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Multivariate Composition Distribution in Free‐Radical Multicomponent Polymerization, 1
Author(s) -
Tobita Hidetaka
Publication year - 2003
Publication title -
macromolecular theory and simulations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.37
H-Index - 56
eISSN - 1521-3919
pISSN - 1022-1344
DOI - 10.1002/mats.200350009
Subject(s) - component (thermodynamics) , distribution (mathematics) , mathematics , distribution function , composition (language) , chain (unit) , gaussian , polymerization , multivariate normal distribution , statistical physics , combinatorics , polymer , thermodynamics , multivariate statistics , chemistry , physics , computational chemistry , mathematical analysis , statistics , quantum mechanics , organic chemistry , linguistics , philosophy
Statistical multicomponent polymerization is a typical example of a Markovian process for which the generating function approach can be applied. Up to the present, generating functions have been used mainly to obtain analytical solutions. However, recent advances of computer software capable of handling symbolic calculations can throw new light on the old mathematical technique. After formulating the equations representing the instantaneous composition distribution of polymers for a given chain length, r , the illustrative numerical calculations are conducted by using the symbolic calculator. For a multicomponent polymerization consisting of more than two components, the second component distribution is dependent on the composition of the first component ( F 1 ), which is represented by the conditional probability given r and F 1 , $Comp\left( {F_2 \left| {F_1 ,r} \right.} \right)$ . It is found that $Comp\left( {F_2 \left| {F_1 ,r} \right.} \right)$ is well approximated by the Gaussian distribution with the variance following the relationship, $r\sigma _{2;1,r}^2 = A + B/r$ , as in the case of the first component distribution $Comp\left( {F_1 \left| r \right.} \right)$ , where A and B are the constants. With the knowledge of chain length distribution, it is now possible to conduct the full analysis of multivariate distribution of chain length and compositions for multicomponent free‐radical polymerization.Bivariate distribution of composition F 1 and F 2 for chain length r = 100 in a three‐component system.