(181ar) Accelerated Design of Molecular Additives for Polymer Crystallization

The overwhelming preponderance of plastics in use today are derived from petroleum sources and are related to polyolefins. Of these, fully 2/3 are semicrystalline, that is, comprise morphologies that are partially crystalline and partially noncrystalline. The morphology and properties of the semicrystalline material can be manipulated empirically at the time of fabrication through processing variables such as temperature or pressure, or through the use of additives that promote a particular crystal polymorph or morphology. Some additives serve as nucleating agents, increasing the number density of crystallites, selecting a preferred crystalline polymorph, or altering the propensity for lamellar crystallites to grow in particular orientations. At present, the industrial design and selection of nucleating agents, clarifiers, and additives is still largely a matter of empiricism. First, it remains a challenge to establish by experiment alone the mechanism by which a particular
nucleating agent works. Second, even promising nucleating agents may prove difficult to work with, due to complications involving particle size or shape, dispersion, or thermal stability. State-of-the-art molecular simulations offer an alternative strategy by which to avoid the experimental difficulties at the screening stage, while at the same time revealing the most promising structures for the development of the desired morphology in the lab.

Molecular simulations have the potential to serve as a basis for the design of additives for polymer crystallization, by embedding these simulations within a model-based optimization. A challenge of finding the solution of these optimizations is that the number of potential degrees of freedom that define the class of potential molecular additives is large and include both continuous and discrete variables. These optimizations become especially challenging to solve numerically when the molecular simulations are stochastic and the optimization objective is inherently stochastic, due to the high computational cost of individual molecular simulations, and that many molecular simulations would be required to determine average values of the optimization objective with sufficiently small variance.

In this work, we examine the computationally expensive and inherently stochastic optimization of molecular additives for the objective of minimizing the induction time of polymer crystal nucleation on a substrate based on non-equilibrium molecular dynamics (NEMD) simulation. The large number of degrees of freedom is addressed by separating the molecular additive design problem into three steps. In the first step, the number of degrees of freedom in the model-based molecular additive design problem is reduced by
parameterizing the molecular additives in terms of a united-atom force field (UAFF) model with unknown parameters. This step is supported by recent work that explored heterogeneous nucleation of n-alkanes on known additives by using a UAFF model [1, 2]. In the second step, a numerical algorithm searches over the unknown parameters in the UAFF model to optimize the design objective. In the third step, a numerical algorithm searches for specific molecules that are well described by the UAFF model for parameters determined in the second step. Most of the presentation focuses on developing numerical algorithms and presenting results for the second step in this strategy, as the first step has already been demonstrated for our materials of interest [1, 2] and the second step is much more computationally challenging than a well-formulated third step.

The nucleation induction time serves as the objective function in the numerical optimization in the second step. It has a very high level of stochasticity— so much so that many NEMD simulations would be needed to compute a small standard error on the mean nucleation induction time. For example, an estimate for the mean nucleation induction time with sufficiently low variance for decision purposes took about four days on a many-processor
computer [1, 2]. Based on past experience with similar materials optimizations (e.g., see [3] and citations therein), the numerical optimization would require more than a hundred objective function evaluations, implying that such a direct approach for computing the solution to an optimization over the expected value of the objective function would require more than a year.

We explore two alternative indirect approaches to solving the numerical optimization that are much more computationally efficient. One approach reformulates the optimization in the form of a maximum-likelihood estimation that enables the stochasticity to be handled more efficiently. Maximum-likelihood estimation is a well-established method for estimating the parameters in a model in which data contain noise [4]. The vast majority of maximum-likelihood estimation applications are in cases where the model is deterministic and the stochasticity only appears in the measured data. In recent years, maximum-likelihood estimation has been applied in cases where the experimental data and the model are both stochastic, such as when the model is described by a Chemical Master equation (e.g., [5, 6]). Although the purpose of our optimization is not parameter estimation per se, the optimization can be reformulated to employ the same mathematical framework, in which the maximum-likelihood estimation has stochasticity in both the data and model (in our case, as an output of an NEMD simulation used in the computation of the optimization objective function).

An alternative indirect approach to solving the optimization in the second step is to employ a multi-step algorithm that is a variant of stochastic response surface methodology. Response surface methodology was originally developed by George E.P. Box and K.B. Wilson for solving optimizations in which the model is deterministic and the stochasticity only appears in the data [7]. The methodology was later extended to include the case in which the model has some form of stochasticity (e.g., [3, 8, 9] and citations therein). We employ a mathematical formulation that is a variation on stochastic response surface methodologies. The variance in the estimate of the stochastic objective function is explicitly tracked when making decisions on how many and where function evaluations should be made to constuct a stochastic response surface. The stochastic response surface is fed to a fast intermediate optimization, which is then used in further refinement of the stochastic response surface. The overall algorithm is most directly an extension of algorithms that were adapted and applied to the optimal design of multiscale materials systems (e.g., see [3, 9] and citations therein).

The approaches are applied to the optimization of molecular additives for the objective of minimizing the mean induction time for the nucleation of polyethylene on the additive. The nucleation induction time was computed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS, Sandia National Laboratory, https://lammps.sandia.gov/). The
molecular additive was characterized by the Stillinger-Weber force field. The computation of nucleation induction times has been reported previously for many sets of force field parameters for tetrahedral [1] and graphene-like [2] materials; these computational data offer a challenging test case for the assessment of optimization algorithms for this type of molecular design problem. We compare the two indirect optimization approaches, and demonstrate efficient sampling of the computationally expensive stochastic calculations to
determine the most favorable additive properties for polymer crystallization. The demonstration of the effectiveness of the two stochastic optimization approaches for accelerating the design of new molecular additives for this application suggests that the approaches have promise for other applications in which the model is in the form of a computationally expensive simulation that generates a stochastic output that is the objective of the design optimization.

References
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