(484e) Mathematical Modeling of Bivariate Distributions of Polymer Properties Using 2-D Probability Generating Functions

The process of addition of monomer units to form polymer chains in a polymerization reaction is governed by probabilistic issues related to changing process conditions. As a result, a mixture of polymer chains with different size and/or structure will compose the final reaction product. Hence, polymer samples usually present distributions of the different molecular properties (i.e. molecular weight distribution (MWD), copolymer composition distribution (CCD), short (SCBD) and long (LCBD) chain branching distribution, particle size distribution (PSD), etc.). Information about these distributions is very important because most of the processing and end-use properties of polymers depend on them. In many cases, a proper characterization of a polymer sample will require simultaneous information on more than one property distribution. For instance, MWD ? CCD is important for copolymer systems; MWD ? SCBD and/or LCBD are needed in the case of branched polymers. This requirement must be taken into account in the development of advanced mathematical models of polymer systems. Therefore, it is extremely important to count with a methodology that allows the joint modeling of multiple distributions in different polymer processes.

Up to day, different methods for the calculation of single distributions have been reported. The MWD has been one of the more extensively studied. Works about modeling of other polymer properties distributions, such as the CCD and the PSD, have also been published. However, the joint prediction of multiple distributions is an area of limited development. In some of the reported works in this line, the numerical fractionation technique has been used for predicting the bivariate molecular weight ? long chain branching distribution.^[1] This method consists in dividing the total population of polymer chains into classes according to the number of branching. However, with this method the reconstruction of the MWD at high monomer conversions and high branching content requires a large number of classes in order to reduce approximation errors. This implies increasing the model size and hence the computational load. In recent works, 2-D sectional grid methods and Monte Carlo methods have been employed for computing molecular weight ? copolymer composition distributions^[2] and molecular weight ? long chain branching distributions.^[3] These two approaches provided good quality performances, but the first one has been reported to require special computational skills to overcome its numerical complexity, and the second one can be computationally expensive.^[3] Other recent approaches include the combination of border density functions with the h-p-Galerkin method,^[4] and the Markov chains method,^[5] for the prediction of the bivariate molecular weight ? long or short chain branching distributions.

In this work, we present a new approach for the prediction of bivariate distributions of polymer properties, based on the transformation of population mass balances by means of probability generating functions (pgfs). In previous works, our research group has developed the pgf technique as a comprehensive numerical tool for the prediction of the MWD in free radical polymer processes.^[6,7] The pgf method was developed as a general modeling tool that can be equally applied to different systems. It has provided excellent results in terms of accuracy, easy of implementation and computational effort in models for simulation and optimization activities. In its previous state of the art, this technique employed univariate pgfs, which allowed modeling a single distribution. Here, we present an extension of this technique to 2-D pgfs, in order to model bivariate distributions.

The pgf method is based on the transformation of the set of infinite population mass balances governing a polymer process into the pgf domain, obtaining a finite set of equations in which the dependent variable is the pgf transform of the distribution. This is followed by the solution of the transformed equations and the inversion of the pgf transforms in order to recover the desired distribution. Focus of this work is placed on the numerical inversion of the pgf transforms, which is the critical stage of the method. A two-step procedure, adapted from the Laplace transform inversion theory,^[8] was employed for the inversion of the 2-D pgf transforms. This step procedure allowed the use of well-established numerical inversion algorithms of univariate pgfs in each of the inversion stages.

Two different inversion algorithms of univariate pgfs^[6,7] were thoroughly tested for the proposed inversion scheme of 2-D pgfs. For this purpose, pgf transforms were obtained from known distributions by applying the 2-D pgf definition. Then, these transforms were inverted using the proposed procedure, obtaining the recovered distributions that were compared against the original ones. A set of actual bivariate polymer distributions covering a very wide range of shapes was employed for this analysis. At the same time, criteria for selecting appropriate values of the method parameters were developed. Very good results were obtained, showing that the inversion scheme provides excellent accuracy.

Finally, as an illustrative example of the overall pgf technique, the methodology was applied to model the molecular weight-copolymer composition distribution in a free radical copolymerization process.

References

[1] P. Pladis, C. Kiparissides, Chem. Eng. Sci. 1998, 53, 3315.

[2] A. Krallis, D. Meimaroglou, C. Kiparissides, Chem. Eng. Sci. 2008, 63, 4342.

[3] D. Meimaroglou, A. Krallis, V. Saliakas, C. Kiparissides, Macromolecules 2007, 40, 2224.

[4] R. A. Hutchison, Macromol. Theory Simul. 2001, 10, 144.

[5] L. Christov, G. Georgiev, Macromol. Theory Simul. 1995, 4, 177.

[6] M. Asteasuain, C. Sarmoria, A. Brandolin, Polymer 2002, 43, 2513.

[7] M. Asteasuain, A. Brandolin, C. Sarmoria, Polymer 2002, 43, 2529.

[8] P. P. Valkó, J. Abate, Appl. Numer. Math. 2005, 53, 73.