Limiting polymer size distribution

Experimental studies show, however, that these limiting approximations must be used with caution. For example, with some emulsion polymerization systems the mean number of radicals per particle may run from one-half to several depending on the size of the particle (I). Assuming that the polymerization process is stationary with known rates of radical arrival and termination, Stockmayer (6) and O Toole (3) have shown how to calculate not only the mean number of radicals but the entire number distribution as well. Until now, no methods of the same generality seem to exist for calculating the polymer size distribution. 

A method is presented here which yields the polymer size distribution for arbitrary rates of radical arrival and termination. Furthermore, from this analysis one can see when each of the limiting cases is applicable. The computations are all carried out under stationary conditions with the rates of radical arrival, propagation, and termination constant. Under transient conditions the computations would be much more difficult. For the limiting cases, however, the moments of the polymer size distributions under transient conditions can be found (4). 

Following Ref. 4, the polymer size distribution in the Smith-Ewart limit D/B f oo can be developed from the probability equations ... 

To find the polymer size distribution in the opposite limiting case, where D/B 0, we start with phenomenological equation... 

Gas AntisolventRecrystallizations. A limitation to the RESS process can be the low solubihty in the supercritical fluid. This is especially evident in polymer—supercritical fluid systems. In a novel process, sometimes termed gas antisolvent (GAS), a compressed fluid such as CO2 can be rapidly added to a solution of a crystalline soHd dissolved in an organic solvent (114). Carbon dioxide and most organic solvents exhibit full miscibility, whereas in this case the soHd solutes had limited solubihty in CO2. Thus, CO2 acts as an antisolvent to precipitate soHd crystals. Using C02 s adjustable solvent strength, the particle size and size distribution of final crystals may be finely controlled. Examples of GAS studies include the formation of monodisperse particles (<1 fiva) of a difficult-to-comminute explosive (114) recrystallization of -carotene and acetaminophen (86) salt nucleation and growth in supercritical water (115) and a study of the molecular thermodynamics of the GAS crystallization process (21). 

It is found that the viscosity of a paste made from a fixed polymer/plasticiser ratio depends to a great extent on the particle size and size distribution. In essence, in order to obtain a low-viscosity paste, the less the amount of plasticiser required to fill the voids between particles the better. Any additional plasticiser present is then available to act as a lubricant for the particles, facilitating their general mobility in suspension. Thus in general a paste polymer in which the pastes have a wide particle size distribution (but within the limit set by problems of plasticiser absorption and settling out, so that particles pack efficiently, will... 

In the early work on the thermolysis of metal complexes for the synthesis of metal nanoparticles, the precursor carbonyl complex of transition metals, e.g., Co2(CO)8, in organic solvent functions as a metal source of nanoparticles and thermally decomposes in the presence of various polymers to afford polymer-protected metal nanoparticles under relatively mild conditions [1-3]. Particle sizes depend on the kind of polymers, ranging from 5 to >100 nm. The particle size distribution sometimes became wide. Other cobalt, iron [4], nickel [5], rhodium, iridium, rutheniuim, osmium, palladium, and platinum nanoparticles stabilized by polymers have been prepared by similar thermolysis procedures. Besides carbonyl complexes, palladium acetate, palladium acetylacetonate, and platinum acetylac-etonate were also used as a precursor complex in organic solvents like methyl-wo-butylketone [6-9]. These results proposed facile preparative method of metal nanoparticles. However, it may be considered that the size-regulated preparation of metal nanoparticles by thermolysis procedure should be conducted under the limited condition. 

Chemical composition of packings. Today, a wider variety of different support materials is available from which to choose. Silica is still widely used, though preparative grades often possess a relatively wide particle size distribution as compared to polymer-based supports. One serious limitation of silica-based supports is the low stability of silicas to alkaline pH conditions, which limits use of caustic solutions in sanitization and depyrogenation. Polymer-based supports, which include poly(styrene-divi-nyl benzene)- or methacrylate-based materials, are widely available and have gained increased acceptance and use. Nonfunctionalized poly(styrene-divinyl... 

A consideration of the preceding equations indicates that high polymer (i.e., large values of X and Xw) will be produced only if p is close to unity. This is certainly what one expects from the previous discussions in Sec. 3-5. The distributions described by Eqs. 2-86, 2-88, and 2-89 have been shown in Figs. 2-9 and 2-10. The breadth of the size distribution Xw/X [also referred to as the polydispersity index (PDI)] has a limiting value of two as p approaches unity. 

Intraparticle diffusion limits rates in triphase catalysis whenever the reaction is fast enough to prevent attaiment of an equilibrium distribution of reactant throughout the gel catalyst. Numerous experimental parameters affect intraparticle diffusion. If mass transfer is not rate-limiting, particle size effects on observed rates can be attributed entirely to intraparticle diffusion. Polymer % cross-linking (% CL), % ring substitution (% RS), swelling solvent, and the size of reactant molecule all can affect both intrinsic reactivity and intraparticle diffusion. Typical particle size effects on the... 

The Rh, Ru, and Pt ionomers of perfluoro- carbonsulfonic acid polymers have been formed and reduced to investigate the formation of metal particles within the ionic domains of these materials 88). The particle size distributions peak in the 2.5 to 4.0 nm range. The reduced ionomers catalyze the CO oxidation with the activity sequence Ru > Rh > Pt. Diffusion limitations occur in the cases of the Rh and Ru, but not the Pt, ionomer catalysts. 

Because the mechanisms are based on pore flow and size exclusion (cf. Section 2.2), the polymer material itself does not have direct influence on flux and selectivity in U F. The U F membranes usually have an integrally asymmetric structure, obtained via the NIPS technique, and the porous selective barrier (pore size and thickness ranges are 2-50 nm and 0.1-1 (im, respectively) is located at the top (skin) surface supported by a macroporous sublayer (cf. Section 2.4.2). However, the pore-size distribution in that porous barrier is typically rather broad (Figure 2.6), resulting in limited size selectivity. 

Although numerical solutions are required when the number of radicals in a particle is arbitrary, we can get analytic solutions for the limiting cases from a previous study by Saidel and Katz (4). The details are worked out in the Appendix. In the limit as D/B f oo (zero or one radical in a particle), the stationary size distribution of dead polymer is... 

In the other limiting situation D/B J, 0 (a large number of radicals in a particle) the size distribution of dead polymer is given by... 

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