Three-dimensional model for chemoresponsive polymer gels

Olga Kuksenok,Victor Yashin, Anna C. Balazs

Chemical Engineering DepartmentUniversity of Pittsburgh

Pittsburgh, PA

R. Yoshida et al, Chaos, 9 (1999) 260

Chemo-Responsive BZ Gels: Introduction

Inspiration: experiments on BZ responsive gel:N-isopropylacrylamide (NIPAAm)

Covalently bonded Ru(bpy)3Swollen in water solution of MA, NaBr03 and HN03

BZ reaction causes periodic swelling-deswelling

Aim: design gel-based system sensitive to mechanical impact

Mechanical energy is converted into chemical signal

Challenge: develop an efficient 3D model for BZ chemo-responsive gels

Ru(bpy)33+Ru(bpy)3

2+ BZ

Model (V. Yashin, A. Balazs, Science 314, 798, 2006)

Evolution equations

Reaction kinetics : modified Oregonator

φ uvolume fraction of polymer, concentration of dissolved reagent, .

;)( p

dtd v⋅∇−= φφ);,,()( φε vuGv

dtdv p +⋅∇−= v

).,,(1

)1(1

)()( φφ

φφ

vuFuuudtdu pp +⎥

⎦

⎤⎢⎣

⎡−

∇−⋅∇+⎥⎦

⎤⎢⎣

⎡−

⋅∇+⋅∇−= vv

concentration of oxidized metal-ion catalyst0)1( )()( ≡−+ sp vv φφ

Polymer velocity Solvent velocity

∇⋅+∂∂

≡ )( p

tdtd v

v

vuG )1()1( 2 φφ −−−=;)1()1()1()1( 2

222

φφφφ

−+−−

−−−−=ququvfuuF

Model: Calculate Energy and Stress Tensor

Define energy density of deformed gel

Elastic energyInteraction energy

Strain invariants where

is crosslink density, : hydrating effect of oxidized catalyst

Define stress tensor

Force balance equation

is polymer-solvent friction coefficient

BIσ)))

000),(φφφ vcvP +−=

φχφφχχφφπ vosm*2

10 ))()1ln(( +++−+−=,2/),(),( 000 φφφπφ vcvvP osm +=

)(),( 331 IUIIUU FHel +=

[ ])1()1()()1ln()1( *3 φχφφφχφφ −−−+−−= vIU FHFH

( )2/131

00 ln32

IIvcUel −−=

;ˆtr1 B=I ;ˆdet3 B=I .ˆˆˆ TFFB ⋅=

0* >χ0c

))((ˆ )()(10

spuDTv vvσ −=⋅∇ − φζ

2/300 )/)(()( φφφζφζ =

3D gLSM Model: Define Elements

)(mrn

Define 3D element

Define local coordinate

system

Define shape functions

Faces at

Coordinates within element

:coordinatesof node

Perform all integration in local coordinate system

Sample consists of elements

),,()( kji=m

∑=

=8

1

)()(n

nnN mrmr

),,( ζηξ

)1)(1)(1(81

nnnnN ζζηηξξ +++=

8..1=n

1,1,1 ±=±=±= ζηξ.

ζ

ξ

η7

3n

1

2 3

4

6

85

1n)3,(1,2 mF

)1,(1,2 mF)2,(1,2 mF

2n

)1()1()1( −×−×− LLL

Total force acting on node n of element m

Calculate spring-like forces

Elastic energy stored in springs between NN and NNN nodes

3D gLSM model: Calculate Forces

.

);(/)( 1,1 mrmF nn U ∂−∂= .)(13

1 ∑= mUU Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

Δ== ′′− − − ∑∑∫ ∫ ∫ 22

2001

1

1

1

1

1 100

1 )()(())()((24

))(,,,(16

)( mrmrmrmrmrm nnNNN

nnNN

vcdddIvcU ζηξζηξ

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛−′+−′′Δ

= ∑∑′

′′

′)()(

00,1 )()()()(),(

12)(

mm

mrmrmrmrmFNNN

nnNN

nnn nnwvc

1),( =′ nnw 2),( =′ nnwif boundary face, if internal face

(m)(m)(m) nnnTOTAL FFF ,2,1, +=

)(,1 mF n

Calculate force acting on node due to isotropic pressure

Summation over elements containing node Pressure within element

Osmotic and elastic contributions

Calculate nodal velocities

Update nodal coordinates Calculate new volumes of elements

3D gLSM model (Cont.)

[ ](m)(m)(m)(m)rnnn

n FFMdt

d,2,1 +=

(m)rn

.

( )∑′

′′+′′+′′′′=m

mmnmmnmmnmmmF )()()()()()())(),((41)( 332211,2 SSSvPn φ

000 2/)())(),(())(),(( φφφπφ mmmmm ′+′′=′′ vcvvP osm

n

m′ n

(m)nF ,2

∫ ∫ ∫− − −=

11

11

11

)(det)( ζηξ dddJV mm

3D gLSM model

Update polymer volume fraction within element

Update values of u and v

More details in “Three-dimensional Model for Chemo-responsive Polymer Gels Undergoing the BZ Reaction, O. Kuksenok, V.V.Yashin and A. C. Balazs,PRE (2008)

)(/),( 03 mm Vtt φφ Δ=Δ+

[ ]))(),(),(()()()(),( 0 mmmmmmm φε vuGTvtvttv +−Δ+=Δ+

[ ]))(),(),(()()()()()(),( 210 mmmmmmmmm φvuFTTTututtu +++−Δ+=Δ+

. ( ) tttT ΔΔ+−= /)(/),(1)(0 mmm φφ

[ ]∫∈

−⋅∇=)(

)(1 ))(1/()()(

)(1)(

mr

mmmvrm

mV

p udV

T φ

[ ]∑=

∇−+∇∇−=kjil

lll uuT,,

22 )(~))(1()(~)()( mmmmm φφ

(Use finite element)

(Use finite difference)

Model Validation

No-diffusion and instantaneous pressure equilibration limitSimplified equations: 2 variables

Evolution of degree of swelling

),(2 lim

0

3/1

000 vvc osm φπ

φφ

φφ

=⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛

λ

timet ,

⎥⎦

⎤⎢⎣

⎡ −= = )(lim

lim lim

)()( φε

φφ

φφvvG

dtdv

vdtd

)(lim1 φφ

φ vvFdtdu

dtdu

=+−

−=

10000 =Λ

Parameters:f=0.8, L=2Same i.c.

3/10 )/( φφλ =

Sample size 12 X12 X 12, f=0.68

No-flux b. c

Time evolution of average characteristics

Largest swelling corresponds to maximum

3D gLSM: Example of Regular Periodic Oscillations

t

>< v

τ

><φ~

><u>< v><φ

4166.0max =v0008.0min =v

)(a

)(c

)(b

)( f)(e

)(d

3D gLSM: Example of Non-Regular Oscillations

Larger sample size (24 X 24 X 24 ), f=0.68

Strong effect of diffusion Non-regular oscillations

Realization depends on initial perturbation

)(a )(b

)(c )(d

Effect of Increasing Stoichiometric Factor f

Larger sample size (24 X 24 X 24 ), f=0.9

Strong effect of diffusionOscillations become regular

Oscillations do not depend on initial perturbation

)(a )(b

)(c )(d

Diagram of States

Three possible cases: steady state, regular and non-regular oscillations

stφLφ

><φ~

Steady- state

Regular oscillations

*2f *

24f **24f f*

12f*6f

Non-regular oscillations

τ

Diagram of States (Cont’d).

For smaller samples :If steady-state; if regular oscillations

increases with increase in size

For larger samples : If steady-state; if regular oscillations

For non-regular oscillations

Period (regular oscill.)For smaller samples

increases withincrease in

For larger samplesis smaller

*Lff >*

Lff ≤*

Lf LLL ××

)12( ≤L

)24( =L*

Lff ≤ **Lff >

***LL fff <<

τ

τL

)12( ≤L

)24( =Lτ

Effect of Flux of through Surface, f=0.68

Larger sample ( )

Flux of u through boundaryOutside sample

Oscillations are regular

Evolution of average swelling

Dramatic effect of flux through the boundaries

0=u

u

24=L

><λ

Flux

No Flux

)(a

)(b

)(c

Conclusions

Developed first 3D computational model to study dynamics ofchemo-responsive gels in three dimensions

Performed first simulations of BZ gels in 3D

Future plansMechano-chemical transduction in BZ gels in 3D

BZ gel coating sends global signal of local impact