Stockmayer distribution

A careful examination of the Stockmayer and the percolation distributions reveals that both theories gives the same type of distribution [110]. In terms of the two exponents in Eqs. 52 and 53, the percolation calculation yields t=2.2 and 0 0.44, and the Stockmayer distribution yields r 2.5 and o 0.50. These differences in the exponents appear to be small in a double logarithmic plot, but they cause significant differences in the absolute values for w(x) when 3-4 decades in the degree of polymerization are covered. Another point is that the cut-off function could be calculated analytically in the FS-theory to be a single exponential function [110], while the percolation theory could only make a guess about its shape [7]. 

Stockmayer distribution function of chemical composition. Macromolecules 20 2624— 2628. 

The Stockmayer distribution is an extension of the Flory distribution for copolymers. When Equation 5.4 is integrated over all chemical compositions, it is reduced to... 

Figure 5.10 The Stockmayer distribution for a copolymer made with a single-site catalyst, a) Three-dimensional plot, b) Bird s eye view. (See insert for the color representation of the figure.)...

Soares and Hamielec extended the Stockmayer distribution for copolymers containing LCBs, where the mechanism of LCB formation is terminal branching via macromonomer incorporation [89, 90]. The resulting trivariate distribution is given by the expression... 

The quantities k , k,p, kpp, and kp, are the rate constants of the four basic propagation reactions of copolymerization. The Stockmayer distribution function takes into account only a chemical polydispersity resulting fi om the statistical nature of copolymerization reactions. This means that all units of all chains are formed under identical conditions. If a monomer is removed from the reacting mixture at a rate which changes the monomer concentration ratio, the monomer concentration will drift, forming a copolymer which varies in the average composition and is broader in the chemical distribution. No such chemical polydispersity can be described by the Stockmayer distribution. Therefore, Eq. (84) has to be restricted in its application to random copolymers synthesized at very low conversions or under azeotropic conditions. For azeotropic copolymers, the feed monomer concentrations [a ] and are chosen in such a way that the second factor on the right-hand side of the basic relation of copolymerization kinetics... 

The application of the Stockmayer distribution function is especially advantageous in the framework of continuous thermodynamics. When cloud-point and shadow curves are calculated, the double integrals in Eqs. (40)-(42) can be evaluated analytically (in contrast the double integrals in Eqs. (50)-(52) can be computed... 

In this case too, the distribution function of the shadow phase II remains the Stockmayer type (Eq. (84)) if the copolymer in the initial phase I obeys the Stockmayer distribution [34,49],... 

In the preceding chapters, chemical polydispersity and its influence on the demixing equilibrium were discussed by applying the generalized Stockmayer distribution function only. With respect to chemical composition of the copolymer this function leads to a symmetric and rather narrow distribution. However, many synthetic... 

Some results obtained by the different approaches are compared below. The systems considered contain a sample of copolyfethene vinyl acetate) which is characterized by = 15500 g mol" and = 40800gmol and B = 0.677, 6b = 0.232. Applicability of the generalized Stockmayer distribution function, Eq. (84), is assumed for the divariate case and, hence, analytical integrability of the double integrals in the scalar equations for ipJJ and is obtained. The... 

With the generalized Stockmayer distribution, Eq. (84), applied to both copolymers, the double integrals in Eqs. (140)-(143) can be solved analytically. The results are the same as given in Eq. (90), i.e., the shadow phase is also characterized by a Stockmayer distribution having the same values of k and as for the distribution functions in the initial homogeneous phase [66]. 

For copolymerizations, the Flory distribution is extended to the so-called Stockmayer distribution (Soares and McKenna, 2012 Stockmayer, 1945). For a single CLD for a copolymerization involving the comonomers A and B, the corresponding copolymer composition CLD is defined by... 

Dirac delta function porosity of catalyst particle parameter for Stockmayer distribution... 

The bivariant Stockmayer distribution has been used for copolymers. In this case, the product of a Schulz-Flory distribution with respect to molar mass (or segment number) and of a Gaussian distribution with respect to chemical composition y is formed. If the copolymer consists of two kinds of monomers the chemical composition is defined as the segment fraction of one of the monomers within the copolymer. For the segment number and chemical composition the Stockmayer distribution is given by ... 

This latter distribution has a relative maximum for x x , in contrast to the general Stockmayer distribution, which decreases monotonically (see Figure 3.2). 

Pgf values of the various distributions with respect to the counts of monomer units, bonds, or unreacted functional groups can be obtained from Eq. (98) by setting equal to one the molecular weight of the species in question and to zero the molecular weights of all the other species. Analytical inversion by computing derivatives with respect to dummy Laplace variables on s = 0 is feasible with the simplest chemical systems, for which the resulting recurrence formulae are not too complex, as happens with the Stockmayer distribution. 

Once the and composition were known, the aforementioned procedure for computing the Stockmayer distribution [14] was applied for one of the copolymerization reaction studied. An illustration of the method is shown in Figure 12.3 [17]. 

Ratzsch et al. [3, 6] could demonstrate that the integrals occurring in (12)-(14) can be solved analytically under certain circumstances, namely if the Stockmayer distribution function [56] is used ... 

The high fractionation effectivity allows correct estimation of the initial copolymer distribution (15) according the molecule weight if only five firactions are formed (Fig. 23). Copolymers distributed according to the Stockmayer distribution function (15) are characterized by a relatively small polydispersity with respect to the chemical composition. In contrast, copolymers showing a distribution given in... 

For example, the calculated fractionation data for four CPF runs are collected in Table 1, where the initial polymer distribution was a Stockmayer distribution (15). For the fractionation, four CPF runs were simulated in which the obtained gel fractions were directly used as new feed phase. 

Table 1 Calculated fractionation data for CPF, where the original copolymer has a Stockmayer distribution (15)...

---