Constant segment density model

The constant segment density model is, of course, only an approximation at best. It would be expected that in general the segment density would be a function of the distance from the surface of the particle. The precise form adopted by the segment density distribution function should depend upon the steric layer properties. These properties will be determined by such factors as the chemical nature of the surface and the polymer, the quality of the solvency of the dispersion medium, the surface coverage, and the mechanism of attachment of the polymer chains to the surface. Some of these expectations have been confirmed by the recent experimental determinations of the segment density distribution functions for several different systems. 

Constant segment density models. We begin this discussion by reconsideration of the constant segment density model so as to be able to demonstrate later that this approach leads to the same result as is predicted by the Fischer theory. 

The normalized segment density distribution functions in the interpenetrational domain (Lssteric layer) for the constant segment density model Ph = p = p h = 1/i-s since Jo (,dx= 1. This leads to... 

Constant segment density model. The way by which the Deijaguin integration is performed for spheres is amply illustrated by the constant segment density model. In the interpenetration domain, equation (12.43) given above leads to... 

Comparison of theory with experiment. It will be shown in Section 13.3.2.1 that the flat plate potentials can be used to calculate the osmotic disjoining pressures in concentrated monodisperse sterically stabilized dispersions. Evans and Napper (1977) have compared the theoretical predictions using the above equations with those measured by Homola and Robertson (1976) for polystyrene latex particles stabilized by poly(oxyethylene) of molecular weight ca 2 000 in aqueous dispersion media. The elastic repulsion in the interpenetrational-plus-compressional domain was estimated from the following expression for the constant segment density model... 

Fig. 12.5. The disjoining pressure as a function of the interparticle distance of separation for spheres sterically stabilized by poly(oxyethylene) curve 1, the experimental results of Homola and Robertson (1976) curve 2, constant segment density model. The crosses (x) show the theoretical results for a softened elastic potential (after Evans and Napper, 1977).

The constant segment density model. Because the perturbation approach introduces considerable complexity into the mixing term, only the simplest model will be considered here. This assumes that the segment density in the steric layer is constant whereupon the following relationships hold Ph = P h = l/h and A< = = 1/ s. Substitution of these values into equations... 

To compare the predictions of cell model calculations with the results of the experiments of Cairns et al. (1976) on poly(methyl methacrylate) latices stabilized by poly(12-hydroxystearic add) in n-dodecane, Cairns et al. (1981) adopted a simple model for the pair potential Uij. This was the constant segment density model of Fischer (1958), as elaterated by Ottewill and Walker (1968). They also considered the modification of that model by Doroszkowski and Lamboume (1971 1973). The latter approach, which allows in a completely arbitrary way for a supposed redistribution of stabilizer segments on close approach of the opposing surface, yields the potential... 

The local mobility A in general depends on the environment. In the simplest model (local coupling approximation), A is assumed to be a constant related to an effective diffusion coefficient D . This assumption is valid when the segment density fluctuations are small compared to the average densities o. Another simple modd is to assume that A depends on the segment density as A =Ao< (r,t), where Aq is a constant. The formal theory of time evolution of density variables with general kinetic coeffidents was developed by Kawasaki and Sddmoto, but this is rather difficult to implement in practice. 

Based on the excess osmotic pressure established by the difference in the concentration of the adsorbed polymer chains between the overlap region and the unaffected region of an isolated pair of particles in combination with the Flory-Huggins polymer solution theory [34-38], for constant polymer chain segment density, a more realistic model developed for calculating AG is shown below [6, 21, 39] ... 

FP does not grow with distance from its nucleus, but attains a value where rod attachment is balanced by frustration allowing steady (constant density) outward growth [668]. This is consistent with our experimental [311] and molecular modeling [661] results, where we found that aromatic polyamide FPs behave as surface fractals and their internal segmental density was coarsely uniform and far lower than in the case of dendrimers. 

In this model the star polymer is divided into three regions. The segment density is constant of order unity near the center further out there is a region with unswollen polymer [Eq. (9b)] and the corona is characterized by swollen arms [Eq. (7b)]. 

To obtain a closed expression for A2, suitable for all values of z, two types of theories have been developed by several authors in recent years. The first type of theory is based on the uniformly expanded chain model and on a spherically symmetrical distribution of segments about the molecular center of mass. The segment distribution is taken to be a spherical cloud of constant density in Flory s first theory 101), a Gaussian function about the center of mass in Fi.ory and Kkigbaum s (103 ) and in Orofino and Flory s (204) theories, and a sum of N different Gaussian functions in Isihara and Koyama s theory (132 ). All of these theories may be summarized in the following type of equation given by Orofino and Flory,... 

The genesis of the UCST curve for polymer-solvent systems is usually ascribed to enthalpic interactions between the mixture components, which are relatively insensitive to pressure for these constant density systems. The UCST curve can be modeled well with a liquid solution model that adequately accounts for specific interactions between the segments of polymer and the solvent. Examples of interactions are hydrogen bonding and polar interactions. The model also needs to account for the combinatorial entropy of mixing solvent molecules with the many segments that make up a single polymer chain (Prausnitz, 1969). 

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