Coagulation equations with gelation
Smoluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernel K/sub i/j models the bonding mechanism. For different classes of kernels we derive criteria for the occurrences of gelation, and obtain critical exponents in the pre- and postgelation stage in terms of the model parameters; we calculate bounds on the time of gelation t/sub c/, and give an exact postgelation solution for the model K/sub i/j = (ij)/sup ..omega../ (..omega..>1/2) and K/sub i/j = ..cap alpha../sup i/+j (..cap alpha..>1). For the model K/sub i/j = i/sup ..omega../+j/sup ..omega../ (..omega..<1, without gelation) initial solutions are given. It is argued that the kernel K/sub i/japprox. (ij)/sup ..omega../ with ..omega..approx. =1-1/d (d is dimensionality) effectively models the sol-gel transformation is polymerizing systems and approximately accounts for the effects of cross-linking and steric hindrance neglected in the classical theory of Flory and Stockmayer (..omega.. = 1). For all ..omega.. the exponents, tau = ..omega..+3/2 and sigma = ..omega..-1/2, ..gamma.. = (3/2-..omega..)/(..omega..-1/2) and ..beta.. = 1, characterize the size distribution, at the slightly below the gel point, under the assumption that scaling is valid.
- Research Organization:
- Institut voor Theoretische Fysica, Fijksuniversiteit Utrecht, The Netherlands
- OSTI ID:
- 5589438
- Journal Information:
- J. Stat. Phys.; (United States), Vol. 31:3
- Country of Publication:
- United States
- Language:
- English
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