Ensemble dependence of critical Casimir forces in films with Dirichlet boundary conditions

In a recent study [Phys. Rev. E 94, 022103 (2016)] it has been shown that, for a fluid film subject to critical adsorption, the resulting critical Casimir force (CCF) may significantly depend on the thermodynamic ensemble. Here we extend that study by considering fluid films within the so-called ordinary surface universality class. We focus on mean-field theory, within which the order parameter (OP) profile satisfies Dirichlet boundary conditions and produces a nontrivial CCF in the presence of external bulk fields or, respectively, a nonzero total order parameter within the film. Additionally, we study the influence of fluctuations by means of Monte Carlo simulations of the three-dimensional Ising model. We show that, in the canonical ensemble, i.e., when fixing the so-called total mass within the film, the CCF is repulsive for large absolute values of the total OP, instead of attractive as in the grand canonical ensemble. Based on the Landau-Ginzburg free energy, we furthermore obtain analytic expressions for the order parameter profiles and analyze the relation between the total mass in the film and the external bulk field.

Figure 1

The total mass as a function of the bulk field $B$ at the bulk critical temperature ($x=0$), determined from the (nonperturbative) numerical solution of the unconstrained ELE in Eq. (18).

Figure 2

OP profiles $m(\zeta ,x)$ across the film in the grand canonical ensemble obtained for zero scaled bulk field, $B=0$. The exact result from Eq. (36) (colored lines) is compared with the numerical solutions (black dashed lines) of the nonlinear MFT for several values of $x$ below the film critical point, i.e., in the range $x<x_c^f$ [see Eq. (34)].

Figure 3

The numerical solution $m\left(\zeta \right)$ (full curves) of the nonlinear MFT in the grand canonical ensemble is compared with $m_0\left(\zeta \right)$ [Eq. (23); dashed curves] for $x=0$ and $B\in \{2,4,6,8,10,12\}$. The discrepancy between the numerical and the truncated perturbative results grows as $B$ increases, and its maximum occurs at the midpoint $\zeta =1/2$ of the film.

Figure 4

The numerical solution $\tilde{m}\left(\zeta \right)$ (continuous curves) of the nonlinear MFT in the canonical ensemble is compared with ${\tilde{m}}_0\left(\zeta \right)$ [Eq. (26), dashed curves] for $x=0$ ($a$) and $x=x_c^f$ [see Eq. (34)] ($b$) for various values of the imposed mass $\mathcal{M}$.

Figure 5

Phase diagram and equation of state of a film with Dirichlet boundary conditions obtained numerically within nonlinear MFT. The color code indicates the value of the mass as a function of the scaled bulk field $B$ and the scaled reduced temperature $x$. Solid lines are iso-$\mathcal{M}$ lines for the film, while overlayed dashed lines correspond to the bulk system. The cross ($\times$) indicates the film critical point [$x=x_c^f=-{\pi }^2$, Eq. (34)] and the dot ($•$) the bulk critical point ($x=0$).

Figure 6

Mass $\mathcal{M}$ at the bulk critical point $x=0$ as a function of the scaled bulk field $B$. The solid blue straight line represents the lowest-order MFT result [Eq. (25)], while the black dots provide the numerical solution of the nonlinear MFT.

Figure 7

Mass $\mathcal{M}$ in a film with Dirichlet boundary conditions for $B=0$ as a function of $x$, as given by the exact MFT expression in Eq. (40) (red solid curve). The asymptotic behavior described in Eq. (43) (dashed curve) and in Eq. (44) (dotted curve) agree with the exact expression for $x=x_c^f\approx -9.87$ and for large values of $-x$, respectively.

Figure 8

(a) A section of the equation of state around the film critical point in the phase diagram. The critical isotherm corresponding to $t^f=0$ [see Eq. (46)] is shown by the dashed black curve, while symbols represent numerical data. (b) Test of the scaling hypothesis [Eq. (47)] for data contained in panel (a). The points corresponding to the “supercritical” regime ($t^f<0$) collapse onto the scaling function ${\mathcal{U}}_{-}$, while data for the “subcritical” regime ($t^f>0$) collapse onto ${\mathcal{U}}_{+}$.

Figure 9

The reduced scaling functions ${\mathrm{\Xi }}^{\left(\mathrm{gc}\right)}/{\mathrm{\Delta }}_0$ (a) and ${\mathrm{\Xi }}^{\left(\mathrm{c}\right)}/{\mathrm{\Delta }}_0$ (b) for the CCF in the grand canonical and canonical ensembles, respectively. Solid lines indicate the results obtained numerically within nonlinear MFT, while dashed lines show the analytical results of linear MFT as given in Eqs. (76) and (83). In both cases, the CCF is computed according to Eqs. (57) and (60) based on the stress tensor. For illustrative purposes, we have chosen two representative temperatures: $x=1$ (lower, red curves) and $x=20$ (upper, blue curves).

Figure 10

The reduced scaling functions ${\mathrm{\Xi }}^{\left(\mathrm{gc}\right)}/{\mathrm{\Delta }}_0$ (a) and ${\mathrm{\Xi }}^{\left(\mathrm{c}\right)}/{\mathrm{\Delta }}_0$ (b) for the CCF in the grand canonical and canonical ensemble, respectively, obtained numerically from nonlinear MFT as function of the scaled bulk field $B$ and the scaled temperature $x$. The cross ($\times$) indicates the film critical point [$x=x_c^f=-{\pi }^2$, Eq. (34)], and the dot ($•$) the bulk critical point ($x=0$). In (b), the cross and the dot are in white for better visibility. Note the different color codes.

Figure 11

The reduced scaling functions ${\mathrm{\Xi }}^{\left(\mathrm{gc}\right)}/{\mathrm{\Delta }}_0$ (a) and ${\mathrm{\Xi }}^{\left(\mathrm{c}\right)}/{\mathrm{\Delta }}_0$ (b) for the CCF in the grand canonical and canonical ensemble, respectively, obtained numerically from the nonlinear MFT as function of the scaled mass $\mathcal{M}$ and scaled temperature $x$. The cross ($\times$) indicates the film critical point [$x=x_c^f=-{\pi }^2$, Eq. (34)] and the dot ($•$) the bulk critical point ($x=0$).

Figure 12

MC simulation data (a). The magnetization $\phi$ per spin as a function of the bulk field $\mu$ for several values of the inverse temperature: $\beta =0.17,0.18,0.19048,0.20003,0.21185,0.22322,0.25657$. (b) These bulk magnetic fields ${\mu }_{\phi }$ as function of the inverse temperature $\beta$ which render the values $\phi =0.1,0.2,0.3,0.4$ of the mean magnetization in the grand canonical ensemble. The four values of $\phi$ considered in (b) are indicated in (a) by horizontal dotted lines. The value ${\mu }_{\phi }=0$ corresponds to $\phi =0$.

Figure 13

MC results for the CCF scaling function ${\mathrm{\Xi }}^{\left(\mathrm{gc}\right)}$ for the grand canonical ensemble as function of the temperature scaling variable $t{[L/{\xi }_{+}^{\left(0\right)}]}^{1/\nu }$ and the bulk magnetic field $\mu$. The cross ($\times$) indicates the film critical point of the Ising model [43].

Figure 14

MC results in the grand canonical ensemble. (a) Value ${\mu }_{\phi }$ of the bulk magnetic field, at which the mean magnetization takes the values $\phi =0.1,0.2,0.3,0.4$, as a function of the temperature scaling variable $t{[L/{\xi }_{+}^{\left(0\right)}]}^{1/\nu }$. In the considered temperature region, $\phi =0$ corresponds to ${\mu }_{\phi }=0$. (b) The scaling function ${\mathrm{\Xi }}^{\left(\mathrm{gc}\right)}$ of the grand canonical CCF as a function of the temperature scaling variable $t{[L/{\xi }_{+}^{\left(0\right)}]}^{1/\nu }$, computed along the lines $\{t{[L/{\xi }_{+}^{\left(0\right)}]}^{1/\nu },{\mu }_{\phi }\}$ in (a) for the mean magnetizations $\phi =0,0.1,0.2,0.3,0.4$.

Figure 15

MC results in the canonical ensemble. (a) Auxiliary function $g_C$ [Eq. (105)] as a function of the inverse temperature $\beta$ for $\phi =0.1,0.2,0.4$. (b) Scaling function ${\mathrm{\Xi }}^{\left(\mathrm{c}\right)}$ of the canonical CCF [Eq. (106)] as function of the scaled temperature $t{[L/{\xi }_{+}^{\left(0\right)}]}^{1/\nu }$ for three mean magnetizations $\phi =0,0.1,0.2,0.4$.