Travis. H. Larsen and Eric M. Furst, PRL, 100(14), 2008 · Microrheology of the liquid-solid transition during gelation Travis. H. Larsen and Eric M. Furst, PRL, 100(14), 2008 Caroline

Microrheology of the liquid-solid transition during gelation

Travis. H. Larsen and Eric M. Furst, PRL, 100(14), 2008

Caroline E. Wagner

McKinley Gels Summer Reading GroupAugust 24 2017

(Also: T. H. Larsen, Schultz, K. and Furst, E. M. KARJ, 20(3), 2008)

Motivation: Characterizing gelation at smaller length scales

2

So far we have discussed several macrorheological experimental techniques for determining the physical characteristics of the liquid/gel transition

Chambon, F. and Winter, H. H. JoR (1987)

Winter, H. H. and Mours, M. Adv Polym Sci (1999)

~'~ nG G / 1tan G G


Relaxation modulus ( )G tLong time residual stress

plateau of solid-like material

Rapid relaxation of stresses of liquid-like material

Sample-spanning network

Larsen and Furst argue that similar methods should be developed for microrheology, as macrorheological measurement may not always be possible / sufficient. For instance:

• Micrheology is more suitable when sample volumes are very limited (e.g. biological samples)

• Microstructural changes may be sufficiently small to not be reflected in the measured macrorheology

• Macrorheological measurements may not be suitable for studying very rapid gelation kinetics

Overview of microrheology

3

1𝜇m beads

1mm

50mm

a1i

x

1it i

t1i i

t t

Camera frame rate 1/f

2x

3

Average of N steps

Average of N-2 steps

Experimental notes• Fewer statistical points at larger lag times.. Image for longer than

you need data!• Heterogeneity at scales larger than the particles can result in a

spectrum of particle walks and hence the MSD may not be representative of the medium properties

log( )

2og( )l x

1

1

1Superdiffusion Diffusion

• Brownian motion (viscous liquid)

Subdiffusion

Elastic solid

x3it

it

Overview of microrheology

4

What rheological information can the MSD provide us with?

Mason, T. G. and Weitz, D. A. PRL (1995)Squires, T. M. and Mason, T. G. Ann Rev Flu Mech (2010)

m VDf

Fluid resistance

RfRandom forces

Generalized Langevin Eq: ( ) ( ) ( ) ( )t

Ro

V t f t t dm V

( )tProbe resistance:

Take Laplace transform:( )

( )

(0)( ) RV fms

ms

sV

s

Velocity autocorrelationfunction:

2(0) (0)(0)

)(

()

( )RV fms

ms

s VV V

s

Helpful identity:

0 if random forces and velocity are uncorrelated

NkBT (equipartition) Assume inertia small compared to fluid resistance

2 22 2r ((0) ( )

2) (

2)V V t rs s

s s

Simplify:2 2

2( )

( )

BNk T

ss r s

Use Stokes identity: ( ) 6 ( )s s a 2 2( )

3 ( )

BNk T

sas r s

Modulus: 2

( ) ( )3 ( )

BNk T

G s s sas r s

Compliance:

23 ( )1( )

( )B

a r sJ s

Nk TsG s

Analytic continuation:s i

*( ) ( )G s G

Inverse Laplace transform

23( ) ( )

B

aJ t r t

Nk T

2 2

3

( ) /( )( ) ~

/B

r t atJ t

k T a

Both J(t) and <r2(t)> are in the time domain!

Another way of thinking about it:

G*(𝜔) is in frequency domain but <r2(t)> is in the time domain

This conversion is sensitive to the range of times over which you have experimental MSD data (which is finite)

Experimental systems in Larsen and Furst

5

1) Physically-crosslinked gel

2) Chemically-crosslinked gels

Observe transient gelation for a fixed peptide concentrationOzbas, B. et al. PRL (2004)

• 20 amino acid long peptide (MAX1) assumes a folded 𝛽-hairpin conformation in response to pH or ionic strength trigger.

• Folded peptides self-assemble to form a network

Polyacrylamide Bis-acrylamide x-linker

• Extent of cross-linking in gel is controlled by concentration of bis-acrylamide

Observe steady state fluid structure for various concentrations of cross-linker

i) Polyacrylamide

i) Heparin (KARJ paper only)

Maleimide-functionalizedHigh MW heparin (HMWH)

Bis-thiol PEG

Observe transient gelation for fixed polymer and x-linker concentrations

• Perform microrheology after sample has equilibrated for 6 hours (ensure equilibrium using UV-vis spectroscopy)

Experimental results for peptide and acrylamide

6

Breakdown of expected MSD scaling

7Jones, R. A. L. Soft Condensed Matter (2002)

MSD during the pre-gel phase

8

• As clusters form during the very early phases of gelation, the relaxation time and hence it appears as though the only effect is an increase in viscosity

log( )

2og( )l x 1

• The longest relaxation time is associated with Rouse-like fluctuations of the clusters. As they grow, increases. For subdiffusive motion is observed, while for normal diffusion is recovered

• As the clusters continue to grow, continues to increase, and subdiffusive motion is observed for longer and longer lag times.

L

L

1/L

f

L

L

L

L

L

MSD during the post-gel phase

9

log( )

2og( )l x

L

L

• Beyond the establishment of a sample-spanning cluster at the critical point, the network continues to develop, forming a viscoelastic solid with relaxation time . Increasing connectivity results in a decreasing .

• For the same Rouse-like subdiffusive dynamics persist, while for , the motion of the particles is arrested by the network and a plateau in the MSD is observed.

• At the most advanced stages of gelation, the particle motion is arrested for all lag times

L L

L

L

Collapsing onto master curves

10

Introduce shift factors:2 2( ) ( )

shiftr b r

shifta

MSD:

Lag time:

0~1/e

b J

~1/L

a

Shift onto curve at lowest extent of gelation

Shift onto curve at highest extent of gelation

Percolation theory and scaling near the critical gel point

11

cp p

cp p

cp p

ct t

ct t

ct t

c c

c c

t t p p

t p

Distance fromcritical point


Scaling of static properties

Scaling of dynamic properties

max

0~ ( )k

cp p

~ ( )c

zeG p p

~max

yL

( )

0~G ~k z y

e maxy z k

0 ~ zeJ

( ) nG t St

@max

t

~ ~ ~z n yne maxG /n z y

Gel fraction (or probability of being in the infinite cluster)

Weight averaged MW (or degree of polymerization)

~ )(c

G p p

(~ )w c

DP p p

Typical cluster radius ~ )(cp p

~( ) nJ t t

Percolation theory and scaling near the critical gel point

12

0 ~ zeJ

Equilibrium compliance

~ yL

Longest relaxation time

/n z y

Acrylamide

Peptide 0.6 0.02n

0.55 0.03n

2( ()~ ) ~ nJ r t tt

Consistent with Rouse dynamics (?)2/3n

Origin of the n=2/3 scaling

13J. E. Martin, Adolf, D. and Wilcoxon, J. P. PRL (1988)

Based on a percolation model for gelation

Lattice dimension: dBond probability: p

Cluster size distribution: (1 / ) /( ) f z

d d m MeN m m

z-average cluster mass (or number):

Fractal dimension: df

Cluster radius: R

Cluster mass (or # of monomers): m

Radius of cluster of mass Mz: 𝜉

Goal: compute relaxation times for a solution or melt of branched polymers

First consider dynamics of single cluster

Relaxation time of a polymer chain : 2~ /r tR D

Assuming self-similar Brownian dynamics, relaxation timesof a linear or branched polymer of m monomers: ~ (1 )

Rjj j m

Cluster diffusion coefficient: ~ btD R

Relaxation time of shortest mode: 0~

m

(2 )/

0~( / ) (1 )fb d

jj m j m

Shear relaxation modulus: )

1

/ /(2

0~ ~ / (( ) )j f

mt d b

m RG dt m t tt e

Zimm

Rouse

Stokes-Einstein: 2~ / dt BD k T R

/2 ( )~ / fd d

mb d G t m t

0~ /t BD k T m

Individual monomer friction

~ ~f fd d

zM m R

0~ / fd

B fRk T b d

/(2 )( )~ / f fd d

mG t m t

Theta solvent:

1/2( )~ /mG t m t

1/ 2fd


J. E. Martin, Adolf, D. and Wilcoxon, J. P. PRL (1988)




d d m MeN m m



Cluster radius: R




Need to understand diffusion in the reaction bath – requires a size dependent viscosity

~ ~f fd d

zM m R

Assume clusters larger than correlation length feel bulk viscosity

Assume clusters smaller than correlation length feel a finite viscosity independent of (since diverges)

b

0

/( )~ ( / )~ kb

R f R R

Describe diffusion of a cluster by Stokes Einstein relationship in d dimensions: 2( ) /6 dt BR k T RD

Hence we obtain: 2 /( )~1/ d ktR RD

ZimmRouse

~1/ fd

tD R 2~1/ d

tD R

2 /2f

d k dd

Use percolation estimate 2

2f

dd

0 1.35k

RouseZimm

and 0.9

6

20 /

dk


J. E. Martin, Adolf, D. and Wilcoxon, J. P. PRL (1988)




d d m MeN m m



Cluster radius: R




For a single branched polymer:

~ ~f fd d

zM m R

/ /(2 ) (

1

)/~ ~ / ~ /( ) j f f f

mdt d b k d

mde tG t m t mt

2

f f

f

kd d

b d

For the macroscopic sample:

( )/( ) ln / lnf fd k d

mH t d t m t d t

and

Relaxation time spectrum

/( )( ) ( ) ( ) ( )~1/ d d km

H t N m H t H t t

nd

d k

ZimmRouse

1n

0k 1.35k

0.67n

Actually the percolation scaling for acrylamide isn’t perfect…

16

0z

1.4y

og( 0 0l ) p c

c

p p

p

since

For 1/L

f

log( )

2og( )l x

1/ f

1

L

a

bSince this is normal diffusion 2x C

2 ( )shift

x C a

2 ( )shift

x C b b a 0

~1/ ~ y ke

b J

Theory has

But if “only” changing0

0~1/ ~ k

eb JSo shifting along the abscissa gives

0,~ ~ka b

3 1.7 1.4k y z

0 ~ zeJ

~ yL

~( ) nJ t ty z k

/n z y

Experimental results for heparin

17No longer Rouse dynamics?

Origin of the heparin scaling n≈0.4

18

J. E. Martin, Adolf, D. and Wilcoxon, J. P. PRL (1988)Stauffer, D. et al. Adv Polym Sci (1982)

Percolation model for gelation



Critical lattice dimension d=6

Flory-Stockmayer/ mean-field model for gelation

Bethe lattice

nd

d k

06

/ 02

dk k

1n

polyacrylamide heparin

High MW between crosslinks

When long linear chains form during the gelation process, they can contribute Rouse relaxation modes, which could tend to reduce 0.5n

Travis. H. Larsen and Eric M. Furst, PRL, 100(14), 2008 · Microrheology of the liquid-solid transition during gelation Travis. H. Larsen and Eric M. Furst, PRL, 100(14), 2008 Caroline

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