Statistical Mechanics and Multi- Scale Simulation Methods ChBE Prof. C. Heath Turner Lecture 00 Some materials adapted from Prof. Keith E. Gubbins: Some materials adapted from Prof. David Kofke:

Course Textbook and Supplements Textbook: A.R. Leach, “Molecular Modelling: Principles and Applications”, 2 nd 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).

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

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

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.

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

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

SIMULATION SIZES and METHODS TIME (s) LENGTH (m) Classical Methods Mesoscale Methods Continuum Methods Semi-Empirical Methods Ab Initio Methods

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.

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)

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. 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) 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(10 3 ) 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.

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, 1×10 6 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 CCl 4 molecules confined in a model MCM-41 silica pore. From F.R. Hung, F.R. Siperstein, K.E. Gubbins.

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. Phase equilibrium between a lamellar surfactant-rich phase and a continuous surfactant-poor phase in supercritical CO 2, from a lattice MC simulation. From N. Chennamsetty, K.E. Gubbins. Advantages Can be used to study structural features of complex systems with O( ) 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.

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. 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) 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.

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

“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 Connections Between the Scales

Surfactant C 8 E 4 Self Assembly of Surfactants on Surfaces Length Scales bond length: O(100pm) chain length: ~ 2nm Micelle diameter : ~4nm Micelle length: O(  m) diameter : O(10nm) length: O(cm)

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

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

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

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

MOLECULAR SIMULATIONS (Example) Two Main Classes: 1.Monte Carlo – equilibrium properties (very efficient) 2.Molecular Dynamics – equilibrium and dynamic properties How does it work?? 1.Describe how the molecules interact… 2.Set up the system… a)temperature b)volume c)number of molecules 3.Initialize the system… 4.Integrate the equations of motion (according to F=m  a)

Thermodynamic Property Prediction T  TcTc  T TcTc

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