 
Summary: Theoretical Framework for Microscopic Osmotic Phenomena
Theoretical Framework for Microscopic Osmotic Phenomena
Paul J. Atzberger
Department of Mathematics
University of California at Santa Barbara.
Peter R. Kramer
Department of Mathematical Sciences
Rensselaer Polytechnic Institute.
(Dated: April 19, 2007)
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 lengthscale associated
with the solutewall interactions or solute molecular interactions. In each of these cases, the assump
tions 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
