Biskup M, Chayes L. and Kotecky R.

Date: 2015-10-07; view: 554.

Text 3. A proof of the Gibbs-Thomson formula in the droplet formation regime. The problem (fragment 2) (770 characters)

The description of equilibrium droplets for systems with coexisting phases is one of the outstanding achievements of classical thermodynamics. Standard treatments of the subject highlight various formulae relating the linear size of the droplet to a specific pressure difference. One of these, called the Gibbs-Thomson formula, concerns the difference between the actual pressure outside the droplet and the ambient pressure of the system without any droplets. (Or, in the terminology used in classical textbooks, «above a curved interface» and «above a planar interface», respectively.) The standard reasoning behind these formulae is based primarily on macroscopic concepts of pressure, surface tension, etc. But, notwithstanding their elegance and simplicity, these derivations do not offer much insight into the microscopic aspects of droplet equilibrium. The goal of the present paper is to give a self-contained derivation of the Gibbs-Thomson formula starting from the first principles of equilibrium statistical mechanics.

While straightforward on the level of macroscopic thermodynamics, an attempt for a microscopic theory of droplet equilibrium immediately reveals several technical problems. First of all, there is no obvious way ‑ in equilibrium ‑ to discuss finite-sized droplets that are immersed in an a priori infinite system. Indeed, the correct setting is the asymptotic behavior of finite systems that are scaling to infinity and that contain droplets whose size also scales to infinity (albeit, perhaps, at a different rate). Second, a statistical ensemble has to be produced whose typical configurations will feature an equilibrium droplet of a given linear size. A natural choice is the canonical ensemble with a tiny fraction of extra particles tuned so that a droplet of a given size is induced in the system. A difficulty here concerns the existence of a minimal droplet size as will be detailed below. Finally, for the specific problem at hand, the notions of pressure «above a curved interface» and «above a planar interface» have to be reformulated in terms of microscopic quantities which allow for a comparison of the difference between these pressures and the droplet size.

Some of these issues have been addressed by the present authors. Specifically, we studied the droplet formation/dissolution phenomena in the context of the canonical ensemble at parameters corresponding to phase coexistence and the particle density slightly exceeding the ambient limiting rarefied density. It was found that, if V is the volume of the system and _N is the particle excess, droplets form when the ratio (_N)(d+1) / d / V is of the order of unity. In particular, there exists a dimensionless parameter 1, proportional to the thermodynamic limit of the above ratio, and a non-trivial critical value 1c, such that, for 1 < 1c, all of the excess will be absorbed into the (Gaussian) fluctuations of the ambient gas, while if 1 > 1c, a mesoscopic droplet will form. Moreover, the droplet will only subsume a fraction _1 < 1 of the excess particles. This fraction gets smaller as 1 decreases to 1c, yet the minimum fraction _1c does not vanish. It is emphasized that these minimum sized droplets are a mesoscopic phenomenon: The linear size of the droplet will be proportional to V1 / (d+1) _ V1 / d and the droplet thus occupies a vanishing fraction of the system. Or, from another perspective, the total volume cannot be taking arbitrary large if there is to be a (fixed-size) droplet at all.

The droplet formation/dissolution phenomena have been the subject of intensive study in last few years. The fact that d / (d + 1) is the correct exponent for the scale on which droplets first appear. We note that the existence of a minimal droplet size seems to be ultimately related to the pressure difference «due» to the presence of a droplet as expressed by the Gibbs-Thomson formula. Indeed, from another perspective, the formation / dissolution phenomena can be understood on the basis of arguments in which the Gibbs-Thomson formula serves as a foundation. Finally, we remark that although the generation of droplets is an inherently dynamical phenomenon (beyond the reach of current methods) it is possible that, on limited temporal and spatial scales, the equilibrium asymptotics is of direct relevance.

The remainder of this paper is organized as follows. In the next subsection we will present an autonomous derivation of the Gibbs-Thomson formula based on first principles of statistical mechanics. Aside from our own (modest) appreciation of this approach, this is worthwhile in the present context because the rigorous analysis develops precisely along these lines. In the section, we will restrict our attention to the 2D Ising lattice gas, define explicitly the relevant quantities and present our rigorous claims in the form of mathematical theorems. The proofs will come in Section 3.