A theoretical model of polyelectrolyte gels is presented to study continuous and discontinuous volume phase transitions induced by changing salt concentration in the external solution. Phase diagrams are constructed in terms of the polymer–solvent interaction parameters, external salt concentration, and concentration of fixed charges. Comparisons with previous experiments for an ionized acrylamide gel in mixed water–acetone solvents are made with good quantitative agreement for a monovalent salt (NaCl) but fair qualitative agreement for a divalent salt (MgCl_{2}), using a simple set of parameters for both cases. The effective polymer–solvent interactions vary with the volume fraction of acetone in the mixed solvent, leading to either continuous or discontinuous volume transitions. The presence of divalent ions (Mg^{2+}) in addition to monovalent ions in the external solution reduces the critical salt concentration for the discontinuous transition by several orders of magnitude. Moreover, a secondary continuous transition is predicted between two highly swollen states for the case of a divalent salt. The present model may be further extended to study volume phase transitions of polyelectrolyte gels in response to other stimuli such as temperature, pH and electrical field.

## Introduction

A polyelectrolyte gel consists of a charged polymer network with ionizable groups on the polymer chains and mobile ions in a fluid solvent [1]. Due to the presence of ionizable groups and mobile ions, polyelectrolyte gels are responsive to many external stimuli such as temperature [2,3], *p*H [4–6], ionic strength [7], electrical fields [8,9], and light [10,11]. Of particular interest is the volume phase transition, when the gel volume changes enormously in response to a small change of the external environment [12]. A discontinuous volume transition was first predicted theoretically in 1968 by Dusek and Patterson [13]. The first experimental observation of a volume phase transition in a gel was reported in 1978 by Tanaka [14]. Since then, many gel systems have been developed with characteristic volume phase transitions for a wide range of applications, such as sensors and actuators [15], artificial muscles [16], and drug delivery [17,18]. Recent developments of ionic skin [19] and ionic cables [20] have used polyelectrolyte gels as effective ionic conductors, opening doors to new applications in wearable electronics and soft robotics.

Many theoretical models have been developed to understand and predict volume phase transitions in polyelectrolyte gels. Katchalsky et al. [21] first extended the Flory–Rehner model for neutral polymer gels to describe swelling of polyelectrolyte gels. Following a similar approach, Dusek and Patterson [13] predicted the possibility of a discontinuous volume change of a gel based on an analogy of the coil–globule transition of polymers in a solution. Later, Shibayama and Tanaka [12] noted a theoretical analogy between the volume phase transition of a gel and the liquid–gas transition of a van der Waals fluid. While the classical mean-field theory of gels provided a qualitative understanding of the volume phase transition [22,23], improvements in quantitative agreement between the theory and experiments have been made by various modifications to the theoretical formulation [24–26]. To study the kinetics of swelling and inhomogeneous phenomena such as coexisting phases in a polyelectrolyte gel, a continuum field theory was developed by Hong et al. [27], which couples large deformation and electrochemistry in the constitutive relations along with a system of governing equations and boundary conditions. Alternatively, multiphasic models have also been developed for polyelectrolyte gels, where the polymer network, the solvent and the ions are treated as distinct phases within the framework of mixture theory [28–31].

Volume phase transitions in gels induced by temperature and *p*H have been studied extensively [5,12]. Less attention has been paid to the effect of salt concentration or salinity on volume phase transitions. Ohmine and Tanaka [32] reported an experimental study on salt-induced volume phase transitions in an ionized acrylamide gel, along with a qualitative explanation based on the classical theory. Their experiments showed intriguing effects due to the valence of the salt ions and composition of a mixed solvent. A direct comparison between the experiments and theoretical predictions would elucidate the capability and limitations of a specific model for a quantitative description of polyelectrolyte gels in both the swollen and collapsed phases. In this paper, we present such a direct comparison based on the continuum theory by Hong et al. [27], slightly reformulated for equilibrium free swelling of a polyelectrolyte gel immersed in a salt solution with a mixed solvent. The theory naturally leads to a balance of osmotic pressures due to elasticity, solvent, and ions in an isotropic, homogeneous state. Under the condition of electroneutrality, the ion concentrations in the gel satisfy the Donnan equilibrium with a Donnan potential, recovering the classical theory for a highly swollen state. Both the salt concentration and solvent composition in the external solution can be varied to induce a discontinuous volume phase transition of the polyelectrolyte gel. Comparisons with the experiments of Ohmine and Tanaka [32] show good quantitative agreement for a monovalent salt (NaCl) but fair qualitative agreement for a divalent salt (MgCl_{2}), which may be further improved by calibrating the model parameters for a specific combination of salt and solvent.

The remainder of this paper is organized as follows: Section 2 presents the theoretical model, including the equations of state for both the polyelectrolyte gel and the external solution as an ionic liquid. Section 3 presents an equilibrium analysis of free swelling, and Sec. 4 discusses the volume phase transition in a general form. Direct comparisons with experiments are presented in Sec. 5 for both monovalent and divalent salts, followed by a short summary in Sec. 6.

## Theory

### Kinematic Variables.

*t*is denoted by $x(X,t)$. The kinematics of deformation is described by the deformation gradient tensor as usual

where $J=detF$. Similarly, the nominal concentration, $Ca$, is defined as the number of mobile particles of species $a$ per unit volume of the polymer network in the gel. The true concentration in the current configuration is then $ca=Ca/J$. The superscript $a$ is used here for all the mobile particles in the gel, including the neutral solvent molecules and the mobile ions. Together, $F$, $D\u0303$, and $Ca$ constitute a set of kinematic variables that describe the current state of the polyelectrolyte gel.

where $\upsilon a$ is the molecular volume of each mobile particle *a*.

### Constitutive Relations.

where $Wgnet$, $Wgsol$, $Wgion$, and $Wgpol$ are the free energy terms associated with stretching of the polymer network, mixing the polymer and the solvent, mixing the solvent and ions, and polarization of the gel, respectively. Note that the free energy density in Eq. (2.4) is per unit volume in the reference configuration (i.e., the dry polymer network); the free energy per unit volume of the gel in the current configuration is simply $Wg/J$.

where $I=tr(FTF)$, $N$ is the effective number density of polymer chains in the dry state, $kT$ is the temperature in the unit of energy with the Boltzmann constant $k$ and the absolute temperature *T*.

*χ*is the Flory–Huggins parameter for polymer–solvent interactions. Equations (2.5) and (2.6) are identical to those for neutral polymer gels in previous studies [34,35].

where the superscript *b* is used here for all the mobile ions, excluding the solvent molecule. Equation (2.7) is slightly different from that in Hong et al. [27], where zero chemical potential was assumed at a reference ion concentration. Here, the free energy of mixing is derived from the statistical mechanics definition of entropy for an ideal solution [36], without explicitly identifying a reference ion concentration. This change leads to some subtle differences in the swelling behavior, as discussed later. By Eq. (2.7), we assume that the free energy due to the mixing between the ions and the polymer network is negligible.

where $\epsilon g$ is the effective dielectric constant of the gel. Due to the different dielectric constants of the polymer and the solvent, the effective dielectric constant of the gel in general depends on the concentration of the solvent [37], but is assumed to be a constant here for simplicity.

where Π* _{g}* is a Lagrange multiplier.

*e*is the elementary charge, $za$ is the valence of the mobile species $a$, and $\Phi $ is the electric potential so that $E\u0303=\u2212\u2207\Phi $. Substituting Eq. (2.9) into Eqs. (2.10) and (2.11), we obtain

which is essentially the osmotic pressure due to polymer–solvent mixing as discussed later (see Eq. (2.22)).

With Eq. (2.18) or Eq. (2.19), the ion concentrations in a polyelectrolyte gel can be obtained explicitly in terms of the electrochemical potentials and the volume swelling ratio ($J=detF$).

which includes four parts: the elastic part due to stretching the polymer network, an osmotic part due to polymer–solvent mixing, another osmotic part due to ionic interactions, and the Maxwell stress due to polarization. In terms of the true stress, $\sigma =PFT/J$, the two osmotic parts are isotropic, yielding two osmotic pressures for the solvent and the mobile ions, respectively. The last term in Eq. (2.22) is the nominal Maxwell stress, with which the true Maxwell stress can be obtained in its usual form: $\sigma Maxwell=\epsilon g(E\u2297E\u2212|E|2I/2)$, with the true electrical field $E=HE\u0303$.

Therefore, the constitutive relations for the polyelectrolyte gel consist of Eqs. (2.15), (2.19) and (2.22), with which the electrical displacement, ion concentrations, and nominal stress can be determined for given deformation gradient ($F$), electrical field ($E\u0303$), and electrochemical potentials ($\mu a$); the solvent concentration can be obtained directly from Eq. (2.16).

### External Solution.

which implies that the entire space is occupied by the solvent molecules and ions in the external solution (i.e., $\u2211a\upsilon aca=1$).

This expression is slightly different from that for the gel in Eq. (2.7) because the fluid mixture occupies the entire volume of the external solution but only a fraction of the volume in a gel containing the polymer network.

where $\epsilon s$ is the effective dielectric constant of the external solution.

where Π* _{s}* is a Lagrange multiplier.

which is independent of the reference configuration or deformation but includes a hydrostatic term undetermined by the constitutive relation as expected for an incompressible fluid.

Therefore, by changing the ion concentrations in the external solution, the chemical potential of solvent and the electrochemical potential of the ions change simultaneously, which would then lead to the change of the equilibrium state of the gel. Since the chemical and electrochemical potentials are set by the external solution in equilibrium with the gel, the only unknown quantities to be determined for the gel are the deformation gradient and the electrical potential.

Note that, based on the statistical mechanics for an ideal solution [36], the chemical potential of each mobile species (including solvent and ions) is zero only when it is not mixed with any other species. Indeed, by Eq. (2.35), the chemical potential of solvent is zero for a pure solvent with zero ion concentrations (i.e., $\u2211b\u2260sc0b=0$). For ions, however, Eq. (2.34) does not vanish even when $\upsilon bc0b=1$ and $cs=0$, because the free energy function in Eq. (2.25) is an approximation for cases with low ion concentrations (e.g., $\upsilon bcb\u226a\upsilon scs$) and thus the resulting chemical potential is inaccurate for the case with a high ion concentration.

## Equilibrium Analysis of Free Swelling

It is well known that an electrical double layer forms at the interface between the polyelectrolyte gel and the external solution (Fig. 1), with a characteristic thickness proportional to the Debye length [27], which is often much smaller than the size of the gel and the external solution. As a result, the fields of electrical potential, stress, and solvent/ion concentrations are all inhomogeneous near the interface. On the other hand, at a distance much larger than the Debye length from the interface, both the gel and the external solution are electrically neutral, and the fields are homogeneous [27]. Here, we ignore the inhomogeneous fields near the interface and focus on the homogeneous part of the gel. Thus, by electroneutrality the ion concentrations in the gel are related to the concentration of fixed charges as

where $Cfix$ is the nominal concentration of the fixed charges on the polymer network. For a strong polyelectrolyte gel, $Cfix$ is a constant, as is assumed in the present study. For a weak polyelectrolyte gel, $Cfix$ may change when the degree of dissociation of the ionizable groups changes in response to the solvent composition or ion concentration [25,37].

For the case of homogeneous swelling, Eq. (3.6) gives a constant electric potential $\Phi $, which depends on the fixed charge concentration, the ion concentration of external solution, and the volume swelling ratio *J*. Under this condition, the nominal electric field $E\u0303$ vanishes and the nominal electric displacement $D\u0303$ is zero in the gel.

where the true ion concentrations ($ca=Ca/J$) are used instead of the nominal concentrations and the approximation is made for $J\u226b1$ (i.e., highly swollen gels).

Solving Eq. (3.9), we obtain the linear stretch $\lambda $ of a polyelectrolyte gel in equilibrium with an external solution, which depends on four dimensionless parameters: the effective number density of polymer chains $N\upsilon $, the Flory–Huggins parameter $\chi $, the nominal concentration of fixed charges $\upsilon Cfix$, and the normalized ion concentration in the external solution $\upsilon c0$. Once the stretch is determined, Eq. (3.6) is used to determine the electric potential in the gel. Then, Eqs. (3.4) and (3.5) are used to find the nominal concentrations of the ions in the gel. For example, Fig. 2 shows the results for free swelling of a polyelectrolyte gel with $N\upsilon =10\u22123$, $\upsilon Cfix=0.02$, and $\chi =0.5$ in an external solution of varying ion concentrations. It can be seen that increasing the ion concentration in the external solution leads to a decrease in the linear stretch and thus shrinking of the gel volume. This is a result mainly due to the decrease in the chemical potential of the solvent in the external solution, as given by Eq. (2.35). Meanwhile, the electrochemical potentials of the ions increase, leading to an increase of the ion concentrations in the gel. However, since the ion concentrations are much lower than the solvent concentration ($\upsilon C+/\u2212\u226a\upsilon Cs\u2248\lambda 3\u22121$), the volume change of the gel is dominated by the change in the solvent concentration. Moreover, with the negative fixed charges in the gel, the electric potential of the gel is negative and increases as the ion concentrations increase in the external solution. These results are similar to those obtained by Hong et al. [27]

by which the osmotic pressures due to elasticity and ions are balanced. The osmotic pressure due to ions in Eq. (3.11) is proportional to $Cfix$, similar to that in Ref. [32] for $c0\u226aCfix$.

where the osmotic pressure due to ions is proportional to $Cfix2$, similar to that in Ref. [32] for $c0\u226bCfix$. However, this simplification is problematic, because the condition $\lambda 3\u226b1$ typically does not hold at high ion concentrations. Hong et al. [27] predicted that the equilibrium swelling ratio is independent of the ion concentration for the limit of a highly concentrated solution. Equation (3.13) predicts that the equilibrium swelling ratio decreases with increasing ion concentration. In contrast, by Eq. (3.12), the equilibrium swelling ratio may increase with increasing ion concentration. Indeed, experiments by Okay and Sariisik [40] found that the equilibrium swelling ratio of a poly(acrylamide-co-sodium acrylate) hydrogel increased with increasing salt (NaCl) concentration in highly concentrated solutions, qualitatively consistent with Eq. (3.12). It should be noted that, with relatively high ion concentrations in the gel, the approximation in Eq. (2.16) may not be applicable, and a more accurate solution can be obtained by solving a system of equations simultaneously.

## Volume Phase Transition

The Flory–Huggins parameter *χ* characterizes the polymer–solvent interactions in the gel, which depends on temperature [12,26] and solvent composition [41,42]. A change of *χ* has a profound effect on the swelling behavior of polyelectrolyte gels. As shown in Fig. 3(a), the equilibrium stretch *λ* is determined as a function of the ion concentration in the external solution by solving Eq. (3.9) for different values of *χ* ($N\upsilon =10\u22123$ and $\upsilon Cfix=0.02$). It is found that when *χ* is small (a good solvent), e.g., $\chi =0.1$, 0.3, and 0.5, the gel is swollen with a relatively large stretch ($\lambda >2$) over the entire range of ion concentration, and the equilibrium stretch decreases continuously with increasing ion concentration. However, when *χ* is larger, e.g., $\chi =0.7$ (a poor solvent), a discontinuous jump of the equilibrium stretch is predicted at a critical ion concentration, where a transition occurs from a highly swollen gel ($\lambda >3$) to a collapsed phase ($\lambda <2$). When *χ* is even larger, e.g., $\chi =0.9$, the equilibrium stretch is small ($\lambda \u223c1$) over the entire range of the ion concentration, and thus the gel stays in the collapsed phase. A phase diagram is constructed in Fig. 3(b) in terms of the normalized ion concentration $\upsilon c0$ and *χ*, where the two phases (highly swollen and collapsed) are separated by a set of transition lines. The line of the discontinuous transition (the thick solid line) is obtained by plotting the critical ion concentration at the discontinuous transition (if it occurs) versus *χ*. Apparently, for $N\upsilon =10\u22123$ and $\upsilon Cfix=0.02$, the discontinuous transition occurs only within a narrow range of *χ* around 0.7. For smaller values of *χ*, the transition is continuous as indicated by the dashed lines, each corresponding to a constant stretch (e.g., $\lambda =$ 3, 2.5, and 2).

To understand the discontinuous transition, we plot the normalized stress–stretch relations (Eq. (3.9)) in Fig. 4(a) for different ion concentrations in the external solution with dimensionless parameters $\chi =0.7$, $N\upsilon =10\u22123$, and $\upsilon Cfix=0.02$. The equilibrium stretch for free swelling corresponds to the stretch with zero stress, i.e., $P(\lambda )=0$. When the ion concentration is relatively high (e.g., $c0/Cfix=0.02$), there is only one solution for the equilibrium stretch and it is in the collapsed phase. When the ion concentration is relatively low (e.g., $c0/Cfix=0.001$), there are three possible solutions for the equilibrium stretch, one in the collapsed phase ($\lambda <2$), one in the highly swollen phase ($\lambda >3$), and another in between. To determine which one of these solutions is thermodynamically stable, we compute the mechanical work required to stretch the gel from the collapsed phase to an arbitrary $\lambda $ as

where $\lambda 1$ is the stretch in the collapsed phase with $P(\lambda 1)=0$. As shown in Fig. 4(b), for a relatively low ion concentration, the mechanical work has two local minima and one local maximum, corresponding to the three possible solutions with zero stress in Fig. 4(a). The stretch with the local maximum mechanical work is an unstable solution. The stretch with the lower mechanical work of the two local minima is thermodynamically stable, while the stretch at the other local minimum is metastable. At a critical ion concentration ($c0/Cfix=0.0052$), the two local minima are equal in the mechanical work, and the two phases may coexist. If the ion concentration is greater ($c0/Cfix>0.0052$) or lower, the stable equilibrium stretch corresponds to the collapsed or highly swollen phase, respectively. The discontinuous transition at the critical ion concentration is thus a first-order phase transition. Alternatively, the critical ion concentration for the discontinuous volume phase transition can also be determined directly from the stress–stretch relations (Fig. 4(a)) by the Maxwell rule [43,44], equivalent to the energetic consideration.

Interestingly, Fig. 4(a) predicts that a discontinuous volume transition may also be induced by mechanical stress (or pressure) for a fixed ion concentration in the external solution. When the ion concentration is greater than the critical concentration (e.g., $c0/Cfix>0.0052$), the gel is in the collapse phase at zero stress. Applying a hydrostatic tension increases the gel volume and induces a discontinuous volume transition to the highly swollen phase at a critical stress. The critical stress can be determined from the stress–stretch diagram by the Maxwell rule. On the other hand, when the ion concentration is lower than the critical concentration (e.g., $c0/Cfix<0.0052$), the gel is in the highly swollen phase at zero stress. In this case, the gel volume decreases under a hydrostatic compression and undergoes a discontinuous transition to the collapsed phase at a critical pressure. At the critical ion concentration, the critical pressure (or stress) is zero. Such stress-induced discontinuous volume phase transition has not received much attention and may deserve further studies.

The discontinuous volume phase transition depends sensitively on the concentration of fixed charges in the polyelectrolyte gel. As shown in Fig. 5(a), with increasing concentration of fixed charges, the transition line shifts up and the range of *χ* values with a discontinuous transition expands. Figure 5(b) shows the upper and lower limits of *χ* versus the concentration of fixed charges, which may be considered as a diagram of phase transitions. For a polyelectrolyte gel with properties (*χ* and $\upsilon Cfix$) between the two lines, a discontinuous volume phase transition is expected at a critical ion concentration. Below the lower (blue) line, a smooth transition is expected. Above the upper (red) line, the gel is in the collapsed phase with no transition. Interestingly, there exists a critical point where the two lines intersect, with $\chi \u22480.63$ and $\upsilon Cfix\u22480.012$ for the case $N\upsilon =10\u22123$. The location of the critical point depends on $N\upsilon $. Therefore, a necessary condition for the discontinuous volume phase transition in polyelectrolyte gels requires a relatively large fix charge concentration ($\upsilon Cfix>0.012$) along with a poor solvent ($\chi >0.63$). In general, it is known that the presence of fixed charges (by ionization) promotes the discontinuous volume phase transition in many gel systems [12,22].

The effects of the composition dependent interaction parameter on the volume phase transition of polyelectrolyte gels are illustrated in Fig. 6. Most interestingly, the minimum concentration of fixed charges required to have a discontinuous volume phase transition can be reduced to nearly zero with $\chi 1=0.4$, as shown in Fig. 6(d). As discussed by Erman and Flory [45], the composition dependent polymer–solvent interaction is essential for uncharged polymer networks ($Cfix=0$) to have the possibility of a discontinuous volume phase transition. For ionic gels, the degree of ionization becomes the dominant factor in phase transitions [22]. As shown in Fig. 6(a) for polyelectrolyte gels with $N\upsilon =10\u22123$, $\upsilon Cfix=0.02$, and $\chi 0=0.9$, the equilibrium stretch undergoes a discontinuous transition for $\chi 1=0.4$ and 0.6, but no transition for $\chi 1=0.2$ or 0. The critical ion concentration for the discontinuous transition depends on both $\chi 0$ and $\chi 1$. As a result, the transition line in Fig. 6(b) shifts up as $\chi 1$ increases. A diagram of phase transitions in terms of the two interaction parameters is presented in Fig. 6(c) for $N\upsilon =10\u22123$ and $\upsilon Cfix=0.02$, where the discontinuous transition is predicted over a range of $\chi 0$ values depending on $\chi 1$. The present results suggest that both the degree of ionization (concentration of fixed charges) and the composition-dependent polymer–solvent interaction are important for the discontinuous volume phase transition in polyelectrolyte gels.

## Comparison With Experiment

_{2}. The acetone concentration of the mixed solvent plays an important role in the swelling and volume phase transition of the gels [14]. Here, we treat the mixed solvent as one pure solvent with an effective solvent–polymer interaction [41,42] depending on the volume fraction of acetone ($\phi $), namely,

for *i* = 0 and 1. When $\phi =0$, the solvent is pure water and the polymer–water interaction is characterized by $\chi iw$. When $\phi =1$, the solvent is pure acetone and the polymer–acetone interaction is characterized by $\chi ia$. In between, a simple rule of mixture is assumed with the parameter $\chi iaw$ as the potential coupling for the mixed solvent. It is assumed that the solvent in the gel has the same volume fraction of acetone as in the external solution. As a result, the gel may undergo a discontinuous volume phase transition upon changing the acetone concentration in the external solution as observed by Tanaka [14], even without adding any salt. As shown in Fig. 5(a), for $\upsilon c0\u21920$, a discontinuous transition occurs at a critical value of $\chi $ that depends on the concentration of fixed charges in the gel.

Consider an external solution of NaCl with the water–acetone mixed solvent, characterized by the volume fraction of acetone ($\phi $) and the salt concentration ($c0$). There exist four types of mobile ions in the gel and the external solution: Na^{+}, Cl^{−}, OH^{−}, and H^{+}. In addition to the salt ions (Na^{+} and Cl^{−}), dissociation of the ionizable group –COOH in the acrylamide gel leaves fixed charges –COO^{−} on the polymer network and mobile counter ions (H^{+}) in the gel, while self-ionization of water yields OH^{−} and H^{+} ions. Strictly, each hydrogen ion (H^{+}) immediately protonates another water molecule to form a hydronium ion, H_{3}O^{+}. Here we do not distinguish H^{+} and H_{3}O^{+} under the assumption that its concentration is relatively low compared to the concentration of neutral solvent. With a given salt concentration ($c0$), the concentrations of the Na^{+} and Cl^{−} ions in the external solution (far away from the gel) are identically $c0$. The concentrations of H^{+} and OH^{−} ions in the external solution are determined by the *p*H value. Assuming a neutral solution with *p*H = 7, as the case in the experiments by Ohmine and Tanaka [32], the concentrations of H^{+} and OH^{−} are equal and fixed at $c1=10\u22124NA$ /m^{3} (i.e., 10^{(−7)} mol/l), where $NA=6.022\xd71023$ is the Avogadro constant. Taking $\upsilon \u223c10\u221228$ m^{3} as a representative volume, we have $\upsilon c1\u223c6\xd710\u22129$, which is a negligibly small portion of the total volume $(\u2211a\upsilon aca=1)$.

^{+}and Cl

^{−}ions are

^{+}and OH

^{−}ions are

which is same as Eq. (3.6) but with $ce=c0+c1$ as the effective ion concentration in the external solution. Moreover, the interaction parameter $\chi $ as given in Eq. (5.1) depends on both the volume fraction of acetone ($\phi $) in the mixed solvent and the volume fraction of polymer ($1/J$) in the gel. Similarly, the nominal stress is obtained in the same form as Eq. (4.3), which equals zero for free swelling. Therefore, for each external solution with given values of $\phi $ (acetone) and $c0$ (NaCl), the equilibrium stretch $\lambda $ can be determined by solving Eq. (4.3).

To compare with the experiment, two dimensionless parameters are needed for the gel, $\upsilon Cfix$ and $N\upsilon $, in addition to the interaction parameters ($\chi iw$, $\chi ia$, and $\chi iaw$). According to Ohmine and Tanaka [32], the concentration of the –COO^{−} group was about 0.1 mol/l in the initial state of the gel with $\lambda 0=201/3=2.7$ (the volume fraction of polymer was 0.05), which can be converted to $\upsilon Cfix\u223c0.1$ with $\upsilon \u223c10\u221228$ m^{3}. To determine the other parameters, we first consider the case with pure water as the solvent ($\phi =0$), with which we determine $N\upsilon $ and $\chi iw$ by fitting the theoretical prediction to the experimental data in Ref. [32]. Then, we consider the case with zero salt concentration ($c0=0$) but varying $\phi $, with which we determine $\chi ia$ and $\chi iaw$.

with which we obtain $\chi 0w\u2212\chi 1w\u223c0.1$. For simplicity, we ignore the effect of the gel composition on the solvent–polymer interaction by setting $\chi 1w=0$ so that $\chi 0w\u223c0.1$. With these parameters ($\upsilon Cfix=0.1$, $N\upsilon =1.06\xd710\u22123$, $\chi 0w=0.1$, and $\chi 1w=0$), the theoretical prediction by Eq. (4.3) is in reasonable agreement with the data for the volume swelling ratio ($\rho =\lambda 03/\lambda 3$) of the acrylamide gel in a NaCl solution with pure water solvent ($\phi =0$) and varying salt concentration from 10^{−7} to 1 mol/l, as shown in Fig. 7(a).

Next consider the case with zero salt concentration ($c0=0$) but varying $\phi $ for the water–acetone mixed solvent. In this case, the gel was highly swollen ($\lambda >7$) when $\phi <65%$ and then underwent a discontinuous transition to the collapsed phase ($\lambda <2$) for $\phi >65%$ [32]. As shown in Fig. 5, the critical value of $\chi $ for the discontinuous transition is ∼1.55 when $\upsilon Cfix=0.1$ and $c0=0$. Assuming $\chi iaw=0$ and $\chi 1a=0$ in Eq. (5.2), we obtain $\chi 0a=2.3$ so that $\chi =1.55$ for the mixed solvent with $\phi =65%$. As a result, the effective interaction parameter $\chi $ varies linearly from 0.1 to 2.3 with the volume fraction of acetone for the mixed solvent. By the phase transition diagram in Fig. 5(b), when $\upsilon Cfix=0.1$, a discontinuous volume phase transition is predicted for $0.98<\chi <1.55$, corresponding to $0.4<\phi <0.65$, in excellent agreement with the experiment (Fig. 7(a)).

Using the parameters determined above, we calculate the equilibrium stretch by Eq. (4.3) as a function of the NaCl concentration for mixed solvents with various volume fractions of acetone and compare directly with the experimental data by Ohmine and Tanaka [32] in terms of the relative volume ratio ($\rho =\lambda 03/\lambda 3$ with $\lambda 0=2.7$) in Fig. 7(a). The overall agreement is remarkable, considering the fact that only a few data points have been used to determine the parameters along with a few assumptions. However, a notable discrepancy between the theoretical and experimental results is for the highly swollen phase at relatively low salt concentrations. As predicted by Eq. (3.11), the theoretical result is independent of volume fraction of acetone at the limit of $c0\u21920$ and $\lambda \u226b1$ ($\phi <0.65$). In contrast, the experimental data at the same limit shows a slight variation of the relative volume ratio from 0.02 to 0.04 with $0<\phi <0.6$. This may be explained by the change of the fixed charge concentration ($\upsilon Cfix$) in the gel. It was noted that the ionizable group (–COOH) is a relatively weak acid and its dissociation in general depends on the dielectric constant of the solvent [37]. The relative dielectric constants of water and acetone at room temperature (∼25 °C) are 78 and 21, respectively. For the water–acetone mixed solvent, the effective dielectric constant would depend on the volume fraction of acetone. As a result, dissociation of –COOH would vary with the volume fraction of acetone, yielding different fixed charge concentrations. By Eq. (3.11), decreasing the fixed charge concentration ($\upsilon Cfix$) would decrease the equilibrium stretch and increase the relative volume ratio.

_{2}. The presence of divalent mobile ions (Mg

^{2+}) makes the calculation slightly more complicated. Deep in the external solution, the ion concentrations are different for Mg

^{2+}and Cl

^{−}, $c0$ and $2c0$, respectively. The concentrations of H

^{+}and OH

^{−}are still the same, $c1=10\u22127$ mol/l with

*p*H = 7. The chemical potential of the solvent is then

^{2+}and Cl

^{−}ions in the external solution, and they are

^{+}and OH

^{−}are identical:

^{2+}and Cl

^{−}ions in the gel are

^{+}and OH

^{−}ions in the gel remain the same as given in Eqs. (5.8) and (5.9). Again, the Donnan equilibrium is satisfied for all of the ion species. The condition of electroneutrality in the gel (far away from the interface) requires that

^{2+}). Two limiting cases may be considered. First, when $c0\u226ac1\u226aCfix$, Eq. (5.20) approaches the same form as Eq. (5.11) for $c0\u21920$. Second, when $c0\u223cc1\u226aCfix$, Eq. (5.20) is approximately

Thus, Eqs. (5.20) and (5.23) can be solved simultaneously to obtain $\gamma $ and $\lambda $ (or $J=\lambda 3$). Similar to the cases with monovalent ions, multiple solutions may exist when a discontinuous transition occurs. The energetic analysis or the Maxwell rule can then be used to determine the thermodynamically stable solution for the equilibrium state.

_{2}concentration for various volume fractions of acetone. To compare with the experimental data by Ohmine and Tanaka [32], the relative volume ratio ($\rho =\lambda 03/\lambda 3$ with $\lambda 0=2.7$) is shown in Fig. 7(b). In this case, the quantitative agreement is poor as illustrated by the data for $\phi =60%$, where the critical salt concentration for the discontinuous transition differs by several orders of magnitude. Such a discrepancy requires further studies. Nevertheless, the theoretical results are qualitatively consistent with the experiment as the critical salt concentration for the discontinuous transition when $\phi =40\u221260%$ is much lower for the divalent salt (MgCl

_{2}) than for the monovalent salt (NaCl). It is also consistent in which no discontinuous transition occurs for $\phi =0\u221220%$. The critical volume fraction of acetone for discontinuous transitions to occur was between 25% and 30% in the experiments, but slightly larger than 30% from the theoretical results. The discrepancy between the theoretical and experimental results for the highly swollen phase at relatively low salt concentrations remains, which may be partly resolved by considering different fixed charge concentrations as discussed for the case of NaCl. Interestingly, the theoretical results for $\phi \u226440%$ show a secondary continuous transition at a very low concentration of MgCl

_{2}($c0\u223c10\u221210$ mol/l), which did not exist for the case of NaCl. This may be explained theoretically by the two limiting cases for Eq. (5.20). When $c0\u226ac1\u226aCfix$ and $c0\u21920$, Eq. (5.20) approaches the same limit as Eq. (5.11) for the monovalent salt, and as a result, Eq. (5.23) approaches Eq. (3.11) for the highly swollen state with the same equilibrium stretch ($\lambda \u223c9.7$), independent of the salt. However, another limit exists for the case of MgCl

_{2}when $c0\u223cc1\u226aCfix$ and Eq. (5.21) can be used to simplify Eq. (5.23) to obtain

which is slightly different from Eq. (3.11). With $N\upsilon =1.06\xd710\u22123$ and $\upsilon Cfix=0.1$, we obtain $\lambda =6.9$ and thus $\rho =0.06$, in close agreement with the numerical results in Fig. 7(b) for $c0\u223c10\u22127$ mol/l and $\phi \u226440%$. Evidently, it is the presence of divalent ions (Mg^{2+}) in addition to monovalent ions in the polyelectrolyte gel and the external solution that leads to the two highly swollen states and the continuous transition. This limit was also noted by Ohmine and Tanaka [32] with an osmotic pressure due to ions similar to that in Eq. (5.24).

To summarize, the comparisons with the experiments in Fig. 7 show quantitative agreement for the monovalent salt (NaCl) and qualitative consistency for the divalent salt (MgCl_{2}), using a simple set of parameters for both cases. It is possible to further improve the quantitative agreement by changing the parameters for different acetone concentrations and different salts. For example, for NaCl in pure water ($\phi =0%$) as the solvent, an excellent agreement can be achieved (Fig. 8(a)) by increasing the concentration of fixed charge to $\upsilon Cfix=0.3$ and changing the other parameters accordingly: $N\upsilon =3.18\xd710\u22123$, $\chi 0w=0.1$, and $\chi 1w=0.6$. Similarly, an excellent agreement can be achieved (Fig. 8(b)) for NaCl in a mixed solvent ($\phi =60%$) by slightly increasing $N\upsilon $ so that the equilibrium stretch at $c0\u223c10\u22127$ mol/l is 8.3 instead of 9.7 by Eq. (3.11). As another example, for the case of MgCl_{2} in a mixed solvent with $\phi =60%$, the critical salt concentration for the discontinuous volume transition can be brought to the same level as the experiment (Fig. 8(b)) by changing the polymer–solvent interaction parameters to $\chi 0w=0.43$ and $\chi 0a=1.55$. As suggested by Safronov et al. [48], the interaction parameter may depend on the type of the counterions. Meanwhile, to fit the equilibrium stretch ($\lambda \u223c8.7$) at $c0\u223c10\u22127$ mol/l, we obtain $N\upsilon =0.66\xd710\u22123$ for $\upsilon Cfix=0.1$ by using Eq. (5.24) for the divalent salt instead of Eq. (3.11). The quantitative agreements in Fig. 8 demonstrate the capability of the present model for accurately describing both the continuous and discontinuous volume phase transitions of polyelectrolyte gels in response to changing salt concentration in the external solution.

## Summary

In this paper, we present a theoretical model of polyelectrolyte gels to study continuous and discontinuous volume phase transitions induced by changing salt concentration in the external solution. Phase diagrams are constructed in terms of the polymer–solvent interaction parameters, external salt concentration, and concentration of fixed charges. Direct comparisons with previous experiments for an ionized acrylamide gel in mixed water–acetone solvents show good quantitative agreement for a monovalent salt (NaCl) but fair qualitative agreement for a divalent salt (MgCl_{2}), using a simple set of parameters for both cases. The quantitative agreement may be further improved for both cases by calibrating the model parameters for a specific combination of salt and solvent composition. The theory in its present form naturally leads to a balance of osmotic pressures due to elasticity, solvent, and ions in an isotropic, homogeneous state, which can be readily extended to study anisotropic swelling (e.g., under constraint) and inhomogeneous phenomena (e.g., coexisting phases). Under the condition of electroneutrality, the ion concentrations in the gel satisfy the Donnan equilibrium with a Donnan potential that depends on the volume swelling ratio (or the volume fraction of polymer). The present model may be further extended to study volume phase transitions of polyelectrolyte gels in response to other stimuli such as temperature, *p*H, electrical fields, and pressure (or stress in general).

## Acknowledgment

The authors gratefully acknowledge the financial support of this work by National Science Foundation through Grant No. CMMI-1538658.