Colligative properties of solutions: II. Vanishing concentrations

TL;DR

This paper investigates the behavior of colligative properties at vanishing concentrations, revealing critical thresholds for phase separation and showing that freezing-point depression is primarily a surface phenomenon.

Contribution

It introduces a scaling framework for solutions at low concentrations and identifies critical parameters governing phase transitions and surface effects.

Findings

Existence of a critical concentration threshold for phase separation.

Phase separation occurs abruptly at a critical chemical potential.

Freezing-point depression is primarily a surface phenomenon.

Abstract

We continue our study of colligative properties of solutions initiated in math-ph/0407034. We focus on the situations where, in a system of linear size $L$, the concentration and the chemical potential scale like $c=\xi/L$ and $h=b/L$, respectively. We find that there exists a critical value $\xit$ such that no phase separation occurs for $\xi\le\xit$ while, for $\xi>\xit$, the two phases of the solvent coexist for an interval of values of $b$. Moreover, phase separation begins abruptly in the sense that a macroscopic fraction of the system suddenly freezes (or melts) forming a crystal (or droplet) of the complementary phase when $b$ reaches a critical value. For certain values of system parameters, under ``frozen'' boundary conditions, phase separation also ends abruptly in the sense that the equilibrium droplet grows continuously with increasing $b$ and then suddenly jumps in size to subsume the entire system. Our findings indicate that the onset of freezing-point depression is in fact a surface phenomenon.