## Salvatore Torquato

### Research Focus

A common theme of our research work in statistical mechanics and soft condensed matter theory is the search for unifying and rigorous principles to elucidate a broad range of physical phenomena. We have contributed to the following diverse fields: (1) Liquid, Glasses and Crystals; (2) Fundamental Aspects of Statistical Physics; (3) Particle Packings; (4) Self-Assembly Theory: Classical Ground and Excited States; (5) Random Heterogeneous Media; (6) Optimal Design of Materials; and (7) Cancer Modeling.

**1. Liquids, Glasses and Crystals**

We are interested in the applications of statistical mechanics to elucidate our fundamental understanding of the molecular theory of disordered condensed states of matter, such as liquids and glasses. We have made seminal contributions to our understanding of the well-known hard- sphere model, which has been invoked to study local molecular order, transport phenomena, glass formation, and freezing behavior in liquids. Other notable research advances concern the theory of water, simple liquids, and general statistical-mechanical theory of condensed states of matter. We have recently become interested in crystal structures and their symmetries.

2. Fundamental Problems in Statistical Physics

We have pioneered the “reconstruction” and “realizability” problems of statistical mechanics and their solutions, g2-invariant processes”, and our basic understanding and characterization of point processes. Often theoretical and computational optimization techniques are employed. Seminal theoretical results were obtained for “hyperuniform” systems, i.e., point patterns that do not possess infinite-wavelength density fluctuations. Such point distributions have connections to integrable quantum systems and number theory. We have used the hyperuniformity concept to view crystals, quasicrystals and special disordered structures in a unified manner.

3. Particle Packings

Packing problems, such as how densely or randomly objects can fill a volume, are among the most ancient and persistent problems in mathematics and science. Packing problems are intimately related to condensed phases of matter, including classical ground states, liquids and glasses. Despite its long history, there are many fundamental conundrums concerning packings of spheres that has remained elusive, including the nature of random packings and whether such packings can ever be denser than ordered packings, especially in high spatial dimensions. The latter problem has direct relevance in communications theory. We have recently been interested in the densest packings of nonspherical particle shapes, including ellipsoids, superballs, and polyhedra.

4. Self-Assembly Theory: Classical Ground and Excited States

While classical ground states are readily produced by slowly freezing liquids in experiments and computer simulations, our theoretical understanding of them is far from complete. There are many open and fascinating questions. For example, to what extent can we control classi- cal ground states? Can ground states ever be disordered? We are leading a program to shed light on these fundamental aspects of classical ground states and their corresponding excited states using the tools and machinery of statistical mechanics. We have recently devised inverse statistical-mechanical methodologies to find optimized interaction potentials that lead sponta- neously and robustly to unusual target many-particle configurations, including low-coordinated crystal ground states and disordered ground states. One can regard such approaches as “tar- geted” self-assembly. The particular experimental systems that could achieve the optimally designed interactions include colloids and polymers, since their interactions can be tuned.

**5. Random Heterogeneous Media**

Random heterogeneous media abound in nature and synthetic situations, and include compos- ites, thin films, colloids, packed beds, foams, microemulsions, blood, bone, animal and plant tissue, sintered materials, and sandstones. This area dates back to the work of Maxwell, Lord Rayleigh and Einstein, and has important ramifications in the physical and biological sciences. The effective properties are determined by the ensemble-averaged fields that satisfy the gov- erning partial differential equations. The properties depend, in a complex manner, upon the random microstructure of the material via various n-point statistical correlation functions, in- cluding those that characterize percolation and clustering. Two decades ago, rigorous progress in predicting the effective properties had been hampered because of the difficulty involved in characterizing the random microstructures. We broke this impasse using statistical-mechanical techniques.

6. Optimal Design of Materials

A holy grail of materials science is to have exquisite knowledge of structure/property relations to design material microstructures with desired properties and performance characteristics. We have been at the forefront of this exciting area. Specifically, we have posed this task as an optimization problem in which an objective function involving a set of physical properties is extremized subject to constraints. The resulting optimal microstructures are often surprising in nature. Combining such modeling techniques with novel synthesis and fabrication methodologies may make optimal design of real materials a reality in the future.

**7. Cancer Modeling**

The aim of the project is to show that we can model the growth (proliferation and invasion) of brain tumors using concepts from statistical physics, materials science and dynamical systems, as well as data from novel oncological experiments. We expect the work not only to increase our understanding of tumorigenesis, but to provide insight into novel ways to treat malignant brain tumors. This work will now be done in conjunction with the new Princeton Physical Sciences-Oncology Center.

### Selected Recent Publications

- S. Torquato, T. M. Truskett and P. G. Debenedetti, "Is Random Close Packing of Spheres Well Defined?,'' Physical Review Letters, 84, 2064 (2000).

- A. R. Kansal, S. Torquato, G. R. Harsh, E. A. Chiocca, and T. S. Deisboeck, "Simulated Brain Tumor Growth Dynamics using a Three-Dimensional Cellular Automaton,'' Journal of Theoretical Biology, 203, 367 (2000).

- T. M. Truskett, P. G. Debenedetti and S. Torquato, "Thermodynamic Implications of Confinement for a Water-Like Fluid,'' Journal of Chemical Physics, 114, 2401 (2001).

- S. Torquato and F. H. Stillinger, "Multiplicity of Generation, Selection, and Classification Procedures for Jammed Hard-Particle Packings,'' Journal of Physical Chemistry B 105, 11849 (2001).

- S. Torquato and F. H. Stillinger, "Controlling the Short-Range Order and Packing Densities of Many-Particle Systems,'' Journal of Physical Chemistry B, 106, 8354 (2002).

- J. R. Errington, P. G. Debenedetti, and S. Torquato, "Cooperative Origin of Low-Density Domains in Liquid Water,'' Physical Review Letters, 89, 215503 (2002).

- S. Torquato, S. Hyun and A. Donev, "Multifunctional Optimal Composite Microstructures: Simultaneous Transport of Heat and Electricity,'' Physical Review Letters, 89, 266601 (2002).

- S. Torquato and F. H. Stillinger, "Local Density Fluctuations, Hyperuniformity, and Order Metrics," Physical Review E, 68, 041113 (2003).

- A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Stillinger, R. Connelly, S. Torquato, and P. M. Chaikin, "Improving the Density of Jammed Disordered Packings using Ellipsoids," Science, 33, 990 (2004).

- A. Donev, F. H. Stillinger, P. M. Chaikin, and S. Torquato, "Unusually Dense Crystal Ellipsoid Packings," Physical Review Letters, 92, 255506 (2004).

- A. Donev, F. H. Stillinger and S. Torquato, "Unexpected Density Fluctuations in Disordered Jammed Hard-Sphere Packings," Physical Review Letters, 95, 090604 (2005).

- J. H. Conway and S. Torquato, "Packing, Tiling and Covering with Tetrahedra," Proceedings of the National Academy of Sciences, 103, 10612 (2006).

- S. Torquato and F. H. Stillinger, "New Conjectural Lower Bounds on the Optimal Density of Sphere Packings," Experimental Mathematics, 15, 307 (2006).

- J. L. Gevertz and S. Torquato, "Modeling the Effects of Vasculature Evolution on Early Brain Tumor Growth," Journal of Theoretical Biology 243, 517 (2006).

- M. C. Rechtsman, F. H. Stillinger and S. Torquato, "Synthetic Diamond and Wurtzite Structures Self-Assemble with Isotropic Pair Interactions," Physical Review E, 75, 031403 (2007).

- Y. Jiao, F. H. Stillinger and S. Torquato, "Modeling Heterogeneous Materials via Two-Point Correlation Functions: Basic Principles," Physical Review E, 76, 031110 (2007).

- S. Torquato and F. H. Stillinger, "Toward the Jamming Threshold of Sphere Packings: Tunneled Crystals," Journal of Applied Physics, 102, 093511 (2007).

- S. Torquato and F. H. Stillinger, ``New Duality Relations for Classical Ground States," Physical Review Letters, 100, 020602 (2008).

- Y. Jiao, F. H. Stillinger and S. Torquato, "Optimal Packings of Superdisks and the Role of Symmetry," Physical Review Letters, 100, 245504 (2008).

- R. D. Batten, F. H. Stillinger and S. Torquato, "Classical Disordered Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous Materials," Journal of Applied Physics, 104, 033504 (2008).

- M. C. Rechtsman, H-C. Jeong, P. M. Chaikin, S. Torquato and P. J. Steinhardt, "Optimized Structures for Photonic Quasicrystals," Physical Review Letters, 101, 073902 (2008).

- M. C. Rechtsman, F. H. Stillinger and S. Torquato, "Negative Poisson's Ratio Materials via Isotropic Interactions," Physical Review Letters, 101, 085501 (2008).

- S. Torquato, A. Scardicchio and C. E. Zachary, "Point Processes in Arbitrary Dimension from Fermionic Gases, Random Matrix Theory, and Number Theory," Journal of Statistical Mechanics: Theory and Experiment, P11019 (2008).

- S. Torquato, "Inverse Optimization Techniques for Targeted Self-Assembly," Soft Matter, 5, 1157 (2009).

- R. D. Batten, F. H. Stillinger and S. Torquato, "Novel Low-Temperature Behavior in Classical Many-Particle Systems," Physical Review Letters, 103, 050602 (2009).

- S. Torquato and Y. Jiao, ``Dense Packings of the Platonic and Archimedean Solids," Nature, 460, 876 (2009).

- Y. Jiao, F. H. Stillinger, and S. Torquato, "A Superior Descriptor of Random Textures and Its Predictive Capacity," Proceedings of the National Academy of Sciences, 106, 17634 (2009).

- J. Gevertz and S. Torquato, "Growing Heterogeneous Tumors in Silico," Physical Review E, 80, 051910 (2009).

- M. Florescu, S. Torquato and P. J. Steinhardt, "Designer Disordered Materials with Large, Complete Photonic Band Gaps," Proceedings of the National Academy of Sciences, 106, 20658 (2009).

- S. Torquato and Y. Jiao, "Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra," Physical Review E, 81, 041310 (2010).

- R. D. Batten, F. H. Stillinger, and S. Torquato, "Phase Behavior of Colloidal Superballs: Shape Interpolation from Spheres to Cubes," Physical Review E, 81, 061105, (2010).

- S. Torquato, "Optimal Design of Heterogeneous Materials," Annual Review of Materials Research, 40, 101 (2010).