1

CE 530 Molecular Simulation

Lecture 1

David A. Kofke Department of Chemical Engineering

SUNY Buffalo kofke@eng.buffalo.edu

2

Multi-Scale Modeling

F.R. Hung, K.E. Gubbins, and S. Franzen, Chemical Engineering Education, Fall 2004

10-15

10-12

10-9

10-6

10-3

100

10-10 10-9 10-8 10-7 10-6 10-5 10-4

(nm) (µm)

(fs)

(ps)

(ns)

(µs)

(ms)

Length/m

Time/s

Mesoscale methodsAtomistic SimulationMethods

Semi-empiricalmethods

Ab initiomethods

Monte Carlo (MC)Molecular Dynamics (MD)

Tight-bindingMNDO, INDO/S

Continuum

Lattice Monte CarloBrownian DynamicsDissipative Particle Dynamics

MethodsBased on SDSC Blue Horizon (SP3)1.728 Tflops peak performanceCPU time = 1 week / processor

10-15

10-12

10-9

10-6

10-3

100

10-10 10-9 10-8 10-7 10-6 10-5 10-4

(nm) (µm)

(fs)

(ps)

(ns)

(µs)

(ms)

Length/m

Time/s

Mesoscale methodsAtomistic SimulationMethods

Semi-empiricalmethods

Ab initiomethods

Monte Carlo (MC)Molecular Dynamics (MD)

Tight-bindingMNDO, INDO/S

Continuum

Lattice Monte CarloBrownian DynamicsDissipative Particle Dynamics

MethodsBased on SDSC Blue Horizon (SP3)1.728 Tflops peak performanceCPU time = 1 week / processor

3

Electronic Scale: ab initio

¡ 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 102 atoms •  Can study only fast processes,

on order of 10 ps •  Approximations necessary to

solve equations

¡ Basic idea •  Calculate properties from first principles by solving Schrödinger

equation numerically.

4

Electronic Scale: semi-empirical

¡ Pros •  Can handle bond breaking/

formation processes •  Can handle larger systems, of

order 103 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)

¡ Basic idea •  Use simplified versions of equations from ab initio methods (e.g.,

treat only valence electrons); include fitting parameters.

5

Classical Atomistic Scale: Molecular Simulation

¡ Pros •  Can handle larger systems, of

order 106 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

¡ Basic idea •  Use empirical or ab initio derived force fields, and sample atom

configurations to determine thermophysical properties.

6

Meso-Scale Modeling

¡ Pros •  Can handle larger systems, of

order 109 “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

¡ Basic idea •  Average out faster degrees of freedom and/or treat large groups

of atoms as single entities with effective interactions.

7

Continuum Modeling

¡ 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

¡ Basic idea •  Assume that matter is continuous and treat system properties as

fields. Numerically solve balance and constitutive equations.

8

Multi-Scale Modeling

F.R. Hung, K.E. Gubbins, and S. Franzen, Chemical Engineering Education, Fall 2004

10-15

10-12

10-9

10-6

10-3

100

10-10 10-9 10-8 10-7 10-6 10-5 10-4

(nm) (µm)

(fs)

(ps)

(ns)

(µs)

(ms)

Length/m

Time/s

Mesoscale methodsAtomistic SimulationMethods

Semi-empiricalmethods

Ab initiomethods

Monte Carlo (MC)Molecular Dynamics (MD)

Tight-bindingMNDO, INDO/S

Continuum

Lattice Monte CarloBrownian DynamicsDissipative Particle Dynamics

MethodsBased on SDSC Blue Horizon (SP3)1.728 Tflops peak performanceCPU time = 1 week / processor

10-15

10-12

10-9

10-6

10-3

100

10-10 10-9 10-8 10-7 10-6 10-5 10-4

(nm) (µm)

(fs)

(ps)

(ns)

(µs)

(ms)

Length/m

Time/s

Mesoscale methodsAtomistic SimulationMethods

Semi-empiricalmethods

Ab initiomethods

Monte Carlo (MC)Molecular Dynamics (MD)

Tight-bindingMNDO, INDO/S

Continuum

Lattice Monte CarloBrownian DynamicsDissipative Particle Dynamics

MethodsBased on SDSC Blue Horizon (SP3)1.728 Tflops peak performanceCPU time = 1 week / processor

9

What is Molecular Simulation? ¡ Molecular simulation is a computational “experiment” conducted on a molecular model.

¡ 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

10 to 100,000 or more atoms are simulated(typically 500 - 1000)

10

What is a Molecular Model? ¡ A molecular model postulates the interactions between molecules

¡ 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

1.5

1.0

0.5

0.0

-0.5

-1.0

2.01.81.61.41.21.0

A typical two-body, spherical potential (Lennard-Jones model)Energy

Separation

u(r) = 4ε σr

⎛ ⎝

⎞ ⎠

12

−σr

⎛ ⎝

⎞ ⎠

6⎡

⎣ ⎢ ⎤

⎦ ⎥

11

Why Molecular Simulation? ¡ Molecular simulation is the only means for accurately determining

the thermophysical properties of a molecular model system

Theory Experiment

Simulation

model and treatment

test treatment test model

12

Example Use of Molecular Simulation 1. ¡ 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

13

Example Use of Molecular Simulation 2. ¡ Particles move at constant velocity until collision with wall ¡ Pressure as a momentum flux

•  P=(momentum to wall)/area-time F/A [=] (mL/t2)/L2 [=] (mL/t)/L2-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

•  sum over long time to, or over many time origins, to get precise average ¡ Click for Ideal-gas simulation

xpp 2=Δ

(px,py)

(-px,py) ∑=Δ=

ot

to

pAt

P0

1

14

Example Use of Molecular Simulation 3. ¡ 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 compressibility

•  Cannot compare to experiment to resolve quality of these formulas 3

2

)1(1

ηηηρ

−++= kTPc

2

2

)1(321

ηηηρ

−++= kTPv

15

General Uses of Molecular Simulation 1. ¡ Abstract models include only the most important qualitative aspects

•  Examples Hard spheres Lattice models Point dipoles, quadrupoles

•  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

Energy

Separation

16

General Uses of Molecular Simulation 2. ¡ 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

18

Water and Aqueous Solutions ¡ GEMC simulation of water/methanol mixtures

•  Excellent agreement between simulation and experiment

•  Strauch and Cummings, Fluid Phase Equilibria, 86 (1993) 147-172; Chialvo and Cummings, Molecular Simulation, 11 (1993) 163-175.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Expt.GEMC

y

x

19

Phase Behavior of Alkanes ¡ Panagiotopoulos group

•  Histogram reweighting with finite-size scaling

20

Alkane Mixtures ¡ Siepmann group

•  Ethane + n-heptane

(Supercritical) Ethane + n-Heptane

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1Xethane

P [M

Pa]

Exp. at 366 KTraPPE-EH at 366 KExp. at 450 KTraPPE-EH at 450 K

21

Kinematic Viscosity Index of Squalane

0.1

1

10

100

0.0001 0.001 0.01 0.1 1

Calc. 311K

Calc. 372 K

Expt. (Newtonian) viscosity

Pred. transition to shear-thinning

ν (1

0-6 m

2 /s)

γ (1012 s-1)Figure 1: Moore, Cui, Cummings and Cochran

Moore, J. D., Cui, S. T., Cummings, P. T. and Cochran, H. D., AIChE Journal, 43 (1997) 3260-3263

Predicted: 103±18

Experiment: 116±30

squalane (C30H62)

Viscosity

Shear rate (Image of alkanes)