Binary nucleation kinetics. II. Numerical solution of the birth–death equations

Abstract

We numerically solve the complete set of coupled differential equations describing transient binary nucleation kinetics for vapor-to-liquid phase transitions. We investigate binary systems displaying both positive and negative deviations from ideality in the liquid phase and obtain numerical solutions over a wide range of relative rates of monomer impingement. We emphasize systems and conditions that either have been or can be investigated experimentally. In almost every case, we find behavior consistent with Stauffer’s idea that the major particle flux passes through the saddle point with an orientation angle that depends on the rates of monomer impingement. When this is true, the exact numerical steady state nucleation rates are within 10%–20% of the predictions of Stauffer’s analytical theory. The predictions of Reiss’ saddle point theory also agree with the numerical results over a wide range of relative monomer impingement rates as long as the equilibrium vapor pressures of the two pure components are similar, but Stauffer’s theory is more generally valid. For systems with strong positive deviations from ideality, we find that the saddle point approximation can occasionally fail for vapor compositions that put the system on the verge of partial liquid phase miscibility. When this situation occurs for comparable monomer impingement rates, we show that the saddle point approximation can be rescued by evaluating an appropriately modified nucleation rate expression. When the two impingement rates differ significantly, however, the major particle flux may bypass the saddle point and cross a low ridge on the free energy surface. Even in these rare cases, the analytical saddle point result underpredicts the numerical result by less than a factor of 10. Finally, we study the transition from binary to unary nucleation by progressively lowering the vapor concentration of one component. Both Reiss’ and Stauffer’s rate expressions fail under these conditions, but our modified rate prescription remains within 10%–20% of the exact numerical rate.