Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009

1. Statistical Mechanics and Multi-Scale Simulation MethodsChBE 591-009 Prof. C. Heath Turner Lecture 00 • Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu • Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm

2. Course Textbook and Supplements Textbook: A.R. Leach, “Molecular Modelling: Principles and Applications”, 2nd edition, Prentice-Hall (2001) Supplementary Texts: • C. J. Cramer, “Essentials of Computational Chemistry: Theories and Models,” Wiley, Chichester (2002) • D. A. McQuarrie, “Statistical Mechanics,” Harper & Row, New York (1976) • C. G. Gray and K. E. Gubbins “Theory of Molecular Fluids,” Clarendon Press, Oxford (1984). • M. P. Allen and D. J. Tildesley, “Computer Simulation of Liquids,” Clarendon Press, Oxford (1987) • D. Frenkel and B. Smit, “Understanding Molecular Simulation,” second edition, Academic Press, San Diego (2002) • J. T. Yates, Jr. and J. K. Johnson, “Molecular Physical Chemistry for Engineers,” University Science Books, Sausalito (2007). • S. M. Binder, “Introduction to Quantum Mechanics,” Elsevier Academic Press, Boston (2004).

3. What this course will cover: • Commonly used theoretical and simulation methods at the electronic, atomistic, and meso scales. • Statistical mechanics of fluids, soft matter • Applications to fluids, interfaces, polymers, surfactants, colloids, biological systems, metals

4. Goals of this Course: Provide you with the background and skills needed to: • Understand the use of theory and simulation in research on fluids and soft matter • Be able to read the simulation literature and evaluate it critically • Identify problems in soft matter amenable to simulation, and decide on appropriate theory/simulation strategies to study them

5. Course Organization: • In addition to the three lectures each week, tutorials will be held periodically in order to introduce students to the web site and web-based applications • You will work problems using web-based modules that will illustrate the different theoretical and simulation approaches, for a variety of problems • No formal exams. You will be asked to complete a term paper project on a topic related to the course. There will be a range of possible topics to choose from.

6. “Chemical Waste Disposal and Computational Technology…” …Which one keeps getting more expensive and which one keeps getting less? Common Simulation Applications: • Toxic Materials • Explosive Materials • High T/P Experiments • Expensive Experiments

7. Research Tools Laboratory Equipment (UA) Shared Equipment (TACC) ~13,000 AMD Opteron processors ~$60,000,000 Garbage IN = Garbage OUT ~100 AMD Opteron processors

8. SIMULATION SIZES and METHODS TIME (s) 100 Continuum Methods 10-2 Mesoscale Methods 10-4 Classical Methods 10-6 10-8 Semi-Empirical Methods 10-10 10-12 Ab Initio Methods 10-14 10-16 10-5 10-4 10-3 10-7 10-6 10-10 10-9 10-8 LENGTH (m)

9. Simulations are Needed for “Small” Systems • The laws that govern the behavior of macroscopic systems often break down for nano-sized systems, such as micro- or meso-porous solids, micellar solutions, colloidal systems, and nano-structured materials. • Examples: • Fick’s Law of Diffusion • Fourier’s Law of heat flow • Kelvin’s and Laplace’s equations for vapor pressure over curved surfaces • The hydrodynamic equations.

10. Ab Initio and DFT Calculations (Quantum Mechanics) Calculate atomic properties by solving the Schrödinger equation for a small system. • Advantages • Can simulate processes that involve bond breaking, bond formation, or electronic rearrangement (e.g. chemical reactions). • Can (in principle) obtain essentially exact properties without any experimental inputs. • Disadvantages • Can handle only small systems, ~200 atoms. • Can only study fast processes, usually ~100 ps. • Approximations are usually necessary to solve the equations. Electron localization function for (a) an isolated ammonium ion and (b) an ammonium ion with its first solvation shell, from ab initio molecular dynamics. From Y. Liu, M.E. Tuckerman, J. Phys. Chem. B 105, 6598 (2001)

11. Semi-empirical Methods Use simplified versions of equations from ab initio methods, e.g. only treat valence electrons explicitly; include parameters fitted to experimental data. • Advantages • Can also handle processes that involve bond breaking/formation, or electronic rearrangement. • Can handle larger and more complex systems than ab initio methods, often of O(103) atoms. • Can be used to study processes on longer timescales than can be studied with ab initio methods, of about O(10) ns. • Disadvantages • Difficult to assess the quality of the results. • Need experimental input and large parameter sets. Structure of an oligomer of polyphenylene sulfide phenyleneamine obtained with the PM3 semiempirical method. From R. Giro, D.S. Galvão, Int. J. Quant. Chem. 95, 252 (2003)

12. Molecular Simulations(Statistical Mechanics) Use empirical force fields, together with semi-classical statistical mechanics (SM), to determine thermodynamic (MC, MD) and transport (MD) properties of systems. Statistical mechanics solved ‘exactly’. • Advantages • Can be used to determine the microscopic structure of more complex systems, 1×106 atoms. • Can study dynamical processes on longer timescales, up to several ms. • Disadvantages • Results depend on the quality of the force field used to represent the system. Properties Measured: heat capacity, phase equilibrium, solvation, PVT behavior, diffusion coefficients, surface tension, solubility Structure of solid Lennard-Jones CCl4 molecules confined in a model MCM-41 silica pore. From F.R. Hung, F.R. Siperstein, K.E. Gubbins.

13. Mesoscale Methods Introduce simplifications to atomistic methods to remove the faster degrees of freedom, and/or treat groups of atoms (‘blobs of matter’) as individual entities interacting through effective potentials. • Advantages • Can be used to study structural features of complex systems with O(108-9) atoms. • Can study dynamical processes on timescales inaccessible to classical methods, even up to O(1) s. • Disadvantages • Can often describe only qualitative tendencies, the quality of quantitative results may be difficult to ascertain. • In many cases, the approximations introduced limit the ability to physically interpret the results. Phase equilibrium between a lamellar surfactant-rich phase and a continuous surfactant-poor phase in supercritical CO2, from a lattice MC simulation. From N. Chennamsetty, K.E. Gubbins.

14. Continuum Methods Assume that matter is continuous and treat the properties of the system as field quantities. Numerically solve balance equations coupled with phenomenological equations to predict the properties of the systems. • Advantages: • Can in principle handle systems of any (macroscopic) size and dynamic processes on longer timescales. • Disadvantages: • Require input (viscosities, diffusion coeffs., eqn of state, etc.) from experiment or from a lower-scale method that can be difficult to obtain. • Cannot explain results that depend on the electronic or molecular level of detail. Temperature profile on a laser-heated surface obtained with the finite-element method. From S.M. Rajadhyaksha, P. Michaleris, Int. J. Numer. Meth. Eng. 47, 1807 (2000)

15. Connections Between the Scales • “Upscaling”: • Using results from a lower-scale calculation to obtain parameters for a higher-scale method. This is relatively easy to do; deductive approach. Examples: • Calculation of phenomenological coefficients (e.g. viscosities, diffusivities)from atomistic simulations for later use in a continuum model. • Fitting of force-fields using ab initio results for later use in atomistic simulations. • Deriving potential energy surface for a chemical reaction, to be used in atomistic MD simulations • Deriving coarse-grained potentials for ‘blobs of matter’ from atomistic simulation, to be used in meso-scale simulations

16. Connections Between the Scales • “Downscaling”: • Using higher-scale information (often experimental) to build parameters for lower-scale methods. This is more difficult, due to the non-uniqueness problem. For example, the results from a meso-scale simulation do not contain atomistic detail, but it would be desirable to be able to use such results to return to the atomistic simulation level. Inductive approach. Examples: • Fitting of two-electron integrals in semiempirical electronic structure methods to experimental data (ionization energies, electron affinities, etc.) • Fitting of empirical force fields to reproduce experimental thermodynamic properties, e.g. second virial coefficients, saturated liquid density and vapor pressure

17. Self Assembly of Surfactants on Surfaces Length Scales diameter : O(10nm) length: O(cm) Surfactant C8E4 bond length: O(100pm) chain length: ~2nm Micelle diameter : ~4nm Micelle length: O(m)

18. Self Assembly of Surfactants on Surfaces Time Scales Self assembly on Surfaces: O(s) and larger molecular motion: (ps to ns) lifetime of micelles: O(s)

19. Self Assembly of Surfactants on Surfaces Mapping mesoscale method (BD,DPD) full atomistic simulation (MD)

20. Self Assembly of Surfactants on Surfaces Course of the Simulation full atomistic simulation (MD) Get coarse grained interaction potentials for mesoscale simulation. Equilibrate the system on the mesoscale. mesoscale method (BD,DPD) Compute mesoscale properties. Compute molecular level properties full atomistic simulation (MD) Refine interaction potentials.

21. Force Polymer Crack Propagation Force Crack Propagation in Glassy Polymers Within the same Simulation: Process Zone process zone: molecular dynamics surrounding: continuum fracture mechanics model.

22. MOLECULAR SIMULATIONS(Example) • Two Main Classes: • Monte Carlo – equilibrium properties (very efficient) • Molecular Dynamics – equilibrium and dynamic properties How does it work?? • Describe how the molecules interact… • Set up the system… • temperature • volume • number of molecules • Initialize the system… • Integrate the equations of motion (according to F=ma)

23. Thermodynamic Property Prediction Tc g T Tc T r

24. Computational Modules • Molecular Dynamics Simulations • ► NAMD ◄ • NPT, NVT, NVE ensemble • Constraints • Steered molecular dynamics • Free energy calculations Electronic Structure Calculations ► Gaussian03 ◄ Ab initio methods Density functional theory (DFT) Semiempirical methods • Molecular Dynamics / Monte Carlo Simulations • ► Etomica / Java Applets ◄