Eyring–Polanyi Rate Theory for the Homogeneous Nucleation of Organic Crystals from Solution

Using elementary concepts of chemical reaction kinetics, the rates of primary homogeneous organic crystal nucleation from supersaturated solutions are modeled as nucleation with first order kinetics from large solute density fluctuations (LSDFs). Solute density fluctuations are modeled as diffusively driven many-body collisions of weakly interacting solvated solute molecules. The first order rate constant is a system-specific supersaturation-independent rate constant for nucleation in LSDFs. It is shown for several solute-solvent systems that the temperature-dependence of this nucleation rate constant exhibits Arrhenius behavior. The activation enthalpy (Δ<i>H</i> ^⧧) and activation entropy (Δ<i>S</i> ^⧧) for homogeneous nucleation is determined from an Eyring-Polanyi analysis of temperature-dependent nucleation rates. The steps of the Eyring-Polanyi analysis are described in detail for the homogeneous nucleation of l-histidine from aqueous solutions. The analysis is also applied to temperature-dependent homogeneous nucleation rates of salicylic acid in four different solvents. For all systems, the supersaturation- and the temperature-dependence of the primary homogeneous nucleation rates are completely reproduced by reference to temperature-dependent solubility data through the activation parameters Δ<i>H</i> ^⧧ and Δ<i>S</i> ^⧧. Δ<i>H</i> ^⧧ is for all examined systems approximately 12 times the enthalpy of solution determined from solubility data, suggesting that nucleation from LSDFs resembles, at the molecular level, a reverse dissolution process. Within the temperature ranges used for measuring nucleation rates, the Gibbs energy of activation Δ<i>G</i> ^⧧ does not vary strongly, resulting in an inverse correlation between enthalpies and entropies of activation. The Eyring-Polanyi framework thus provides, for the first time, a method for semiquantitative predictions of homogeneous nucleation rates from temperature-dependent solubilities.