Polymer particle number density

To illustrate how the effect of the adsorption on the modulus of the filled gel may be modelled we consider the interaction of the same HEUR polymer as described above but in this case filled with poly(ethylmetha-crylate) latex particles. In this case the particle surface is not so hydrophobic but adsorption of the poly (ethylene oxide) backbone is possible. Note that if a terminal hydrophobe of a chain is detached from a micellar cluster and is adsorbed onto the surface, there is no net change in the number of network links and hence the only change in modulus would be due to the volume fraction of the filler. It is only if the backbone is adsorbed that an increase in the number density of network links is produced. As the particles are relatively large compared to the chain dimensions, each adsorption site leads to one additional link. The situation is shown schematically in Figure 2.13. If the number density of additional network links is JVL, we may now write the relative modulus Gr — G/Gf as... 

This defines the number density p and hence concentration. This is determined by the particle radius. Analogous behaviour exists for polymers and there is a concentration at which polymer coils just touch ... 

The ratio of the polymer deposited on the surface of the particles to that generating micelles was determined by fitting the experimental results. If the number of polymer chains in each micelle is m (which is taken as 50 in the present calculations1), the number density of the micelles is... 

Consider a cylindrical soft particle, that is, an infinitely long cylindrical hard particle of core radius a covered with an ion-penetrable layer of polyelectrolytes of thickness d in a symmetrical electrolyte solution of valence z and bulk concentration (number density) n. The polymer-coated particle has thus an inner radius a and an outer radius b = a + d. The origin of the cylindrical coordinate system (r, z, cp) is held fixed on the cylinder axis. We consider the case where dissociated groups of valence Z are distributed with a uniform density N in the polyelectrolyte layer so that the density of the fixed charges in the surface layer is given by pgx = ZeN. We assume that the potential i/ (r) satisfies the following cylindrical Poisson-Boltz-mann equations ... 

The origin of the spherical polar coordinate system (r, 9, cp) is held fixed at the center of one particle and the polar axis (9 = 0) is set parallel to E. Let the electrolyte be composed of M ionic mobile species of valence zt and drag coefficient A,-(/ = 1, 2,. . . , M), and let nf be the concentration (number density) of the ith ionic species in the electroneutral solution. We also assume that fixed charges are distributed with a density of pflx. We adopt the model of Debye-Bueche where the polymer segments are regarded as resistance centers distributed in the polyelectrolyte... 

Equation (4Ib) is valid when there are no polymer particles flowing into the reactor with all the particles nucleated within the reactor. It is assumed that density changes can be neglected and that particles follow the streamlines. These are reasonable assumptions in view of the small size of particles and the small density difference between particle and water. When two or more CSTRs are employed in series, however, one must remember that the total residence time of a polymer particle is made up of different times in each reactor in the train. The relative amounts of time spent in each reactor will not matter when the volumetric growth rate of a particle is the same in each. This would require that the temperature, monomer concentration, and average number of radicals per particle he the same for each reactor, an unlikely possibility. This idealization is useful, however, when illustrating the effect of increasing the number of CSTRs in series on the breadth of the particle size distribution. 

It should be pointed out that there is no direct physical relation between the phenomenon of fractionated crystallization and the number and the size of spherulites in the pure polymer. Whereas the occurrence of fractionated crystallization is related to the ratio between the number densities of dispersed polymer particles and primary nuclei, the size and the number of spherulites are additionally influenced by the cooling rate and the crystallization temperature. There is, therefore, also no relation between the fractionated crystallization and the type of the arising crystalline entities (complete spherulites, stacks of lamellae,...) both in the pure and in the blended material. There is, finally, no relation between the scale of dispersion which is necessary for the occurrence of fractionated crystallization and the spherulite size in the unblended polymer. 

We argue that the above features of star dynamics are generic for soft systems of the core-shell type for which stars serve as prototype. Support for this comes from the dynamic light scattering (DLS) investigation of large block copolymer micelles, where all three relaxation modes, i.e., cooperative, structural and selfdiffusion are observed [188]. In particular, the star model discussed above applies to core-shell particles with a small spherical core relative to the chain (shell) dimensions. For a surface number density a = f / (47i r ) the polymer layer thickness under good solvent conditions is L ... 

The number of cells per unit area equals the pore density, N. Note that the size of the cell Is not constant, and that the number of the polymer particles and pore particles are No(l - L) and N R, respectively. The number of ways, Nq(1-L)Si ... 

The fit shown in Fig. 18 is restricted by a number of experimental parameters such as electron density of the polymers, and the concentration of the particles. It must be noted that absolute intensities have been used here. Hence, the number density of the particles is fixed and cannot be used as a fit parameter. These constraints lead to the elucidation of the radial structure of the particles with a resolution of a few nm. 

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