Benjamin J. McCoy

Analytical solution for a population balance equation with aggregation and fragmentation

Lage (2002) and Patil and Andrews (1998), denoted here as LPA, have derived a particular solution for the population balance equation (PBE) with simultaneous aggregation(coalescence) and fragmentation (breakage), but for the special case where the total number of particles is constant. The more general reversible case, when either fragmentation or coalescence can dominate, has numerous applications, and is thus of considerable importance (McCoy and Madras, 1998; McCoy and Madras, 2002; Sterling and McCoy, 2001, McCoy, 2001). We wish to comment that a more general solution is available when the number of particles is not constant. The discrete-distribution solution with arbitrary ratio of aggregation and fragmentation rate coefficients was presented by Blatz and Tobolsky (1948) and Family et al. (1986), and later extended by Browarzik and Kehlen (1997) to continuous distributions. Aizenman and Bak (1979) showed that the exponential distribution is of the form of Flory’s (1963) most probable distribution. Cohen (1992) presented a combinatorial approach to derive the similarity solution, and Vigil and Ziff (1989) provided moment solutions that focused on the evolution to steady state distributions. Spicer and Pratsinis (1996) numerically simulated the evolution to a steady-state similarity solution for different forms of the breakage stoichiometry and rate coefficients. McCoy and Madras (1998, 2002) based their solution on moment analysis, which was validated by substitution into the PBE. LPA have suggested applying a straightforward Laplace transform method to solve the PBE, which we explore in this comment.

Distribution kinetics of polymer crystallization and the Avrami equation

Cluster distribution kinetics is adopted to explore the kinetics of polymer crystallization. Population balance equations based on crystal size distribution and concentration of amorphous polymer segments are solved numerically and the related dynamic moment equations are also solved. The model accounts for heterogeneous or homogeneous nucleation and crystal growth. Homogeneous nucleation rates follow the classical surface-energy nucleation theory. Different mass dependences of growth and dissociation rate coefficients are proposed to investigate the fundamental features of nucleation and crystal growth. A comparison of moment solutions with numerical solutions examines the validity of the model. The proposed distribution kinetics model provides a different interpretation of the familiar Avrami equation.

Temperature effects during Ostwald ripening

Temperature influences Ostwald ripening through its effect on interfacial energy, growth rate coefficients, and equilibrium solubility. We have applied a distribution kinetics model to examine such temperature effects. The model accounts for the Gibbs–Thomson influence that favors growth of larger particles, and the dissolution of unstable particles smaller than critical nucleus size. Scaled equations for the particle size distribution and solution concentration as functions of time are solved numerically. Moments of the distribution show the temporal evolution of number and mass concentration, average particle size, and polydispersity index. Parametric and asymptotic trends are plotted and discussed in relation to reported observations. Temperature programming is proposed as a potential method to control the size distribution during the phase transition. We also explore how two crystal polymorphs can be separated by a temperature program based on different interfacial properties of the crystal forms.

Ostwald ripening in two dimensions: Time dependence of size distributions for thin-film islands

Thin film phase transitions can involve monomer deposition and dissociation on clusters (islands) and island coalescence. We propose a distribution kinetics model of island growth and ripening to quantify the evolution of island size distributions. Incorporating reversible monomer addition at island edges, cluster coalescence, and denucleation of unstable clusters, the governing population dynamics equation can be transformed into difference-differential or partial differential forms with an integral expression for coalescence. The coalescence kernel is assumed proportional to xνx′μ, where x and x′ are masses of coalescing clusters. The deposition and dissociation rate coefficients are proportional to cluster mass raised to a power, xλ. The equation was solved numerically over a broad time (θ) range for various initial conditions. The moment form of the equation algebraically yields asymptotic long-time expressions for the power law dependence of decreasing number concentration (θ−b) and increasing average cluster size (θb). When coalescence is negligible the power is b = 1/(1 − λ + d−1), where d = 2 or 3 is the dimensionality. If coalescence dominates the asymptotic time dependence, the power is b = 1/(1 − ν − μ).

Continuous distribution theory for Ostwald ripening: comparison with the LSW approach

A numerical solution for a general population balance equation (PBE) for Ostwald ripening is compared with the usual approach developed by Lifshitz-Slyozov-Wagner (LSW). The PBE incorporates denucleation for unstable particles smaller than the critical nucleus size and reversible growth or dissolution of stable particles. The PBE theory shows how supersaturation decays to equilibrium, and (unlike LSW) how the particle size distribution (PSD) and its moments evolve to a final monodisperse state. The LSW model is known to correctly depict time dependence of particle number concentration and average particle size, but misrepresents the PSD higher moments.

Temperature effects for isothermal polymer crystallization kinetics

We adopt the cluster size distribution model to investigate the effect of temperature on homogeneous nucleation and crystal growth for isothermal polymer crystallization. The model includes the temperature effects of interfacial energy, nucleation rate, growth and dissociation rate coefficients, and equilibrium solubility. The time dependencies of polymer concentration, number and size of crystals, and crystallinity (in Avrami plots) are presented for different temperatures. The denucleation (Ostwald ripening effect) is also investigated by comparing moment and numerical solutions of the population balance equations. Agreement between the model results and temperature-sensitive experimental measurements for different polymer systems required strong temperature dependence for the crystal-melt interfacial energy.

Lattice-Boltzmann simulation of coalescence-driven island coarsening

A two-dimensional lattice-Boltzmann model (LBM) with fluid-fluid interactions was used to simulate first-order phase separation in a thin fluid film. The intermediate asymptotic time dependence of the mean island size, island number concentration, and polydispersity were determined and compared with the predictions of the distribution-kinetics model. The comparison revealed that the combined effects of growth, coalescence, and Ostwald ripening control the phase transition process in the LBM simulations. However, the overall process is dominated by coalescence, which is independent of island mass. As the phase transition advances, the mean island size increases, the number of islands decrease, and the polydispersity approaches unity, which conforms to the predictions of the distribution-kinetics model. The effects of the domain size on the intermediate asymptotic island size distribution, scaling form of the island size distribution, and the crossover to the long-term asymptotic behavior were elucidated.

Effect of tetralin on the degradation of polymer in solution

The effect of a hydrogen-donor solvent tetralin on thermal degradation of poly(styrene-allyl alcohol) in liquid solution was investigated in a steady-state tubular flow reactor at 1000 psig at various tetralin concentrations, polymer concentrations, and temperatures. The experimental data were interpreted with continuous- mixture kinetics, and rate coefficients determined for the specific and random degradation processes.