Multiresolution methods for materials modeling via coarse-graining

Multiscale modeling of physical systems often requires the use of multiple types of simulations to bridge the various length scales that. need to be considered: for example, a density-functional theory at the electronic scale will be combined with a molecular-dynamics simulation at the atomistic level, and with a finite-element method at the macroscopic level. An improvement to this scheme would be a method which is capable of consistently simulating a system at multiple levels of resolution without passing from one simulation type to another, so that different simulations can be studied at a common length scale by appropriate coarse-graining or refinement of a given model. We introduce the wavelet transform as the basis for a new coarse-graining framework. A family of orthonormal basis, the wavelet transform separates data sets, such as spatial coordinates or signal strengths, into subsets representing local averages and local differences. The wavelet transform has several desirable properties for coarse-graining: it is hierarchical, compact, and has natural applications to approximating physical data sets. As a hierarchical method, it can be used to rescale a Hamiltonian to a desired length scale, and at the same time also rescales the particles of the system by creating "blocked" particles in the spirit of renormalization group (RG) calculations. The wavelet-accelerated Monte Carlo (WAMC) framework performs a Monte Carlo simulations on a small system which will be transformed into a block particle to obtain the probability distribution of the blocked particle; a Monte Carlo simulation is then performed on the resulting system of blocked particles. This method, which can be repeated as needed, can achieve significant speed-ups in computational time, while obtaining useful information about the thermodynamic behavior of the system. We show how statistical mechanics can be formulated using the wavelet transform as a coarse-graining technique. For small systems in which exact enumerations of all states is possible, we illustrate how the method recovers reasonably good estimates for physical properties (errors no more than 10%) with several orders of magnitude fewer operations than are required for an exact enumeration. In addition, we illustrate that errors introduced by the wavelet transform vanish in the neighborhood of fixed points of systems as determined by RG theory. Using scaling results from simulations at different length scales, we estimate the thermodynamic behavior of the original system without performing simulations on the full original system. In addition, we make the method adaptive by using fluctuation properties of the system to set criteria under which further coarse graining or refinement of the system is required. We demonstrate our method for the Ising universality class of problems. 2 We also examine the applicability of the WAMC framework to polymer chains. Polymers are quintessential examples of the need for simulations at multiple scales: at one end, we can study short chains using quantum chemistry methods; yet polymers can have relaxation times on the order of seconds or longer, and molecular weights of 106 or more. Even with modern computational resources, simulating behavior at long times or for long chains is still prohibitively expensive. While many approaches have been developed for studying such systems, many of these are specific to particular polymer chemistries, or fundamentally change the basic model of the system on an ad hoc basis. We also demonstrate how the WAMC framework can be adapted to study coarse-grained polymer chains represented as interacting lattice and off-lattice random walks. These walks can incorporate many of the same interactions as traditional "off-lattice" polymer models: excluded volume, stiffness, and non-bonded pair interactions. Coarse-graining the chain using the wavelet transform leads to each segment of the chain being replaced by a bead located at the center of mass of the segment. Interactions along the contour-such as stiffness potentials-are directly handled as well and incorporated as an internal configuration energy. Non-bonded interactions, such as excluded volume and non-bonded pair interactions, must be handled differently; we discuss possible approaches for handling these terms hierarchically within the WAMC framework. We present the details of the implementation of this algorithm, its performance for basic thermodynamic properties, as well as its connections to other effective coarse-grained models such as freely-jointed and Gaussian chains. In the development of our coarse-grained models, we have also discovered that the coarsegrained degrees of freedom-bond lengths, bond angles, and torsion angles-have distributions which are much more complicated than are typically employed in coarse-grained simulation techniques. This coupling of behavior is observed even when the model studied is as simple as a freely-jointed chain. In addition, Monte Carlo simulations have allowed us to establish the existence of numerical scaling laws for the overlap probabilities which we invert to determine the intraand intermolecular potentials as a function of the number of repeat units as well as the ratio of repeat unit (or bead) size to the bond length. These results are compared to the results obtained from analytical derivations based on the freely-jointed chain which show qualitative agreement between the two approaches. Consequently, we can use take the potentials determined by a simulation performed at one resolution of coarse-graining and derive from the scaling laws new potentials which describe the behavior at another resolution. This allows us to "tune" the WAMC algorithm to obtain results more efficiently than would be possible with algorithms that operate at fixed levels of coarse-grained resolution. Thesis Supervisor: Gregory C. Rutledge Title: Professor of Chemical Engineering Thesis Supervisor: George Stephanopoulos Title: Arthur D. Little Professor of Chemical Engineering