Rounding of Phase Transitions in Cylindrical Pores

Phase transitions of systems confined in long cylindrical pores (capillary condensation, wetting, crystallization, etc.) are intrinsically not sharply defined but rounded. The finite size of the cross section causes destruction of long range order along the pore axis by spontaneous nucleation of domain walls. This rounding is analyzed for two models (Ising or lattice gas and Asakura-Oosawa model for colloid-polymer mixtures) by Monte Carlo simulations and interpreted by a phenomenological theory. We show that characteristic differences between the behavior of pores of finite length and infinitely long pores occur. In pores of finite length a rounded transition occurs first, from phase coexistence between two states towards a multidomain configuration. A second transition to the axially homogeneous phase follows near pore criticality.

Figure 1

(a) Magnetization of Ising strips for $L=480$, $D=10$ plotted vs field $H$ at $T=1.5$ (■), 1.6 (▴), 1.7 (●), 1.8 (★), 1.9, and 2.0. Runs with decreasing $H$ are shown as full curves, with increasing $H$ as broken curves. A detailed analysis shows that the hysteresis disappears at $T_0=1.9$ in this case. (b) Distribution $P(M)$ vs $M$ for $H=0$ and $T=1.8–2.2$ from bottom to top at $M=0$. The inset is a typical snapshot at $T=2.1$ containing multiple domains stretched in the $y$ direction by a factor $\approx 4$.

Figure 2

(a) Correlation length $\xi$ (on a logarithmic scale) plotted vs $T$ for Ising strips of width $D=10$. Monte Carlo results (shown with error bars) were extracted for a system of $L=10000$, recording the wave-vector-dependent susceptibility $\chi (\overrightarrow{k})=N⟨|M(\overrightarrow{k}){|}^2⟩$ where $M(\overrightarrow{k})=\sum _{}^{}{S}_j\exp (i\overrightarrow{k}\times {\overrightarrow{r}}_j)$ for $\overrightarrow{k}$ oriented in the long direction and $k=k_{\min }=2\pi /L$, using the formula quoted in the text. Transfer matrix results were computed from the exact formula [Eq. (4.39) of [22] ] for the $D\times \infty$ system. The broken curve shows the approximation $\xi \approx \exp (D\sigma /T)$ where $\sigma$ is the exactly known interfacial tension of the Ising model. The value of $\xi$ at $T_c$ [21], ${\xi }_c=(4D/\pi )$, is shown as a dot. (b) Weight $W$ of the central peak for strips of width $D=10$ plotted vs temperature for $L=1000$, 480, 240, 120, 80, 60 from left to right. The symbols indicate $\xi =L$, $L/3$, and $L/9$, respectively. The inset shows a plot of $T_0(L,D)$ vs $L$ (note logarithmic scale) for two choices of $D$.

Figure 3

(a) Barrier $\Delta F/T$ against nucleation of interfaces in Ising strips plotted vs $T$. Several choices of $L$ and $D$ are shown, as indicated. (b) Barrier $\Delta F/T$ against nucleation of interfaces in the AO model confined to cylindrical pores of diameter $D=12$ plotted vs inverse polymer reservoir packing fraction $1/{\eta }_p^r$.

Figure 4

(a) Distribution $P({\eta }_c)$ of the number of colloids in a cylinder of diameter $D=12$ and length $L=180$ (all lengths are measured in units of the colloid radius) for ${\eta }_p^r=1.075$, 1.10, 1.115, 1.118, 1.20 from top to bottom at $⟨{\eta }_c⟩\approx 0.175$. Above the plot we show a typical snapshot (cross section through the cylinder) at ${\eta }_p^r=1.10$ containing multiple domains. (b) Phase diagram of the AO model in a cylindrical pore of diameter $D=12$ and lengths $L=30$, 60, and 180, as indicated. The full curve is the bulk coexistence curve [39]. Note that the points shown near $⟨{\eta }_c⟩\approx 0.16$ to 0.17 mark ${\eta }_{p0}^r(D,L)$ for three choices of $L$.