Dynamical Nucleation Theory

The following schemafic summarizes how nucleation theories such as Dynamical Nucleation Theory (DNT), the GDE, and radiative forcing from Mie calcula-... 

The dependence of / on S is known from homogeneous nucleation theory discussed earlier in this chapter. Thus, there are four differential equations in five unknowns. A, M, S,N, and /, with a relationship between / and S. The set of equations for the dynamics of the stable aerosol is remarkable because the calculation of the important moments, A and N, does not require the detemiination of the size distribution of the stable particles. 

On physical grounds, loa = 0 because there is no toss of particles by growth from the upper end of the distribution. The term /,/ is the particle current (lowing into the lower end of the spectrum. When homogeneous niicleation takes place, this term is important. For vj = v, the critical particle size, /,j, is the particle current of homogeneous nucleation theory. Hence the dynamic equation for the number concentration is... 

Figure 2. Temperature dependence of the nucleation rate in superheated water at atmospheric pressure. Data of dynamic experiments 1 — [ ], 2 - [ ], 3 - fj, 4 - [ ], 5 - [ ]. The solid line shows calculation by homogeneous nucleation theory.

Recent developments in the application of molecular dynamics promise a resolution of many questions in nucleation theory. A review of simulation methods in nucleation theory and of their results is given in [2.39]. Consequently, a survey will not be undertaken here. However, some of the results of SCHIEVE and co-workers [2.40-42] will be discussed here as they throw light on several questions of importance in terms of modelling aerosol growth from vapor condensation. 

The application of the dynamic SCF theory [97] or EPD [29,31,109] to the collective dynamics of concentration fluctuations and the relation between the dynamics of collective concentration fluctuations and the single chain dynamics is an additional, practically important aspect. We have merely illustrated the simplest possible case—the early stages of spontaneous phase separation within purely diffusive dynamics. In applications the hydrodynamic effects [110,111], shear and viscoelasticity [112] might become important. Even deceptively simple situations—like nucleation phenomena in binary polymer blends—still pose challenging questions [113]. Also the assumption of local equilibrium for the chain conformations, which allows us to use the SCE free energy functional, has to be questioned critically. Methods have been devised to incorporate some of these complications [76,96,99, 111, 112] but the development in this area is still in its early stages. 

These rates are used to calculate as a function of time, which in turn are the source rates for the aerosol dynamic equations. Notice that k2 and k4 have a negative temperature dependance. This is characteristic for recombination reactions and in the language of classical nucleation theory imply that as the temperature increases the nucleation rate should drop. A more thorough discussion of this point can be found in reference 12. 

Mandell, McTague, and Rahman used molecular dynamics to study crystal nucleation in a three-dimensional Lennard-Jones system representing supercooled liquid argon. These workers have observed the formation of solid clusters at a rate which they believe is consistent with classical nucleation theory, and a similar calculation for liquid rubidium also produced crystal nucleation. 

Recently Rao, Berne, and Kalos have reported Monte Carlo and molecular dynamics studies of the homogeneous nucleation of a Lennard-Jones fluid (argon). Several features of this study are noteworthy. Perhaps most importantly, the authors conclude from their dynamical study that growth of the nucleation microclusters takes place under essentially adiabatic conditions. That is, collisions between the microcluster and the vapor are not sufficiently frequent to maintain the droplet at a constant temperature. This result suggests that the isothermal assumption common in conventional nucleation theories... 

This paper presents the physical mechanism and the structure of a comprehensive dynamic Emulsion Polymerization Model (EPM). EPM combines the theory of coagulative nucleation of homogeneously nucleated precursors with detailed species material and energy balances to calculate the time evolution of the concentration, size, and colloidal characteristics of latex particles, the monomer conversions, the copolymer composition, and molecular weight in an emulsion system. The capabilities of EPM are demonstrated by comparisons of its predictions with experimental data from the literature covering styrene and styrene/methyl methacrylate polymerizations. EPM can successfully simulate continuous and batch reactors over a wide range of initiator and added surfactant concentrations. 

---