Branching moment distribution

In order to derive the balance equations for the branching moment distributions, the following steps have to be performed [31) ... 

The resulting terms have to be expressed in terms of the branching moment distributions and are added to the system of equations. 

Propagation step In this case the degree of branching is not altered, so this is a simple step from this point of view. The contribution to the balance of oth branching moment distributions of living chains in reduced or pseudo-distribution notation follows, by multiplication with t and taking the summation over the branching index t ... 

The first branching moment distribution follows by multiplication of Eqs. (44)-(46) with the number of branches i, and subsequent summation ... 

The second branching moment distribution follo vs by multiplication of the TDB propagation terms in the 2D equations vith the squared number of branches and subsequent summation ... 

Here the functions D and D(, are in fact polydispersities of the branching moment distributions and in principle are to be determined as functions of chain length n. Inserting these closure relationships in Eqs. (65), (66), and (70), (71), reduces them to ... 

The 1D population balance for the pseudo-distribution or first branching moment distribution follows by taking the first branching moment of Eq. (78) ... 

In fact, 0 and T represent the moments of the branching distributions of living and dead chains at the given chain length, n. Thus, the branching polydispersity D follows from the second branching moment according to ... 

We conclude that the pseudo-distribution approach can be appUed successfiilly, provided a good approximation can be made for the branching distribution at given chain length from the branching moments. The method is valid for batch reactors as well, in contrast to, for example, the pgf-cascade method (Section 9.7), which is restricted to steady state reactors. It is to be preferred over classes methods in cases, like the metallocene one, where the second distribution dimension may assume high values as well. 

This problem has been introduced in the discussion of the classes approach. For reaction equations and a full set of population balances, see Tables 9.5 and 9.6. Here, we address the more general problem of more than one TDB per chain [9]. This occurs as a consequence of insertion of TDB chains created by disproportionation or of recombination termination. We start with the full 3D set of Table PVAc2 and then reduce it to a ID formulation by developing the TDB and branching moment expressions. The (N, M)th branching-TDB moments or pseudo distributions for living and dead chains are defined by ... 

Performing the corresponding summations on the equations in Table 9.6, one obtains the (N, JVf)th moment formulation of Table 9.11. Some of the summation terms in these equations will not be evaluated for the general (N, M) case, but we will determine them by assigning values to N and M. Since we will not address branching, we take M = 0 here, but in principle this can be treated in a similar way. We will focus now on the TDB moment distributions and successively derive the model equations for the zeroth, first, and second moments, or N = 0,1, and 2. Solving the model thus essentially means solving the population balances of the real concentration distributions and P and the pseudo-distributions and... 

This implies that under the condition of a maximum of one TDB per chain, the set of population balance equations of the TDB branching moment variant of the model is solvable without requiring any additional closure assumption. The results obtained with the pseudo-distribution model are identical to those obtained with the classes model shown before (see Figure 9.6). 

Equation (31) applies to monodisperse systems. For polydisperse systems Rg reflects a high-order moment of the distribution, the ratio of the 8th to the 6th moment of the distribution in mean size. For this reason Rg will correlate with the largest sizes of a distribution. There are several advantages to Rg as a measure of size over the end-to-end distance. For branched, star and ring structures the end-to-end distance has no clear meaning while Rg retains its meaning. Further, Rg is directly measured in static scattering measurements so it maintains a direct link to experiment. 

The first moment of the distribution is Pt0T the total, cumulative molar concentration of polymeric material. As the molecular weight of polymeric species increases, branching and crosslinking reactions yield a thermoset resin. Chromatography analysis of epoxy resin extracts confirms the expected population density distribution described by Equation 4, as is shown in Figure 2. Formulations and cure cycles appear in Table II. 

Sole (22) has calculated the moments of the distributions of the three principal orthogonal components of obtained by decomposing along the three principal axes of inertia of the chain, for certain star and comb molecules in addition to linear ones and rings he finds that branching or ring closure decreases the high asymmetry found for linear chains. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

Graessley and his co-workers have made calculations of the effects of branching in batch polymerizations, with particular reference to vinyl acetate polymerization, and have considered the influence of reactor type on the breadth of the MWD (89, 91, 95, 96). Use was made of the Bamford and Tompa (93) method of moments to obtain the ratio MJMn, and in some cases the MWD by the Laguerre function procedure. It was found (89,91) that narrower distributions are produced in batch (or the equivalent plug-flow) systems than in continuous systems with mixing, a result referrable to the wide distribution of residence times in the latter. 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

An extensive treatment of this subject has been ven very recently by Lichti el a . (1980), and a brief summary was given in an earlier paper (Lichti el ai, 1978). The model assumed for this treatment is a three-state model in which i is 0, I, or 2. An earlier paper (Lichti et al., 1977) applied a similar treatment to a two-state model in which i is 0 or 1. The treatment allows for the possibility that mutual termination may result in either combination or disproportionation. It also allows for the possibility of transfer to monomer. It has not, however, been possible to make allowance for branching and cross-linking. Prediction of the full distribution of molecular sizes, and not merely of particular moments of the distribution, has been achieved. The conclusion has been reached that compartmen-talization of the reaction leads to a broadening of the molecular-weight distribution. 

In order to incorporate the molecular weight averages and long chain branching effects, a similar approach has been taken. Following the development of Hamielec [j>], expressions for the moments of the molecular weight distribution for the class of particles born between times t and t+dt were developed. 

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