Curvature corrections to the surface tension of fluid drops: Landau theory and a scaling hypothesis

Abstract

The leading curvature correction $\ensuremath{\delta}$ to the surface tension of a large spherical fluid drop at equilibrium is defined by $\ensuremath{\sigma}={\ensuremath{\sigma}}_{\ensuremath{\infty}}(\frac{1\ensuremath{-}2\ensuremath{\delta}}{R}+\ensuremath{\cdots}$, where $R$ is the drop radius. When the two coexisting phases are symmetric, $\ensuremath{\delta}$ is shown to vanish exactly at all temperatures $T$. By using Landau theory and expanding in powers of $\frac{1}{R}$, we derive an explicit expression for $\ensuremath{\delta}$ in terms of the planar interfacial profile. In the presence of asymmetry and within Landau theory, $\ensuremath{\delta}(T)$ approaches a constant value ${\ensuremath{\delta}}_{c}$ (of the order of atomic dimensions), as $T\ensuremath{\rightarrow}{T}_{c}^{\ensuremath{-}}$. We propose a scaling hypothesis for the exact critical behavior of $\ensuremath{\delta}(T)$ beyond Landau theory, when fluctuations are important. There are two terms: One arises from field mixing and goes as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}}$, depending on bulk relevant exponents only [$t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$]; the other arises from true, non-Ising asymmetry and goes as ${|t|}^{{\ensuremath{\theta}}_{5}\ensuremath{-}\ensuremath{\nu}}$, where ${\ensuremath{\theta}}_{5}$ is the leading asymmetric correction-to-scaling exponent. Observation of the critical behavior of $\ensuremath{\delta}(T)$ would provide information about the irrelevant exponent ${\ensuremath{\theta}}_{5}$.