(200l) Modeling of 2D Distributions of Polymer Properties Using 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 need 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.

The prediction of the distribution of a single property, in most cases the MWD, has been studied extensively. However, the treatment of joint distributions with more than one independent coordinate leads to problems for which solution techniques are scarce. Reported methods can be divided in two groups: stochastic [1-5] and deterministic [2, 6-8] methods. Stochastic approaches are mainly represented by the Monte Carlo technique. The advantage of this approach is that it is relatively simple to implement and can provide very detailed information on the microstructure of the polymer and the topological architecture of the chain, which is not available with deterministic methods. However, a major drawback of this technique is the high computational cost required to obtain accurate results, even in modern parallelized systems. The deterministic methods require less computer time and storage capacity stochastic methods. They are also more suitable for identification and optimization, where it is necessary to have smooth and differentiable mathematical structures. These methods are based on the solution of the infinite population kinetic balances describing the polymer system.

In previous works [9,10] we presented a deterministic 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). The pgf method was developed as a general modeling tool that with potentiality to be applied to different systems. It provided excellent results in terms of accuracy, simplicity of implementation and computational effort, in models for simulation and optimization activities. In particular, modeling of the bivariate MWD – CCD in copolymerization systems was considered.

A feature of the 2D distributions to which the pgf method has been successfully applied so far is that the two independent variables are of the same order of magnitude. However, the pgf technique presented numerical problems with distributions in which the independent variables had values within very different ranges. Typical examples of these distributions are the MWD – CBD. These molecular properties are very important, for example, for highly branched polymers like low density polyethylene or for linear copolymers of ethylene obtained in coordination polymerization. In this work, we present an update of the pgf technique that overcomes this drawback.

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. Difficulties presented by the previous version of the pgf technique were due to the numerical methods that were employed for the inversion of the pgf. Here we present two new inversion methods of 2D pgf of higher accuracy that allow applying the pgf method for modeling bivariate distributions with independent variables of different size.
A two-step procedure is used for the inversion of the 2D pgf transforms, which consist in inverting each dimension stepwise using univariate inversion algorithms in each step. Two univariate inversion methods are considered. The first one is a highly accurate algorithm that takes into account complex transformed variables. The second method uses only real values of the transformed variable for the inversion and is less computationally demanding. Based on these univariate algorithms, two 2D pgf inversion methods were constructed. The first of them employs the complex-variable method for inverting the two dimensions. On the other hand, the second method employs the complex-variable algorithm for inverting the independent dimension of smallest value, and the real-variable one for the largest independent dimension. The two proposed 2D pgf inversion methods were thoroughly tested. For this purpose, pgf transforms were obtained from known distributions by applying the 2D 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 with both methods. The first one showed excellent accuracy, but it required for the inversion a large number of pgf values that may result in a model whose size might be prohibitory large in systems of high molecular weight. The second method was less accurate than the other one. However, precision was still very good employing a reasonable amount of computational resources.

Finally, as representative examples of the 2D pgf technique, the methodology was applied to model the MWD – SCBD in a mixed-metallocene polymerization of ethylene and the MWD – CCD in the copolymerization of methyl methacrylate and vinyl acetate.

References

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