For a number of polymerization models, we have developed equations that enable one to follow in time the moments of the polymer size distribution and the conversion. The exact and approximate solutions for the leading moments of the size distribution are presented for polymerization with initiation, propagation, and termination by combination. Approximate solutions for polymerizations with the additions of terminotion by disproportionation, chain transfer to monomer, and chain transfer to solvent show how these reactions decrease the average polymer size. From a simple model that exhibits autoacceleration o concomitant increase in the average polymer size also occurs.
Although the general kinetic equations of radical polymerization carry complete knowledge of the polymer size distribution as well as the rate of polymerization, they are tractable for relatively few polymerizations (1 to 5 ) .
Usually in a size distribution study of polymerization, the equations are simplified by the assumption of a stationary state (6).
To make a large number of polymerizations amenable to analysis without such a limiting assumption, we focus our attention on the moments of the size distribution rather than on the size distribution itself. As a workicg matter, we are often content with knowing just the leading moments. This presupposes, of course, that we are describing a unimodal distribution. Although experimental evidence on this point is lacking, we expect on physical grounds that most polymerizations yield unimodal distributions.
In developing equations in the moments we choose for analytical convenience to work with the continuous instead of the exact discrete kinetic equations, that is, we take the polymer size (chain length) to be a continuous rather than a discrete variable. Zeman and Amundson (2, 3 ) have previously used the continuous approach, but here we develop the equations in the framework given by Hulburt and Katz (7) in their study of particulate systems. Consequently, we can write the continuous kinetic equations directly from the pertinent chemical reactions.
Among the reactions which may occur in polymerization, we consider the following:
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Initiation, where initiator I decomposes into the primary radical Ro a! I + 2Ro 2. Propagation, where monomer M reacts with radical Ri containing i monomer units, to form a radical with i + 1 units B Ri + M-+ Ri+l i = 0,1, 2 , . . .
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Chain transfer to monomer, where monomer M reacts with Ri to form terminated polymer Pi and radical R1