Stockmayer’s bivariate distribution

Stockmayer s bivariate distribution [43] describes, instantaneously, the joint distributions of chain length and chemical composition of hnear polymers made with coordination catalysts ... 

A similar approach can be used to predict the bivariate distribution of MWD and CCD for binary copolymers of propylene and a-olefins [1]. Stockmayer s bivariate distribution can be conveniently used to calculate the instantaneous MWD and CCD of binary, linear copolymers, such as the ones formed by copolymerizing propylene and a-oletos ... 

I- 00. Consequently, linear binary copolymers described by Stockmayer s bivariate distribution also obey Flory-Schultz s distribution for chain length. 

Stockmayer s bivariate distribution is an analytical expression describing the weight distribution of the kinetic chain length and the chemical composition for linear binary copolymers. This distribution quantifies the deviation from the average comonomer composition and molecular weight due to the statistical nature of copolymerization. 

Stockmayer s bivariate distribution for Unear binary copolymers can be expressed by the simple equations,... 

Two main approaches have been proposed to model Crystaf fractionation (1) models based on Stockmayer s bivariate distribution [58,80], and (2) models based on the distribution of chain crystallizabilities using Monte Carlo simulation [57,81]. 

Fig. 48 Comparison between experimental Crystaf profiles and Stockmayer s bivariate distribution [58]...

Fig. 8.21. Chemical composition distribution for several chain lengths, as described by Stockmayer s bivariate distribution. Distribution parameters r = 1000, Fa = 0.5, pirg = 1.

Fig. 26. Top bivariate distribution of chain sizes and compositions for poly(MMA-co-BA) obtained at high conversions. Bottom corresponding contour map. The random copolymer was fractionated by SEC and the average molecular weight and average composition of each fractions were determined by MALDI-TOF MS and NMR, respectively. The mol fraction of MMA was calculated according to the Stockmayer s theoretical prediction. Reproduced from [142] with permission...

For the case of linear binary copolymers, the instantaneous bivariate chain length and chemical composition distribution, w(r, y), is described by Stockmayer s distribution [Eqs. (16)-(19)] [34]. 

Similarly, the bivariate molecular weight and chemical composition distributions of binary copolymers made with multiple-site catalysts can be described as a weighted superposition of Stockmayer s distributions [39]. If we consider only the chemical composition component of the distribution, as described by Eq. (22), the distribution of polymer made with a multiple-site catalyst becomes Eq. (28). 

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