Micromechanical theory for pH-dependent polyelectrolyte multilayer capsule swelling

Hollow polyelectrolyte multilayer capsules made of ionizable segments ( e. g., carboxylic acid, amine, pyridine) have an equilibrium size that is independent of salt concentration and pH in a wide pH range but swell to a larger size at lower and higher pH. Besides, at the edges of the stability plateau their equilibrium size increases with increasing ionic strength. We set up a micromechanical theory in which we combine an expansive force due to ion entropy with an elastic contractive force and derive from balance of forces a simple expression for the equilibrium capsule size. The theory describes the influence of pH on size ( including the wide plateau region as well as the steep increase in size at the edge of the pH plateau) almost quantitatively without fitting parameters as well as the increase in size with increasing ionic strength. This effect is contrary to what would be expected when the polymer charge would directly depend on solution pH instead of local pH within the shell and is a phenomenon comparable to the osmotic regime found for ionizable ("annealed") polyelectrolyte brushes at very low ionic strength, for which it is also observed that the brush expands when the salt concentration is increased.