THE SIMULATION OF THE SWELLING DYNAMICS OF GELSBY THE STRESS-DIFFUSION COUPLING MODEL

Tatsuya Yamaue 1,2 and Masao Doi 11 Department of Computational Science and Engineering,

Nagoya University, Nagoya 464-8601, Japan.2 Biorheo-project, CREST, JST.

Phone : +81-52-789-3595e-mail : yamaue@stat.cse.nagoya-u.ac.jp

The dynamics of gels has been described by the collective diffusion model of gel networks, [1] whichwell reproduces the swelling phenomena in one axis, such as spherical gels. But the collective diffusionmodel of gel networks can't reproduce the swelling phenomena in two or three axes.[2] The dynamics of polymer gels should be discussed from the stand point of the dynamics of polymersolutions. Recently, several models, which are based on the two fluids model, have been proposedas dynamics of gels. We have formulated the dynamics and the simulationscheme for large deformation of gels using the stress-diffusion coupling model, inwhich the continuity of solvent and the coupling between solvent diffusionand network stress are considered. [3] Here, using the linearized stress-diffusion coupling model of gels,we have analized the free swelling process of long rod/large disk gels and the 1 dim.swelling process of gels clumped between glass plates. (Fig.1) [4] Furthermore,we have developed the gel dynamics simulator named MUFFIN-GelDyna [6] as one of opensource simulation softwares belonging to OCTA system (Open Computational Tools for Advancedmaterial technology). [5] GelDyna is the finite-element-method (FEM) simulator and cancalculate the large deformation dynamics of gels, accompanying a change of external stimuli, suchas, temperature, pressure, and a load of forces, such as gravity, andsurface forces. As boundary conditions, permeable surfaces, impermeable surfaces, fixed surfaces orsurfaces moving with constant velocity, and load of surface forces, are implemented. Here,using GelDyna, we have simulated the free swelling process of 2D slab gels, 3D long rod gels, and3D large disk gels, and have reproduced experimentally observed swelling ratios by the stress-diffusioncoupling model.

Fig1. Thinkness of gels clumped between glass plates related to time for various width 4.2, 5.0, 9.3, 14.8, 19.3, 29.4, 35.8, 53.1 (mm)

26.0 (mm). These results nicely reproduce the experimental results. [4]

[1] T.Tanaka and D.J.Fillmore, J.Chem.Phys 70, (1979), 1214.[2] Y.Li and T.Tanaka, J.Chem.Phys 92, (1990), 1365.[3] T.Yamaue, T.Taniguchi and M.Doi, AIP Conference Proceedings 519, (2000), 584.[4] A.Suzuki and T.Hara, J.Chem.Phys. 114, 5012 (2001).[5] What is OCTA, M.Doi, http://octa.jp (2002).[6] MUFFIN user’s manual, T.Yamaue, T.Taniguchi, et.al., http://octa.jp (2002).

Tatsuya YamaueBiorheo-project, CREST, Japan Science and Technology Agency.Dept. of Computational Science and Enginnering, Nagoya Univ.

Masao DoiDept. of Computational Science and Enginnering, Nagoya Univ.

IPa03THE SIMULATION OF

THE SWELLING DYNAMICS OF GELS

BY THE STRESS-DIFFUSION COUPLING

MODEL

GelSympo.2003, 18-20 Nov., 2003, Tokyo Univ.

AbstractThe dynamics of gels has been described by the collective diffusion of polymer networks related to the solvent. The corrective diffusion model of polymer networks described by the Tanaka-Fillmore's equation (TF equation) nicely reproduces the swelling phenomena of a spherical gel. But the TF equation can't reproduce general anisotropic deformations of gels sush as the free swelling phenomena of long cylindrical and large disk-like gels. The dynamics of gels should be discussed from the stand point of the dynamics of polymer solutions. The stress-diffusion coupling model, which are based on the two fluids model, have been proposed as dynamics of polymer solutions and blends, where the continuity of solvent and the mechanical coupling between the solvent diffusion and the polymer stress are considered.

Actually, in general anisotropic deformations of gels such as a deformation under an external stress, the solvent and polymer network of gels can move together by the shear relaxation without pressure gradients. For cylindrical and disk-like gels, the TF equation was extended by taking the shear relaxation and separating the time-scale of the solvent diffusion and that of the shear relaxation of polymer networks. The dynamics of gels should be generally described by the relative motion between the solvent and the polymer of gels with the mechanical coupling between the solvent diffusion and the polymer network stress.

Recently, the anitropic swelling kinetics of thin-plate gels with rectangular surfaces under mechanical constraint was experimentally investigated. In this system, the top and bottom surfaces of gels were chemically clamped on the glass plates, and the gels could swell and shrink only along the thickness direction between two swollen states by the change of osmotic pressure due to the temperature jumps.

Here, we have formulated the dynamics of gels using the stress-diffusion coupling model, and have theoretically calculated the swelling process of thin-plate gels with rectangular surfaces under mechanical constraint using the linearized stress-diffusion cooupling model of gels based on the two fluids model and the linear elasticity theory.

Simple diffusion of networks in immobile solvent is assumed

Equation of motion

ς : the friction coefficient between the network and the solvent

j

iji

xtu

∂

σ∂=

∂∂

ζ

∂

∂−

∂

∂+

∂

∂+

∂

∂= ij

k

k

i

j

j

iij

k

kij x

u

x

u

x

uG

x

uK δδσ

32

ζ+=

1)G

34

K(D

kk

i2

ki

k2

i

xxu

Gxx

u)

3G

K(tu

∂∂∂

+∂∂

∂+=

∂∂

ζ

fDtf

Δ=∂∂

u•∇=f volume change

Linear elasticity model

SwellingSwelling of

Spherical Gelshas been wellexplained byTF theory

Pvsv

p∇−

SP vv //Photo: Swelling Process of Spherical NIPA T.Tanaka Lab. MIT.

The Collective Diffusion Model of Gel Networks. (Tanaka-Filmore’s Equation, 1979)

T.Tomari and M.Doi:

Macromolecules 28 (1995),

8334-8343.

TF Equation can’t describe these problems.

- Patterns and plateau period in deswelling gels.(Matsuo-Tanaka, 1988)- One-dim. swelling and deswelling under mechanical constraint.(Suzuki-Hara, 2001)

- Swelling of long rod and large disk shaped gels. (Li-Tanaka, 1990)

glass

gel clamped to the plates

SP vv //

Pvsvp∇−

Inkwaterink vv //

Gels SP vv //

p∇−Pv

The solvent and network can move together.

0tu

xi

j

ij =∂∂

ζ−∂

σ∂0v

tu

x sii

j

ij =

−∂∂

ζ−∂

σ∂

Stress Diffusion Coupling Model for Gelsbased on two fluid model

0])1([ =−+⋅∇ sp vv φφ

: Motion of solvent related to polymer

: Motion of polymer related to solvent

: Incompressibility condition

Boundary conditionsfor solvent

for polymer

permeable wall (dirichlet) Impermeable wall (neumann)

poutp outpp = bcv0)(

,,

=⋅∇→

=

np

vv nbcns

deformable boundary(neumann)

bcfForce balance

€

(σ − pI)•n = fbc

gel fixed to the boundary (dirichlet)

bcp vv =

(1)

(2)

(3)

€

ζ (vp − vs) = −φ∇p +∇ ⋅σ

€

ζ (vs − vp ) = −(1−φ)∇p

Linearized Equations

δ

∂∂

−∂

∂+

∂∂

+δ∂∂

=σ ijk

k

i

j

j

iij

k

kij x

u32

x

u

xu

Gxu

K

ikk

i2

ki

k2

xp

xxu

Gxxu

)3G

K(∂∂

=∂∂

∂+

∂∂∂

+

iisi x

puv

∂

∂−−=−−

ζφ

φ2)1(

))(1( &

0xv

)1(xu

i

si

i

i =∂∂

φ−+∂∂

φ&

u•∇=f

fDtf

Δ=∂∂

)( pDtp

Δ∇=∂∂

∇ )34

()1( 2

GKD +−

=ζφ

: linearized stress tensor

: mechanical balance

: solvent flux

: incompressibility(mass balance)

: volume change

(4)

(5)

(6)

(7)

(8)

Experiment(A.Suzuki and T.Hara, JCP, V.114 (2001), 5012)

Widths of glass surfaces in experimenta0=4.2, 5.0, 9.3, 14.8, 19.3, 29.4, 35.8, 58.9 mmb0=26.0 mmc0=0.9 mm

Swelling of thin-plate gels, clamped on glass surfaces

Initial thicknesses of glass surfaces in experimenta0=9.5 mmb0=26.0 mmc0=0.142, 0.402, 0.915, 1.363, 1.774 mm

Size of gelsWidth a0 × b0 （Size of glass surfaces）Initial thickness c(t=0) = c0 , Thickness c(t)

glass

gels

solvent

Single exponent relaxation process

Experiment(A.Suzuki and T.Hara, JCP, V.114 (2001), 5012)

by changing the width of glass surfaces - Swelling is a single exponent relaxation process- Larger width Larger relaxation time

c(t)/c0

c(t)/c0time (min)

time (min)

by changing the initial thickness of gels- time evolution of swelling ratio is independent on the initial thickness of gels

Size : large

€

τ a0−2 + b0

−2( ) = 38.33[min /mm2]

relaxation timeτ

€

a0−2 + b0

−2( )−1

（swelling）

τ∝

€

a0−2 + b0

−2( )−1

Swelling analysis by the linearlized stress-diffusion coupling model- Model and Boundary Conditions -

a0, b0

2c(t)z

x, y

pressure p0

€

p(x = ±a02,y,z, t) = p0

p(x,y = ±b02,z,t) = p0

∂p∂x(x = 0,y,z,t) = 0

∂p∂y(x,y = 0,z,t) = 0

€

ux (x = 0,y,z, t) = 0ux (x,y,z = ±c,t) = 0uy (x,y = 0,z, t) = 0uy (x,y,z = ±c,t) = 0uz (x,y,z = 0,t) = 0

€

p = p(x,y,z,t)ux = ux (x,y,z,t)uy = uy (x,y,z,t)uz = uz (x,y,z,t)

boundary condition

€

p = p(x,y, t)ux = ux (x,y,z,t)uy = uy (x,y,z,t)uz = uz (z, t)

€

ca0, cb0

<<1

lowest order

What are solved

permeable surfaces

clamped on glass surfaces

€

∂p∂x

= (K +43G)∂

2ux∂x 2

+G(∂2ux∂z2

+∂2ux∂y 2

) + (K +G3)∂ 2uy∂x∂y

∂p∂y

= (K +43G)∂ 2uy∂y 2

+G(∂ 2uy∂z2

+∂2uy∂x 2

) + (K +G3) ∂

2ux∂x∂y

0 = (K +43G)∂

2uz∂z2

+ (K +G3)(∂

2ux∂x∂z

+∂2uy∂y∂z

)

€

∂ ˙ u x∂x

+∂˙ u y∂y

+∂ ˙ u z∂z

=(1− φ)2

ζ(∂

2 p∂x 2 +

∂ 2 p∂y 2 )

(9)

(10)

(11)

(12)

(13)

Mechanical balance on the glass plates (free surfaces)

Mechanical balance between pressure and polymer stress in gels

Equations of motion (by eliminating the velocity of solvent)

€

dx0

a0 / 2∫ dy0

b0 / 2∫ (σ zz − p) |z= ±c= dx0

a0 / 2∫ dy0

b0 / 2∫ [(K −23G)(∂ux

∂x+∂uy∂y) + (K +

43G)∂uz

∂z− p]z= ±c = 0

Swelling analysis by the linearlized stress-diffusion coupling model

- equations of motion -

eq.(12) and clamped boundary conditions on glass surfaces

€

(K +43G){∂uz

∂z(z,t) − ∂uz

∂z(z = ±c,t)}+ (K +

G3){∂ux∂x(x,y,z,t) +

∂uy∂y(x,y,z,t)} = 0 (14)

sum is independent on x and y.

€

A(z,t) ≡ ∂ux∂x(x,y,z,t) +

∂uy∂y(x,y,z,t)

€

∂p∂x

=G(∂2ux∂x 2

+∂ 2ux∂y 2

+∂ 2ux∂z2

)

∂p∂y

=G(∂ 2uy∂x 2

+∂ 2uy∂y 2

+∂ 2uy∂z2

)

eqs.(10,11) leads with eq.(15)(15)

solve the z-dependency

by the clamped conditions on glass surfacesand the symmetry about z -> -z.

(16)

(17) (18)

(19)(21)

(22)

Swelling analysis by the linearlized stress-diffusion coupling model- separation of z-dependency -

€

∂p∂x

= −GUx

∂p∂y

= −GUy

€

ux (x,y,z, t) = {c(t)2 − z2}Ux (x,y, t)uy (x,y,z, t) = {c(t)2 − z2}Uy (x,y, t)

€

a(t) ≡ ∂Ux

∂x(x,y,t) +

∂Uy

∂y(x,y,t) (23)

eq.(15) by separating z-dependency,

(24)

(21,22,23)→poisson’eq. for pressure.

Solve eq.(24) by Fourier transform with B.C.x

yB.C. on

permeable surfacesp=p0 (25)

(31)

Swelling analysis by the linearlized stress-diffusion coupling model- poisson eq. for pressure -

€

p(x, y,t) = p0 − Cm,n ˜ a (t)cos((2m −1)πa0

x)cosn=1

∞

∑m=1

∞

∑ ((2n −1)πb0

y)

€

Cm,n = (−1)m+n−2 16(2m −1)(2n −1)π 2 {(

(2m −1)πa0

)2 + ((2n −1)πb0

)2}−1

€

p(x,y,t) = p0 1−1Dτ

Cm,n cos((2m −1)π

a0x)cos

n=1

∞

∑m=1

∞

∑ ((2n −1)πb0

y)

exp(− t

τ)

€

A(z,t) = {c(t)2 − z2}a(t)( )

€

˜ a (t) ≡ ∂2 p∂x 2 +

∂ 2 p∂y 2 = −Ga(t)

€

∂ ˙ u z∂z z= ±c

=(1− φ)2

ζ˜ a (t)

€

(K +43

G)∂uz

∂z z= ±c

= p0 −64

{(2m −1)(2n −1)π 2}2 {((2m −1)πa0

)2 + ((2n −1)πb0

)2}−1

n=1

∞

∑m=1

∞

∑

a (t)

€

˜ ˙ a (t) = −τ−1 ˜ a (t)

€

˜ a (t) = ˜ a (0)exp(− tτ

)

τ−1 ≡ D 64{(2m −1)(2n −1)π 2}2 {((2m −1)π

a0

)2 + ((2n −1)πb0

)2}−1

n=1

∞

∑m=1

∞

∑

−1

€

D ≡(1−φ)2

ζ(K +

43G)

€

˜ a (0) =p0

Dτ

€

c(t)c0

=1+ p0(K +43G)−1(1− exp(− t

τ))

from eq. of motion (13),

from eq.(9),

(26)

(27)

(28)

from eqs.(26,27),

the relaxation time :τfrom eqs.(27,28),(29)

Single exponent relaxation processeqs.(26,28) (30)

Swelling analysis by the linearlized stress-diffusion coupling model

- the relaxation time and the swelling ratio -

0

20

40

60

80

100

120

140

160

0 100 200 300 400 500 600

tau D

[m

m^2]

(a0^-2+b0^-2)^-1 [mm^2]

tau

€

˜ a (t) = ˜ a (0)exp(− tτ

)

τ−1 ≡ D 64{(2m −1)(2n −1)π 2}2 {((2m −1)π

a0

)2 + ((2n −1)πb0

)2}−1

n=1

∞

∑m=1

∞

∑

−1

Results- the relaxation time : τ -

τ

€

a0−2 + b0

−2( )−1

€

D = 0.026[mm2 /min]the collective diffusion constant (by experiment)

τ× D [mm^2]

€

a0−2 + b0

−2( )−1

Reproduce linearity

experimenttheory

1

1.1

1.2

1.3

1.4

1.5

0 2000 4000 6000 8000 10000 12000 14000 16000

C/C

0

time [min]

swelling ratio

4.25.09.3

14.819.329.435.853.1

Results- swelling process and width of gels -

experiment theory

width:large

€

c(t)c0

=1+ p0(K +43G)−1(1− exp(− t

τ))

Single exponent relaxation process

c(t)/c0

time (min)

c(t)/c0

time (min)

width:large

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

0 2000 4000 6000 8000 10000 12000 14000 16000

C/C

0

time [min]

swelling ratio

0.1420.4020.9151.3631.774

Results- swelling process and initial thickness of gels -

experiment theory

€

c(t)c0

=1+ p0(K +43G)−1(1− exp(− t

τ))

swelling process is independent on the initial thickness

c(t)/c0

time (min)

c(t)/c0

time (min)

Results- pressure and width of gels -

p/p0 =1.0(red),0.0(white)a0[mm] b0=26.0[mm]

4.2

9.3

19.3

35.8

time [min] 0 3200 6400 9600 12800 16000

Summary・Theoretical analysis of the anisotropic swelling process of gels by the linearized stress-diffusion coupling model.・We have nicely reproduced the experimental results.　→ Availability of the stress-diffusion coupling model for the general anisotropic dynamics of gels

Next：・By applying the MUFFIN/GelDyn simulator to this system, validate the simulator in comparison with the thoretical results. ・Theoretical analysis of long rod and large disk gels. (relaion of stress-diffusion coupling and Li-Tanaka’s theory?)・Applications：　　electrolyte gels (artificial muscles,DDS), coating and gelation.

AcknowledgementWe acknowledge support from the Japan Science and Technology Agency (CREST-JST).

References1) A. Suzuki and T. Hara: J. Chem. Phys. 114 (2001) 5012.

2) T. Tanaka and D. J. Fillmore: J. Chem. Phys. 70 (1979) 1214.

3) K.Sekimoto and K.Kawasaki: Physica A154 (1989), 384.

4) T. Tomari and M. Doi: Macromolecues 28 (1995) 8334.

5) M. Doi: Dynamics and Patters in Complex Fluids,

A. Onuki and K. Kawasaki Eds. (Springer, 1990) p.100.

6) M. Doi and A. Onuki: J. Phys. II France 2 (1992) 1631.

7) Y. Li and T. Tanaka: J. Chem. Phys 92 (1990) 1365.

8) T.Yamaue, T.Taniguchi and M.Doi: Progress of Theoretical Physics Supplement 138 (2000), 416.

9) T.Yamaue, T.Taniguchi and M.Doi: AIP Conference Proceedings 519 (2000), 423.

10) T.Yamaue, T.Taniguchi and M.Doi: Transactions Material Research Society of Japan 25[3] (2000), 767.11) T.Yamaue and M.Doi: in preparation.