Theory of freezing point depression in charged porous media

Freezing in charged porous media can induce significant pressure and cause damage to tissues and functional materials. We formulate a thermodynamically consistent theory to model freezing phenomena inside charged heterogeneous porous space. Two regimes are distinguished: free ions in open pore space lead to negligible effects of freezing point depression and pressure. On the other hand, if nanofluidic salt trapping happens, subsequent ice formation is suppressed due to the high concentration of ions in the electrolyte. In this case our theory predicts that freezing starts at a significantly lower temperature compared to pure water. In one dimension, as the temperature goes even lower, ice continuously grows until the salt concentration reaches saturation, all ions precipitate to form salt crystals, and freezing completes. Enormous pressure can be generated if initial salt concentration is high before salt entrapment. We show modifications to the classical nucleation theory due to the trapped salt ions. Interestingly, although the freezing process is enormously changed by trapped salts, our analysis shows that the Gibbs-Thompson equation on confined melting point shift is not affected by the presence of the electrolyte.

Figure 1

Free ions vs nanofluidic salt trapping. (a) When the freezing pore is well connected to the reservoir, the ions move through the pathway and not constrained inside the pore. This case is referred to as “free” ions. (b) When the freezing pore is only connected via bottlenecks, geometric constraints make the solvated ions difficult to transport through the bottleneck. (c) When heterogeneous freezing happens and the freezing pore is blocked by ice formed in its larger neighbors, only a thin ($\lesssim \mathrm{nm}$ size) layer of lubricating liquid layer exists as channels to connect to the reservoir. Transport of the ions deviates from bulk transport behavior, and they will be hindered due to the highly charged surface of the narrow liquid channels. (d) The ion and water channels on a membrane can actively control these transport processes, hence controlling the salt concentration inside.

Figure 2

The free energy for the case of freezing with free ions, as a function of $r$ in a pore of radius $R=5$ nm, with surface charge density $q=1{\mathrm{nm}}^{-2}$ and bulk water permittivity. Legend subscripts denotes the total free energy ($F_{\mathrm{tot}}$), the electrolyte contribution ($F_{\mathrm{ele}}$), the ice freezing enthalpy contribution ($F_{\mathrm{solid}}$), and the surface tension term ($F_{\mathrm{surf}}$)m (a–c) at temperature $T=270$ K, (d–f) at $T=230$ K. (a, d) $d=1$, (b, e) $d=2$, and (c, f) $d=3$. Counterion recombination with the pore surface charge is neglected. The equilibrium solution is given by the global minimum of $F_{\mathrm{tot}}$ at $r^{*}$. If $r^{*}=0$ no freezing happens; otherwise $r^{*}>0$ freezing starts in the pore. For $d=1$ the pore almost freezes completely even at $T=272$ K, showing very minimal freezing point depression. For $d=2,3$ the competition between surface tension and freezing enthalpy overshadows the contribution from OCP. (a, d, e, f) Frozen pores and (b, c) unfrozen pores.

Figure 3

The free energy for the case of freezing with nanofluidic trapped ions as a function of $r$ in a pore of radius $R=5$ nm, with surface charge density $q=1{\mathrm{nm}}^{-2}$ and assuming saturated water permittivity $\epsilon =10$. Initial salt concentrations for all subplots are $c_0=1$ M. Legend subscripts denotes the total free energy ($F_{\mathrm{tot}}$), the electrolyte contribution ($F_{\mathrm{ele}}$), the ice freezing enthalpy contribution ($F_{\mathrm{solid}}$), and the surface tension term ($F_{\mathrm{surf}}$) (a, b, c) at temperature $T=270$ K, (d, e, f) at $T=250$ K. Counterion recombination with pore surface charge is neglected. The equilibrium solution is given by the global minimum of $F_{\mathrm{tot}}$ at $r^{*}$. If $r^{*}=0$ no freezing happens, otherwise $r^{*}>0$ freezing starts in the pore. The discontinuity in $F_{\mathrm{tot}}$ and $F_{\mathrm{ele}}$ happens at salt saturation. Beyond the saturation point $F_{\mathrm{tot}}$ has no contribution from $F_{\mathrm{ele}}$. (d, e, f) Frozen pores, (b, c) unfrozen pores. Notice that in (a), even though salt crystallization lead to a global minimum at the frozen state, the homogeneous nucleation barrier is sufficiently large that the unfrozen state is likely to be observed.

Figure 4

The salt concentration dependence of free energy. Similar to Fig. 3, here we show the case of freezing with nanofluidic trapped ions as a function of $r$ in a pore of radius $R=5$ nm, with surface charge density $q=1{\mathrm{nm}}^{-2}$ and assuming saturated water permittivity $\epsilon =10$. Initial salt concentrations are (a) $c_0=2$ M, (b) $c_0=3$ M, (c) $c_0=4$ M. Legend subscript denotes the total free energy ($F_{\mathrm{tot}}$), the electrolyte contribution ($F_{\mathrm{ele}}$), the ice freezing enthalpy contribution ($F_{\mathrm{solid}}$), and the surface tension term ($F_{\mathrm{surf}}$). All subplots are at $T=260$ K for $d=1$. Counterion recombination with pore surface charge is neglected. The equilibrium state is given by the minimum of $F_{\mathrm{tot}}$ before salt crystallization at $r^{*}$. If $r^{*}=0$ no freezing happens, otherwise $r^{*}>0$ freezing starts in the pore. Due to salt crystallization the global minima of (a–c) are all frozen states; however, the homogeneous nucleation barrier can be significant for the metastable states to be observed instead, in which case only (a) here is labeled as fully frozen.

Figure 5

The critical nuclei radius ($r_0$) and energy barrier for nucleation ($\mathrm{\Delta }{G}_0$) from classical nucleation theory, the simplified model [using Eq. (60)], and numerical solutions [based on Eq. (61)] for $d=2$ and $d=3$. The concentration is 1 mol/L salt, the surface charge density is 0.1 ${\mathrm{nm}}^{-2}$, and the pore radius is $R=5$ nm. The critical nuclei radius and the energy barrier are both increased in the presence of trapped salt. The numerical solution is truncated when trapped salt makes freezing thermodynamically unfavorable (neglecting saturation of salt).