Theoretical Framework for Microscopic Osmotic Phenomena

TL;DR

This paper develops a theoretical framework to understand osmotic pressure in microscopic systems where classical laws may not apply, highlighting differences in pressure notions and providing numerical examples with polymers.

Contribution

It introduces a general theoretical approach to account for size and interaction effects on osmotic pressure at microscopic scales, extending classical theory.

Findings

Classical van't Hoff Law may not hold at microscopic scales.

Distinction between hydrostatic and mechanical pressure is necessary.

Numerical results for polymer confinement illustrate the framework.

Abstract

The basic ingredients of osmotic pressure are a solvent fluid with a soluble molecular species which is restricted to a chamber by a boundary which is permeable to the solvent fluid but impermeable to the solute molecules. For macroscopic systems at equilibrium, the osmotic pressure is given by the classical van't Hoff Law, which states that the pressure is proportional to the product of the temperature and the difference of the solute concentrations inside and outside the chamber. For microscopic systems the diameter of the chamber may be comparable to the length-scale associated with the solute-wall interactions or solute molecular interactions. In each of these cases, the assumptions underlying the classical van't Hoff Law may no longer hold. In this paper we develop a general theoretical framework which captures corrections to the classical theory for the osmotic pressure under more general relationships between the size of the chamber and the interaction length scales. We also show that notions of osmotic pressure based on the hydrostatic pressure of the fluid and the mechanical pressure on the bounding walls of the chamber must be distinguished for microscopic systems. To demonstrate how the theoretical framework can be applied, numerical results are presented for the osmotic pressure associated with a polymer of N monomers confined in a spherical chamber as the bond strength is varied.