Dynamic model of monovalent-divalent cation exchange in polyelectrolyte gels

A multicomponent model that imposes conservation laws and constitutive relations for polymer chains, water, and ions was investigated by determining the transient changes in a negatively charged gel exposed to a solution containing both mono- and divalent cations. Association of ion exchange with gel volume is achieved by imposing a linear relation between the polymer-solvent interaction parameter and concentration of divalent cations adsorbed onto the polymer chains. Semiquantitative agreement with measurements made on sodium polyacrylate gels is demonstrated in three aspects: (1) dynamics of gel swelling and deswelling, (2) ion partitioning coefficient, and (3) effect of cross-link density. These results imply that the multicomponent coarse-grained continuum modeling approach can be useful for quantitative predictions over macroscopic length and timescales including the description of volume transitions exhibited by these systems.

Figure 1

Polymer network volume fraction as a function of the 1D spatial coordinate $x$. (a) At initial state, the polymer charged groups are neutralized with sodium ions and the solvent contains 40 mM NaCl, 0.3 mM $\mathrm{Ca}{\mathrm{Cl}}_2$ at $\mathrm{pH}=7$. (b) Volume fraction profile of the polymer network at a nonequilibrium instance $\left(t/{\tau }_w=1.2\right)$. (c) Volume fraction of the polymer network profile at equilibrium $\left(t/{\tau }_w>20\right)$. Model parameters are listed in Sec. S3 of the Supplemental Material [38].

Figure 2

(a) Degree of swelling as a function of time for different initial $\mathrm{Ca}{\mathrm{Cl}}_2$ concentrations in the solution. Model parameters are listed in Sec. S3 of the Supplemental Material [38]. (b) Calculation of the equilibrium degree of swelling as a function of initial $\mathrm{Ca}{\mathrm{Cl}}_2$ in the salt solution. Gray area indicates the region of volume transition. (c) Measured data showing the degree of swelling of NaPA gels as a function of dimensionless time. (d) Equilibrium degree of swelling of NaPA gels measured after 5 days ($\frac{t}{{\tau }_w}=360$).

Figure 3

(a) Transient change of the Flory-like interaction parameter averaged within the gel interior (x/L<0.0125) as a function of time for multiple values of the initial calcium ion concentration in the external bath solution, $c_{\mathrm{Ca}}$. (b) Equilibrium values of the interaction parameter as a function of $c_{\mathrm{Ca}}$. Gray region indicates the region of volume transition. Dashed black line is a linear fit to data:$\overline{\chi }=2.9{\left[\mathrm{Ca}{\mathrm{Cl}}_2\right]}^2+2.34\left[\mathrm{Ca}{\mathrm{Cl}}_2\right]-11.48$.

Figure 4

(a) Partition of calcium ions as a function of time for different values of initial concentration of calcium in the external bath solution [color code similar to Figs. 2 and 3]. (b) Calcium ion partition as a function of $K_{\mathrm{Ca}}\left(=\frac{k_{\mathrm{Ca}}^{-}}{k_{\mathrm{Ca}}^{+}}\right)$ upon starting with 0.5 mM $\mathrm{Ca}{\mathrm{Cl}}_2$ in the external bath solution (other parameters are similar to previous graphs). (c) Calcium (circles) and sodium (triangles) partition at last time step of calculation (approximately “equilibrium”). (d) Ion partition coefficient measured for NaPA gels brought in equilibrium with an aqueous solution at $\mathrm{pH}=5.5$, containing 40 mM NaCl, and different concentrations of $\mathrm{Ca}{\mathrm{Cl}}_2$ [5].

Figure 5

(a) Model calculation of the degree of swelling as a function of initial calcium ion concentration in the external bath solution for different values of the elastic modulus $G_0$. $k_{\mathrm{Ca}}^{-}={10}^{-2}$ was used. Other parameters are the same as in Fig. 2. (b) Measurement of the degree of swelling as a function of calcium ion concentration in NaPA gels prepared with different cross-link densities. Average number of monomer units per cross-linker is 160, 320, 800, and 1600, respectively.