Phase separation in swelling and deswelling hydrogels with a free boundary

We present a full kinetic model of a hydrogel that undergoes phase separation during swelling and deswelling. The model accounts for the interfacial energy of coexisting phases, finite strain of the polymer network, and solvent transport across free boundaries. For the geometry of an initially dry layer bonded to a rigid substrate, the model predicts that forcing solvent into the gel at a fixed rate can induce a volume phase transition, which gives rise to coexisting phases with different degrees of swelling, in systems where this cannot occur in the free-swelling case. While a nonzero shear modulus assists in the propagation of the transition front separating these phases in the driven-swelling case, increasing it beyond a critical threshold suppresses its formation. Quenching a swollen hydrogel induces spinodal decomposition, which produces several highly localized, highly swollen phases which coarsen and are then ejected from free boundary. The wealth of dynamic scenarios of this system is discussed using phase-plane analysis and numerical solutions in a one-dimensional setting.

Figure 1

Schematic of the cross section of a laterally bounded gel film that is attached to a rigid substrate (shown here in the reference state). The gel can exchange solvent with a bath through its upper surface.

Figure 2

The phase diagram when $\mathcal{G}=0.01$ in terms of dimensionless quantities. The dashed line defined by $F=0$ determines the equilibrium solvent fraction for the free-swelling case as a function of $\chi$. For values of $({\varphi }_f,\chi )$ to the right of this curve, spatially homogeneous states have a positive chemical potential $\mu =F(C;\chi ,G)>0$, and negative to the left. The thick black line given by $F^{\prime }=dF/dC=0$ represents the spinodal curve and marks the boundary of the region where homogeneous states are linearly unstable (LU; shaded red). The thin black lines are defined by (49) and represent the binodal curves. The metastable region (M) is shaded in blue.

Figure 3

Numerical simulations of the full model (44) showing the onset and suppression of surface-induced phase separation for weak elasticity and strong elasticity, respectively. The dimensionless shear moduli are given by (a) $\mathcal{G}=0$, (b) 0.01, (c) 0.02, and (d) 0.05. The parameter values are $\chi =1$, $\omega ={10}^{-6}$, $\beta =0$, and $\mathcal{Q}=0.01$. The system has a miscibility gap when $\mathcal{G}\le 0.019$.

Figure 4

Evolution of the (a) position of the interfacial layer $\overline{Z}=\overline{S}\left(t\right)$ that separates solvent-rich and solvent-poor layers in the hydrogel and (b) the solvent volume fraction at the surface ($\overline{Z}=0$) and substrate ($\overline{Z}=1$). The position of the interfacial layer is implicitly defined in (50). The parameter values are $\chi =1$, $\omega ={10}^{-6}$, $\beta =0$, and $\mathcal{Q}=0.01$.

Figure 5

The effect of stopping the flux of solvent into the gel. Parameters and conditions are the same as for Fig. 3 except that the flux was stopped at $t=200$. In (a), the phase separation solvent profile settles into an equilibrium shape with a fixed front $\overline{S}\left(t\right)$. In (b), a constant chemical potential $\mu =0$ is imposed as a boundary condition at $x=0$ for $t\ge 200$. A second, depletion front forms as the liquid drains out of the gel into the bath at $x=0$.

Figure 6

Phase separation upon quenching a swollen hydrogel with a no-flux boundary condition imposed at the free surface $\overline{Z}=0$. The initial condition is a noisy homogeneous state calculated with $\chi =1$. Simulations are carried out at $\chi =1.82$ with $\omega ={10}^{-6}$, $\beta =0$, and $\mathcal{G}=0.01$. The initial exponential growth of the perturbations gives way to progressively slower coarsening resulting in a state with two spikes for which no further motion could be detected. Further details are given in the main text.

Figure 7

Phase separation upon quenching a swollen hydrogel with the boundary condition $\mu =0$ imposed at the free surface $\overline{Z}=0$. The initial condition is the homogeneous equilibrium solution at $\chi =1$ (without noise). Simulations are carried out using $\chi =1.82$ with $\omega ={10}^{-6}$, $\beta =0$, and $\mathcal{G}=0.01$. The penultimate image is not the final state, as the single peak eventually vanishes.

Figure 8

(a) Graphs of $F$ for $\chi =1$ and increasing values of $\mathcal{G}$. (b)–(e) Phase planes of (53) when (b) $\chi =1$, $\mathcal{G}=0.05$, $\mu =0$; (c)–(e) $\chi =1$, $\mathcal{G}=0.01$ with ${\mu }_1<\mu <\overline{\mu }$, $\mu =\overline{\mu }$, and $\overline{\mu }<\mu <{\mu }_2$, respectively. The values $\overline{\mu }$, ${\mu }_1$, and ${\mu }_2$ are explained in the text. Fixed points are emphasized by small circles, trajectories emanating from or asymptoting into fixed points are indicated by solid lines, all other trajectories by dashed lines.

Figure 9

Numerical simulation of a forced-swelling scenario that is stopped when $t=200$ after which a no-flux condition is imposed. The parameters are the same as in Fig. 6 except that $\omega ={10}^{-4}$ and $\gamma \left(C\right)={(1+C)}^{-4}$.