# The Materials Complexity Frontier: Applied Math and Computational

The Materials Complexity Frontier: Applied Math and Computational Challenges S.J.L. Billinge Department of Applied Physics and Applied Mathematics Columbia University, CMPMS, Brookhaven National Laboratory Complicated Problems Complex materials will lie at the heart of the solutions to many of societys most pressing problems

Sustainable Energy Environmental remediation Health Complex materials Photovoltaics with improved efficiency Nanoparticles in the light collecting layer High energy density batteries Electrodes

Electrolytes Fuel cells for transportation applications Electrodes Electrolytes

Catalysts Hydrogen storage Sequestration Functionalized mesoporous materials Image credits: 10.1126/science.1185509 U. Uppsala The Nanostructure Problem We want to engineer materials at the nanoscale

But we cant even solve the atomic structure at the nanoscale: The nanostructure problem The comprehension problem The nanostructure problem is a concrete example of what I call The Comprehension Problem Data -> Knowledge -> Robust Understanding The Crystal Structure Problem

Problem: Here is a crystal, what is its structure? Solution: 1. Give it to your grad student 2. She puts it on the x-ray machine 3. Pushes the button

1. Machine tells you the structure Crystallography is largely a solved problem From LiGaTe2: A New Highly Nonlinear Chalcopyrite Optical Crystal for the Mid-IR L. Isaenko, et al., J. Crystal Growth, 5, 1325 1329 (2005) The Nanostructure Problem

Problem: Here is a nanoparticle, what is its structure? Solution: 1. Give it to your grad student 2. She puts it on the x-ray machine 3. Pushes the button

Complex materials Photovoltaics with improved efficiency Nanoparticles in the light collecting layer High energy density batteries Electrodes Electrolytes Fuel cells for transportation applications

Electrodes Electrolytes Catalysts Hydrogen storage Sequestration

Functionalized mesoporous materials Image credits: 10.1126/science.1185509 U. Uppsala The Nanostructure Problem Problem: Here is a nanoparticle, what is its structure?

Solution: 1. Give it to your grad student 2. She puts it on the x-ray machine 3. Pushes the button Structure Solution from PDF Example: C60 60 atoms => n(n-1)/2 = 1770 pair-vectors We know the lengths (not the directions) of ~18 unique distances

We have an imperfect measure of the multiplicities of those distances We dont have any symmetry information to help us Is the problem well conditioned or ill conditioned? Is there a unique solution? 60 atoms C60

~64 atoms Ultra-small CdSe NPs Successology Problem Well posed problem: Information in the PDF data ILL POSED Problem!

Degrees of freedom in the model Bits of information Structure Solution Complex Modeling Solution c = a + ib complex number mixes real and imaginary parts m = e + it complex

modeling mixes experiment and theory in a coherent computational framework Billinge and Levin, Science 2007 Compl- exoscale Computing

Joking aside Main applied math challenges: Heterogeneous data Reliability of results Tolerating systematic errors/aberrations Main computational challenges: Expensive forward calculations (e.g., DFT!), now in a regression loop Complex Modeling infrastructure: Diffpy-CMI

Official release of Diffpy-CMI v0.1 (Complex Modeling Infrastructure) on 3/31/14 (284 downloads) www.diffpy.org Robustness (degeneracy/convergence) of modeling results HPC enabled brute-force search of structure solution phase-space of CdSe Quantized growth nanoparticles

How does the addition of Small Angle Scattering Data affect the results? SAXS data from NSLS X9B on CdSe particles dissolved in toluene tetrahedral CdSe model has SAXS residuum 0.008 clusters generated from PDF optimization have poor fit to the SAXS data validate uniqueness of

the tetrahedral structure models. Materials modelers, please dont get too smug Ultra-stable Au144 nanocluster Now do all that in quasi-real time: in-situ, in-operando experiments We can see precursor species in solution We can measure Nanoparticle structural parameters => Lets do in-situ studies of synthesis

collaboration with the group of Bo Iversen (Aarhus) Image credit Christoffer Tyrsted Or spatially resolved Computed tomography nano-diffraction 103 more data than a regular ct-scan Data Rates at modern synchrotron facilities

Courtesy Ray Osborne, APS, ANL These data rates are not exceptionally high But there are complex data assessment, reduction and modeling steps that must be carried out in quasi-real time requiring particular architecture and access mode. Solving this problem will revolutionize how scattering scientists do their experiments. Imagine a doctor interpreting an MRI scan from thousands of frames of the raw nuclear spin relaxation data.

To make significant progress in Complex Materials 1. Use theoretical/computational and experimental data synergistically 2. Validate robustness of solutions 3. Increase reproducibility of all computational results 4. Couple data acquisition, assessment, reduction and modeling more closely 5. ..automate discovery using data analytics approaches These are applied math, software and architectural issues: Compl-exoscale computing!

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