1 CE 530 Molecular Simulation Lecture 1 David A. Kofke Department of Chemical Engineering SUNY Buffalo kofke@eng.buffalo.edu
2 Time/s Multi-Scale Modeling Based on SDSC Blue Horizon (SP3) 1.728 Tflops peak performance CPU time = 1 week / processor Continuum Methods 10 0 (ms) (µs) 10-3 10-6 Atomistic Simulation Methods Mesoscale methods Lattice Monte Carlo Brownian Dynamics Dissipative Particle Dynamics (ns) 10-9 Semi-empirical methods Monte Carlo (MC) Molecular Dynamics (MD) (ps) (fs) 10-12 10-15 Ab initio methods Tight-binding MNDO, INDO/S 10-10 10-9 10-8 10-7 10-6 10-5 10-4 (nm) (µm) F.R. Hung, K.E. Gubbins, and S. Franzen, Chemical Engineering Education, Fall 2004 Length/m
Electronic Scale: ab initio 3 Basic idea Calculate properties from first principles by solving Schrödinger equation numerically. Pros Can handle bond breaking/ formation processes Can be improved systematically, allowing assessment of quality Can in principle obtain exact properties from input of only atoms in system Cons Can handle only small systems, on order 10 2 atoms Can study only fast processes, on order of 10 ps Approximations necessary to solve equations
Electronic Scale: semi-empirical 4 Basic idea Use simplified versions of equations from ab initio methods (e.g., treat only valence electrons); include fitting parameters. Pros Can handle bond breaking/ formation processes Can handle larger systems, of order 10 3 atoms Can be used for longer time scales, on order of 10 ns Cons Difficult to assess the quality of the result Need experimental input and large parameter sets Parameters for one behavior may not be best for another (non-transferable)
Classical Atomistic Scale: Molecular Simulation Basic idea Use empirical or ab initio derived force fields, and sample atom configurations to determine thermophysical properties. 5 Pros Can handle larger systems, of order 10 6 atoms Can be used for longer time scales, on order of 1 µs Can provide a broad range of properties all consistent to same molecular model Cons Tradeoff between quality of model and size/time accessible to simulation Some behaviors occur on time scales still inaccessible (e.g., diffusion in solids, many chemical reactions, protein folding, micellization) Lose electronic properties, rxns
Meso-Scale Modeling 6 Basic idea Average out faster degrees of freedom and/or treat large groups of atoms as single entities with effective interactions. Pros Can handle larger systems, of order 10 9 atoms Can be used for longer time scales, on order of seconds Cons Often provides only qualitative information; difficult to assess correctness of quantitative data Approximations limit ability to physically interpret results; key information is lost or averaged out
Continuum Modeling 7 Basic idea Assume that matter is continuous and treat system properties as fields. Numerically solve balance and constitutive equations. Pros Can handle systems of any macroscopic size and time scale Input properties often accessible from experiment Cons Requires specification of constitutive model Requires data from experiment or lower-level method Cannot explain molecular origins of behavior
8 Time/s Multi-Scale Modeling Based on SDSC Blue Horizon (SP3) 1.728 Tflops peak performance CPU time = 1 week / processor Continuum Methods 10 0 (ms) (µs) 10-3 10-6 Atomistic Simulation Methods Mesoscale methods Lattice Monte Carlo Brownian Dynamics Dissipative Particle Dynamics (ns) 10-9 Semi-empirical methods Monte Carlo (MC) Molecular Dynamics (MD) (ps) (fs) 10-12 10-15 Ab initio methods Tight-binding MNDO, INDO/S 10-10 10-9 10-8 10-7 10-6 10-5 10-4 (nm) (µm) F.R. Hung, K.E. Gubbins, and S. Franzen, Chemical Engineering Education, Fall 2004 Length/m
What is Molecular Simulation? Molecular simulation is a computational experiment conducted on a molecular model. 10 to 100,000 or more atoms are simulated (typically 500-1000) 9 Many configurations are generated, and averages taken to yield the measurements. One of two methods is used: Molecular dynamics Monte Carlo Integration of equations of motion Ensemble average Deterministic Stochastic Retains time element No element of time Molecular simulation has the character of both theory and experiment Applicable to molecules ranging in complexity from rare gases to polymers to metals
What is a Molecular Model? 10 A molecular model postulates the interactions between molecules Energy 1.5 1.0 0.5 0.0 A typical two-body, spherical potential (Lennard-Jones model) σ u(r) = 4ε r 12 σ r 6-0.5 More realistic models require other interatomic contributions Intramolecular stretch, bend, out-of-plane bend, torsion, +intermolecular terms Intermolecular van der Waals attraction and repulsion (Lennard-Jones form) electrostatic multibody -1.0 1.0 1.2 1.4 1.6 1.8 2.0 Separation
Why Molecular Simulation? 11 Molecular simulation is the only means for accurately determining the thermophysical properties of a molecular model system Theory model and treatment Experiment test treatment Simulation test model
Example Use of Molecular Simulation 1. 12 Ideal gas equation-of-state macroscopic model P = ρrt P = pressure (bar) ρ = molar density (moles/liter) R = gas constant (0.08314 bar-liter/mol-k) T = absolute temperature (K) molecular model U(r) = 0 no molecular interactions macroscopic model can be derived exactly from molecular model deviation of ideal gas EOS with experiment indicates failing of molecular model no need for simulation nevertheless, instructive to consider its application
Example Use of Molecular Simulation 2. 13 Particles move at constant velocity until collision with wall Pressure as a momentum flux (p x,p y ) P=(momentum to wall)/area-time F/A [=] (ml/t 2 )/L 2 [=] (ml/t)/l 2 -t [=] p/a-t elastic collisions with container walls momentum transfer per collision sum over collisions for unit of time pressure is not given exactly, but as an average P t= 0 Δp sum over long time t o, or over many time origins, to get precise average Click for Ideal-gas simulation = 1 At o t o Δp = 2 p x (-p x,p y )
Example Use of Molecular Simulation 3. 14 Interacting particles hard spheres particles move at constant velocity until collision with another disk or a wall elastic collisions cannot solve for exact EOS Approximate EOS Percus-Yevick virial 2 1+ 2η + 3η P v = ρkt 2 (1 η) compressibility 2 1+ η + η P c = ρkt 3 (1 η) Cannot compare to experiment to resolve quality of these formulas
General Uses of Molecular Simulation 1. 15 Abstract models include only the most important qualitative aspects Examples Hard spheres Lattice models Point dipoles, quadrupoles Energy Separation Lessons Attraction is needed to condense, but not to freeze Molecular diffusion is coupled to molecular convection No analytic equation of state can describe the critical region Volume fraction takes the role of mole fraction in describing macromolecular systems Quadrupole moments raise the triple point relative to the critical point In these applications, molecular simulation is a tool to guide theory
General Uses of Molecular Simulation 2. 16 Realistic models include the greatest feasible detail to give quantitative predictions and explanations Features Lennard-Jones forms Multisite Point charges, polarizable Very specialized Applications Biochemical systems (1 atm, 25ºC) Processes inside of zeolites Alkane critical properties with chain length Fits of individual properties (e.g., density, liquid enthalpy) to experiment for a few systems In these applications, molecular simulation has the potential to guide, and in some instances replace, experiment
17
Water and Aqueous Solutions 18 GEMC simulation of water/methanol mixtures Excellent agreement between simulation and experiment 1.0 0.8 y 0.6 0.4 0.2 Expt. GEMC 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x Strauch and Cummings, Fluid Phase Equilibria, 86 (1993) 147-172; Chialvo and Cummings, Molecular Simulation, 11 (1993) 163-175.
Phase Behavior of Alkanes 19 Panagiotopoulos group Histogram reweighting with finite-size scaling
Alkane Mixtures 20 Siepmann group Ethane + n-heptane 10 8 Exp. at 366 K TraPPE-EH at 366 K Exp. at 450 K TraPPE-EH at 450 K P [MPa] 6 4 2 0 0 0.2 0.4 0.6 0.8 1 X ethane
Kinematic Viscosity Index of Squalane squalane (C 30 H 62 ) 21 ν (10-6 m 2 /s) 100 Viscosity 10 1 Calc. 311K Calc. 372 K Expt. (Newtonian) viscosity Pred. transition to shear-thinning Predicted: 103±18 Experiment: 116±30 0.1 Moore, 0.0001 J. D., Cui, S. T., Cummings, 0.001 P. 0.01 Shear rate 0.1 1 T. and Cochran, H. D., AIChE Journal, γ (10 12 s -1 ) 43 (1997) 3260-3263 Figure 1: Moore, Cui, Cummings and Cochran (Image of alkanes)