Why Statistical Physics Should Be A Foundation For Materials Design
Author: Marc Miskin
James Franck Institute and Department of Physics, The University of Chicago, USA.
The fact that major epochs in human history have been named after materials like bronze, steel, silicon, and stone expresses both how important materials are for technology and how long it can take before a new material is discovered. Even today, the timespan to convert materials' discoveries into functioning technologies takes upwards of 20 years. In part this is because creating technologically useful materials requires selecting a wide range of parameters to optimize the material’s performance. While tools like statistical physics are useful for describing a material’s behavior given a set of parameters, it remains unknown how to generally invert these relationships to target desired behavior. This task is called materials design and it is a new concept at the forefront of materials research.
Recently, several methods have emerged across disciplines that draw upon optimization and simulation to create computer programs that tailor material responses to specified behaviors. However, so far the methods developed either involve black-box techniques, in which the optimizer operates without explicit knowledge of the physical laws that underpin the material’s behavior, or require carefully tuned algorithms with applicability limited to a narrow subclass of materials.
Our recent publication titled “Turning Statistical Physics Models into Materials Design Engines”  presents a new perspective for material design. In contrast to prior approaches, it is broad enough to be applied without modification to any system that is well described by statistical mechanics and also retains much of the key insight that is at the heart of statistical physics. In short, our formalism allows a user to transform the capacity to predict material behavior into an optimizer that tunes it.
We achieved this by examining the fundamental relationship between microstate configurations and material properties. Statistical physics poses the idea that materials are intrinsically statistical objects: the properties a bulk material has are best calculated by averaging over all the possible configurations for the material's microscopic parts. Our insight was that design programs should focus on tailoring materials at the level of micro-states themselves, rather than simply focusing on the bulk emergent properties.
For instance, suppose that the pressure and temperature affect the stiffness of a given material and the goal is to set these two parameters to make the stiffest material. The black-box approach is to view this as an optimization problem with pressure and temperature as inputs and stiffness as an output. Yet this view completely ignores the micro-states. A better perspective is to treat the control parameters as means to alter the likelihood of micro-states. To design the material, an algorithm should tweak the parameters so that the material is more likely to be in micro-states with the target behavior.
This idea is the kernel of our approach: we built a formalism out of this concept and tackle materials' design at the level of micro-state information. This gave us a program that is broad enough to address the range of materials that are well described by statistical physics, and we achieve this with a boost in efficiency, thanks to the extra information extracted from the microstate configurations.
To test our approach, we constructed test problems that a good materials optimization scheme should be able to address by itself. Material scientists need optimizers that can solve problems where the search landscape has little variation between candidate materials, juggle multiple potentially competing physical effects, operate in high dimensional search spaces, tune the processing conditions that a material is subject to, and operate when on real-world scale optimization problems. We then translated these challenges into physical test problems, and compared our approach against optimization schemes that we have used successfully for materials design in the past. Given the criterion that the best optimizer is the one that has to make the fewest guesses to arrive at the material that performs a target function, our optimizer outperformed all of our old standards.
Probably the two most fascinating solutions presented in the paper  are the polymer folding problem and the directed self-assembly problem. In the polymer folding problem, we asked our optimizer to tune the interaction strengths between 6 beads attached to each other along a linear chain (Figure 1). Because the interactions are attractive, when they are strong enough the chain will fold itself up into a compact shape. The goal here was to make the chain fold into a specific shape: an octahedron. Its an interesting problem because it's well known that simply making all the interactions large will not produce an octahedral geometry. Instead, the interactions needed to be developed into three separate families to generate octahedron and it took hard work from the colloidal self-assembly community to show that this worked. So it was very exciting for us when our optimizer not only produced a virtually identical motif, but managed to yield the result in the span of hours.
Figure 1: Given a polymer of 6 beads each of different color, how should the strength of each short-ranged interaction be picked so that the polymer self-folds into an octahedron geometry? Each image shows a typical polymer configuration obtained at each stage in an optimization using our new approach. The optimizer essentially starts from random, chain like geometries and after ~200 cycles transforms them into the target shape.
There is a similar story behind our directed self-assembly problem. In this case, the material is a polymer made of two types of beads. The goal is to pattern a substrate with a thin strip that has an affinity for a particular one of those two beads (Figure 2). By setting the strip width and the strength of affinity just right, it is possible to make the polymer self-assemble into stripes containing only one polymer type followed by a stripe of the other polymer type and so on. This idea holds serious promise as a next-generation manufacturing technology for semiconductors because the sizes of these stripes are on the order of nanometers. By using the polymer stripes as stencils or masks, it is possible to make next generation circuits or hard drive media with features significantly smaller than what current processing techniques allow. What we found was that not only can our optimizer produce solutions to the problem of tailoring interaction strengths for this kind of directed self-assembly, but that it does so between 5-130X faster than approaches we had tried in the past. To put this into context, solving a directed self assembly problem in the past took us roughly 1 week. Now we can solve them in just under 12 hours.
Figure 2: Given melt of polymer chains each made from half a-type (red) and half b-type (blue) beads, how should one tune the interactions between the a and b beads and a substrate so that the polymer melt self-assembles into equally spaced stripes of a and b? On the top is an image of the original polymer melt configuration for randomly seeded interaction parameters. On the bottom shows the structures that result from using our algorithm to elicit self-assembly. Note the substrate has been colored based on the affinity for each type.
Speaking broadly, materials by design is a radical shift in how we transform bulk matter into useful technology. Historically materials have been either discovered by accident or appropriated from nature to perform technological functions. What we're after is the capacity to systematically identify which materials produce a target response. The benefit of this paradigm is that increasingly complex materials could be cooked up automatically to meet specific technological demands. Our algorithm is a small part of this growing field, but the hope is that it will inspire others to consider their expertise on materials within the new context of design.
Our experience in the past has been that it can be difficult to get started on building design engines for a new material. If the problem isn't posed the right way or the optimizer isn't appropriately tailored, it may require a substantial investment of time to construct an optimization scheme that actually works. What we tried to show in this paper is that our formalism works broadly over a range of very different physical problems without any need for additional modifications. It works out of the box, so to speak, for designing any material that can be simulated using a statistical physics approach. Our hope is that this robustness will translate into a reduction in the time researchers need to spend building design algorithms, and free them up to focus on the task of making exotic materials.
Some of our recent work along these lines can be found here:
 Marc Z. Miskin, Gurdaman Khaira, Juan J. de Pablo, Heinrich M. Jaeger, "Turning statistical physics models into materials design engines", Proceedings of the National Academy of Sciences, 113, 34-39 (2016). Full Text.
 Marc Z. Miskin, Heinrich M. Jaeger, "Adapting granular materials through artificial evolution", Nature Materials, 12, 326-331 (2013). Abstract.
 Marc Z. Miskin, Heinrich M. Jaeger, "Evolving design rules for the inverse granular packing problem", Soft Matter, 10, 3708-3715 (2014). Abstract.
 Jian Qin, Gurdaman S. Khaira, Yongrui Su, Grant P. Garner, Marc Miskin, Heinrich M. Jaeger, Juan J. de Pablo, "Evolutionary pattern design for copolymer directed self-assembly", Soft Matter, 9, 11467–11472 (2013). Abstract.