(164i) Efficient Modeling of the Bivariate Molecular Weight Distribution – Copolymer Composition Distribution in SAN Copolymerization Using Parallel Computing

Styrene-co-acrylonitrile (SAN) is a multipurpose copolymer whose applications primarily involve transparent structural materials as well as a blend component. SAN copolymers are typically manufactured by solution, suspension, or emulsion processes. Industrially useful products are single-phase transparent materials. The most important molecular parameters for the solid-state properties and processing performance of this copolymer are the weight fraction of acrylonitrile (w_AN) and the molecular weight distribution (MWD). Increasing molecular weights and w_AN contents improve toughness and solvent resistance but impact negatively on cost and processability; besides, excessively high w_AN levels are not good for color properties [1].

The typical free-radical manufacturing processes of SAN yield polymers with a broad MWD. Additionally, there is a composition distribution superimposed on the MWD. Compositional drift is larger for polymerizations that are run to high conversion and do not have the azeotropic composition (76:24 w/w S/AN). The average composition or molecular weight does not uniquely define the expected properties of the material. Therefore, knowledge about the copolymer composition distribution (CCD) and the MWD is important to determine the source of variation in SAN copolymer properties. For instance, composition drift towards high acrylonitrile-containing fractions can lead to undesirable yellow color, and excessively broad composition drift can cause opacity and brittleness in the material due to phase separation. A multidimensional characterization that simultaneously measures the MWD and the CCD provides insights into structure-property relationships and the control of the polymerization process [1].

Reversible deactivation radical polymerization (RDRP) is a new family of polymerization techniques that have the capability of synthesizing polymers with relatively uniform molecular structures in fairly industrially-friendly operating conditions. This technique allows manufacturing polymers with more uniform MWD or CCD than typical free-radical processes. One of the most versatile RDRP techniques is activators regenerated by electron transfer (ARGET) atom transfer radical polymerization (ATRP). ARGET ATRP has several advantages, such as a wide range of polymerizable monomers, the low cost and availability of reagents, and the possibility of achieving high-molecular-weight polymers. Moreover, it tolerates low levels of oxygen [2]. The synthesis of SAN by ARGET ATRP allows enhancing the properties of this copolymer by achieving a more uniform molecular structure of the population of chains in a polymer sample [3].

Mathematical modeling is a powerful tool for gaining insight into a process, improving production times, and optimizing process operation. However, modeling polymer processes is challenging. A mathematical model capable of predicting a univariate distribution of polymer properties, like the MWD, involves an extremely large (infinite in theory) system of coupled, stiff differential equations for the population of chains characterized by their length (i.e., number of monomer units). Different methods have been proposed for tackling this task. The computation of multivariate distributions, for instance, the joint MWD-CCD, is even more complex in formulation and model size. For this reason, a reduced number of approaches have been reported [4]. One of the main problems when solving these models is not only the complexity in their formulation but also the efficiency concerning computation times, data processing, and CPU and RAM memory requirements. A computational technique that has the potential to speed up the execution of mathematical models is parallel computing. To solve a particular problem using parallel computing, the problem needs to be divided into independent sections, so that each section can be solved concurrently with the others. Not all the methods proposed for calculating multivariate distributions meet this condition. Besides, developing an algorithm designed for parallel computing generally requires more effort and time than one designed for serial computing. The use of the appropriate programming language is very important to facilitate this process.

A mathematical model of the ARGET ATRP copolymerization of SAN that could compute the bivariate MWD-CCD efficiently in terms of CPU resources and time would be a powerful tool for the understanding and operation of this process. The probability generating function (pgf) technique [5, 6] is a very efficient method for modeling multivariate distributions and is particularly suitable for parallel computing. The pgf method has been used previously by the authors for modeling univariate and multivariate polymer distributions [5, 7]. This technique is based on solving the transformed pgf equations of the infinite population balances that describe a certain process and subsequently inverting the resulting pgf to recover the desired distribution. In this work, a model of the ARGET ATRP copolymerization of SAN developed by the authors [8, 9] is used to study the evolution of the joint MWD-CCD along the reaction path under different operating conditions. In addition to the bivariate MWD-CCD, the model calculates the overall MWD, the overall CCD, average molecular weights and composition, and conversion. The detailed mathematical model is programmed in Julia, which is an open-source programming language easy to use and that performs very fast calculations. The pgf is used to process the infinite population balances of the system. The code developed in Julia allowed predicting the distributed copolymer properties in a very efficient manner. The application of the pgf technique results in a mathematical model consisting of 2 352 904 independent blocks of 50 pgf equations each. Therefore, the total model size is 117 645 200 differential equations. The solution of a model of this size, using an implicit DAE solver as required by the stiff system of equations, is infeasible in a regular PC. On the other hand, the parallelized implementation of the solution of the independent blocks in Julia, using a laptop equipped with an Intel Core i5-8250U processor and 12 GB RAM, demanded about 6 h CPU time. The MWD-CCD is computed on a very fine grid of molecular weights and composition, which results in a very smooth surface for the bivariate distribution.

The model allows obtaining very detailed knowledge of the evolution of the copolymer microstructure during the reaction for different operating conditions. Under azeotropic conditions (76:24 w/w S/AN), the model predicts a MWD-CCD that is narrow in composition around the azeotropic value of 24 % w AN for all molecular weights; the distribution in molecular weight is also narrow. Along the reaction path, the MWD-CCD remains narrow; the mean value of the CCD remains the same and the MWD evolves towards higher molecular weights. This uniformity in the MWD-CCD would allow us to expect good final properties in the copolymer. For an initial condition richer in AN (10:90 w/w S/AN), the MWD-CCD has a shape similar to that in the previous case despite the non-azeotropic conditions up to 40 % conversion, with a CCD of about 40 % w AN. However, thereafter a new population of chains develops that is much richer in AN, lower in molecular weight, and with a significant composition drift. As a result, the final MWD-CCD is extremely non-uniform and would impact negatively on the copolymer properties. However, the polymer microstructure would be quite uniform if the reaction is stopped at 40 % conversion. The type of multimodal, broad MWD-CCD predicted by the model for non-azeotropic conditions has been observed experimentally in emulsion free-radical copolymerization of SAN [10].

These results show that the pgf model parallelized in Julia provides a very detailed description of the SAN molecular structure synthesized by ARGET ATRP in reasonable CPU times, which makes it a powerful tool for the process operation.

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