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Thermodynamics and hydrodynamics of chemical gels II. Gels in binary solvents
Ken Sekimoto
To cite this version:
Ken Sekimoto. Thermodynamics and hydrodynamics of chemical gels II. Gels in binary solvents. Jour- nal de Physique II, EDP Sciences, 1992, 2 (9), pp.1755-1768. �10.1051/jp2:1992232�. �jpa-00247764�
Classification Physics Abstracts
82.70G 62.20 05.70L
Thermodynandcs and hydrodynandcs of chendcal gels
II. Gels b1 bblary solvents
Ken Sekimoto
Department of Applied Physics, Nagoya University, Nagoya 464, Japan (Received 23 March 1992, accepted in final form 5 June 1992)
Abstract. We have developed a framework of the thermodynamics and hydrodynamics of chemically crosslinked gels in a binary miscible solvent. This work is an extension of a previous
one on gels in
a single solvent [Sekimoto K., J. Phys. II France1 (1991) 19]. Introduction of a composition degree of freedom of the solvent leads to both thermodynamic chemo-mechanical coupling and kinematical cros~coupling between permeation and mutual difusion of solvent.
Using an appropriate thermodynamic potential for gelin a solvent bath, we also discussed (local) equilibrium boundary conditions at the surface of a gel and at the interface between a shrunken phase domain and a swollen phase domain in
a gel undergoing a volume phase transition.
1. l~~troduction.
In a previous paper [I], the author developed a framework of thermodynamics and hydrody-
namics of chemically cross-linked gels. In that work, gels were considered as composed of two
mutually interpenetrating continua: a fluid continuum and an elastic contiuum. In gels, as in polymer solutions, there occurs mutual (cooperative) diffusion between the two components.
However, unlike polymer solutions, an elastic continuum of gel can retain shear strain in equi-
librium. In case of spat1al phase coexistence between swollen phases and shrunken phases, an equilibrium coexistence condition [2] is imposed on the interface between these two phases.
The motion of such interfaces and the permeation of solvent through gel were studied in [3].
These theories were developed on the assumption of local equilibrium.
In the present paper, we generalize the previous framework to include the case of gels
immersed in binary solvents. Actually most experiments on gels have been done with multi- component solvents. A typical example of a gel with a multi-component solvent is N-isopropyl polyacrylamide (NIPA) gel immersed in an alchol-water mixture [4, 5]. Another example of a
gel in a multicomponent solvent is the case of NIPA-sodium acrylate (SA) copolymerized-gel
immersed in pure water [6, 7] in which the network of gel has ionizable (SA) groups and the solvent contains counterions. Up to present, these experimental data, however, have not been
analyzed from general view-point of a multi-component solvent. The former system [4, 5] has been analysed using the sc-called single liquid approximation (SLA) [8] in which one assumes that the volumeric ratio of the binary solvent in gel is the
same as the volumeric ratio of the external solution surrounding the gel sample. These authors concluded that the SLA is not even qualitatively sufficient to reproduce their experimental data. In the case of ionic gels [6, 7], an important contribution to free energy comes from the electrostatic interaction between ions; for this special case, a theoretical model based on donnan-equilibrium theory developed
many years ago by Flory [9] shows that and there the electrostatic interaction gives rise to
an additional osmotic pressure in a gel. The framework we develop below is oriented to gels
with nonionic binary solventsj the inclusion of the case of ionic gel into our framework will be postponed to future studies. In the rest of this section we first describe what new qualita- tive features enter when the solvent becomes multi-component in water, and then we give the
outline of our paper.
As compared with gels in a pure solvent, gels in a binary solvent have an additional degree
of freedom~ the local composition of solvent. Both the interaction energy and mixing entropy in the gel depend on this composition. This new degree of freedom yields sc-called chemc- mechanical coupling, through which the swelling ratio of gel may depend on the composition of solvent and vice versa. Very recently Leibler and the author [10] have studied the effect of chemc-mechanical coupling using Flory's model of nonionic gels [9]. We calculated the equilibrium phase diagram both by considering explicitly the solvent composition~ and also
by using the SLA. Comparison of both results has shown us how and when the SLA yields qualitatively incorrect conclusions. As for nonequilibrium conditions, that is, hydrodynamics, the local composition of solvent is changed through two transport processes which are either
competitive or constructive, depending on the situation. One process is permeation of solvent,
in which the composition profile is convected at the mean flow velocity of solvent. The other is internal [mutual] diffusion of solvent components, which takes place without accompanying
the mean flow of solvent. In the presence of the composition degree of freedom of the solvent, the definition of osmotic pressure must make clear the condition on the flux(es) of solvent
component(s) under which the deformation of gel occurs. As a consequence of this, some argument is needed to establish the boundary conditions on the gel surface. It is the purpose of the present paper to give a proper account on these points, and to discuss applications to
various actual situations of interest.
The outline of the present paper is as follows. In the next section (Sect. 2) we discuss the thermodynamics of gels with a binary solvent. The works by Jowhnson and Alexander [11, 12] are the starting point of that section; they have studied solid-liquid and solid-solid
phase coexistence of compressible multicomponent systems. We consider here the problem
under slightly different conditions: (I) we impose an incompressibility constraint, which is
an appropriate assumption for gels unless sound waves are considered. The definitions of osmotic pressure and the exchange chemical potential of the solvent components are mutually related, the details of which are described in Appendix B. (2) We introduce what we call the grand potential of the gel immersed in an infinite amount of bulk solution. This grand potential is utilized instead of the free energy for macroscopic but finite system. The details of the derivation is given in Appendix A. (3) Thermodynamic relations, such as Gibbs-Duhem
relation, are expressed in such forms as to be convenient for developing the framework of
hydrodynamics. (4) The equilibrium coexistence condition at the interface between a swollen
phase and a shrunken phase is derived from the grand potential of the incompressible gel, and the result is cast into a form of the generalized Maxwell's construction rule. In section 3, we derive hydrodynamic equations of motion of gels saturated by binary solvent. We discuss in
some detail the quasi-static case in which mechanical balance has been establised in a short
time; short compared with the time required for the permeation or the interdiffusion of solvent components. Under this condition we derive the surface boundary condition which connects the mechanical pressure po of external solution and the Lagrange multiplier p which appears as
a result of the incompressibility constraint inside the gel. The last section (Sect. 4) is devoted to the discussion of two additional points. First concerns the rapidity of attaining chemical balance as compared with osmotic balance, Second concerns the validity of local equilibrium assumption, which we utilized throughout the present paper.
Throughout this paper we assume that the gel monomers and the two solvent components
are incompressible and, therefore, have constant specific volumes, Vm, VA and flB> respectively,
The generalization to the compressible case can be done with no difficulty since the recipe has been established [I]. The generalization to the case of solvent with more than two components is also straightforward. These are not explored in the present paper to avoid unnecessary
complexity of the description.
2. Thermodynamics of gels with binary solvent.
We assume that the gel is prepared in a homogeneous and isotropic state. To describe the de-
formation of the solid part of the gel, we take a macroscopically uniform and homogeneous state of the gel as a reference state. Each material point of the solid continuum (a monomer in a twc- fluid description) is distinguished by the reference Cartesian coordinate X e (X~,X~,X~l'~
at which the material point filaced in the reference state. Deformed states of gel are then defined by the mapping from X
- x(X) describing the spatial position x % (z~, z~,z~l'~ of each material point X. We introduce the distortion matix F by the definition
~ ~
(~)p fizP
" ~p " @ (~'~)
The entropy S of a homogeneously deformed gel is a function of energy E, mass of B-solvent,
MB, mass of monomers, Mm, and the distortion F. Throughout this paper, we assume the
incompressibility of each constituents of gel and, unless stated otherwise, we chose the mass of A-solvent MA as the dependent variable which is determined through the relation,
~)~
= MAVA + MBVB + MmVm, (2.2)
Pm where we have defined
J % det(F), (2.3)
and where p[ is the constant mass density of solid portion of gel in the reference state. VA, flB and flm are, respectively, the specific volumes of A-solvent, B-solvent and monomers. The
left hand side of (2.2) is the [envelope] volume occupied by gel. If we introduce the volume fractions of monomers, A-solvent, and B-solvent as #m, #A, and #B, respectively, then (2.2) simply becomes,
I m #m + #A + ~B. (2.4)
The volume fraction of monomers #m and the distortion F are related via the relation, #mJ =
#mo, where #mo is the volume fraction of monomer in the reference state. Next we introduce the thermodynamic quantities T, pm, pB and n~ by the following equation,
dS e dE ~~ dMB ~~ dmm ~~'f~~~ dF(, (2.5)
T T T pmT
where T is the temperature and pB is the exchange chemical potent1al, and H~ is osmotic
stress matrix of the Piola-Kirchhoff type. We call pB the exchange chemical potential based on the following observation: suppose that we have a gel with total monomer mass Mm, and that
we somehow fix the gel so that it has a homogeneously deformed (or undeformed) state with
a given F. Under these conditions (dmm = 0 and dF
= 0) and at a fixed temperature dT = 0, the change of the Helmholz free energy of the gel, A % E TS, is solely due to the exchange
of solvent molecules. From (2.5), we have d(E TS) =pBdMB. Through the incompressibility requirement (2.2), dMB and dMA are mutually dependent; VAdMA+VBdMB " 0. Thus pB deserves the name of exchange chemical potential. The physical meaning of the osmotic stress is described in detail later in this section (see the paragraph following (2.14) below). As we have noted before (see Appendix A of [I]), pm is a quantity defined by (2.5), and it should not be identified as the chemical potent1al of monomers of the gel.
Hereafter we adopt the Einstein's convention for all repeated indices except for the suffices m, A and B. Using the extensivity properties of the entropy function, we can obtain the
fundamental relation and Gibbs-Duhem relation among these thermodynamic quantities (see Eqs. (2.3 2.9) of [I]). We could point out a certain parallelism between an incompressible gel
in a binary incompressible solvent and a compressible gel in a single compressible solvent. The
analogy holds only for thermodynamics; it cannot be generalized for hydrodynamics. From the extensivity of the entropy function we can derive the following thermodynamic relations
among the mass densities of monomers pm(= J~~ pi) and of B-solvent pB, the energy per mass ofgel, e, and the entropoy per mass of gel, s:
Pe " PST + PBPB + pmpm + trH (2.6)
0 = psdT + pBdpB + pmd(pm ~~~ + d(H~)(, (2.7)
Pm
~
Td(Ps) = d(Pe) »BdPB btm £ldPm (II~)ld()), (2.8)
where p is the total mass density. The Gibbs-Duhem relation holds locally even in gels under
non-homogeneous deformation, as is explicitly written in (2.7). This reflects the assumption that the gel under consideration can be (hypothetically) divided into small but macroscopic
sub-volumes within which thermodynamic laws of homogeneous gels can be applied. Non- classical critical phenomena in gels are, therefore, beyond the scope of the present description.
In the above equations the Cauchy osmotic stress tensor H has been introduced which is related
to H~ by
("~)SF/
= J III- (2.9)
In considering thermodynamics, it is more convenient to consider the Helmholz free energy A % E TS than to consider the entropy S. Allowing for inhomogeneous deformations of the
gel, we introduce the free energy density of gel per unit reference volume of gel, a', by
A = / dv'a', (2.10)
Mm/pj
where dV' is the volume element of gel in the reference space, and Mm /p[ is the total volume of gel in the reference state. We have from (2.5) the following relation,
da'(T, #BJ,F) = pJsdT + ~~ )d(#BJ) + (n~)(dF(. (2.II VB
The appearance of #BJ as in the papers by Johnson and Alexander [11, 12] is natural since the
mass of B-solvent per unit reference volume ofgel is proportional to #B/#m «#BJ, rather than
#B. The Helmliolz free energy density, however, should be modified when we consider a gel
immersed in a large bath of solvent in equilibrium, because what we can then directly control is the composition of the solvent bath (and temperature). Thus we need rather an expression
of what we may call grand potential per unit reference volume ofgel, g'(T, #BJ, F; #Bo), where
#Bo is the volume fraction of B-solvent in the solvent bath. The relation to Helmholz free energy is given by
g' = «' JOO ~° J(#B #Bo), (2.12)
B
where ao(T,#Bo) is the (Helmholz) free energy density of the external solution, and pBo +
flB3ao/fi#Bo. The derivation of (2.12) is given in Appendix A. llrom f2.ll) and (2.12), the
change of the grand potential per unit reference volume of gel under fixed temperature T and fixed composition of external solution, #Bo, or equivalently pBo, is given as
fig' = ~~~ ~~°~
J6#B + (H~)(6F(
VB
[ao + (#B #Bo) ~~° #Bd]J(F~~)$6F(, (2.13)
VB VB
where we have used the geometrical identity
bJ = J(F~~)$6F(. (2.14)
From the expression (2.13), we may distinguish two different kinds of reversible mechanical work due to deformation SF: (a) One is the work due to the deformation allowing free exchange
of A and B solvent molecules across the surface of gel immersed in a solvent bath, and (b)
the other is a rather hypothetical work due to the deformation during which we fix somehow
the amount of B-molecules per unit reference volume of gel. The former process (a) requires
from (2.13) the chemical equilibrium condition, pB # pBo, while in the latter process (b), the constraint is represented by the relation 6(#BJ) = 0. For both processes the reversible work
bg' has the same expression,
>g' = i(n~)i i«o <~o j°iJ(F-i)ii>Fj. (2.15)
The quantity within the curly braket
on the right hand side is the conjugate thermodynamic
force of deformation. The magnitude of this force is generally different for the precesses (a)
and (b), and according to the Le Chatelier-Braun principle, the force in the process (b) will be greater [stifler] than that in the precess (a). The quantity
To + loo #Bo~~°], (2.16)
VB
which appeared on the right hand side of (2.15) can be regarded as the osmotic pressure of the A-component in the external solution, while we see from (2.ll) that n~ is the osmotic force measured with a fixed B-content per monomer, I.e., 6(#B/#m) «6(#BJ) = 0. Thus from
(2.15) we see that the net osmotic force of gel immersed in external solution is expressed as
the difference of the osmotic forces of gel and that of the external solution with respect to the transport of A-solvent molecules. The apparent asymmetry of formalism with respect to A- and B-components of solvent comes from our choice of independent thermodynamic variables;
JOURNAL DE PHYSIQUE II -T 2. N'9, SEPTEMBER 1992 6s
T we had chosen as an independent variable the volume fraction of A-component instead of that of B-component, we would have diferent definitions of cheInical potentials and osmotic stress tensor. Of course this asymmetry has no e@ects on any predictions or calculations. We note also that the terms 'chemical'or 'mechanical'used above do not have definitive meaning because the definitions of chemical potential and osmotic strest are partly dependent on the choice of
independent variables. The relations among these quantities are sumrnerized in Appendix B.
In an illustrating figure (Fig. I), we summarize the two thermodynamic processes, one related to the exchange of solvent components and the other related to deformation of gel. These
two concern independent degrees of freedom of a gel in binary solvent. However, there are
thermodynamic couplings between these; changes in the composition of solvent in gel leads to the change of osmotic strew, and inversely, deformation of gel alters the exchange chendcal
potient1al pB. '
permeable semi-permeable
container membrane
, gel
~~~
A-solvent
A-sojvent B-solvent
a) b)
Fig. 1. Schematically shown are the twcuthermodynamic processes that characterize chemical gels
in binary solvent. (a) Component difusion occurs through the exchange of solvent components A and B between the inside of the gel and the external solvent. For this process to occur, the gel might
be clamped by a rigid but permeable container. (b) Deformation of the gel occurs accompanying th~~
permeation of solvent (A and/or B component) under the incompressibility requirement. For this process to occur, the gel might be jacketed by a semi-permeable membrane ±hrough which only one component of the solvent, say A-solvent, can flow.
As one of the simplest example, the equilibrium isotropic state of a freely suspended gel
in a binary solvent with composition #Bo is determined by two conditions, pB " »Bo and
fl( = -To 6v~. This is a generalization of the osmotic equilibrium condition (9] of a gel in
single solveni. For general anisotropic and nonhomogeneous deformation of a gel, (2.15[yields
the boundary condition on the surface of the gel; we use sc-called Nanson's formula, which
gives the relation between an area element NpdL in the reference space and its image in the actual space nvd« by the mapping X - x(X),
n~d« = J(F~~)$NpdL, (2.17)
where n~ and Np are the unit normal vectors of the area elements, respectively, in the actual space and in the reference space. Suppose we apply an external osmotic force l~~s on the surface of a gel, for example, by displacing porous plates attached to the surface. The virtual work
