Stockmayers Bivariate Distribution

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 [16] derived this distribution with the aid of some approximations from a general theory of chain copolymerization described earher by Simha and Branson [17]. Stockmayer s distribution has been foimd to be a useful tool for understanding chain microstructures of several copolymers [18-22]. 

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

This distribution function provides important insights on the detailed composition distribution of linear binary copolymers. We will use it frequently to interpret Crystaf and Tref fractionation results in the next sections of this review. Therefore, it is useful to discuss some of its main characteristics in this section. 

Integrating Eq. 5 over all chain lengths, one obtains the equation describing the CCD component of Stockmayer s distribution, independently of chain length  

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For long linear chains the second condition is supported by the Stockmayer bivariate distribution (8,9) which shows the bivariate distribution of chain length and composition is the product of both distributions, and the compositional distribution is given by the normal distribution whose variance is inversely proportional to chain length. 

Stockmayer, W.H., Bivariate Distribution of chain lenghts and compositions, /. Chem Phys, 13, 199 (1945). 

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

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...

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. 

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]...

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]. 

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.

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). 

The bivariant Stockmayer distribution has been used for copolymers. In this case, the product of a Schulz-Flory distribution with respect to molar mass (or segment number) and of a Gaussian distribution with respect to chemical composition y is formed. If the copolymer consists of two kinds of monomers the chemical composition is defined as the segment fraction of one of the monomers within the copolymer. For the segment number and chemical composition the Stockmayer distribution is given by ... 

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