Morphological Thermodynamics of Fluids: Shape Dependence of Free Energies

We examine the dependence of a thermodynamic potential of a fluid on the geometry of its container. If motion invariance, continuity, and additivity of the potential are satisfied, only four morphometric measures are needed to describe fully the influence of an arbitrarily shaped container on the fluid. These three constraints can be understood as a more precise definition for the conventional term extensive and have as a consequence that the surface tension and other thermodynamic quantities contain, aside from a constant term, only contributions linear in the mean and Gaussian curvature of the container and not an infinite number of curvatures as generally assumed before. We verify this numerically in the entropic system of hard spheres bounded by a curved wall.

Figure 1

The expansion coefficients as defined in Eq. (1) of the grand potential of a hard-sphere fluid. For each value of $\eta$, the four thermodynamic coefficients can be used to calculate thermodynamic quantities for arbitrarily shaped systems. The solid line of the inset shows the relative error of $\gamma$ at a cylinder with radius $R_c$ calculated using the thermodynamic coefficients at $\eta =0.3$. This error can be compared to the numerical relative error for a sum rule [18, 19] (dashed line), which gives an estimate for the accuracy of our DFT data. Both errors are of the same order of magnitude such that our numerical data are in agreement with the prediction of Eq. (1). The fluid was modeled via Rosenfeld's FMT [5].

Figure 2

Curvature expansion coefficients for the contact density of a hard-sphere fluid as a function of the packing fraction $\eta$. Only the additive contributions ${\rho }_P^c$, ${\rho }_H^c$, and ${\rho }_K^c$ contribute. The inset shows the relative error for the contact density at a spherical container with radius $5R$ between a direct DFT calculation and two approaches: For the solid curve all shown curvature expansion coefficients are used, for the dashed one only additive contributions. The error, which is due to the numerical inaccuracies, is in both cases roughly the same. This shows that nonadditive terms do not contribute to the thermodynamic quantity ${\rho }^c$. Similar results were found for $\gamma$ (Fig. 1).

Figure 3

If the external potential of a hard container is perturbed by an additional potential as, e.g., $V_{\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}}(u)=V_0\exp (-u/\lambda )$ the contact density ${\rho }^c$ is no longer a thermodynamic quantity for $V_0\ne 0$. In this case nonadditive contributions such as ${\rho }_{H^2}^c$ are necessary to describe the dependence on the curvatures of the wall. Here we show ${\rho }_K^c$ and ${\rho }_{H^2}^c$ in a logarithmic plot for $\lambda =R$ and a packing fraction $\eta =0.3$. It is striking that even very small perturbations lead to ${\rho }_{H^2}^c\ne 0$ which is a clear indication that additivity of the contact density ${\rho }^c$ is destroyed. In contrast, ${\rho }_K^c$ contributes to the contact density for all $V_0$.