Theory of size dependent deliquescence of nanoparticles:: Relation to heterogeneous nucleation and comparison with experiments

In this paper, we develop a thermodynamic theory for the deliquescence behavior of soluble crystals in an atmosphere of solvent vapor. In this endeavor, we have focused on studying possible free energy barriers that could impede deliquescence. Our aim was to construct a theory general enough to treat both macroscopic and nanosized crystals. Toward this end, as a first attempt, we focused on a theory capable of describing the qualitative features of the results of recent experimental measurements, especially in the nanometer range where interfacial effects are bound to play a role. However, we have also opted for simplicity, and with this in mind, the surface thermodynamics that we have used are of the simplest type, ignoring crystal shape, rigorously defined dividing surfaces, curvature dependence of surface tension, and the presence of surface excess (adsorption). We do however include the effects of disjoining pressure. Nevertheless, we are able to describe several of the observed features and to calculate free energy surfaces traversed by the path of a deliquescing system. Analyses of these paths enable us to define two types of deliquescence, nucleate and activate, that occur respectively with and without a free energy barrier. A most important experimental behavioral feature that the theory cannot yet comfortably describe is the apparent existence, for nanosized and micron sized crystals, of ranges of vapor saturation ratio within which there is a continuum of deliquescent states such that a film of solution coexists in equilibrium with the core crystal. Within our thermodynamic theory, such coexistence can only be achieved using draconian measures such as the choice of interfacial tensions that have an unphysical behavior. Because, in the case of micron sized crystals, surface effects cannot be responsible for the coexistence of core and film, this together with the difficulty encountered in fashioning a thermodynamic theory, incorporating surface phenomena, that allows such coexistence, suggests that apparent nonprompt deliquescence must be due to some other factor such as the state of the initial core crystals. The measurements on small crystals involved (NH4)(2)SO4-H2O and NaCl-H2O systems and were performed using a tandem differential mobility analyzer. Aside from the failure to predict continuous deliquescence, our first results are promising, and a more sophisticated thermodynamic theory should provide a more thorough description of the observed features of deliquescence.