Interfaces

Abstract

As has been analyzed in Chap. 9, in a discontinuous phase transition two phases of different average density may coexist in equilibrium. In the various examples of coexistence considered in Sects. 9.1 and 9.2, the average densities of the phases are uniform, i.e., the conditions of mechanical equilibrium and chemical equilibrium at a given temperature have been derived from the free energy of each bulk phase (in the thermodynamic limit), and so the density at coexistence changes discontinuously from one phase to the other. For instance, in the liquid–vapor transition the densities at coexistence are uniform, $\rho_{L} (T)$ and $\rho_{V} (T)$, according to (9.18). This is only an approximation since, as may be observed experimentally, there are two kinds of surface effects to be considered. The first one is the effect of the walls of the container, which is not specific to the system and which gives rise to the well-known meniscus effects, which may be neglected at points far away from the wall. The second effect is that, due to the gravitational field, the denser phase (the liquid) occupies the lower part of the container, the less dense phase (the vapor) the upper part, and the local equilibrium density $\rho_{1} ({\mathbf{r}}) = \rho_{1} (z)$ will change continuously (albeit abruptly) from one phase to the other. This profile and the discontinuous profile of Chap. 9 are shown schematically in Fig. 11.1 Note that for the continuous profile $\rho_{1} (z) \simeq \rho_{L} (T)\left( {z < z_{0} } \right),\rho_{1} (z) \simeq \rho_{V} (T)\left( {z > z_{0} } \right)$ and in the interface $\left( {z \simeq z_{0} } \right)$ the gradient of the local density is $\rho^{\prime}_{1} (z) = d\rho_{1} (z) /dz \ne 0$. The consequences of the existence of this transition region or interface between the two coexisting phases are studied in this chapter.