From density to interface fluctuations: The origin of wavelength dependence in surface tension

The height-height correlation function for a fluctuating interface between two coexisting bulk phases is derived by means of general equilibrium properties of the corresponding density-density correlation function. A wavelength-dependent surface tension $\gamma \left(\mathbf{q}\right)$ can be defined and expressed in terms of the direct correlation function $c(\mathbf{r},{\mathbf{r}}^{\prime })$, the equilibrium density profile ${\rho }_0\left(\mathbf{r}\right)$, and an operator which relates density to surface configurations. Neither the concept of an effective interface Hamiltonian nor the difference in pressure is needed to determine the general structure of the height-height correlations or $\gamma \left(\mathbf{q}\right)$, respectively. This result generalizes the Mecke-Dietrich surface tension ${\gamma }_{\mathsf{MD}}\left(q\right)$ [Phys. Rev. E 59, 6766 (1999)] and modifies recently published criticism concerning ${\gamma }_{\mathsf{MD}}\left(q\right)$ [Tarazona, Checa, and Chacón, Phys. Rev. Lett. 99, 196101 (2007)].

Figure 1

The random surface $\mathbf{s}$ is generated by random displacements $u\left(\mathbf{R}\right)$ of ${\mathbf{s}}_0\left(\mathbf{R}\right)$ along its normal vector ${\mathbf{n}}_0\left(\mathbf{R}\right)$ [see Eq. (1)]. For simplicity, the surface patch ${\mathbf{s}}_0$ is drawn spherically. For each spatial point $\mathbf{r}$ we write $d\equiv d\left(\mathbf{r}\right)$ for its (normal) distance to the equilibrium interface ${\mathbf{s}}_0$ [see Eq. (3)]. Random deviations of the equilibrium density ${\rho }_0\left(\mathbf{r}\right)$ are caused by the random displacements $u\left(\mathbf{R}\right)$ of the interface as well as by a local change in volume [see Eqs. (4, 6)].