Modeling Gel Swelling Equilibrium in the Mean Field: From Explicit to Poisson-Boltzmann Models

We develop a double mean-field theory for charged macrogels immersed in electrolyte solutions in the spirit of the cell model approach. We first demonstrate that the equilibrium sampling of a single explicit coarse-grained charged polymer in a cell yields accurate predictions of the swelling equilibrium if the geometry is suitably chosen and all pressure contributions have been incorporated accurately. We then replace the explicit flexible chain by a suitably modeled penetrable charged rod that allows us to compute all pressure terms within the Poisson-Boltzmann approximation. This model, albeit computationally cheap, yields excellent predictions of swelling equilibria under varying chain length, polymer charge fraction, and external reservoir salt concentrations when compared to coarse-grained molecular dynamics simulations of charged macrogels. We present an extension of the model to the experimentally relevant cases of $p\mathrm{H}$-sensitive gels.

Figure 1

A schematic of the (a) macroscopic gel, (b) single-chain, and (c) PB model of a macroscopic gel in contact with a reservoir. The plot sketches a typical radial density profile.

Figure 2

Comparison between the three models. The equilibrium swelling length $R_{\mathrm{eq}}$ as a function of (a) the charge fraction $f$ along the gel polymer backbone for $c_{\mathrm{salt}}=0.01\mathrm{mol}{\mathrm{L}}^{-1}$ and $N\approx 80$; (b) the salt bath concentration $c_{\mathrm{salt}}$ for $f=0.5$ and $N\approx 60$. As can be seen in Fig. 3, the quality of the predictions of the single-chain model and the PB model for other parameter combinations are also very good.

Figure 3

The swelling equilibria of the single-chain cell model and the PB model compared to the periodic gel simulations. The results are compared for a wide exploration of in total 60 parameter combinations with $N\approx 40$, 60, 80, $f\in \{0.125,0.25,0.5,1\}$ and $c_{\mathrm{salt}}\in \{0.01,0.02,0.05,0.1,0.2\}\mathrm{mol}{\mathrm{L}}^{-1}$. The linear function has the form $y(x)=x$.

Figure 4

The swelling equilibria as a function of $p{K}_a-p\mathrm{H}$ for different salt concentrations $c_{\mathrm{salt}}\in \{0.01,0.1,0.2\}\mathrm{mol}{\mathrm{L}}^{-1}$ for $N=59$ and $K_a={10}^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$.