Molecular Weight Distribution Functions

Tung, L. H., Method of calculating molecular-weight-distribution functions from gel permeation chromatogram, /. Appl. Polym. Sci., 10, 375, 1966. 

More recently, the same author [41] has described polymer analysis (polymer microstructure, copolymer composition, molecular weight distribution, functional groups, fractionation) together with polymer/additive analysis (separation of polymer and additives, identification of additives, volatiles and catalyst residues) the monograph provides a single source of information on polymer/additive analysis techniques up to 1980. Crompton described practical analytical methods for the determination of classes of additives (by functionality antioxidants, stabilisers, antiozonants, plasticisers, pigments, flame retardants, accelerators, etc.). Mitchell... 

Molecular weight distribution function for the case where the length of the growth stage is short compared to the residence time in reactor. (Reprinted with permission from Chemical Reactor Theory, by K. G. Denbigh and J. C. R. Turner. Copyright 1971 by Cambridge University Press.)... 

Hence, is the statistics average Molecular weight calculated from the number Molecular weight distribution function, N(M). Mn is the statistical average Molecular weight calculated from the weight... 

Molecular weight distribution function, W (M). is the statistical average Molecular weight... 

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). 

One advantage of the procedure delineated above is that it permits the complete molecular weight distribution function to be calculated, sometimes analytically, whatever the termination mechanism. Of course, the lower moments of the distribution function can also be readily calculated ... 

Precipitation chromatography, as proposed by Baker and Williams and improved in our laboratory, has a systematic error caused by the nonuniformity of the fractions. This error can be eliminated if the fraction nonuniformity is determined by a special kind of mixing experiment. Another interfering effect which is produced in anionic polymerizates by traces of impurities can be eliminated by testing the BW fractions by GPC. Exactly determined molecular weight distribution functions of samples which are synthesized under well defined kinetic conditions provide much information about elementary reactions of the polymerization process. This is demonstrated by some examples of the anionic polymerization of styrene and methyl methacrylate. 

The locus of failure depends on parameters such as the state of crosslinking of PUa before contact and the characteristics of the HTPB used (molecular weight distribution, functionality, etc.) but also on testing conditions (peel rate, air or liquid). It can be related to the properties of the material close to the interface. 

Figure 9.12 Molecular weight distribution function for situations in which the duration of the growth stage is short compared to the residence time in the reactor (e.g., free-radical polymerization). The steady-state operating conditions in CSTR A and CSTR B correspond to relative monomer concentrations of 0.1 and 0.354, respectively. The conditions in the batch reactor involve a transition from a relative monomer concentration of 1.0 to one of 0.1. [Adapted with permission of The Royal Society of Chemistry from the contribution of K. G. Denbigh to Trans. Faraday Soc., 43,648 (1947). Copyright 1947, The Royal Society of Chemistry.]...

The principle of equal reactivity states that all functional groups of the same species are equivalent, that is, the reactivity of all functional groups on the polymers is the same irrespective of their molecular weight and structure [1]. In other words, there is an intrinsic reactivity of the polycondensation. We shall derive the molecular weight distribution function of nonlinear polymers under the assumption of equal reactivity. 

The molecular weight distribution function of the sol part is therefore given by replacing a by a in (3.19). The weight fraction of the sol in the postgel regime is then calculated to be... 

Fig. 3.10 The molecular weight distribution function Wm and the gel fraction w for polycondensation reaction of trifunctional monomers R A3 plotted against the reactivity.

Molecular weight distribution function of the binary mixtures R A/ /R B ... 

To obtain the molecular weight distribution, the fractions of known mass must be characterized by a molecular weight technique. The integral and differential molecular weight distribution may be evaluated using a method (154) that assumes no particular molecular weight distribution function for the fraction or treatments that assume a particular model for the molecular weight distribution of the fractions (155). Fractionation may also be achieved by crystallization from dilute solution (156). 

An exact molecular weight distribution function has been determined for at least one commercial polymer (unpublished work by Schoon quoted by Hofmann, 1964) (Fig. 6.3). 

A very elegant version of this approach is the so-called continuous thermodynamics [56-59]. It can be considered as a reformulation of the classical thermodynamic relationships that allows for using continuous molecular weight distribution functions W M) rather than the mole fractions of discrete polymer components. Using this approach, e.g., (30) becomes ... 

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