CHE 597BCH 795N MultiScale Modeling of Fluids and Soft Matter

1
CHE 597B/CH 795NMulti-Scale Modeling of Fluids
and Soft Matter

• Instructors Stefan Franzen and Keith E. Gubbins
• Lecture 4 Introduction to Semi-Classical
  Statistical Mechanics

2
Multi-Scale Modeling of Fluids and Soft
MatterOutline

• 1. 01/13/04 Introduction. Electronic,
  atomistic, mesoscale modeling examples L 1
• 2. 01/16/04 Ab initio methods. Schrödinger eq.
  B-O approx. L2,3
• 3. 01/21/04 Ab initio methods. MO theory, basis
  sets L 2,3
• 4. 01/23/04 Ab intio methods. DFT and
  applications L 3
• 5. 01/26/04 Statistical mechanics. Canonical
  ensemble. Partition function, thermo GG3
• 6. 01/28/04 Semi-classical SM. Factoring the
  distribution function Q GG3
• 7. 02/02/04 Factoring the distribution function
  Q GG3
• 8. 02/04/04 Distribution functions and
  correlation functions GG3
• 9. 02/09/04 Uniqueness theorem. Grand canonical
  ensemble GG3
• 10. 02/11/04 Force fields. Contributions to
  intermolecular forces S1,4 GG 2 L 4
• 11. 02/16/04 Composite force fields L 4
• 12. 02/18/04 Parameterization of force
  fields L 4
• 13. 02/23/04 Atomistic simulation. General
  features L 6 GQ 1 AT 1
• 14. 02/25/04 Monte Carlo simulation. Metropolis
  method L 8 GQ1 AT4
• 15. 03/01/04 Monte Carlo. Isobaric, grand, Gibbs
  ensembles L 8  FS 5, 7
• 16. 03/03/04 Monte Carlo. Reactive MC, Reverse
  MC, free energies FS 7,8, Notes
• 03/08/04 03/12/04 SPRING BREAK

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Multi-Scale Modeling of Fluids and Soft
MatterOutline

• 17. 03/15/04 Molecular dynamics. Finite
  difference methods L 7 AT 3 FS 4
• 18. 03/17/04 Molecular dynamics. Constraint
  dynamics L 7 AT3 FS4
• 19. 03/22/04 Mesoscale methods. Lattice MC,
  Brownian dynamics FS17
• 20. 03/24/04 Mesoscale methods. Dissipative
  Particle Dynamics FS 17
• 21. 03/29/04 Mesoscale methods. Examples Notes
• 22. 03/31/04 The interface surface tension,
  adsorption Notes
• 23. 04/05/04 Density functional theory of
  interfaces Notes
• 24. 04/07/04 No Class (Easter Break)
• 25. 04/12/04 Adsorption, fluids in pores Notes
• 26. 04/14/04 Surfactant solutions Notes
• 27. 04/19/04 Colloids Notes
• 28. 04/21/04 Biological systems Notes, L10
• 29. 04/26/04 Biological systems Notes, L12
• 30. 04/28/04 Special topics Notes
• ATAllan Tildesley FSFrenkel and Smit
  GGGrayGubbins GQGubbinsQuirke LLeach
  SStone..

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Theory and Simulation Scales
Continuum
Methods
Based on SDSC Blue Horizon (SP3) 512-1024
processors 1.728 Tflops peak performance CPU time
1 week / processor
TIME/s
100
Atomistic SimulationMethods
Mesoscale methods
10-3
(ms)
Lattice Monte Carlo
Brownian dynamics
Dissipative particle dyn
10-6
(ms)
Semi-empirical
10-9
(ns)
methods
Monte Carlo
molecular dynamics
10-12
(ps)
Ab initio
methods
tight-binding
MNDO, INDO/S
10-15
(fs)
10-10
10-9
10-8
10-7
10-6
10-5
10-4
(mm)
(nm)
LENGTH/meters
NC State University 2002
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CHE 597B/CH 795NMulti-Scale Modeling of Fluids
and Soft Matter

• Text
• A.R. Leach, Molecular Modeling Principles and
  Applications, 2nd edn., Prentice-Hall, ISBN
  0-582-38210-6 (2001)
• Supplementary Texts
• Cramer, C.J., Essentials of Compuational
  Chemistry Theories and Models, Wiley,
  Chichester (2002)
• McQuarrie, D.A., Statistical Mechanics, Harper
  Row, New York (1976)
• Gray, C.G. and Gubbins, K.E., Theory of
  Molecular Fluids, Clarendon Press, Oxford
  (1984).
• Stone, A.J., The Theory of Intermolecular
  Forces, Clarendon Press, Oxford (1996)
• Allen, M.P. and Tildesley, D.J., Computer
  Simulation of Liquids, Clarendon Press, Oxford
  (1987)
• Frenkel, D. and Smit, B., Understanding
  Molecular Simulation, second edition, Academic
  Press, San Diego (2002)
• Gubbins, K.E. and Quirke, N., eds., Molecular
  Simulation and Industrial Applications, Gordon
  Breach, Amsterdam (1996)

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Statistical Mechanics of Fluids A Brief History

• Noninteracting particles (1920 to 1950s)
• (a) Ideal gas (U0) JANAF, API Tables
• (b) Theory of solids Debye, Einstein
• (c) Electrons in metals (ideal gas of
  fermions)
• (d) Photon gas (Bose-Einstein gas)
• (e) Adsorption on solids at low pressure
• Virial expansion (1930s to 1960s) Ursell, 1927
  Mayer 1937
• (a) equation of state for gases
• (b) adsorption on solids
• (c) dilute solutions

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Statistical Mechanics of Fluids A Brief History

• Cell Lattice Theories of Liquids (1930s to
  present)
• (a) Mean filed theories (Lennard-Jones-Devonshi
  re, Guggenheim, etc.)
• (b) Ising model, decorated lattice models
• (c) Renormalization group (K. Wilson, 1971)
• Integral Equation Theories (1930s to present)
• (a) Born-Green-Yvon-Kirkwood theories
• (b) Percus-Yevick theory (PY)
• (c) Hypernetted Chain theory (HNC)
• (d) Mean Spherical Approximation (MSA)
• (e) Reference Interaction Site Model (RISM)

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Statistical Mechanics of Fluids A Brief History

• Perturbation Theories (1950s to present)
• (a) Cluster expansions in the pertubing
  potential
• (b) Density functional theories (1970s to
  present)
• liquids, solids
• surface properties
• Atomistic Simulation (1953 to present)
• (a) Monte Carlo Metropolis, Rosenbluth, Teller
  and Teller, 1953
• (b) Molecular Dynamics Alder Wainwright 1957

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The Semi-Classical Approximation

• For many applications in chemistry, engineering,
  materials science and biology, we can treat
  translational, configurational and rotational
  degrees of freedom classically this implies
  ?ekT. Electronic and vibrational energies must
  usually be treated quantally. This is the
  Semi-Classical Approximation
• Partition function QQclassQquant
• Probability distribution law

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The Semi-Classical Approximation

• The semi-classical approximation is usually
  accurate provided the temperature is not very low
  and the molecules are not very light (H, He).
  When quantum effects are not negligible we can
  often correct using the Wigner-Kirkwood expansion
  for Q.
• This approximation implies that the
  intermolecular potential energy ? is independent
  of electronic and (some or all) vibrational
  states of the molecules. This can be checked by
  spectroscopy.

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The Semi-Classical Approximation

• The semi-classical treatment will be a good
  approximation provided that the spacing between
  energy levels is small enough compared to kT.
  Specifically, the conditions are
• Translational
• Here ? is the dimensionless de Broglie
  wavelength (or de Boer parameter)
• Rotational KE
• where Be is the rotational constant for the
  molecule in cm-1

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The Semi-Classical Approximation

• Vibrational
• where ?v h?/k is the characteristic vibrational
  temperature
• Symmetry effects These arise from the boson or
  fermion character of the particles, and are
  nearly always negligible for our applications.
  The criterion that these effects be negligible is
  that the thermal de Broglie wavelength,
  ?th/(2?mkT)1/2, be small compared to the
  inter-particle spacing. The hard cores in real
  molecules prevent the molecules getting close
  enough to notice their fermion/boson character
  (except He see GG p. 9 for brief discussion and
  references)

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Table 1.1. Quantum Effects in Liquids at their
Normal Boiling Point
For polyatomics ?v is the smallest ?v, and ?r
is the largest value.
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Quantum Corrections Wigner-Kirkwood Expansion

• Wigner-Kirkwood expand the partition function Q
  about its classical value in powers of h
  neglecting fermion/ boson effects, only even
  powers of h occur in the series. For linear
  molecules

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Quantum Corrections Wigner-Kirkwood Expansion

• Here lt?2gt mean squared torque and ltF2gt is mean
  squared force on a molecule
• Similar eqns. hold for other molecular symmetries
  and are given in
• GG Appendix 3D
• J.G. Powles and G. Rickayzen, Mol.
  Phys., 38, 1875 (1979)

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Rigid Molecule Approximation

• For small molecules such as N2, O2, CH4, H2O,
  NH3, C2H4 it is reasonable to treat the molecule
  as a rigid body when considering intermolecular
  forces. Then
• ??(r1,r2,r3?1,?2,?3,)
• where ri center of mass location of mol. i
  xi, yi, zi
• and ?I orientation of mol. i ?i, ?i,
  ?i (nonlinear molecules) or ?i, ?I (linear
  molecules). In this approximation ? does not
  depend on the vibrational state, bond angles or
  bond lengths. Note that ?d?i 4? for linear and
  8?2 for nonlinear molecules.

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Body-fixed, Space-Fixed Intermolecular Frames

• Space-fixed axes (X,Y,Z) are fixed in space and
  dont rotate when the molecule rotates.
  Body-fixed axes (x,y,z) are fixed in the body of
  the molecule, and rotate as the molecule rotates.
  The molecular orientation angles (? ?, ?, ?)
  specify the orientation of the body-fixed axes
  relative to the space fixed axes. The usual
  choice of definition of these angles is that of
  Euler (see below).

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Space- and Body-Fixed Axes
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Space- and Body-Fixed Axes
(a) Spherical molecules u(r) depends only on
separation. (b) For nonspherical molecules
u(r,?1,?2) depends on vector r and on molecular
orientations. Here ?I represents the orientation
of the body-fixed axes x,y,z relative to the
space-fixed axes X,Y,Z.
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Intermolecular Frame
21
Euler Angles
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Rigid Molecule Approximation

• To what extent do the vibrational states
  influence intermolecular forces for small
  molecules? This is hard to test experimentally,
  but we can answer the inverse question. To what
  extent do the intermolecular forces influence the
  vibrations? This is readily answered by
  measuring vibration frequencies spectroscopically
  in dilute gases and again in a dense state, or in
  a liquid solvent. Such changes are generally
  small, of order 0.1 to 0.5, and so are usually
  ignored for small, more or less rigid, molecules.
  For larger molecules, particularly chain-like
  molecules (alkanes, polymers, most surfactants,
  proteins, etc.), bending and torsional motions do
  have an important influence on intermolecular
  forces and should be accounted for.

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Vibration Frequency Shifts for Substances
Dissolved in Liquid Argon at 90-100K, and in CCl4
at Room Temperature
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Pairwise Additivity Approximation

• For an isolated pair of molecules in some
  specific configuration, specified by r, ?1, ?2,
  there will be some definite (isolated) pair
  potential u(r,?1, ?2) and corresponding
  intermolecular force F(r,?1, ?2), where

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Three-Body Forces
The isolated 12 pair of molecules will have
electron clouds indicated by the solid lines, and
will interact with the (isolated) pair potential
u(r12 ,?1,?2). When a third molecule 3 is
brought near the 12 pair their electron clouds
will be perturbed as shown by the dashed lines.
Now the total potential energy of the 3 body
system will no longer be the sum of the three
isolated pair potentials, but will contain also a
3-body potential u(12, 13, 23), where
12r12,?1,?2, etc.
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Multi-Body Potentials
In general, the intermolecular potential energy
is written as a sum of isolated two-body
interaction terms, u(ij), a sum of three-body
interaction terms, u(ijk), that represent the
additional potential energy contribution due to
the presence of a third body (in the absence of
4th, 5th, etc. bodies, a sum of 4-body terms,
u(ijkl), and so on. Here i denotes ri,?1,?2.
Note that in this way of writing the potential
energy, u(ij) is the potential energy due to the
interaction between a pair of molecules with
coordinates i and j in the absence of any other
molecules. Similarly, u(ijk) is the potential
energy of interaction due to a third body that is
in addition to the sum of the isolated pair
interactions and so on. In practice, only the
first few terms of this series are considered.
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Multi-Body Potentials

• Types of Multi-Body Potentials
• Repulsion Can usually be neglected the
  molecules hard cores prevent them getting close
  enough
• Dispersion A theory of 3-body dispersion
  interactions was worked out by Axilrod and Teller
  in 1943, and extended to molecules by Stogryn in
  1970. These forces contribute a few percent to
  the internal energy for liquids, e.g. about 5
  for liquid argon.
• Electrostatic Exactly additive (Coulombs Law)
• Induction Many-body forces can be significant
  when polar or charged species are present and
  molecules are highly polarizable, e.g. 10 of
  total potential energy.

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Pairwise Additivity Approximation

• In most calculations we shall assume pairwise
  additivity of the potential energy,
• When using semi-empirical force fields for U it
  is usual to fit two or more parameters in the
  equation for the pair potential to experimental
  data. If the data is for a dense gas or a
  condensed phase, then the pair potential becomes
  an effective pair potential, since the parameter
  fitting will now have to embrace any 3-body and
  higher potential effects. In this case the
  parameter values will depend on the properties
  chosen for fitting and on the state condition,
  particularly the density.

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Sensitivity to Multi-Body Forces

• Some physical properties are especially sensitive
  to three-body and higher body forces. Examples
  are
• Third virial coefficients
• Surface tension

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Effect of 3-Body Forces on Third Virial
Coefficient
Third virial coefficient of argon from experiment
(circles), calculated using 2 body BFW potential
(lower dashed curve), and calculated including
various 3-body potentials. Dotted curve is using
Axilrod-Teller 3 body term, dash-dot and solid
curves using higher order contributions to 3 body
potentials.
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Surface Properties of Liquid Argon at the Triple
Point