Nanoparticle collisions in the gas phase in the presence of singular contact potentials

Collisional growth and ionization is commonplace for gas phase nanoparticles (i.e., in aerosols). Nanoparticle collisions in atmospheric pressure environments occur in the mass transfer transition regime, and further attractive singular contact potentials (which arise when modeling nanoparticles as condensed matter and for which the potential energy approaches -∞ when two entities contact) often have a non-negligible influence on collision processes. For these reasons collision rate calculations for nanoparticles in the gas phase are not straightforward. We use mean first passage time calculations to develop a simple relationship to determine the collision rate in the gas phase, accounting for the influences of both the transition regime and singular contact potentials (specifically the non-retarded van der Waals and image potentials). In the presented analysis, methods to determine the degree of enhancement in collision rate due to attractive singular potentials in the continuum (diffusive) regime, η(C), and the degree of enhancement in the free molecular (ballistic) regime, η(FM), are first reviewed. Accounting for these enhancement factors, with mean first passage time calculations it is found that the collision rate for gas phase nanoparticles with other gas phase entities can be determined from a relationship between the dimensionless collision rate coefficient, H, and the diffusive Knudsen number, Kn(D), i.e., the ratio of the mean collision persistence distance to the collision length scale. This coincides with the H(Kn(D)) relationship found to appropriately describe collisions between entities interacting via a hard-sphere potential, but with η(C) and η(FM) incorporated into the definitions of both H and Kn(D), respectively. The H(Kn(D)) relationship is compared to the predictions of flux matching theory, used prevalently in prior work for collision rate calculation, and through this comparison it is found that at high potential energy to thermal energy ratios, flux matching theory predictions underestimate the true collision rate. Finally, a series of experimental measurements of nanoparticle-nanoparticle collision rates are compared to the determined H(Kn(D)) expression, considering that nanoparticles interact via non-retarded van der Waals potentials. Very good agreement is found with collision rates inferred from experiments, with almost all measured values from four separate studies within 25% of model predictions.