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Theory of phase transition in polymer gels

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Responsive Gels: Volume Transitions I

Part of the book series: Advances in Polymer Science ((POLYMER,volume 109))

Abstract

A variety of unique phenomena are encountered in gels which arise from coupling between phase transition and elasticity. On the basis of a Ginzburg-Landau theory with a tensor order parameter, macroscopic and bulk instabilities, phase transitions in anisotropically deformed gels, and scattering amplitudes in various situations are considered. Basic dynamic equations of network and solvent are presented to analyze swelling processes, critical dynamics, and dynamics in anisotropic gels. A surface mode of uniaxial gels is also described which becomes unstable when the degree of anisotropy is increased eventually resulting in periodic folding of the surface. We propose a theory of dynamic light scattering which takes account of the frequency-dependence of the elastic moduli originating from network relaxation. As a result, the time correlation function of the density fluctuations has a slowly decaying component and the effect is enhanced near the spinodal point.

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Abbreviations

B:

dimensionless parameter representing the magnitude of the logarithmic term in F el

c:

parameter characterizing uniaxial extension

C:

coefficient in F inh

CV :

specific heat of gel at constant volume

C ∏ :

specific heat of gel at constant osmotic pressure

Dh :

diffusion constant given by Kawasaki's formula

Eij :

tensor defined by Eq. (4.38) for homogeneous gel

F:

free energy change after mixing of solvent and an initially, unstrained polymer network

Fel :

free energy due to elastic deformations

Finh :

free energy due to large scale inhomogeneities

Fion :

free energy due to ions

Fmix :

mixing free energy

ℱ=(ℱ1, ℱ2, ℱ3):

force density −∇·∏

g:

dimensionless free energy defined by Eq. (2.27)

hk :

Fourier component of height fluctuation of gel surface

J(\(\hat q\)):

dimensionless parameter dependent on \(\hat q\) defined by Eq. (4.43)

kB :

Boltzmann constant

K:

bulk modulus

p:

degree of inhomogeneity in the crosslink density

p 0 :

pressure of solvent outside gel

\(\hat q\) :

direction of wave vector q

S(q):

structure factor

S(q, t):

time correlation function of density fluctuations

T:

absolute temperature

Ts :

spinodal temperature

u=(u1, u2, u3):

displacement vector X−x around a homogeneous state

v:

inverse of φ

v 1 :

volume of one solvent molecule

V:

total volume of gel

V 0 :

total volume of gel in the relaxed state

v:

average velocity

v f :

solvent velocity

v g :

network velocity

x=(x1, x2, x3):

Cartesian coordinates of affinely deformed, homogeneous gel

x0=(x 01 , x 02 , x 03 ):

Cartesian coordinates in the relaxed state

X=(X1, X2, X3):

Cartesian coordinates of deformed gel

α:

elongation ratio of isotropic gel

β 0 :

dimensionless parameter defined by Eq. (2.30)

γi :

dimensionless parameter defined by Eq. (2.29)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta }\) :

parameter characterizing uniaxial extension and being equal to c −3/2

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } _c\) :

critical value of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta }\) being dependent of ε

ε:

K/μ+1/3 in isotropic gel and (K+μ/3)/μc2 in uniaxial gel

ζ:

polymer-solvent friction constant

ηs :

solvent viscosity

μ:

shear modulus

μs :

chemical potential of solvent inside gel

μ 0s :

chemical potential of solvent outside gel

ν 0 :

crosslink number density in the relaxed state

νi :

counter ion density in the relaxed state

νs :

crosslink density in deformed homogeneous gel

ξth :

thermal correlation length

∏:

osmotic pressure

∏ij :

stress tensor due to polymer

ρf :

solvent mass density

ρg :

polymer mass density

σ:

surface tension of gel-solvent interface

φ:

volume fraction of polymer

φ 0 :

volume fraction of polymer in the relaxed state

φc :

volume fraction at criticality

χ:

polymer-solvent interaction parameter

ω 0 :

high crossover frequency below which the Biot mass coupling is negligible

ωc :

low crossover frequency of transverse sounds in gel

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  1. Department of Physics, Kyoto University, 606, Kyoto, Japan

    Akira Onuki

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  1. Akira Onuki
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K. Dušek

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© 1993 Springer-Verlag

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Onuki, A. (1993). Theory of phase transition in polymer gels. In: Dušek, K. (eds) Responsive Gels: Volume Transitions I. Advances in Polymer Science, vol 109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56791-7_2

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  • DOI: https://doi.org/10.1007/3-540-56791-7_2

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