Gerald Wilemski

On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion

Differential equations governing the time evolution of distribution functions for Brownian motion in the full phase space were first derived independently by Klein and Kramers. From these so-called Fokker-Planck equations one may derive the reduced differential equations in coordinate space known as Smoluchowski equations. Many such derivations have previously been reported, but these either involved unnecessary assumptions or approximations, or were performed incompletely. We employ an iterative reduction scheme, free of assumptions, and calculate formally exact corrections to the Smoluchowski equations for many-particle systems with and without hydrodynamic interaction, and for a single particle in an external field. In the absence of hydrodynamic interaction, the lowest order corrections have been expressed explicitly in terms of the coordinate space distribution function. An additional application of the method is made to the reduction of the stress tensor used in evaluating the intrinsic viscosity of particles in solution. Most of the present work is based on classical Brownian motion theory, but brief consideration is given in an appendix to some recent developments regarding non-Markovian equations for Brownian motion.

Binary nucleation kinetics. I. Self-consistent size distribution

Using the principle of detailed balance, we derive a new self‐consistency requirement, termed the kinetic product rule, relating the evaporation coefficients and equilibrium cluster distribution for a binary system. We use this result to demonstrate and resolve an inconsistency for an idealized Kelvin model of nucleation in a simple binary mixture. We next examine several common forms for the equilibrium distribution of binary clusters based on the capillarity approximation and ideal vapor behavior. We point out fundamental deficiencies for each expression. We also show that each distribution yields evaporation coefficients that formally satisfy the new kinetic product rule but are physically unsatisfactory because they depend on the monomer vapor concentrations. We then propose a new form of the binary distribution function that is free of the deficiencies of the previous functions except for its reliance on the capillarity approximation. This new self‐consistent classical (SCC) size distribution for bin...

The Kelvin equation and self-consistent nucleation theory

Issues of self‐consistency are reviewed for several unary equilibrium size distributions based on the capillarity approximation. Some apparent difficulties of interpretation are resolved. In terms of the kinetic approach to nucleation theory, the influence of self‐consistency on the nucleation rate is shown to arise entirely from differences in the dimer evaporation rates for nearly all versions of classical theory. The nucleation rate behavior of the Kelvin model is explored. In this model, the Kelvin equation is used to prescribe all cluster evaporation rates. Nucleation rates predicted by the Kelvin model are quantitatively similar to those of the self‐consistent classical (SCC) theory, but not to other simple versions of the classical theory. This behavior arises entirely from the relatively close coincidence of the SCC and Kelvin dimer evaporation rates. This means that, for the distribution‐based versions of classical theory, the SCC model is the closest analogue of the Kelvin model. Because the Kel...

Fabrication of low-density foam shells from resorcinol-formaldehyde aerogel

Resorcinol-formaldehyde (RF) aerogel chemistry has been used with encapsulation techniques to fabricate low-density, transparent, foam shells. To accomplish this, the gelation time was reduced from several hours to several minutes by the addition of acid following base-catalyzed RF particle growth. However, additional “annealing” of the gel for at least 20 h was needed to maximize crosslinking and minimize swelling in exchange solvents. Increasing the molar ratio of formaldehyde to resorcinol from 2 to 3 also helped to increase crosslinking. Densification of the foam shells due to dehydration during curing was greatly reduced by judicious choice of immiscible oil phases and by saturating the exterior oil phase during the annealing stage. Shells have been produced with diameters of about 2 mm, wall thicknesses ranging from 100 to 200 μm and foam densities approaching 50 mg/cc.

Binary nucleation kinetics. III. Transient behavior and time lags

Transient binary nucleation is more complex than unary because of the bidimensionality of the cluster formation kinetics. To investigate this problem qualitatively and quantitatively, we numerically solved the birth–death equations for vapor‐to‐liquid phase transitions. Our previous work [J. Chem. Phys 103, 1137 (1995)] showed that the customary saddle point and growth path approximations are almost always valid in steady state gas phase nucleation and only fail if the nucleated solution phase is significantly nonideal. The current work demonstrates that in its early transient stages, binary nucleation rarely, if ever, occurs via the saddle point. This affects not only the number of particles forming but their composition and may be important for nucleation in glasses and other condensed mixtures for which time scales are very long. Before reaching the state of saddle point nucleation, most binary systems pass through a temporary stage in which the region of maximum flux extends over a ridge on the free e...

Some issues of thermodynamic consistency in binary nucleation theory

The ‘‘generalized’’ Kelvin equations used to determine the critical nucleus composition in some binary nucleation theories are shown to be thermodynamically inconsistent. Also, the surface tension calculated by the method of Flageollet‐Daniel, Garnier, and Mirabel [J. Chem. Phys. 78, 2600 (1983)] is shown to be thermodynamically consistent with respect to the dynamic surface tension, contrary to the recent assertion of Spiegel, Zahoransky, and Wittig.

Experimental evidence for internal structure in aqueous–organic nanodroplets

The spatial distribution of species within an aerosol droplet influences how it interacts with its environment. Despite the ubiquity of multicomponent nanodroplets in natural and technological aerosols, there are no published measurements of their internal structure. Here, we report the first experimental results for structure in aqueous organic nanodroplets based on small angle neutron scattering by high number density aerosols. For H(2)O-n-butanol droplets, fitting of the diffraction patterns confirms the picture of an aqueous core containing approximately 3 mol% alcohol covered by a shell of densely packed alcohol molecules.

Nucleation near the spinodal: Limitations of mean field density functional theory

We investigate the diverging size of the critical nucleus near the spinodal using the gradient theory (GT) of van der Waals and Cahn and Hilliard and mean field density functional theory (MFDFT). As is well known, GT predicts that at the spinodal the free energy barrier to nucleation vanishes while the radius of the critical fluctuation diverges. We show numerically that the scaling behavior found by Cahn and Hilliard for these quantities holds quantitatively for both GT and MFDFT. We also show that the excess number of molecules Deltag satisfies Cahn-Hilliard scaling near the spinodal and is consistent with the nucleation theorem. From the latter result, it is clear that the divergence of Deltag is due to the divergence of the mean field isothermal compressibility of the fluid at the spinodal. Finally, we develop a Ginzburg criterion for the validity of the mean field scaling relations. For real fluids with short-range attractive interactions, the near-spinodal scaling behavior occurs in a fluctuation dominated regime for which the mean field theory is invalid. Based on the nucleation theorem and on Wangs treatment of fluctuations near the spinodal in polymer blends, we infer a finite size for the critical nucleus at the pseudospinodal identified by Wang.

Nucleation rates of water and heavy water using equations of state.

The original formula of Gibbs for the reversible work of critical nucleus formation is evaluated in three approximate ways for ordinary and heavy water. The least approximate way employs an equation of state to evaluate the pressure difference between the new and old phases. This form of the theory yields a temperature dependence for the nucleation rate close to that observed experimentally. This is a substantial improvement over the most commonly used (and most approximate) form of classical theory.

A structural phase diagram for model aqueous organic nanodroplets

Our density functional theory calculations predict that model aqueous organic nanodroplets have either well mixed or core-shell structures, depending on the state of the metastable binary vapor and that, furthermore, there is a broad transition region in the phase diagram where both structures can occur at the same vapor state.