Monodispersed approximation model

The simplest case for modeling particle dissolution is to assume that the particles are monodisperse. Under these conditions, only one initial radius is required in the derivation of the model. Further simplification is possible if the assumption is made that mass transport from a sphere can be approximated by a flat surface or a slab, as was the case for the derivation for the Hixson-Crowell cube root law [70], Using the Nernstian expression for uniaxial flux from a slab (ignoring radial geometry or mass balance), one can derive the expression... 

Recent experimental studies (1-3), on systems of sterically stabilized colloidal particles that are dispersed in polymer solutions, have highlighted the role played by the free polymer molecules. These experiments are particularly relevant because the systems chosen are model dispersions in which the particles can be well approximated as monodisperse hard spheres. This simplifies the interpretation of the data and leads to a better understanding of the intcrparticle forces. DeHek and Vrij (1, 2) have added polystyrene molecules to sterically stabilized silica particles dispersed in cyclohexane and observed the separation of the mixtures into two phases—a silica-rich phase and a polystyrene-rich phase—when the concentration of the free polymer exceeds a certain limiting value. These experimental results indicate that the limiting polymer concentration decreases with increasing molecular weight of... 

Fig. 9.3. The scaling function R (N,Cp)/R (A, Cj, — 0) (excluded volume limit, monodisperse) hh function of s = CpRg N,Cp = 0). The full line is the renormalization group result. (See Chap. 18.) The broken lijies give approximations as motivated bv the blob model...

In the case of monodisperse foam, another approximation formula for the interface mean curvature was obtained in [156] by using the deformation theory [430] and the rounded dodecahedron model namely,... 

From the above summary of the application of hydrodynamic theories to amylose, it can be seen that discrepancies exist between the calculated and experimental parameters, expecially when the exponent in the Mark-Houwink equation is high, that is, a 1. Errors can arise in the experimental determinations of the various parameters and also in using the incorrect mathematical averages in the theoretical relations. (The hydro-dynamic theories are based on monodisperse polymer systems, a criterion which is rarely satisfied, and it is necessary to introduce somewhat arbitrary corrections for heterogeneity.) Furthermore, the model on which the theory is based, or the mathematical approximations introduced during the subsequent treatment, may be incorrect. 

Most often the average molecular weight is used in relation to rheological properties (Berry and Fox, 1968), although this is only an approximation. When model blends are prepared by mixing monodisperse polymers, the molecular weight dependences are complicated. For example, the melt viscosity is less than the value predicted assuming t] is determined by My, (Watanabe et al., 1995 Monfort et al., 1984 Plazek et al., 1991). 

A simple example is one where the hydrophobic interactions result in < i for all > 1, but the packing constraints on the chains and heads result in a minimum energy for a finite value of N = M (i.e., m < for N Af). (Below, a curvature energy model is discussed this model can also be used to motivate, but not to calculate in detail, a study of micellar sizes and shapes.) If this minimum is deep n rises sharply compared to ksT around N = Af), the distribution of micelles will be nearly monodisperse. In this approximation, one can consider monomers and micelles of aggregation number Af only. At small values of 4>s (or equivalently, at small values of fi). Pi Pm (Af > 1) almost all the surfactant exists as monomers and the number of micelles is exponentially small. The requirement that all amphiphiles have the same chemical potential in equilibrium, Eq. (8.2) and the definition of the CMC (where Pi = Pi, Pm = Pmc ft = = c) allows us to calculate... 

The thermodynamic model of micellization, presented here, describes the association of any amphiphilic molecules, including low molecular weight surfactants or polymeric amphiphiles. The physical origin of the minimum in the free energy, as a function of p, is specified by the molecular architecture and the interactions between amphiphilic molecules involved in the assembly, and will be discussed in the corresponding sections. An extension of the model for the case of a continuous distribution of micelles with respect to aggregation number (polydispersity of the aggregates) involves the value of d Fp/dp. If this quantity is small in the vicinity of p = Po, then the micelle distribution is wide, and vice versa [37]. The approximation of micelle monodispersity is essential for application of the numerical SCF model which is discussed in Sect. 9. 

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