Princeton University
fhs@princeton.edu
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Frank Stillinger
Activities

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Complete Publications List
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Accomplishments
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Editorial Boards And Professional Committees
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Invited Lectures
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Coauthors
Postdoctoral Collaborators
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Honors and Awards
• Address/Personal/Employment
Principal Research Interests
Molecular theory of water and aqueous solutions, phase transition theory, glass phenomena, quantum chemistry, packing geometry problems. Over 325 publications.
Principal Research Accomplishments
• Introduction of capillary wave theory for liquid interfaces.
• Development of statistical mechanical theory for water and aqueous solutions, and its implementation in computer simulations.
• Discovery of the exact Stillinger-Lovett second moment condition for electrolyte distribution functions.
• First formulation of the "Stillinger cluster" approach to nucleation.
• Development of the Lubachevsky-Stillinger algorithm for computer simulation of rigid-particle packings.
• Mathematical theory for "potential energy landscapes" of condensed-matter phenomena, including introduction of "inherent structures", their basins, and statistical distributions.
• Introduction of the first classical statistical mechanical model for "inverse melting".
• Construction of "designed potentials" for spontaneous assembly of particles in selected crystal structures, with application to colloids.
• Invention and numerical study of a model for chiral amplification in the pre-biotic environment.
SELECTED PUBLICATIONS
- "Hidden Structure in Liquids," F.H. Stillinger and T.A. Weber,
Phys. Rev. A 25, 978-989 (1982).
- "Structure of the Interface Between Coexisting Fluid Phases,"
F.H. Stillinger, Int. J. Quantum Chem., Quantum Chem. Symp. No. 16,
(Interscience, 1982), pp.137-147.
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"Computer Simulation of Local Order in Condensed Phases of Silicon,"
F.H. Stillinger and T.A. Weber, Phys. Rev. B 31, 5262-5271 (1985);
errata: Phys. Rev. B 33, 1451 (1986).
- "Supercooled Liquids, Glass Transitions, and the Kauzmann Paradox,"
F.H. Stillinger, J. Chem. Phys. 88, 7818-7825 (1988).
- "Pseudoharmonic Oscillators and Inadequacy of Semiclassical
Quantization," F.H. Stillinger and D.K. Stillinger, J. Phys. Chem. 93,
6890-6892 (1989).
- "Spontaneous Development of Optical Activity During Freezing,"
J.G. Harris and F.H. Stillinger, J. Chem. Phys. 92, 6140-6151 (1990).
- "Geometric Properties of Random Disk Packings," B.D. Lubachevsky
and F.H. Stillinger, J. Stat. Phys. 60, 561-583 (1990).
- "Computational Study of Transition Dynamics in 55-Atom Clusters,"
F.H. Stillinger and D.K. Stillinger, J. Chem. Phys. 93, 6013-6024
(1990).
- "Planck's-Constant Expansion for Bound States," F.H. Stillinger,
Phys. Rev. A 43, 3317-3324 (1991).
- "Disks vs. Spheres: Contrasting Properties of Random Packings," B.D.
Lubachevsky, F.H. Stillinger, and E.N. Pinson, J. Stat. Phys. 64,
501-524 (1991).
- "Toy Model for Protein Folding," F.H. Stillinger, T. Head-Gordon,
and C. Hirshfeld, Phys. Rev. E 48, 1469-1477 (1993).
- "Generic Dimer-Buckling Model for Semiconductor Surfaces. Dynamical
Simulations," D.K. Stillinger and F.H. Stillinger, Phys. Rev. B 48,
15047-15056 (1993).
- "Translation-Rotation Paradox for Diffusion in Fragile Glass-Forming
Liquids," F.H. Stillinger and J.A. Hodgdon, Phys. Rev. E 50, 2064-
2068 (1994).
- "A Topographic View of Supercooled Liquids and Glass Formation,"
F.H. Stillinger, Science 267, 1935-1939 (1995).
- "Theoretical Metastability of Semiconductor Crystallites in High-Pressure
Phases, with Application to Beta-Tin Structure in Silicon," L.E. Brus,
J.A.W. Harkless, and F.H. Stillinger, J. Am. Chem. Soc. 118, 4834-
4838 (1996).
-
"Constraints, Metastability, and Inherent Structures in Liquids,"
D.S. Corti, P.G. Debenedetti, S. Sastry, and F.H. Stillinger,
Phys. Rev. E 55, 5522-5534 (1997).
- "Modeling the Interaction of Hydrogen with Silicon Surfaces," D. Kohen,
J.C. Tully, and F.H. Stillinger, Surface Science 397, 225-236 (1998).
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"Enumeration of Isobaric Inherent Structures for the Fragile Glass
Former o-Terphenyl," F.H. Stillinger, J. Phys. Chem. B 102, 2807-
2810 (1998).
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"Signatures of Distinct Dynamical Regions in the Energy Landscape of
a Glass-Forming Liquid," S. Sastry, P.G. Debenedetti, and F.H.
Stillinger, Nature 393, 554-557 (1998).
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"Resolving Vibrational and Structural Contributions to Isothermal
Compressibility," F.H. Stillinger, P.G. Debenedetti, and S. Sastry,
J. Chem. Phys. 109, 3983-3988 (1998).
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"Structural Precursor to Freezing in the Hard-Disk and Hard-Sphere
Systems," T.M. Truskett, S. Torquato, S. Sastry, P.G. Debenedetti, and
F.H. Stillinger, Phys. Rev. E 58, 3083-3088 (1998).
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"Model Studies of Nonadiabatic Dynamics," D. Kohen, F.H. Stillinger, and
J.C. Tully, J. Chem. Phys. 109, 4713-4725 (1998).
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"Free Volume in the Hard Sphere Liquid," S. Sastry, T.M. Truskett, P.G.
Debenedetti, S. Torquato, and F.H. Stillinger, Mol. Phys. 95, 289-297
(1998).
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"Spontaneous Patterns in Disk Packings," B.D. Lubachevsky, R.L. Graham, and F.H. Stillinger, in Bridges: Mathematical Connections in Art, Music, and Science, Reza Sarhangi, editor (Gilliland Printing, Arkansas City, Kansas, 1998), pp. 181-193.
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"Exponential Multiplicity of Inherent Structures," F.H. Stillinger,
Phys. Rev. E 59, 48-51 (1999).
- "Inherent Structure of a Molten Salt," R.A. La Violette, J.L. Budzien, and F.H. Stillinger, J. Chem. Phys. 112, 8072-8078 (2000).
- "Møller-Plesset Convergence Issues in Computational Quantum Chemistry," F.H. Stillinger, J. Chem. Phys. 112, 9711-9715 (2000).
- "Triangle Distribution and Equation of State for Classical Rigid Disks," D.K. Stillinger, F.H. Stillinger, S. Torquato. T.M. Truskett, and P.G. Debenedetti, J. Stat.Phys. 100, 49-72 (2000).
- "Improving the Density of Jammed Disordered Packings Using Ellipsoids," A. Donev, I. Cisse, D. Sachs, E.A. Variano, F.H. Stillinger, R. Connelly, S. Torquato, and P.M. Chaikin, Science 303, 990-993 (2004).
- "Dynamic Heterogeneity and Non-Gaussian Behavior in a Model Supercooled Liquid," M.S. Shell, P.G. Debenedetti, and F.H. Stillinger, J. Phys.: Condens. Matter 17, S4035-S4046 (2005).
- "On the Realizability of Pair Correlation Functions," O.U. Uche, F.H. Stillinger, and S. Torquato, Physica A 360, 21-36 (2006).
- "Packing Hyperspheres in High-Dimensional Euclidean Spaces," M. Skoge, A. Donev, F.H. Stillinger, and S. Torquato, Phys. Rev. E 74, 041127 (2006). Erratum: Packing Hyperspheres in High-dimensional Euclidean Spaces, Phys. Rev. E 75, 029901 (2007).
- "Computational Probes of Molecular Motion in the Lewis Wahnström Model for Ortho-Terphenyl," T.G. Lombardo, P.G. Debenedetti, and F.H. Stillinger, J. Chem. Phys. 125, 174507 (2006).
- "Configurational Entropy of Binary Hard-Disk Glasses: Nonexistence of an Ideal Glass Transition," A. Donev, F.H. Stillinger, and S. Torquato, J. Chem. Phys. 127, 124509 (2007).
- "Toward the Jamming Threshold of Sphere Packings: Tunneled Crystals," S. Torquato and F.H. Stillinger, J. Appl. Phys. 102, 093511 (2007).
- "A Water-Explicit Lattice Model of Heat-, Cold-, and Pressure-Induced Protein Unfolding," B.A. Patel, P.G. Debenedetti, F.H. Stillinger, and P.J. Rossky, Biophys. J. 93, 4116-4127 (2007).
- "Negative Thermal Expansion in Single-Component Systems with Isotropic Interactions," M.C. Rechtsman, F.H. Stillinger, and S. Torquato, J. Phys. Chem. A 111, 12816-12821 (2007). [G. Scoles special issue]
ADDRESS - PERSONAL - EMPLOYMENT
EDITORIAL BOARDS AND PROFESSIONAL COMMITTEES
- Editorial Board, Journal of Chemical Physics, 1968-1970
- Editorial Board, Journal of Statistical Physics, 1969-1978
- Chair, Division of Chemical Physics, American Physical Society 1975-1976
- Advisory Panel, Heat Division, National Bureau of Standards, 1975-1978
- Editorial Board, Physical Review A, 1978-1983
- Advisory Panel, Chemistry Division, National Science Foundation, 1980-1983
- National Research Council Panel, "Mathematical Research in Materials Science," 1991-1993, National Research Council Report "Mathematical Research in Materials Science. Opportunities and Perspectives" (National Academy Press, Washington, D.C., 1993).
- Advisory Panel, Mathematics and Physical Sciences Directorate, National Science Foundation, 1992-1994
- Resource Advisory Committee, North Carolina Supercomputing Center, 1992-1998
- Chair, National Research Council Panel, "Mathematical Challenges from Theoretical/Computational Chemistry," 1994-1995, National Research Council Report "Mathematical Challenges from Theoretical/Computational Chemistry" (National Academy Press, Washington, D.C., 1995).
- National Research Council Panel, "Containing the Threat from Illegal Bombings," 1996-1998, National Research Council Report "Containing the Threat
from Illegal Bombings. An Integrated National Strategy for Marking,
Tagging, Rendering Inert, and Licensing of Explosives and Their
Precursors" (National Academy Press, Washington, D.C., 1998).
- Panel Member, National Research Council Workshop "Summary Report
of a Workshop on Actions for the Mathematical Sciences" (National
Academy Press, Washington, D.C., 1997).
- Editorial Board, Proceedings of the National Academy of Sciences (USA), 1998-2000.
- "Risk Assessment of Radon in Drinking Water", 1999 (National Research Council Study Monitor)
- "Classifying Drinking Water Contaminants for Regulatory Consideration", 2001 (National Research Council Study Monitor)
- "Arsenic in Drinking Water: 2001 Update", 2001 (National Research Council Study Monitor)
- "Assessment of the Practicality of Pulsed Fast Neutron Analysis (PFNA) for Aviation Security", 2002 (National Research Council Study Monitor)
- "Interim Report of the Committee on Advanced Energetic Materials and Manufacturing Technologies", 2002 (National Research Council Report Reviewer)
- "Materials and Society: From Research to Manufacturing", 2002 (National Research Council Report Reviewer)
- "Assuring Sound Science in Support of Everglades Restoration: An Assessment of the Critical Ecosystems Studies Initiative", 2002 (National Research Council Study Monitor)
- "NEON: Addressing the Nation’s Environmental Challenges", 2003 (National Research Council Report Review Coordinator)
- "Advanced Energetic Materials", 2003 (National Research Council Report Monitor)
- "Making the Nation Safe from Fire", 2003 (National Research Council Report Monitor)
- "Endangered and Threatened Species in the Platte River Basin", 2004 (National Research Council Report Monitor)
- "California Agricultural Research Priorities. Pierce's Disease," 2004 (National Research Council Report Reviewer)
- "Re-engineering Water Storage in the Everglades: Risks and Opportunities," 2004 (National Research Council Monitor)
- "Restructured Upper Mississippi River -- Illinois Waterway Feasibility Study, Third Report," 2005 (National Research Council Monitor)
- "Drawing Louisiana's New Map: Addressing Land Loss in Coastal Louisiana," 2005 (National Research Council Monitor)
- "Progress Toward Restoring the Everglades: The First Biennial Review, 2006", 2006 (National Research Council Report Monitor)
- "Plans and Practices for Groundwater Protection at the Los Alamos National Laboratory. Final Report", 2007 (National Research Council Report Monitor)
- "The Mississippi River and the Clean Water Act: Progress, Challenges, and Oportunities", 2007 (National Research Council Report Monitor)
- "Base Map Inputs for Floodplane Mapping", 2007 (National Research Council Report Monitor)
- "Progress Toward Restoring the Everglades: The Second Biennial Review, 2008", 2008 (National Research Council Report Monitor)
- "Nutrient Control Actions for Improving Water Quality in the Mississippi River Basin and Northern Gulf of Mexico", 2008 (National Research Council Report Monitor)
RECENT INVITED LECTURES
- "Tutorial on Rugged Landscapes and Inherent Structures", Gordon
Conference on Chemistry and Physics of Liquids, Plymouth, NH, Aug.5,
1997.
- "Some Remarks about Glass Transitions", Edmund A. DiMarzio 65th
Birthday Symposium, NIST, Gaithersburg, MD, Nov. 6-7, 1997.
- "Supercooling and Glass Formation", Princeton Materials Institute,
Princeton University, Princeton, NJ, Dec. 3, 1997.
- "Hijacking Solid State Concepts for Liquids", Benjamin Widom 70th/
Leo Kadanoff 60th Birthdays Symposium, Rutgers University Statistical
Mechanics Meeting, New Brunswick, NJ, Dec. 14-16, 1997.
- "Insights and Predictions for Liquids from the Inherent Structure
Viewpoint", Symposium on Dynamics on Rugged Energy Landscapes:
Clusters, Liquids, and Proteins, American Physical Society Meeting,
Los Angeles, CA, Mar. 16-20, 1998.
- "Resolving Vibrational and Inherent Structural Contributions to
Water Properties", Symposium on the Chemistry and Physics of Water,
American Physical Society Meeting, Los Angeles, CA, Mar. 16-20, 1998.
- "Inherent Structure Representation for Condensed Matter Phenomena",
Symposium celebrating the 20th anniversary of the formation of the
Theoretical Chemistry Division, American Chemical Society Meeting,
Boston, MA, Mar. 16-20, 1998.
- "Hard Particles, Hard Problems," 81st Semiannual Statistical Mechanics Meeting, H.L. Frisch 70th Birthday Symposium, Rutgers University, Dec. 15, 1998.
- "Separation of Vibrational and Structural-Shift Contributions to Condensed-Phase Thermodynamic Properties," American Physical Society, Atlanta, GA, March 26, 1999.
- "Unusual Properties of the Classical Gaussian Core Model," 87th Semiannual Statistical Mechanics Meeting, H. Reiss 80th Birthday Symposium, May 19-21, 2002.
- "Polymers, Gaussian Core Model Dualities, and Inverse Melting," Onsager Medal Lecture, Norwegian University of Science and Technology, (NTNU), Trondheim, Norway, Sept. 19, 2002.
- "Hard Sphere Packing Properties and Problems," George Stell Retirement Symposium, SUNY-Stony Brook, April 24, 2004.
- "Potentials, Structures, and Self Assembly," David Chandler 60th Birthday Symposium, University of California, Berkeley, January 8, 2005.
- "Inherent Structure View of Interface Properties," Pacifichem 2005, Honolulu, HI, Dec. 19, 2005.
- "Hard Sphere Pair Correlation via Scaled Particle Theory," 2006 Mini Statistical Mechanics Meeting, University of California, Berkeley, CA, Jan. 15, 2006.
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"Inventing Particle Interactions for Targeted Self-Assembly,"
Slides, Mechanical Engineering Department, Yale University,
April 6, 2006.
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"Configurational Polytopes near the Jamming Limit," Princeton Center for Theoretical Physics workshop, April 12, 2007.
COAUTHORS
Coauthor List:
The following alphabetical arrangement identifies all coauthors from the complete publication list, up through publication number 335. The bracketed numbers following the listed names identify the relevant coauthored publications.
Abrahams, Sidney
C. [113]
Arrecis, Julio
[197]
Bair, Harvey E.
[113]
Batten, Robert D.
[335]
Ben-Naim, Arieh
[34, 44, 56, 111, 114]
Brus, Louis E.
[223]
Budzien, Joanne
L. [249]
Buff, Frank P. [1,
11, 16, 25]
Burton, Joshua [302]
Chatterjee, Swaroop [313, 314, 328]
Cisse, Ibrahim [278]
Chaikin, Paul M. [278, 281, 292, 309]
Chakravarty, Charusita [298, 318]
Chen, Yingzi [258]
Connelly, Robert
[277, 278, 279, 284, 317]
Corti, David S. [224, 230]
Cotter, Martha A. [55, 59, 62]
Crawford, Jenness [273]
David, Carl W.
[96, 108, 123]
Debenedetti, Pablo G.
[224, 230, 231, 234, 235, 236, 238, 242, 243, 245, 246, 247, 251, 252, 254,
255, 259,
261, 262, 266, 271, 272, 276, 280, 291, 295, 297, 298, 312, 313, 314, 318,
323, 324, 328, 330]
DiMarzio, Edmund A. [19, 20]
Donev, Aleksandar
[274, 277, 278, 279, 281, 284, 287, 288, 290, 292, 293, 302, 305, 309, 311,
317, 319, 321]
Dyre, Jeppe C. [246]
Eroles, Juan M. [256]
Fan, Yizhong [193]
Feeney, Melissa R. [271]
Festa, Roger R. [131]
Fredrickson, Glenn H. [160]
Frisch, Harry L. [6, 9, 14]
Gay, David M. [199, 202]
Geiger, Alfons [98, 100]
Glotzer, Sharon C. [246]
Graham, Ronald L. [225, 239]
Haddon, Robert C.
[113, 119]
Hankins, David
[46, 49]
Harkless, John A.W. [221, 223]
Harris, Jonathan G. [179, 180, 183, 189, 196]
Head-Gordon, Teresa
[197, 199, 200, 202, 203, 204, 205, 207, 208, 218]
Heeger, David [292]
Helfand, Eugene
[12, 22, 41]
Herrick, David R.
[74, 75, 78, 82, 87]
Hirshfeld, Catherine L. [207]
Hodgdon, Jennifer A. [206, 212, 213, 214, 222]
Hopkins, Adam B. [336]
Hynninen, Antti-Pekka [307]
Inati, Souheil [292]
Jiao, Yang [320, 327, 333, 337]
Julian, Maureen M. [131]
Kansal, Anuraag R. [267, 268]
Kirkwood, John G. [2, 3, 4]
Kohen, Daniela
[232, 237, 248]
Kornegay, Robert L. [19, 24]
Lombardo, Thomas G. [312]
LaViolette, Randall A. [149, 152, 155, 157, 158, 159, 163, 249]
Lemberg, Howard L. [76, 77, 80, 83]
Lewis, Catherine P. [261]
Losonczy, Miklos
[64, 69]
Lovett, Ronald A.
[25, 37, 38, 42, 191]
Lubachevsky, Boris D. [186, 194, 211, 215, 225, 239, 283]
Lynden-Bell, Ruth M. [328]
Man, Weining [192, 309]
Maranas, Janna K. [258]
Moskowitz, Jules W. [46, 49, 64, 69]
Panagiotopoulos, Athanassios Z. [307]
Patel, Bryan A. [323, 324, 330]
Percus, Jerome K. [193]
Pinson, Elliot N. [194]
Rahman, Aneesur
[54, 57, 60, 65, 66, 67, 68, 72, 80, 92, 98, 100]
Rechtsman, Mikael C.
[301, 306, 307, 316, 325, 334]
Rice, Stuart A. [241]
Roberts, Christopher J. [243, 245]
Root, Leslie J. [169, 173, 176, 181]
Rossky, Peter J. [323, 330]
Rudd, Walter G.
[35, 43]
Ruskai, Mary Beth
[143, 185]
Russel, Matthew T. [292]
Sachs, David [278]
Sakai, Hajime [263, 264, 270]
Salsburg, Zevi W.
[24, 32, 35, 43, 45]
Sastry, Srikanth
[224, 230, 231, 234, 235, 236, 238, 246]
Scardicchio, Antonello [329]
Schrøder, Thomas B. [246]
Schweizer, Kenneth S. [137, 139, 140]
Shell, M. Scott
[276, 280, 295, 297]
Shen, Vincent K. [266]
Skoge, Monica [311]
Stillinger, Dorothea K.
[70, 178, 187, 188, 193, 209, 228, 251, 252, 258, 304]
Sturgeon, Kathy S. [198]
Sullivan, Matthew T. [292]
Suplinskas, Raymond J. [28]
Svensson, C. [113]
Torquato, Salvatore
[236, 238, 251, 252, 256, 260, 263, 264, 265, 267, 268, 270, 273, 274, 275, 277,
278, 279,
281, 282, 284, 285, 286, 287, 288, 290, 292, 293, 294, 296, 300, 301,
302, 303, 305, 306,
307, 308, 309, 310, 311, 315, 316, 317, 319, 320, 321, 322, 325, 326, 327, 329, 332,
333, 334, 335, 336, 337]
Truskett, Thomas M. [236, 238, 243, 251, 252, 256, 259, 261]
Tully, John C.
[232, 237]
Uche, Obioma U. [282, 285, 300, 308, 315]
Utz, Marcel [247, 255]
Variano, Evan A.
[278]
Wasserman, Zelda [94]
Washington, Gary E. [169]
Weber, Thomas A.
[71, 95, 101, 110, 115, 116, 117, 118, 120, 122, 123, 124, 125, 126, 127, 129,
133, 134,
135, 136, 138, 141, 142, 144, 145, 146, 148, 150, 153, 154, 159, 160, 162, 164,
165,
166, 168,
175, 177, 184, 192, 210]
Weeks, John D. [216]
White, Ronald J.
[47, 50, 51, 52]
Widom, Benjamin
[61]
Wojtowicz, Peter J. [2]
Wright, Margaret H. [199, 202]
Yu, A.P. [43]
Zachary, Chase E. [332]
POSTDOCTORAL COLLABORATORS (BELL LABS)
(1) Edmund A. DiMarzio, 1962-1963.
(2) Raymond J. Suplinskas, 1964-1965.
(3) Ronald A. Lovett, 1966-1967.
(4) Arieh Ben-Naim, 1967-1968.
(5) Ronald J. White, 1968-1970.
(6) Martha A. Cotter, 1970-1972.
(7) David R. Herrick, 1973-1975.
(8) Howard L. Lemberg, 1973-1975.
(9) Kenneth S. Schweizer, 1981-1983.
(10) Randall A. LaViolette, 1984-1986.
(11) Leslie J. Root, 1986-1988.
(12) Jonathan G. Harris, 1988-1990.
(13) Teresa Head-Gordon, 1990-1992.
(14) Jennifer A. Hodgdon, 1992-1994.
(15) Daniela L. Kohen, 1995-1997.
HONORS AND AWARDS
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Frank (red tie) received the
Onsager Medal from Rektor Eivind Hiis Hauge (blue tie) and gave the 2002 Onsager Lecture on Sept. 19, 2002 at the Norges Teknisk-Naturvitenskapelige Universitet in Trondheim, Norway.
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- Welch Foundation Lecturer, 1974.
- Elliott Cresson Medal, Franklin Institute, 1978.
- University of Maryland Distinguished Lecturer, 1981.
- Distinguished Member of Technical Staff (DMTS), Bell Laboratories, 1982-2001.
- Karcher Lecturer, University of Oklahoma, 1984.
- Trumbull Lecturer in Chemistry, Yale, 1984.
- Elected to National Academy of Sciences, 1984.
- Washburn Award Lecturer, University of Nebraska, 1986.
- Hildebrand Prize, American Chemical Society, 1986.
- Gucker Lecturer, University of Indiana, 1987.
- W. A. Noyes Lecturer, University of Texas, 1988.
- Langmuir Award, American Physical Society, 1989.
- Regents Professorship, UCLA, 1990.
- Meek Industrial Lecturer, Ohio State University, 1990.
- Peter J. Debye Award, American Chemical Society, 1992.
- McElvane Lecturer, University of Wisconsin, 1992.
- Pitzer Lecturer, UC Berkeley, 1996.
- Visiting Research Collaborator, Princeton Materials Institute,
Princeton University, 1996-2001.
- Visiting Research Collaborator, Department of Chemistry,
Princeton University, 2001-2003.
- Senior Scientist, Department of Chemistry,
Princeton University, 2003-present.
- Onsager Lecture and Medal, Norwegian University of Science and Technology, Trondheim, Norway, 2002.
- Over 300 invited lectures at universities, conferences, government and industrial laboratories.
Last Updated 5-28-09.