Moments of a molecular weight distribution

Distributions discussed in this section will be limited to homopolymers. The fcth moment of a PCLD, Pj, is defined as 

Moments of virtually all the various MWDs can be related simply to the PCLD moments in the case of homopolymers. The relations between MWD and PCLD moments are somewhat more complicated in the case of copolymers. 

The relation between the average molecular weights and the moments of the PCLD is 

The average degrees of polymerization (DP) are specified solely in terms of the moments  

From the definition of the moments, equation (3.1), it is apparent that physical significance can be attached to the lower moments of the PCLD. The zeroth moment represents the total molar concentration of polymer, the first moment is the total number of monomer elements/volume in the polymer, while the second moment is the weighted sum of the number of polymers present/volume.  

---

Moments. The nth moment of a molecular weight distribution is defined as follows ... 

The prediction of the MWD of emulsion polymers proved to be a relatively intractable problem even after the advent of the Harkins-Smith-Ewart theory. Perhaps the most successful early attack on the problem was that of Katz, Shinnar and Saidel (2). They considered only two microscopic events entry and bimolecular termination by combination. Their theory resulted in a set of partial integrodifferential equations, whose numerical solution provided the lower moments of the molecular weight distribution function. Other attempts to predict the MWD of emulsion polymers include those of Parts and Wat ter son (3 ), Sundberg and Eliassen (4), Min and Ray (5) and Gardon (6). 

The viseosity average moleeular weight A/y, which will be discussed later in Section 3.3, is the only average listed in these tables that is not a simple ratio of sueeessive moments of the molecular weight distribution. 

Complete description of a molecular weight distribution implies a knowledge of all its moments. The central tendency, breadth, and skewness may be summarized by parameters calculated from the moments about zero U, U[, U 2, and These moments also define the molecular weight averages Mn, and M. Note that A/n and My, can be measured directly wiihoul knowing the distribution but it has not been convenient to obtain A j of synthetic polymers as a direct measurement of a property of the sample. Thus, some information about the breadth of the number... 

Therefore, for Gaussian molecules, the above parameters are functions of moments of the molecular weight distribution tiq a M,, and Jg a Mg.Mj+i/M. Otherwise, the mass dependence should be slightly different for qg and a large deviation from a combination of various average molecular weights is expected for the steady-state compliance. 

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. 

The development takes into account transfer to monomer, transfer to polymer, and terminal double bond polymerization. For the vinyl acetate system where transfer to monomer is high, the generation of radicals by transfer to monomer is much greater than the generation of radicals by initiation, so that essentially all radicals present have terminal double bonds hence, effectively all dead polymer molecules also have a terminal double bond. Thus, for vinyl acetate polymerization, the terminal double bond polymerization can be significant, and has been built into the development. The equations for the moments of the molecular weight distribution and the average number of branches per polymer molecule are as follows ... 

Effect of band broadening for a polymer with polydispersity 2 on the measured moments of the molecular weight distribution by light scattering, where Dj is the slope of log molecular weight and elution volume and (tb is the peak variance caused by band broadening (39). 

The number of pseudocomponents can be remarkably decreased by choosing them in a such a way that they represent relevant moments of the molecular weight distribution. These moments are related to the molecular weight averages Mn, and, by ... 

Taking into account that each pseudocomponent is characterized by two properties (its mole fraction X2pj and its molecular weight M pj), n pseudocomponents can reproduce 2n moments of a polymer distribution. This means, that, using only two pseudocomponents, one can exactly reproduce A/ , M, and M. It also means that the mole fractions of the two components have to add to unity (A/ ). Using three pseudocomponents, even two additional moments of the molecular weight distribution can be covered. 

For the purpose of illustration and simplicity, it is assumed here that we are dealing with a monodisperse system. This simph fication avoids having to define the moment of the molecular weight distribution that may be involved. The required moment to be used has been changed several times during the course of the development. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

An infinite set of moments is theoretically necessary to describe a molecular weight distribution but as a practical matter, knowing moments 0, 1, and 2 is usually adequate. The initial condition for all the moments is ii = 7o at T = 0. Solution gives... 

---