The Mathematics of Soft Matter

Combinatorial and graph algorithms play a critical enabling role in numerous scientific applications. The irregular memory access nature of these algorithms makes them one of the hardest algorithmic kernels to implement on parallel systems. To address the challenges, ExaGraph, the co-design center on combinatorial algorithms, was established to design and develop methods and techniques for efficient implementation of key combinatorial (graph) algorithms chosen from a set of exascale applications, targeting accelerator-enabled pre-exascale and exascale systems. I will present a brief overview of the latest work on multi-GPU systems for two prototypical graph problems — graph clustering and influence maximization — and demonstrate substantial gains in performance.

Interesting processes in nature such as the formation of a snowflake to a chemical reaction often involve large energy barriers. These are also molecular phenomena, and therefore, molecular simulations have become integral tools to investigate these processes. However, the large free energy barriers make it difficult to sample a statistically significant number of events. In our research, we study such processes and think about methodologies that can address the sampling issues. In addition, we are interested in using the data generated to tease out the molecular details that govern the process. In using heterogeneous ice nucleation as a case study, I will illustrate some of our successes and current challenges. We hope that the current challenges can be tackled with machine learning. I hope the workshop can nucleate collaborations toward this.

Understanding the molecular-scale origins of structure and phase transitions in multicomponent, hierarchically structured soft matter phases is a fundamental challenge. One application affected by these phenomena is liquid-liquid extraction, a predominant low-energy separations technique. This technique is a go-to process for materials recycling, including the recovery of rare earth elements. The complex solutions encountered in these applications feature structure at the nanoscale, which is understood to affect the subtle energetic balanced used to control the separation process. Further, a fundamental limitation to liquid-liquid extraction are undesirable liquid-liquid phase transitions that occur on sufficient loading of, e.g., lanthanide ions into extractant-containing organic phases.  Here, we discuss ongoing work to understand how solution structure and phase behavior are fundamentally connected in a way that can be understood by applying critical phenomena theory. By combining small angle X-ray scattering with molecular dynamics simulations, we demonstrate how organic phase structure over a wide range of process-relevant composition space is dominated by critical fluctuations. Scaling relations provided by critical phenomena theory provide a quantitative connection between aggregation in the organic phase and its phase behavior, deepening our understanding of both. Making use of these relationships will inform the design of more efficient separations processes and, more broadly, explore how critical phenomena manifest in complex solutions whose structure and thermodynamics are governed by combinations of a variety of intermolecular interactions that span length and energy scales.

Many complex macroscopic phenomena have their origins in microscopic, molecule and atom level interactions. Using techniques like molecular dynamics and machine learning, connections can be made between these two scales of observation. In this talk, I will showcase the power of these techniques in four cases. First, the solvation properties of carbon dioxide in ionic liquids are examined using one dimensional and two dimensional infrared spectroscopy. Second, the structure of the hydrated proton is elucidated by combining molecular dynamics with infrared and Raman spectroscopy. Third, molecular dynamics is used to examine the behavior of nanoparticles when interacting with biomolecules. Fourth, machine learning is used to develop design rules for creating sustainable nanotechnologies. In my new position at Haverford College, I am extending these techniques to study more complex related systems.

Soft materials including polymers, gels, bilayers, colloids, and molecular electrolytes are mostly amorphous structures spanning nanometers to microns, and even millimeters, and possess an extraordinarily broad spectrum of timescales. Their conformations can be modified with external stimuli such as temperature, ionic strength and external fields. Because of chemical versatility and ability to create structures with tailored features at different length scales, they are attractive materials to construct biomimetic materials and futuristic devices. I will discuss models to describe these challenging materials which are molecularly heterogeneous and larger than molecular heterogeneities in the medium are required to access their functions. 

Key biochemical processes, such as DNA replication, transcription, and repair, are initiated by molecular recognition events that involve binding proteins to other proteins and DNA. While the binding mechanisms are mainly unknown, we know that slow molecular fluctuations play a crucial role in the binding process as they define the leading pathway, following the Monod-Wyman-Changeaux model. Thus, a critical goal is to develop theoretical models that identify the slow, leading fluctuations inside noisy Molecular Dynamics (MD) simulations trajectories. Unfortunately, methods such the time-lagged Independent Component Analysis (tICA) identify slow fluctuations, but they do not provide an equation of motion.

To address this problem, we derived a Langevin equation in the coarse-grained coordinates of the amino acids. We started from a nonlinear Langevin equation in the lab-frame representation and transformed the equations of motion to a canonical body-centered representation, in which we linearize the internal dynamics. The resulting equation of motion includes rotation, translation, internal fluctuations, and coupling terms between these motions, all with inertia. We observe emerging underdamped modes and coupling between the protein’s conformation and its global diffusion, which may provide biologically important contributions to the dynamics. When coupling terms are negligible, the formalism reduces to the Langevin Equation for Protein Dynamics or LE4PD, whose modes have similarities with tICA. In those conditions, the LE4PD accurately identifies slow fluctuations and describes their dynamics, predicting time-correlation functions according to simulations.

The scattering of X-rays from liquid interfaces has led to a remarkable advance in our understanding of molecular and atomic ordering at liquid-vapor and liquid-liquid interfaces. These interfaces are soft in the sense that thermal fluctuations are important for their structure and functionality. These fluctuations have influenced the X-ray scattering determination of interfacial structure not only because molecules at liquid interfaces generally have greater dynamic disorder than those at solid interfaces, but also because of the challenge in understanding the X-ray diffuse scattering produced by this disorder. X-ray surface diffuse scattering has been used to measure the height-height correlation function of capillary waves at liquid interfaces with Å-scale sensitivity over a range of in-plane length scales from nanometers to micrometers. Today, however, distortions of liquid interfaces on the nanoscale that go beyond capillary waves are a topic of interest. These include perturbations of the interface that occur as ions are transported across liquid-liquid interfaces. This may happen, for example, as the result of electric fields that drag ions across interfaces or from dynamical processes of interfacial molecular aggregation that are catalyzed by the presence of the ions. We will discuss these issues from the perspective of an experimentalist using X-ray scattering to study liquid interfacial processes.

In post-combustion carbon capture, CO2 is separated from N2, a process that is performed conventionally with aqueous amines to effectively bind CO2. However, these solvents interact strongly with CO2 and must be scrubbed with large amounts of heat to regenerate the solvent by removing CO2, but the regeneration of the solvent is an energetically expensive process. Passive non-porous polymeric membranes offer an alternative, cost-effective technology for CO2 capture. Noncovalent interactions allow the CO2 molecules to diffuse through the membrane while N2 molecules do not interact with the polymeric matrix and do not permeate the membrane, which results in a lower energy cost separation. The interaction of CO2 with the polymer is one of the most important components, where CO2–philicity can be enhanced by introducing in the material certain functional groups (usually, Lewis bases).   We have performed computational studies on individual families of functional groups introduced in the repeating units of polymeric materials. Recently, we developed a novel molecular fingerprinting method based on persistent homology that can encode the geometrical and electronic structure of molecules for chemical applications. We have demonstrated its applicability on interactions between functional groups of materials and CO2/N2 gas molecules. Our quantum chemical calculations were performed on a small number of molecules (~220) for the generation of meaningful data. We have used these data in order to train a machine learning model for high-throughput virtual screening by predicting the properties of larger molecular databases (more than 130,000 entries) where quantum chemical data are not available.

Recycling of metals, such as rare earths, into valuable material relies on ion specific separation, basis of the hydrometallurgy [1]. Most of efficient methods known for separating ions are based on equilibria between complex fluids, typically between aqueous and organized organic phases.  Understanding the driving forces of the ion transfer is therefore a crucial issue to understand the properties of liquid-liquid interfaces between organic and aqueous phases, but also to assess the chemical potentials of the compounds involved. Here, we propose multi-scale approaches for calculating thermodynamics properties of ions in aqueous and organic phases directly comparable to the experiments ones only by taking into account the molecular properties of the solutes with no adjustable parameters.Based on the osmotic equilibrium method, activities and activity coefficients for aqueous electrolyte solutions composed of nitrate lanthanide salts have been successfully calculated [2]. In the meantime, thermodynamics and elastic properties in organic phase have been deduced from umbrella-sampling molecular dynamics simulations. We demonstrated that molecular complexes formed during solvent extraction self-assemble as reverse micelles, and therefore induce a supramolecular organization. In most of the cases, water molecules play an essential role in the stability of such aggregates in this non polar medium [3]. We also pointed out that the length of the solvent’s aliphatic chains has a minor effect on the elastic properties of the polar core of the aggregate, i.e., the spontaneous packing parameter and the effective bending rigidity [4-5]. Coupling these molecular properties with a mesoscopic water/oil interface model based on the microemulsion theory allows for accessing all the thermodynamic properties needed for chemical engineering, e.g., activity coefficients, association constants, ternary phase diagrams [6]. References: [1] Th. Zemb, C. Bauer, P. Bauduin, L. Belloni, C. Déjugnat, O. Diat, V. Dubois, J.-F. Dufrêche, S. Dourdain, M. Duvail, C. Larpent, F. Testard, S. Pellet-Rostaing, Colloid Polym. Sci., 2015, 293, 1.[2] M. Bley, M. Duvail, Ph. Guilbaud, J.-F. Dufrêche, J. Phys. Chem. B, 2017, 121, 9647.[3] Y. Chen, M. Duvail, Ph. Guilbaud, J.-F. Dufrêche, Phys. Chem. Chem. Phys., 2017, 19, 7094.[4] M. Duvail, S. van Damme, Ph. Guilbaud, Y. Chen, Th. Zemb, J.-F. Dufrêche, Soft Matter, 2017, 13, 5518.[5] S. Stemplinger, M. Duvail, J.-F. Dufrêche, J. Mol. Liq., 2022, 348, 118035.[6] S. Gourdin-Bertin, J.-F. Dufrêche, M. Duvail, Th. Zemb, Solv. Extr. Ion Exch., 2022, 40, 28.

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality d. The key observation is that nonlinear O(d)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars — scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.