Molecular Theory and Modeling Chemical Engineering, 698D Edward J. Maginn University of Notre Dame Notre Dame, IN 46556 USA ­c 1997 Edward Maginn 2 Contents Preface iii 1 Outline and Scope 1 1.1Definitions................................................ 1 1.2 Synopsis ................................................. 1 1.3 Outline of Statistical Mechanics Section . ............................. 2 1.4 Outline of Molecular Simulation Section . ............................. 3 2 Introduction 5 2.1HistoricalPerspective.......................................... 5 2.2LinkBetweenMicroscopicandMacroscopic............................. 6 2.3PracticalProblems........................................... 6 2.3.1 MaterialDesign......................................... 6 2.3.2 Thermophysical Property Estimation ............................. 9 2.4ReviewofSomeMathematicsandStatistics.............................. 9 2.4.1 Probability . ........................................ 9 2.4.2 Counting............................................ 12 2.4.3 Distributions . ........................................ 14 2.4.4 Stirling’s Approximation . .................................. 17 2.4.5 Lagrange Multipliers ...................................... 18 3 Elementary Concepts in Mechanics and Phase Space 21 3.1GeneralizedCoordinates........................................ 21 3.2 Review of Classical Mechanics . .................................. 22 3.2.1 NewtonianMechanics..................................... 22 3.2.2 LagrangianMechanics..................................... 24 3.2.3 HamiltonianMechanics.................................... 25 3.3 Phase Space and Ergodicity ....................................... 26 3.4 Probability Density and the Liouville Equation ............................ 27 3.4.1 Probability Density ....................................... 28 3.4.2 Evolution of Probability Density – Engineering Approach .................. 29 3.5 Ergodicity and Mixing in Phase Space ................................. 30 3.5.1 Ergodic Flow . ........................................ 31 3.5.2 MixingFlow.......................................... 33 4 Equilibrium Ensembles 35 4.1BasicPostulates............................................. 35 4.2 Microcanonical Ensemble ....................................... 35 i ii CONTENTS 4.2.1 Connection With Thermodynamics . ............................. 37 4.2.2 Second Law of Thermodynamics . ............................. 38 4.2.3 Third Law of Thermodynamics . ............................. 40 4.3 Canonical Ensemble . ........................................ 40 4.3.1 Relationship Between the Canonical Ensemble and Thermodynamics ............ 43 4.3.2 Using the Canonical Ensemble ................................. 45 4.4ElementaryStatisticalMechanicsofFluids.............................. 46 4.4.1 Thermophysical Properties of Ideal Gases ........................... 51 4.4.2 Thermodynamic Properties from Ensemble Averages . .................. 53 4.5OtherEnsembles............................................ 58 4.5.1 Grand Canonical Ensemble . .................................. 58 4.5.2 Isothermal–Isobaric(NPT)Ensemble............................. 62 4.6EquivalenceofEnsembles-Preliminaries............................... 62 4.6.1 EquivalenceofEnsembles-AnExample........................... 64 5 Application of Ensemble Theory: Mean Field Theories 65 5.1 Introduction and Motivation ...................................... 65 5.2MeanFieldApproximations...................................... 65 5.3RegularSolutionTheory........................................ 66 5.4Quasi–ChemicalApproximation.................................... 71 5.4.1 QCAAssumptions....................................... 71 5.4.2 Outline of QCA Derivation . .................................. 71 5.5vanderWaalsEquationofState.................................... 73 5.6Solids:EinsteinModel......................................... 79 5.7 Adsorption: Lattice Gas Models . .................................. 80 5.8PolymerChains............................................. 87 5.8.1 MeanFieldPolymerFormulation............................... 88 6 Intermolecular Interactions 93 6.1 Introduction . ............................................. 93 6.2ConfigurationalEnergy......................................... 94 6.3TypesofIntermolecularInteraction................................... 95 6.4Short–RangeRepulsionInteraction................................... 96 6.5DispersionInteractions......................................... 97 6.6 Composite Potential Energy Functions for Non–Polar Molecules . .................. 98 7 Distribution Functions in Classical Monatomic Liquids 101 7.1 Introduction and Physical Interpretations . .............................101 7.2 Development of Equations Describing Distribution Functions . ..................102 7.3 The Pair Distribution Function . ..................................104 ´Ö µ 7.3.1 Physical Interpretation of g .................................104 ´Ö µ 7.4 Experimental Measurement of g ...................................107 7.5 Thermodynamics From the Radial Distribution Function .......................108 7.5.1 InternalEnergy.........................................109 7.5.2 Pressure .............................................110 7.5.3 Compressibility Equation . ..................................111 7.5.4 PotentialofMeanForce–TheReversibleWorkTheorem..................111 ´Ö µ 7.6 Getting Thermodynamics From g ..................................112 7.6.1 Low Density Limit of the Pair Distribution Function . ..................114 CONTENTS iii 7.6.2 IntegralEquationApproach–BGYEquation.........................115 7.6.3 AnotherApproach–TheDirectCorrelationFunction.....................117 7.6.4 Solving the OZ Equation – The Percus–Yevick and Hyper–netted Chain Approximations . 118 7.6.5 ThePYSolutionForHardSpheres..............................120 8 Introduction to Molecular Simulation Techniques 125 8.1ConstructionofaMolecularModel...................................125 8.2 Model System Size and Periodic Boundary Conditions . .......................129 9 Monte Carlo Methods 133 9.1 Historical Background . ........................................133 9.2MonteCarloasanIntegrationMethod.................................134 9.3ImportanceSampling..........................................138 9.3.1 MarkovChains.........................................138 9.3.2 The Metropolis Monte Carlo Algorithm ............................140 9.3.3 FlowofCalculationsinMetropolisMonteCarlo.......................142 9.3.4 Example: Canonical Ensemble MC of a Simple Liquid . ..................143 9.3.5 Metropolis Method: Implementation Specifics . .......................144 9.3.6 OtherTechnicalConsiderations................................145 9.4 Application of Metropolis Monte Carlo: Ising Lattice . .......................146 9.5 Grand Canonical Monte Carlo Simulations . .............................147 9.6GibbsEnsembleMonteCarlo......................................150 9.6.1 GeneralizationoftheGibbsTechnique.............................153 9.6.2 AdditionalComputationalDetailsforMonteCarloSimulations...............154 10 Molecular Dynamics Methods 157 10.1 Introduction . .............................................157 10.2FormulationoftheMolecularDynamicsMethod...........................157 10.3SomeGeneralFeaturesoftheEquationsofMotion..........................159 10.4 Difference Methods for the Integration of Dynamical Equations . ..................159 10.4.1 Algorithmic Considerations in MD . .............................160 10.4.2 Gear Predictor–Corrector Methods . .............................161 10.4.3 The Verlet Algorithm ......................................162 10.4.4 The Leap–Frog Verlet Algorithm . .............................163 10.5MolecularDynamicsofRigidNon-LinearPolyatomicMolecules...................164 10.5.1ConstraintDynamics......................................167 10.5.2 General Comments on the Choice of Algorithms .......................170 10.6MolecularDynamicsInOtherEnsembles...............................170 10.6.1 Stochastic Methods .......................................171 10.6.2 Extended System Methods . ..................................171 10.6.3 Constraint Methods .......................................172 10.7StructuralInformationFromMD(andMC)..............................173 10.7.1 Pair Distribution Functions . ..................................173 10.7.2MolecularOrientation.....................................175 10.8 Dynamical Information From Equilibrium MD ............................175 10.9TransportCoefficientsFromCorrelationFunctions..........................178 10.9.1TransportCoefficientsFromMD................................179 10.9.2ComputingCorrelationFunctionsFromMDRuns......................181 10.9.3 General Considerations For Writing an MD Code .......................182 iv CONTENTS 10.9.4StructuringtheProgram....................................183 Preface If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms-little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information