Schedule for: 24w5159 - Mathematics of Multiscale and Multiphysics Phenomena in Materials Science

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Meet and Greet in TCPL Lounge (PDC 2nd Floor).

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Nonlocal interaction arises naturally as a generic feature of model reduction for multiscale processes. Recently, there has been much interest in models associated with a finite horizon parameter that characterizes the effective range of nonlocal interactions. For such models defined on a bounded domain, an important issue is their proper mathematical formulation near or at the domain boundary. This lecture offers an overview of related works. We discuss a few possibilities for imposing suitable local or nonlocal boundary conditions to complement nonlocal interactions prescribed in the interior of the domain. We also present recent development of the mathematical and numerical analysis of the resulting problems and applications to the coupling of different models.

We consider a class of nonconvex energy functionals that lies in the framework of the bond-based peridynamic model of continuum mechanics. The energy densities are functions of a nonlocal strain that describes deformation based on pairwise interaction of material points. We apply variational analysis to investigate the consistency of the effective behavior of these nonlocal nonconvex functionals with established classical and peridynamic models in two different regimes. In the regime of small displacement, we show the model can be effectively described by its linearization, deriving the linearized bond-based peridynamic functional as a variational limit of nonlinear functionals. In the regime of vanishing nonlocality, the effective behavior of the nonlocal nonconvex functionals is characterized by an integral representation with quasiconvex integrand. We demonstrate that the density vanishes on matrices whose singular values are less than or equal to one. These results confirm that the localization, in the context of variational convergence, of peridynamic-type energy functionals exhibit behavior quite different from classical hyperelastic energy functionals. This is a joint work with James M. Scott.

We present high-order variational Lagrangian finite element methods for compressible fluids using a discrete energetic variational approach. Our spatial discretization is mass/momentum/energy conserving and entropy stable. Fully implicit time stepping is used for the temporal discretization, which allows for a much larger time step size for stability compared to explicit methods, especially for low-Mach number flows and/or on highly distorted meshes. Ample numerical results are presented to showcase the good performance of our proposed scheme. This is a joint work with Prof. Chun Liu from IIT.

We study the Nakazawa-Ohta ternary inhibitory system, which describes domain morphologies in a triblock copolymer as a nonlocal isoperimetric problem for three interacting phase domains. We consider global minimizers on the two-dimensional torus, in a droplet regime where some species have vanishingly small mass but the interaction strength is correspondingly large. In this limit there is splitting of the masses, and each vanishing component rescales to a minimizer of an isoperimetric problem for clusters in 2D. Depending on the relative strengths of the coefficients of the interaction terms we may see different structures for the global minimizers, ranging from a lattice of isolated simple droplets of each minority species to double-bubbles or core-shells. These results have led to a new type of partitioning problem that I will also introduce. These represent joint work with S. Alama, X. Lu, and C. Wang, as well as with M. Novack and S. Vriend.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The behavior of materials involve physics at multiple length and time scales: electronic, atomistic, domains, defects etc. The engineering properties that we observe and exploit in application are a sum total of all these interactions. Multiscale modeling seeks to understand this complexity with a divide and conquer approach. It introduces an ordered hierarchy of scales, and postulates that the interaction is pairwise within this hierarchy. The coarser-scale controls the finer-scale and filters the details of the finer scale. Still, the practical implementation of this approach is computationally challenging. This talk introduces the notion of neural operators as controlled approximations of operators mapping one function space to another and explains how they can be used for multiscale modeling. We demonstrate the ideas with examples, and highlight the open mathematical issues.

The integration of machine learning (ML) into the traditional modeling workflows is replacing decades-old ad hoc approximations (e.g., in constitutive laws) leading to new models that far outstrip their predecessors in accuracy and transferability. "Pure" ML approaches are rarely successful but remarkable results can be achieved when integrated with domain knowledge. My talk will focus on the atomistic scale where the development of reduced-order (coarse-grained) interaction laws (e.g. interatomic potentials) has made immense progress. I will outline how a combination of modelling, analysis and approximation theory tools lead to an end-to-end justification of a practical class of hybrid ML models. A key point is that systematic coarse-graining leads to many-body interaction; I will therefore also show how such interactions can be parameterized efficiently.

Numerical simulations using self consistency field theory (SCFT) have been a powerful tool to study soft materials like polymers. However, SCFT simulations are a complex and computationally costly process and exploring the vast design space of polymers via SCFT is impractical. We will discuss in this talk our recent efforts to leverage SCFT with Machine Learning (we design specific architectures of convolutional neural networks and generative adversarial networks) to accelerate the exploration of parameter space and to effectively predict polymer structures and for the inverse design problem.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Grain boundary (GB) migration occurs through the motion of discrete steps along the GB. Unlike at free surfaces, steps at internal interfaces in crystalline materials commonly also have dislocation character. These line defects, constrained to the interface, are disconnections; characterized by a combination of a step height and Burgers vector, dictated by (bi)crystallography. I will show both MD and experimental evidence for these defects and explore their basic features, as related to GB migration. I will then describe a continuum equation of motion and show applications of this based on front tracking and phase field methods. I will demonstrate how GB motion during microstructure evolution gives rise to grain rotation and how cyclic annealing or stressing can accelerate grain growth.

In the 1950s, one thought that the length of grain boundaries was the main effect of the evolution of grain boundaries. Nowadays, in material science, we seek the effect of the grain boundaries' length and the lattice misorientations on the evolution. In this talk, I will explain a new mathematical model of grain boundary motion, including dynamic lattice misorientations and triple junction drag. Using the energetic variational approach, we derive a system of geometric differential equations to describe the motion of such grain boundaries. Disappearance events of grain boundaries, so-called critical events, are the main difficulty in analyzing the model. To overcome this difficulty, an empirical distribution of the relative length with a given lattice misorientation and normal, called a grain boundary character distribution(GBCD, for short), was considered. To understand the relationship between the model and the previous study of GBCD, I next explain a stochastic model of the evolution of grain boundaries. This talk is based on the joint work with Yekaterina Epshteyn (The University of Utah)and Chun Liu (Illinois Institute of Technology).

Although grain boundaries are easy to define in a continuum setting – they are the common boundary shared by a pair of adjacent crystals - they are extremely challenging to precisely define, or even characterize, on the atomic scale. Decades of careful work have resulted in broad and deep understanding of the atomistic structure of grain boundaries. Conventional approaches, however, are often limited to special kinds of grain boundaries in special kinds of systems. In this talk I would like to tell you about a new statistical structural description of grain boundaries suitable for studying realistic systems such as those described by thermodynamic ensembles, including ones far from their “ground states”. This characterization is robust in the sense that it is primarily determined by macroscopic degrees of freedom, and is only minimally affected by thermal noise, differences in local density, the presence of defects, and other variations in microscopic degrees of freedom.

Sintering, a pivotal technology in additive manufacturing, transforms ceramic and metallic powders into solid objects. To achieve products with customized properties, a deep understanding of microstructure evolution during sintering is crucial. Our approach ensures thermodynamic consistency, deriving the driving force for particle motion from the system’s free energy. As a result, our proposed phase-field-micromechanics model guarantees microstructure evolution that minimizes the system’s energy. We rigorously validate this model against recent theoretical benchmarks. Subsequently, we employ it to simulate the microstructure evolution of polycrystalline powder particles, shedding light on the mechanisms governing crystallite growth. Additionally, we analyze how grain boundary structure and orientation impact sintering kinetics

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A grand challenge problem in engineering of polycrystals is to develop prescriptive process technologies capable of producing an arrangement of grains that provides for a desired set of materials properties. One method by which the grain structure is engineered is through grain growth or coarsening of a starting structure. During grain growth, an initially random grain boundary arrangement reaches a state that is strongly correlated to the interfacial energy density. Grain growth can be considered as the evolution of a large metastable network, and can be modeled by a set of deterministic local evolution laws for the growth of individual grains combined with stochastic models to describe their interactions. However, despite tremendous progress in formulating models of grain growth, existing descriptions do not fully account for various grain growth mechanisms, detailed grain topologies, and the effects of different time scales on microstructural evolution. Thus, to develop a predictive and prescriptive theory, an investigation of a broad range of statistical measures of microstructure is needed and must be obtained using experiments, simulations, data analytics, and mathematical modeling. This talk will focus on experimental studies that use thin metallic films as the platform. Recent hardware and software advances have removed bottlenecks to large scale ex situ and in situ data acquisition via (1) automated grain boundary segmentation in micrographs, (2) low thermal mass microelectromechanical systems and (3) integrated hardware-software drift correction and data management solutions. These innovations render thin films a key integrated experimental platform for current and future grain growth studies. The experimental advances together with advances in data analytics, simulations and theory are expected to enable microstructure by design.

An understanding of microstructural dynamics under non-isothermal conditions is crucial to materials design. During a so-called ‘cyclic heat treatment’ of shape memory alloys, dissolution of semi-coherent precipitates generates dislocations, thereby raising the stored strain energy within grains. In this work, we employ synchrotron high-energy x-ray diffraction microscopy (HEDM) to map in 3D and time the heterogeneous microstructure over three orders-in-magnitude of length-scale, including the precipitates, grains, and intra-granular dislocation densities. Correspondingly, we use phase field simulations to bridge the spatiotemporal gaps in the HEDM study and propose a mechanism of strain-energy-driven growth. The joint experiments and simulations reveal a significant diversity in grain shapes, sizes, and dislocation densities that are generated over the course of the non-isothermal anneal. Our data also reveal a new mode of grain growth, involving the macroscopic translation of grain centers over time. Broadly, our efforts highlight a complexity in the microstructural dynamics brought about by stored strain energy, not predicted by conventional theories nor metallographic analyses.

This talk reports on the recent findings about the rigorous periodic homogenization of four coupled PDE systems that model heterogeneous composite materials. Specifically, it focuses on the following models: (i) a suspension comprising magnetizable rigid particles in a non-conducting carrier viscous Newtonian fluid, (ii) strongly coupled magnetorheological fluid, (iii) nonlinear dielectric elastomer, and (iv) high-contrast dielectric elastomer. The effective or homogenized response for these composites, which involves PDEs whose coefficients depend on the composite's geometry, the periodicity of the original microstructure, and the coefficients characterizing the initial heterogeneous material, will be demonstrated. The corresponding cell problems, along with the key concepts for justification, will also be mentioned. Additionally, various aspects of the underlying PDEs, such as nonlinearity and high contrast, will be examined.

Thermoelectric effects are widely used in industry. Yet, in contrast to other coupled field phenomena, such as thermoelasticity or piezoelectricity, there seems to be some confusion about the proper form of constitutive laws. In this talk I will describe thermodynamically consistent constitutive laws that retain their form under homogenization. In addition I will address the question of characterizing all subclasses of constitutive laws retaining their form under homogenization.

We will describe a model of dislocation mechanics based (crystal)plasticity of unrestricted geometric and material nonlinearity that, when exercised on a sufficiently fine scale, can rigorously predict fields of arbitrary dislocation distributions in finite bodies of arbitrary anisotropy, and when exercised at larger scales of resolution adequate for meso/macro scale structural response, suitably adapting established macroscale phenomenology related to kinetics of plastic flow, makes predictions up to finite strains of size and rate-dependent mechanical behavior, texture, and mesoscale dislocation microstructure evolution in polycrystalline aggregates and single crystals. The phenomenology used to go to the mesoscale can be systematically improved as the need arises, as can the geometric fields involved along with their governing equations. The framework will be demonstrated by results with a focus on effects not predictable within linear dislocation statics or dynamics, or geometrically linear or nonlinear phenomenological plasticity theories. In particular, we will show calculations of static and dynamic finite deformation stress fields of individual dislocations, including their annihilation and the production of Mach cones in intersonic dislocation motion. At the mesoscale, we shall recover the dramatically different experimentally observed size effects in compression and shear inferred from micropillar experiments that all strain gradient plasticity models overestimate and fail to predict. Similarly, additively manufactured nanolaminates show kink banding for sufficiently small lamination widths, which is again a strict test for continuum dislocation dynamics and strain gradient plasticity models at finite strains. We will show the recovery of such predictions by our model. The model raises significant and challenging mathematical questions.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

What do these two themes have in common? Both are treated variationally, both deal with energies of different dimensionalities, and concepts of geometric measure theory prevail in both. Phase Separation in Heterogeneous Media: Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations using a variational approach based on the gradient theory of phase transitions, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed, now leading to an isotropic interfacial energy. In the subcritical case with moving wells, where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains. This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), USA), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA). Learning Training Schemes for Image Denoising: Due to their ability to handle discontinuous images while having a well-understood behavior, regularizations with total variation (TV) and total generalized variation (TGV) are some of the best known methods in image denoising. However, like other variational models including a fidelity term, they crucially depend on the choice of their tuning parameters. A remedy is to choose these automatically through multilevel approaches, for example by optimizing performance on noisy/clean image training pairs. Such methods with space-dependent parameters which are piecewise constant on dyadic grids are considered, with the grid itself being part of the minimization. Existence of minimizers for discontinuous parameters is established, and it is shown that box constraints for the values of the parameters lead to existence of finite optimal partitions. Improved performance on some representative test images when compared with constant optimized parameters is demonstrated. This is joint work with Elisa Davoli (TU Wien, Austria), Jose Iglesias (U. Twente, The Netherlands) and Rita Ferreira (KAUST, Saudi Arabia)

Area minimization among a suitable class of 2D surfaces is the basic variational model describing the interfaces in films/foams. In this talk we will discuss a modification of this paradigm in which surfaces are replaced with regions of small but positive volume. The model captures features of real films/foams, such as Plateau borders, that cannot be described by zero volume objects. We will also discuss the PDE approximation of this problem via the Allen-Cahn equation and its relation to Plateau's laws, which govern the possible singularities.

In this talk, we consider the robustness and optimal design of a class of ''geometry invariant resonant cavities'' that are made from "Epsilon Near Zero" (ENZ) materials that have a core-shell structure: the shell is made from a material whose dielectric permittivity is near zero at the frequency at which the device operates, while the core is a dielectric inclusion. We'll present theorems on the complex analytic dependence of the resonance on the near-zero permittivity, in a neighborhood of zero-- this demonstrates the robustness of such resonators to the presence of losses, and their broadband effects around the ENZ frequency. In the 2 dimensional Transverse Magnetic (TM) setting, the leading order behavior of the resonance turns out to be independent of the geometry of the ENZ shell, but only its area. However, the leading order correction depends on shell-geometry-- we investigate the shape optimization problem that optimizes the quality of the resonance, and present a convex relaxation which is a free boundary problem of Alt-Caffarelli type. In the fully three dimensional setting, the picture is substantially richer, and we describe an interesting class of resonances that are still geometry independent at leading order, but lead to an interesting vectorial version of the classical ``overdetermined problem of J. Serrin''. This project represents joint work with Robert V. Kohn (Courant), inspired by discussions with Nader Engheta (UPenn).

The incommensurate stacking of multilayered two-dimensional heterostructures is a challenging problem from a theoretical perspective and an intriguing avenue for manipulating their physical properties. I will derive a continuum model to compute the mechanical relaxation for an arbitrary number of layers by parametrizing the displacement by the local configuration (disregistry) rather than real space, thus bypassing the need for the standard supercell approximation and giving a true aperiodic atomistic configuration. We note that the displacement of a heterostructure of p layers is not generally periodic in real two-dimensional space for p>2, but the displacement is periodic on a 2(p-1) dimensional torus as a function of its disregistry. We will show that the regularity of the displacement has a major effect on electronic properties.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Miguel Ayala, McGill University, "Validated Numerical Techniques for Geometric Analysis of Carbon Nanotubes". Henry Brown, Temple University, "Extrapolation of completely monotone functions." Chi-An Chen, IIT, "Lagrangian Particle Scheme for Regularized Kimura Equations". Carson Collins, UCLA, “Free boundary regularity for a tumor growth model". Samuel Matthews, McGill University, "Molecular Modeling of sII Gas Hydrate Interfacial Structures and Processes". Matthew Patrick, Columbia University, "In situ Grain Growth Experiments: Dynamic Data Collection, Processing, & Analysis". Rawan Tarabeh, Technion, “Mullins’ nonlinear grooving solution”.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In previous literature the passage from the kinetic Doi-Onsager model to the Ericksen-Leslie equations has been made using the Hilbert expansion method. Classical kinetic theory offers an alternate way to pass from kinetic to fluid equations: the moment method. However, in the case of the Doi-Onsager model, the moment method is not directly applicable, due to the lack of conservations, (or collision invariants). In this talk, I will show that suitable generalized conservation relations (aka generalized collision invariants) hold and make the moment method applicable in spite of the lack of conservation relations in the strict sense. It could lead to a better understanding of the structural properties of the Doi-Onsager model and open the way to rigourous convergence proofs that would require less regularity than the Hilbert method. This talk is based on ''From kinetic to fluid models of liquid crystals by the moment method'' (P. Degond, A. Frouvelle, J-G. Liu, Kinetic and Related Models 15 (2022), pp. 417-465).

One way to glue objects together at the nanoscale or microscale is by ligand-receptor interactions, where short sticky hair-like ligands stick to receptors on another surface, much like velcro on the nanoscale. Such interactions are common in biology, such as white blood cells, virus particles, cargo in the nuclear pore complex, etc, and they are also useful in materials science, where coating colloids with single-stranded DNA creates particles with programmable interactions. In these systems, the ligand-receptor interactions not only hold particles together, but also influence their dynamics. How do such particles move? Do they “roll” on each others’ surfaces, as is commonly thought? Or could they slide? And does it matter? In this talk I will introduce our modelling and experimental efforts aimed at understanding the coarse-grained dynamics of particles with ligand-receptor interactions. Our models predict these interactions can change the particles' effective diffusion by orders of magnitude. Our experiments, using DNA-coated colloids, verify this dramatic dynamical slowdown, but also show other dynamical features not yet captured by our model, which suggest new avenues for exploration.

We propose a model for the debonding of a thin elastic film from a rigid substrate when subject to gravity and exposed to a liquid environment. The model combines nonlinear elasticity with the classical work by Flory and Rehner, together with the variational description of fracture. The theory presented is validated by laboratory experiments. The problem is motivated from the design of the synthetic polymers that coat various medical implants, in order to estimate how thin the gel coating needs to be in order to be stable against debonding from the mechanical substrate. After analyzing the variational problem for a fully bonded film, we present that of the partially attached one, formulated as an obstacle problem. The minimization of the (inequality) constraint problem is achieved by means of the Gamma convergence approach. We calculate the energy release rate leading to debonding and derive the maximum thickness of the film that makes it stable against swelling and debonding forces. We present a finite element study to validate the theoretical prediction of the energy release rate and compare the results with the laboratory experiments.

I will discuss the origin of contact angle hysteresis of capillary drops from micro-scale surface roughness. This will motivate discussion and theoretical results on some macroscopic toy models which can describe the large-scale effects of the pinning phenomenon.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In science, different kinds of shortcuts are often used to study big problems, especially in computational modeling of matter. By combining various scientific models, better results can be achieved, work can be done more efficiently, and methods can be applied to more areas. In our research, we explore how to mix different ways of studying chemicals on computers to better understand materials and living molecules. This area is growing rapidly and contributes to various studies by providing basic knowledge and predictions. Special attention is given to molecular theory of solvation which allows us to accurately introduce the effect of environment on complex nano-, bio- or macromolecular system. The uniqueness of that approach is that it can be naturally coupled with the full range of computational chemistry approaches (QM, MM, and Coarse Grained).

In this talk we present a second order in time approximation for a sixth-order Cahn-Hilliard type equation which models the dynamics of phase transitions in ternary oil-water-surfactant systems. For its spatial discretization, we decompose this nonlinear sixth-order parabolic equation into a mixed formulation comprising a system of three second order (in space) equations, one of which is parabolic while the other two equations are algebraic. We discuss the key properties of unconditional stability and unique solvability for our scheme and demonstrate its numerical performance through results of selected computational experiments.

Working Session and Panel discussion to discuss and exchange ideas on: a) Perspective on current fundamental questions and challenges in their respective areas, as well as ideas on possible approaches to tackle them; b) Successes and challenges of cross-disciplinary work: for example, one topic that will be discussed is the improvement of communication skills (reduction of scientific language barriers) among, e.g. mathematicians, materials scientists, and engineers; c) Training next-generation applied mathematics workforce: as a part of the discussion, we will exchange ideas about interdisciplinary training which includes close interactions/collaboration among mathematicians, materials scientists, and engineers, as well as discussing important questions of increasing diversity and broadening participation within the field. d) Current and new funding opportunities

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.