Direct calculation of the critical Casimir force in a binary fluid

We show that critical Casimir effects can be accessed through direct simulation of a model binary fluid passing through the demixing transition. We work in the semi-grand-canonical ensemble, in slab geometry, in which the Casimir force appears as the excess of the generalized pressure, $P_{\perp }-n\mu$. The excesses of the perpendicular pressure, $P_{\perp }$, and of $n\mu$, are individually of much larger amplitude. A critical pressure anisotropy is observed between forces parallel and perpendicular to the confinement direction, which collapses onto a universal scaling function closely related to that of the critical Casimir force.

Figure 1

Typical configuration for the binary fluid in slab geometry. Particles of type A (blue) and type B (orange). The image is shown for density $n=0.7$ and dimensions $L_{\perp }=5$ and $L_{\left|\right|}=60$; see text.

Figure 2

(a) pressure anisotropy $P_{\perp }-P_{\left|\right|}$ in slab geometry with $L_{\perp }=5$ for $n=0.6,0.7$. Dashed lines mark the noncritical pressure anisotropy, estimated from the high and low pressure limits and using Eq. (8); see text. (b) pressure anisotropy and estimated noncritical pressure anisotropy for $L_{\perp }=6$. (c) pressure anisotropy for a one component Lennard-Jones fluid for $L_{\perp }=5$; see text. (d) finite-size scaling of the critical part of the pressure anisotropy for the systems in (a) and (b). Data are compared to an estimate of the universal scaling function $\theta +\mathrm{\Theta }$ computed from a simulation of an Ising model (solid line) with $L_{\perp }=9$ and 20 and $L_{\left|\right|}=60$ [19].

Figure 3

Excess part of the confinement pressure $P_{\text{ex},\perp }$ (symbols), and excess chemical potential multiplied by the density, $n{\mu }_{\text{ex}}$ (lines), for $L_{\perp }=5,6$ and $n=0.6$ (a) and $n=0.7$ (b).

Figure 4

(a, b) Symbols correspond to the excess part of the generalized pressure ${\tilde{P}}_{\text{ex},\perp }$ with $L_{\perp }=5,6$ and for $n=0.6$ (a) and $n=0.7$ (b). Dashed lines are estimates of the noncritical part of ${\tilde{P}}_{\text{ex},\perp }$. (c, d) Scaling of ${\tilde{P}}_{\text{ex},\perp }$ and the scaling function $\theta \left(x_t\right)$ obtained from Ising model simulation in Ref. [8] (full line).