Steady electrical and micro-rheological response …rspa.royalsocietypublishing.org/content/royprsa/466/2113/213.full.pdf · Steady electrical and micro-rheological response functions

Proc. R. Soc. A (2010) 466, 213–235doi:10.1098/rspa.2009.0286

Published online 7 October 2009

Steady electrical and micro-rheological responsefunctions for uncharged colloidal inclusions

in polyelectrolyte hydrogelsBY ALIASGHAR MOHAMMADI1 AND REGHAN J. HILL1,2,*

1Department of Chemical Engineering, McGill University, Montreal,Quebec H3A 2B2, Canada

2McGill Institute for Advanced Materials, McGill University, Montreal,Quebec H3A 2A7, Canada

The electric-field-induced response of an uncharged colloidal sphere embedded in aquenched polyelectrolyte hydrogel is calculated from a model where the polymer networkis treated as an elastic, porous skeleton saturated with an aqueous electrolyte. We presentexact analytical solutions for the steady response to a uniform electric field, as well asthe steady susceptibility, defined as the ratio of the particle displacement to the strengthof an optical or magnetic force. Even though the particle is uncharged, it attains a finiteelectric-field-induced displacement owing to hydrodynamic coupling with electroosmoticflow. The steady susceptibility decreases with increasing charge and decreasing electrolyteconcentration; in general, charge imparts a small correction to the classical theory for anuncharged linearly elastic continuum.

Keywords: micro-rheology; electrical micro-rheology; polyelectrolyte hydrogel;steady susceptibility; colloidal inclusions; hydrogel–colloid composites

1. Introduction

Hydrogels are water-saturated polymer networks that have widespreadapplications in drug delivery (Qiu & Park 2001) and tissue engineering (Brandlet al. 2007); they have also been identified as promising candidates for syntheticmuscles (Calvert 2004; Bar-Cohen 2007). In the recent years, there has been muchinterest in micro-rheological characterization of hydrogels, particularly those ofbiological origin (Waigh 2005).

Micro-rheology probes viscoelastic properties of the micro-structure fromthe dynamics of embedded colloidal particles. The principal advantages ofmicro-rheology over macro-scale rheological methods include small samplesize (MacKintosh & Schmidt 1999; Gardel et al. 2005), wide frequencyrange (Schnurr et al. 1997) and the ability to directly probe micro-scalecharacteristics of soft matter (Schnurr et al. 1997; MacKintosh & Schmidt 1999;Meyer et al. 2006).

*Author for correspondence ([email protected]).

Received 27 May 2009Accepted 11 September 2009 This journal is © 2009 The Royal Society213

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214 A. Mohammadi and R. J. Hill

While two-point micro-rheology probes non-local (bulk) characteristics(Crocker et al. 2000; Levine & Lubensky 2001b), single- and two-point methodsare both sensitive to electrostatic, chemical and steric interactions between theparticle and the matrix (McGrath et al. 2000; Valentine et al. 2004; Ehrenberg &McGrath 2005). Such interactions are generally undesirable in micro-rheologybecause they complicate an otherwise simple conversion of experimental datato bulk rheological characteristics. However, with an increasing interest inhydrogel–colloid composites (Schexnailder & Schmidt 2009), understanding howsuch interactions affect particle dynamics may furnish novel diagnostics, in asimilar manner that micro-electrophoresis and electroacoustics, for example,have become routine for studying colloidal dispersions (O’Brien & White 1978;O’Brien 1990).

Micro-rheological techniques are generally classified as active or passive (Waigh2005). In active methods, probe particles are driven by magnetic (Freundlich &Seifriz 1923; Ziemann et al. 1994) or optical (Valentine et al. 1996) forces, whereasthe dynamics in passive experiments are entirely due to thermal fluctuations.Electrical forces have received much less attention because of complicatingelectrokinetic influences, such as diffuse double-layer dynamics and electroosmoticflow. Accordingly, few experiments have been reported (Mizuno et al. 2000,2001, 2004) and our understanding of the electric-field-induced displacementis poor.

Hill & Ostoja-Starzewski (2008) undertook the first theoretical study ofelectric-field-induced particle displacement. They calculated the steady electric-field-induced response of a charged, spherical colloid embedded in incompressible,uncharged polymer gels, showing that sub-nanometre displacements prevail undertypical experimental conditions. Wang & Hill (2008) extended their model tocompressible, but still uncharged, polymer skeletons, predicting displacementsthat are large enough to register with optical microscopy.

However, many hydrogels are charged, and even ideally uncharged gels(e.g. polyacrylamide) become weakly charged owing to chemical reactions, e.g.hydrolysis (Kizilay & Okay 2003). Fixed charge is well known to impactswelling and other responses to external stimuli (Skouri et al. 1995). Note alsothat the principal biological subjects of micro-rheology (e.g. F-actin networks)are charged.

In this paper, we take a first step towards quantifying the effect of polymercharge on the particle response to steady electrical and non-electrical forces.While our analysis is limited to uncharged inclusions in charged skeletons, itprovides a simple physical and mathematical framework to furnish the exactanalytical solutions. In principle, an experimental test of the theory could beundertaken using quenched polyelectrolyte hydrogels with an electrolyte whosepH is tuned to the isoelectric point of the immobilized inclusions. Interestingly,our theory predicts that particles with a dielectric constant that is much higherthan water have a relatively strong response to electrical forcing when embeddedin charged hydrogels.

We extend earlier theoretical analyses of spherical particles in chargedviscoelastic matrices (Schnurr et al. 1997; Levine & Lubensky 2000), providinga more comprehensive interpretation of active and passive micro-rheology.Electrostatic interactions arising from changes in the density of the polymerskeleton upon deformation are expected to increase the effective rigidity and,

Proc. R. Soc. A (2010)



Polyelectrolyte hydrogel micro-rheology 215

therefore, attenuate the particle response. In previous theoretical treatments ofactive and passive micro-rheology, electrostatic influences have been implicitlylumped into the effective elastic constants, e.g. Poisson ratio and Young’smodulus (Levine & Lubensky 2000, 2001a). By separating the electrostaticpenalty of compression from the intrinsic elastic energy, we seek to quantify howelectrostatic screening owing to added electrolyte—and indeed from the chargedmatrix itself—attenuates the particle response.

Modern micro-rheological instruments, including contemporary electrophoresisdevices (Minor et al. 1997), adopt oscillatory forcing. Nevertheless, smallparticles often respond in a quasi-steady manner. We pursue the quasi-steady response here to provide a sound understanding of the physics, in asimilar manner to classical analyses of the Stokes hydrodynamic mobility andthe Smoluchowski electrophoretic mobility, among other quasi-steady responsefunctions. The frequency-dependent dynamic response may be more relevant tofuture interpretation of experiments, as amplitude attenuation and phase lag areexpected due to draining and inertial influences. These are beyond the scope ofthis paper, but will be addressed elsewhere.

Note that a significant body of literature on the dynamics of polyelectrolytehydrogels has emerged from the studies of articular cartilage (Lai et al. 1991;Gu et al. 1998). Lai et al. (1991) and Gu et al. (1998) derived the dynamicalequations for charged, soft tissue based on multi-phase continuum theoriesthat account for a charged porous solid, solvent (water) and added salt. Suchtheories are not suitable for the hydrogel–colloid composites considered in thispaper, because they enforce local electroneutrality. Rather, these theories areappropriate on (macroscopic) scales larger than the Debye screening length,generally � 100 nm in aqueous electrolytes.

Li et al. (2004a,b, 2006) developed a so-called multi-physic, multi-effect modelthat adopts the Poisson equation to handle electrostatics. Similarly, Hill &Ostoja-Starzewski’s (2008) electrokinetic model for uncharged hydrogels withcharged spherical inclusions extends the two-fluid model of Levine & Lubensky(2000, 2001a) by including electrokinetic influences. Our work extends theseelectrokinetic models to polyelectrolyte hydrogels by including the electrostaticforces owing to fixed charge on the polymer. In our parametric analysis, weapproximate the polyelectrolyte as quenched, meaning that the fixed chargedensity is independent of pH, added salt and polymer concentration (Raphael1990; Guo & Ballauff 2000, 2001). Nevertheless, such influences can be accountedfor with models or experimental measurements of polyelectrolyte charge for aspecific polymer architecture.

2. Theoretical model

Our continuum model for a polyelectrolyte hydrogel comprises three phases:a charged, soft, porous solid (polymer network), solvent (water) and ions(counterions and added salt) (Hill et al. 2003; Hill & Ostoja-Starzewski 2008).The porous medium is modelled as a compressible linear elastic solid witha continuous uniform distribution of fixed charge, and the solvent as anincompressible Newtonian fluid. The ionic charge is either mobile or fixed to thepolymer skeleton. Mobile ions include M species of counterions of the fixed charge,





and N species of ions from added salt. Electrostatics are governed by thePoisson equation

−ε◦εs∇2ψ = ρm + ρf , (2.1)

where ψ , ε◦ and εs are, respectively, the electrostatic potential, vacuumpermittivity and solvent dielectric constant. The mobile and fixed chargedensities are ρm = ∑M+N

j=1 njzje and ρf = ∑Mj=1 nf

j zfj e, where e, nj , nf

j , zj andz fj are, respectively, the elementary charge, jth mobile ion number density,

jth fixed charge number density, jth mobile ion valence and jth fixed chargevalence. The flux of jth mobile ion is given by the well-known Nernst–Planckequation

j j = −Dj∇nj − zjeDj

kBTnj∇ψ + nju, (2.2)

where, at steady state, conservation demands

∇ · j j = 0. (2.3)

Here Dj , u, kB and T are, respectively, the jth ion diffusion coefficient, fluidvelocity, Boltzmann constant and absolute temperature. Under steady conditions,the fluid velocity and ion fluxes are relative to a stationary porous skeleton.

Fluid momentum conservation is achieved via a linearized Navier–Stokesequation with electrical and Darcy drag forces

∇ · σh − η�−2u − ρm∇ψ = 0, (2.4)

where σh = −pI + 2ηeh is the Newtonian fluid stress tensor with eh = (1/2)[∇u +(∇u)T] the (fluid) rate of strain tensor and I the identity tensor. Here, η, p and� are, respectively, the fluid shear viscosity, pressure and Brinkman screeninglength of the porous skeleton (Brinkman 1947). The Brinkman screening lengthis the square root of the Darcy permeability; in the polymer physics literatureit is taken to be the mesh size of a polymer gel. The second and third terms onthe right-hand side of equation (2.4) are, respectively, the hydrodynamic dragexerted by the polymer on the fluid, and the electrostatic body force acting onthe fluid. Fluid mass conservation demands

∇ · u = 0, (2.5)

because we assume the volume fraction of solvent approaches one (Lai et al. 1991;Levine & Lubensky 2001a).

Static equilibrium of the poroelastic skeleton demands

∇ · σe + η�−2u − ρf∇ψ = 0 (2.6)

with linear elastic stress tensor

σe = E1 + ν

[e + ν

1 − 2ν(∇ · v)I

]. (2.7)





Here, ee = (1/2)[∇v + (∇v)T], v, ν and E are, respectively, the polymer elasticstrain tensor, displacement, Poisson ratio and Young’s modulus. The secondand third terms in equation (2.6) are the hydrodynamic drag of the fluidand electrostatic body force acting on the polymer, respectively. Substitutingequation (2.7) into equation (2.6) gives an equation of static equilibrium:

E2(1 + ν)

[∇2v + 1

1 − 2ν∇(∇ · v)

]+ η�−2u − ρf∇ψ = 0. (2.8)

The density of the polymer skeleton is expressed as (Landau & Lifshitz1986) ρp = ρ◦

p/(1 + ∇ · v) = ρ◦p[1 − ∇ · v + (∇ · v)2 − · · · ], where ρ◦

p and ρp are thereference and deformed porous solid densities, respectively. Therefore, the fixedcharge density under small-strain deformation is

ρf = ρf◦(1 − ∇ · v), (2.9)

where ρf◦ is the equilibrium fixed charge density of the polymer skeleton. Notethat eqn (66) of Lai et al. (1991) approaches our equation (2.9) as the watervolume fraction φ◦

w → 1, which is often the case for the viscoelastic subjects ofmicro-rheology.

A spherical polar coordinate system (r , θ , φ) is adopted to solve the foregoingmodel equations. When an applied electric field E is directed along the polaraxis ez , with the origin centred on the particle, the boundary conditions forthe electrostatic potential are ε◦εp(∇<ψ) · n̂ − ε◦εs(∇>ψ) · n̂ = 0 at r = a, ψcontinuous at r = a, ψ → −E · r as r → ∞, and ψ finite at r = 0. Here, n̂is an outward unit normal to the particle surface (for a spherical particlen̂ = er); subscripts < and >, respectively, distinguish the particle and the solventsides of the interface. Boundary conditions for the polymer displacement arev = Z at r = a and v → 0 as r → ∞. Other boundary conditions (see Hill &Ostoja-Starzewski 2008) are u = 0 at r = a, j j · n̂ = 0 at r = a, ∇p → 0 at r → ∞,ρm → ρm◦ as r → ∞, ρf → ρf◦ as r → ∞. Note that ρm◦ is the equilibrium densityof mobile charge, and bulk electroneutrality demands ρm◦ = −ρf◦.

To determine the particle displacement under the influence of a weak electricfield, a perturbation methodology is adopted (O’Brien & White 1978; Hill et al.2003). The fields are calculated for equilibrium conditions, i.e. in the absence ofan electric field and external force. For this equilibrium base state, the solution ofthe governing equations is simply ρf = ρf◦ = −ρm = −ρm◦ = constant, ψ = ψ◦ =constant, p = p◦ = constant, u = u◦ = 0, and v = v◦ = 0, where the superscript ◦denotes the equilibrium base state. Then, in the presence of an electric field, themodel equations are linearized with ρm = ρm◦ + ρm′, . . . , ψ = ψ◦ + ψ ′, . . . , v = v′,where primed quantities denote the perturbations from equilibrium. Linearizedequations for the perturbations, which are generally valid for weak electricfields E = |E| � κkBT/e and small particle displacements Z = |Z | � a, are

−ε◦εs∇2ψ ′ = ρm′ − ρf◦∇ · v′, (2.10)

∇ · (−Dj∇n ′j − zj e

Dj

kBTn◦

j ∇ψ ′ + n◦j u

′) = 0 ( j = 1, . . . , M + N ), (2.11)





η∇2u ′ − ∇p′ = η�−2u ′ + ρm◦∇ψ ′, (2.12)

E2(1 + ν)

[∇2v′ + 1

1 − 2ν∇(∇ · v′)

]= −η�−2u ′ + ρf◦∇ψ ′ (2.13)

and ∇ · u ′ = 0, (2.14)

where ρm′ = ∑M+Nj=1 n ′

j zj e. The foregoing M + N + 8 independent scalar equationscan be solved analytically to ascertain ψ ′, p′, u ′, v′ and n ′

j . From the resultingforce on the inclusion, the superposition methodology detailed in the next sectiongives the particle displacement in response to an applied force.

(a) Particle displacement

The general solution for the perturbations is obtained from two independentsub-problems. In the so-called Z -problem, the particle is displaced a distance Zin the absence of an applied electric field (E = 0). This is equivalent to a uniformtranslation of the far field with the particle fixed at the origin, giving boundaryconditions v′ = 0 at r = a and v′ → −Z as r → ∞. In the so-called E-problem,the particle is fixed at the origin (Z = 0) in the presence of an external electricfield E. The boundary conditions for the polymer displacement are then v′ = 0at r = a and v′ → 0 as r → ∞.

The total force on the particle F is the sum of the forces FZ and FE fromthe foregoing Z - and E-problems. To satisfy the particle equation of motion atsteady state, FZ = −FE . Because the perturbed problem is linear, the forces canbe written (O’Brien & White 1978)

FE = f EE and FZ = f ZZ ,

where f E and f Z are independent of E and Z . Accordingly, the electrical responseis defined as

Z/E = −f E/f Z ,

where f E and f Z are from equations (2.10)–(2.14) and their boundary conditions.It is expedient to separate the forces acting on the colloidal particle into electrical,hydrodynamic, and elastic (or mechanical-contact) contributions. However, thereis no electrical force, so the hydrodynamic and elastic forces are furnished byintegrals of the hydrodynamic and elastic stress.

(i) Z-problem

In this problem, the fluid is at rest, so there is no force from the deviatoric fluidstress. However, in contrast to uncharged hydrogels (Hill & Ostoja-Starzewski2008; Wang & Hill 2008), there exists a hydrostatic force owing to the gradientsof osmotic pressure. The mechanical-contact force is due to polymer displacementwhere, in contrast to uncharged hydrogels, the displacement is coupled to theelectrostatic potential.





It is well known that a vector field can be decomposed into irrotational andsolenoidal parts, so

v′ = −Z + ∇ × A + ∇φ, (2.15)

where A (with ∇ · A = 0) and ∇φ (with ∇ × ∇φ = 0) decay as r → ∞. Linearityand symmetry require (Landau & Lifshitz 1987)

A = ∇f (r) × Z , (2.16)

so equation (2.11) becomes

∇2n ′j = − zj e

kBTn◦

j ∇2ψ ′. (2.17)

Multiplying both sides of equation (2.17) by zj e, and summing over the mobileions ( j = 1, . . . , M + N ) gives

∇2ρm′ = −2I e2

kBT∇2ψ ′, (2.18)

where I = (1/2)∑M+N

j=1 z2j n◦

j is the ionic strength. The sum includes counterionsof the fixed charge, so the ionic strength is non-zero in the absence of added salt.

Substituting equation (2.15) into equations (2.10) and (2.13), taking thedivergence and curl of the resulting equation (2.13), and noting that u ′ = 0, gives

ε◦εs∇2ψ ′ = −ρm′ + ρf◦∇2φ, (2.19)

−∇p′ = ρm◦∇ψ ′, (2.20)

∇[∇2∇2f (r)] = 0 (2.21)

and ∇2∇2φ = (1 − 2ν)(1 + ν)

(1 − ν)

ρf◦

E ∇2ψ ′. (2.22)

From an exact analytical solution of these equations, the hydrodynamic (osmoticpressure) and mechanical-contact forces are

Fh,Z = −4πaE(βa)2(1 − 2ν)

Da(2 + εp/εs + Da)[4(1 − ν) + (1 − 2ν)(κ/D)2] + (5 − 6ν)(2 + εp/εs)Z

(2.23)and

Fe,Z = −4πaE[3(Da + 1)(2 + εp/εs) + 2(Da)2 + (κa)2](1 − ν)/(1 + ν)

Da(2 + εp/εs + Da)[4(1 − ν) + (1 − 2ν)(κ/D)2] + (5 − 6ν)(2 + εp/εs)Z ,

(2.24)

respectively, where

D2 = κ2 + β2(1 − 2ν)(1 + ν)/(1 − ν) (2.25)





with κ2 = 2I e2/(kBTε◦εs) and β2 = ρf◦2/(ε◦εsE). Note that κ−1 is the well-known

Debye screening length, and β−1 is a new length scale whose physical significancewill be discussed below. Recall, I includes contributions from the added salt andcounterions of the fixed charge. Therefore, if the polymer bears a finite charge,κ−1 and D−1 are finite in the absence of added salt.

Finally, summing the forces above gives

f Z = −12πaE[(Da)2 + (Da + 1)(2 + εp/εs)](1 − ν)/(1 + ν)

Da(2+εp/εs +Da)[4(1−ν)+ (1−2ν)(κ/D)2]+ (5−6ν)(2+εp/εs). (2.26)

As the fixed charge density vanishes, D → κ and it is readily verified thatwe recover the same formula f Z = −12πaE(1 − ν)/[(1 + ν)(5 − 6ν)] as Hill &Ostoja-Starzewski (2008) and Lin et al. (2005) for uncharged hydrogels.

(ii) E-problem

In this problem, the polymer displacement, fluid velocity and pressure are allnon-zero. As the fluid velocity is divergence-free,

u ′ = ∇ × B, (2.27)

where B = ∇g(r) × E. Similarly to the Z -problem, the polymer displacementtakes the form

v′ = ∇ × C + ∇ϕ, (2.28)

where C = ∇k(r) × E, and the vector fields C and ∇ϕ vanish as r → ∞.We now obtain

ε◦εs∇2ψ ′ = −ρm′ + ρf◦∇2ϕ, (2.29)

∇2ρm′ = − 2Ie2

kBT∇2ψ ′, (2.30)

−∇2p′ = ρm◦∇2ψ ′, (2.31)

η∇2(∇ × u ′) = η�−2(∇ × u ′), (2.32)

∇2∇2ϕ = ρf◦

E(1 + ν)(1 − 2ν)

(1 − ν)∇2ψ ′ (2.33)

and ∇2(∇ × v′) = −2η�−2 (1 + ν)

E (∇ × u ′). (2.34)

From an exact analytical solution of these equations for the polymer displacement,and fluid velocity and pressure, the hydrodynamic and mechanical-contactforces are

Fh,E = 2πa3ρf◦[− �2

a2

(a2

�2+ 3

a�

+ 3)

+ B]E (2.35)





and

Fe,E = 2πa3ρf◦(βa)−2 (1 − ν)

(1 + ν)(1 − 2ν)

{B

[[(κa)2 + (Da + 1)(2 + εp/εs)]

+ 2(1 + ν)

(1 − ν)[(Da)2 + (Da + 1)(2 + εp/εs)]

]− [(κa)2 + 2[(Da)2

+ 3(Da + 1)]]}E, (2.36)

where

B =4(1 − ν)[(Da)2 + 2Da + 2] + 4(1 + ν)(1 − 2ν)(a/�)(β�)2 + (1 − 2ν)[(κa)2

+2(κa)2/(Da) + 2][4(1 − ν) + (κ/D)2(1 − 2ν)][(Da)2 + (Da)(2 + εp/εs)] + (5 − 6ν)(2 + εp/εs)

.

(2.37)

The hydrodynamic force comprises viscous, dynamic- and osmotic-pressure terms.Summing the forces above gives

f E = −2πρf◦a�2{

(ν − 1)

(β�)2(2ν − 1)(ν + 1)[(κa)2 − 3B[(Da)2 + (Da + 1)(2 + εp/εs)]

+ 2[(Da)2 + 3(Da + 1)]] +(

a2

�2+ 3

a�

+ 3)}

. (2.38)

(b) Dielectric contrast

Surprisingly, the electrical response from equations (2.26) and (2.38) dependson the particle dielectric constant εp. This is contradictory to the expectationsfrom O’Brien & White’s (1978) well-known analysis of the electrostaticboundary conditions for electrophoresis of colloidal particles in Newtonianelectrolytes. They proved that the electrophoretic mobility of a chargedcolloidal particle is independent of the particle dielectric constant. Moreover,Hill & Ostoja-Starzewski (2008) identified a close connection between thesteady electrical displacement of charged inclusions embedded in uncharged,incompressible hydrogels. Accordingly, for uncharged, compressible hydrogels,the particle displacement is also independent of the dielectric constant(Wang & Hill 2008).

To help verify the distinctly different behaviour with a charged polymerskeleton, we follow O’Brien & White (1978) and introduce ψ ′1, p′1, u ′1 andv′1 as solutions of the X -problem (with X = Z or E) with particle dielectricconstant ε1

p. The electrostatic boundary conditions at the particle surface r = aare εs∂ψ ′1/∂r |> = ε1

p∂ψ ′1/∂r |< with continuous ψ ′1. Here, subscripts < and >

distinguish, respectively, the particle and the solvent sides of the interface.Similarly, ψ ′2, p′2, u ′2 and v′2 denote solutions of the same problem, butwith a particle dielectric constant ε2

p and boundary conditions εs∂ψ ′2/∂r |> =ε2p∂ψ ′2/∂r |< and continuous ψ ′2.





As the equations governing the perturbations are linear, differences (denotedwith the symbol � below) owing to changing εp are the solutions of thesame equations, but with electrostatic boundary conditions εs∂�ψ ′/∂r |> =ε2p∂�ψ ′/∂r |< + q and continuous �ψ ′, where q = −(ε1

p − ε2p)∂ψ ′1r/∂r |<. All other

differences at the particle surface vanish, and all the differences vanish as r → ∞.Note also that the governing equations are independent of whether X = Z or E .Our solutions for ψ ′ yield

q = − (ε1p − ε2

p

) ∂ψ̂1

∂r

∣∣∣∣∣<

X cos θ , (2.39)

where θ is the azimuthal angle with X directed along the polar axis. Accordingly,the differences are the perturbations that arise from endowing an originallyuncharged particle embedded in a uniform, unperturbed hydrogel with thenon-uniform surface charge density given by equation (2.39).

With an uncharged polymer skeleton, the surface charge q cannot give riseto a force, because excess negative charge on one side is compensated for by anequal excess of positive charge on the other side. Overall, the osmotic pressureretains fore–aft symmetry, so there is no net force on the particle. In a chargedpolymer skeleton, however, negative surface charge on one side of the particlerepels (attracts) a negatively (positively) charged skeleton, while positive chargeon the other side attracts (repels) a negatively (positively) charged skeleton. Thus,the particle experiences a net elastic (mechanical-contact) force whose magnitudeis expected to increase with the (dipole) strength of the surface charge densitygiven by equation (2.39).

The force on the particle in the X -problem can be calculated from the far-fielddecay of the pressure p, velocity u and displacement v. In the electrophoreticmobility problem, as O’Brien & White have shown, the far-field decays of velocityand pressure are independent of ψ . Therefore, the force is independent of εp.However, for the problem addressed in this work, p and v are coupled to ψ inthe far field.

By introducing a function Φj defined by

nj(r) = n◦j exp

[− ezj

kBT(ψ + Φj + E · r)

], (2.40)

O’Brien & White show that the far-field decays of p and u in the electrophoreticmobility problem are independent of the potential ψ and ion densities nj . We have

∇2ψ ′ − κ2ψ ′ = e2

ε◦εskBT

M+N∑j=1

n◦j z

2j (Φj + E · er ) − ρf◦

ε◦εs∇ · v′, (2.41)

η∇2u ′ − ∇p′ = η�−2u ′ + ρm◦∇ψ ′, (2.42)

∇ ·(

ezjn◦j

Dj

kBT∇Φj + ezjn◦

jDj

kBTE + n◦

j u)

= 0, (2.43)

E2(1 + ν)

[∇2v′ + 1

1 − 2ν∇(∇ · v′)

]= −η�−2u ′ + ρf◦∇ψ ′ (2.44)

and ∇ · u ′ = 0, (2.45)

where the boundary conditions for Φj are (∇Φj + E) · n̂ = 0 at r = a and Φj → 0as r → ∞.





Table 1. The principal set of dimensional parameters for the results presented in this work. Notethat the ionic strength I includes contributions from ions of the added salt and counterions of thefixed charge. The accompanying dimensionless parameters are ν ≈ 0.2 (Poisson ratio), κa ≈ 100,βa ≈ 10, Da ≈ 100, a/� ≈ 100 and εp/εs ≈ 0.02.

temperature, T 298 Kelectric-field strength, E 25 V cm−1

particle radius, a 500 nmparticle dielectric constant, εp 1.6water dielectric constant, εs 80water viscosity, η 10−3 kg m−1 s−1

ionic strength, I 3.78 × 10−3 mol l−1

fixed charged density, ρf◦ 1.68 × 104 C m−3 (=1.05 × 10−4e nm−3)Brinkman screening length, � 5 nmYoung’s modulus, E 1 kPa

Equation (2.45) with the curl of equation (2.42) shows that u is independentof ψ . Thus, with a uniformly charged polymer skeleton at equilibrium, the fluidvelocity disturbance is the same as for pressure-driven flow past a sphericalinclusion in a uniform Brinkman medium (Brinkman 1947). However, the far-field fluid velocity u → −η−1�2ρf◦E rather than −η−1�2∇p. Note that the far-fielddecay of p must be obtained from equation (2.42) with the knowledge of ψ .Similarly, ψ influences v through equations (2.41) and (2.44). These couplingsarise from terms involving the fixed charge ρf◦, so the force on the particleand, hence, its displacement depend on εp through the far-field decay of ψ .Quantitative influences are examined below.

3. Electrical response

We examine the general features of the perturbed fields with the representativeparameters in table 1. As noted previously, the ionic strength includes counterionsof the fixed charge and ions from added salt. Accordingly, the Debye lengthκ−1 involves a sum over all mobile ions. The perturbations to the electrostaticpotential, pressure, ion concentrations, fluid velocity and the particle and polymerdisplacement are proportional to the electric-field. This is the only way theelectric-field strength enters the problem. Note that only four of the five identifieddimensionless parameters are independent, because D is related to κ and βby equation (2.25). Generally, κ can be considered a measure of the mobileion concentration (counterions and added electrolyte), with β a measure of thefixed charge concentration, and D a measure of the total ion concentration.More detailed, quantitative parametric studies—based on the five independentdimensionless variables—are undertaken below.

Streamlines of the fluid velocity (from right to left), and isocontours of theelectrostatic potential (proportional to the free charge density) are shown infigure 1a. The polymer displacement and isocontours of the pressure, which,recall, has osmotic and hydrodynamic contributions, are shown in figure 1b.





–3 –2 –1 0 1 2 3

−2

−1

0

1

2

3

x/a

y/a

(a)

–3 –2 –1 0 1 2 3x/a

(b)

Figure 1. (a) Streamlines of fluid flow (right to left), and equispaced isocontours of electrostaticpotential (solid vertical lines) and fixed charge density (grey lines). (b) Displacement vectors andequispaced isocontours of pressure (grey lines). The maximum and minimum of the net fixedcharge density are, respectively, at the front of (left) and behind (right) the particle, opposite tothose of the pressure. Here, the electric field is directed (left to right) along the polar axis, andthe ratio of the particle displacement to the electric-field strength Z/E ≈ −2.0 × 10−3 nm V−1 cm;other parameters are listed in table 1.

Note that the fixed charge on the polymer is positive, and the electric field isdirected from left to right. Although the particle has zero charge, it is displaced(right to left) in the direction of the undisturbed electroosmotic flow.

An accumulation and depletion of free charge, respectively, is evident at thefront of and behind the particle. Compression of the polymer skeleton at the frontincreases the electrostatic energy of the (positive) fixed charge, thereby increasingthe effective elastic restoring force. Accordingly, the apparent elastic modulus islarger than the intrinsic value E for an uncharged skeleton. These observationsare consistent with the modelling of articular cartilage by Sun et al. (2004), whichrevealed a higher apparent Young’s modulus in unconfined compression tests thanunder shear.

The particle displacement in figure 1 is co-linear with the undisturbedelectroosmotic flow. Thus, even though the polymer experiences an electricalforce (left to right), the polymer displacement in close proximity to the particlereflects the hydrodynamic drag exerted by the fluid on the particle and polymer.Alternatively, the particle can be considered as responding to the electric field asif it bears the same signed charge as the counterions of the fixed charge. Clearly,the hydrodynamic drag of the polymer is expected to play an important role intransferring the electrical charge on the counterions to the particle. Note alsothat the electroosmotic flow exerts a force on the particle whose magnitude isproportional to the mobile charge density. The pressure isocontours in figure 1bare similar to those of the perturbed fixed charge density, but with opposite sign.This reflects the O(ρf◦Ea) hydrodynamic pressure dominating the O(ρf◦kBT/e)osmotic contribution.





In the following parametric studies, Young’s modulus, Poisson ratio andthe Brinkman screening length (hydrodynamic permeability) are implicitlyspecified as independent of the fixed charge density and ionic strength. We alsoneglect annealing influences on the fixed charge, i.e. we assume that the fixedcharge is quenched and, thus, independent of pH, electrolyte concentration andelectrostatic potential.

Note that counterion condensation places a practical upper limit on theeffective fixed charge density. According to Manning’s well-known theory, thefixed charge density ρf◦ is limited to values with less than one elementary chargeper Bjerrum length of polymer contour (Manning 1969). For a representativehydrogel comprising 5 per cent polymer with monomer molecular weight100 g mol−1, and 10 per cent charged monomers (Tong & Liu 1993; Okay &Durmaz 2002), the maximum fixed charge density ρf◦ ∼ 107 C m−3 (equivalentto ≈ 0.062e nm−3). This is consistent with values reported for articular cartilage atphysiological pH, e.g. 2 × 107 C m−3 from Lai et al. (1991). With Young’s modulusE ∼ 1 kPa and particle diameter ∼1 μm, βa � 104.

We term the ratio of the particle displacement to the electric-field strength theelectrical response function

Z/E = −ρf◦�2E−1Z ∗E (a/�, κa, βa, εp/εs, ν), (3.1)

where Z ∗E is a dimensionless function—given explicitly by the ratio of

equations (2.26) and (2.38)—of the five indicated dimensionless parameters. Thedimensional prefactor in equation (3.1) is the scaling of Z/E that prevails forincompressible skeletons (ν = 0.5).

Independent calculations with ν = 0.5 show that Z ∗E = 3 for incompressible

skeletons, i.e.Z/E = −3ρf◦�2E−1 (ν = 0.5). (3.2)

This formula can be derived by summing the fluid and polymer equationsof motion (with ∇ · v′ = 0) giving 0 = −∇p′ + ∇2w and ∇ · w = 0, where −p′Iand ∇w + (∇w)T are the isotropic and deviatoric stresses, respectively (Hill &Ostoja-Starzewski 2008). Here, w = μv + ηu with μ = E/3 the polymer shearmodulus when ν = 0.5. The boundary conditions for the E-problem are w = 0 atr = a and w → −�2ρf◦E as r → ∞. Therefore, by analogy with the well-knownproblem of Stokes flow past an impenetrable sphere, the solution yields a forceFE = −6πa�2ρf◦E. Balancing this with the elastic restoring force FZ = −2πEaZin the Z -problem gives equation (3.2).

Let us now consider practically relevant situations where ν < 0.5. FollowingWang & Hill (2008), we adopt ν = 0.2 as a representative value for charged anduncharged hydrogels, e.g. ν ≈ 0.15 for agarose (Freeman et al. 1994) and ν ≈0.2 for articular cartilage (Jurvelin et al. 1997). In addition, we fix the ratioof dielectric constants εp/εs = 0.02, which is representative of a wide variety ofinclusions in aqueous electrolytes. Accordingly, figure 2 shows how Z ∗

E varies withκa for various βa and several values of a/�. These plots reveal three physicallydistinct regions of the parameter space, each of which is examined below.

Firstly, when κa � 1 and β � κ, the scaled displacement Z ∗E plateaus to

a larger value than in the high κa limit where electrostatic interactions arescreened by the added electrolyte. Thus, the skeleton of very weakly chargedpolymers (vanishing β) is softened in the absence of added salt. For example,





10−2 100 102 10410−2 100 102 104

2.4

2.6

2.8

3.0

3.2

βa = 0.001

0.1

0.5

1

3

10 100 1000

ZE*

ZE*

(a)

2

3

4

5

6

7

8βa = 0.001

0.1

0.5

1

310

1001000

(b)

0

20

40

60

80

100βa = 0.001

0.1

0.5

1

310

100 1000

κa

(c)

100

101

102

103

104

βa = 0.001

0.1

1

3

10

30

1001000

κa

(d)

Figure 2. Dimensionless electrical response Z ∗E versus κa with ν = 0.2, εp/εs = 0.02: (a) a/� = 1,

(b) 10, (c) 100 and (d) 1000.

this yields a decreasing particle displacement with increasing concentration ofadded electrolyte (increasing κ). This unexpected result may be due to thegradient of fixed charge density that accompanies dilation. This would induce anaccompanying electrostatic dipole moment that enhances the local electric-field,which, in turn, enhances electroosmotic flow. As discussed above, the particledisplacement is generally attributed to viscous drag on the particle. Thus, instriking contrast to charged inclusions dispersed in uncharged media, electricalpolarization increases the particle displacement. This is only possible when theDebye length κ−1 is much larger than the characteristic length scale for dilation,i.e. when κa � 1. Because this mechanism depends on the intrinsic elasticity ofthe gel (E and ν), the softening effect vanishes as ν → 0.5.

Next, when κa � 1 and β � κ, the scaled displacement Z ∗E plateaus to a

smaller value than in the high κa limit. For example, increasing the added saltconcentration increases the particle displacement. Here, the polymer skeleton iselectrostatically stiffened by the fixed charge, and this stiffening is evidently more





10–2 10–1 100 101

a/102 103

100

101

102

100

300500

(a)

104 105 106 10710–4

10–3

10–2

10–1

100

101

= 10 Pa

102

103

104

ρ f (C m–3)

(b)

ZE*

–Z/E

(nm

V–1

cm

)

βa = 0.001

Figure 3. (a) Dimensionless electrical response Z ∗E versus a/�: κa = 100, ν = 0.2, and εp/εs = 0.02.

Note that Z ∗E is independent of βa when β � κ. The scaling highlighted by equation (3.3) is evident

at all values of βa when a/� � 1. (b) Dimensional electrical response Z/E (nm V−1 cm) versus fixedcharge density ρf◦ (C m−3) for various values of Young’s modulus E (Pa): Is = 10−2 mol l−1 withother parameters listed in table 1.

influential than the accompanying enhancement of electroosmotic flow. Again,this influence vanishes as ν → 0.5, because, under these conditions, the skeletonincompressibility is independent of polymer charge.

Finally, when κa � 1 and κ � β, the scaled displacement Z ∗E plateaus to a

value that depends only on a/� when ν and εp/εs are fixed. In this regime,viscous shear stresses on the inclusion scale as τ ∼ ηU /� when a/� � 1, where thecharacteristic fluid velocity beyond a Brinkman screening length � of the particlesurface is the velocity of the undisturbed electroosmotic flow, U = −�2η−1ρf◦E .Thus, the hydrodynamic drag force on the particle F ∼ τa2 ∼ −ρf◦E�a2. Next,balancing this force with the intrinsic elastic restoring force of the hydrogel, whichis ∼ − EaZ when the particle is displaced a distance Z , gives

Z/E ∼ −ρf◦�aE−1 (a/� � 1). (3.3)

Accordingly, in contrast to intrinsically incompressible skeletons, we findZ ∗

E ∼ a/�. This scaling is highlighted in figure 3a where Z ∗E is plotted as a

function of a/� for several values of βa with κa = 100. As expected from thepreceding analysis for incompressible skeletons, the foregoing scaling vanisheswhen ν = 0.5 (figure 4). Similarly to the earlier studies of charged inclusionsin uncharged skeletons (Hill & Ostoja-Starzewski 2008; Wang & Hill 2008), theparticle displacement is independent of particle size when ν = 0.5, but otherwiseincreases in proportion to the particle radius a. Note, however, that whilethe absolute displacement increases with a, the displacement remains a smallfraction of a. This fraction increases with charge density ρf◦, hydrodynamicpermeability � and compliance E−1. As noted above, these parameters aregenerally not independent (see Sasaki et al. 1995; Sasaki 2006, and the referencestherein), but are coupled according to the polymer architecture and gel synthesis.





0 0.1 0.2ν ν

0.3 0.4 0.5100

101

102

100

200

300

500

1000

ZE*

(a)

−1 −0.95 −0.9 −0.85 −0.80

5

10

15

βa = 0.001

100

200

300

500

(b)

βa = 0.001

Figure 4. Dimensionless electrical response Z ∗E versus Poisson ratio ν: κa = 100, a/� = 100

and εp/εs = 0.02.

The scaled particle displacements Z ∗E in figure 2 partially obscure how the

dimensional displacement depends explicitly on hydrogel charge. Therefore,figure 3b shows a representative plot of the dimensional electrical response Z/Eas a function of the fixed charge density ρf◦ for several values of Young’smodulus E. With E = 1 kPa, for example, Z ≈ 1.08 nm with E = 100 V cm−1 andρf◦ = 105 C m−3. Obviously, more compliant gels with a higher charge densityyield larger particle displacements. Nevertheless, while these displacements arewithin the range of detection using optical tweezers with back-focal-planeinterferometry (Gittes & Schmidt 1998), the particle displacements are generallymuch smaller than for charged colloidal inclusions (with typical surface chargedensities) in uncharged hydrogels with a comparable intrinsic Young’s modulus.Thus, if the response of a charged particle in an uncharged gel (Wang & Hill2008) were naively superposed with the response of an uncharged particle in acharged gel, the displacement would tend to reflect the particle charge. Clearly,this important problem deserves future attention, as the most general problem ofpractical significance involves charged particles in charged hydrogels.

Poisson ratios of hydrogels are almost exclusively in the range 0 ≤ ν ≤ 0.5.Accordingly, figure 4a shows how Z ∗

E varies with ν in this range for severalrepresentative values of βa. Generally, there is a rapid change in Z ∗

E as ν → 0.5,but this sensitivity vanishes as ν → 0. Z ∗

E is insensitive to ν when ν → 0 forall βa; when βa is small, however, there exists a maximum in Z ∗

E when ν ≈ 0,but this vanishes with increasing βa. In the thermodynamic limit ν → −1, thestrain tensor of a linearly elastic medium is symmetric with vanishing deviatoricterms (Landau & Lifshitz 1986). Thus, any stress is accompanied by dilation inthe absence of shear/extension. Similarly to uncharged hydrogels (Wang & Hill2008), the particle displacement vanishes as ν → −1. Figure 4b shows the resultsfor several values of βa with −1 ≤ ν ≤ −0.8.

Finally, figure 5 shows how the scaled displacement Z ∗E depends on the

dielectric constants of the particle and electrolyte. Recall, this dependenceis absent for particles in uncharged media, i.e. electrophoresis, and charged





0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5100

101

102

103

100

101

102

103

βa = 0.001

1

3

10

100

ZF*

(a)

βa = 0.0011

3

10

100

(b)

0

10

20

30

40

50

60

βa = 0.001

50

100

1000

(c)

/p s/p s /p s

Figure 5. Dimensionless electrical response Z ∗E versus εp/εs with ν = 0.2, a/� = 100: (a) κa = 0.01,

(b) 1 and (c) 100. The limiting values of Z ∗E as εp/εs → ∞ are furnished by equations (3.4) and (3.5).

inclusions in uncharged hydrogels. However, the scaled particle displacement withpolyelectrolyte gels is particularly sensitive to the particle dielectric constantwhen κa and βa are small. Moreover, the forces and particle displacementincrease with particle dielectric constant, approaching finite limits as εp/εs → ∞.Accordingly, for uncharged metallic inclusions (εp → ∞), we have

f Z = −12πaE(Da + 1)(1 − ν)/(1 + ν)

Da[4(1 − ν) + (1 − 2ν)(κ/D)2] + (5 − 6ν)(3.4)

and

f E = −2πρf◦a�2[

(ν − 1)

(β�)2(2ν − 1)(ν + 1)

{(κa)2 − 3B(Da + 1)

+ 2[(Da)2 + 3(Da + 1)]} +(

a2

�2+ 3

a�

+ 3)]

(3.5)

with

B =4(1 − ν)[(Da)2 + 2Da + 2] + 4(1 + ν)(1 − 2ν)(a/�)(β�)2

+(1 − 2ν)[(κa)2 + 2(κa)2/(Da) + 2]Da [4(1 − ν) + (κ/D)2(1 − 2ν)] + (5 − 6ν)

. (3.6)

4. Micro-rheological response (steady susceptibility)

Using mode-coupling theory (MCT), Nägele (2003) identified a breakdown of thewidely adopted generalized Stokes–Einstein relation (GSER) (Mason & Weitz1995; Mason et al. 1997) for a charged-colloidal sphere in a dispersion of charged-stabilized colloidal particles. Earlier theoretical studies of the susceptibility havenot explicitly considered the influence of charge. Rather, such influences have beenlumped into the effective shear and bulk moduli for a linearly elastic continuum.Continuum theories include the GSER, which is exact for incompressible elasticskeletons; Levine and Lubensky’s approximate solution of a two-fluid continuummodel (Schnurr et al. 1997; Levine & Lubensky 2000, 2001a)—recently solved





exactly in the course of studying electrical influences (Wang & Hill in press);and the theory of Fu et al. (2008) accounting for slip at the particle surface.Explicit neglect of charge is generally justified by compressibility, i.e. a Poissonratio ν < 0.5, increasing particle displacements by an amount that is less thanthe experimental uncertainty. Nevertheless, in experiments where small changesin susceptibility can be accurately measured, it will be invaluable to interpretsuch changes in terms of the accompanying changes in charge density rather thanadopting effective properties.

It is customary to write the particle displacement as

Z = αF , (4.1)

where Z , F and α are, respectively, the particle displacement, appliedforce and steady susceptibility. For uncharged, elastic, compressible matrices(Schnurr et al. 1997),

α−1 = 6πμa4(1 − ν)/(5 − 6ν), (4.2)

where μ = E/[2(1 + ν)] is the shear modulus, often reported as the zero-frequencystorage modulus G ′. In the thermodynamically accessible range of Poisson’s ratio(−1 ≤ ν ≤ 0.5), the factor 4(1 − ν)/(5 − 6ν) in equation (4.2) has a maximumvalue of one when ν = 0.5, and decreases monotonically to 8/11 when ν = −1.

Equation (4.2) motivates writing

α = (6πμa)−1Z ∗F (βa, κa, ν, εp/εs), (4.3)

where the dimensional prefactor is the scaling that prevails for uncharged,incompressible skeletons (ν = 0.5) with shear modulus μ = E/[2(1 + ν)] = E/3. Wewill refer to this limit as the GSER. Note also that Z ∗

F obtained directly fromequation (2.26) recovers equation (4.2) when βa → 0.

The polymer displacement and isocontours of the electrostatic potential areshown in figure 6a with the parameters listed in table 1. Isocontours of thepressure (not shown) are qualitatively the same as those of the electrostaticpotential (see equation (2.20)), and the perturbed fixed charge density (notshown) is similar to that in figure 1. The electrostatic potential is clearlyperturbed at the front and rear of the particle, and the accompanying increasein electrostatic energy with dilation increases the skeleton’s resistance todeformation, thereby decreasing the particle susceptibility.

Figure 6b–d shows the effect of various parameters on the response ofan uncharged particle in a charged hydrogel. To distinguish the influences of theadded salt from the polymer counterions, we consider the ionic strength of theadded salt Is and βa as independent variables rather than κa and βa.

In figure 6b, the influence of βa (scaled fixed charge density) is shown forseveral values of Is; our results (solid lines) are compared with the GSER andequation (4.2) (Schnurr et al. 1997). At high ionic strength, the displacementplateaus to the value expected for uncharged, compressible polymer networks(equation (4.2)). More precisely, when κa � βa, the screening of electrostaticinteractions by the added electrolyte eliminates electrostatic resistance todeformation. However, at low ionic strength, the displacement plateaus tothe value for uncharged, incompressible skeletons (GSER). When κa � βa,electrostatic stiffening yields an incompressible skeleton without affecting the





−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x/a

y/a

(a)

10–2 10–1 100 101 1020.95

1.0

1.05

1.10

1.15

1.20

1.25

Is = 0 (mol l–1)

10–310–1

βa

(b)

(c) (d)

10–4 10–3 10–2 10–1 1000.95

1

1.05

1.1

1.15

1.2

1.25

βa = 0.10.5

1

3

10

50

Is (mol l–1)

ZF*

ZF*

ν0 0.1 0.2 0.3 0.4 0.5

βa = 1

3

10

Figure 6. Steady micro-rheological response. (a) Displacement vectors and equispaced isocontoursof the electrostatic potential (grey lines) when the uncharged spherical inclusion embedded in acharged hydrogel is displaced a distance Z by a force F in the absence of an applied electric-field.The maximum and minimum of the electrostatic potential are, respectively, at the front (left) andrear of the particle. Parameters are listed in table 1. (b)–(d) show the scaled susceptibility Z ∗

Fversus (b) βa, (c) Is , and (d) ν with Is = 10−2 mol l−1. Z ∗

F is compared to the GSER (dash-dottedlines) and (4.2) (dashed lines). The response is practically independent of εp.

shear modulus. Clearly, the transition from the low to the high electrolyteconcentration regimes depends on the fixed charge density and intrinsic stiffnessof the uncharged polymer skeleton. These findings are consistent with ourdiscussion of the electrical response and with the independent studies of articularcartilage (Sun et al. 2004).

Figure 6c shows how Z ∗F varies with Is for the various values of βa. The

response increases with the accompanying change in κa, and plateaus to thevalue for uncharged, compressible skeletons. Again, counterions screen the fixedcharge, thereby increasing the effective compressibility.





Figure 6d shows the effect of Poisson ratio for the various values of βa.With increasing βa, Z ∗

F asymptotes to the value for uncharged, incompressibleskeletons. On the other hand, with decreasing βa, Z ∗

F plateaus to the value forcompressible hydrogels. Note that, in contrast to the electrical susceptibility, Z ∗

Fis practically independent of the particle dielectric constant εp. Accordingly, toan excellent approximation,

Z ∗F (βa, κa, ν, εp/εs) ≈ Z ∗

F (βa, κa, ν). (4.4)

From figure 6, the maximum variation in the response with κa or βa isapproximately 15 per cent. Moreover, with changes in βa and κa, the responseis bounded by the limits for uncharged skeletons. The resulting absolute changein displacement is rather small under the forces typically used in micro-rheology,but within the typical limits of detection. For example, the displacement resultingfrom a 1 pN force, which is representative of active micro-rheology (Ziemann et al.1994; Valentine et al. 1996), yields a displacement in the range 0.26–0.30 nm withthe parameters adopted in figure 6. Such displacements are consistent with theexperiments of Di Cola et al. (2007), who established consistency of micro- andmacro-rheology for highly charged linear polyelectrolytes. However, with recenttechnological advances, F ∼ 1 nN forces can be achieved (Uhde et al. 2005a,b;Kollmannsberger & Fabry 2007). For example, a F ∼ 1 nN force produces aZ ∼ 50 nm displacement, which is well within the range of digital-imaging opticalmicroscopy.

5. Summary

The electric-field-induced response of an uncharged spherical particle embeddedin a charged hydrogel was studied theoretically. A three-phase electrokineticmodel (solvent, mobile ions and charged polymer) for the quenchedpolyelectrolyte hydrogel was presented as an extension of Hill & Ostoja-Starzewski’s (2008) model for uncharged skeletons. Linear perturbation andsuperposition were used to derive the exact analytical solutions for the steadyresponse to a steady electric field and external force.

Noteworthy is that the uncharged particles are displaced by an electric field.This is primarily due to electroosmotic flow, with secondary influences attributedto the polarization of the diffuse double layer and electrostatic stiffening, thelatter of which is apparent when the underlying uncharged polymer skeletonis compressible. Accordingly, the electrical response is sensitive to the fixedcharge density and ionic strength. Overall, increasing the fixed charge densityincreases the particle displacement because of the enhanced electroosmoticflow. Moreover, the response generally increases with increasing ionic strength,owing to increasing compressibility from the screening of electrostatic repulsionamong fixed charges.

In contrast to the electrophoretic mobility of colloidal particles dispersedin Newtonian electrolytes, the electrical particle displacement depends on thedielectric constant of the inclusion. Increasing the particle dielectric constantincreases the electric-field-induced displacement.

Finally, our theory captures the influence of charge on the static susceptibilitywidely used to interpret active and passive micro-rheology experiments. Wequantified the roles of fixed charge and ionic strength, showing that the response





is bounded by the compressible (upper) and incompressible (lower) limits for theuncharged polymer skeleton. While the influences of charge are most significantfor the electrical response, which involves electroosmotic flow and electricalpolarization of diffuse double layers, we demonstrated that charge is unlikelyto significantly impact the present interpretations of classical micro-rheology.

R.J.H. gratefully acknowledges support from the Natural Sciences and Engineering ResearchCouncil of Canada and the Canada Research Chairs Program; and A.M. thanks the McGill Facultyof Engineering for the generous financial support through a McGill Engineering Doctoral Award(the Hatch Graduate Fellowships in Engineering), and M. Wang for helpful discussions.

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