B. Nowakowski

A kinetic theory of nucleation in liquids

Abstract A kinetic theory of nucelation is developed in which the rate of dissociation of the nuclei is calculated on the basis of a mean first passage time analysis. In contrast to the classical theory of nucleation, the theory does not employ the macroscopic concept of interfacial tension and the principle of detailed balance. It involves, instead, the interaction between a molecule and a cluster. An equation is derived for the critical radius, which in the limit of large clusters yields the Kelvin equation as well as an expression for the macroscopic surface tension. Numerical calculations assuming an interaction potential between two molecules which combines the dispersive attraction with the rigid core repulsion are performed. The predictions of the theory are consistent with the classical results in the limit of large critical clusters, i.e., for small supersaturations. For small critical clusters the present theory provides much higher rates of nucleation than the classical one. This difference can be attributed to the fact that the classical theory uses the macroscopic interfacial tension, which overpredices the surface energy of small clusters, and consequently provides lower values for the nucelation rate. The present paper extends the approach initiated in the previous paper ( G. Narsimhan and E. Ruckenstein, J. Colloid Interface Sci. 128, 549, 1989 ) to a more realistic interaction potential.

A kinetic approach to the theory of nucleation in gases

A theory of nucleation in gases is developed in which the rate of evaporation of the molecules from the cluster is calculated on the basis of a diffusion equation in the energy space. In contrast to the classical theory of nucleation, the theory does not employ macroscopic thermodynamics. Numerical calculations were performed by assuming an intermolecular potential which combines the dispersive attraction with the rigid core repulsion. The predictions of the theory are consistent with the classical theory in the limit of small supersaturations, i.e., for large critical clusters. For small critical clusters the present theory provides much higher rates of nucleation than the classical one. This difference can be attributed to the use of the macroscopic surface tension in the classical theory, which overpredicts the surface energy of small clusters, and consequently provides lower values for the nucleation rate. The theoretical results are compared with experimental data for nucleation of supersaturated vap...

Homogeneous nucleation in gases : a three-dimensional Fokker-Planck equation for evaporation from clusters

A theory of homogeneous nucleation in gases is developed, based on independently obtained expressions for the rates of evaporation from and condensation on the clusters. The rate of evaporation was calculated using a diffusion equation in the energy space, derived from the three‐dimensional Fokker–Planck equation averaged with respect to the distribution of the angular variables. The obtained rates of nucleation are compared with those obtained previously by the authors on the basis of a diffusion equation in the energy space and a unidimensional Fokker–Planck equation, as well as with the predictions of the classical theory. The present approach predicts rates of nucleation which are closer to the classical theory than those provided by the former approaches. Nevertheless, the difference between the present approach and the classical result is significant for small clusters, i.e., for high supersaturations.

Brownian coagulation of aerosol particles by Monte Carlo simulation

Abstract A Monte Carlo method of simulation of Brownian coagulation of spherical aerosol particles in a bath gas is presented. Brownian motion of the particles is followed within “a boundary distance” which is of an order of the apparent mean free path of the particles. Outside this distance diffusion according to the solution of the Fokker-Planck equation is assumed. A conjunction of both the fluxes of coagulating particles, simulated and calculated by use of the analytical expression, at the boundary distance is applied to obtain the resulting coagulation constant. Results of the simulation of the coagulation of diethylhexyl sebacate droplets in nitrogen are also presented.

Particle dynamics simulations of Turing patterns

The direct simulation Monte Carlo method is used to reproduce Turing patterns at the microscopic level in reaction-diffusion systems. In order to satisfy the basic condition for the development of such a spatial structure, we propose a model involving a solvent, which allows for disparate diffusivities of individual reactive species. One-dimensional structures are simulated in systems of various lengths. Simulation results agree with the macroscopic predictions obtained by integration of the reaction-diffusion equations. Additional effects due to internal fluctuations are observed, such as temporal transitions between structures of different wavelengths in a confined system. For a structure developing behind a propagating wave front, the fluctuations suppress the induction period and accelerate the formation of the Turing pattern. These results support the ability of reaction-diffusion models to robustly reproduce axial segmentation including the formation of early vertebrae or somites in noisy biological environments.

Nonequilibrium molecular velocity distribution in binary reactive gaseous mixture

The Boltzmann equation for reacting and diffusing components of binary mixture is solved by means of the perturbative Chapman–Enskog method. It is assumed that all molecules have identical mechanical properties and the system is in mechanical equilibrium. Typical reaction schemes consisting of bimolecular processes only are considered. Corrections of two orders to the distribution functions perturbed by the chemical reactions are calculated. The first-order correction gives a modification of the reaction rate in the reaction-diffusion equation for concentrations. The correction to the diffusion coefficient is obtained in the second-order approximation. Moreover, the solution at this order yields a term proportional to the square of the concentration gradient. These corrections are calculated for the molecular model of reactive hard spheres. For low activation energies the magnitude of the relative correction to the diffusion coefficient can be larger than that to the reaction rate.

Molecular-dynamics simulations and master-equation description of a chemical wave front: Effects of density and size of reaction zone on propagation speed

We compare the master-equation description and molecular-dynamics simulations of a chemical wave front. We find that the front propagation speed depends on the number of particles in the reaction zone. For the master equation the dependence follows the well-known logarithmic prediction obtained when introducing a cutoff into the macroscopic reaction-diffusion equation. The molecular-dynamics simulations reveal that the logarithmic law is compromised for dense fluids.

Reaction-diffusion approach to prevertebrae formation: Effect of a local source of morphogen

Periodic structure formation is an essential feature of embryonic development. Many models of this phenomenon, most of them based on time oscillations, have been proposed. However, temporal oscillations are not always observed during development and how a spatial periodic structure is formed still remains under question. We investigate a reaction-diffusion model, in which a Turing pattern develops without temporal oscillations, to assess its ability to account for the formation of prevertebrae. We propose a correspondence between the species of the reaction scheme and biologically relevant molecules known as morphogens. It is shown that the model satisfactorily reproduces experiments involving grafting of morphogen sources into the embryos. Using a master equation approach and the direct simulation Monte Carlo method, we examine the robustness of the results to internal fluctuations.

Time-dependent cluster distribution in nucleation

Abstract The time-dependent size distribution function of clusters, as well as the relaxation and lag times, are calculated numerically in the framework of the recently developed kinetic theories of nucleation in liquids (E. Ruckenstein and B. Nowakowski, J. Colloid Interface Sci.137, 583 (1990)), and gases (B. Nowakowski and E. Ruckenstein, J. Chem. Phys.94, 1397 (1991)). The basic approach used in the calculation is the same as that employed for the classical theory. Differences in the treatment occur as a result of the fact that, in contrast to the classical theory, the rates of condensation and evaporation are calculated independently. The size distribution function of clusters approaches monotonically the steadystate distribution. The nucleation rate relaxes monotonically to the steady-state value for clusters greater than the critical cluster x ∗ ; for x ∗ the nucleation rate exhibits an initial overshoot that exceeds the stationary value and then tapers to the steady-state rate from above. The relaxation times calculated for typical systems are of the order of 1 μs both in liquids and gases. The time lags were found to be of the order of, but larger than, the relaxation time and were found to increase with the size of the cluster. In contrast to the steady-state nucleation rates, which differ by orders of magnitude, the relaxation and lag times calculated in the framework of the kinetic theories are comparable to those obtained on the basis of the classical theory.

Effect of chemical reaction on diffusion of diluted gas: Simulations by means of two Monte Carlo methods

Two Monte Carlo methods are used to simulate a dynamics of molecules of a foreign gas A, highly diluted in a carrier gas C. Diffusion in the presence of bimolecular reaction A+C→products is studied. Nonequilibrium corrections to the reaction rate constant and diffusion coefficient are calculated for a wide range of molecular mass ratio mA/mC and activation energies of the reaction. Some differences between simulation results of the two methods are observed. The deviation of the velocity distribution from the Maxwellian form is quantified by means of the fourth order cumulants. The nonequilibrium effects are most significant in the Lorentz range, that is for mA≪mC. The simulation results prove that the theoretical predictions based on a perturbative solution of the Boltzmann equation are valid for not too small values of mA/mC, but are not correct in the Lorentz range.