Implications of interface conventions for morphometric thermodynamics

Several model fluids in contact with planar, spherical, and cylindrical walls are investigated for small number densities within density functional theory. The dependence of the solid-fluid interfacial tension on the curvature of spherical and cylindrical walls is examined and compared with the corresponding expression derived within the framework of morphometric thermodynamics. Particular attention is paid to the implications of the choice of the interface location, which underlies the definition of the interfacial tension. We find that morphometric thermodynamics is never exact for the considered systems and that its quality as an approximation depends sensitively on the choice of the interface location.

Figure 1

The reference surface (solid line) at position $X$ is determined by the external potential. It consists of the set of all reachable locations of the centers of the fluid particles which are closest to the geometrical wall surface (dash-dotted line) at position $X_g$. Depending on the wall-fluid interaction potential, the geometrical wall surface and the reference surface can be distinct. The interface of area $\mathcal{A}$ (dashed line), with respect to which the interfacial tension $\gamma$ in Eq. (16) is defined, may differ from the reference surface. The parameter $\delta$ measures the offset of the interface with respect to the reference surface. This sketch refers to the cases of spherical or cylindrical walls for which $X\equiv R$. However, the concept involving the parameter $\delta$ is valid also for other geometries, in particular for a planar wall with $X\equiv z=0$. If the dashed line runs within the interior of the reference surface, $\delta$ is taken to be negative. The fluid volume $\mathcal{V}$, which refers to the set of all points being not closer to the geometrical wall surface than the interface, is shaded in gray.

Figure 2

The interfacial tension $\gamma$ of hard spheres with diameter $L$ for the convention $\delta /L=0$ as a function of the radius $R$ of the reference surface forming the boundary of a spherical (a) and a cylindrical (b) hard body. Polynomials (lines) as predicted by morphometric thermodynamics [up to order $1/{(R+\delta )}^2$ for spherical walls and up to order $1/(R+\delta )$ for cylindrical walls] have been fitted to the numerical data (dots). In panel (a) the fit shows good agreement with the data, whereas panel (b) clearly shows deviations from the predicted behavior. This means that for the cylindrical wall the curvature expansion of $\gamma$ Eq. (3) does not terminate after the first-order term, in contradiction to the prediction from morphometric thermodynamics. The hard spheres interact among each other via the square-shoulder potential $U\left(r\right)$ in Eq. (4) with $\beta U_0=\infty$. The chemical potential is chosen as $\beta \mu =-2.768839$, which corresponds to the low packing fraction $\eta =\frac{\pi }{6}{\varrho }_{\text{eq}}^{\text{bulk}}{L}^3\approx 0.02656$.

Figure 3

Comparison of the simulation results of Ref. [4] (squares $\square )$ with the corresponding data of the present DFT study (crosses $\times )$ for the packing fraction $\eta \approx 0.02656$. The latter have been obtained by translating the data displayed in Fig. 2 (for $\delta =0)$ into the convention $\delta /L=-0.5$ by using Eq. (17). In contrast to Fig. 2, the plot of the interfacial tension in the case of a cylindrical wall (b) almost coincides with a straight line.

Figure 4

Dependence of the interfacial tension $\gamma$ for hard spheres at a cylindrical hard wall on the radius $R$ of the reference surface and on the shift parameter $\delta$. The contribution of the planar wall ${\gamma }_0\left(\delta \right)$, which corresponds to the limit $R\to \infty$, has been subtracted. The numerical data (red dots) are obtained within the convention $\delta /L=0$ (a) and translated to the convention $\delta /L=-0.5$ (b) via Eq. (17). The three straight lines represent the terms $n=1,2,3$ of the curvature expansion in Eq. (18). In the case $\delta /L=0$ the coefficient ${\gamma }_2\left(\delta \right)$ is the largest one; i.e., for small $R$ the contribution $\beta {\gamma }_2\left(\delta \right)L^4/{(R+\delta )}^2$ is the dominant one so that a deviation from the leading behavior $\sim 1/(R+\delta )$ becomes obvious for $R/L\lesssim 3$. Within the convention $\delta /L=-0.5$ the leading coefficient ${\gamma }_1\left(\delta \right)$ is the largest one, i.e., the leading term $\beta {\gamma }_1\left(\delta \right)L^3/(R+\delta )$ is the dominant one. Here $U\left(r\right)$ is given by Eq. (4) with $U_0\to \infty$ and $\beta \mu =-3.88$ so that $\eta \approx 0.01$. Note that at $(R+\delta )/L=1$ the plots render the values of the dimensionless coefficients $\beta {\gamma }_n\left(\delta \right)L^2$.

Figure 5

Reduced coefficients ${\gamma }_n\left(\delta \right),n\in \{0,1,2,3\},$ characterizing the curvature expansion in Eq. (18) as function of the shift parameter $\delta$ and as obtained from Eq. (25). The data correspond to a hard sphere fluid at $\beta \mu =-3.88$, so that $\eta \approx 0.01$, exposed to hard spherical (a) or cylindrical (b) walls.

Figure 6

Reduced coefficients ${\gamma }_n\left(\delta \right),n\in \{0,1,2,3\},$ characterizing the curvature expansion in Eq. (18) as function of the shift parameter $\delta$ and as obtained from Eq. (25). The data correspond to a fluid with a square-shoulder pair potential $U\left(r\right)$ [Eq. (4)] with $\beta U_0=0.1$ and $\beta \mu =-3.95$ so that $\eta \approx 0.01$. The fluid is exposed to hard spheres (a) or hard cylinders (b).

Figure 7

Same as Fig. 6 for a fluid with a Yukawa pair potential [Eq. (5)] with $\beta U_0=0.1$, $\beta \mu =-3.94$, and $L_c/L=5$ so that $\eta \approx 0.01$.

Figure 8

Same as Fig. 7 for a fluid with a Lennard-Jones pair potential [Eq. (6)] with $\beta U_0=0.1$, $\beta \mu =-3.92$, and $L_c/L=5$ so that $\eta \approx 0.01$.

Figure 9

Reduced coefficients ${\gamma }_n\left(\delta \right),n\in \{0,1,2,3\},$ characterizing the curvature expansion in Eq. (18) as function of the shift parameter $\delta$ and as obtained from Eq. (25). Panels (a), (b), and (c) belong to the same systems as in Figs. 6, 7, and 8, respectively, with an additional soft part $V$ of the external substrate potential: (a) $V$ given by Eq. (B1), (b) $V$ given by Eq. (B2), and (c) $V$ given by Eq. (B4). The data correspond to the following choice of system parameters: $\beta V_p^{sq}=\beta V_p^{yu}=\beta V_p^{LJ}=0.1$ (see Appendix pp2), $L_w/L=1$, and $L_{cw}/L=5$.