Particle polymer volume distributions

Kiparissides, et al. (8) developed mathematical models of two levels of sophistication for the vinyl acetate system a comprehensive model that solved for the age distribution function of polymer particles and a simplified model which solved a series of differential equations assuming discrete periods of particle nucleation. In practice, the simplified model adequately describes the physical process in that particle generation generally occurs in discrete intervals of time and these generation periods are short in duration when compared with operation time of the system. The simplified model is expanded here for a series of m reactors. The total property balances for number of particles, polymer volume, conversion, and area of particles, are written as ... 

Polymer Particle Balances (PEEK In the case of multiconponent emulsion polymerization, a multivariate distribution of pjarticle propierties in terms of multiple internal coordinates is required in this work, the polymer volume in the piarticle, v (continuous coordinate), and the number of active chains of any type, ni,n2,. .,r n (discrete coordinates), are considered. Therefore... 

Figures 7 and 8 show typical particle size distributions for vinyl acetate emulsions produced in a single CSTR. A large number of particles,are quite small with 80 to 90% being less than 500 A in diameter. The large particles, though fewer in number, account for most of the polymer mass as shown by the cumulative volume distributions. Data are also presented on Figures 7 and 8 for the number of particles based on diameter measurements (N ), the average number of free radicals per particle, and the steady state conversion.

If/m is the maximum packing density of the particles, which is defined as the volume fraction at which the particles touch one another, so that flow is not possible, then the actual particle volume firaction/used in injection molding is lower than/m by 5-10 vol%. This means that in a well-dispersed suspension, the particles are separated from one another by a thin layer of polymer with a thickness of about 50 nm dming the molding, so that the mixture is able to flow. Therefore, the volume fraction of particles / is determined by the particle size and distribution and the particle shape. In practice, the volume firaction of ceramic powders is determined from viscosity measurements by using a capillary rheometer. Data for the relative viscosity, i.e., the viscosity of the mixture divided by the viscosity of the unfilled polymer versus particle concentration can be well fitted by the following equation [209] ... 

An ideal filler would be low in cost compared to the PVC, have perfectly spherical particles with a small but uniform particle size and distribution, be easily dispersed within the polymer matrix, have a refractive index that could be chosen to either increase opacity or provide clarity, and have a low densily or specilie gravity compared to PVC. Since PVC compounds compete on a cosl/volume basis against other polymers, PVC s high density compared to polyolefins or polycarbonates often places the final formulation at a disadvantage when measured solely on eost per unit of weight. Fillers, while providing rigidity and reinforcement, will also lower the formulation costs. 

The term /3(m, v) represents the coalescence rate constant of two colloidal particles of volume u and v. Note that the initial particle growth occurs mainly by particle aggregation and, to a smaller extent, by polymerization of the adsorbed monomer in the polymer-rich phase [58]. Thus, knowledge of analytical expressions for the coalescence rate constant is of profound importance to the solution of the population balance model (Equation 4.46), describing the time evolution of the primary particle size distribution. Such expressions have been derived by Kiparissides et al. [57, 59]. 

Table 1. Composition of the simulated dense systems (Afp = number of polymer chains of 100 units Nf = number of randomly distributed spherical filler particles = diameter of the particles ifi = volume fraction of filler)...

Most percolation studies rely on Monte Carlo simulations, whose properties of interest are critical cluster size, bond densities (i.e. number of bonds that lead to network per site), particle volume in the matrix, particle orientation and network geometry at the threshold. In most cases simulations treat the percolation in a statistical manner which means that the particles are randomly distributed in the matrix and network pathways are formed simply by increasing the particle volume fraction in the composite. Although such an approach has merit and provides valuable insight on the network formation, it is far away from reality, especially when polymers are concerned. Addition of particles in a polymer matrix is mainly performed via solution or melt mixing, which means that both particles and polymer chains are in motion and interact with each other. More advanced theoretical approaches do take into consideration the thermodynamic interactiOTis between the composite constituents (particle-particle, particle-polymer and... 

All the above-mentioned models do not account for the existence of interparticle interactions. Recently, Tronc et al., Fiorani et al., and Dormann and Fiorani proposed a model for describing the time decay of TRM in a series of y-FejOj particles dispersed in a polymer, with different interparticle distances and then with different interactions strengths. The model accounts for the particle volume distribution as well as for the existence of interparticle interactions. 

In the present approach, the simulations focus on the ways the filler particles change the distribution of the end-to-end vectors of the polymer chains making up the elastomeric network, fi-om the fact that the filler excludes the chains firom the volumes it occupies. The changes in the polymer chain distributions fi om this filler excluded volume effect then cause associated changes in the mechanical properties of the elastomer host matrix. Single polymer chains are... 

K parameter in tydization or parameterin semibatdi control A parameter in free volume H particle volumetric growth rate or parameter in metallocene polymerization with branching parameter in fiee-volume equation Pm monomer density Pp polymer density p branching density p 6, ) CTOSS-fink density distribution pa 0, ) additional cross-Unk density distribution pj 0, ) cyclization density Pcs,a(0/ ) additional secondary cyclization density Pcp(0) primary cyclization density Paui(0) instantaneous secondary cyclization density Pt(0) instantaneous cross-link density reaction radius of the reacting species r ratio of reaction rates as defined by eqn [56] number fraction of type-i radicals monomer volume fraction polymer volume fraction chain length r number fraction < > s chain length s number fraction 0 present conversion... 

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