Modeling & Simulation of Glass Structure

VCG Lecture 21

John Kieffer Department of Materials Science and Engineering University of Michigan

1 Virtual Glass Course — 07 [email protected] Overview …

 Historical perspective  Simulation methodologies  Theoretical background  Monte Carlo simulation  Molecular dynamics simulation  Reverse Monte Carlo  Force fields  Information retrieval  Statistical mechanical formalisms  Structural analyses  Dynamics  Application examples

2 Virtual Glass Course — 07 [email protected] Force Fields

 Types of interactions between two molecules  Electrostatic interactions  Between charges (if ions) and between permanent dipoles, quadrupoles and higher multipole moments – May be represented through partial charges  Embedding terms (metals)  Induction interactions  Between, for example, a permanent dipole and an induced dipole  Dispersion interactions  Long-range attraction that has its origin in fluctuations of the charge distribution  Valence (overlap) interactions  Covalent bonds  Repulsive interaction at short range which arises from exclusion principle  Residual valence interactions  Specific chemical forces giving rise to association and complex formation - e.g., hydrogen bonding  Types of interactions within a molecule  Bond stretch, bond-angle bending, torsional potential

r q 

3 Virtual Glass Course — 07 [email protected] Force Fields

 Non-bonded interactions are typically long-range and involve a large number of particles. Yet, contributions to the total energy and forces are considered as pair- wise additive

Non-bonded (electrostatic)

+ -

Non-bonded (van der Waals) + Johannes Diderik Van der Waals 1837-1923 4 Virtual Glass Course — 07 [email protected] Metals

0.2  Example: Morse force field  (r) 0  Approximate physical behavior

 Computationally efficient -0.2

-0.4 Morse Harmonic  2 a  r  r0   a  r  r0   r   D 0 e  2 e  -0.6

 a  r  r   a  r  r   D e 0 e 0  2 0   -0.8

-1 0.5 1 1.5 2 2.5

r 

5 Virtual Glass Course — 07 [email protected] Van der Waals interactions

 The most popular model of the van der Waals non-bonded interactions between atoms is the Lennard-Jones potential.

12 6      U ( r )  4       r  r   U(r)



Collision typical cutoff diameter at 2.5  Sir John Edward  minimum at r=1.122 Lennard-Jones 1894-1954

r = rij 6 Virtual Glass Course — 07 [email protected] Metals

 Embedded atom method

1 E  F    r t   i     ij  2 i i j  i

  f r i   ij  j  j 



7 Virtual Glass Course — 07 [email protected] Metals

 Embedded atom method

  r re 1 f r  f e e

  r re 1  r   e e



      F    E 1  ln      6       c   e   e  e  e 

8  Virtual Glass Course — 07 [email protected] Embedded atom method

      F    E 1  ln      6      c   e   e  e  e 

1 2  B       E c   Ec = cohesive energy Other factors are model parameters that are determined based on B = bulk modulus  lattice parameter, bulk modulus,  = atomic volume shear modulus, cohesive energy, vacancy formation energy, and average electron density

9 Virtual Glass Course — 07 [email protected] Formation of dislocations in metals

V. Bulatov, LLNL

10 Virtual Glass Course — 07 [email protected] Simulation of Laser Ablation on Metal Surface

QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.

G.H. Gilmer, LLNL 3·106 atoms x 140 ps

11 Virtual Glass Course — 07 [email protected] Embedded atom method

 Simple closed form  Computationally efficient  Based on density functional theory concept  Accurate for highly symmetric structures (FCC, BCC)  Accurate for spherical electron orbitals

 Increasingly inaccurate for complex atomic electron configurations and low-symmetry structures

12 Virtual Glass Course — 07 [email protected] Range of interactions

Short-range interactions: Long-range interactions: 1/2·N·NC force calculations 1/2·N·(N-1) force calculations

13 Virtual Glass Course — 07 [email protected] Ionic Materials

 Coulomb interactions are fundamental to atomistic systems  Coulomb interactions have long-range effects

N N 1 q iq j  CC    4   r 0 i j  i ij

 14 Virtual Glass Course — 07 [email protected] Ionic Materials

 Degree of charge localization can vary  Charge polarization effects can be accounted for analytically

N N N N   q   CD 1 q i DC 1 j      j  r ij     i  r ij 4   r 3 4   r 3  0 i j  i ij  0 i j  i ij

  N N      DD 1 3     i j      r   r     5  i ij  j ij  3  4    r r  0 i j  i  ij ij 

15 Virtual Glass Course — 07 [email protected] Ionic Materials

 Repulsion due to electron orbital overlap  Repulsion due to interactions between nuclei … handled by empirical function

N NC  z  BM z j  i  j  rij  ij   A 1  i  e   ij   n n i j  1  i j 



16 Virtual Glass Course — 07 [email protected] The trouble with Coulomb interactions

5

CC  (r ) ij 4

3

N N 1 q iq j  CC    4   r 2 0 i j  i ij

1

 0 0 1 2 3 4 5 r ij

17 Virtual Glass Course — 07 [email protected] The trouble with Coulomb interactions

Conditionally convergent … I.e., depends on how one adds it up …

18 Virtual Glass Course — 07 [email protected] Long-range interactions: Ewald summation method

Total potential energy of one ion at the reference point. Summation decomposed into two parts, one to be evaluated in real space, the other in reciprocal space. …

 b

   1   2  a      1 a b 

2   3 2        r  r        l         r  r   q e      l  l      2

 19 Virtual Glass Course — 07 [email protected] Reciprocal space

2   a   4    a

iGr  a   c G e G   iGr     G e G

 G 2 c e iGr  4   e iGr 2  G  G  c G  4   G G G G



 20 Virtual Glass Course — 07 [email protected]  Reciprocal space

iGr     G e G

3 2 2    r  rl      r  r l    q l    e l l

 21 Virtual Glass Course — 07 [email protected]

 Reciprocal space

 e iGr e  iGr d r    G  G cell

 2   iGr 3 2    r  r   iGr  e d r  q   e l e d r    l    cell cell  l 

 22 Virtual Glass Course — 07 [email protected]

 Reciprocal space

 e iGr e  iGr d r    G  G cell

 2   iGr 3 2    r  r   iGr  e d r  q   e t e d r    t    cell space  t 

 23 Virtual Glass Course — 07 [email protected]

 Reciprocal space

 2  iGr  iGr 3 2    r  r   iGr  e e d r     q   e t e d r G  G   t    cell space  t 

2 3 2  iGr  iG         q   e t e d r G  t    t space    2 2  iGr t  G 4   G 4   G    q t e e  S G e  t  



24 Virtual Glass Course — 07 [email protected] Reciprocal space

2 4   2 iG  r  G 4    a   S G G e 

4  2   S G G  2 e  G 4  At r = 0: a    b Self-energy correction  1 2   4  r 2  r dr  2 q   b    t   0  

2 4   2  G 4  1 2  1   S G G e  2 q t     

25 Virtual Glass Course — 07 [email protected]  Real space

 2

 r  l   1 1  r    q    r dr  dr  2  l     r r r  l  l l 0 rl 

q   l erfc  r 2   l  r l l r l 

26 Virtual Glass Course — 07 [email protected] Convergence in real and reciprocal space

4  2 1 2 q   S G G  2 e  G 4   2 q    l erfc  r i    i     l   r l l 5

1 CC S(G)  (r ) ij 4

0.5  3

0

2

-0.5

1

-1 0

0 5 10 15 20 25 0 1 2 3 4 5 G r ij 27 Virtual Glass Course — 07 [email protected] Molecular Simulation

 Massively parallel molecular dynamics  Spatial domain decomposition - large systems  Replicated data - long simulation times

28 Virtual Glass Course — 07 [email protected] Spatial Decomposition and Ewald Sum

1 2 > 1

Requires larger G

4  2 1 2 q   S G G  2 e  G 4   2 q    l erfc  r i    i     l   G l rl 29 Virtual Glass Course — 07 [email protected] Particle-Mesh Ewald Method

 iG r  i h x  k y  lz  S G  q e t  q e t t t    t  t t t

r r j j f ij m+1 f ij

m r r i i f[u] interpolation function

n n+1  iG u   i h u  k v  lw  S G  f q e  q e    u     t t t 30 Virtual Glass Course — 07 [email protected] Example: fracture in brittle materials

Quic kTime™ and a GIF decompressor are needed to see this picture.

31 Virtual Glass Course — 07 [email protected] Overview …

 Historical perspective  Simulation methodologies  Theoretical background  Monte Carlo simulation  Molecular dynamics simulation  Reverse Monte Carlo  Force fields  Information retrieval  Statistical mechanical formalisms  Structural analyses  Dynamics  Application examples

32 Virtual Glass Course — 07 [email protected] Ionic materials

 Fundamentally correct  Accurate for strongly ionic compounds, including many ceramics and salts …  Long-range interactions cause computational expense

 Increasingly inaccurate when partially covalent interactions and charge transfer occurs

33 Virtual Glass Course — 07 [email protected] Questions?

34 Virtual Glass Course — 07 [email protected] “Universal” force fields

 Goal is to develop force field that can be to describe intra- and inter-molecular interactions for arbitrary molecules  UFF (Universal force field)  A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard and W. M. Skiff, "UFF, a Full Periodic-Table Force-Field for Molecular Mechanics and Molecular-Dynamics Simulations," J. Am. Chem. Soc, 114, 10024-10035 (1992)  Used for many nanoscale carbon structures (buckyballs, nanotubes)  COMPASS (Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies)  Sun, H; Rigby, D. "Polysiloxanes: Ab Initio Force Field And Structural, Conformational And Thermophysical Properties", Spectrochimica Acta (A) ,1997 ,53 , 130, and subsequent references  http://www.accelrys.com/cerius2/compass.html  AMBER (Assisted Model Building with Energy Refinement)  Both a force field and a simulation package for biomolecular simulations  http://amber.scripps.edu

2 2 V n E total   K r r  req    K q r  rq    1  cos n     2 bonds angles dihedrals     A ij B ij q i q j C ij D ij        12 6    12 10  R R  R R R i  j  ij ij ij  H  bonds  ij ij  35 Virtual Glass Course — 07 [email protected]

 “Universal” force fields

 CHARMM (Chemistry at HARvard Macromolecular Mechanics )  Both a force field and a simulation package for biomolecular simulations  B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus, “A Program for Macromolecular Energy, Minimization, and Dynamics Calculations, J. Comp. Chem. 4 (1983) 187-217 (1983)  http://www.charmm.org/  Others  GROMOS (http://www.igc.ethz.ch/gromos)  Jorgensen‟s Optimized Potential for Liquid Simulation (OPLS) (http://zarbi.chem.yale.edu)  DREIDING (Mayo, S. L., Olafson, B. D. & Goddard, W. A. Dreiding - a Generic Force- Field for Molecular Simulations. Journal of Physical Chemistry 94 (1990) 8897-8909 )  Specific systems  Alkanes and related systems – SKS: B. Smit, S. Karaborni, and J.I. Siepmann, `Computer simulation of vapor- liquid phase equilibria of n-alkanes', J. Chem. Phys. 102 , 2126-2140 (1995); `Erratum', 109 , 352 (1998) – TraPPE: Siepmann group (http://siepmann6.chem.umn.edu)

36 Virtual Glass Course — 07 [email protected] Latest trends: Charge transfer

 Charge localization depends on environment  Chemical nature of surrounding atoms  Local structure

 E   0   1 2 IP  EA Electronegativity i      q  i i 0

 2  0  E Atomic hardness J ii     IP  EA   q 2  i i 0

IP = first ionization potential; EA = electron affinity 37 Virtual Glass Course — 07  [email protected] Charge transfer

0 0 2 Energy of an isolated atom: E i q  E i 0   i q i  1 2 J ii q i  

Energy of an N atoms:  N N N 0 0 2 E q1 , , q N    E i 0    i q i  1 2 J ii q i   q i  q j J ij i i j  i

N N N 0 1   E i 0    i q i    q iq j J ij 2 i i 1 j  1 

38 Virtual Glass Course — 07  [email protected] Charge transfer

5

J (r) ij 4

1 J r   r  r d r d r 3 ij    i  i  j  j  i j r i  r j  r

2

Spherically symmetric Slater-type 1  orbitals are assumed for (r) 0 0 0.5 1 1.5 2 2.5 3 3.5 4 r ij

39 Virtual Glass Course — 07 [email protected] Charge transfer

Chemical potential of electrons associated with atom i:

N  E 0   i   i   q j J ij  q i j  1

Equilibrium requires 1 = 2 = 3 = … = N, which yields N–1 equations. In addition, charge neutrality requires that  N

 q j  0 j  1

40 Virtual Glass Course — 07  [email protected] Charge transfer

Solving the system of equations yields the equilibrium charges

     1  1 q1 0     0 0  J 21  J 11 J 2 N  J 1 N q 2  1   2                    J  J J  J q  0   0  N 1 11 NN 1 N  N   1 N  Equilibrium charges depend on the chemical nature of species and on the geometry of the configuration, by virtue of the fact that  1 J r   r  r d r d r ij    i  i  j  j  i j r i  r j  r

41 Virtual Glass Course — 07 [email protected]

 -18 4 10 1 repulsive charge transfer Covalently bonded r fe s n a tr e g r a h C

-18 function

materials 2 10 0.5 ) (J )

ij 0 0

(r 

-18 -2 10 Coulomb covalent attractive

-18 -4 10

total

-18 -6 10 0 0.1 0.2 0.3 0.4 0.5 0.6 r (nm) ij

N N C  1 N C 0.4 3 B 0.3    ij   ik  q ijk     torque

i  1 j  1 k  j  1 0.2

 Directional character of bonds 0.1 0

 q ijk  -0.1

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5  Covalent bonding term (empirical), e.g.,

 1  (     r )    ij  rij  ij rij  bij a ij    A e i j ij ij   e 1  e ij ij  ij      42 Virtual Glass Course — 07 [email protected] Reactive force fields … N N C  1 N C  3 B      q     ij ik   ijk   Charge redistribution i  1 j  1 k  j  1

N C  1 o b ij rij  a ij  q  q  e  1  e i i  ij    j  1  Flexible coordination

6 2 2   (q  q )  A ( Z  Z ) Z ijk Z 0 i Z 0  q  C e 0 e 0  ijk   Z 0 Z  3 0 N ( r  a ) b Z  ( e ij Z Z  1)  1 i  j  i

L. P. Huang and J. Kieffer, J. Chem. Phys. 118, 1487 (2003) 43 Virtual Glass Course — 07 [email protected] Force field parameterization

 Reproduce known structures (static properties)  density at ambient pressure  equation of state (density vs. pressure)  mechanical properties (elastic moduli, strength, etc.)  Reproduce dynamic properties  melting temperatures  vibrational spectra  Compare with results from ab initio calculations  bonding energies  Interatomic forces

44 Virtual Glass Course — 07 [email protected] Crystalline silica polymorphs – one parameterization

0 GPa (P4 2 2) -quartz at 0 GPa 1 1 -cristobalite 14 GPa (P4 2 2) a = 4.9 Å 1 1 b = 4.9 Å c = 5.4 Å

16 GPa (C2221) 20 GPa (P41212) X-I phase High-P quartz at 25 GPa a‟ = 6.7 Å a = 4.2 Å b‟ = 5.7 Å b = 4.2 Å c‟ = 6.1 Å c = 6.0 Å a‟

c‟

25 GPa (P42/mnm) stishovite 40 GPa (Pnnm) post-stishovite a = 4.2 Å a = 3.7 Å b = 4.2 Å b = 4.5 Å c = 2.8 Å c = 2.8 Å

Ideal high-P quartz with bcc anion sub-lattice

Huang, Durandurdu and Kieffer, 45 Nature Materials (in press) Virtual Glass Course — 07 [email protected] Parameter Optimization: Crystal Structures

(Hexyl-POSS)

J. Chem. Soc. Dalton Trans. 1994, 3123-3128 291K, NVT, 64 molecules 46 Virtual Glass Course — 07 [email protected] Parameter Optimization: IR Spectrum

J. Chem. Soc. Dalton Trans. 1994, 312347 -3128 Virtual Glass Course — 07 [email protected] Parameter Optimization: Fitting to Electronic Structure Calculations

High POSS-NaCl Potential Surface Intermediate Low

48 Virtual Glass Course — 07 [email protected] Fitting Results

Mulliken point shell EEM

Si 1.094 3.417 2.685 0.7

O -0.65 -2 -1.545 -0.46

H -0.119 -0.417 -0.3675 -0.01~0.01

(E)2 0.107583 0.104569 0.0541699 0.00427882

49 Virtual Glass Course — 07 [email protected] Charge Equilibration Method

50 Virtual Glass Course — 07 [email protected] Questions?

51 Virtual Glass Course — 07 [email protected] QM Methods

 Doing a QM simulation or calculation means including the electrons explicitly.

 With QM methods, we can calculate properties that depend upon the electronic distribution, and to study processes like chemical reactions in which bonds are formed and broken.

 The explicit consideration of electrons distinguishes QM models and methods from classical force field models and methods.

52 Virtual Glass Course — 07 [email protected] Different QM Methods

 Several approaches exist. The two main ones are:  Molecular orbital theory  Came from chemistry, since primarily developed for individual molecules, gases and now liquids.  Two “flavors” – Ab initio: all electrons included (considered exact) – Semi-empirical: only valence electrons included

 Density functional theory  Came from physics and materials science community, since originally conceived for solids.  All electrons included via electronic density (considered exact).

53 Virtual Glass Course — 07 [email protected] Fundamentals

 The operator that returns the system energy is called the Hamiltonian operator H.

Eigenvalue equation Time-independent for system energy H = E Schrodinger Equation

2 2 2 2 2  2  2 e Z k e e Z k Z l H     i    k        2 m 2 m r r r  i e k k i k ik i j ij k  l kl Kinetic Kinetic Potential Potential Potential energy of energy of energy of energy of energy of electrons nuclei electrons electrons nuclei  & nuclei

54 Virtual Glass Course — 07 [email protected] Born-Oppenheimer Approximation

Need to simplify!  Neutrons & protons are >1800 times more massive than electrons, and therefore move much more slowly.  Thus, electronic “relaxation” is for all practical purposes instantaneous with respect to nuclear motion.  We can decouple the motion, and consider the electron-electron interactions independently of the nuclear interactions. This is the Born- Oppenheimer approximation.  For nearly all situations relevant to soft matter, this assumption is entirely justified.

(Hel+Vn)el(qi;qk) = Eelel(qi;qk)

The Electronic Schrodinger Equation

55 Virtual Glass Course — 07 [email protected] QM Methods

 Goal of all QM methods in use today:

Solve the electronic Schrodinger equation for the ground state energy of a system and the wavefunction that describes the positions of all the electrons.

 The energy is calculated for a given trial wavefunction, and the “best” wavefunction is found as that wavefunction that minimizes the energy.

56 Virtual Glass Course — 07 [email protected] QM Methods

 Solving SE is not so easy! Anything containing more than two elementary particles (i.e. one e- and one nucleon) can‟t be solved exactly: the “many-body problem”.

 Even after invoking Born-Oppenheimer, still can‟t solve exactly for anything containing more than two electrons.

 So -- all QM methods used today are APPROXIMATE after all, even if considered “exact”! That is, they provide approximate solutions to the Schrodinger equation.  Some are more approximate than others.

57 Virtual Glass Course — 07 [email protected] Molecular Orbital Theory

 MOT is expressed in terms of molecular wave functions called molecular orbitals.  Most popular implementation: write molecular orbital as a linear combination of atomic orbitals (LCAO):

K   a  Eq. 2.68 in Leach i   i  K = # atomic orbitals   1

 Many different ways of writing “basis set”, which leads to many different methods and implementations of MOT. 

58 Virtual Glass Course — 07 [email protected] Molecular Orbital Theory

 Dozens of approaches for writing basis sets (e.g. in terms of Gaussian wavefunctions, or as linear combos of Gaussians).

 Different implementations retain different numbers of terms.

 Semi-empirical MOT methods consider only valence electrons.

 Some methods include electron exchange.

 Some methods include electron correlation.

59 Virtual Glass Course — 07 [email protected] Density Functional Theory

 A different approach for solving Schrodinger‟s equation for the ground state energies of matter.  Based on theory of Hohenberg and Kohn (1964) which states that it is not necessary to consider the motion of each individual electron in the system. Instead, it suffices to know the average number of electrons at any one point in space.

 The HK theorem enables us to write Eel as a functional of the electron density .

 2 M     Z   r2    1  A  d r  V r  r    r      2 X C  1  i  i  i i  i  2  r  r  A  1 1 A 1 2   To perform a DFT calculation, one optimizes the energy with respect to the electron probability density, rather than with respect to the electronic wave function.

For a given density, the lowest energy is the best one. 60 Virtual Glass Course — 07 [email protected] Density Functional Theory

 In the commonly used Kohn-Sham implementation, the density is written in terms of one-electron molecular orbitals called “Kohn-Sham orbitals”.

 This allows the energy to be optimized by solving a set of one-electron Schrodinger equations (the KS equations), but with electron correlation included. This is a key advantage of the DFT method - it‟s easier to include electron correlation.

 Different choices of basis sets, how many terms to use, of what type, contribute to difficulty of calculation. More than a few hundred light atoms is still too time-consuming, even on big computers.

 For molecules or systems with large numbers of electrons, pseudopotentials are used to represent the wavefunctions of valence electrons, and the core is treated in a simplified way.

61 Virtual Glass Course — 07 [email protected] Applications of ab initio computations using DMol3

62 Virtual Glass Course — 07 [email protected] Electronic band structure of POSS cubes functionalized with n

benzene molecules (n = 0-8) …

-10 -8 -6 -4 -2 0 2 4 6 8 10 Energy (eV) 63 Virtual Glass Course — 07 [email protected] Electronic band structure of POSS cubes functionalized with acene molecules (benzene, naphtalene, anthracene, tetracene, and pentacene)

Pure poss -10 -8 -6 -4 -2 0 2 4 6 8 10 Energy (eV) 64 Virtual Glass Course — 07 [email protected] Electron densities of acene-functionalized POSS

Band Gap Molecule Pure Acene (eV) (eV)

P-POSS 0.999 1.335

T-POSS 1.486 1.597

A-POSS 2.295 3.268

N-POSS 3.325 3.348

B-POSS 4.842 13.776

POSS 8.4 -

HOMO LUMO 65 Virtual Glass Course — 07 [email protected] Electron densities of silica nanotubes

HOMO LUMO 66 Virtual Glass Course — 07 [email protected] Classical vs. ab initio methods

Classical ab initio

 No electronic properties  Electronic details included

 Phenomenological potential  Potential energy surface energy surface (typically 2- calculated directly from body contributions) Schrodinger equation (many body terms included  Difficult to describe bond automatically) breaking/formation  Describes bond  Can do up to a billion breaking/formation particles  Limited to several hundred atoms with significant dynamics

67 Virtual Glass Course — 07 [email protected] Editor‟s Note: Lecture 21 ended here – 4/2/07

Questions?

68 Virtual Glass Course — 07 [email protected] Overview …

 Historical perspective  Simulation methodologies  Theoretical background  Monte Carlo simulation  Molecular dynamics simulation  Reverse Monte Carlo  Force fields  Information retrieval  Statistical mechanical formalisms  Structural analyses  Dynamics  Application examples

69 Virtual Glass Course — 07 [email protected] Simulation Observables: Thermodynamics

In a classical, many body system, the temperature is defined from the equipartition theorem in terms of the average kinetic energy per degree of freedom:

1 2 1 m v x  k B T 2 2

In a simulation, we use this equation as an operational definition of T.

Thus the lower the temperature, the lower the kinetic energy, and the slower the average velocity.

70 Virtual Glass Course — 07 [email protected] Simulation Observables: Thermodynamics

N 2 m v ( t ) T ( t )   i i k ( 3 N  N )  Thermodynamic quantities i 1 B c  Calculate T, , p, E etc. depending on ensemble



Nc = # of constraints (e.g. linear momentum)

3N - Nc = # of degrees NVE of freedom

time steps 71 Virtual Glass Course — 07 [email protected] Simulation Observables: Thermodynamics

N N N 1 2 1 Pressure P   m iv i    f ij rij 3V 6V i 1 i 1 j  1 NVE

P is obtained  from the virial thm of Clausius.

Virial = exp. value of the sum of the products of the particle coords and the forces acting on them.

W=xifxi=-3NkBT

time steps 72 Virtual Glass Course — 07 [email protected] Measuring quantities in molecular simulations

•Suppose we wish to measure in an experiment the value of a property, A, of a system such as pressure or density or heat capacity.

•The property A will depend on the positions and momenta of the N atoms or molecules that comprise the system. The instantaneous value of A is:

N N A(p (t),r (t)) = A(p1x,p1y,p1z, …, x1,y1,z1,…t).

•Over time, A fluctuates due to interactions between the atoms.

73 Virtual Glass Course — 07 [email protected] Measuring quantities in molecular simulations

•In any experiment, the value that we measure is an average of A over the time of the measurement (a time average).

•As the time, t, of the measurement increases to infinity, the “true” average value of A is attained:

1  A  lim  A p N ( t ), r N ( t )dt     t  0

Note that strictly speaking this is true only for ergodic systems in equilibrium.

 74 Virtual Glass Course — 07 [email protected] Measuring quantities in molecular simulations

•To calculate properties in a molecular simulation, we average a property A over successive configurations in the trajectory generated by the equations of motion.

n steps 1 A  lim  A p N ( t ), r N ( t ) n   steps n steps t  0

•The larger nsteps is, the more accurate the estimate of the true time-averaged value of A.

 75 Virtual Glass Course — 07 [email protected] Ensemble averages

 In statistical mechanics we learn that experiments and simulations may be performed in different ensembles.

 An ensemble is a collection of individual systems that can be studied en masse to yield macroscopic properties of the entire collection.

 Every equilibrium configuration we generate via MD represents a unique microstate of the ensemble.

76 Virtual Glass Course — 07 [email protected] Ensemble averages  There are many different thermodynamic ensembles, each of which is characterized by thermodynamic variables that are fixed in the simulation.  NVE - microcanonical  NVT - canonical  NPT - isobaric-isothermal  VT - Gibbs

N = number of particles, V = volume, E = total energy, T = temperature, P = pressure,  = chemical potential

• The simplest MD simulation has constant N, V and E. The NVE ensemble is the microcanonical ensemble; all states are equally likely.

• In NVE MD, solve F=ma for an isolated system, one for which no exchange of energy or particles with the outside.

77 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NVT

Velocity Rescaling 2  Recall KE = mv /2 = 3NkBT/2 in an unconstrained system.  Simple way to alter T: rescale velocities every time step by a factor (t):  If temperature at time t is T(t) and v is multiplied by , then the temperature change is

1 N 2 m (  v ) 2 1 N 2 m v 2  T   i i   i i 2 i  1 3 N k B 2 i  1 3 N k B

2  T  T ta rg e t  T (t )  (   1)T (t )

  T ta rg e t / T (t )

78 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NVT

Velocity Rescaling  This method does not generate configurations in the canonical ensemble.

 Why? Because system is not coupled to an external reservoir, or heat bath.

 Thus not a good method for controlling temperature, since it‟s unknown how rescaling affects properties of the system.

 Instead, use a “proper” thermostat.  Using a weakly coupled thermostat can help eliminate drift in total energy due to algorithm inaccuracies and round-off error.

 Several types of thermostats; choose wisely depending on your problem!

79 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NVT

Stochastic collision method: Andersen thermostat  Occasionally replace a randomly chosen particle‟s velocity with a velocity chosen randomly from a Maxwell-Boltzmann distribution at the desired T. 1/2   m 1 2 p (v ix ) =   exp (- m v ix ) 2  k B T  2 k B T

 This replacement represents stochastic collisions with the heat bath.

 Equivalent to the system being in contact with a heat bath that randomly emits thermal particles that collide with the atoms in the system and change their velocity. 

80 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NVT

Extended system method: Nose’-Hoover thermostat  Introduced by Nose‟ and developed by Hoover in 1985.

 Considers the thermostat to be part of the system (hence „extended system‟).

 Based on extended Lagrangian approach in which an additional dynamical variable or degree of freedom s, representing the reservoir, imposes constraint of constant T.

extended reservoir system physical system

81 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NVT

Extended system method: Nose’-Hoover thermostat

 Reservoir has potential energy Ures and kinetic energy Kres.

reservoir Ures = (f+1)kBT ln s physical system 2 Kres = (Q/2)(ds/dt)

 ( t )  sp s Q

 f = # of dof in physical system, and T is the desired temperature. Q has dimensions energy x (time)2, and is considered the fictitious “mass” of the additional degree of freedom.   The magnitude of Q controls the coupling between the physical system and the reservoir, and thus influences the temperature fluctuations.

82 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NVT

Extended system method: Nose’-Hoover thermostat  Q controls the energy flow between system and reservoir.  If Q is large, the energy flow is slow; the limit of infinite Q corresponds to NVE MD since there is then no exchange of energy between system and reservoir.  If Q is small, then the energy oscillates unphysically, causing equilibration problems.

 Nose‟ suggested Q = CfkBT, where C can be obtained by performing trial simulations for a series of test systems and observing how well the system maintains that desired T.  Upshot: equations of motion are modified and t is multiplied by Q.

Read in Frenkel and Smit for details if interested.

83 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NVT

Extended system method: Nose’-Hoover thermostat

 Best and most used thermostat today for NVT simulations.

 Ensures that average total KE per particle is that of the equilibrium isothermal ensemble.  Reproduces canonical ensemble in every respect (provided only the total energy of the extended system is conserved.)  If momentum also conserved, NH still works only if COM fixed.  Otherwise, must use the method of “NH chains”.

 More complicated than other methods, but can be used with velocity Verlet and predictor-corrector methods.  Dynamics NOT independent of Q, but better than Andersen.

84 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NPT

 Just as one might like to specify T in an MD simulation, one might also like to maintain the system at a fixed pressure P.

 Simulations in the isobaric-isothermal NPT ensemble are the most similar to experimental conditions.

 Certain phenomena may be more easily achieved under conditions of constant pressure, like certain phase transformations.

85 Virtual Glass Course — 07 [email protected] Molecular Dynamics Simulation: NPT

Two types:  Andersen Barostat  Allows box size to fluctuate but not change shape.  Appropriate for liquids, which do not support shear stress.  Change volume by rescaling positions: r = r+r

 Rahman-Parinello Barostat  Allows box size and shape to change.  Use only for solids.  Necessary for simulating phase transformations.

Read in Frenkel and Smit for details if interested.

86 Virtual Glass Course — 07 [email protected] Revisit the equations of motion …

N N N 1 L  m rÝ T rÝ   r Recall Lagrangian  i i i    ij  2 i 1 i 1 j  i

  L   L  L   L   Ý  m rÝÝ    m ir i   i i  t  rÝ  r Ý  t  rÝ  i  i  r i  i    L   r ij     f i  r  r i j  i ij r ij   

ÝÝ m ir i  f i 

87 Virtual Glass Course — 07 [email protected] Flexible simulation boxes

 Mechanical deformation  Structural transitions

88 Virtual Glass Course — 07 [email protected] Raman-Parinello algorithm

Basis vectors are dynamical c variables:

h  a b c 

b 

a

89 Virtual Glass Course — 07 [email protected] Raman-Parinello algorithm

N N N 1 1 T 1 L  m sÝ T G sÝ  W Tr hÝ hÝ   r  p  Tr  G  i i i    ij    2 2   2 i 1 i 1 j  i

  L   L   L   L      Ý  t  sÝ   s  t  h   h  i i

Raman-Parinello Nosé-Hoover algorithm  

N 1 1 T Q L  m s&T G s& f 2  W T r h& h&  f&2  i i i   2 i  1 2 2 N N 1   r  p  T r  G  g  1 k T ln f    ij      B 2  i  1 j  i 90 Virtual Glass Course — 07 [email protected] Simulation Observables: Thermodynamics

Thermodynamic Response Functions: Heat Capacity At the end (less error?)

  E  1 2 C V     E   E     2  T V k B T

- 2 On the fly  NVE: Perform simulations at many different E for the same N &

V, and measure T. Calculate Cv from derivative of E(T) (numerically or fitting polynomial and then differentiating).  NVT: (1) Fix N and V, perform simulations at different T and

measure E. Then calculate Cv from derivative of E(T).

(2) Calculate Cv from fluctuations in energy.

92 Virtual Glass Course — 07 [email protected] Simulation Observables: Thermodynamics

Thermodynamic Response Functions: Adiabatic Compressibility

1  V  2  S      P   P   V  P S

 NVE: (1) Fix N and E, perform simulations at different V and measure P.

Calculate S from derivative of V(P).  (2) Calculate S from fluctuations in pressure.

 NVT: Calculate S from fluctuations in pressure.

93 Virtual Glass Course — 07 [email protected] Simulation Observables: Thermodynamics

Thermodynamic Response Functions: Isothermal Compressibility

2 1  V  1 V  V         T 2 V  P T k B T V

 NVT: Fix N and T, perform simulations at different V and measure P. Calculate T from derivative of V(P).   NPT: Fix N and T, perform simulations at different P and measure V. Calculate T from derivative of V(P).

 NPT: Calculate T from fluctuations in volume.

 VT: Calculate T from fluctuations in N.

94 Virtual Glass Course — 07 [email protected] Questions?

95 Virtual Glass Course — 07 [email protected]