Interfacial free energy of a hard-sphere fluid in contact with curved hard surfaces

Using molecular-dynamics simulation, we have calculated the interfacial free energy $\gamma$ between a hard-sphere fluid and hard spherical and cylindrical colloidal particles, as functions of the particle radius $R$ and the fluid packing fraction $\eta =\rho {\sigma }^3/6$, where $\rho$ and $\sigma$ are the number density and hard-sphere diameter, respectively. These results verify that Hadwiger's theorem from integral geometry, which predicts that $\gamma$ for a fluid at a surface, with certain restrictions, should be a linear combination of the average mean and Gaussian surface curvatures, is valid within the precision of the calculation for spherical and cylindrical surfaces up to $\eta \approx 0.42$. In addition, earlier results for $\gamma$ for this system [Bryk et al., Phys. Rev. E 68, 031602 (2003)] using a geometrically based classical density functional theory are in excellent agreement with the current simulation results for packing fractions in the range where Hadwiger's theorem is valid. However, above $\eta \approx 0.42$, $\gamma \left(R\right)$ shows significant deviations from the Hadwiger form indicating limitations to its use for high-density hard-sphere fluids. Using the results of this study together with Hadwiger's theorem allows one, in principle, to determine $\gamma$ for any sufficiently smooth surface immersed in a hard-sphere fluid.

Figure 1

Top: Interfacial free energy between a hard-sphere fluid and a hard cylindrical colloidal particle as a function of $\eta$ for several values of the radius $R$. Bottom: Same as the top panel except for the spherical surface. The inset shows the value of the excess volume $v_N$ for the spherical surface as a function of packing fraction for $R=\infty$, 5, 1, and 0.5. The symbols in this panel and in the inset are as indicated in the top panel legend.

Figure 2

The calculated values of $h$ and $\kappa$ for the spherical and cylindrical walls. Solid symbols are the results from the present MD simulations and the open symbols are the values calculated from DFT [27]. The solid and dashed lines are the values of $h$ and $\kappa$ from SPT [Eq. (4)]. Note that the symbols for the various values of $h$ are difficult to resolve from one another because they are nearly coincident.

Figure 3

Top: Plot of the first non-Hadwiger coefficient $a_2\left(\eta \right)$ in the weighted least-squares fit to a quadratic polynomial in 1/R of $\gamma$ for the hard-sphere fluid at a cylindrical colloidal particle. Bottom: Same as the top panel, but for $a_3\left(\eta \right)$ in the cubic polynomial fit of $\gamma$ at a spherical colloidal particle. For clarify, the insets show the data for $\eta <0.42$ on a smaller scale.