Moments of molecular weight distribution

M0 total number of polymer molecules XP (zeroth moment of molecular weight distribution). 

Z-average molecular weight, M, Represents the second moment of the weight distribution for a sample and is mathematically... 

Finally we define a quantity known as the kth moment of the distribution. In terms of molecular weight. 

The breadth of the molecular weight distribution is often discussed in terms of the dispersity (Z>) and is expressed in terms of the moments as shown in eq. 15 ... 

This distribution is known as the Schultz-Flory or most probable distribution.2S The moments of the molecular weight distribution are ... 

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. 

Also, the zeroth moment of the differential molecular weight distribution, DMWD, may be obtained by integration of the simplified equation ... 

The moments of the molecular weight distribution are defined as either... 

The present section analyzes the above concepts in detail. There are many different mathematical methods for analyzing molecular weight distributions. The method of moments is particularly easy when applied to a living pol5mer polymerization. Equation (13.30) shows the propagation reaction, each step of which consumes one monomer molecule. Assume equal reactivity. Then for a batch polymerization. 

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... 

Advanced computational models are also developed to understand the formation of polymer microstructure and polymer morphology. Nonuniform compositional distribution in olefin copolymers can affect the chain solubility of highly crystalline polymers. When such compositional nonuniformity is present, hydrodynamic volume distribution measured by size exclusion chromatography does not match the exact copolymer molecular weight distribution. Therefore, it is necessary to calculate the hydrodynamic volume distribution from a copolymer kinetic model and to relate it to the copolymer molecular weight distribution. The finite molecular weight moment techniques that were developed for free radical homo- and co-polymerization processes can be used for such calculations [1,14,15]. 

C. H. Bamford and H. Tompa, J. Polymer Sci.j 10, 345 (1953), first derive the moments of the distribution for the case of chain transfer to polymer. They then obtain the molecular weight distribution from these moments by appropriate mathematical methods. Their procedure should be applicable to a wide variety of polymerization mechanisms. 

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