Moments in polymerization kinetics

Our earlier discussion showed that moments are a straightforward and valuable means of characterizing a PCLD. Bamford and Tompa [16] suggest the use of moments in the analysis of polymerization kinetics since for some systems the set of equations describing the polymerization can be reduced to a small finite set. Hulburt and Katz [45], following Bamford et al [27], have shown that a distribution can be reconstructed using a finite number of moments. This stems from the observation made by Ray [31] that a distribution is generally characterized by only a finite number of moments. The example discussed in section 3.4.4 showed that a Poisson distribution can be completely characterized by only the first moment of the PCLD. 

Just how many moments are necessary to provide a unique determination of the differential PCLD One example has been shown where the number is finite, but more complex polymerization mechanisms certainly demand more information. Consider, for example, a radical chain group polymerization with a termination reaction as well as a number of transfer reactions. Given all the kinetic parameters (in this case the rate constants and activation energies) it is possible to calculate the distribution. So if kinetic parameters are required for the characterization, measuring one or more moments, enables the rate equation to be used to determine a set of kinetic parameters, provided N N. Confidence in the accuracy of the moments dictates whether more moments are essential for complete characterization of the distribution. 

It has been shown by Bamford and Tompa [16] that generalized Laguerre polynomials provide a suitable method of obtaining the PCLD from the moments. 

To do this, the PCLD is expanded in an orthogonal series of special functions, in this case Laguerre polynomials. The coefficients of the series are given in terms of the moments of the original CLD, as illustrated below. 

Any function which ranges from 0 to oo (and satisfying certain conditions of continuity) can be expressed in terms of these polynomials. In particular, P can be expressed as 

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