Ψs). (3.26)

Taking the curl and divergence of (3.26) separately give,

∇2∇2Ψs = 0 and ∇2∇2φs = 0, (3.27)

so, from axisymmetry (Lamb 1945) and relations between biharmonic and

harmonic functions (Fung & Tong 2001), the general solutions are

φs = (A′r3 + Ar +B′ +Br−2)E cos θ, (3.28)

Ψs = (C ′r3 + Cr +D′ +Dr−2)E sin θ, (3.29)

where the constants A–D and A′–D′ are specified by satisfying (3.26) and the

boundary conditions.

The polymer displacement is

v = [(3A′ + 2C ′)r2 + (A+ 2C) + 2D′r−1 − 2(B −D)r−3](E · er)er

+[(A′ + 4C ′)r2 + (A+ 2C) + (B′ +D′)r−1

+(B −D)r−3](E · eθ)eθ, (3.30)

so, because v → 0 as r → ∞ and A′ = C ′ = (A + 2C) = 0, it follows that v

decays as r−1 in the far field. To satisfy (3.26),

(4µ+ 2λ)B′ − (η/`2)CE = 2µD′, (3.31)

where B′ and D′ give

ZE1 = B′ +D′ and ZE

2 = D′ −B′. (3.32)

23

Finally, B′, D′ and (B−D), i.e., three independent constants, are determined

by satisfying (3.31) and matching to the radial and tangential components of

the inner polymer displacement field.

3.3.2 Inner solution and matching

The boundary-layer solution for the fluid velocity is independent of the

polymer deformation and, after matching, gives radial and tangential compo-

nents (Hill 2006b)

ur = 2a1(`/a)exp[−(r − a)/`]− 1E cos θ

−[2a2/(κa)]exp[−κ(r − a)]− 1E cos θ, (3.33)

uθ = a1 exp[−(r − a)/`]− a2 exp[−κ(r − a)] + a3E sin θ, (3.34)

where

a1 =3κ`2σ

2η[(κ`)2 − 1]− CE

a3, a2 =

3κ`2σ

2η[(κ`)2 − 1], a3 =

CE

a3,

and

CE =3σ(a`)2

2η(κ`+ 1).

Note that the surface charge density σ and ζ-potential are related by the

Debye-Huckel approximation σ = εoεsκζ when κa 1.

With standard boundary-layer scaling arguments, we establish the follow-

ing dominant balances for the radial and tangential components of the polymer

equation of static equilibrium:

(2µ+ λ)∂2vr∂r2

+

(µ+ λ

a sin θ

)∂2(vθ sin θ)

∂r∂θ= −(η/`2)ur, (3.35)

µ∂2vθ∂r2

= −(η/`2)uθ. (3.36)

24

These provide the following radial and tangential components of the inner

displacement field:

vr = − 2η

µ`2(a1`

3/a) exp[−(r − a)/`]

+[a2/(κ3a)] exp[−κ(r − a)] + C2E cos θ, (3.37)

vθ = − η

µ`2a1`

2 exp[−(r − a)/`]

+(a2/κ2) exp[−κ(r − a)] + C1E sin θ, (3.38)

where the constants

C1 = −`2a1 − a2/κ2 and C2 = −a1(`3/a)− a2/(κ

3a) (3.39)

ensure v = 0 at r = a. Note that linear and quadratic terms have been ne-

glected to permit matching when κa→∞. Accordingly, the inner components

of v as r →∞ are

vr →

2η`

µa4CE +

3σ[(κ`)2 + (κ`) + 1]

κ2aµ(κ`+ 1)

E cos θ, (3.40)

vθ →ηCE

µa3+

3σ

2µκ

E sin θ, (3.41)

and the outer components as r → a are

vr → [2D′/a− 2(B −D)/a3]E cos θ, (3.42)

vθ → −[(B′ +D′)/a+ (B −D)/a3]E sin θ. (3.43)

Equating the inner and outer displacement fields gives

B′ =3aσ

E(1 + ν)(1− 2ν)

(5− 6ν)

×`(`/a)− (`/a)2

(κ`+ 1)+ κ−1

[(κ`)2 + (κ`) + 1

κa(κ`+ 1)− 1

]+

a

κ`+ 1

, (3.44)

D′ = −3a2σ(1 + ν)

2E(κ`+ 1)+

6aσ

E(1 + ν)(1− ν)

(5− 6ν)

×`(`/a)− (`/a)2

(κ`+ 1)+ κ−1

[(κ`)2 + (κ`) + 1

κa(κ`+ 1)− 1

]+

a

κ`+ 1

, (3.45)

25

so writing (3.22) in terms of B′ and D′ gives

Z/E = −2(D′/a)(5/6− ν)/(1− ν), (3.46)

or, after simplifying for κa 1 and `/a 1,

Z/E = 2εoεsζE−1(1 + ν) +εoεsζκa(1 + ν)(1− 2ν)

2E(κ`+ 1)(1− ν). (3.47)

In terms of Lame constants µ and λ, (3.47) is

Z/E = εoεsζµ−1 +

εoεsζκa

2(κ`+ 1)(λ+ 2µ). (3.48)

The first term in (3.47) is the electric-field-induced displacement derived

by Hill & Ostoja-Starzewski (2008) for particles embedded in incompressible

skeletons (ν = 0.5). In the incompressible limit, the displacement is indepen-

dent of the particle size and hydrogel permeability. In striking contrast, (3.47),

which is valid for all thermodynamically admissible Poisson ratios, reveals a

particle displacement that increases linearly with particle size, and decreases

with increasing permeability.

To quantify the role of compressibility, and to explore the possibility of ex-

perimentally testing the theory, let us consider a representative example where

a particle with surface potential ζ = −3kT/e ≈ −75 mV (at room tempera-

ture) and radius a = 5 µm is embedded in a gel with Poisson’s ratio ν = 0.2,

Young’s modulus E = 100 Pa and Brinkman screening length ` = 10 nm. Fur-

thermore, with an electrolyte concentration corresponding to κ−1 ≈ 10 nm,

κa = 500 and (3.47) gives Z/E ≈ −6.0 nm/(V cm−1). For comparison, the

same particle in an incompressible gel (ν = 0.5) with the same E moves only

Z/E ≈ −0.16 nm/(V cm−1). With an electric field E = 10 V cm−1, the 60 nm

displacement in a compressible skeleton could be registered using optical mi-

croscopy (Ziemann et al. 1994; Cicuta & Donald 2007) and digital particle

tracking (Hoffman et al. 2006). However, measuring the 1.6 nm displacement

in an incompressible gel presents a much greater challenge, which is likely to

26

be met only with much more specialized equipment, involving interferome-

try (Allersma et al. 1998) and, possibly, feedback-controlled nano-positioning.

It is interesting to note that, when κ→∞ (high ionic strength) with `/a 1

and ν < 0.5, the displacement is

Z/E ≈ εoεsζ(a/`)(1 + ν)(1− 2ν)

2E(1− ν)=εoεsζ(a/`)

2(λ+ 2µ). (3.49)

With the same parameters as the previous example, (3.49) gives Z/E ≈

−12 nm/(V cm−1) at room temperature, so with E = 10 V cm−1, the particle

moves Z ≈ −120 nm. However, it should be remembered that the ζ-potential

decreases with increasing κ at constant surface charge density, and the elastic

modulus should increase with decreasing hydrodynamic permeability (Candau

et al. 1982).

3.4 Numerically exact results

Our numerical calculations are performed with the same characteristic

scales for length, velocity, and displacement as Hill & Ostoja-Starzewski (2008):

κ−1, u∗ = εsεo(kT/e)2/(ηa) and ηu∗/E = εsεo(kT/e)

2/(Ea), respectively. Also,

because the particle displacement remains inversely proportional to Young’s

modulus, our numerical calculations of Z/E are presented, for convenience,

with E = 1 kPa. Note, however, that many polymeric networks have a lower

modulus (Lin et al. 2004; Ziemann et al. 1994; Yamaguchi et al. 2005; Schnurr

et al. 1997), so particle displacements can be significantly larger than indicated

with E = 1 kPa. Clearly, the displacement for arbitrary Young’s modulus can

be obtained from the figures by a trivial rescaling of the ordinate axes.

The general structures of the electrolyte flow, polymer displacement, and

electrostatic potential are shown in figure 3–1. The flow (for clarity, stream-

lines are shown only in the bottom-half of each panel) and displacement (vec-

tors shown only in the top-half of each panel) are superposed on isocontours of

the total electrostatic potential in panel (a), perturbed electrostatic potential

in panel (b), and perturbed electrostatic potential without the applied field in

27

panel (c). All fields are axisymmetric about the polar axis, which coincides

with the rectangular cartesian x-axis shown.

The particle has a negative surface charge (and therefore a positive layer of

counter charge), and it is subjected to an electric field directed along the polar

axis, i.e., from left to right along the cartesian x-axis. Accordingly, the flow

is from left to right and the particle is displaced to the left. Also the polymer

deformation is dominated by its response to the particle displacement—not

its coupling to the electroosmotic flow—so its displacement is predominantly

in the opposite direction of the flow.

Finally, the electrostatic potential isocontours in panel (a) reflect the mag-

nitude of the applied electric field E relative to κζ ∼ κkT/e. The electric field

E = 0.01κkT/e ≈ 25 V cm−1 adopted here is not only representative of readily

achieved experimental conditions, but is also consistent with our linearization

of the perturbations to equilibrium. While panel (a) emphasizes the equilib-

rium potential within a Debye length of the particle surface, panel (c) reveals

intricate features of the polarized double layer that are not clearly evident from

panel (b). Finally, because the pressure (not shown) is also dipolar in the far

field, the streamlines tend to be perpendicular to the electrostatic potential

isocontours in panel (c) beyond a Debye length of the particle surface.

The particle displacement is plotted in figure 3–2 as a function of Pois-

son’s ratio for various values of κa and scaled ζ-potential. The dimensional

parameters (see the figure caption) facilitate direct comparisons with Hill &

Ostoja-Starzewski’s numerical calculations (Hill & Ostoja-Starzewski 2008,

figure 2) with ν = 0.5. As κa increases, the other independent dimensionless

parameter κ` = (`/a)κa = 0.01κa increases accordingly. As is customary,

we have plotted the results with constant ζ-potential, so the surface charge

density σ increases with κa according to the non-linear Poisson-Boltzmann

equation (Overbeek 1943). For example, when κa 1, σ ≈ εoεsa−1ζ, and

when κa 1, σ ≈ εoεsκζ. The ostensible increase in the displacement with

28

−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15

κx

κy

(a) ψ = ψ0 − rE cos θ + ψ(r)E cos θ

−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15

κx

κy

(b) −rE cos θ + ψ(r)E cos θ

−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15

κx

κy

(c) ψ(r)E cos θ

Figure 3–1: Streamlines (black lines, with flow from left to right), polymerand particle displacement (vectors, not to scale), and electrostatic potentialisocontours (colored lines, with potential increasing from blue to red) on aplane that contains the polar axis (all fields are axisymmetric): a = 500 nm,κ−1 = 100 nm; ζ ≈ −100 mV; ` = 5 nm, E = 1 kPa, ν = 0.2; E ≈ 25 V cm−1.

29

increasing κa can therefore be attributed, in part, to the accompanying in-

crease in surface charge.

When κa = 0.1 (results not shown), the numerical calculations coincide

with (3.2), which, recall, reflects a direct balance between the bare Coulomb

force and elastic restoring force of the polymer skeleton. Under these con-

ditions, the particle displacement is extraordinarily small. When κa & 1,

however, the displacement is considerably larger. Comparing the numeri-

cal (solid lines) and boundary-layer theory (dashed lines) demonstrates that

(3.48) provides satisfactory predictions of the displacement when κa & 10 and

|ζ| < kT/e. It is noteworthy that the boundary-layer theory, which, recall,

also rests on the Debye-Huckel approximation (|ζ| < kT/e), is also reliable

when |ζ| > kT/e if κa is sufficiently large. Therefore, (3.48) furnishes accu-

rate predictions of the displacement for sufficiently large particles embedded

in any (uncharged) hydrogel at any reasonable ionic strength.

As identified in the introduction, negative Poisson’s ratios are not relevant

for hydrogel skeletons. Nevertheless, for completeness, it is interesting to note

that a negative Poisson’s ratio indicates an isotropic continuum that, under

homogeneous axial strain, adopts an equally signed transverse strain. More

specifically, the axial and transverse strain are equal when ν = −1, and,

therefore, the material changes density without changing shape. In general,

such materials can only support an isotropic state of strain, which is clearly

evident when writing the strain in terms of the stress (Landau & Lifshitz

1986). Therefore, because the strain tensor must be linear in the electric

field or displacement vector, the displacement must be zero when ν = −1, as

confirmed by the numerics and boundary-layer theory.

In the range of Poisson’s ratios appropriate for hydrogel skeletons (0 ≤

ν < 0.5), the displacement achieves a maximum that depends on κa. The

maximum displacement is achieved at a Poisson’s ratio that approaches zero

with increasing κa, and the sensitivity of the displacement to Poisson’s ratio—

as measured qualitatively by the ratio of the maximum displacement to the

30

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Poisson’s ratio ν

-0.018

-0.016

-0.014

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0

Z/E

, [nm

/(V

/cm

)]

1

2

4

6

(a) κa = 1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Poisson’s ratio ν

-0.05

-0.04

-0.03

-0.02

-0.01

0

Z/E

, [nm

/(V

/cm

)]

1

2

4

6

46

(b) κa = 10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Poisson’s ratio ν

-0.28

-0.24

-0.20

-0.16

-0.12

-0.08

-0.04

0

Z/E

, [nm

/(V

/cm

)]

1

2

4

6

(c) κa = 100

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Poisson’s ratio ν

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Z/E

, [nm

/(V

/cm

)]

1

2

4

6

(d) κa = 1000

Figure 3–2: The ratio Z/E as a function of Poisson’s ratio ν for various scaledreciprocal double-layer thickness κa = 1, 10, 100 and 1000, and scaled ζ-potentials −ζe/(kT ) = 1, 2, 4 and 6: NaCl at T = 298 K; a/` = 100; andE = 1 kPa. Solid lines are numerically exact calculations (a = 500 nm and` = 5 nm), and dashed lines are the boundary-layer theory in (3.48). Notethat the displacement is inversely proportional to E .

31

displacement when ν = 0.5—increases significantly with κa. This important

result, which is also captured by the boundary-layer theory in (3.47), produces

significantly larger particle displacements in compressible skeletons than in in-

compressible ones when κa is large. In other words, finite compressibility

permits large particles to undergo relatively large electric-field-induced dis-

placements. The displacement is plotted in figure 3–3 as a function of κa for

various ζ-potentials with ν = 0.2. As highlighted in the introduction, this

Poisson’s ratio is representative of the values ascertained by several indepen-

dent experiments reported in the literature involving hydrogels under drained

conditions. Again, note that the particle radius a = 500 nm and Brinkman

screening length ` = 5 nm are fixed, so κ` = (`/a)κa = 0.01κa. This way

of plotting the results clearly identifies the ranges of κa and ζ-potential over

which the analytical theories for small (dash-dotted lines) and large (dashed

lines) κa are accurate. Furthermore, in the parameter space where the particle

displacements are large and, therefore, most easily measured, the boundary-

layer theory in (3.47) is reliable.

To draw comparisons between the displacement and electrophoretic mo-

bility, which Hill & Ostoja-Starzewski (2008) demonstrated are very closely

connected when ν = 0.5, figure 3–4 presents the scaled particle displacement

−(Z/E)Ee/(2εoεskT ) as a function of the scaled ζ-potential for various re-

ciprocal double-layer thickness κa with `/a = 0.01 and ν = 0.2. In addi-

tion to testing the boundary-layer approximation (dashed lines, right panel),

this figure clearly identifies maximums in the displacement due to polariza-

tion and relaxation. These processes are well known from their influences on

the electrophoretic mobility (O’Brien & White 1978) and incremental pore

mobility (Hill 2006d). Note also that, due to finite polymer compressibility

(ν = 0.2), the scaled displacements in figure 3–4 (right panel) are significantly

larger than the corresponding scaled electrophoretic mobility (Hill & Ostoja-

Starzewski 2008, figure 5). Even the general shape of the collection of curves is

different from the incompressible limit. In fact, the overall trends bear a much

32

10-2

10-1

100

101

κa

-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0

Z/E

, [nm

/(V

/cm

)]

1

2

4

6

8

(a)

100

101

102

103

κa

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Z/E

, [nm

/(V

/cm

)]

1

2

4

6

8

(b)

Figure 3–3: The ratio Z/E as a function of the scaled reciprocal double layer-layer thickness κa for various scaled ζ-potentials −ζe/(kT ) = 1, 2, 4, 6 and8: NaCl at T = 298 K; a/` = 100; E = 1 kPa; and ν = 0.2. Solid lines arenumerically exact calculations (a = 500 nm and ` = 5 nm), and the dashedand dash-dotted lines are, respectively, analytical theories for large and smallκa. Note that the displacement is inversely proportional to E .

33

closer resemblance to the incremental pore mobility (Hill 2006d), thereby pro-

viding a valuable clue toward understanding how the compressibility of the

polymer skeleton influences the particle displacement.

Note that the shape of the electrophoretic mobility versus ζ-potential and

κa relationship (and hence that of the scaled displacement when ν = 0.5) prin-

cipally reflects the variation of the electric-field-induced force on a fixed parti-

cle, since the balancing hydrodynamic drag is, to a first approximation (i.e., in

the absence of electroviscous effects), equal to the Stokes drag force −6πηaU ,

where U is the particle velocity. By direct analogy, the displacement of a par-

ticle embedded in an incompressible matrix reflects the electric-field-induced

force on a fixed particle, since the elastic restoring force is exactly the value

given by (3.1) with ν = 0.5. As demonstrated by Hill & Ostoja-Starzewski

(2008), the hydrodynamic coupling of the fluid and an incompressible polymer

skeleton leads to the same state of stress on the particle as in electrophoresis.

When the polymer is compressible, however, the net force due to the

electric field when the particle is fixed at the origin is evidently dominated by

the hydrodynamic and electrical contributions

f e,E + f d,E = 4π(η/`2)CEE → 6πεoεsζκa2

(κ`+ 1)E as κa→∞, (3.50)

not those arising from the electroosmotic-flow-induced distortion of the poly-

mer skeleton. Accordingly, Hill’s interpretation of (3.50), which does not in-

volve polymer deformation (Hill 2006b), also provides an appealing interpre-

tation of the particle displacement in compressible polymer skeletons. More

specifically, Hill (2006b) showed that the force represented by (3.50) exceeds

the bare electrical force when κa 1 and κ` 1. This is due to an ad-

verse pressure gradient (increasing pressure in the direction of electroosmotic

flow) that must develop to sustain a Darcy (pressure driven) flow in the far

field. Since this force increases with the square of the particle radius a, and

the elastic restoring force is linear in a as shown in (3.1), it follows that the

34

0 2 4 6 8 10

-ζe/(kT)

0

1

2

3

4

5

6

7

scal

ed d

ispl

acem

ent

0.01 0.10.2

0.4

1

2

(a)

0 2 4 6 8 10

-ζe/(kT)

0

20

40

60

80

100

scal

ed d

ispl

acem

ent

3

610

15

30

50

100

1000

(b)

Figure 3–4: The scaled displacement −(Z/E)Ee/(2εoεskT ) as a function of thescaled ζ-potential −ζe/(kT ) for various scaled reciprocal double-layer thick-ness κa = 0.01, 0.1, 0.2, 0.4, 1 and 2 (left panel); and κa = 3, 6, 10, 15, 30,50, 100 and 1000 (right panel): KCl at T = 298 K; a/` = 100; and ν = 0.2.Solid lines are numerically exact calculations (a = 500 nm and ` = 5 nm) andthe dashed lines (κa ≥ 10) are the boundary-layer theory in (3.47).

particle displacement should increase linearly with a when κa 1. This is

indeed verified by our numerics and boundary-layer theory.

Let us briefly address the influence of the polymer hydrodynamic perme-

ability, which, recall, earlier calculations (Hill & Ostoja-Starzewski 2008, e.g.,

figure 4) showed it has a very weak influence when ν = 0.5. The particle

displacement is plotted in figure 3–5 as a function of the scaled Brinkman

screening length κ` with ν = 0.2. In striking contrast to the incompress-

ible limit, but consistent with Hill’s interpretation of the force represented by

(3.50), the displacement undergoes a significant transition from a high plateau

when κ` is small (low permeability) to a much lower plateau when κ` is large

(high permeability). Again, when the permeability is low, the electrical force is

accentuated by an adverse pressure gradient. However, when the permeability

35

10-3

10-2

10-1

100

101

102

103

104

κl

10-3

10-2

10-1

100

-Z/E

, [nm

/(V

/cm

)]

10

100

1000

1

0.1 & 0.01

Figure 3–5: The ratio Z/E as a function of scaled Brinkman screening lengthκ` for various scaled reciprocal double-layer thicknesses κa = 0.01, 0.1, 1, 10,100 and 1000: NaCl at T = 298 K; ζe/(kT ) = −1; E = 1 kPa; and ν = 0.2.Solid lines are numerically exact calculations (a = 500 nm) and the dashedlines are the boundary-layer theory in (3.47). Note that the displacement isinversely proportional to E .

is high, the weak coupling of the polymer and fluid leads to a balance be-

tween the electrical and particle-displacement-induced elastic restoring force

as shown in (3.1). It should also be noted that the boundary-layer theory

in (3.47) is valid only when κa 1 and a/` 1, so, as expected, the ap-

proximation breaks down in figure 3–5 when κ` > κa. The small discrepancy

between the boundary-layer and numerical solutions when κa 1 and κ` 1

reflects a subtle but minor shortcoming of the boundary-layer approximation.

Specifically, the leading-order boundary-layer approximation for κa 1 and

` a in the inner region does not include a small but finite tangential pres-

sure gradient. A similar limitation was identified by Hill & Saville (2005)

when comparing numerically exact calculations of the electrophoretic mobility

of soft colloidal spheres with analytical theories in the literature for large κa.

36

3.5 Summary

We generalized the electrokinetic model of Hill & Ostoja-Starzewski (2008)

to calculate the electric-field-induced displacement of a charged, spherical col-

loid embedded in an electrolyte-saturated compressible polymer skeleton. The

fluid velocity and polymer displacement fields were calculated to linear order

in perturbations from an equilibrium state governed by the non-linear Poisson-

Boltzmann equation. Using linear superposition, we expressed the particle dis-

placement in terms of asymptotic coefficients that describe the far-field decays

of the fluid pressure and polymer displacement. Because the polymer skeleton

is compressible, two asymptotic coefficients are necessary to correctly quan-

tify the polymer distortion and, hence, to compute the electric-field-induced

particle displacement. In addition to numerically exact solutions of the full

model, we derived an analytical boundary-layer solution for the limit in which

the inclusion radius is larger than the Debye and Brinkman screening lengths.

Our theory reveals an electric-field-induced particle displacement that is

a sensitive and, in general, complicated function of the Poisson ratio and

hydrodynamic permeability of the polymer skeleton, the size and charge of

the inclusion, and the concentration of the electrolyte. However, the parti-

cle displacement remains inversely proportional to Young’s modulus (or shear

modulus) of the polymer. More importantly, polymer compressibility yields

a particle displacement that has a qualitatively different dependence on the

particle size than for incompressible polymer skeletons. Specifically, when

κa 1, the electric-field-induced displacement increases linearly with the

particle size, rather than tending to a size-independent value. Fortunately,

in the parameter space where experimentally measurable particle displace-

ments are expected, our boundary-layer approximation furnishes a reliable

and convenient alternative to numerical solutions of the full model. To our

best knowledge, experiments have not been reported in the literature, so we

hope our theory will stimulate future experimental work in this area.

37

CHAPTER 4Dynamic electric-field-induced response of charged spherical

colloids in uncharged hydrogels

Embedding colloidal particles in polymeric hydrogels often endows the

polymer skeleton with appealing characteristics for microfluidics and biosens-

ing applications. This chapter provides a rigorous foundation for interpreting

active electrical microrheology and electroacoustic experiments on such mate-

rials. In addition to viscoelastic properties of the composites, these techniques

sense physicochemical characteristics of the particle-polymer interface. We

extended the steady multi-phase electrokinetic model in the previous chapter

to calculate the dynamic response of charged spherical colloids embedded in

uncharged hydrogels when subjected to harmonically oscillating electric fields.

The frequency response depends on the particle size and charge, ionic strength

of the electrolyte, and elastic and hydrodynamic characteristics of the polymer

skeleton. Our calculations capture the transition from quasi-steady compress-

ible to quasi-steady incompressible dynamics as the frequency passes through

the reciprocal draining time of the gel. At higher frequencies, the dynamics

are dominated by hydrodynamic viscous and inertial forces, with the response

eventually becoming equal to the classical dynamic electrophoretic mobility

measured using modern electroacoustic instruments. We establish the connec-

tion between the electroacoustic signal for hydrogel composites and the single-

particle dynamic electrophoretic mobility. Finally, we provide an approximate

analytical theory that captures the elasticity of the hydrogel at the ultrasonic

frequencies used in commercially available electroacoustic instruments. This

agrees very well with our numerics over a wide range of the experimentally

accessible parameter space.

38

4.1 Introduction

Hydrogels are an important class of soft matter that have gained widespread

application in drug delivery (Qiu & Park 2001; Lin & Netters 2006; Peppas

et al. 2000), tissue engineering (Khademhosseini & Langer 2007; Barndl et al.

2007; Drury & Mooney 2003), advanced materials (Peppas et al. 2006; Edding-

ton & Beebe 2004; Chaterji et al. 2007), and molecular separations (Wang

et al. 1993; Kim & Park 1998). Novel characteristics can be achieved by

immobilizing organic and inorganic colloidal particulates in the polymer skele-

ton. For example, embedding gold or gold-coated silica nanoparticles into a

thermally responsive hydrogel induces light-wavelength-sensitive swelling to

achieve optically active microfluidic flow control (Sershen et al. 2005). In

biosensing, immobilizing silica nanoparticles in polyacrylamide hydrogels and

applying an electric field increase the otherwise diffusion-limited flux of un-

charged macromolecules across the composite membrane (Matos et al. 2006).

This flux enhancement can be attributed to electroosmotic flow (Hill 2006d),

and theoretical predictions from continuum electrokinetic theory are in good

agreement with the available experiments (Hill 2007). Other applications in-

clude delivering growth factors for bone regeneration (Chung et al. 2007),

improving the contrast of ultrasound imaging for early tumor detection (Liu

et al. 2006; Dayton & Ferrara 2002), and absorbing infrared energy for certain

cancer treatment (Loo et al. 2005). Note also that polystyrene nanoparticles

have been dispersed in neutral polyacrylamide hydrogels to increase the storage

modulus and produce mechanoelectrical effects for artificial tactile perception

and psycho-sensorial materials (Thevenot et al. 2007).

Advances in the development and application of hydrogel-colloid com-

posites could benefit from a quantitative characterization of the microstruc-

ture. This motivates the present theoretical study, where we investigate the

response of hydrogel-colloid composites to dynamic electric fields. Our the-

ory provides a first step toward quantifying the electrical, hydrodynamic and

mechanical interactions under dynamic conditions. It therefore establishes a

39

rigorous foundation for interpreting electrical microrheology and electroacous-

tic experiments. We show that electrophoretic microrheology (Mizuno et al.

2000; Kimura & Mizuno 2007) is appropriate at low frequencies, whereas the

response to electric fields at high frequencies is best probed using electroacous-

tics, namely the electrokinetic sonic amplitude (ESA) (O’Brien 1988, 1990;

Hunter 1998).

In microrheology, the storage and loss moduli of soft materials, such as hy-

drogels and polymer solutions, can be obtained from the frequency-dependent

susceptibility of probe particles in an external field (Ziemann et al. 1994;

Breuer 2005). One advantage of applying a dynamic electric field is that the

viscoelastic properties of the matrix and the physicochemical characteristics of

the probe particles can be measured simultaneously. Note that physicochem-

ical properties, such as the surface charge or ζ-potential of the inclusions, are

important for electroosmotic pumping (Yao & Santiago 2003; Yao et al. 2003;

Matos et al. 2006; Hill 2007) and micromixing (Matos et al. 2008). Another

advantage of dynamic experiments is that the spectral response provides con-

siderably more information than a steady experiment (Hunter 2001; Russel

et al. 1989).

Microrheology is often adopted for materials that are too fragile for bulk

rheology measurements (MacKintosh & Schmidt 1999; Cicuta & Donald 2007;

Breuer 2005). Passive microrheology, such as diffusing wave spectroscopy

(DWS) (Pine et al. 1988; Mason & Weitz 1995) or one- and two-particle mi-

crorheology (Levine & Lubensky 2000), measures thermal fluctuations in the

displacement of probe particles. In active microrheology, the probe particles

typically respond to applied magnetic (Ziemann et al. 1994) or optical (Valen-

tine et al. 1996; Yamaguchi et al. 2005) forces.

Mizuno et al. (2000, 2001, 2004) and Kimura & Mizuno (2007) have ap-

plied electric fields in a novel heterodyne light scattering technique to mea-

sure the dynamic electrophoretic mobility of nanoparticles in dilute lamellar

phases. Their experiments simultaneously measure the dynamic mobility and

40

diffusion coefficient of probe particles at frequencies from less than 1 Hz to

about 50 kHz. Microrheology techniques can also be applied to assess the sur-

face characteristics of colloidal particles. With optical tweezers, Galneder et al.

(2001) measured the ζ-potential of phospholipid-bilayer-coated silica beads by

monitoring electrophoretic forces. Electrical microrheology can also be ap-

plied to hydrogel-colloid composites, so simultaneous characterization of the

viscoelastic and physicochemical properties of the composite microstructure

could be achieved.

The upper frequency limit of electrical microrheology is about 50 kHz (Mizuno

et al. 2000). Higher frequencies have been achieved with Newtonian colloidal

dispersions using electroacoustics, typically operating between 0.3 MHz and

11 MHz (Hunter 1998). Electroacoustics has also been successful for deter-

mining the size and ζ-potential of colloidal particles in Newtonian electrolytes.

It includes the colloid vibration potential (CVP), arising from external sound

waves, and the electrokinetic sonic amplitude (ESA), generated by oscillat-

ing electric fields. These techniques are independent of the suspension opti-

cal properties, and are therefore particularly well suited for opaque and con-

centrated dispersions. Modern electroacoustic theories (O’Brien 1988, 1990)

connect the macroscopic electric-field-induced pressure disturbances or sound-

wave-generated electrical potentials to the dynamic mobility and polarizability

of dispersed colloidal particles. Theoretical calculations of the dynamic elec-

trophoretic mobility for dilute (Mangelsdorf & White 1992; Preston et al. 2005;

O’Brien 1988) and concentrated (Ahualli et al. 2006; Rider & O’Brien 1993;

O’Brien et al. 2003) suspensions have been successfully compared with exper-

iments, which has undoubtedly facilitated successful commercialization of the

technology (Hunter 1998).

In this work, we connect the macroscopic electroacoustic response of

hydrogel-colloid composites to the dynamic electrophoretic mobility of a sin-

gle colloidal particle embedded in a hydrogel matrix. Our analysis demon-

strates that the electroacoustic response of a hydrogel-colloid composite could

41

be measured using electroacoustic instruments currently available for colloidal

dispersions. More importantly, we identify the frequency range where the

electroacoustic signal is particularly sensitive to the elasticity of the hydrogel

skeleton. This could facilitate novel experiments to monitor the kinetics of

gelation and other developments of the microstructure. Noteworthy is that

the dynamic electrophoretic mobility at MHz frequencies is particularly high

when with relatively stiff polymer skeletons. This is in striking contrast to the

amplitude of the particle displacement measured in electrical microrheology,

which tends to be measurable only when the frequency and elastic modulus

are low.

Previous theories for the steady electric-field-induced displacement of

spherical colloidal particles in uncharged hydrogels (Hill & Ostoja-Starzewski

2008; Wang & Hill 2008) reveal that the colloid displacement reflects a simple

balance between the electric Coulomb force and the elastic restoring force of

the gel when the particle radius a is much smaller than the Debye screening

length κ−1. This situation prevails with small particles and low electrolyte

concentrations. Otherwise, when a is much larger than κ−1, the displacement

quantifies how electroosmotic flow, arising from the diffuse layer of counterions

that envelops each inclusion, interacts with the polymer skeleton. The later is

considerably more challenging to compute, but is analytically tractable.

Wang & Hill (2008) recently showed that compressibility of the hydrogel

skeleton, as quantified by Poisson’s ratio ν, can have a significant influence on

the particle displacement Z when a charged inclusion is subjected to a steady

electric field E. When κa 1, for example,

Z/E = 2εoεsζE−1(1 + ν) +εoεsζκa(1 + ν)(1− 2ν)

2E(κ`+ 1)(1− ν)(κa 1, ` a), (4.1)

where ζ is the well-known ζ-potential, E and ` are Young’s modulus and

Brinkman screening length (Brinkman 1947) of the polymer skeleton (`2 is the

Darcy permeability), and εo and εs are the vacuum permittivity and dielectric

constant of the gel. With an incompressible skeleton (ν = 0.5), the second

42

term on the right-hand side of (4.1) vanishes and the displacement is indepen-

dent of the particle size. For compressible skeletons with ν ∼ 0.2, however, the

displacement increases linearly with κa. Therefore, in the experimentally ac-

cessible parameter space, the particle displacement in compressible hydrogels

can be an order of magnitude larger than in incompressible hydrogels.

Note that the apparent compressibility of a hydrogel, i.e., the compos-

ite compressibility of the fluid and the polymer, depends on the draining

time (Schnurr et al. 1997; Hill & Ostoja-Starzewski 2008)

τd ∼ (1− 2ν)(η/E)(a/`)2, (4.2)

where η is the fluid viscosity. If the experimental time scale τc < τd, the

fluid is unable to escape the polymer network, so the hydrogel appears incom-

pressible; otherwise, the skeleton has time to drain and adopt its equilibrium

(compressible) state of strain. Consistent with scaling theory (Geissler &

Hecht 1980, 1981), the Poisson ratio of hydrogel skeletons is generally found

to be in the range 0–0.25, so the draining time is indeed finite. In addition

to the draining time, other important time scales affect the response. For

example, balancing the O(ηu∗a−1) viscous hydrodynamic stresses with the

O(µu∗τva−1) elastic stresses identifies a viscous time scale τv = ηµ−1. Here,

u∗ is the characteristic velocity, and µ is the shear modulus of the polymer

skeleton. Moreover, balancing the O(ρfu∗τ−1f ) inertial stresses with the fore-

going viscous stresses identifies a fluid inertia time scale τf = ρfa2η−1, where

ρf is the fluid density. At frequencies greater than the reciprocal viscous time,

the dynamics are the same as in the absence of polymer, i.e., at high enough

frequencies the dynamic mobility of an inclusion in a hydrogel becomes equal

to its mobility in a Newtonian electrolyte. Note that the ion diffusion time

τi = (a+ κ−1)2D−1 (DeLacey & White 1981), where D is a characteristic ion

diffusivity, provides a time scale for accessing dynamics of polarization and re-

laxation of the diffuse double layer, but this has a relatively weak influence on

43

particle dynamics. Although the foregoing time scales are helpful for under-

standing qualitative aspects of the dynamics, quantitative transitions between

these characteristic times must be established by calculating the frequency

spectrum of the colloid displacement.

In this work, frequency spectra are calculated using an electrokinetic

model in which the fluid and hydrogel skeleton are coupled by Darcy drag.

When electrical forces are negligible, i.e., in the absence of an electric field

and surface charge, the hydrodynamic and elastic equations of motion couple

to form a so-called two-fluid model. This yields a response function for probing

particles subjected to a known external force.

Levine & Lubensky (2001) derived an approximate response function that

is valid when fluid inertia can be neglected, typically at frequencies up to sev-

eral kHz. However, much higher frequencies are important for electrophoretic

microrheology and electroacoustics. Therefore, to correctly interpret such ex-

periments, an exact solution of the full two-fluid model is required. An exact

solution of this model is also necessary for calculating the dynamic electric-

field-induced response. More specifically, the two-fluid model provides far-

field boundary conditions for accurately calculating the electric-field-induced

particle response, which is governed by the much more complex multi-phase

electrokinetic model addressed in this work.

This chapter is arranged as follows. In § 4.2 we solve the two-fluid model

analytically for an uncharged spherical colloid in uncharged hydrogels. This

sets a foundation for solving the full multi-phase electrokinetic model in § 4.3.

An asymptotic analysis for the far-field is undertaken in § 4.4, which provides

boundary conditions to facilitate accurate numerical solutions of the full elec-

trokinetic model. Following O’Brien (1988, 1990), § 4.5 establishes the link

between electroacoustics and the single-particle response for colloids immo-

bilized in hydrogels. An analytical boundary-layer approximation, valid for

frequencies ω with a−2D ω κ2ηρ−1f is presented in § 4.6. Numerical

44

and analytical solutions of the two-fluid and multi-phase electrokinetic mod-

els are presented in § 4.7, where we undertake a detailed parametric study.

Section 4.7.1 compares our exact solution of the two-fluid model with the ap-

proximation of Levine & Lubensky (2001) and the generalized Stokes-Einstein

relation (GSER). Our numerical solutions of the full multi-phase electrokinetic

model are presented in § 4.7.2. This section examines the spectrum of the dy-

namic response Z/E, which is of prime importance in electrical microrheology.

Section 4.7.3 examines the dynamic electrophoretic mobility µd = −iω(Z/E)

from the perspective of electroacoustics, focusing on how the elasticity of the

polymer gel distinguishes particle dynamics from those already established for

colloids dispersed in Newtonian electrolytes. Finally, our analytical theory for

the mobility at the ultrasonic frequencies encountered in electroacoustic ex-

periments is compared with numerically exact solutions of the full model. We

conclude with a summary in § 4.8.

4.2 Two-fluid model and response for uncharged colloids

Consider an uncharged spherical colloid with radius a and density ρp em-

bedded in an uncharged hydrogel with Young’s modulus E , Poisson’s ratio

ν, and Darcy permeability `2. The particle is subjected to a harmonically

oscillating external force F exp(−iωt), where ω is the angular frequency and

i =√−1. In microrheology, such a force arises from optical or magnetic

fields, which are generally decoupled from the fluid and polymer. The particle

responds by undergoing a displacement Z exp(−iωt) that reflects the hydrody-

namic and elastic forces as determined from the fluid velocity u and polymer

displacement v. Accordingly, the response function α(ω) ≡ Z/F is obtained

by satisfying the particle equation of motion.

45

In the absence of electrical influences, the two-fluid model in the frequency

domain for harmonic dynamics comprises (e.g., Levine & Lubensky 2001)

−iωρfu = −∇p+ η∇2u− (η/`2)(u+ iωv), (4.3a)

0 = ∇ · u, (4.3b)

0 = µ∇2v + (µ+ λ)∇(∇ · v) + (η/`2)(u+ iωv), (4.3c)

where p is the pressure, and the first and second Lame constants, λ and µ,

respectively, are related to Young’s modulus E and Poisson’s ratio ν by λ =

Eν/[(1 + ν)(1− 2ν)] and µ = E/[2(1 + ν)]. In a frame of reference that moves

with the sphere, the boundary conditions are

u = v = 0 at r = a, (4.4)

u→ −iωY , v → Y as r →∞, (4.5)

where Y = −Z.

In general, the first and second Lame constants for the polymer skeleton

are complex and frequency dependent (Larson 1999). However, their deter-

mination requires specific knowledge of the polymer and gelation. Therefore,

for simplicity, the first and second Lame constants are specified as real con-

stants. Note that time derivatives appear via the factor −iω; polymer inertia

is neglected because of its low mass fraction; and the fluid and polymer are

coupled by the Darcy drag force (η/`2)(u+ iωv).

4.2.1 Fluid velocity and polymer displacement fields

Inspired by the method of Markov (2005), we construct the fluid velocity

and polymer displacement fields as

u = ∇Φ1 + ∇×Ψ1 + ∇×Ψ2 − iωY , (4.6a)

v = m∇Φ1 + ∇Φ2 +M1∇×Ψ1 +M2∇×Ψ2 + Y , (4.6b)

where Φ1 and Φ2 are scalar functions; Ψ1 and Ψ2 are vector functions; and m,

M1 and M2 are constants. Physically, Φ1 can be attributed to the pressure in

46

the incompressible fluid; Φ2 represents a compressional wave; and Ψ1 and Ψ2

represent shear waves.

Fluid incompressibility (4.3b) requires

∇2Φ1 = 0, (4.7)

so taking the divergence of (4.3c), and substituting (4.6a) and (4.6b) gives

∇2[∇2(mΦ1 + Φ2) + (η/`2)(λ+ 2µ)−1Φ1 + iω(η/`2)(λ+ 2µ)−1(mΦ1 + Φ2)] = 0.

(4.8)

Next, eliminating Φ1 by setting m = −(iω)−1 gives

∇2Φ2 + k2Φ2 = 0, (4.9)

where

k2 = iω(η/`2)(µ+ 2λ)−1. (4.10)

Taking the curl of (4.3a) and (4.3c), and substituting (4.6a) and (4.6b) gives

∇×∇×2∑j=1

η∇2 + [iωρf − (η/`2)− iω(η/`2)Mj]Ψj = 0, (4.11a)

∇×∇×2∑j=1

µMj∇2 + [(η/`2) + iω(η/`2)Mj]Ψj = 0, (4.11b)

or

∇2Ψj +K2jΨj = 0 (j = 1, 2), (4.12)

where

K2j = [iωρf − (η/`2)− iω(η/`2)Mj]/η (j = 1, 2). (4.13)

For Ψ1 and Ψ2 to be distinct, M1 and M2 must be roots of the quadratic

iω(µ/`2)M2j + [iω(η/`2) + (µ/`2)− iωρf (µ/η)]Mj + (η/`2) = 0. (4.14)

The wave numbers k correspond to the propagation of compressional waves,

and K1 and K2 are associated with shear waves. Note that all the foregoing

47

wave numbers can also be obtained from the Fourier representation of the

governing equations (Levine & Lubensky 2001).

With the prevailing axisymmetric spherical geometry, the Laplace equa-

tion (4.7) and Helmholtz equations (4.9) and (4.12) are easily solved analyti-

cally. The vector potential Ψj can be written as Ψj = Ψjeφ (j = 1,2) (Lamb

1945; Markov 2005; Oestreicher 1951; Temkin & Leung 1976), where eφ is one

of the mutually orthogonal unit basis vectors (er, eθ, eφ) for spherical polar

coordinates (r, θ, φ) with respect to the polar axis ez such that ez ·er = cos θ.

Since the fluid velocity and polymer displacement must be linear with respect

to Y , and vanish as r →∞, we have

Φ1 = A1r−2Y cos θ, (4.15a)

Φ2 = A2h(kr)Y cos θ, (4.15b)

Ψj = Bjh(Kjr)Y sin θ (j = 1, 2), (4.15c)

where h(x) = −x−2(x+ i) exp(ix) is the spherical Hankel function of the first

kind, which represents an outward propagating wave, and Aj and Bj (j = 1, 2)

are constants to satisfy the boundary conditions at r = a. To ensure vanishing

far-field disturbances, Im(k) > 0, Im(K1) > 0, and Im(K2) > 0.

The radial and tangential components of the fluid velocity and polymer

displacement are

ur = [−2A1r−3 + 2

2∑j=1

Bjr−1h(Kjr)− iω]Y cos θ, (4.16a)

uθ = A1r−3 +

2∑j=1

BjKj[(Kjr)−1h(Kjr) + h′(Kjr)]− iω(−Y sin θ), (4.16b)

vr = [−2mA1r−3 + A2kh′(kr) + 2

2∑j=1

MjBjr−1h(Kjr) + 1]Y cos θ, (4.16c)

vθ = mA1r−3 + A2r

−1h(kr) +2∑j=1

MjBjKj[(Kjr)−1h(Kjr) + h′(Kjr)] + 1

×(−Y sin θ). (4.16d)

48

Note that the prime on the spherical Hankel function denotes its first deriva-

tive, and the constants Aj and Bj (j = 1, 2) are chosen to satisfy the no-slip

boundary conditions at r = a.

4.2.2 Force and response function

The force f exerted on the sphere by the fluid and polymer is calculated

from knowledge of the fluid velocity and polymer displacement. Integrating the

hydrodynamic and elastic surface tractions over the particle surface (Landau

& Lifshitz 1987) gives

f = (4/3)πa2−iωρfA1Y/a2 + [iω(η/`2)− λk2]A2Y h(ka) + 2f1(a) + 2f2(a)

= −4πiωρfA1Y, (4.17a)

where

f1(r) cos θ = ηur,r + µvr,r, (4.17b)

f2(r) sin θ = ηuθ,r + µvθ,r. (4.17c)

Note that the subscripts “r” following commas denote differentiation with

respect to r. Equation (4.17a) is the same as obtained by applying Gauss’s

divergence theorem to the volume enclosed by the particle surface and a large

concentric sphere. With no-slip boundary conditions at r = a,

A1 = a3(Θ + Γ)(2H)−1, (4.18a)

where

Θ = ω(M1 −M2)[2i(β1β2)2(b+ i)− 2b2(β1 + i)(β2 + i)

+b2(β21 + iβ1 − 1)(β2

2 + iβ2 − 1)], (4.18b)

Γ = 2(b2 + 3ib− 3)[β21(1 + iωM1)(β2 + i)

−β22(1 + iωM2)(β1 + i)], (4.18c)

H = i(β1β2)2(b2 + 2ib− 2)(M1 −M2)

+b2[β21(M2 −m)(β2 + i)− β2

2(M1 −m)(β1 + i)], (4.18d)

49

with b = ka and βj = Kja (j = 1, 2).

Finally, in a stationary reference frame, also taking into account particle

and fluid inertia (see Appendix 4.A), the particle equation of motion is

F + 4πiωρfA1Z = ω2Vp(ρf − ρp)Z, (4.19)

where Vp = (4/3)πa3 is the particle volume, and, recall, Z = −Y . Our exact

solution of the two-fluid model for the response function is therefore

α(ω) ≡ Z/F = [ω2Vp(ρf − ρp)− 4πiωρfA1]−1. (4.20)

This is compared with the approximation of Levine & Lubensky (2001) and

the GSER in § 4.7.1. To evaluate A1 in (4.20), quadratic equation (4.14) is

solved for M1 and M2, which provide wave numbers K1 and K2 from (4.13).

Next, k is obtained from (4.10). Note that Im(k) > 0 and Im(Kj) > 0

(j = 1, 2). After evaluating b, β1 and β2, equations (4.18b)–(4.18d) give Θ,

Γ, and H, which provide A1 from (4.18a). Several of the foregoing steps must

be performed using multiple precision algebra (Granlund 2007; Fousse et al.

2007; Enge et al. 2007) to avoid significant round-off errors. The integrals in

the approximation of Levine & Lubensky (2001) are easily evaluated using

standard numerical quadrature.

4.3 Multi-phase electrokinetic model

In general, the particle surface charge is screened by a diffuse layer of

electrolyte- and counter-ions, whose bulk concentration determines the Debye

length κ−1 and surface potential ζ for a given surface charge density σ. How-

ever, when κa 1, the external force on the particle equals the bare Coulomb

force F = σ4πa2E, and there is vanishing electroosmotic flow. Therefore, the

ratio of the particle displacement to the electric field strength is simply

Z/E = σ4πa2α(ω). (4.21)

50

As highlighted by scaling analysis (Schnurr et al. 1997) and the approximate

response function of Levine & Lubensky (2001), in (4.21) the hydrogel com-

pressibility affects Z/E by at most 25%. When κa & 1, however, an electroki-

netic model is necessary to capture the influence of electroosmotic flow and

polarization of the diffuse double layer, which together modify the phase and

amplitude of the effective Coulomb force on the particle.

4.3.1 Governing equations and boundary conditions

Our multi-phase electrokinetic model augments the two-fluid model con-

sidered in § 4.2 with an electrical body force on the fluid, a Poisson equation

linking the electrostatic potential to the free-charge density, and electrolyte-

ion conservation equations to account for ion diffusion, electromigration and

convection. With harmonic time dependence, e.g., an applied electric field

E exp (−iωt), the full electrokinetic model is

0 = εoεs∇2ψ +N∑j=1

njzje, (4.22a)

−iωnj = −∇ · jj, (4.22b)

−iωρfu = η∇2u−∇p− (η/`2)(u+ iωv)−N∑j=1

njzje∇ψ, (4.22c)

0 = ∇ · u, (4.22d)

0 = µ∇2v + (λ+ µ)∇(∇ · v) + (η/`2)(u+ iωv), (4.22e)

with ion fluxes

jj = nju−Dj∇nj − zjeDjnj(kT )−1∇ψ. (4.22f)

Note that ψ is the electrostatic potential, and nj is the concentration of the

jth ion species with valence zj and diffusivity Dj. The fundamental charge

and thermal energy are e and kT , respectively, and the diffusivities of the N

ion species are related to their limiting conductances Λj (Speight 2005) by

Dj = (kTΛj)/(e2|zj|). Note that the Debye length is κ−1 =

√kTεsεo/(2Ie2),

51

where the ionic strength I = (1/2)∑N

j=1 z2jn∞j with n∞j the bulk concentration

of the jth ion species.

Among the principal assumptions underlying this model are a linearly

elastic hydrogel skeleton that is isotropic and homogeneous (Hill & Ostoja-

Starzewski 2008), and does not hinder ion diffusion and electromigration (Hill

2006d). Furthermore, the colloidal particle is assumed to be rigid, and the

polymer displacement and fluid velocity are assumed to be continuous across

the particle-hydrogel interface.

Accordingly, in a reference frame that moves with the particle, no-slip

boundary conditions at the particle surface r = a require

u = v = 0. (4.23a)

Other possibilities, such as slipping or the opening of a crack at the particle-

hydrogel interface significantly complicate the problem, and are not pursued

here. An impenetrable and non-conducting particle demands

ψ> − ψ< = 0, (4.23b)

εpεo(er ·∇<)ψ − εsεo(er ·∇>)ψ = σ, (4.23c)

jj · er = 0, (4.23d)

where εp is the particle dielectric constant, er is the outward unit normal, and

the subscripts “<” and “>” distinguish the particle and hydrogel sides of the

interface.

Far from the particle, disturbances to the equilibrium electrostatic po-

tential, ion concentrations, fluid velocity, and polymer displacement vanish.

Therefore, as r →∞,

ψ → −E · r and nj → n∞j , (4.24a)

u→ iωZ and v → −Z. (4.24b)

52

Note that these boundary conditions cannot be directly applied in numerical

computations, because the slowly decaying, oscillating disturbances yield nu-

merical instabilities. We remedy this with an asymptotic analysis detailed in

§ 4.4.

4.3.2 Solution methodology

The equations in § 4.3.1 are solved by linearizing perturbations from an

equilibrium base state governed by the non-linear Poisson-Boltzmann equa-

tion. This methodology is widely adopted for calculating the steady and

dynamic electrophoretic mobilities of colloidal particles (O’Brien & White

1978; Hill et al. 2003a; Mangelsdorf & White 1992) and a variety of other

electrokinetic phenomena. The perturbation approach is accurate when the

applied electric field is sufficiently weak, i.e., |E| κζ. Under these condi-

tions, which are often achieved in experiments, the methodology is much more

computationally efficient than solving the full non-linear model (Masliyah &

Bhattacharjee 2006).

The equilibrium base state (ψ0 and n0j) prevails in the absence of external

stimuli. In this work, perturbations to equilibrium are induced by Y = −Z

and E. Accordingly, ψ and nj are constructed as

ψ = ψ0 + ψ′ = ψ0 −E · r + ψ′′ and nj = n0j + n′j, (4.25)

where linearity and axisymmetry demand

n′j = nj(r)X · er and ψ′′ = ψ(r)X · er, (4.26)

with X ∈ Y ,E. Note that fluid velocity u and polymer displacement v are

perturbed quantities whose constructions are given below.

It is expedient to linearize the perturbations and construct the solution by

superposing two sub-problems with either Y = 0 or E = 0, and the particle

fixed at the origin.

53

As is well known, the equilibrium ion concentrations are

n0j = n∞j exp[−zjeψ0/(kT )], (4.27)

and the Poisson-Boltzmann equation with spherical symmetry is

εoεsL0ψ0 = −

N∑j=1

zjn0je, (4.28)

with boundary conditions

ψ0 = ζ or εsεoψ0,r = −σ at r = a, (4.29a)

ψ0 → 0 as r →∞. (4.29b)

We solve (4.28) using a standard finite difference method with an adaptive

grid (Hill et al. 2003a) that ensures ψ0 decays as exp(−r)/r when (r − a)

κ−1 (Shkel et al. 2000; Verwey & Overbeek 1948).

Following our earlier work (Wang & Hill 2008), the fluid velocity and

compressible polymer displacement are constructed as

u = ∇×∇× [f(r)X]− iωY (4.30)

= −iωY + (−r−1f,r − f,rr)X + (−r−1f,r + f,rr)X · erer,

v = g1(r)X + g2(r)X · erer + Y , (4.31)

so taking the curl of the fluid momentum equation (4.22c) gives

−iωρfL1f,r = ηL2f,rrr − (η/`2)[L1f,r − iω(g1,r − r−1g2)]

−N∑j=1

zjer−1njψ0

,r − n0j,r[ψ − r(E/X)], (4.32)

where

L0(·) = (·),rr + 2r−1(·),r, (4.33a)

L1(·) = (·),rr + 2r−1(·),r − 2r−2(·), (4.33b)

L2(·) = (·),rr + 4r−1(·),r − 4r−2(·). (4.33c)

54

The X and X · erer components of the linear elasticity equation (4.22e) are

0 = µ(g1,rr + 2r−1g1,r + 2r−2g2) + (µ+ λ)(r−1g1,r + r−1g2,r + 2r−2g2)

+(η/`2)(−f,rr − r−1f,r + iωg1), (4.34)

0 = µ(g2,rr + 2r−1g2,r − 6r−2g2) + (µ+ λ)

×(g1,rr + g2,rr − r−1g1,r + r−1g2,r − 4r−2g2)

+(η/`2)(f,rr − r−1f,r + iωg2), (4.35)

and the perturbed Poisson equation (4.22a) and ion-conservation equations

(4.22b) are

εoεsL1ψ = −N∑j=1

zjnje, (4.36)

−iωnj = n0j,r[2r

−1f,r + iω(Y/X)] +DjL1nj + zjeDj(kT )−1

×n0j,r[ψ,r − (E/X)] + ψ0

,rnj,r + n0jL1ψ + njL0ψ

0. (4.37)

The boundary conditions for ψ and nj at r = a are

ψ,r − (E/X)− (εp/εs)[ψ/a− (E/X)] = 0, (4.38a)

zjeDj(kT )−1njψ0,r + n0

j [ψ,r − (E/X)]+Djnj,r = 0, (4.38b)

and no-slip at r = a requires

f,r = −(iωa/2)(Y/X) and f,rr = −(iω/2)(Y/X), (4.38c)

g1 = −(Y/X) and g2 = 0. (4.38d)

Finally, vanishing of the disturbances as r →∞ requires

ψ → 0 and nj → 0, (4.39a)

f,r → 0 and f,rr → 0, (4.39b)

g1 → 0 and g2 → 0. (4.39c)

55

4.3.3 Simplification for incompressible hydrogels

For incompressible hydrogels, the second term in the elasticity equation

(4.22e) is singular because λ → ∞ as ν → 1/2. However, similarly to Hill &

Ostoja-Starzewski (2008), the displacement can be expanded as a power series

in a small parameter ε = 1− 2ν, i.e.,

v = v0 + εv1 + . . . , (4.40)

which, after substituting into (4.22e) and collecting terms of like order in ε,

gives at O(ε−1)

∇ · v0 = 0, (4.41a)

and at O(1)

µ[∇2v0 + ∇(∇ · v1)] + (η/`2)(u+ iωv0) = 0. (4.41b)

Note that −µ∇ · v1 in (4.41b) can be replaced by a pressure, so (4.41a) and

(4.41b) are equivalent to the Stokes equations with a body force (η/`2)(u +

iωv0).

Since ∇ · v0 = 0, the leading-order displacement may be constructed as

v0 = ∇×∇× [g(r)X] + Y . (4.42)

Taking the curl of (4.41b) and (4.22c), the fluid momentum and incompressible

polymer elasticity equations become

−iωρfL1f,r = ηL2f,rrr − (η/`2)(L1f,r + iωL1g,r)

−N∑j=1

zjer−1njψ0

,r − n0j,r[ψ − r(E/X)], (4.43)

0 = µL2g,rrr + (η/`2)(L1f,r + iωL1g,r) (4.44)

with boundary conditions

g,r = (a/2)(Y/X) and g,rr = (1/2)(Y/X) at r = a (4.45)

g,r → 0 and g,rr → 0 as r →∞. (4.46)

56

It is expedient to write (4.43) and (4.44) as

L2

f,rrr

g,rrr

+ML1

f,r

g,r

=

η−1∑N

j=1 zjer−1njψ0

,r − n0j,r[ψ − r(E/X)]

0

,

(4.47)

where

M =

iωρfη−1 − `−2 −iω`−2

ηµ−1`−2 iωηµ−1`−2

. (4.48)

Functions f(r) and g(r) are decoupled by diagonalizing M as

M = R

λ1 0

0 λ2

R−1, (4.49)

where λ1 and λ2 are the eigenvalues of M, and the columns of R are the

corresponding eigenvectors. Substituting M into (4.47) and introducing h1(r)

h2(r)

≡ R−1

f(r)

g(r)

(4.50)

give

L2

h1,rrr

h2,rrr

+L1

λ1h1,r

λ2h2,r

= R−1

η−1∑N

j=1 zjer−1njψ0

,r − n0j,r[ψ − r(E/X)]

0

,

(4.51)

which replaces (4.32), (4.34) and (4.35) above for compressible hydrogels. Note

that f,r in the perturbed ion-conservation equations (4.37) is expressed as a

linear combination of h1 and h2 according to (4.50).

4.3.4 Force and dynamic electrokinetic response

The force and particle response are written in terms of asymptotic coef-

ficients that characterize the far-field decays of the fluid velocity and polymer

displacement. Because the electrical body force vanishes as r → ∞, u and v

have the forms

uX → −iωY + CXr−3X − 3CXr−3X · erer, (4.52a)

vX → Y + ZXr−3X − 3ZXr−3X · erer, (4.52b)

57

where X ∈ E,Y . Recall, (4.52a) and (4.52b) emerge from the two-fluid

model presented in § 4.2. Since the fluid and polymer skeleton move together

in the far field, their respective asymptotic coefficients CX and ZX defined

here are related by ZX = −(iω)−1CX . Accordingly, the force and particle

displacement can be written in terms of CX alone.

Electrical, hydrodynamic, and elastic forces are exerted on the particle for

each E and Y sub-problem. The corresponding stress tensors are the Maxwell

stress

Tm = εoεs∇ψ∇ψ − (1/2)εoεs(∇ψ ·∇ψ)I, (4.53)

Newtonian hydrodynamic stress

Tf = −pI + η[∇u+ (∇u)T ], (4.54)

and linear elastic stress

Te = λ(∇ · v)I + µ[∇v + (∇v)T ], (4.55)

where I is the identity tensor.

The total force is

fX =

∫r=a

(Tm + Tf + Te

)· erdA, (4.56)

so applying Gauss’s divergence theorem to a volume that encloses the particle

surface and a large concentric sphere with radius r →∞ gives

fX =

∫r→∞

(Tm + Tf + Te

)· erdA−

∫ r→∞

r=a

∇ ·(Tm + Tf + Te

)dV. (4.57)

From the governing equations, the divergence of the foregoing stresses are

∇ · Tm = −ρe∇ψ, (4.58a)

∇ · Tf = −iωρfu+ (η/`2)(u+ iωv) + ρe∇ψ, (4.58b)

∇ · Te = −(η/`2)(u+ iωv), (4.58c)

58

where ρe =∑N

j=1 njzje is the free charge density. Accordingly, the force is

fX =

∫r→∞

(Tm + Tf + Te

)· erdA+ iωρf

∫ r→∞

r=a

udV

=

∫r→∞

(Tm + Tf + Te

)· erdA

+iωρf

[∫r→∞

(u · er)rdA−∫r=a

(u · er)rdA

]. (4.59)

Note that the first integral on the left-hand side of (4.59) is finite only if the

stress decays as r−2. The only such term involves the fluid pressure, so with

the no-slip boundary conditions at r = a,

fX = −∫r→∞

perdA+ iωρf

∫r→∞

(u · er)rdA. (4.60)

Substituting the velocity given by (4.52a) into the second integral on the

right-hand side of (4.60) gives∫r→∞

(u · er)rdA = −(8/3)πCXX − iω

∫r→∞

(Y · er)rdA, (4.61)

and, as r →∞,

p→∫ r

[η∇2u+ iωρfu− (η/`2)(u+ iωv)− ρe∇ψ] · erdr′. (4.62)

Since ∇2u ∼ r−5 and ρe∇ψ decays exponentially, the r−2 decaying and grow-

ing contributions give

p = [iωρfCXr−2 − (η/`2)(CX + iωZX)r−2](X · er) + ω2ρfr(Y · er)

= iωρfCXr−2(X · er) + ω2ρfr(Y · er), (4.63)

and, finally, the force on the particle is

fX = −4πiωρfCXX. (4.64)

Note that the radially growing term in (4.63) cancels the surface integral on

the right-hand side of (4.61) when evaluating the force. Equation (4.64) has

59

the same form as for a Newtonian fluid, and is valid for both compressible and

incompressible polymer skeletons.

Superposing the E and Y problems with Y = −Z, and correctly ac-

counting for fluid and particle inertia (see Appendix 4.A), give

−4πiωρfCEE + 4πiωρfC

YZ = ω2VpZ(ρf − ρp), (4.65)

where Vp = (4/3)πa3 is the particle volume. Accordingly, the electric-field-

induced dynamic response is

Z/E = iCE/[iCY + ωa3(ρp − ρf )/(3ρf )]. (4.66)

Comparing this with (4.20) reveals that the equivalent bare Coulomb electrical

force is −4πiωρfCEE.

4.4 Far-field asymptotic analysis

In the numerical computation of CE and CY , the far-field boundary condi-

tions in § 4.3 cannot be applied directly, because more information concerning

the functional forms of the solutions as r →∞ is required to avoid numerical

instabilities.

Beyond the equilibrium double layer, i.e., where r a+κ−1, the equilib-

rium electrostatic potential ψ0 and ion concentrations n0j decay rapidly (expo-

nentially fast) to their far-field values (ψ0 → 0 and n0j → n∞j ) as r →∞, and

the equations governing the linearized perturbations simplify to two decoupled

sets. The first set comprises the Poisson and ion-conservation equations:

εsεo∇2ψ′ +N∑j=1

zjn′je = 0, (4.67a)

iωn′j + zjeDjn∞j (kT )−1∇2ψ′ +Dj∇2n′j = 0, (4.67b)

and the other is identical to (4.3a)–(4.3c) in § 4.2.

In contrast to the full electrokinetic model, these equations have analyti-

cal solutions. We start by considering the first set of equations involving the

60

perturbed potential ψ′′ and ion concentrations n′j, and then establish the con-

nection between the asymptotic forms of f,r, g1 and g2 given in § 4.3.2 and

the exact solution of the two-fluid model in § 4.2. A separate analysis for

incompressible skeletons follows.

4.4.1 Far-field decays of ψ and nj

Equations (4.67a) and (4.67b) can be written

εsεoL1ψ +N∑j=1

zjnje = 0, (4.68a)

DjL1nj − zje2Djn∞j (εsεokT )−1

N∑k=1

zknk + iωnj = 0, (4.68b)

or

L1

n1

n2

...

nN

ψ

+ P

n1

n2

...

nN

ψ

= 0, (4.69)

where

P =e2

εsεokT

iωεsεokTe2D1

− n∞1 z21 −n∞1 z1z2 . . . −z1n

∞1 z2 − n∞1 z1zN 0

−n∞2 z2z1iωεsεokTe2D2

− n∞2 z22 . . . −n∞2 z2zN 0

......

. . ....

...

−n∞N zNz1 −n∞N zNz2 . . . iωεsεokTe2DN

− n∞N z2N 0

z1kT/e z2kT/e . . . zNkT/e 0

.

(4.70)

Similarly to the incompressible problem, matrix P can be diagonalized as

P = Q

γ1 0 . . . 0

0 γ2 . . . 0...

.... . .

...

0 0 . . . γN+1

Q−1, (4.71)

61

where γk (k = 1, 2, . . . , N + 1) are the eigenvalues of P, and the kth column

of Q is the corresponding eigenvector. By setting

χ1

χ2

...

χN

χN+1

≡ Q−1

n1

n2

...

nN

ψ

, (4.72)

we obtain a set of simpler equations

L1χk + γkχk = 0 (k = 1, 2, . . . , N + 1), (4.73)

where χk = χk(r). Using standard techniques (Lamb 1945; Markov 2005;

Oestreicher 1951; Temkin & Leung 1976; MacRobert 1967), the solutions of

(4.73) are

χk(r) = Dkh(√γkr) when γk 6= 0 (4.74a)

χk(r) = Dkr−2 when γk = 0, (4.74b)

where Dk are unknown constants. From (4.70), P has only one γk = 0, and to

ensure decaying solutions as r →∞, Im(√γk) > 0 for all γk 6= 0.

Equation (4.73) demonstrates that χk are decoupled in the far field.

Therefore nj and ψ can be constructed by inverting (4.72). Dividing the

far-field expressions for χk by their first derivative eliminates the constants

Dk, and therefore the boundary conditions for χk as r →∞ are

χk − [h(√γkr)/h

′(√γkr)]χk,r = 0 at r = rmax when γk 6= 0, (4.75)

χk + (r/2)χk,r = 0 at r = rmax when γk = 0, (4.76)

where rmax is the maximum radial extent of the numerical calculations where

a ≤ r ≤ rmax. Note that the kth column of Q corresponding to γk = 0

has only one non-zero entry, which equals one, and the dipole strength of the

62

electrostatic potential is

DX = r2maxχk(rmax) when γk = 0 as rmax →∞. (4.77)

4.4.2 Far-field decays of f,r, g1 and g2

Since the electrical body force vanishes far from the particle, i.e., where

r a+ κ−1, the fluid velocity and polymer displacement in § 4.2 can be used

to construct f,r(r), g1(r) and g2(r) in the far field. These provide boundary

conditions at r = rmax for the numerical solution in the region where a ≤ r ≤

rmax with rmax a+ κ−1.

From § 4.3.2, the radial and tangential components of the fluid velocity

are

ur = [−2r−1f,r − iω(Y/X)]X cos θ, (4.78a)

uθ = [−r−1f,r − f,rr − iω(Y/X)](−X sin θ), (4.78b)

and the radial and tangential components of the polymer displacement are

vr = [g1 + g2 − (Y/X)]X cos θ, (4.78c)

vθ = [g1 − (Y/X)](−X sin θ). (4.78d)

Equating these to the exact solution in § 4.2 with X = Y gives

f∞,r = A1r−2 −B1h(K1r)−B2h(K2r), (4.79a)

g∞1 = mA1r−3 + A2r

−1h(kr)

+2∑j=1

MjBjKj[(Kjr)−1h(Kjr) + h′(Kjr)], (4.79b)

g∞2 = −3mA1r−3 + A2k[(kr)−1h(kr)− h′(kr)]

+2∑j=1

MjBjKj[(Kjr)−1h(Kjr)− h′(Kjr)], (4.79c)

where superscripts “∞” denote far-field asymptotic solutions. Similarly to the

numerical calculation of electrophoretic mobilities in Newtonian electrolytes,

the foregoing far-field asymptotic solutions each differ from their exact solution

63

by a multiplicative constant. Therefore, the boundary conditions at r = rmax

for the numerical computation of f,r(r), g1(r) and g2(r) are

f,r − (f∞,r /f∞,rr)f,rr = 0, (4.80a)

f,rr − (f∞,rr/f∞,rrr)f,rrr = 0, (4.80b)

g1 − (g∞1 /g∞1,r)g1,r = 0, (4.80c)

g2 − (g∞2 /g∞2,r)g2,r = 0, (4.80d)

and the asymptotic coefficient

CX = A1f,r(rmax)/f∞,r (rmax) as rmax →∞. (4.81)

4.4.3 Far-field analysis for incompressible hydrogels

The far-field decay of the fluid velocity and polymer displacement for

incompressible hydrogels is handled in a similar manner to § 4.3.3. As r →∞,

the combined fluid momentum and O(1) linear elasticity equations are

∇2

u

v0

+ ∇

−η−1p

∇ · v1

+ M

u

v0

= 0, (4.82)

where M is defined in (4.48). Diagonalizing M according to (4.49), and left-

multiplying by R−1, which is defined in (4.49), yield two decoupled equations

∇2wj + ∇qj + λ1wj = 0 (j = 1, 2), (4.83)

where w1

w2

= R−1

u

v0

and

q1

q2

= R−1

−η−1p

∇ · v1

. (4.84)

An incompressible fluid and polymer skeleton require ∇ · wj = 0 (j = 1, 2),

so the solutions are

wj = ∇Φj + ∇×Ψj +W j (j = 1, 2), (4.85)

64

where Φj are scalar potentials, Ψj = Ψjeφ are vector potentials, and W 1

W 2

= R−1

−iωY

Y

. (4.86)

Substituting the relations above into (4.83) and taking the curl yields

∇2Ψj + λjΨj = 0 (j = 1, 2), (4.87)

and incompressibility requires ∇2Φi = 0, so

Φj = A′jr−2Y cos θ, (4.88a)

Ψj = B′jh(Kjr)Y sin θ, (4.88b)

where A′j and B′j (j = 1, 2) are constants to match the boundary conditions at

r = a, and Kj =√λj with Im(Kj) > 0 to ensure vanishing polymer displace-

ment and fluid velocity as r → ∞. The radial and tangential components of

wj = ∇×∇× [hj(r)Y ] +W j are

wjr = [−2A′jr−3 + 2B′jr

−1h(Kjr) + (Wj/X)]X cos θ, (4.89a)

wjθ = A′jr−3 +B′jKj[(Kjr)−1h(Kjr) + h′(Kjr)] + (Wj/X)

×(−X sin θ). (4.89b)

Matching at r = rmax gives

h∞j,r = A′jr−2 −B′jh(Kjr) as rmax →∞, (4.90)

where the four constants A′j and B′j (j = 1, 2) are obtained from the boundary

conditions at r = a with X = Y as h∞1,r(a)

h∞2,r(a)

= R−1

−iωa/2

a/2

(4.91a)

and h∞1,rr(a)

h∞2,rr(a)

= R−1

−iω/2

1/2

. (4.91b)

65

Finally, the far-field boundary conditions for hj at r = rmax are

hj,r − (h∞j,r/h∞j,rr)hj,rr = 0 and hj,rr − (h∞j,rr/h

∞j,rrr)hj,rrr = 0 (j = 1, 2) (4.92)

and the asymptotic coefficient

CX =2∑j=1

R1jA′jhj,r(rmax)/h∞j,r(rmax) as rmax →∞, (4.93)

where R11 and R12 are elements of R.

4.5 Connection to electroacoustics

Here we show the close connection between the electroacoustic properties

of hydrogel-colloid composites and the dynamic electric-field-induced response

of a single particle addressed in § 4.3. Following O’Brien (1988, 1990), our

analysis is for composites with arbitrary colloid concentration. The macro-

scopic momentum, mass, and charge conservation equations, and suspension

constitutive equations from O’Brien (1990) can be directly applied to hydrogel-

colloid composites, since the elasticity of the polymer does not invalidate the

macroscopic equations. However, elasticity changes an integral that underlies

the electroacoustic reciprocal relation, which is crucial for subsequent simpli-

fications of the governing equations.

Electroacoustic signals originate from perturbations to an equilibrium

base state. To linear order, these perturbations satisfy

εsεo∇2ψ′ = −ρe′, (4.94a)

−iωn′j = −∇ · jj, (4.94b)

−iωρfu = ∇ · Tf − ρe′∇ψ0 − ρe0∇ψ′ − (η/`2)(u+ iωv), (4.94c)

0 = ∇ · u, (4.94d)

0 = ∇ · Te + (η/`2)(u+ iωv), (4.94e)

where the perturbed ion fluxes are

jj = −Dj∇n′j − zjeDj(kT )−1n′j∇ψ0 − zjeDj(kT )−1n0j∇ψ′ + n0

ju. (4.94f)

66

The primes denote perturbed quantities and the superscripts “0” denote the

equilibrium base state. The local current density in a harmonically oscillating

external field is (O’Brien 1988, 1986; DeLacey & White 1981)

i =N∑j=1

jjzje+ iωεoεs∇ψ′, (4.95)

and satisfies ∇ · i = 0.

Following O’Brien (1988, 1990), consider an integral over a representative

volume V enclosed by a surface A,

V −1

∫A

[u1 · Tf

2 − iωv1 · Te2 − i1ψ′2 −

N∑j=1

kT

n0j

n′j2(jj1 − n0ju1)

]· ndA (4.96)

for two systems “1” and “2”, where n is an outward unit normal. From

the linearly perturbed equations and Gauss’s divergence theorem, the integral

above is

V −1

∫Vh

[2ηef1 : ef2 − 2iωµee1 : ee2 + (η/`2)(u1 + iωv1) · (u2 + iωv2)

+N∑j=1

kT

Djn0j

(jj1 − n0ju1) · (jj2 − n0

ju2)

]dV

−iωV −1

∫Vh

[λ(∇ · v1)(∇ · v2) + ρfu1 · u2 + εoεs(∇ψ′1) · (∇ψ′2)

+N∑j=1

kT

n0j

n′j1n′j2

]dV

+V −1∑p

∫Ap

[u1 · Tf

2 − iωv1 · Te2 − i1ψ′2

−N∑j=1

kT

n0j

n′j2(jj1 − n0ju1)

]· ndA, (4.97)

where ef = (1/2)[∇u+ (∇u)T ] and ee = (1/2)[∇v + (∇v)T ]. Note that∑

p

indicates a sum over all the particles enclosed by A, Ap is the particle surface

within and intersecting A, and Vh is the hydrogel volume excluding particles

enclosed by A. According to O’Brien (1990), exchanging indices “1” and “2”

in (4.97) does not change the integral.

67

If the surface A is large enough to contain many particles, and its radius

of curvature is everywhere greater than the length scales associated with fluc-

tuations of the perturbed quantities, it is a “macroscopic boundary” and can

be divided into portions that are small enough to neglect their curvature, but

large enough to average the perturbed quantities (O’Brien 1979). Also, for an

incompressible fluid, we need to ensure that the size of A is smaller than the

sound wave length (O’Brien 1988). With these constraints, and an assumption

of statistical homogeneity, (4.96) is

〈u〉1 ·∇〈p〉2 + 〈i〉1 · 〈E〉2, (4.98)

where the angled brackets denote volume averaged quantities,

〈·〉 ≡ V −1

∫V

(·)dV. (4.99)

Note that only terms that grow with r contribute to the integral in (4.96) over

the macroscopic surface A, and consequently the second and fourth terms in

(4.96) are negligibly small. Surprisingly, although the polymer displacement v

and elastic stress tensor Te enter the integral in (4.96), they do not contribute

to (4.98).

Since disturbances to the hydrogel-colloid composite can only be intro-

duced through external electric fields and sound waves (pressure gradients),

the electroacoustic constitutive relations are the same as for colloidal disper-

sions, i.e.,

〈U〉 = α∇〈p〉+ µd〈E〉 and 〈i〉 = γ∇〈p〉+K∗〈E〉, (4.100)

where α, µd, γ and K∗ are composite transport properties: K∗ is the complex

conductivity and µd is the particle dynamic electrophoretic mobility. Note

that 〈U〉 is the particle velocity averaged over all particles in V (O’Brien

et al. 2003); it is connected to the average fluid velocity 〈u〉 via the mass

68

averaged momentum

〈ρ0〉u ≡ 〈ρu〉 = ρf〈u〉+ (ρp − ρf )φ〈U〉, (4.101)

where 〈ρ0〉 is the equilibrium composite density, ρ is the position dependent

local density, and φ is the particle volume fraction.

Setting 〈E〉1 = ∇〈p〉2 = 0 for the two systems that differ only by the

boundary conditions due to external fields, and noting that (4.98) is indepen-

dent of an exchange of indices, it follows that

〈u〉2 ·∇〈p〉1 = 〈i〉1 · 〈E〉2. (4.102)

Therefore, from the macroscopic momentum conservation equation (O’Brien

1990, § 3), u = 0 if ∇〈p〉 = 0, and together with (4.100)–(4.102) we have

γ = ρ−1f (ρp − ρf )φµd. (4.103)

This is the same as O’Brien’s formula for colloidal dispersions (O’Brien 1990,

§ 5), so his subsequent analysis for particulate suspensions is applicable to

hydrogel-colloid composites. Accordingly, the elasticity introduced by the

polymer skeleton only affects the electroacoustic response through the dy-

namic electrophoretic mobility

µd = −iω(Z/E), (4.104)

with all other macroscopic relations the same as for Newtonian colloidal dis-

persions.

One concern with electroacoustic characterization of hydrogel-colloid com-

posites is whether the pressure fluctuations in an ESA experiment are measur-

able. Therefore, in § 4.7 we compare the dynamic electrophoretic mobilities

of particles in hydrogels and Newtonian fluids at the operating frequencies of

commercial electroacoustic instruments. Noteworthy is that the mobility for

69

hydrogel composites has a comparable amplitude to those of Newtonian dis-

persions, but with characteristics that are sensitive to the shear modulus of the

polymer skeleton. Moreover, in the relevant frequency range, the response for

particles with κa 1 can be captured with the following analytical solution

of the electrokinetic model.

4.6 High-frequency boundary-layer approximation

In this section we derive an approximate expression for the frequency de-

pendent response Z/E when κa 1 and a `. The dynamics of the diffuse

double layer are calculated using the surface conduction model of O’Brien

(1986), which is valid when ω a−2D. With a ∼ 1 µm and D ∼ 10−9 m2 s−1,

the model is valid at frequencies beyond the kHz range where colloidal dy-

namics are accessed as dynamic electrophoretic mobilities in electroacoustic

experiments.

We solve this problem using the decomposition and superposition tech-

niques outlined in § 4.3. Briefly, the approximate asymptotic coefficients for

the E and Y sub-problems are calculated, and the response Z/E is obtained

by the superposition leading to (4.66). For the Y sub-problem with κa 1,

we neglect the influence of surface charge, so the full model reduces to the

two-fluid model addressed in § 4.2. In this case, CY = A1, where A1 is given

in (4.18a). For the E sub-problem, we calculate the polymer displacement

and fluid velocity inside the thin double layer, and match the inner and outer

solutions to obtain CE.

In our earlier publication (Wang & Hill 2008), we derived a boundary-layer

approximation for the steady response Z/E given in (4.1). The contribution

that depends on the particle size, and permeability and compressibility of the

hydrogel, arises from terms in the inner solution that are O[(κa)−1] smaller

than the leading-order terms as κa → ∞. Note that hydrogel compressibil-

ity is important only if ω . τ−1d . Moreover, hydrogel-colloid composites with

a ∼ 1 µm, ` ∼ 10 nm, E ∼ 1 kPa and ν ∼ 0.2 have a reciprocal draining

time τ−1d ∼ 1 kHz that is comparable to the lower frequency limit of O’Brien’s

70

surface conduction model. Therefore, under conditions where the surface con-

duction model is valid, the hydrogel can also be considered incompressible.

Accordingly, with ω & τ−1d it is reasonable to neglect the O[(κa)−1] terms in

the inner solution, which tremendously simplifies the resulting expression for

Z/E.

As shown by O’Brien (1988), perturbations to the equilibrium ion con-

centrations in the inner and outer regions are negligible at the frequencies of

interest for electroacoustics, so they do not enter the fluid momentum equation.

Therefore, the perturbed electrostatic potential ψ′ satisfies Laplace’s equation

outside the equilibrium diffuse double layer. Accordingly, in the outer region

ψ′ = −rE cos θ − a3Pr−2E cos θ, (4.105)

where P is the dipole strength. Analysis of conduction within a thin surface

layer (O’Konski 1960) gives

P =(1− iω′)− (2Du− iω′εp/εs)

2(1− iω′) + (2Du− iω′εp/εs), (4.106)

where ω′ = ωεsεo/σ∞, σ∞ =

∑Nj=1 z

2jn∞j Dje

2/(kT ) is the bulk conductiv-

ity, and Du = σs/(σ∞a) is the Dukhin number (Lyklema 1995). The con-

nection between the surface conductivity σs and the ζ-potential is addressed

below (Hunter 2001; Lyklema 1995).

Adopting standard boundary-layer scaling (Pozrikidis 1996), the fluid mo-

mentum and linear elasticity equations (radial and tangential directions) in the

inner region where κ(r − a) 1 are

−p,r − ρe0ψ′,r = 0, (4.107a)

−a−1p,θ + ηuθ,rr − ρe0a−1ψ′,θ − (η/`2)(uθ + iωvθ) = 0, (4.107b)

(2µ+ λ)vr,rr + (µ+ λ)(a sin θ)−1(vθ sin θ),rθ

+(η/`2)(ur + iωvr) = 0, (4.107c)

µvθ,rr + (η/`2)(uθ + iωvθ) = 0, (4.107d)

71

with fluid continuity equation

vr,r + (a sin θ)−1(uθ sin θ),θ = 0. (4.107e)

Note that the equilibrium charge density ρe0 = −κ2εsεoζ exp[−κ(r− a)] when

|ζ| kT/e and κa 1 (Hunter 2001), and fluid inertia has been neglected

inside the double layer, since it is important only beyond the GHz range (Hill

et al. 2003b). For simplicity, we addresses the inner problem when |ζ| kT/e

before considering higher ζ-potentials.

Since p and ψ′ both vary on the length scale of the particle size a, to lead-

ing order they are radially invariant in the inner region. Therefore, (4.107a)

gives p = pE cos θ, where p is a constant. By setting uθ = uθ(y)E sin θ and

vθ = vθ(y)E sin θ, where y = r − a, the tangential equations (4.107b) and

(4.107d) become

ηuθ,yy − (η/`2)(uθ + iωvθ) + a−1p+ ζκ2εsεo(1 + P ) exp(−κy) = 0, (4.108)

µvθ,yy + (η/`2)(uθ + iωvθ) = 0. (4.109)

With boundary conditions vθ = uθ = 0 at y = 0, and finite vθ and uθ as

y →∞, the solutions are

uθ = iωεsεoζ(1 + P )(µ− iωη)−1[exp(−κy)− 1]

−µc1η−1 exp(−sy) + pµ[ηa(µ− iωη)s2]−1

+µκ2εoεsζ(1 + P )[η(µ− iωη)(s2 − κ2)]−1 exp(−κy), (4.110)

vθ = −εsεoζ(1 + P )(µ− iωη)−1[exp(−κy)− 1]

+c1 exp(−sy)− p[a(µ− iωη)s2]−1

−κ2εoεsζ(1 + P )[(µ− iωη)(s2 − κ2)]−1 exp(−κy), (4.111)

where s2 = (µ− iωη)(µ`2)−1 with Re(s) > 0. The no-slip boundary conditions

at y = 0 require

c1 = p[(µ− iωη)s2a]−1 + κ2εsεoζ(1 + P )[(µ− iωη)(s2 − κ2)]−1. (4.112)

72

For the inner solution to be finite as y → ∞, the terms that are linear and

quadratic in y must be neglected. Integrating (4.107e) gives ur = ur(y)E cos θ,

where

ur = −2a−1

∫ y

0

uθ(y′)dy′

= 2εsεoζ(1 + P )[κa(µ− iωη)]−1iω + µκ2[η(s2 − κ2)]−1[exp(−κy)− 1]

−2µc1(ηsa)−1[exp(−sy)− 1]. (4.113)

Note that setting ω = 0 recovers the steady solutions derived in earlier works (Wang

& Hill 2008; Hill 2006b).

Finally, integrating (4.107c) gives vr = vr(y)E cos θ, where

vr = c2 exp(−ny)− a1(n2 − κ2)−1 exp(−κy)

−a2(n2 − s2)−1 exp(−sy) + a3[n−2 − (n2 − κ2) exp(−κy)]

+a4[n−2 − (n2 − s2) exp(−sy)] (4.114)

with n2 = −iωη[`2(2µ+ λ)]−1 and Re(n) > 0,

a1 = −(µ+ λ)κεsεoζ(1 + P )[a(2µ+ λ)(µ− iωη)]−1

×[1 + κ2(s2 − κ2)−1], (4.115a)

a2 = (µ+ λ)sc1[a(2µ+ λ)]−1, (4.115b)

a3 = −ηεsεoζ(1 + P )[κa`2(2µ+ λ)(µ− iωη)]−1

×iω + µκ2[η(s2 − κ2)]−1, (4.115c)

a4 = µc1[`2sa(2µ+ λ)]−1, (4.115d)

and c2 is chosen to ensure vr = 0 at y = 0.

73

In the outer region, the governing equations are the same as the two-fluid

model in § 4.2, so

u = AE1 ∇(r−2E · er) +2∑j=1

BEj ∇× [h(Kjr)E × er], (4.116a)

v = mAE1 ∇(r−2E · er) + AE2 ∇[h(kr)E · er]

+2∑j=1

MjBEj ∇× [h(Kjr)E × er], (4.116b)

where k, m, Kj and Mj (j = 1, 2) are also the same as in § 4.2. Note that

the asymptotic coefficient CE = AE1 is determined by matching the inner and

outer solutions. From (4.3a), the pressure in the outer region is

p = iωρfAE1 r−2E cos θ − iω(η/`2)AE2 h(kr)E cos θ, (4.117)

so

p = iωρfAE1 a−2 − iω(η/`2)AE2 h(ka). (4.118)

Matching the inner solutions (as y →∞)

uθ = −iω(µ− iωη)−1εsεoζ(1 + P )E sin θ and ur = 0, (4.119a)

vθ = (µ− iωη)−1εsεoζ(1 + P )E sin θ and vr = 0, (4.119b)

to the outer solutions (4.116a) and (4.116b) (as r → a) gives

AE1 = a3εsεoζ(1 + P )(−iωηµ−1ΘE + ΓE)[(µ− iωη)H]−1, (4.120a)

where, from b and βj (j = 1,2) in § 4.2,

ΘE = −µη−1ib2(M1 −M2)(β1 + i)(β2 + i)

+(b2 + 2ib− 2)[M1β21(β2 + i)−M2β

22(β1 + i)], (4.120b)

ΓE = (b2 + 2ib− 2)[β21(β2 + i)− β2

2(β1 + i)]. (4.120c)

Note that the matching is achieved by neglecting asymptotically small terms

as κa→∞ and `/a→ 0.

74

Next, we need to establish the connection between the Dukhin number

Du and ζ-potential for hydrogels to determine P . This relation is not the

same as for Newtonian electrolytes, because the polymer skeleton modulates

the convective transport of ions (O’Brien 1986). Following a similar procedure

for obtaining (4.110), the tangential velocity in the double layer as κa→∞ is

uθ = iωεsεo(µ− iωη)−1(ψ0 − ζ)∇sψ′, (4.121)

where ∇s is the tangential gradient operator.

Although (4.121) is derived for |ζ| kT/e, the vanishing of the equilib-

rium electrostatic potential ψ0 outside the double layer permits (4.121) to be

generalized for any ζ-potential.

From the definition of Du and σs (O’Brien 1986; Lyklema 1995), and

using (4.121), for symmetrical z-z electrolytes with |ζ| kT/e,

Du = (z2 + 3m)(2κa)−1[ζe/(kT )]2, (4.122)

where m = −2iωεsεo[3D(µ − iωη)]−1(kT/e)2, and D = D1 = D2 is the ion

diffusivity. For general electrolytes and higher ζ-potentials (O’Brien 1986),

Du =n∞i z

2i Di

√2∑N

j=1 z2jn∞j Dj

(1 +

3mi

z2i

)exp[−eziζ/(2kT )]

κia(4.123)

when exp[−eziζ/(2kT )] 1. Here, the subscripts “i” refer to the counter ion

with highest charge, and

mi = −2iωεsεo[3Di(µ− iωη)]−1(kT/e)2, (4.124)

κ2i = (z2

i e2n∞i )(εsεokT )−1. (4.125)

Finally, superposing the Y and E sub-problems, and accounting for fluid

and particle inertia (see Appendix 4.A), the dynamic response is

Z/E = εsεoζ(1 + P )(µ− iωη)−1G, (4.126)

75

Table 4–1: Parameters for the results shown in figure 4–1.

Particle radius, a 500 nmFluid viscosity, η 8.904× 10−4 Pa sPolymer Young’s modulus, E 1 kPaPolymer Poisson’s ratio, ν 0.2Fluid density, ρf 997 kg m−3

Particle density, ρp 2000 kg m−3

where

G =−iωηµ−1ΘE + ΓE

(Θ + Γ)/2 + ω2H(ρp − ρf )/(3ρf ). (4.127)

Recall, this approximation is valid for frequencies ω a−2D, ω 2πτ−1d , and

ω κ2ηρ−1f with κa 1 and `/a 1.

4.7 Results

4.7.1 Response functions for an uncharged particle

The response function α(ω) for an uncharged particle is shown in figure 4–

1. Our exact analytical solution derived in (4.20) is compared with the approx-

imation of Levine & Lubensky (2001) and the GSER where α(ω) = [6πa(µ−

iωη)]−1. The figure also highlights several characteristic frequencies (recip-

rocal time scales) identified in the introduction. Note that ωB = η−1(2µ +

λ)(`/a)2(π2/4), ω∗ satisfies |β(ω∗)| = 1 where β(ω) = 4a2ω2ρf/[(µ − iωη)π2],

ωd = 2π(1 − 2ν)−1(E/η)(`/a)2, and ωv = 2πµ/η. The first two of these were

adopted by Levine & Lubensky (2001), and the other two are, respectively,

the reciprocal draining time and reciprocal viscous time introduced in § 4.1.

The results with ` = 1 and 100 nm are representative of tightly and loosely

coupled fluid and polymer. Other parameters are summarized in table 4–1.

Recall, the first and second Lame constants (µ and λ) of the hydrogel skeleton

are taken to be real constants.

The transition from quasi-steady compressible to incompressible dynam-

ics is evident from the plateau seen in figure 4–1(a) at intermediate frequencies

76

10-2

100

102

104

106

108

1010

ω, (rad/s)

-50

0

50

100

150

200

250

300

350R

e[α(

ω)]

, (m

/N)

103

104

105

106

107

108

10910

-3

10-2

10-1

100

101

102

103

|Re[

α(ω

)]|,

(m/N

)ω

B ωd

ωv ω∗

(a) ` = 1 nm, Re(α)

10-2

100

102

104

106

108

1010

ω, (rad/s)

0

40

80

120

160

Im[α

(ω)]

, (m

/N)

103

104

105

106

107

108

10910

-3

10-2

10-1

100

101

102

103

|Im[α

(ω)]

|, (m

/N)

ωB ω

d

ωv

ω∗

(b) ` = 1 nm, Im(α)

10-2

100

102

104

106

108

1010

ω, (rad/s)

-50

0

50

100

150

200

250

300

350

Re[

α(ω

)], (

m/N

)

ωB

ωd

ωv ω∗

(c) ` = 100 nm, Re(α)

10-2

100

102

104

106

108

1010

ω, (rad/s)

0

40

80

120

160

Im[α

(ω)]

, (m

/N)

ωB

ωd

ωv

ω∗

(d) ` = 100 nm, Im(α)

Figure 4–1: Response function α(ω) as a function of angular frequency ωfor different Brinkman screening lengths: ` = 1 nm [panels (a) and (b)] and` = 100 nm [panels (c) and (d)]. Other parameters are listed in table 4–1.Solid lines are exact solutions of the two-fluid model (4.20), dashed lines arethe approximation of Levine & Lubensky (2001), and the dash-dotted linesare the GSER. Several characteristic frequencies are identified (see text fordetails): ωv ≈ 2.9×106 rad s−1 and ω∗ ≈ 8.8×106 rad s−1 for all panels; ωB ≈12 rad s−1, ωd ≈ 47 rad s−1 [panels (a) and (b)]; and ωB ≈ 1.23× 105 rad s−1,ωd ≈ 4.70× 105 rad s−1 [panels (c) and (d)].

77

with ` = 1 nm. When ` = 100 nm, however, the draining time τd is compa-

rable to the viscous time τv, so the low-frequency plateau (compressible elas-

tic regime) in figure 4–1(c) transits to the high-frequency viscous dominated

regime without an intermediate (incompressible elastic) plateau.

The real parts of the quasi-steady compressible and quasi-steady incom-

pressible elastic plateaus differ by at most 25%, as predicted by Schnurr et al.

(1997). This relatively small change is often used to justify neglecting com-

pressibility when interpreting optical and magnetic tweezers microrheology

experiments (Schnurr et al. 1997; Ziemann et al. 1994). However, if the exter-

nal force is accompanied by electroosmotic flow, the effect of compressibility

is much more significant when κa & 1 (Wang & Hill 2008). This is explored

when we address the electric-field-induced response.

Levine and Lubensky’s approximation is valid when fluid inertia can be

neglected. Accordingly, it agrees well with the exact solution when ω ω∗.

As seen in the insets of figures 4–1(a) and 4–1(b), which have logarithmic

axes, Levine and Lubensky’s theory yields increasingly large relative errors

at higher frequencies. However, the absolute displacement is practically zero

at such high frequencies, so the errors are of minor concern for magnetic and

optical microrheology, but have important consequences for electroacoustics.

Earlier studies suggest the GSER is valid at frequencies between ωB and

ω∗ (Levine & Lubensky 2001, 2000). However, our exact results in figure 4–

1 show that the GSER is a good approximation only when the quasi-steady

incompressible elastic plateau is present. At higher frequencies, the GSER de-

viates from the exact solution at frequencies well below ω∗, because it does not

account for fluid or particle inertia. Consequently, our calculations reveal that

the frequency range of validity for the GSER is considerably narrower than

previously expected. Also, the characteristic frequencies ωB and ω∗ intro-

duced by Levine & Lubensky (2001) are similar here to ωd and ωv adopted in

this work; they are practically equivalent for mapping out the experimentally

accessible parameter space.

78

Our exact solution of the two-fluid model is essential for the following

numerical and analytical solutions of the multi-phase electrokinetic model. As

described in § 4.4, the two-fluid model provides far-field boundary conditions

for the fluid velocity and polymer displacement fields, and it is also the ba-

sis of our analytical approximation for the high-frequencies encountered in

electroacoustics.

4.7.2 Numerical solution of the multi-phase electrokinetic model

The model presented in § 4.3 is solved numerically by adopting κ−1, u∗ =

εsεo(kT/e)2/(ηa) and ηu∗/µ = εsεo(kT/e)

2/(µa) as the characteristic scales

for length, fluid velocity and polymer displacement, respectively; and similarly

to (4.66), the dimensional response Z/E is obtained from the dimensionless

asymptotic coefficients CE and CY as

Z/E = iCE[εoεs(kT/e)/(µκa)]/[iCY − (ωη/µ)(κa)3(ρf − ρp)/(3ρf )]. (4.128)

Separate programs were written to compute the response for compressible and

incompressible hydrogels. Asymptotic coefficients are extracted from the far-

field decay of the perturbations, and the dynamic response Z/E is obtained

from the superposition in (4.66) or (4.128). Note that the asymptotic analysis

detailed in § 4.4 permits Z/E to be calculated over an extraordinarily wide

range of frequencies, from as low as 0.01 Hz to higher than 1 GHz. An algo-

rithmic description of the computational methodologies and external libraries

used in our programs is provided in Appendix 4.B.

A representative spectrum of Z/E for a colloidal particle in a compressible

hydrogel is presented in figure 4–2. The computations are validated, in part, by

the steady boundary-layer results for compressible and incompressible hydro-

gels shown in (4.1) (Hill & Ostoja-Starzewski 2008; Wang & Hill 2008). Note

that the response (real part) undergoes a distinct transition from quasi-steady

compressible to quasi-steady incompressible elastic plateaus as the frequency

passes through the reciprocal draining time τ−1d ≈ 150 Hz. The transition is

79

slow, however, spanning several frequency decades, with the steady compress-

ible asymptote realized at extremely low frequencies ∼ 10−2 Hz. Note that

the small discrepancy between the steady (horizontal line) and low-frequency

dynamic asymptote in figure 4–2 is due to small errors in the boundary-layer

approximation (4.1), which, in this example, is about 2% smaller than the

numerically exact value (Wang & Hill 2008).

The quasi-steady compressible and incompressible elastic responses differ

by an order of magnitude. At higher frequencies, |Z/E| decays to zero be-

cause the viscous and inertial stresses dominate the particle response. Note

that in active microrheology, most experimentally accessible frequencies are

in the transition from the quasi-steady compressible to incompressible elastic

regimes, so these dynamic calculations are essential for correctly interpreting

such experiments. For the hydrogel-colloid composite in figure 4–2, an applied

electric field E = 20 V/cm with frequency ω/(2π) = 1 Hz induces a parti-

cle displacement with amplitude Z ≈ 4 nm, which could be resolved using

back-focal-plane interferometry (Allersma et al. 1998). The sub-nanometer

displacements at higher frequencies (& 10 kHz) are clearly too small to detect

using any direct measurement of particle displacement. Instead, the parti-

cle velocity −iωZ should be measured, as routinely undertaken at ultrasonic

frequencies in electroacoustic experiments.

The high-frequency regime is examined in figure 4–3, where the absolute

values of the real and imaginary parts of Z/E for particles in a hydrogel (solid

and dashed lines) are compared with their counterparts (Z/E)∗ = −µd∗/(iω)

for the same particles dispersed in the electrolyte without polymer. Accord-

ingly, quantities with superscripts “∗” are from numerically exact solutions of

the standard electrokinetic model (Mangelsdorf & White 1992), as calculated

by the MPEK software package (Hill et al. 2003a). Important characteris-

tic frequencies in § 4.1, including the reciprocal draining time τ−1d , reciprocal

viscous time τ−1v , and reciprocal inertia time τ−1

f are identified. Again, the

80

10-2

100

102

104

106

108

frequency, (Hz)

0

0.05

0.1

0.15

0.2

0.25

0.3

-Z/E

, [nm

/(V

/cm

)]

10-2

100

102

104

106

10810

-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

Figure 4–2: Representative frequency spectrum of Z/E for a charged colloidalsphere embedded in an uncharged, compressible electrolyte-saturated hydro-gel: NaCl at T = 298 K; a = 500 nm; κa = 500; −ζe/(kT ) = 3; ` = 5 nm;ρp = 1050 kg m−3, ν = 0.2; and E = 800 Pa. Solid and dashed lines are thereal and imaginary parts of Z/E from numerically exact solutions of the fullmulti-phase electrokinetic model; and dash-dotted lines are from (4.1) withν = 0.2 and 0.5.

transition from quasi-steady compressible to quasi-steady incompressible elas-

tic dynamics is clearly evident as the frequency passes through τ−1d ≈ 150 Hz.

More importantly, this figure highlights the transition from quasi-steady elas-

tic to viscous dynamics as the frequency passes through τ−1v ≈ 370 kHz. At

frequencies beyond τ−1f ≈ 3.6 MHz, both the real and imaginary parts of Z/E

equal their (Z/E)∗ counterparts. Therefore, it is only at these ∼MHz frequen-

cies that the electroacoustic response of the hydrogel composite is the same as

for its respective colloidal dispersion (Hunter 1998; O’Brien 1988). Moreover,

at lower frequencies, the electroacoustic response probes the shear modulus of

the polymer skeleton and the size and charge of the inclusions.

Having identified several qualitative features of a typical frequency spec-

trum, let us explore the influences of various parameters. First, figure 4–4

shows how Poisson’s ratio, increasing from 0 to 0.5 with fixed shear modulus

µ ≈ 0.333 kPa, affects the response. Recall, the spectrum for the incom-

pressible hydrogel (ν = 0.5) was calculated using an independently developed

81

10-2

100

102

104

106

108

frequency, (Hz)

10-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

τv

-1

τd

-1

τf

-1

Figure 4–3: Comparison of Z/E for a compressible hydrogel with the response(Z/E)∗ = −µd∗/(iω) for a classical Newtonian dispersion. Parameters are thesame as in figure 4–2. Solid and dashed lines are the real and imaginary partsof Z/E from numerically exact solutions of the full multi-phase electrokineticmodel. Dash-dotted and dash-double-dotted lines are the real and imaginaryparts of (Z/E)∗ = −µd∗/(iω) calculated from the MPEK software package(Hill et al. 2003a).

program based on the theory in § 4.3.3 and § 4.4.3. Comparing the spectra for

ν = 0.5 and ν < 0.5 provides an important consistency check on our numeri-

cal computations, since the methodologies for compressible and incompressible

skeletons are distinct. In general, Z/E can vary with Poisson’s ratio by up

to an order of magnitude at frequencies below the reciprocal draining time.

The response is clearly very sensitive to Poisson’s ratio as ν → 0.5 at these

frequencies. At higher frequencies, however, Z/E is independent of Poisson’s

ratio with fixed shear modulus µ, because the compressible polymer skeleton

is hydrodynamically coupled to the incompressible fluid.

Next, figure 4–5 shows how Young’s modulus affects the response spec-

trum. As expected from the steady displacement (Hill & Ostoja-Starzewski

2008; Wang & Hill 2008), the response is indeed inversely proportional to

the elastic modulus at frequencies below the reciprocal viscous time τ−1v . In

addition, the elastic modulus changes both the draining and viscous times.

Accordingly, the spectra in figure 4–5 overlap at frequencies below τ−1v when

82

10-2

100

102

104

106

108

frequency, (Hz)

0

0.1

0.2

0.3

0.4

-Z/E

, [nm

/(V

/cm

)]

10-2

100

102

104

106

10810

-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

0

0.1

0.2

0.3

0.4

0.5

Figure 4–4: Frequency spectra of Z/E for various Poisson ratios ν = 0, 0.1,0.2, 0.3, 0.4, and 0.5 with a fixed shear modulus µ = E/(2 + 2ν) ≈ 333 Pa.All other parameters are the same as in figure 4–2. Solid and dashed lines thereal and imaginary parts of Z/E from numerically exact solutions of the fullmulti-phase electrokinetic model.

multiplying Z/E by E and dividing the frequency ω/(2π) by E . Noteworthy

from the perspective of electroacoustics is that the real part of Z/E (solid

lines) is sensitive to E at ultrasonic frequencies, whereas the imaginary part

(dashed lines) is practically independent of E .

The influences of the scaled ζ-potential ζe/(kT ) and scaled reciprocal

double layer thickness κa on Z/E are examined in figure 4–6. Note that our

computational methodology is stable and accurate for all κa & 1. When κa .

1, however, electro-osmotic flow is extremely weak and the dynamic response

can be approximated by (4.21). Situations of practical significance most often

occur when κa & 1, so figure 4–6 presents spectra for six values of κa in the

range 1–500. When κa is large, electroosmotic flow significantly affects the

particle displacement in the quasi-steady compressible elastic regime in the

same manner as for steady electric fields (Wang & Hill 2008). Consequently,

the displacement at frequencies below the reciprocal draining time increases

with κa relative to the respective quasi-steady elastic plateaus (incompressible

regimes).

83

100

102

104

106

108

frequency, (Hz)

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

| Z/E

|, [

nm/(

V/c

m)]

0.1 kPa

1 kPa

10 kPa

Figure 4–5: Frequency spectra of Z/E for various Young’s moduli E = 100 Pa,1 kPa, and 10 kPa with Poisson ratio ν = 0.2. All other parameters are thesame as in figure 4–2. Solid and dashed lines the real and imaginary partsof Z/E from numerically exact solutions of the full multi-phase electrokineticmodel.

The response Z/E is also affected by polarization and relaxation of the

diffuse double layer. This is especially evident for particles with thick double

layers and high ζ-potentials, a situation where the back-field of the polarized

double layer is strong (Gibb & Hunter 2000). For example, in figure 4–6(a)

with κa = 1 and |ζ| & 2kT/e, the real part of Z/E increases with frequency,

and the imaginary part changes sign between approximately 2 and 30 kHz.

These changes occur at frequencies higher than the reciprocal diffusion re-

laxation time τ−1i , which represents the maximum frequency that the diffuse

double layer is capable of following the external field (DeLacey & White 1981).

The increase in the real part of Z/E indicates that the back-field decreases

with increasing frequency, thereby reducing the so-called retardation experi-

enced by the particle. The sign change of the imaginary part of Z/E indicates

that the double layer polarization lags the applied field when the frequency is

higher than τ−1i . For particles with large κa, the back-field is weak, because

relaxation via diffusion across a thin double layer is fast. In the panels with

κa = 5, 10, and 50, maximums in the real part of Z/E with respect to |ζ| are

84

100

102

104

106

108

frequency, (Hz)

0

0.005

0.01

0.015

0.02

-Z/E

, [nm

/(V

/cm

)]

-1

-2

-4

-6

-8

(a) κa = 1

100

102

104

106

108

frequency, (Hz)

0

0.005

0.01

0.015

0.02

0.025

0.03

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-7

10-6

10-5

10-4

10-3

10-2

10-1

| Z/E

|, [

nm/(

V/c

m)]

-1

-2

-4

-6 -8

(b) κa = 5

100

102

104

106

108

frequency, (Hz)

0

0.01

0.02

0.03

0.04

0.05

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

-1

-2

-4

-6

-8

(c) κa = 10

100

102

104

106

108

frequency, (Hz)

0

0.05

0.1

0.15

0.2

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

-1

-2

-4

-6

-8

(d) κa = 50

100

102

104

106

108

frequency, (Hz)

0

0.05

0.1

0.15

0.2

0.25

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

-1

-2

-4

-6

-8

(e) κa = 100

100

102

104

106

108

frequency, (Hz)

0

0.1

0.2

0.3

0.4

0.5

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

-1

-2

-4

-6

-8

(f) κa = 500

Figure 4–6: Frequency spectra of Z/E for various scaled ζ-potentials−ζe/(kT ) = 1, 2, 4, 6, and 8; and various scaled reciprocal double-layerthicknesses κa = 1, 5, 10, 50, 100, and 500: KCl at T = 298 K; a/` = 100(a = 500 nm and ` = 5 nm); ρp = 1050 kg m−3; ν = 0.2; and E = 1 kPa. Solidand dashed lines the real and imaginary parts of Z/E from numerically exactsolutions of the full multi-phase electrokinetic model.

85

evident. These can lead to ambiguity in determining the ζ-potential from the

steady response (Hill & Ostoja-Starzewski 2008; Wang & Hill 2008). However,

by measuring the frequency spectrum of Z/E (or the mobility µd = −iωZ/E),

it may be easier to unambiguously ascertain the correct ζ-potential. This ap-

proach has been useful for interpreting electroacoustic measurements of the

dynamic mobility (Hunter & O’Brien 1997).

Finally, figures 4–7 and 4–8, respectively, show the influence of hydrogel

permeability `2 for small and large values of κa. The Brinkman screening

length ` has a significant influence on the hydrodynamic coupling between

the fluid and polymer skeleton. Earlier studies (Hill & Ostoja-Starzewski

2008; Wang & Hill 2008) demonstrate that the polymer displacement at steady

state is practically independent of ` for incompressible hydrogels, but varies

significantly for compressible hydrogels due to an adverse electroosmotic-flow-

induced pressure gradient, particularly when κa is large. Note that the Brinkman

screening length also affects the draining time.

When κa is small (figure 4–7), the Brinkman screening length is most ef-

fective in changing reciprocal draining time τ−1d . Accordingly, as the Brinkman

screening length increases, the frequency range exhibiting a quasi-steady in-

compressible elastic response decreases, and eventually disappears, with Z/E

transferring directly from the quasi-steady compressible plateau to the vis-

cous and inertial stress dominated regimes. When κa is large (figure 4–8),

the Brinkman screening length also significantly changes the amplitude of the

quasi-steady compressible plateau. Similarly to the steady displacement (Hill

& Ostoja-Starzewski 2008; Wang & Hill 2008), decreasing the permeability

increases the magnitude of the adverse tangential pressure gradient, which, in

turn, increases the particle displacement.

4.7.3 High-frequency boundary-layer approximation and applica-tion to electroacoustics

The amplitude of the dynamic electrokinetic response Z/E was demon-

strated above to become extraordinarily small at high frequencies. The results

86

100

102

104

106

108

frequency, (Hz)

0

0.001

0.002

0.003

0.004

0.005

-Z/E

, [nm

/(V

/cm

)]

1 10

1 10

50

50

100

1005

5

(a) κa = 1, ζe/(kT ) = −1

100

102

104

106

108

frequency, (Hz)

0

0.004

0.008

0.012

0.016

-Z/E

, [nm

/(V

/cm

)]

1 10

1 10

50

50

100

100

5

5

(b) κa = 1, ζe/(kT ) = −4

100

102

104

106

108

frequency, (Hz)

0

0.002

0.004

0.006

0.008

0.01

0.012

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-710

-610

-510

-410

-310

-210

-1

| Z/E

|, [

nm/(

V/c

m)]

1 10

1 10

50

50

100

1005

5

(c) κa = 10, ζe/(kT ) = −1

100

102

104

106

108

frequency, (Hz)

0

0.01

0.02

0.03

0.04

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-710

-610

-510

-410

-310

-210

-1

| Z/E

|, [

nm/(

V/c

m)]

110

1 10

50

50

100

100

5

5

(d) κa = 10, ζe/(kT ) = −4

Figure 4–7: Frequency spectra of Z/E for various Brinkman screening lengths` = 1, 5, 10, 50, and 100 nm with scaled ζ-potentials −ζe/(kT ) = 1 and4, and scaled reciprocal double-layer thickness κa = 1 and 10. All otherparameters are the same as in figure 4–6. Solid and dashed lines the real andimaginary parts of Z/E from numerically exact solutions of the full multi-phase electrokinetic model.

87

100

102

104

106

108

frequency, (Hz)

0

0.01

0.02

0.03

0.04

0.05

0.06

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-7

10-6

10-5

10-4

10-3

10-2

10-1

| Z/E

|, [

nm/(

V/c

m)]1

10

50

100

5

(a) κa = 100, ζe/(kT ) = −1

100

102

104

106

108

frequency, (Hz)

0

0.05

0.1

0.15

0.2

0.25

0.3

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-8

10-6

10-4

10-2

100

| Z/E

|, [

nm/(

V/c

m)]

1

10

50100

5

(b) κa = 100, ζe/(kT ) = −4

100

102

104

106

108

frequency, (Hz)

0

0.05

0.1

0.15

0.2

0.25

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

1

1050 100

5

(c) κa = 1000, ζe/(kT ) = −1

100

102

104

106

108

frequency, (Hz)

0

0.2

0.4

0.6

0.8

1

-Z/E

, [nm

/(V

/cm

)]

100

102

104

106

10810

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

| Z/E

|, [

nm/(

V/c

m)]

1

10 50 100

5

(d) κa = 1000, ζe/(kT ) = −4

Figure 4–8: Frequency spectra of Z/E for various Brinkman screening lengths` = 1, 5, 10, 50, and 100 nm with scaled ζ-potentials −ζe/(kT ) = 1 and 4,and scaled reciprocal double-layer thickness κa = 100 and 1000. All otherparameters are the same as in figure 4–6. Solid and dashed lines the real andimaginary parts of Z/E from numerically exact solutions of the full multi-phase electrokinetic model.

88

below demonstrate that the dynamic mobility µd = −iωZ/E is large at the fre-

quencies used in commercial electroacoustic instruments. In § 4.5 we showed

that O’Brien’s macroscopic electroacoustic equations (O’Brien 1990) can be

applied to hydrogel-colloid composites. Accordingly, a close connection was

established between the electroacoustic signal (pressure fluctuations) in ESA

measurements and the dynamic mobility. This motivated the derivation in

§ 4.6 leading to the analytical approximation (4.126) for the high frequencies

encountered in electroacoustic experiments.

The dynamic electrophoretic mobilities for representative hydrogel-colloid

composites from (4.126) and numerically exact computations are presented in

figure 4–9. Because our calculations neglect interactions, they are suitable

for composites with low particle volume fractions. Spectra are shown with

Young’s modulus spanning three decades. Note that the spectrum with a finite

(real part) plateau at low frequencies is the mobility for the same particles

dispersed in a Newtonian electrolyte (without polymer); this was calculated

using the MPEK software package (Hill et al. 2003a). At this large value of

κa = 500, the analytical approximation (4.126) compares extremely well with

the numerically exact calculations. Note also that the real parts (dash-dotted

and solid lines) depart very slightly when the amplitude vanishes at lower

frequencies, and the imaginary parts are practically identical at all frequencies.

Noteworthy is that the real part of µd can be distinguished from µd∗

only at MHz frequencies when Young’s modulus of the skeleton is greater

than about 10 kPa. However, the imaginary part is very sensitive to the

elastic modulus at about 1 MHz, suggesting that, at a given fixed frequency,

or in a narrow range of frequencies, an electroacoustic experiment could probe

the kinetics of polymer gelation and aging on time scales less than a second.

Recall, commercial electroacoustic instruments operate at frequencies between

0.3 MHz and 11 MHz (Hunter 1998). Clearly, to probe the elastic modulus of

hydrogel skeletons with lower moduli, a wider frequency range—extending to

lower frequencies—is required. Nevertheless, many important hydrogels have

89

103

104

105

106

107

108

109

frequency, (Hz)

-8

-6

-4

-2

0

2

4

6

mob

ility

, [(µ

m/s

)/(V

/cm

)]

0.1 kPa 1 kPa 10 kPa

Figure 4–9: Frequency spectra of the dynamic electrophoretic mobility µd =−iωZ/E for Young’s moduli E = 100 Pa, 1 kPa, and 10 kPa with Poisson’sratio ν = 0.2. Other parameters are the same as in figure 4–2. Solid anddashed lines are the real and imaginary parts of µd = −iωZ/E from numeri-cally exact solutions of the full multi-phase electrokinetic model. Dash-dottedlines are the real parts of µd from the analytical approximation (4.126); theimaginary parts of µd from (4.126) are not shown because they overlap thenumerically exact solution. Note that the spectrum with a finite (real) low-frequency plateau is the mobility µd

∗ = −iω(Z/E)∗ for the same particlesdispersed in a Newtonian electrolyte (without polymer), calculated using theMEPK software package (Hill et al. 2003a).

an elastic modulus greater than 10 kPa, e.g., reverse thermoresponsive poly(N-

isopropylacrylamide) gels at 8 wt% and 40 C have E ≈ 170 kPa (Takigawa

et al. 1997), and polyacrylamide gels at 0.8% w/v have E ≈ 35 kPa (Takigawa

et al. 1996), so the real and imaginary parts of their electroacoustic responses

at MHz frequencies would be extremely sensitive to changes in the elastic

modulus. Note also that, because the electroacoustic response is prominent

at frequencies much higher than the reciprocal draining time, the response

only reflects changes in the shear modulus, not the accompanying changes in

hydrodynamic permeability.

Figure 4–10 compares numerically exact calculations of the dynamic elec-

trophoretic mobility with our analytical boundary-layer approximation (4.126)

90

with κa = 50 (top panels) and 1000 (bottom panels). The boundary-layer ap-

proximation is accurate when κa = 1000 for all the experimentally accessible

ζ-potentials and frequencies, and even when κa = 50 the approximation de-

viates only slightly from the exact calculations at the highest ζ-potentials

and frequencies. These dynamics are independent of the Brinkman screening

length and Poisson ratio (with fixed shear modulus). Note that (4.126) does

not capture the compressible dynamics at frequencies below the reciprocal

draining time, where the dynamic mobility is vanishingly small.

4.8 Summary

We extended the multi-phase electrokinetic model of Hill & Ostoja-Starzewski

(2008) and Wang & Hill (2008) to calculate the dynamic response of a charged,

spherical colloid embedded in uncharged hydrogels subjected to harmonically

oscillating electric fields. We began by solving the two-fluid model of Levine

& Lubensky (2001) exactly, and compared our analytical solution with two

approximations widely adopted in the microrheology literature. We then de-

veloped a powerful computational methodology to solve the full multi-phase

electrokinetic model by linearly perturbing an equilibrium base state governed

by the non-linear Poisson-Boltzmann equation. The particle response, defined

as the ratio of the displacement to the electric field strength, was obtained

by superposing two simpler sub-problems to satisfy the particle equation of

motion. Compressible and incompressible hydrogel skeletons had to be con-

sidered separately. By adopting an analytical solution in the far field, we

achieved accurate numerical solutions over an extraordinarily wide range of

frequencies, in a wide range of the experimentally accessible parameter space.

In addition, we examined the dynamic electrophoretic mobility, defined as the

ratio of the particle velocity to the electric field strength, and its connection

to electroacoustic diagnostics for characterizing hydrogel-colloid composites.

Noteworthy was an analytical boundary-layer approximation that compares

extremely well with the numerically exact results at the ultra-sonic frequen-

cies adopted in commercial electroacoustic instruments.

91

103

104

105

106

107

108

109

frequency, (Hz)

-8

-7

-6

-5

-4

-3

-2

-1

0

1

mob

ility

, [(µ

m/s

)/(V

/cm

)]

-2

-1

-4

-6

-8

(a) κa = 50, Re(µd)

103

104

105

106

107

108

109

frequency, (Hz)

-3

-2

-1

0

1

2

3

4

5

mob

ility

, [(µ

m/s

)/(V

/cm

)]

-2

-1

-4

-6

-8

(b) κa = 50, Im(µd)

103

104

105

106

107

108

109

frequency, (Hz)

-16

-14

-12

-10

-8

-6

-4

-2

0

2

mob

ility

, [(µ

m/s

)/(V

/cm

)]

-1

-2

-4

-6

-8

(c) κa = 1000, Re(µd)

103

104

105

106

107

108

109

frequency, (Hz)

-6

-4

-2

0

2

4

6

8

mob

ility

, [(µ

m/s

)/(V

/cm

)]

-1

-2

-4

-6

-8

(d) κa = 1000, Im(µd)

Figure 4–10: Frequency spectra of the dynamic electrophoretic mobility µd =−iωZ/E for various scaled ζ-potentials −ζe/(kT ) = 1, 2, 4, 6, and 8: κa = 50(top panels) and 1000 (bottom panels). All other parameters are the sameas in figure 4–6. Solid lines are the real (left panels) and imaginary (rightpanels) parts of µd from numerically exact solutions of the full multi-phaseelectrokinetic model, and dashed lines are µd from the analytical boundary-layer approximation (4.126).

92

The approximate solution of the two-fluid model by Levine & Lubensky

(2001) agrees well with our exact analytical solution when fluid and particle

inertia can be neglected. However, the range of applicability of the gener-

alized Stokes-Einstein relation (GSER) was found to be narrower than pre-

viously thought. The electric-field-induced dynamic response of a colloidal

particle in a hydrogel often exhibits an ostensible transition from quasi-steady

compressible to incompressible elastic dynamics—both characterized by dis-

tinct plateaus in the real part of the frequency spectrum—as the frequency

passes through the reciprocal draining time τ−1d of the hydrogel. At higher

frequencies, when the dynamics are dominated by viscous and inertial forces,

the response is similar to a particle in a Newtonian electrolyte. In general,

the response depends on Poisson’s ratio, Young’s modulus, and Brinkman

screening length of the hydrogel, as well as physicochemical characteristics of

the inclusions, including size and charge. At frequencies above the reciprocal

draining time, the response is practically independent of hydrogel permeabil-

ity and compressibility, since the the fluid and polymer skeleton are strongly

coupled by hydrodynamic drag forces. At frequencies below the reciprocal

draining time, hydrogel compressibility can increase the electric-field-induced

particle displacement by an order of magnitude relative to the displacement in

a perfectly incompressible skeleton with the same shear modulus. Accordingly,

the dynamics of compressible and incompressible hydrogels are qualitatively

different at low frequencies. Note that the response spectrum also reflects po-

larization and relaxation of the diffuse double layer, particularly for inclusions

with thick double layers (small κa) and high ζ-potentials.

The present theory provides a rigorous foundation for interpreting two

classes of electric-field-based diagnostic experiments involving hydrogel-colloid

composites. Such experiments probe both the physicochemical characteris-

tics of the charged inclusions, and the viscoelastic rheology of the hydrogel.

Our calculations demonstrate that the particle displacement at low frequen-

cies could be directly measured using active electrical microrheology. At

93

higher frequencies, however, the particle displacements are too small (sub-

nanometer) to measure directly, so electroacoustic techniques are necessary to

measure instead the dynamic electrophoretic mobility of the inclusions. Ac-

cordingly, we showed that the macroscopic relations for colloidal dispersions

developed by O’Brien (1990) can be directly applied to hydrogel-colloid com-

posites. Our calculations of the dynamic electrophoretic mobility demonstrate

that the strength of electroacoustic signals from hydrogel-colloid composites

are comparable to those from Newtonian electrolytes without a polymer skele-

ton. Accordingly, our calculations suggest that electroacoustic experiments

on hydrogel-colloid composites could be performed using presently available

commercial instruments.

This chapter and chapter 3 provide the complete solution of the multi-

phase electrokinetic model for dynamic and steady electric-field-induced re-

sponses of colloidal particles embedded in hydrogels. The particle responses

are sensitive to the hydrogel viscoelasticity, compressibility and hydrodynamic

permeability, and physicochemical properties of the inclusion. Therefore, char-

acterization techniques based on these responses can be developed. Clearly,

the continuous phase significantly affects particle dynamics. Effects of colloidal

particles on the bulk composite properties are revealed in the next chapter with

a different problem, where the nanoparticle-induced anomalous bulk viscosity

reductions found in polymer-nanocomposite melts are studied.

Appendicies

4.A Point-force representation of a particle in an uncharged hy-drogel matrix

Here we relate the net force on a spherical colloid in an uncharged hydrogel

to the strength of a point force that produces the same far-field disturbances.

The particle undergoes harmonic translation in an otherwise stationary hy-

drogel. The strength of the point force is obtained from reciprocal relations

similar to Hill et al. (2003a). However, in addition to the Lorentz reciprocal

94

relation for fluid in a domain S (Kim & Karrila 1991),∫∂S

(u′ · Tf − u · Tf ′) · ndA =

∫S

(u′ ·∇ · Tf − u ·∇ · Tf ′)dV, (4.129)

a similar reciprocal relation, known in solid mechanics as the Betty theo-

rem (Barber 2003) is required. This is∫∂S

(µ′v′ · Te − µv · Te′) · ndA =

∫S

(µ′v′ ·∇ · Te − µv ·∇ · Te′)dV, (4.130)

provided λ = λ′.

Consider a large domain Ω with boundary ∂Ω and outward unit normal

n that encloses an oscillating sphere centered at position r1 with radius a

(system 1) and a fixed point-force centered at position r2 (system 2). Note that

|r1−r2| a+κ−1. Furthermore, the sphere occupies volume Ω1 and undergoes

oscillatory translation with velocity −iωZ. The corresponding surface and

outward unit normal are denoted ∂Ω1 and n1, respectively.

The divergence of elastic and hydrodynamic stresses for system 1 (∇ ·Tf1

and ∇ · Te1) are given by (4.58b) and (4.58c), respectively. For system 2,

∇ · Tf2 = η∇2u2 −∇p2

= −iωρfu2 + (η/`2)(u2 + iωv2) + f fδ(r2), (4.131a)

∇ · Te2 = µ∇2v2 + (λ+ µ)∇(∇ · v2)

= f eδ(r2)− (η/`2)(u2 + iωv2), (4.131b)

where δ(r) is the Dirac-delta function, and f f and f e are the point forces

exerted on the fluid and elastic medium, respectively.

Applying the Lorentz reciprocal relation to the volume enclosed by ∂Ω1

and ∂Ω gives∫∂Ω

(u1 · Tf2 − u2 · Tf

1) · ndA−∫∂Ω1

(u1 · Tf2 − u2 · Tf

1) · n1dA

=

∫Ω−Ω1

(u1 ·∇ · Tf2 − u2 ·∇ · Tf

1)dV. (4.132)

95

Because u ∼ r−3 as r → ∞, the integral over ∂Ω on the left-hand side of

(4.132) vanishes when Ω is sufficiently large. Therefore, substituting (4.58b)

and (4.131a) into (4.132) gives

−∫∂Ω1

(u1 · Tf2 − u2 · Tf

1) · n1dA

= u1(r2) · f f +

∫Ω−Ω1

[iω(η/`2)(u1 · v2 − u2 · v1) + u2 ·∇ · Tm1 ]dV.(4.133)

Inside the particle, u1 = −iωZ, and since |r1 − r2| a + κ−1, u2(|x −

r1| ≤ a) can be considered constant. Therefore, applying Gauss’s divergence

theorem to the integral over ∂Ω1 on the left-hand side of (4.133) gives∫∂Ω1

(u1 · Tf2 − u2 · Tf

1) · n1dA

= −iωZ ·∫

Ω1

∇ · Tf2dV − u2(r1) ·

∫∂Ω1

Tf1 · n1dA

= −u2(r1) ·[ω2ρfVpZ +

∫∂Ω1

Tf1 · n1dA

], (4.134)

where Vp is the particle volume. Substituting (4.134) into (4.133) gives

f f · u1(r2) = u2(r1) ·[ω2ρfVpZ +

∫∂Ω1

Tf1 · n1dA

]−∫

Ω−Ω1

[iω(η/`2)(u1 · v2 − u2 · v1) + u2 ·∇ · Tm1 ]dV. (4.135)

Similarly, for the elastic displacements, applying the same procedure as above,

but with the Betty theorem, yields

f e ·v1(r2) = v2(r1) ·∫∂Ω1

Te1 ·n1dA−

∫Ω−Ω1

(η/`2)(u1 ·v2−u2 ·v1)dV. (4.136)

where v2(|x− r1| ≤ a) may be considered constant.

Again, since u1(r2) = −iωv1(r2) and u2(r1) = −iωv2(r1), multiplying

(4.136) by −iω and adding (4.135) gives

f ·u1(r2) = u2(r1)·[ω2ρfVpZ +

∫∂Ω1

(Tf1 + Te

1) · n1dA

]−∫

Ω−Ω1

u2 ·∇·Tm1 dV,

(4.137)

96

where f = f e + f f is the total point force. Note that the volume integral on

the right-hand side of (4.137) can be factored to give∫Ω−Ω1

u2 ·∇ · Tm1 dV = −u2(r1) ·

∫∂Ω1

Tm1 · n1dA, (4.138)

because ∇ · Tm is exponentially small when |x − r1| a + κ−1, and u2 can

be considered constant where ∇ · Tm is finite.

Substituting (4.138) into (4.137) yields

f = ω2ρfVpZ +

∫∂Ω1

(Te1 + Tf

1 + Tm1 ) · n1dA. (4.139)

The integral over ∂Ω1 on the left-hand side of (4.139) is the total force on the

sphere, which according to Newton’s second law must equal the acceleration

of its mass, −Vpρpω2Z, so the strength of the point force is

f = ω2VpZ(ρf − ρp). (4.140)

In other words, similarly to bare particles (Mangelsdorf & White 1992) and

particles with polymer coatings (Hill et al. 2003a) dispersed in Newtonian

electrolytes, the acceleration of the mass of fluid displaced by a finite sized

inclusion in a hydrogel must be added to the force on a point particle producing

the same far-field fluid velocity and polymer displacement disturbances. Note

that the foregoing analysis neglects the mass of the polymer.

4.B Numerical solution of the field equations

The field equations are solved according to the outline presented in § 4.3.

First, the Poisson-Boltzmann equation is solved efficiently using the adaptive

mesh algorithm developed by Hill et al. (2003a), and then various equilibrium

quantities and their derivatives are computed. Next, the linearly perturbed

equations are solved. Before this calculation, the equations are transformed to

simplify the numerical methods discussed in § 4.3 and § 4.4. The matrix alge-

bra and eigenvalue calculations involved in the transformations are performed

97

using BLAS and LAPACK routines (Anderson et al. 1999). Completely differ-

ent computational strategies are adopted for calculating the linearized pertur-

bations for incompressible and compressible hydrogel skeletons. Asymptotic

coefficients and physical quantities are then constructed from the numerical so-

lutions. The following discusses in further detail how the perturbed problems

are solved.

For incompressible skeletons, the perturbed equations are transformed to

the decoupled forms outlined in § 4.3, and the resulting differential equations

are discretized using a second-order central difference scheme, and solved using

a banded matrix solver. Solutions are then improved iteratively using a mov-

ing mesh method based on the methodology of Hill et al. (2003a). When the

solution has converged, asymptotic coefficients based on the far-field asymp-

totic analysis are obtained. The far-field solution is calculated using LAPACK.

Our program to compute the response for incompressible skeletons is written

entirely in C.

For compressible skeletons, the perturbed solutions oscillate in space with

several wave lengths, e.g., the construction of the fluid velocity and polymer

displacement in § 4.2 involves three wave lengths. The second-order central

difference scheme with the moving mesh method of Hill et al. (2003a) does not

converge. We therefore modified a general-purpose boundary value problem

software package TWPBVPL (Cash & Mazzia 2006), which solves the differ-

ential equations using fourth-, sixth- and eighth-order methods with hybrid

mesh selection, to solve the linearly perturbed problem. The second-order or-

dinary differential equations (ODEs) presented in § 4.3.2, i.e., (4.36), (4.37),

(4.32), (4.34), and (4.35), are arranged into a set of first-order ODEs

x,r = C · x+ q, (4.141)

where x = [n1, n1,r, . . . , nN , nN,r, ψ, ψ,r, f,r, f,rr, f,rrr, f,rrrr, g1, g1,r, g2, g2,r]T

is a vector of unknown functions, C is a coefficient matrix, and q is a vec-

tor. Note that x, C and q all depend on radial position r. To convert nj

98

(j = 1, 2, . . . , N) and ψ to χk (k = 1, 2, . . . , N +1) in x, we introduce a square

transformation matrix

T =

Q−1

0

0 I

, (4.142)

and a transformed vector of unknowns

y = T · x. (4.143)

Here, Q is given in (4.71), and the tilde denotes a matrix augmentation oper-

ation that increases the size of an n × n matrix M (with elements Mij) to a

2n× 2n augmented matrix

M =

M11 0 M12 0 . . . M1n 0

0 M11 0 M12 . . . 0 M1n

......

......

. . ....

...

Mn1 0 Mn2 0 . . . Mnn 0

0 Mn1 0 Mn2 . . . 0 Mnn

. (4.144)

Equation (4.141) is then transformed to

y,r = (TCT−1) · y + T · q, (4.145)

where the far-field boundary conditions at r = rmax given in § 4.4 can be

directly applied. The original boundary conditions at the particle surface

r = a in matrix form

B · x = β (4.146)

are transformed to

(BT−1) · y = β. (4.147)

The transformed equations (4.145), together with their boundary conditions,

i.e., (4.147) as boundary conditions at r = a and far-field boundary conditions

presented in § 4.4 at r = rmax, are first solved with a second-order central differ-

ence scheme (Ascher et al. 1988) on the non-uniform mesh of the equilibrium

99

solution. With this initial guess, TWPBVPL is used to calculate y by ad-

justing the mesh and applying higher order methods. The far-field boundary

conditions are calculated using multiple precision packages GMP (Granlund

2007), MPFR (Fousse et al. 2007), and MPC (Enge et al. 2007) to avoid

round-off errors, and various matrix operations are performed using LAPACK

and BLAS. When the error tolerance or the maximum number of iterations is

achieved, asymptotic coefficients are extracted. Our program combines codes

written in C and FORTRAN.

100

CHAPTER 5Anomalous bulk viscosity of polymer-nanocomposite melt

Different from chapters 3 and 4, this chapter discusses the influences of

nanoparticles, i.e., colloidal particles of nanometer size (Rotello 2004), on the

bulk viscosity of polymer-nanocomposite melts. This shows, as an example,

how colloid-induced microstructures can change the bulk transport properties

of composites.

Nanoparticles dispersed in polymer melts have recently been shown to

decrease the bulk viscosity. This contradicts expectations based on Einstein’s

well-known theory for effective viscosity of dilute, random dispersions of rigid

spheres in Newtonian fluids. In this chapter, we examine a continuum hydro-

dynamic model where a layer of polymer at the nanoparticle-polymer interface

has a different viscosity and density to the bulk polymer. When the layer thick-

ness is greater than the nanoparticle radius, and the layer viscosity is lower

than that of the bulk polymer, the intrinsic viscosity is comparable to the

unexpectedly large, negative values reported experimentally in the literature.

Accordingly, our continuum hydrodynamic model attributes the bulk viscos-

ity reduction to a lower melt viscosity at the nanoparticle-polymer interface.

Such a reduction can be attributed to the increased free volume and the Rouse

dynamics of polymer chains that interact strongly with the nanoparticles. Our

model also supports Mackay and coworkers’ arguments that nanoparticles can

organize the bulk polymer even when the nanoparticle volume fraction is very

small.

5.1 Introduction

Polymer-nanocomposites formed by dispersing organic or inorganic nanopar-

ticles (NPs) in polymeric matrices often have significantly improved mechani-

cal, thermal, electrical, and magnetic properties (Kickelbick 2003). Purposely

101

tailoring the composite microstructure provides opportunities for developing

a variety of new or substantially improved technological applications, e.g.,

gas- and liquid-based membrane separations, fuel cells, and semiconductor

technologies (Kang et al. 2006; Schmidt & Malwitz 2003; Abetz et al. 2006;

Rotello 2004).

Because of their extraordinarily small sizes, the contribution of nanopar-

ticles to the bulk properties often cannot be explained by classical theories

(Torquato 2002). For instance, Merkel et al. (2002) showed that impenetrable

fumed silica nanoparticles embedded in a glassy polymer matrix unexpectedly

promote gas permeability and reverse selectivity: a phenomenon that cannot

be explained by classical Maxwell-like theories, where nanoparticles obstruct

molecular transport, and therefore decrease membrane permeability (Maxwell

1873). Merkel et al. (2002) interpreted the enhanced permeability and reverse

selectivity on the basis of the theory of Cohen & Turnbull (1959) by proposing

that nanoparticles increase the polymer free volume. In a more detailed quan-

titative theory, Hill (2006a,c) attributed the free-volume changes to polymer

depletion layers established during the membrane casting. These calculations

show that nanoparticle-induced microstructural changes, such as polymer de-

pletion layers, modify the polymer in a manner that is in good quantitative

agreement with the available experiments.

In this work, we use a similar idea to interpret the non-Einstein-like

anomalous bulk viscosity of certain polymer-nanocomposite melts, most no-

tably in the experiments by Mackay and coworkers (Mackay et al. 2003; Tuteja

et al. 2005). The viscosity of particulate suspensions is expected to increase

with the particle volume fraction, as demonstrated in various experimen-

tal (Rutgers 1962) and theoretical (Einstein 1906; Brennen 1975; Pal 2007;

Batchelor 1967) studies. At low volume fractions, interactions between parti-

cles can be neglected, and the bulk viscosity η is expected to increase according

to η = η0[1 + (5/2)φ], where η0 is the solvent viscosity, and φ is the particle

volume fraction (Einstein 1906).

102

However, Einstein’s relation does not hold for many polymer-nanocomposite

melts, even at very low nanoparticle volume fractions. Roberts et al. (2001)

dispersed 0.35 nm silicate nanoparticles into poly(dimethyl siloxane) (PDMS)

melts, and found that the composite bulk viscosity decreases linearly with

the nanoparticle volume fraction. The viscosity reduction was attributed

to the solvent effect of nanoparticles due to their similar sizes as PDMS

monomers (Roberts et al. 2001). Bulk viscosity reduction was also observed in

poly(methyl methacrylate) (PMMA) melts with untethered polyhedral oligomeric

silsesquioxanes (POSS) nanoparticles (Kopesky et al. 2004). In these exper-

iments, the bulk viscosity first decreases, and then increases with the POSS

volume fraction. The explanation is that, at low volume fractions, the POSS

nanoparticles act as plasticizer, increasing the free volume and therefore de-

creasing the bulk viscosity; whereas at higher volume fractions, nanoparticles

agglomerate and form crystallites that increase the bulk viscosity by the hy-

drodynamic mechanisms that underly Einstein’s theory.

A reduction of the polymer-nanocomposite melt viscosity is helpful for

processing. Clearly, an understanding of the underlying mechanism is nec-

essary. However, the foregoing experiments do not reveal whether the bulk

viscosity reduction arises from nano-scale effects, or interactions between the

nanoparticle surface and the polymer, or both. Insight into the origin of the

viscosity reduction can be obtained by studying an ideal system where the

nanoparticles and the polymer are chemically identical, so that the surface

enthalpic interactions are minimized (Russel et al. 1989).

Mackay et al. (2003) studied such an ideal system of polystyrene (PS)

nanoparticles dispersed in linear PS melts. They found that the bulk viscosity

decrease is a unique feature of nano-scale systems, and attributed the viscosity

decrease to the increase in free volume, as evidenced from the nanoparticle-

induced glass transition temperature and polymer configuration changes. Tuteja

et al. (2005) expanded the experimental data sets and revealed conditions for

the reduced bulk viscosity. They concluded that: (1) the polymers have to

103

be entangled, i.e., above the critical molecular mass for entanglement cou-

pling Mc; and (2) the average nanoparticle separation distance must be smaller

than twice the polymer radius of gyration Rg. Consequently, they proposed

that, in tandem to the free-volume changes, nanoparticles affect polymer chain

entanglement dynamics similarly to constraint release (Dealy & Larson 2006),

i.e., the constraints imposed by the surrounding molecules constituting the

entanglement mesh (tube) on a particular polymer chain are released due to

movement of the mesh. In this model, nanoparticles reduce the relaxation time

of the polymer melt, but do not affect the plateau modulus, and consequently

contribute to the bulk terminal viscosity reduction. Using the aforementioned

conditions of bulk viscosity reduction, low-viscosity multifunctional polymer-

nanocomposites with enhanced mechanical, electrical and magnetic properties

have been prepared (Tuteja et al. 2007a). Nanoparticle-induced polymer con-

figuration changes have been partly demonstrated by the recent small angle

neutron scattering (SANS) of Tuteja et al. (2008), where the polymer chains

are found to swell (Rg increases) due to nanoparticles.

There are currently no theories available to successfully quantify the nega-

tive intrinsic viscosity of polymer-nanocomposite melts. Ganesan et al. (2006)

proposed a wavenumber-dependent non-local phenomenological constitutive

equation for melt viscosity based on Rouse dynamics. Their equation is valid

when b < q−1 < L, where q is the velocity disturbance wavenumber, b is

the polymer segment size, and L is a characteristic length scale that controls

the break down of classical Einstein predictions. Solving a Fourier spectral

representation of slow viscous flow, with the viscosity constitutive equation,

yields a bulk viscosity reduction at low particle loading when nanoparticles

are smaller than L. Ganesan et al. (2006) also identified that L ≈ Rg for un-

entangled melts, and L ≈ dt for entangled melts, where dt is the entanglement

tube diameter. Their analytical prediction is consistent with accompanying

computer simulations for unentangled melts with low nanoparticle volume frac-

tions. However, contrary to Tuteja et al. (2005), their model predicts a much

104

smaller bulk viscosity reduction, and demonstrates that chain entanglement is

unnecessary. Consequently, Ganesan et al. (2006) did not compare their quan-

titative theory with the available experimental data of Mackay et al. (2003)

and Tuteja et al. (2005).

In this work, we present a new theory based on a layer of polymer at the

particle-polymer interface that has a viscosity and density that are, in general,

different from the bulk. A simple analytical model, valid for low nanoparticle

volume fractions, reveals that such a layer can significantly reduce the bulk

viscosity. We compare our theory with the experiments of Mackay et al. (2003)

and Tuteja et al. (2005) to ascertain the layer thickness. This provides valuable

insight toward understanding the mechanism of bulk viscosity reduction.

The chapter is arranged as follows. In § 5.2 we present an analytical model

for the reduced bulk viscosity arising from a layer around each nanoparticle

in a dilute, random dispersion. For simplicity, we consider the polymer melt

to be a Newtonian fluid, which is reasonable at the low shear rates encoun-

tered when measuring the steady shear viscosity. Our model satisfies quasi-

steady momentum and mass conservation equations, and allows for slip at the

particle-polymer interface. Section 5.3 presents the results of the analytical

theory, and highlights the importance of parameters that affect the bulk vis-

cosity reduction. The model is used to interpret experimental measurements

in § 5.4, where origins of the bulk viscosity reduction are also discussed. A nu-

merical extension of the analytical theory, which permits a continuous change

in layer properties, is also presented. We conclude with a brief summary in

§ 5.5.

5.2 Theory

Similarly to particulate suspensions, the bulk viscosity of a dilute polymer-

nanocomposite melt can be written

η = η0(1 + [η]φ+ . . .), (5.1)

105

where the intrinsic viscosity (Larson 1999)

[η] ≡ limφ→0

(η/η0 − 1)/φ, (5.2)

characterizes the contribution of each (non-interacting) particle to the bulk

viscosity. For example, the Einstein relation gives [η] = 5/2, and a reduced

bulk viscosity corresponds to [η] < 0. Note that the intrinsic viscosities of

many particulate suspensions have been measured, and, until recently, were

always positive (Rutgers 1962).

It is tempting to hypothesize that the intrinsic viscosity of bubbly suspen-

sions comprised of a low-viscosity dispersed phase in a higher viscosity con-

tinuous phase would be negative. However, Taylor (1932) has shown that the

intrinsic viscosity for spherical drops with arbitrary viscosity ratio asymptotes

to [η] = 1 for inviscid, spherical bubbles. Accordingly, the impenetrability of

a spherical bubble is as influential on the intrinsic viscosity as the vanishing

viscosity of the bubble itself.

Here we present a simple hydrodynamic model to interpret the large,

negative intrinsic viscosities reported by Mackay et al. (2003) and Tuteja et al.

(2005). We introduce a step change in the polymer segment density and

viscosity around each nanoparticle. Our analytical model can be generalized to

handle—in a numerically efficient way—a continuous radial change of density

and viscosity.

The intrinsic viscosity of nanoparticles is calculated from the velocity dis-

turbance produced by a rigid force- and torque-free nanoparticle embedded in

a polymer melt whose undisturbed velocity is u = G · r, where r is position

relative to the particle center, and the velocity-gradient tensor G = E + Ω

comprises the rate-of-strain tensor E and vorticity tensor Ω. For macroscop-

ically incompressible dispersions, E is symmetric and traceless, representing

an extensional flow. For torque-free particles, only the extensional part of the

106

macro-scale velocity gradient contributes to the intrinsic viscosity. It is there-

fore sufficient to compute the disturbance of a fixed sphere at the origin of an

extensional flow where u = E · r.

Our model neglects particle interactions, which may be important at much

lower particle volume fractions than expected for nanoparticles in molecular

fluids. Since the chain dynamics for an entangled melt are often characterized

by the entanglement tube diameter dt (Brochard-Wyart & de Gennes 2000),

we may assume nanoparticles with radius a organize polymer chains in their

close proximity over a distance that is comparable to dt. Then the theory

will be limited to particle volume fractions φ φm(1 + dt/a)−3, where φm

is the maximum random packing volume fraction (' 0.638). For PS melts,

dt ' 10 nm and the nanoparticle concentration when a ' 3 nm should yield

φ 0.008.

We approximate the polymer melt as an incompressible Newtonian fluid

with shear viscosity η and density ρ, which, in general, are presumed to vary

with radial position from the center of each nanoparticle. The Newtonian

stress tensor is T = −pI + η[∇u+ (∇u)T ], where p is the dynamic pressure

and I is the identity tensor. The viscosity and density inside a uniform layer

are denoted ηi and ρi, respectively, and, similarly, outside the layer ηo and ρo.

Newtonian rheology demands the polymer chains to be in equilibrium dur-

ing the deformation, i.e., the chain relaxation time τ should be much shorter

than the characteristic flow-deformation time. In Mackay and coworkers’ ex-

periments (Mackay et al. 2003; Tuteja et al. 2005), the zero-shear viscosity is

extrapolated from the dynamic viscosity in the low-shear Newtonian regime

at frequencies ω less than 0.01 rad s−1. The characteristic relaxation times of

the linear PS melt, obtained from the analysis of Baumgaertel et al. (1990)

with temperature adjusted to 170C from the William-Landau-Ferry (WLF)

equation (Ferry 1980), are τ ∼ 0.03 and 20 s for melts with molecular weights

75 and 393 kDa, respectively. These are clearly much shorter than the char-

acteristic flow-deformation time ω−1 ∼ 100 s. In other words, the Deborah

107

number De ≡ ωτ 1 (Dealy & Larson 2006), so the polymer melt is expected

to flow as a Newtonian fluid.

Spatial and temporal fluid inertia can be neglected due to the small

Reynolds number Re = γa2ρ/η 1 and its product with the Strouhal number

Re Sr 1, where Sr = ω/γ (Batchelor 1967). The characteristic shear strain

rate γ ∼ ωγ, where γ is the characteristic shear strain (Larson 1999). For

the parallel-plate rheometer used by Mackay et al. (2003) and Tuteja et al.

(2005), Soskey & Winter (1984) showed that a linear response is achieved with

γ ≤ 0.2, so we conservatively estimate γ ≈ 0.11 . Note that the reciprocal

shear strain rate γ−1 is the characteristic time for a polymer coil to convect

past a nanoparticle. Consequently, in the experiments of Mackay et al. (2003)

and Tuteja et al. (2005), Re = ωγa2ρo/ηo ∼ 10−22 and Sr = γ−1 ∼ 10−21 when

ω ∼ 0.01 rad s−1 and a ' 3 nm.

The intrinsic viscosity of nanoparticles dispersed in an incompressible

Newtonian fluid can be obtained by solving the quasi-steady mass and mo-

mentum conservation equations with uniform polymer viscosity and density

in a spherically symmetric shell. With the assumptions above, the governing

equations are well-known Stokes equations:

0 = ∇ · u, (5.3a)

0 = −∇p+ η∇2u. (5.3b)

The boundary conditions at the particle surface demand zero radial ve-

locity and, in general, finite tangential slip. Accordingly, at r = a:

u · n = 0, (5.4a)

t− t · nn = kη(u− u · nn). (5.4b)

1 Mackay et al. (2003) and Tuteja et al. (2005) did not report γ.

108

Here, t = T · n is the surface traction, k is a slipping-friction parameter—

hereafter referred to as a reciprocal slipping length—and n is an outward unit

normal. When k = 0, the polymer slips without resistance along the nanopar-

ticle surface, whereas k → ∞ reproduces the conventional no-slip boundary

condition. Note that slipping of polymer at interfaces has been theoretically

discussed (Brochard & de Gennes 1992) and experimentally observed (Migler

et al. 1993) at small scales.

We demand a continuous stress and mass flux at the interface between the

uniform layer and the bulk polymer (r = a+δ). This ensures that a continuum

of uniform shells, each with uniform density and viscosity, yields the solution

of the mass and momentum conservations equations with continuous, radially

varying density and viscosity. Accordingly, at r = a+ δ:

Ti = To, (5.5a)

ρiui = ρouo, (5.5b)

where superscripts “i” and “o” indicate quantities evaluated inside and outside

the interface, respectively.

A continuous mass flux is consistent with the mass conservation equation

∂ρ/∂t+∇ ·(ρu) = 0. Although the polymer melt is considered incompressible

on both sides of the interface, individual chains are compressible, and they

must be compressed or dilated as they convect through the interface or, indeed,

through a region of continuously varying density. With the continuous stress

boundary condition, we neglected the force required to compress or dilate

polymer when it moves across the interface, since this force is much smaller

than the viscous force. For each nanoparticle, the viscous force fv ∼ ηγ(a+δ)2,

and with γ ∼ 10−3 s−1 and η ∼ 105 Pa s in the experiments, fv ∼ 102(a+δ)2 N.

On the other hand, the force fp required to move the polymer across the

interface scales as fp ∼ κ∆Rg[(a+δ)/Rg]2, where κ is the polymer chain spring

constant and ∆Rg is the extent of the chain compression or dilation. Using

the dumbbell model (Larson 1999), the chain spring constant κ = kBT (2R2g)−1

109

with kBT the thermal energy. At the experimental temperature T = 170C,

assuming the chain deformation ∆Rg/Rg ∼ 0.01 and Rg ∼ 20 nm, fp ∼

4(a+δ)2 N, so fp/fv ∼ 0.04 1, which justifies the continuous stress boundary

condition.

As r →∞, vanishing of the disturbance velocity and pressure require

u → E · r, (5.6a)

p → 0. (5.6b)

Linearity of the Stokes equations and boundary conditions require that

u and p are linear in E, so the general solution inside the layer must have the

form (Lamb 1945)

ui =c1

2ηirE:∇∇r−1 +

c2

2ηir5rE:∇∇r−1 + c4E ·∇r−1

+c5r3E ·∇r−1 + c6E:∇∇∇r−1 + c7r

7E:∇∇∇r−1, (5.7a)

pi = c1E:∇∇r−1 + c2r5E:∇∇r−1, (5.7b)

and, similarly, outside the layer

uo =c3

2ηorE:∇∇r−1 + c8E ·∇r−1 + c9E:∇∇∇r−1 + E · r, (5.7c)

po = c3E:∇∇r−1. (5.7d)

Note that (5.7c) and (5.7d) satisfy the far-field boundary conditions (5.6a)

and (5.6b). The nine scalar constants c1, c2, . . . , c9 can be specified to satisfy

two (scalar) mass-conservation equations, i.e., ∇ ·ui = ∇ ·uo = 0, and seven

scalar equations from the boundary conditions at r = a and r = a + δ: there

are two scalar equations from (5.4a) and (5.4b) at r = a, which are linear in

E · r and E:rrr; two from (5.5b) at r = a + δ, which are linear in E · r and

E:rrr; and three from (5.5a) at r = a + δ, which are linear in E, E · rr, and

E:rrrr.

The intrinsic viscosity can be obtained from the strength of the r−3

(quadrupole) decay of the velocity disturbance (Batchelor 1967). It follows

110

that [η] = −(3/2)a−3(ηo)−1c3. Solving the Stokes equations above with the

boundary conditions gives

[η] =(A′χ2 +B′χ+ C ′) + ka(D′χ2 + E ′χ+ F ′)

(Aχ2 +Bχ+ C) + ka(Dχ2 + Eχ+ F ), (5.8)

where a′ = a+ δ, χ = (ηi/ηo)/(ρi/ρo), and

A = −480a13 − 800a10a′3 + 900a6a′7 + 380a3a′10, (5.9a)

B = 960a13 − 400a10a′3 + 300a6a′7 + 890a3a′10, (5.9b)

C = −480a13 + 1200a10a′3 − 1200a6a′7 + 480a3a′10, (5.9c)

D = 96a13 + 400a10a′3 − 672a8a′5 + 450a6a′7 + 76a3a′10, (5.9d)

E = −192a13 + 200a10a′3 − 336a8a′5 + 150a6a′7 + 178a3a′10, (5.9e)

F = 96a13 − 600a10a′3 + 1008a8a′5 − 600a6a′7 + 96a3a′10, (5.9f)

A′ = −1200a10a′3 − 2000a7a′6 + 2250a3a′10 + 950a′13, (5.9g)

B′ = 400a10a′3 + 4000a7a′6 − 2500a3a′10 − 150a′13, (5.9h)

C ′ = 800a10a′3 − 2000a7a′6 + 2000a3a′10 − 800a′13, (5.9i)

D′ = 240a10a′3 + 1000a7a′6 − 1680a5a′8 + 1125a3a′10 + 190a′13, (5.9j)

E ′ = −80a10a′3 − 2000a7a′6 + 3360a5a′8 − 1250a3a′10 − 30a′13, (5.9k)

F ′ = −160a10a′3 + 1000a7a′6 − 1680a5a′8 + 1000a3a′10 − 160a′13.(5.9l)

An important limiting case arises when a → 0. This corresponds to an

instantaneously spherical, deformable drop-like particle with radius δ. The

intrinsic viscosity is

[η] =5χ− 5

2χ+ 3. (5.10)

Note that to calculate the bulk viscosity using (5.1), φ = n(4/3)πδ3 with n

the particle number density.

Equation (5.10) reveals that, when χ→∞ (rigid bubble), [η]→ 5/2, so,

as expected, the shell mimics an impenetrable, rigid sphere. When χ → 0

(inviscid bubble), [η]→ −5/3, which is different from the intrinsic viscosity of

undeformable bubbles with impenetrable surface, i.e., [η] = 1 (Taylor 1932).

111

Brennen (1975) obtained (5.10) using a “cell” model for the intrinsic viscosity

of droplets enclosed by an infinitely deformable membrane in the dilute limit.

This earlier result validates the present model in the liquid-droplet regime

(a→ 0).

It is important to note that the layer around a rigid nanoparticle un-

dergoes continuous deformation, while polymer chains passing through the

interface must be simultaneously compressed or dilated to maintain—to lead-

ing order in the perturbed density and velocity—the equilibrium structure of

the layer. This requires the polymer self-diffusion time to be much shorter

than the characteristic flow deformation time. For PS melts, the self diffu-

sion coefficients Ds ∼ 10−15 and 10−17 m2 s−1 for molecular weights of 75 and

393 kDa, respectively (Green & Kramer 1986). The corresponding self diffu-

sion times a2/Ds ∼ 0.1 and 10 s when a = 10 nm, which are much shorter

than ω−1 and γ−1 in the experiments.

5.3 Intrinsic viscosity from the single-layer model

The intrinsic viscosity from (5.8) depends on three dimensionless param-

eters: the scaled layer thickness δ/a, the scaled reciprocal slipping length

ka, and a layer property parameter χ = (ηi/ηo)/(ρi/ρo). In polymer solu-

tions (de Gennes 1979; Larson 1999), the bulk viscosity increases with the

polymer concentration according to a power law. Considering the similarity

between polymer melts and solutions, it is natural to assume that the viscos-

ity in the layer varies with the polymer segment density as η ∼ ρn (n > 0).

Consequently, χ = (ρi/ρo)n−1, and when ρi < ρo, it follows that χ < 1 if n > 1.

Figures 5–1 and 5–2 present the effect of scaled layer thickness δ/a on the

intrinsic viscosity [η] for χ ≤ 1 and χ ≥ 1, respectively. Evidently, when the

layer properties are identical to those of the bulk polymer melt, i.e., χ = 1, [η]

is independent of the layer thickness. Note that ka = 0 gives [η] = 1, which

mimics the intrinsic viscosity of an inviscid spherical bubble, and ka→∞ gives

[η] → 5/2, which recovers the hard-sphere result from Einstein (1906). With

increasing layer thickness δ/a, the intrinsic viscosity decreases from positive

112

10-3

10-2

10-1

100

101

Scaled layer thickness, δ/a

10-2

10-1

100

101

102

Intr

insi

c vi

scos

ity, [

η]0

10-4

1

0.01

0.5

0.1

(a) ka = 0, perfect slip

10-3

10-2

10-1

100

101

Scaled layer thickness, δ/a

10-2

10-1

100

101

102

Intr

insi

c vi

scos

ity, [

η]

0

10-4

1

0.01

0.5

0.1

(b) ka→∞, no slip

Figure 5–1: Intrinsic viscosity [η] as a function of the scaled layer thicknessδ/a according to (5.8) for polymer-nanocomposite melts with χ = 0, 10−4,0.01, 0.1, 0.5 and 1; ka = 0 (left panel) and ka→∞ (right panel). Solid anddashed lines indicate positive and negative values of [η], respectively.

to negative values when χ < 1, and increases when χ > 1. In both cases, [η]

asymptotes to the results with χ = 1 for small δ/a, and its absolute magnitude

increases with increasing layer thickness. Moreover, comparing results for

ka = 0 and ka→∞ reveals that the intrinsic viscosity is relatively insensitive

to the degree of slip when δ/a is large. Here, the case with χ ≤ 1 shown in

figure 5–1 is more important, because it exhibits negative intrinsic viscosities.

When the magnitude of the intrinsic viscosity is small, the corresponding

layer thickness δ/a is sensitive to χ, but this sensitivity vanishes when the

intrinsic viscosity is large. For example, in figure 5–1(a), the layer thicknesses

corresponding to [η] = −1 with χ = 10−4 and 0.5 are δ/a = 0.03 and 0.5,

respectively, but the layer thicknesses corresponding to [η] = −100, for the

same χ, are δ/a = 3 and 4, indicating a reduced sensitivity to χ at large

[η]. Accordingly, the large, negative intrinsic viscosities [η] ∼ −100 reported

by Mackay et al. (2003) and Tuteja et al. (2005) correspond to large layer

thicknesses, and are therefore relatively insensitive to χ.

Two limiting cases, χ → 0 and χ → ∞, with large δ/a are identified

in figures 5–1 and 5–2, respectively. As χ → ∞, the layer becomes rigid

113

10-3

10-2

10-1

100

101

Scaled layer thickness, δ/a

100

101

102

Intr

insi

c vi

scos

ity, [

η]

11.1

25100

∞

Figure 5–2: Intrinsic viscosity [η] as a function of the scaled layer thicknessδ/a according to (5.8) for polymer-nanocomposite melts with ka = 0 (solidlines) and ka→∞ (dashed lines); χ = 1, 1.1, 2, 5, 100 and ∞.

(infinitely viscous), so nanoparticles behave as rigid spheres with radius a+ δ

in the melt, giving

[η]→ (5/2)(1 + δ/a)3 as χ→∞. (5.11)

Similarly, as χ → 0, nanoparticles behave as instantaneously spherical, de-

formable bubbles with radius a+ δ, giving

[η]→ −(5/3)(1 + δ/a)3 as χ→ 0. (5.12)

Note that these limiting cases are independent of slip at the nanoparticle-

polymer interface.

The influence of the property parameter χ on the intrinsic viscosity is

shown in figure 5–3. These results are for a no-slip nanoparticle-polymer

interface, i.e., ka→∞. As expected, [η] = 5/2 when χ = 1, and the intrinsic

viscosity asymptotes to the values given by (5.12) and (5.11) when χ→ 0 and

∞, respectively.

Figure 5–4 demonstrates how the scaled reciprocal slipping length ka

affects [η] at χ = 0.1 for several values of δ/a. The slipping parameter is

114

10-3

10-2

10-1

100

101

102

103

Dimensionless parameter, χ

10-2

10-1

100

101

102

103

104

Intr

insi

c vi

scos

ity, [

η]

0.1

0.3

1

3

10

Figure 5–3: Intrinsic viscosity [η] as a function of the parameter χ =(ηi/ηo)/(ρi/ρo) according to (5.8) for polymer-nanocomposite melts withka → ∞ (no-slip) and δ/a = 0.1, 0.3, 1, 3, and 10. Solid and dashed linesindicate positive and negative values of [η], respectively.

clearly most influential with thin layers, where the magnitude of [η] is small,

and has negligible effect when δ/a & 1. For the experiments of Mackay et al.

(2003) and Tuteja et al. (2005) where [η] ∼ −100, slipping at the nanoparticle-

polymer interface does not significantly affect [η].

From the results above, the model demonstrates that the layer thickness

must be comparable to or larger than the nanoparticle radius to achieve the

negative intrinsic viscosities measured experimentally. Under these conditions,

the intrinsic viscosity is almost independent of ka and χ when χ . 0.1. It

should therefore be reasonable to interpret the experiments by focusing on the

influence of layer thickness, as captured by the single parameter δ/a.

5.4 Theoretical interpretation of experiments

Here we interpret the experiments of Mackay et al. (2003) and Tuteja

et al. (2005) using the model presented in § 5.2 and § 5.3. To ensure the

polymer molecules are entangled and confined (Tuteja et al. 2005), only the

experiments with entangled polymer melts and Rg/h & 1 are considered. Here

h is the interparticle half gap defined by Tuteja et al. (2005), and Rg/h & 1

indicates the polymers are confined by nanoparticles. Also, since our model

115

10-2

10-1

100

101

102

Scaled reciprocal slipping length, ka

10-3

10-2

10-1

100

101

102

103

104

Intr

insi

c vi

scos

ity, [

η]

0.01

0.1

0.15

1

10

Figure 5–4: Intrinsic viscosity [η] as a function of the scaled reciprocal slippinglength ka according to (5.8) for polymer-nanocomposite melts with χ = 0.1and δ/a = 0.01, 0.1, 0.15, 1, and 10. Solid and dashed lines indicate positiveand negative values of [η], respectively.

does not account for hydrodynamic (and other) interactions, we restrict our

analysis to the lowest nanoparticle volume fraction, i.e., φ = 0.005. From

the estimated viscosity and density in the layers, the layer thicknesses are

obtained by fitting the theory to experiments. Such comparisons shed light

on the mechanisms of bulk viscosity reduction.

We propose two possible origins for the layers. Firstly, these may arise

from changes in free volumes induced by nanoparticles. In this case, polymer

segments are depleted at the nanoparticle surfaces due to excluded volume

effects, and the melt viscosity and density recover their bulk values over a

length scale of the polymer correlation length ξ. For polymer melts, ξ ≈

b ∼ 1 nm (Fleer et al. 1993), where, recall, b is the segment size. Using

this idea, Hill (2006a,c) successfully explained the enhanced permeability and

reverse selectivity of nanocomposite membranes (Merkel et al. 2002). The

segment density profile can be calculated by mean-field theories (de Gennes

1979; Wu et al. 1995) and computer simulations (Daoulas et al. 2005). Here,

as a first approximation, we assume that the polymer segment density inside

a uniform layer is half of the bulk value, i.e., ρi/ρo = 1/2. This is equivalent

116

to assuming a linear polymer density profile. The simple scaling of de Gennes

(1976b) for entangled polymer solutions connects the viscosity and segment

density in the layer by η ∼ ρ15/4. Therefore, the layer property parameter

χ ≈ (ρi/ρo)11/4 ≈ 0.15.

Another possible origin for the layers is the local Rouse viscosity of the

melt in close proximity of the nanoparticles. In this case, the layer has the

Rouse viscosity and bulk polymer density, and the layer thickness is charac-

terized by the tube diameter dt ∼ 10 nm (Dealy & Larson 2006). This is

advanced from the idea of Brochard-Wyart & de Gennes (2000), where they

proposed that nanoparticles smaller than the entanglement tube diameter dt

experience Rouse dynamics in entangled melts, since the nanoparticle motion

only involves simple chain rearrangements. The concept partially explained

the unusually high nanoparticle diffusivity in polymer-nanocomposite melts

(Tuteja et al. 2007b). The Rouse viscosity can be calculated according to

Tuteja et al. (2007b) as ηRouse = ηc(M/Mc)aT , where ηc = 292 Pa s is the PS

melt viscosity at the entanglement critical molecular weight Mc = 32.7 kDa,

and aT is the shift factor given by log(aT ) = −7.65[T (C)−170]/[T (C)−28.1].

Note that the actual PS melt viscosity can be calculated according to ηmelt =

ηc(M/Mc)3.68aT (Tuteja et al. 2005). Evidently, ηRouse ηmelt for entangled

melts at high molecular weight, and this yields χ ≈ 0.11 and 0.0015 for the

the 75 and 396 kDa polymer melts, respectively.

The preceding analysis shows that χ 1 regardless of whether the layer is

attributed to an increase in free volume or a decrease in viscosity due to Rouse

dynamics. Recall from § 5.3, the depletion layer thickness δ/a is insensitive to

χ and ka when the magnitude of the intrinsic viscosity is large. Consequently,

as a first approximation, we accept the surface-slip independent case when

χ = 0 to approximate the layer thickness with a uniform layer.

Table 5–1 presents the effective layer thickness δ and other important

quantities obtained by fitting the analytical model to the experiments of

Mackay et al. (2003) and Tuteja et al. (2005). The best-fit values of δ are: (1)

117

Table 5–1: Summary of the parameters that characterize the experiments ofMackay et al. (2003) and Tuteja et al. (2005) with Rg/h & 1 and φ = 0.005,and theoretical interpretations (providing fitted values for δ) based on (5.8)with χ = 0. Radii of gyration Rg are calculated from Cotton et al. (1974)and values of Rg/h are from Tuteja et al. (2005). Note that with propertemperature adjustment, the PS tube diameter dt ≈ 9.4 nm at 170C (Dealy& Larson 2006).

Sample Rg/h [η] a (nm) Rg (nm)PS 75 kDa/25 kDa NP 0.88 -14 2.1 7.6PS 393 kDa/25 kDa NP 2.02 -118 2.1 17.2PS 393 kDa/52 kDa NP 1.58 -84 2.7 17.2PS 393 kDa/135 kDa NP 1.15 -94 3.7 17.2

Sample δ (nm) δ/a (δ + a)/dt φm(1 + δ/a)−3

PS 75 kDa/25 kDa NP 2.2 1.0 0.46 0.074PS 393 kDa/25 kDa NP 6.7 3.2 0.93 0.009PS 393 kDa/52 kDa NP 7.3 2.7 1.1 0.013PS 393 kDa/135 kDa NP 10.5 2.8 1.5 0.011

much larger than any reasonable estimate of the polymer correlation length

ξ; (2) smaller or comparable to the entanglement tube diameter dt; (3) larger

than the nanoparticle radius a; and (4) smaller than the polymer radius of

gyration Rg. These suggest that the viscosity reduction is more likely due

to the local Rouse dynamics of polymer chains in their entanglement tubes.

Note that to achieve [η] < 0 with nano-scale Rouse dynamics, it is necessary

that ηRouse ηmelt. This indicates that chain entanglement is essential for

bulk viscosity reduction, and is consistent with the rationale of Tuteja et al.

(2005).

The importance of interactions is also revealed in table 5–1 via the ratio

(δ+ a)/dt, which describes the influence of nanoparticles on the entanglement

tube dynamics, and the quantity φm(1+ δ/a)−3, which provides upper bounds

on the nanoparticle volume fraction φ = n(4/3)πa3 for the particles and their

layers not to overlap. If nanoparticles do not perturb the tube dynamics, we ex-

pect δ < (dt/2−a), so the Rouse viscosity only exists in an entanglement tube

with diameter dt that contains the nanoparticle (Brochard-Wyart & de Gennes

118

2000). However, (δ + a)/dt > 1/2 implies strong polymer-nanoparticle inter-

actions, because the thickness of the layer exceeds the entanglement tube

diameter for a pure melt.

Evidently, the experimental data giving (δ + a)/dt > 1/2 are always ac-

companied by small φm(1 + δ/a)−3 close to the actual nanoparticle volume

fraction φ = 0.005. This implies that distorted polymer configurations may

arise from nanoparticle interactions. Moreover, careful examination of data

from Tuteja et al. (2005) reveals that interactions also introduce strong free-

volume effects on the composite, as is evidenced by the large change in Tg.

Note that this is different from the free volumes introduced by excluded vol-

ume effects described above, which give rise to a layer thickness δ ∼ 1 nm,

and consequently a moderate negative intrinsic viscosity [η] ≈ −5. Clearly,

the large negative intrinsic viscosities observed by Mackay et al. (2003) and

Tuteja et al. (2005) are the result of strong polymer-mediated interactions

between nanoparticles and the polymer. Accordingly, we envision no clear

distinction between the bulk polymer and polymer in layers, so extending our

notion of perturbed polymer layers surrounding individual nanoparticles in an

unperturbed bulk polymer, we hypothesize that the polymer is everywhere

perturbed by the nanoparticles.

With weak interactions, the nano-scale Rouse dynamics in the polymer

entanglement tubes give rise to negative intrinsic viscosities much smaller in

magnitude than those with strong interactions. This is evidenced from the PS

75 kDa/25 kDa NP polymer nanocomposite in table 5–1, which shows (δ +

a)/dt < 1/2, φm(1 + δ/a)−3 φ, and the glass transition temperature change

∆Tg = 0.1C (Tuteja et al. 2005). Clearly, weak interactions explain the

thinnest depletion layer and moderate negative intrinsic viscosity achieved by

this sample, which is representative of a single nanoparticle in an unperturbed

entangled polymer melt.

Our uniform layer model can be extended to handle continuous polymer

density and viscosity layers by extending the uniform layer model to multiple

119

Table 5–2: Best-fit polymer correlation lengths ξ ascertained from experi-ments of Mackay et al. (2003) and Tuteja et al. (2005) and the theoreticalinterpretation based on a continuous-layer-profile model with ka = 0. Thepolymer segment densities are from Tuinier & Lekkerkerker (2002), and theviscosity-density relationship is from Colby et al. (1994).

Sample [η] a (nm) Rg (nm) ξ (nm) ξ/aPS 75 kDa/25 kDa NP -14 2.1 7.6 2.3 1.1PS 393 kDa/25 kDa NP -118 2.1 17.2 9.0 4.3PS 393 kDa/52 kDa NP -84 2.7 17.2 9.5 3.5PS 393 kDa/135 kDa NP -94 3.7 17.2 13.9 3.8

layers, and discretizing the continuous profiles accordingly. However, a con-

tinuous change in the layer density and viscosity does not significantly modify

the qualitative pictures emerging from the foregoing analytical theory for uni-

form layers. This is demonstrated in table 5–2, which presents the best-fit

correlation lengths ξ for a continuous polymer segment density described by

the Edwards-de Gennes equation (de Gennes 1979), and the polymer viscos-

ity from the two-parameter scaling theory of Colby et al. (1994) for good

solvents. Since the spatial dependence of nano-scale Rouse viscosity around

nanoparticles in an entanglement tube is not available, we assume, as a first

approximation, that the layer arises from excluded volume effects. Note that

Edwards-de Gennes equation is the ground state dominance approximation of

Doi-Edwards theory (Doi & Edwards 1987), valid for infinitely long chains,

as well as long, but finite, chains near a hard wall (Wu et al. 1995). Here,

the segment density profile is described using the analytical approximate so-

lution from Tuinier & Lekkerkerker (2002). Moreover, the polymer melt in

the layer is considered a polymer solution, due to the similar polymer dynam-

ics (de Gennes 1979). Interestingly, even when the layers are attributed to

excluded volume effects, the resulting best-fit correlation lengths ξ are still

on the order of the entanglement tube diameter dt. This indicates the char-

acteristic layer thickness is qualitatively independent of the layer profile, and

supports the foregoing interpretation based on nano-scale Rouse dynamics.

120

5.5 Summary and conclusions

We proposed a simple hydrodynamic model to interpret the reduced

bulk viscosity of polymer-nanocomposite melts observed by Mackay et al.

(2003) and Tuteja et al. (2005). The model adopts a uniform viscosity and

density layer around nanoparticles, and can be easily extended to continu-

ous layer profiles (as demonstrated by the results in table 5–2). Comparing

the theory with experiments suggests that the reduced bulk viscosity arises

from nano-scale Rouse dynamics experienced by nanoparticles in entanglement

tubes (Brochard-Wyart & de Gennes 2000) when the nanoparticle-polymer

interactions are weak. Our model also confirms that polymer-mediated inter-

actions are indeed crucial for the large bulk viscosity reductions found with

high molecular weight polymer, as first pointed out by Tuteja et al. (2005).

The comparison also suggests that the polymers in samples that exhibit a large

bulk viscosity reduction are strongly perturbed. We further demonstrated that

a continuous variation of polymer density and viscosity around nanoparticles

does not change the qualitative picture emerging from our analytical expres-

sion for a single uniform layer. Our model showed how the microstructure

affects the bulk viscosity in a manner that is consistent with experiments, and

it highlighted how sensitive the bulk viscosity is to interactions between the

nanoparticles and polymer.

121

CHAPTER 6Summary and conclusions

This thesis showed (i) the influences of the continuous phase on the

single-particle and bulk-composite responses by investigating the electric-field-

induced steady and dynamic responses of charged colloidal particles embedded

in uncharged hydrogel matrices and (ii) the influences of colloid-induced mi-

crostructural changes on composite transport properties by theoretically inter-

preting the recently discovered anomalous bulk viscosity reduction in polymer-

nanocomposite melts. The objectives outlined in § 1.2 have been accomplished,

and the theories and results presented in chapters 3–5 contribute to our un-

derstanding of the dynamics of interacting dispersed and continuous phases in

colloid composites. These dynamics are crucial for developing new composite

materials and diagnostic techniques to probe their microstructures.

Chapters 3 and 4 investigated the steady and dynamic electric-field-induced

responses of a charged colloidal particle in an uncharged hydrogel matrix.

These theoretical studies are not only important for understanding the in-

fluences of complex fluids, but also serve as rigorous foundations for electric

microrheology and electroacoustic techniques. A multi-phase electrokinetic

model, generalized from the standard electrokinetic model, was used to de-

scribe the polymer displacement, fluid velocity, ion fluxes and electrostatic

potentials. Computer programs based on MPEK (Hill et al. 2003a) were de-

veloped to solve the multi-phase electrokinetic model accurately with steady

and dynamic electrical forcing. Generally, the electric-field-induced particle

responses depend on both the hydrogel viscoelastic and the particle physico-

chemical properties, making the electric-field-based characterizations distinct

from existing techniques. Chapter 3 showed that the particle displacements

in compressible hydrogels with Poisson’s ratio close to zero can be an order of

122

magnitude larger than those for incompressible hydrogels. This is confirmed

by dynamic studies in chapter 4, where the response spectra for compress-

ible hydrogels present an evident transition from quasi-steady compressible

to quasi-steady incompressible elastic regimes. These results imply that elec-

tric microrheology could be successfully applied to hydrogel-colloid composites

at frequencies below the reciprocal draining time. On the other hand, high

frequency responses can be measured using electroacoustics, and chapter 4

also showed that the electroacoustic signals and the single particle response

are connected by the theory of O’Brien (1988, 1990). Similarly to other elec-

trokinetic phenomena (Hunter 2001; Lyklema 1995), both steady and dynamic

responses exhibit double layer polarization and relaxation at large ζ-potentials.

Moreover, boundary-layer analytical approximations—valuable for interpret-

ing experiments—were derived in chapters 3 and 4 for both the steady and

dynamic forcing.

After studying the influences of a hydrogel matrix on particle responses

in chapters 3 and 4, chapter 5 focused on the effect of colloid-induced mi-

crostructural changes on the bulk viscosity of polymer-nanocomposite melts.

This chapter developed a hydrodynamic model that incorporates a polymer

layer with properties different from the bulk. The model suggested that the

bulk viscosity reductions observed in the experiments of Mackay et al. (2003)

and Tuteja et al. (2005) can be attributed to the Rouse dynamics in poly-

mer entanglement tubes experienced by the nanoparticles, as proposed by

Brochard-Wyart & de Gennes (2000). The model also revealed that the inter-

actions are strong in polymer-nanocomposite melts, and that polymer chains

are likely everywhere disturbed, as evidenced from the large fitted layer thick-

ness and free volume changes. This chapter shows the effect of the inhomo-

geneous layer as a result of colloid-induced microstructural changes, and is an

important first step toward more quantitative theories to interpret intriguing

nano-scale phenomena, such as bulk viscosity reduction.

123

Experimental verifications of the theoretical treatment presented in chap-

ters 3 and 4 using electric microrheology and electroacoustics are recommended

for future investigations. Other possibilities include studying the dielectric re-

laxation spectra of dilute hydrogel-colloid composites theoretically and exper-

imentally, and developing new theories that take particle-particle interactions

into account. For the colloid-induced bulk viscosity reduction, theoretical,

simulation, and experimental studies focused on resolving the polymer-particle

and particle-particle interactions that influence these anomalous behaviors are

recommended.

124

REFERENCES

Abetz, V., Brinkmann, T., Dijkstra, M., Ebert, K., Fritsch, D.,Ohlrogge, K., Paul, D., Peinemann, K. V., Nunes, S. P.,Scharnagl, N. & Schossig, M. 2006 Developments in membrane re-search: from material via process design to industrial application. Ad-vanced Engineering Materials 8, 328–358.

Ahualli, S., Delgado, A. V., Miklavcic, S. J. & White, L. R. 2006Dynamic electrophoretic mobility of concentrated dispersions of sphericalcolloidal particle. On the consistent use of the cell model. Langmuir 22,7041–7051.

Allersma, M. W., Gittes, F., deCastro, M. J., Stewart, R. J. &Schmidt, C. F. 1998 Two-dimensional tracking of ncd motility by backfocal plane interferometry. Biophysical Journal 74, 1074–1085.

Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J.,Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S.,McKenney, A. & Sorensen, D. 1999 LAPACK Users’ Guide, 3rdedn. Society for Industrial and Applied Mathematics.

Ascher, U. M., Mattheij, R. M. M. & Russell, R. D. 1988 Numeri-cal Solution of Boundary Value Problems for Ordinary Differential Equa-tions . Englewood Cliffs: Prentice Hall.

Barber, J. R. 2003 Elasticity , 2nd edn. Dordrecht: Springer.

Barndl, F., Sommer, F. & Goepferich, A. 2007 Rational design of hy-drogels for tissue engineering: impact of physical factors on cell behavior.Biomaterials 28 (2), 134–146.

Barrire, B. & Leibler, L. 2003 Kinetics of solvent absorption and perme-ation through a highly swellable elastomeric network. Journal of PolymerScience Part B: Polymer Physics 41, 166–182.

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics . Cambridge:Cambridge University Press.

Baumgaertel, M., Schausberger, A. & Winter, H. H. 1990 The relax-ation of polymers with linear flexible chains of uniform length. RheologicaActa 29, 400–408.

Biot, M. A. 1941 General theory of three-dimensional consolidation. Journalof Applied Physics 12, 155–164.

Biot, M. A. 1956a Theory of propagation of elastic waves in a fluid-saturated

125

porous solid. I. low-frequency range. The Journal of The Acoustical Soci-ety of America 28, 168–178.

Biot, M. A. 1956b Theory of propagation of elastic waves in a fluid-saturatedporous solid. II. higher frequency range. The Journal of The AcousticalSociety of America 28, 179–191.

Booth, F. & Enderby, J. A. 1952 On electrical effects due to sound wavesin colloidal suspensions. Proceedings of the Physical Society. Section A65, 321–324.

Boudou, T., Ohayon, J., Arntz, Y., Finet, G., Picart, C. & Trac-qui, P. 2006 An extended modeling of the micropipette aspiration exper-iment for the characterization of the Young’s modulus and Poisson’s ratioof adherent thin biological samples: Numerical and experimental studies.Journal of Biomechanics 39, 1677–1685.

Brennen, C. 1975 A concentrated suspension model for the Couette rheologyof blood. Canadian Journal of Chemical Engineering 53, 126–133.

Breuer, K., ed. 2005 Microscale Diagnostic Techniques . Berlin: Springer.

Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowingfluid on a dense swarm of particles. Applied Scientific Research Section A1, 27–34.

Brochard, F. & de Gennes, P.-G. 1977 Dynamical scaling for polymersin theta solvents. Macromolecules 10, 1157–1161.

Brochard, F. & de Gennes, P.-G. 1992 Shear dependent slippage at apolymer/solid interface. Langmuir 8, 3033–3037.

Brochard-Wyart, F. & de Gennes, P.-G. 2000 Viscosity at small scalesin polymer melts. European Physical Journal E 1, 93–97.

Buscall, R. & Ettelaie, R. 2006 Colloidal dispersions in polymer melts.Industrial and Engineering Chemistry Research 45, 6915–6922.

Candau, S., Bastide, J. & Delsanti, M. 1982 Structural, elastic, anddynamic properties of swollen polymer networks. Advances in PolymerScience 44, 27–71.

Cash, J. R. & Mazzia, F. 2006 Hybrid mesh selection algorithms based onconditioning for two-point boundary value problems. Journal of Numeri-cal Analysis, Industrial and Applied Mathematics 1, 81–90.

Chaterji, S., Kwon, I. K. & Park, K. 2007 Smart polymeric gels: Re-defining the limits of biomedical devices. Progress in Polymer Science 32,1083–1122.

Chung, Y.-I., Ahn, K.-M., Jeon, S.-H., Lee, S.-Y., Lee, J.-H. &Tae, G. 2007 Enhanced bone regeneration with BMP-2 loaded functionnanoparticle-hydrogel complex. Journal of Controlled Release 121, 91–99.

126

Cicuta, P. & Donald, A. M. 2007 Microrheology: a review of the methodand applications. Soft Matter 3, 1449–1455.

Cohen, M. H. & Turnbull, D. 1959 Molecular transport in liquids andglasses. Journal of Chemical Physics 31, 1164–1169.

Colby, R. H., Rubinstein, M. & Daoud, M. 1994 Hydrodynamics ofpolymer solutions via two-parameter scaling. Journal de Physique II 4,1299–1310.

Cosgrove, T., ed. 2005 Colloid Science: Theory, Methods and Applications .Ames: Blackwell Publishing.

Cotton, J. P., Decker, D., Benoit, H., Farnoux, B., Higgins, J.,Jannink, G., Ober, R., Picot, C. & Cloizeau, J. D. 1974 Confor-mation of polymer chain in the bulk. Macromolecules 7, 863–872.

Coussy, O. 2004 Poromechanics . Chichester: Wiley.

Daoulas, K. C., Theodorou, D. N., Harmandaris, V. A., Karayian-nis, N. C. & Mavrantzas, V. G. 2005 Self-consistent-field study ofcompressible semiflexible melts adsorbed on a solid substrate and com-parison with atomistic simulations. Macromolecules 38, 7134–7149.

Dayton, P. A. & Ferrara, K. W. 2002 Targeted imaging using ultra-sound. Journal of Magnetic Resonance Imaging 16, 362–377.

Dealy, D. M. & Larson, R. G. 2006 Structure and Rheology of MoltenPolymers. From Structure to Flow Behavior and Back Again. Munich:Hanser.

Debye, P. 1933 A method for the determination of the mass of electrolyticions. Journal of Chemical Physics 1, 13–16.

DeLacey, E. H. B. & White, L. R. 1981 Dielectric response and conduc-tivity of dilute suspensions of colloidal particles. Journal of the ChemicalSociety, Faraday Transactions II 77, 2007–2039.

Doi, M. & Edwards, S. F. 1987 The Theory of Polymer Dynamics . NewYork: Oxford University Press.

Drury, J. L. & Mooney, D. J. 2003 Hydrogels for tissue engineering:scaffold design variables and applications. Biomaterials 24, 4337–4351.

Dukhin, A. S. & Goetz, P. J. 2002 Ultrasound for Characterizing ColloidsParticle Sizing, Zeta Potential, Rheology . Amsterdam: Elsevier.

Dukhin, A. S., Goetz, P. J., Wines, T. H. & Somasundaran, P.2000 Acoustic and electroacoustic spectroscopy. Colloids and Surfaces A:Physicochemical and Engineering Aspects 173, 127–158.

Dukhin, A. S., Ohshima, H., Shilov, V. N. & Goetz, P. J. 1999aElectroacoustics for concentrated dispersions. Langmuir 15, 3445–3451.

Dukhin, A. S., Shilov, V. N., Ohshima, H. & Goetz, P. J. 1999b

127

Electroacoustic phenomena in concentrated dispersions: new theory andCVI experiment. Langmuir 15, 6992–6706.

Eddington, D. T. & Beebe, D. J. 2004 Flow control with hydrogels.Advanced Drug Delivery Reviews 56, 199–210.

Einstein, A. 1906 The theory of the Brownian motion. Annalen Der Physik19, 371–381.

Enderby, J. A. 1951 On electrical effects due to sound wave in colloidalsuspensions. Proceedings of the Royal Society of London Series A 207,329–342.

Enge, A., Pelissier, P. & Zimmermann, P. 2007 MPC: Multiple Preci-sion Complex Library . INRIA.

Engler, A., Bacakova, L., Newman, C., Hategan, A., Griffin, M.& Discher, D. 2004 Substrate compliance versus ligand density in cellon gel responses. Biophysical Journal 86, 617–628.

Ferry, J. D. 1980 Viscoelastic Properties of Polymers . New York: JohnWiley and Sons.

Fleer, G. J., Cohen-Stuart, M. A., Scheutjens, J. M. H. M., Cos-grove, T. & Vincent, B. 1993 Polymers at Interfaces . London: Chap-man and Hall.

Fousse, L., Hanrot, G., Lefevre, V., Pelissier, P. & Zimmermann,P. 2007 MPFR: A multiple-precision binary floating-point library withcorrect rounding. ACM Transactions on Mathematical Software 33, 13.

Freeman, P. M., Natarajan, R. N., Kimura, J. H. & Andriacchi,T. P. 1994 Chondrocyte cells respond mechanically to compressive loads.Journal of Orthopaedic Research 12, 311–320.

Frenkel, J. 1944 On the theory of seismic and seismoelectric phenomena ina moist soil. Journal of Physics-USSR 3, 230–241.

Fung, Y. C. & Tong, P. 2001 Classical and Computational Solid Mechan-ics . Singapore: World Scientific.

Galneder, R., Kahl, V., Arbuzova, A., Rebecchi, M., Radler,J. O. & McLaughlin, S. 2001 Microelectrophoresis of a bilayer-coatedsilica bead in an optical trap: Application to enzymology. BiophysicalJournal 80, 2298–2309.

Ganesan, V., Pryamitsyn, V., Surve, M. & Narayanan, B. 2006Noncontinuum effects in nanoparticle dynamics in polymers. Journal ofChemical Physics 124 (22), 221102.

Geissler, E. & Hecht, A. M. 1980 The Poisson ratio in polymer gels.Macromolecules 13, 1276–1280.

128

Geissler, E. & Hecht, A. M. 1981 The Poisson ratio in polymer gels. 2.Macromolecules 14, 185–188.

de Gennes, P.-G. 1976a Dynamics of entangled polymer solutions: I. theRouse model. Macromolecules 9, 587–593.

de Gennes, P.-G. 1976b Dynamics of entangled polymer solutions. II. in-clusion of hydrodynamic interactions. Macromolecules 9, 594–598.

de Gennes, P.-G. 1979 Scaling Concepts in Polymer Physics . Cornell Uni-versity Press.

Gibb, S. E. & Hunter, R. J. 2000 Dynamic mobility of colloidal particleswith thick double layers. Journal of Colloid and Interface Science 224,99–111.

Granlund, T. 2007 GNU MP: The GNU Multiple Precision Arithmetic Li-brary , 4th edn.

Green, P. G. & Kramer, E. J. 1986 Matrix effects on the diffusion of longpolymer chains. Macromolecules 19, 1108–1114.

Gross, J., Scherer, G. W., Alviso, C. T. & Pekala, R. W. 1997Elastic properties of crosslinked resorcinol-formaldehyde gels and aerogels.Journal of Non-Crystalline Solids 211, 132–142.

Haraguchi, K. & Takehisa, T. 2002 Nanocomposite hydrogels: a uniqueorganic-inorganic network structure with extraordinary mechanical, op-tical, and swelling/de-swelling properties. Advanced Materials 14, 1120–1124.

Haraguchi, K., Takehisa, T. & Fan, S. 2002 Effects of clay con-tent on the properties of nanocomposite hydrogels composed of poly(N-isopropylacrylamide) and clay. Macromolecules 35, 10162–10171.

Hill, R. J. 2006a Diffusive permeability and selectivity of nanocompositemembranes. Industrial and Engineering Chemistry Research 45, 6890–6898.

Hill, R. J. 2006b Electric-field-induced force on a charged spherical colloidembedded in an electrolyte-saturated Brinkman medium. Physics of Flu-ids 18, 043103.

Hill, R. J. 2006c Reverse-selective diffusion in nanocomposite membranes.Physical Review Letters 96, 216001.

Hill, R. J. 2006d Transport in polymer-gel composites: theoretical method-ology and response to an electric field. Journal of Fluid Mechanics 551,405–433.

Hill, R. J. 2007 Electric-field-enhanced transport in polyacrylamide hydrogelnanocomposites. Journal of Colloid and Interface Science 316, 635–644.

129

Hill, R. J. & Ostoja-Starzewski, M. 2008 Electric-field-induced dis-placement of a charged spherical colloid embedded in an elastic Brinkmanmedium. Physical Review E 77, 011404.

Hill, R. J. & Saville, D. A. 2005 ‘Exact’ solutions of the full electrokineticmodel for soft spherical colloids: electrophoretic mobility. Colloids andSurfaces A: Physicochemical and Engineering Aspects 267, 31–49.

Hill, R. J., Saville, D. A. & Russel, W. B. 2003a Electrophoresisof spherical polymer-coated colloidal particles. Journal of Colloid andInterface Science 258, 56–74.

Hill, R. J., Saville, D. A. & Russel, W. B. 2003b High-frequencydielectric relaxation of spherical colloidal particles. Physical ChemistryChemical Physics 5, 911–915.

Hoffman, B. D., Massiera, G., Van Citters, K. M. & Crocker,J. C. 2006 The consensus mechanics of cultured mammalian cells. Pro-ceedings of the National Academy of Sciences of the United States ofAmerica 103, 10259–10264.

Hunter, R. J. 1998 Recent developments in the electroacoustic character-isation of colloidal suspensions and emulsions. Colloids and Surfaces A:Physicochemical and Engineering Aspects 141, 37–65.

Hunter, R. J. 2001 Foundations of Colloid Science, 2nd edn. Oxford: OxfordUniversity Press.

Hunter, R. J. & O’Brien, R. W. 1997 Electroacoustic characterization ofcolloids with unusual particle properties. Colloids and Surfaces A: Physic-ochemical and Engineering Aspects 126, 123–128.

Johnson, B., Bauer, J. M., Niedermaier, D. J., Crone, W. C. &Beebe, D. J. 2004a Experimental techniques for mechanical character-ization of hydrogels at the microscale. Experimental Mechanics 44 (1),21–28.

Johnson, B. D., Beebe, D. J. & Crone, W. C. 2004b Effects of swellingon the mechanical properties of a pH-sensitive hydrogel for use in mi-crofluidic devices. Materials Science and Engineering: C 24, 575–581.

Kang, S. W., Kim, J. H., Char, K., Won, J. & Kang, Y. S. 2006Nanocomposite silver polymer electrolytes as facilitated olefin transportmembranes. Journal of Membrane Science 285, 102–107.

Khademhosseini, A & Langer, R. 2007 Microengineered hydrogels fortissue engineering. Biomaterials 28, 5087–5092.

Kickelbick, G. 2003 Concepts for incorporation of inorganic building blocksinto organic polymers on a nanoscale. Progress in Polymer Science 28,83–114.

130

Kim, J. J. & Park, K. 1998 Smart hydrogels for bioseparation. Bioseparation7, 177–184.

Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and SelectedApplications . Boston: Butterworth-Heinemann.

Kimura, Y. & Mizuno, D. 2007 Microrheology of a swollen lyotropic lamel-lar phase. Molecular Crystals and Liquid Crystals 478, 3–13.

Kopesky, E. T., Haddad, T. S., Cohen, R. E. & McKinley, G. H.2004 Thermomechanical properties of poly(methly methacrylate)s con-taining tethered and untethered polyhedral oligomeric silsesquioxanes.Macromolecules 37, 8992–9004.

Lamb, H. 1945 Hydrodynamics . New York: Dover Publications.

Landau, L. D. & Lifshitz, E. M. 1986 Theory of Elasticity , 3rd edn.Oxford: Pergamon Press.

Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics . Oxford: Perga-mon Press.

Larson, R. G. 1999 The Structure and Rheology of Complex Fluids . NewYork: Oxford University Press.

Levine, A. J. & Lubensky, T. C. 2000 One- and two-particle microrheol-ogy. Physical Review Letters 85, 1774–1777.

Levine, A. J. & Lubensky, T. C. 2001 Response function of a sphere in aviscoelastic two-fluid medium. Physical Review E 63, 041510.

Li, Y., Hu, Z. & Li, C. 1993 New method for measuring Poisson ratio inpolymer gels. Journal of Applied Polymer Science 50, 1107–1111.

Lin, C.-C. & Netters, A. T. 2006 Hydrogels in controlled release for-mulations: Network design and mathematical modeling. Advanced DrugDelivery Reviews 58, 1379–1408.

Lin, D. C., Yurke, B. & Langrana, N. A. 2004 Mechanical proper-ties of a reversible, DNA-crosslinked polyacrylamide hydrogel. Journal ofBiomechanical Engineering-Transactions of ASME 26, 104–110.

Lin, D. C., Yurke, B. & Langrana, N. A. 2005 Use of rigid spher-ical inclusions in Young’s moduli determination: application to DNA-crosslinked gels. Journal of Biomechanical Engineering-Transactions ofASME 127, 571–579.

Liu, J., Levine, A. L., Mattoon, J. S., Yamaguchi, M., Lee, R. J.,Pan, X. L. & Rosol, T. J. 2006 Nanoparticles as image enhancingagents for ultrasonography. Physics in Medicine and Biology 51, 2179–2189.

131

Loo, C., Lowery, A., Halas, N., West, J. & Drezek, R. 2005 Im-munotargeted nanoshells for integrated cancer imaging and therapy. NanoLetters 5, 709–711.

Lyklema, J. 1995 Fundamentals of Interface and Colloid Science. II. Solid-Liquid Interfaces . London: Academic Press.

Mackay, M. E., Dao, T. T., Tuteja, A., Ho, D. L., Van Horn,B., Kim, H.-C. & Hawker, C. J. 2003 Nanoscale effects leading tonon-Einstein-like decrease in viscosity. Nature Materials 2, 762–766.

MacKintosh, F. C. & Schmidt, C. F. 1999 Microrheology. Current Opin-ion in Colloid and Interface Science 4, 300–307.

MacRobert, T. M. 1967 Spherical Harmonics: An elementary treatise onharmonic functions, with applications., 3rd edn. Oxford: Pergamon Press.

Mangelsdorf, C. S. & White, L. R. 1992 Electrophoretic mobility of aspherical colloidal particle in an oscillating electric field. Journal of theChemical Society, Faraday Transactions 88, 3567–3581.

Mangelsdorf, C. S. & White, L. R. 1997 Dielectric response of a dilutesuspension of spherical colloidal particles to an oscillating electric field.Journal of the Chemical Society, Faraday Transactions 93, 3145–3154.

Markov, M. G. 2005 Propagation of longitudinal elastic waves in a fluid-saturated porous medium with spherical inclusions. Acoustical Physics51, S115–S121.

Masliyah, J. H. & Bhattacharjee, S. 2006 Electrokinetic and ColloidTransport Phenomena. Hoboken: Wiley-Interscience.

Mason, T. G. & Weitz, D. A. 1995 Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids. Physical ReviewLetters 74 (7), 1250–1253.

Matos, M. A., White, L. R. & Tilton, R. D. 2006 Electroosmoticallyenhanced mass transfer through polyacrylamide gels. Journal of Colloidand Interface Science 300, 429–436.

Matos, M. A., White, L. R. & Tilton, R. D. 2008 Enhanced mixing inpolyacrylamide gels containing embedded silica nanoparticles as internalelectroosmotic pumps. Colloids and Surfaces B: Biointerfaces 61, 262–269.

Maxwell, J. C. 1873 Treatise on Electricity and Magnetism, vol. 1 . London:Oxford University Press.

Merkel, T. C., Freeman, B. D., Spontak, R. J., He, Z., Pinnau,I., Meakin, P. & Hill, A. J. 2002 Ultrapermeable, reverse-selectivenanocomposite membranes. Science 296, 519–522.

Migler, K. B., Hervet, H. & Leger, L. 1993 Slip transition of a polymermelt under shear stress. Physical Review Letters 70, 287–290.

132

Milner, S. T. 1993 Dynamical theory of concentration fluctuations in poly-mer solutions under shear. Physical Review E 48, 3674–3691.

Mizuno, D., Kimura, Y. & Hayakawa, R. 2000 Dynamic electrophoreticmobility of colloidal particles measured by the newly developed methodof quasi-elastic light scattering in a sinusoidal electric field. Langmuir 16,9547–9554.

Mizuno, D., Kimura, Y. & Hayakawa, R. 2001 Electrophoretic microrhe-ology in a dilute lamellar phase of a nonionic surfactant. Physical ReviewLetters 87, 088104.

Mizuno, D., Kimura, Y. & Hayakawa, R. 2004 Electrophoretic microrhe-ology of a dilute lamellar phase: Relaxation mechanisms in frequency-dependent mobility of nanometer-sized particles between soft membranes.Physical Review E 70, 011509.

O’Brien, R. W. 1979 A method for the calculation of the effective trans-port properties of suspensions of interacting particles. Journal of FluidMechanics 91, 17–39.

O’Brien, R. W. 1986 The high frequency dielectric dispersion of a colloid.Journal of Colloid and Interface Science 113, 81–93.

O’Brien, R. W. 1988 Electro-acoustic effects in a dilute suspension of spher-ical particles. Journal of Fluid Mechanics 190, 71–86.

O’Brien, R. W. 1990 The electroacoustic equations for a colloidal suspen-sion. Journal of Fluid Mechanics 212, 81–93.

O’Brien, R. W., Jones, A. & Rowlands, W. N. 2003 A new formulafor the dynamic mobility in a concentrated colloid. Colloids and SurfacesA: Physicochemical and Engineering Aspects 218, 89–101.

O’Brien, R. W. & White, L. R. 1978 Electrophoretic mobility of a spher-ical colloidal particle. Journal of the Chemical Society, Faraday Transac-tions II 74, 1607–1626.

Oestreicher, H. L. 1951 Field and impedance of an oscillating sphere ina viscoelastic medium with an application to biophysics. The Journal ofThe Acoustical Society of America 23, 707–714.

O’Konski, C. T. 1960 Electric properties of macromolecules V. Theory ofionic polarization in polyelectrolytes. Journal of Physical Chemistry 64,605–619.

Overbeek, J. Th. G. 1943 Theorie der elektrophorese. Kolloid-Beih 54,287–364.

Pal, R. 2007 Rheology of Particulate Dispersions and Composites . Boca Ra-ton: CRC Press.

Peppas, N. A., Bures, P., Leobandung, W. & Ichikawa, H. 2000

133

Hydrogels in pharmaceutical formulations. European Journal of Pharma-ceutics and Biopharmaceutics 50, 27–46.

Peppas, N. A., Hilt, J. Z., Khademhosseini, A. & Langer, R. 2006Hydrogels in biology and medicine: from molecular principles to bionan-otechnology. Advanced Materials 18, 1345–1360.

Pine, D. J., Weitz, D. A., Chaikin, P. M. & Herbolzheimer, E. 1988Diffusing wave spectroscopy. Physical Review Letters 61, 1134–1137.

Pozrikidis, C. 1996 Introduction to Theoretical and Computational FluidDynamics . New York: Oxford University Press.

Preston, M. A., Kornbrekke, R. & White, L. R. 2005 Determinationof the dynamic electrophoretic mobility of a spherical colloidal particlethrough a novel numerical solution of the electrokinetic equations. Lang-muir 21 (22), 9832–9842.

Qiu, Y. & Park, K. 2001 Environment-sensitive hydrogels for drug delivery.Advanced Drug Delivery Reviews 53, 321–339.

Rider, P. F. & O’Brien, R. W. 1993 The dynamic mobility of particlesin a non-dilute suspension. Journal of Fluid Mechanics 257, 607–636.

Roberts, C., Cosgrove, T., Schmidt, R. G. & Gordon, G. V. 2001Diffusion of poly(dimethyl siloxane) mixtures with silicate nanoparticles.Macromolecules 34, 538–543.

Rotello, V. M., ed. 2004 Nanoparticles: Building Blocks for Nanotechnol-ogy . New York: Kluwer Academic/Plenum Publishers.

Russel, W. B., Schowalter, W. R. & Saville, D. A. 1989 ColloidalDispersions . Cambridge: Cambridge University Press.

Rutgers, Ir. R. 1962 Relative viscosity of suspensions of rigid spheres inNewtonian liquids. Rheologica Acta 2, 202–210.

Scherer, G. W. 1992 Bending of gel beams: method for characterizing elas-tic properties and permeability. Journal of Non-Crystalline Solids 142,18–35.

Schmidt, G. & Malwitz, M. M. 2003 Properties of polymer-nanoparticlecomposites. Current Opinion in Colloid and Interface Science 8, 103–108.

Schnurr, B., Gittes, F., MacKintosh, F. C. & Schmidt, C. F. 1997Determining microscopic viscoelasticity in flexible and semiflexible poly-mer networks from thermal fluctuations. Macromolecules 30, 7781–7792.

Sershen, S. R., Mensing, G. A., Ng, M., Halas, N. J., Beebe, D. J.& West, J. L. 2005 Independent optical control of microfluidic valvesformed from optomechanically responsive nanocomposite hydrogels. Ad-vanced Materials 17, 1366–1368.

Shkel, I. A, Tsodikov, O. V. & Record Jr, M. T. 2000 Complete

134

asymptotic solution of cylindrical and spherical Poisson-Boltzmann equa-tions at experimental salt concentrations. Journal of Physical ChemistryB 104, 5161–5170.

Snieder, R. & Page, J. 2007 Multiple scattering in evolving media. PhysicsToday May 2007, 49–55.

Soskey, P. R. & Winter, H. H. 1984 Large step shear strain experimentswith parallel-disk rotational rheometers. Journal of Rheology 28, 625–645.

Speight, J. G. 2005 Lange’s Handbook of Chemistry , 16th edn. New York:McGraw-Hill.

Strubulevych, A., Leroy, V., Scanlon, M. G. & Page, J. H. 2007Characterizing a model food gel containing bubbles and solid inclusionsusing ultrasound. Soft Matter 3, 1388–1394.

Takigawa, T., Morino, Y., Urayama, K. & Masuda, T. 1996 Poisson’sratio of polyacrylamide (PAAm) gels. Polymer Gels and Networks 4, 1–5.

Takigawa, T., Yamawaki, T., Takahashi, K. & Masuda, T. 1997Change in Young’s modulus of poly(N-isopropylacrylamide) gels by vol-ume phase transition. Polymer Gels and Networks 5, 585–589.

Tanaka, T., Hocker, L. O. & Benedek, G. B. 1973 Spectrum of lightscattered from a viscoelastic gel. Journal of Chemical Physics 59, 5151–5159.

Taylor, G. I. 1932 The viscosity of a fluid containing small drops of anotherfluid. Proceedings of the Royal Society of London Series A 138, 41–48.

Temkin, L. & Leung, C. M. 1976 On the velocity of a rigid sphere in asound wave. Journal of Sound and Vibration 49, 75–92.

Thevenot, C., Khoukh, A., Reynaud, S., Desbrieres, J. & Grassl,B. 2007 Kinetic aspects, rheological properties and mechanoelectrical ef-fects of hydrogels composed of polyacrylamide and polystyrene nanopar-ticles. Soft Matter 3, 437–447.

Torquato, S. 2002 Random Heterogeneous Materials: Microstructure andMacroscopic Properties . New York: Springer-Verlag.

Tuinier, R. & Lekkerkerker, H. N. W. 2002 Polymer density around asphere. Macromolecules 35, 3312–3313.

Tuteja, A., Duxbury, P. M. & Mackay, M. E. 2007a Multifunctionalnanocomposites with reduced viscosity. Macromolecules 40, 9427–9434.

Tuteja, A., Duxbury, P. M. & Mackay, M. E. 2008 Polymer chainswelling induced by dispersed nanoparticles. Physical Review Letters 100,077801.

Tuteja, A., Mackay, M. E., Hawker, C. J. & Van Horn, B. 2005

135

Effect of ideal, organic nanoparticles on the flow properties of linear poly-mers: Non-Einstein-like behavior. Macromolecules 38, 8000–8011.

Tuteja, A., Mackay, M. E., Narayanan, S., Asokan, S. & Wong,M. S. 2007b Breakdown of the continuum Stokes-Einstein relation fornanoparticle diffusion. Nano Letters 7, 1276–1281.

Urayama, K., Takigawa, T. & Masuda, T. 1993 Poisson ratio ofpoly(vinyl alcohol) gels. Macromolecules 26, 3092–9096.

Valentine, M. T., Dewalt, L. E. & Ou-Yang, H. D. 1996 Forces ona colloidal particle in a polymer solution: a study using optical tweezers.Journal of Physics: Condensed Matter 8, 9477–9482.

Verwey, E. J. W. & Overbeek, J. Th. G. 1948 Theory of Stability ofLyophobic Colloids . Elsevier, Amsterdam.

Wang, K. L., Burban, J. H. & Cussler, E. L. 1993 Hydrogels as sepa-ration agents. Advances in Polymer Science 110, 67–79.

Wang, M. & Hill, R. J. 2008 Electric-field-induced displacement of chargedspherical colloids in compressible hydrogels. Soft Matter 4, 1048–1058.

Westermeier, R. 2005 Electrophoresis in Practice. Weinheim: Wiley-VCH.

Wu, D. T., Fredrickson, G. H., Carton, J. P., Ajdari, A. &Leibler, L. 1995 Distribution of chain-ends at the surface of a poly-mer melt: compensation effects and surface-tension. Journal of PolymerScience Part B: Polymer Physics 33, 2373–2389.

Yamaguchi, N., Chae, B.-S., Zhang, L., Kiick, K. L. & Furst, E. M.2005 Rheological characterization of polysaccharide-poly(ethylene glycol)star copolymer hydrogels. Biomacromolecules 6 (4), 1931–1940.

Yao, S. H., Hertzog, D. E., Zeng, S. L., Mikkelsen, J. C. & Santi-ago, J. G. 2003 Porous glass electroosmotic pumps: design and experi-ments. Journal of Colloid and Interface Science 268, 143–153.

Yao, S. H. & Santiago, J. G. 2003 Porous glass electroosmotic pumps:theory. Journal of Colloid and Interface Science 268, 133–142.

Ziemann, F., Radler, J. & Sackmann, E. 1994 Local measurement ofviscoelastic moduli of entangled actin networks using an oscillating mag-netic bead micro-rheometer. Biophysical Journal 66, 2210–2216.

136