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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008
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1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation
Part II - lecture 3
Atomistic and molecular methods
2
Content overviewLectures 2-10February/March
Lectures 11-19March/April
Lectures 20-27April/May
I. Continuum methods1. Discrete modeling of simple physical systems:
Equilibrium, Dynamic, Eigenvalue problems2. Continuum modeling approaches, Weighted residual (Galerkin) methods,
Variational formulations3. Linear elasticity: Review of basic equations,
Weak formulation: the principle of virtual work, Numerical discretization: the finite element method
II. Atomistic and molecular methods1. Introduction to molecular dynamics2. Basic statistical mechanics, molecular dynamics, Monte Carlo3. Interatomic potentials4. Visualization, examples5. Thermodynamics as bridge between the scales6. Mechanical properties – how things fail7. Multi-scale modeling8. Biological systems (simulation in biophysics) – how proteins work and
how to model them
III. Quantum mechanical methods1. It’s A Quantum World: The Theory of Quantum Mechanics2. Quantum Mechanics: Practice Makes Perfect3. The Many-Body Problem: From Many-Body to Single-Particle4. Quantum modeling of materials5. From Atoms to Solids6. Basic properties of materials7. Advanced properties of materials8. What else can we do?
3
Additional Reading
Books:
Allen and Tildesley: “Computer simulation of liquids”D. C. Rapaport (1996): “The Art of Molecular Dynamics Simulation”D. Frenkel, B. Smit (2001): “Understanding Molecular Simulation”J.M. Haile (Wiley, 1992), “Molecular dynamics simulation”
Additional lecture notes and publications (see MIT Server website)
4
Overview: Material covered
Lecture 1: Introduction to atomistic modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)
Lecture 2: Basic statistical mechanics (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function, solution techniques: Monte Carlo and molecular dynamics)
Lecture 3: Basic molecular dynamics (advanced property calculation, chemical interactions)
5
II. Atomistic and molecular methods
Lecture 3: Basic molecular dynamicsOutline:1. Review lectures 1&2
1.1 Continuum-atomistic 1.2 Basic statistical mechanics
2. Advanced property calculation 2.1 Radial distribution function (RDF)2.2 Velocity autocorrelation function (VAF)2.3 Cauchy stress tensor – virial stress (VS)2.4 Closure – property calculation
3. Introduction: Chemical interactions4.
Goal of today’s lecture: Review of previous lectures to be up to speed for part IILearn how to characterize whether a system is a solid, liquid, or a gas, and how to determine what atomistic crystal structure it hasHow to calculate diffusivity using an alternative approach, how to calculate viscosity, how to calculate thermal conductivity, and how to calculate the stress tensor inside a materialLearn how to model chemical bonds in a material
6
1. Review lectures 1&2
7
1.1 Continuum-atomistic
8
Atomistic versus continuum viewpointContinuum:
Effective behavior (“constitutive laws”, coefficients, parameters), e.g. differential equationsCan be applied at “any” scale: Same theory for earth, microdevices, computer chip…Assume no underlying discreteness of matter
“Partial differential equations”
Atomistic viewpoint:Explicitly consider discrete atomistic structureSolve for atomic trajectories and infer from these about material properties & behaviorFeatures internal length scales (atomic distance)
“Many-particle system with statistical properties”
2
2
dxcdD
tc=
∂∂
9
Diffusion: Phenomenological description
Ink dotTissue
Ink
conc
entra
tion2
2
dxcdD
tc=
∂∂ 2nd Fick law
Image removed due to copyright restrictions. Please see http://www.uic.edu/classes/phys/phys450/MARKO/dif.gif
10
Continuum model: Empirical parameters
Continuum model requires parameter that describes microscopic processes inside the material
Need experimental measurements to calibrate
Microscopicmechanisms
???
Continuummodel
Length scale
Time scale“Empirical”or experimentalparameterfeeding
11
Atomistic model of diffusionDiffusion at macroscale (change of concentrations): result of microscopic processes, random motion of particles
Atomistic model provides an alternative way to describe diffusion
Enables us to directly calculate the diffusion constant from thetrajectory of atoms
Follow trajectory of atoms and calculate how fast atoms leave their initial position
txpDΔΔ
−=2
Follow this quantity overtime
12
Molecular dynamics: A “bold” idea
Follow trajectories of atoms
( ) ...)()(2)()( 20000 +Δ+Δ+Δ−−=Δ+ ttattrttrttr iiii
“Verlet central difference method”
Positions at t0
Accelerationsat t0
Positions at t0-Δt
mfa ii /=
vi(t), ai(t)
ri(t)
xy
z
N particlesParticles with mass mi
13
Typical MD simulation procedure
Set particlepositions
Assign particlevelocities
Calculate forceon each particle
Move particles bytimestep Dt
Save current positions and
velocities
Reachedmax. number of
timesteps?
Stop simulation Analyze dataprint results
14
Example molecular dynamics
2rΔ
Average square of displacement of all particles
( )22 )0()(1)( ∑ =−=Δi
ii trtrN
tr
Particles Trajectories
Mean Square Displacement function
Courtesy of the Center for Polymer Studies at Boston University. Used with permission.
15
2rΔ
Calculation of diffusion coefficient
1D=1, 2D=2, 3D=3
slope = D
( )22 )0()(1)( ∑ =−=Δi
ii trtrN
tr
Position of atom i at time t
Position of atom i at time t=0
( ))(lim21 2 tr
dtd
dD
tΔ=
∞→
Einstein equation
16
???
SummaryMolecular dynamics provides a powerful approach to relate the diffusion constant that appears in continuum models to atomistictrajectories
Outlines multi-scale approach: Feed parameters from atomistic simulations to continuum models
MD
Continuummodel
Length scale
Time scale“Empirical”or experimentalparameterfeeding
17
Example: MD simulation results
liquid solid
Courtesy of Sid Yip. Used with permission.
18
Atomistic trajectory – information about material state
Courtesy of Sid Yip. Used with permission.
19
JAVA applet
http://polymer.bu.edu/java/java/LJ/index.html
Courtesy of the Center for Polymer Studies at Boston University. Used with permission.
20
Experimentally one measured the same value of D at a slightly different temperature.
Both simulation and experiment were carried out at the same density, but the experiment was conducted at 90K whereas the simulation temperature was 94.4 K.
This is therefore an example of accurate prediction of diffusion in a simple liquid.
Argon
Rahmann, 1960
Mean squared displacement function obtained for liquid argon bymolecular dynamics simulation. The curve is plotted as the meanof 64 simulations; data with the highest deviations are shown as points.
Figure by MIT OpenCourseWare. Adapted from Rahman, A. “Correlationsin the Motion of Atoms in Liquid Argon.” Physical Review 136 (October 19, 1964): A405-A411.
5.0
4.0
3.0
2.0
1.0
1.0 2.0 3.00
0
D = 2.43 x 10-5 cm2 sec-1Mea
n sq
uare
dis
plac
emen
t A2.
Time in 10-12 sec
21
1.2 Basic statistical mechanics
How to calculate properties from atomistic simulation
22
Property calculation: IntroductionHave:
“microscopic information”
Want: Thermodynamical properties (temperature, pressure, stress tensor..)Material state (gas, liquid, solid) and structure, e.g. BCC, FCC, ..…
)(),(),( txtxtxatoms ∀
23
Macroscopic vs. microscopic states
NVPT ,,,
≡
…
1C
2C
3C
NC
Macroscopic state is represented by many microscopic configurations
)(),(),( txtxtx
Microscopic states
Macroscopic state
24
Microscopic states
Microscopic states characterized by pr,
{ } { }iii xmpxr == , Ni ..1=
Hamiltonian (sum of potential and kinetic energy, total energy)
)()(),( pKrUprH +=
∑=
=Ni
i rrU..1
)()( φ ∑=
=Ni i
i
mppK
..1
2
21)(
ip
=
25
Ensembles: Set of microscopic configurations that realize macroscopic states
Result of thermodynamical constraints, e.g. temperature, pressure…
MicrocanonicalCanonicalIsobaric-isothermalGrand canonical
NVE
NpTμTV
NVT
Heat bath (constant temperature)Large system
NVT = canonical ensembleSmall system
Physical realization of canonical ensemble
26
Why one can not use a single microscopic state to calculate macroscopic properties
)(131)(
1
2 tvmNk
tTN
iii
B∑=
=
)(tT
t
Which to pick?
Specific (individual) microscopic states are insufficient to relate to macroscopic properties
Courtesy of the Center for Polymer Studies at Boston University. Used with permission.
27
How to calculate the macroscopic ensemble average of a quantity A from microscopic states
),( prA Property due to specific microstate
drdprprpAAp r∫ ∫>=< ),(),( ρ
• Ensemble average, obtained by integral over all microscopic states
• Proper weight - depends on ensemble ),( rpρ
∑=
=N
iii
B
vmNk
pA1
2131)( Temperature
28
How to solve…
drdprprpAAp r∫ ∫>=< ),(),( ρ
Probability density distribution
Virtually impossible to carry out analytically
Must know all possible configurations
Therefore: Require numerical simulation
Molecular dynamics OR Monte Carlo
29
Monte Carlo schemeConcept: Find simpler way to solve the integral
Use idea of “random walk” to step through relevant microscopic states and thereby create proper weighting (visit states with higher probability density more often)
Monte Carlo schemes: Many applications (beyond what is discussed here; general method to solve complex integrations)
drdprprpAAp r∫ ∫>=< ),(),( ρ
30
Monte Carlo scheme to calculate ensemble average
Similar method can be used to apply to integrate the ensemble averageNeed more complex iteration scheme
E.g. Metropolis-Hastings algorithm
drdprprpAAp r∫ ∫>=< ),(),( ρ ∑><
ii
A
AN
A 1Want:
31
Metropolis-Hastings AlgorithmConcept: Generate set of random microscopic configurationsAccept or reject with certain scheme
Images removed due to copyright restrictions.Please see Fig. 2.7 in Buehler, Markus J. Atomistic Modeling of Materials Failure.New York, NY: Springer, 2008.
Metropolis-Hastings Algorithm: NVT
Images removed due to copyright restrictions.Please see Fig. 2.8 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.
33
Arrhenius law - explanation
E
)()( AHBH −
iteration
⎥⎦
⎤⎢⎣
⎡ −−<
TkAHBHp
B
)()(exp
attempt frequency ⎥
⎦
⎤⎢⎣
⎡ −−
TkAHBH
B
)()(exp
)()( AHBH −
1
0
E.g. = 0.8 most choices for pwill be below, i.e. higherchance for acceptance
0.8 low barrier
high barrier0.1
)(AH
)(BH
)()( AHBH >E
)()( AHBH −
radius
34
Summary: MC scheme
drdprprpAAp r∫ ∫>=< ),(),( ρ ∑
=
><ANi
iA
AN
A..1
1Have achieved:
Note:• Do not need forces, only energies• Only valid for equilibrium processes
35
Molecular dynamics
Ergodic hypothesis:
Ensemble (statistical) average = time average
All microstates are sampled with appropriate probability density over long time scales
∑∑==
=>=<>=<tA Nit
TimeEnsNiA
iAN
AAiAN ..1..1
)(1)(1
MC MD
36
MD versus MC
Obtain ensemble average by random stepping schemeAcceptance/rejection algorithm leads to proper distribution of microscopic statesNo dynamical information about system behavior (only equilibrium processes) – no “real” time
Obtain ensemble average by dynamical solution of equation, that is, time historyTime average equals ensemble average (Ergodichypothesis)Dynamical information about system obtained
Monte Carlo Molecular Dynamics
∑∑==
=>=<>=<tA Nit
TimeEnsNiA
iAN
AAiAN ..1..1
)(1)(1
MC MD
37
Property calculation: Temperature and pressure
Temperature
Pressure( )>⋅−<
Ω= ∑
=
N
iiiii rfvmP
1
2
31
Distance vector multipliedby force vector (scalar product)
Volume
Kinetic contribution
><= ∑=
N
iii
B
vmNk
T1
2131
iii vvv ⋅=2
>⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
−Ω
=< ∑ ∑ ∑= = ≠=
=3..1 ..1 ,..1,
,, |)(21
31
i N aNrri
iii r
rr
rrvvmP
α ββαααα αβ
φ
38
2. Advanced property calculation
Structures – distinguish crystal structures, liquids, gas
Calculate properties such as stress, thermal conductivity, shear viscosity…
39
2.1 Radial distribution function (RDF)
40
MD modeling of crystals – solid, liquid, gas phase
Crystals: Regular, ordered structure
The corresponding particle motions are small-amplitude vibrations about the lattice site, diffusive movements over a local region, and long free flights interrupted by a collision every now and then.
Liquids: Particles follow Brownian motion (collisions)
Gas: Very long free paths
J. A. Barker and D. Henderson, Scientific American, 1981
Figure by MIT OpenCourseWare. After J. A. Barker and D. Henderson.
41
How to characterize material state (solid, liquid, gas)
Application: Simulate phase transformation (melting)
http://www.t2i2edu.com/WebMovie/1Chap1_files/image002.jpg
Solid State- Ordered, dense.- Has definite shape and volume.- Very slightly compressible.
Liquid State- Disordered; usually less dense than a solid.- Definite volume; takes shape of its container.- Slightly compressible.
Gas State- Disordered; much less dense than solids or liquids.- Indefinite shape and volume.- Highly compressible.
Figure by MIT OpenCourseWare.
42
How to characterize material state (solid, liquid, gas)
IdeaMeasure distance of particles to their neighborsAverage over large number of particlesAverage over time (MD) or iterations (MC)
Regular spacing
Neighboring particles found at characteristic distances
Irregular spacing
Neighboring particles found at approximate distances (smooth variation)
More irregular spacing
More random distances, less defined
43
Reference atom
Formal approach: Radial distribution function (RDF)
ρρ /)()( rrg =
Ratio of density of atoms at distance r (in control area dr) by overall density
= relative density of atoms as function of radius
Courtesy of the Department of Chemical and Biological Engineering of the University at Buffalo. Used with permission.
http://rheneas.eng.buffalo.edu/wiki/LennardJones:Background:Lennard_Jones_Radial_Distribution_Function
44
Formal approach: Radial distribution function (RDF)
ρρ /)()( rrg =
The radial distribution function is defined as
Provides information about the density of atoms at a given radius r; ρ(r) is the local density of atoms
ρ1
)()()(
2
2r
r
rrNrg
Δ
Δ
±Ω>±<
=
=drrrg 22)( π Number of particles that lie in a spherical shellof radius r and thickness dr
Local density
Density of atoms (volume)
Volume of this shell (dr)
Number of atoms in the interval 2rr Δ±
45
Radial distribution function
ρ)()()(
2
2r
r
rrNrg
Δ
Δ
±Ω±
=
VN /=ρDensity
Note: RDF can be measured experimentally using x-ray or neutron-scattering techniques
)( 2rr Δ±Ω considered volume
2rΔ
2rΔ
46
Radial distribution function: Solid versus liquid
Interpretation: A peak indicates a particularlyfavored separation distance for the neighbors to a given particleThus, RDF reveals details about the atomic structure of the system being simulatedJava applet:http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html
solid liquid
10.80
1
2
3
4
5
6
7
1.2 1.4Distance
g(R
)
1.6 1.8 2 2.2 2.4 10.80
0.5
1
2
2.5
3
1.5
1.2 1.4Distance
g(R
)
1.6 1.8 2 2.2 2.82.4 2.6
A B
Figure by MIT OpenCourseWare.
47
Radial distribution function: JAVA applet
Java applet:http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html
Images removed due to copyright restrictions.Screenshots of the radial distribution Java applet.
48
Radial distribution function: Solid versus liquid versus gas
Note: The first peak corresponds to the nearest neighbor shell, the second peak to the second nearest neighbor shell, etc.
In FCC: 12, 6, 24, and 12 in first four shells
5
4
3
2
1
00 0.2 0.4 0.6 0.8
g(r)
distance/nm
5
4
3
2
1
00 0.2 0.4 0.6 0.8
g(r)
distance/nm
Solid Argon
Liquid Argon
Liquid Ar(90 K)
Gaseous Ar(90 K)
Gaseous Ar(300 K)
Figure by MIT OpenCourseWare.
49
Notes: Radial distribution function
Pair correlation function (consider only pairs of atoms)Provides structural informationCan provide information about dynamical change of structure, butnot about transport properties (how fast atoms move)
Additional comments:
Describes how - on average - atoms in a system are radially packed around each other Particularly effective way of describing the structure of disordered molecular systems (liquids) In liquids there is continual movement of the atoms and a singlesnapshot of the system shows only the instantaneous disorder it is extremely useful to be able to deal with the average structure
50
2.2 Velocity autocorrelation function (VAF)
51
Autocorrelation functions
Concept: Calculate how property at one point in time is related to property at a later point in time
Time-dependent quantity
Can represent transport information of the system, e.g. viscosity, thermal conductivity, diffusivity
Autocorrelation functions belong to a class of functions called the Green-Kubo relations
A: Dynamical properties
time t0 time t0+t
∑=
+>=<N
iii ttAtA
NtAA
100 )()(1)()0(
52
• The velocity autocorrelation function gives information about the atomic motions of particles in the system: Relation of velocity at one point in time with velocity at another point in time
• Correlation of the particle velocity at one time with the velocity of the same particle at another time
• The information refers to how a particle moves in the system, such as diffusion (by time integration)
∑=
+⋅>=<N
iii ttvtv
Ntvv
100 )()(1)()0(
Velocity autocorrelation function (VAF)
scalar product)()( 00 ttvtv ii +
53
How to calculate the VAF
∑=
>==<N
iii tvtv
Nttvv
1000 )()(1)()0(
∑=
Δ+>=Δ+=<N
iii ttvtv
Ntttvv
1000 )()(1)()0(
∑=
Δ+>=Δ+=<N
iii ttvtv
Ntttvv
1000 )2()(1)2()0(
∑=
Δ+>=Δ+=<N
iii tntvtv
Ntnttvv
1000 )()(1)()0(
…
54
Harmonic oscillator (“crystal”)v
u
t
∑=
>=<N
iii tvv
Ntvv
1
)()0(1)()0(
VAF
t
1t
2t
3t
4t0t
1t 2t 3t 4t0t
k
55
Velocity autocorrelation function
http://www.eng.buffalo.edu/~kofke/ce530/Lectures/Lecture12/sld010.htm
∑=
Δ+>=Δ+=<N
iii tntvtv
Ntnttvv
1000 )()(1)()0(
VAF
t
Damping due to dissipation - solid
Courtesy of the Department of Chemical and Biological Engineering of the University at Buffalo. Used with permission.
56
Velocity autocorrelation function: Solid, liquid, ideal gas
T
VAF
1
Liquid
Dense gas
Ideal gas
Solid
Figure by MIT OpenCourseWare.
57
Velocity autocorrelation function (VAF)Liquid or gas (weak molecular interactions):Magnitude reduces gradually under the influence of weak forces: Velocity decorrelates with time, which is the same as saying the atom 'forgets' what its initial velocity was.
VAF plot is a simple exponential decay, revealing the presence of weak forces slowly destroying the velocity correlation. Such a result is typical of the molecules in a gas.
Solid (strong molecular interactions):Atomic motion is an oscillation, vibrating backwards and forwards, reversing their velocity at the end of each oscillation.
VAF corresponds to a function that oscillates strongly from positive to negative values and back again. The oscillations decay in time.
This leads to a function resembling a damped harmonic motion.
VAF may be Fourier transformed to project out the underlying frequencies of the molecular processes: closely related to the infra-red spectrum of the system, which is also concerned with vibration on the molecular scale.
58
VAF used to calculate diffusion constant
')()0(31 '
0'
dttvvDt
t∫∞=
=
><=
Diffusion coefficient (see additional notes available on MIT Server “Additional notes re. velocity autocorrelation function”):
Diffusion coefficient can be obtained from both VAF and the MSD
59
Green-Kubo relations
')'()0(1
0'
dttTUk t
xyxyB∫∞
=
><= σση
Diffusivity
Shear viscosity
Thermal conductivity
')()0(31 '
0'
dttvvDt
t∫∞=
=
><=
')'()0(1
0'2 dttqq
TUk tB∫∞
=
><=λ
⎟⎠
⎞⎜⎝
⎛ += ∑ ∑= =
N
i
N
iijii rUvm
dtdq
1 1
2 )(21
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
+−Ω
= ∑ ∑= ≠=
=N aN
rryx
yxxy rrr
rruum
..1 ,..1,,, |)(
211
α ββαααα αβ
φσ
60
DiffusivityTime integral over VAF
')()0(31 '
0'
dttvvDt
t∫∞=
=
><=
)()( 00 ttvtv ii +⋅=
Velocity vector of particle i at t0
Velocity vector of particle i at t0+t
61
Shear viscosity
')'()0(1
0'
dttTUk t
xyxyB∫∞
=
><= σση
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
+−Ω
= ∑ ∑= ≠=
=N aN
rryx
yxxy rrr
rrvvm
..1 ,..1,,, |)(
211
α ββαααα αβ
φσ
Time integral over autocorrelation function of shear stresses
shear stresses
Definition of shear stresses:
Force –Fi, x-componentvolume
mass of particle α x/y velocity component of particle α
distance vector betweenα and β, y-component
62
Shear viscosity: Physical meaning
dydvAF xη=Resistance to shearing
Fvx
x
y
shear viscosity
velocity gradient
velocity profile )( yvx
63
Thermal conductivity
')'()0(1
0'2 dttqq
TUk tB∫∞
=
><=λ ⎟⎠
⎞⎜⎝
⎛+= ∑ ∑
= =
N
i
N
iijii rUvm
dtdq
1 1
2 )(21
Time integral over autocorrelation function of quantity q (time derivative of Hamiltonian)
time derivative kinetic
energy of particle i
total potential energy (sum over all pairs of atoms)
potentialenergy
64
Thermal conductivity – physical meaning
T1
T2dxdTA
dtdQH λ−== dt
dQFourier's law:
In 3D formulation (heat flux density)
TAq ∇−= λ Describes rate of heat flow across an interface with different temperatures (temperature gradient)
65
2.3 Virial stress (directional dependent pressure)
66
Stress tensor
“Forces inside the material”
surface normalforce direction
y
x
z
∆P∆P
∆Py
∆P∆A x
∆A ∆Py
Figure by MIT OpenCourseWare. After Buehler.
z
x
yσyy
σxy
σyx
σyz
σxz σxx
ny
nx
Ax
Ay
div σ + f = 0
Figure by MIT OpenCourseWare.
67
Atomic stress tensor: Virial stress
Virial stress:
D.H. Tsai. Virial theorem and stress calculation in molecular-dynamics. J. of Chemical Physics, 70(3):1375–1382, 1979.
Min Zhou, A new look at the atomic level virial stress: on continuum-molecular system equivalence, Royal Society of London Proceedings Series A, vol. 459, Issue 2037, pp.2347-2392 (2003)
Jonathan Zimmerman et al., Calculation of stress in atomistic simulation, MSMSE, Vol. 12, pp. S319-S332 (2004) and references in those articles by Yip, Cheung et al.
Force -Fi
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
+−Ω
= ∑ ∑= ≠=
=N aN
rrji
jiij rrr
rrvvm
..1 ,..1,,, |)(
211
α ββαααα αβ
φσ
x1
x2
Contribution by atoms moving through control volume
Pressure (no directionality)
atom number direction
( )zzyyxxp σσσ ++−=31
>⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
−Ω
=< ∑ ∑ ∑= = ≠=
=3..1 ..1 ,..1,
,, |)(21
31
i N aNrri
iii r
rr
rrvvmP
α ββαααα αβ
φ
Frαβ
Atom α
Atom β
68
2.4 Closure – property calculation
69
( )>⋅+<= ∑=
N
iiiii frvm
VP
1
2
31
><= ∑=
N
iii
B
vmNk
T1
2131
iii vvv ⋅=2
∑∑==
+>=<iN
kkiki
N
i i
ttvtvNN
tvv11
)()(11)()0(
>±Ω±
=<Δ
Δ
ρ)()()(
2
2r
r
rrNrg
( )22 )0()(1)( ∑ =−>=Δ<i
ii trtrN
tr
Temperature
Pressure
Stress
MSD
RDF
VAF
Direct
Direct
Direct
Diffusivity
Atomic structure (signature)
Diffusivity, phase state, transport properties
>⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
+−<Ω
= ∑ ∑≠
=α ββα
ααα αβ
φσa
rrji
jiij rrr
rruum
,,,, |)(
211
Property Definition Application
Summary – property calculation
70
Material properties: Classification
Structural – crystal structure, RDF, VAF
Thermodynamic -- equation of state, heat capacities, thermal expansion, free energies, use RDF, VAF, temperature, pressure
Mechanical -- elastic constants, cohesive and shear strength, elastic and plastic deformation, fracture toughness, use Virial Stress
Vibrational -- phonon dispersion curves, vibrational frequency spectrum, molecular spectroscopy, use VAF
Transport -- diffusion, viscous flow, thermal conduction, use MSD, temperature, VAF (viscosity, thermal conductivity)
71
3. Introduction: Chemical interactions
72
Molecular dynamics: A “bold” idea
( ) ...)()(2)()( 20000 +Δ+Δ+Δ−−=Δ+ ttattrttrttr iiii
Positions at t0
Accelerationsat t0
Positions at t0-Δt
mfa ii /=Forces between atoms… how to obtain?
vi(t), ai(t)
ri(t)
xy
z
N particlesParticles with mass mi
73
Atomic scale – how atoms interact
Atoms are composed of electrons, protons, and neutrons. Electron and protons are negative and positive charges of the same magnitude, 1.6 ×10-19 CoulombsChemical bonds between atoms by interactions of the electrons ofdifferent atoms
“Point” representation
Figure by MIT OpenCourseWare.
no p+ p+
p+
e -
e-
e-
e-
e-
e-
no
no no
no
p+ no
p+
p+
V(t)
y x
r(t) a(t)
Figure by MIT OpenCourseWare. After Buehler.
74
Concept: Interatomic potential
“point particle” representation
Attraction: Formation of chemical bond by sharing of electronsRepulsion: Pauli exclusion (too many electrons in small volume)
r
Electrons
Core
Energy Ur
1/r12 (or Exponential)
Repulsion
Radius r (Distancebetween atoms)
eAttraction
1/r6
Figure by MIT OpenCourseWare.
75
Interatomic bond - model
Attraction: Formation of chemical bond by sharing of electronsRepulsion: Pauli exclusion (too many electrons in small volume)
Harmonic oscillatorr0
φ
r~ k(r - r0)2
Figure by MIT OpenCourseWare.
Energy Ur
1/r12 (or Exponential)
Repulsion
Radius r (Distancebetween atoms)
eAttraction
1/r6
Figure by MIT OpenCourseWare.
76
Interatomic pair potentials
Morse potential
Lennard Jones potential
Harmonic approximation
