Phase separation in fluids exposed to spatially periodic external fields

When a fluid is confined within a spatially periodic external field, the liquid-vapor transition is replaced by a different transition called laser-induced condensation (LIC) [Götze et al., Mol. Phys. 101, 1651 (2003)]. In $d=3$ dimensions, the periodic field induces an additional phase, characterized by large density modulations along the field direction. At the triple point, all three phases (modulated, vapor, and liquid) coexist. At temperatures slightly above the triple point and for low (high) values of the chemical potential, two-phase coexistence between the modulated phase and the vapor (liquid) is observed; by increasing the temperature further, both coexistence regions terminate in critical points. In this paper, we reconsider LIC using the Ising model to resolve a number of open issues. To be specific, we (1) determine the universality class of the LIC critical points and elucidate the nature of the correlations along the field direction, (2) present a mean-field analysis to show how the LIC phase diagram changes as a function of the field wavelength and amplitude, (3) develop a simulation method by which the extremely low tension of the interface between modulated and vapor or liquid phase can be measured, (4) present a finite-size scaling analysis to accurately extract the LIC triple point from finite-size simulation data, and (5) consider the fate of LIC in $d=2$ dimensions.

Figure 1

LIC in the $d=3$ Ising model; plotted in each of the graphs is the free energy $F\left(m\right)$ in units of $k_{B}T$ (vertical axes) versus the magnetization $m$ (horizontal axes). The free energy curves present actual simulation data obtained for system size $L=10$ and $D=\lambda$. (a) Free energy for $J>J_{\mathrm{tr}}$ and $H=0$, i.e., above the triple point. A coexistence between two phases, I and II, is observed. (b) Free energy for $J=J_{\mathrm{tr}}$ and $H=0$, i.e., exactly at the triple point; three-phase coexistence is observed. (c) Free energy measured between the triple and critical points, $J_{\mathrm{cr}}<J<J_{\mathrm{tr}}$, but still using $H=0$. We now observe two common tangent lines: $l_1$ and $l_2$. By choosing $H={\Delta }_1$, where ${\Delta }_1$ is determined from the slope of $l_1$, coexistence between phases I and III can be induced (d). Similarly, from the slope of $l_2$, we obtain $H={\Delta }_2$, at which phases II and III coexist (e).

Figure 2

(a) Phase diagram of LIC for the $d=3$ Ising model. The phase diagram is a symmetric “pitchfork” featuring one triple point and two critical points (indicated by dots). The lines correspond to first-order phase transitions (between phases I and II, I and III, and II and III). (b) LIC in $d=2$ dimensions. In this case, there is no phase III. The phase diagram features only a single line of first-order phase transitions terminating in a critical point.

Figure 3

A sequence of magnetization profiles for varying $J$, going from $J=0.14$ (top) to $J=0.24$ (bottom), in increments of 0.01, obtained from the mean-field theory. For low values of $J$, one observes phase III, which is characterized by large modulations in the magnetization profile. For high values of $J$, phase II is observed, which is characterized by an overall low value of the magnetization and only small modulations. Note the discontinuous change in the profiles as the system crosses the II-III phase boundary (here at $J\approx 0.18–0.19$). The remaining parameters used in the figure are $H=0.03$, $h=0.075$, and $\lambda =10$.

Figure 4

LIC phase diagrams for $d=3$ obtained using mean-field theory (for clarity only the transition lines between the zebra III phase and the I and II phases are shown). In (a) we show results for $h=0.075$ and various values of the field wavelength $\lambda$. In (b) we show results for $\lambda =10$ and various values of the field amplitude $h$.

Figure 5

Finite-size-scaling analysis to locate the LIC triple point in $d=3$ dimensions for $h=0.075$ and $\lambda =10$. (a) Binder cumulant $U_4$ versus the coupling constant $J$ for various system sizes $L$. The curves strikingly intersect at $U_{4,\mathrm{triple}}=2/3$ (horizontal line) of a triple point; the value of $J$ at the intersection yields $J_{\mathrm{tr}}$. (b) The scaled free energy $F\left(m\right)/L$ precisely at the triple point, $J=J_{\mathrm{tr}}$, for various system sizes. Clearly visible are the three minima, corresponding to the coexisting phases, and a free energy barrier that increases linearly with $L$.

Figure 6

Finite-size-scaling analysis to locate the LIC critical point for the $d=3$ Ising model. (a) Cumulant $Q_4$ versus the coupling constant $J$ for various system sizes $L$; the intersection yields $J=J_{\mathrm{cr}}$. (b) Scaling plot of the order parameter $M$ where $t=J/J_{\mathrm{cr}}-1$; by using $J_{\mathrm{cr}}\approx 0.2531$ and $d=2$ Ising values for the critical exponents $\beta ,\nu$ the data for different $L$ collapse onto a single master curve.

Figure 7

Instantaneous magnetization profiles $m\left(z\right)$ obtained for $J=0.275$, $L=14$, and fixed overall magnetization $m=-0.426$. In (a) we observe a coexistence between two domains, while in (b) a coexistence between four domains is seen. The vertical lines indicate the approximate locations of the interfaces. By counting how often the arrangements (a) and (b) occur during a long simulation run, the interface tension can be determined; see Eq. (6).

Figure 8

(a) Variation of $\ln R$ with system size $L$ for $J=0.275$; symbols are simulation data, the curve is a fit to Eq. (6). Note that the data are consistent with ${\lim }_{L\to 0}\ln R=\ln 2$ of the entropy difference. (b) The interfacial tension $\gamma$, in units of $k_{B}T$ per squared lattice spacing, as a function of $J$. In agreement with theoretical expectations [18], $\gamma$ is extremely low and decreases as $J\to J_{\mathrm{cr}}$.

Figure 9

Investigation of the critical behavior using a simulation box with $D=5\lambda$; the simulations are performed at fixed magnetization $m=-0.426$ and $J=J_{\mathrm{cr}}$. (a) The susceptibility profiles $\chi \left(z\right)$ for $L=15,20,25,30$ (from bottom to top). The key point to note from this figure is that $\chi \left(z\right)$ diverges with $L$ only at special values $z=z_{\mathrm{cr}}$. (b) Finite-size-scaling analysis of the average peak height ${\chi }_L$ of the susceptibility profiles. We plot ${\chi }_L$ versus $L$ on double-logarithmic scales. The dashed line corresponds to a power law with exponent $\gamma /\nu =7/4$ of the $d=2$ Ising model.

Figure 10

The correlation function $C(z_{\mathrm{cr}},\Delta z)$ in the critical regime. We show results for $D=5\lambda ,L=30$ (a), and $D=20\lambda ,L=8$ (b). The vertical arrow in (a) marks the amplitude $A$ of the anticorrelations, which conforms to Eq. (8).

Figure 11

The analog of Fig. 6a but for the case $d=2$. Note the logarithmic vertical scale. The data are obtained using fixed $D=\lambda$. The key observation is that the curves for different $L$ do not intersect in a single point, implying the absence of a critical point. This, in turn, is consistent with $d=1$ Ising universality.

Figure 12

The analog of Fig. 5 but for the case $d=2$. The main difference is that we now observe a critical point, as opposed to a triple point. (a) The Binder cumulant as a function of $J$ for $H=0$ and different system sizes $L$. The curves for different $L$ intersect as in Fig. 5, from which one might conclude the presence of a triple point. (b) However, the scaling of the free energy $F\left(m\right)$ is not consistent with a triple point. Plotted is $F\left(m\right)$ at the cumulant intersection, with $F(m=0)$ shifted to zero. We see that the depth of the central minimum $\Delta F_2\to 0$ as $L$ increases, while the depth $\Delta F_1$ of the outer minima appears to be independent of $L$. This type of scaling is consistent with a critical point [28].

Figure 13

Free energy $F\left(m\right)$ for $d=2$ and $J=0.8>J_{\mathrm{cr}}$; the data are obtained using $D=2\lambda$, $H=0$, and two values of $L$ as indicated. The key thing to note from this figure is that the free energy barrier $\Delta F_1$ increases with $L$, indicating a first-order phase transition [28]. Note also the spurious minima $M_1$ and $M_2$: these reflect metastable coexistence states [18] whose role in the thermodynamic limit is negligible.

Figure 14

Computer-generated snapshots obtained using $D=10\lambda$ and $L=120$; white regions correspond to spin up. The snapshots in (a) and (b) are obtained for $J=0.8$ and $m=0$ and show early time and late time snapshots, respectively (the simulation was started with a random spin configuration). Since $J>J_{\mathrm{cr}}$ we observe coarsening of domains (a) until two large domains have formed (b). In (c), we show a snapshot for the case when $J=0.5<J_{\mathrm{cr}}$ and $m=0.52$. In this situation, domains do not coarsen with time, but remain finite in size, reminiscent of the $d=1$ Ising model.