University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Data analysis for different types of simulations

We may be interested in:• Equilibrium properties of the model system.• Structure and properties of the system in a metastable state.• Dynamic processes in the system far from equilibrium.

1. Equilibration, Deborah number, statistical errors2. Visual analysis of simulation results3. Energy (potential, kinetic, total), temperature4. Finding the melting temperature - coexistence simulations5. Calculation of the equation of state of the model material6. Pressure, atomic-level stresses (separate sets of handouts)7. Time and space (velocity and density) correlations (separate set of handouts)8. Mean square displacement, diffusion (separate set of handouts)

Issues relevant to the analysis of the results of atomic-level simulations:

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Equilibration

System is out of (stable or metastable) equilibrium after 1. We change a parameter of the simulation or2. State of the system changes spontaneously

)texp(BA)t(A 0

During equilibration a physical quantity Aoften approaches its equilibrium value A0 as

The ratio of the relaxation time to theobservation time tmax is called theDeborah number.

Lattice density vs density forStillinger-Weber potential for Si[PRB 31, 5262, 1985].

A(t) is a physical quantity averaged over ashort time to get rid of fluctuations, but not oflong-term drift. This drift is described by therelaxation time τ.

(e.g. as a result of a phase transformation)

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Equilibration

)texp(BA)t(A 0

For short τ we just wait for equilibration and startcollecting data for equilibrium parameters of the system.

For very long relaxation times MD is not appropriatetechnique.

For intermediate case we can estimate A0 even if we cannot reach it.

Sometimes finding τ is of interest by itself.

In many cases we do not want to have equilibrium, it is our purpose to study activenon-equilibrium processes.

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Measuring parameters of the system in MD simulations1. Express an observable quantity as a function of the positions and

velocities, A(t) = f(ri(t),vi(t))

2. Perform time averaging over the simulated trajectory,

stepsN

startNtstartsteps)t(A

NN1A

M(A)σ)A(σ

22

222

mm

2 AAAAM1)A(

Usually in simulations the measurements are not independent and this expressionunderestimates the variance (the effective number of independent measurements is lessthan M).

Parameters that can be measured: energy (potential, kinetic, total), T, P, V, …

The variance of the mean for M independent measurements is

Where the variance is

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Visual analysis of the simulation results with atomic resolution

Atomic configurations from MD simulations by E. H. Brandt,

J. Phys.: Condens. Matter 1, 10002 (1989).

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Looking at a “big picture” in large-scale simulations

Simulation of laser ablation of AgChengping Wu

Crack propagation in graphine,

Omeltchenko, Yu, Kalia, Vashishta, Phys. Rev. Lett. 78, 2148, 1997.

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Energy calculation

Total energy per atom

N

1i

N

ijji

N

1ii |))t(r)t(r(|U

N1)t(P

N1)t(P

for two-body interactions

Kinetic energy per atom is usually computed within the integrator

N

1i

2ii )t(vm

N21)t(K

)t(K)t(P)t(E

Temperature. Equipartition of energy: the average energy of every quadratic term inthe energy expression for classical system has the same value, ½ kT. For a simple 3D

one-component system we have Tk23K(t) B

Potential energy per atom is usually computed together with forces

(implemented in F-pair.f90 & Temper.f90)

(implemented in Nord5.f90 & Temper.f90)

(implemented in Temper.f90)

K(t)3k

2TB

(implemented in Temper.f90)Tk23K(t)0)P(TP(t) BThermal potential energy:

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

What is the use of energy calculation?

Check of the total energy conservation.

Energy flow from kinetic to potential energy can indicate the occurrence of aphase transition in the system.

A jump in the caloric curve E(T) points to the first order phase transition.Example: melting with a jump corresponding to the latent heat of melting.

In systems far from equilibrium analysis of the energy flow/redistribution cangive useful information.

A detailed atomic-level analysis of energy-per-atom in a static/quenchedconfiguration can help to identify defects and analyze the structure.

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Analysis of energy-per-atom

Structure of grain boundaries in diamond.Color shows atoms with high potential energy.Figure by Shenderova and Brenner fromhttp://www.mse.ncsu.edu/CompMatSci/

“Mesoscopic” view at energy distribution: 1 billionatoms MD with Lennard-Jones interaction potential.Figure by F. Abraham et al. fromhttp://www.almaden.ibm.com/st/computational_science/msmp/fractures/df/

MD simulation of a Cu surface under 1 keV Arbombardment. Color code shows the average kineticenergy around the atoms (purple: molten zone), Figure byH. M. Urbassek from http://www.physik.uni-kl.de/urbassek/

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Energy development in a target irradiated by ashort laser pulse. Energy is averaged overlayers of material at different depths under thesurface for different times.

Temperature variation of potentialenergy per atom in a modelbicrystal - MD simulation atconstant pressure by T. Kwok, P.S. Ho, and S. Yip, Phys. Rev. B29, 5354 (1984).

Evolution of the energy distribution in simulations far from equilibrium

Jump in energy -indication of the onset of structural changes

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Temperature and velocities

At equilibrium velocity distribution is Maxwell-Boltzmann,

zyx

2z

2y

2x2

3

dvdvdv2kT

vvvmexp

kT2mT,vdN

m/Tk3v B2

If system is not at equilibrium it is often difficult to separate different contributions tothe kinetic energy and to define temperature.

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Atoms are colored by velocity relative tothe left-to-right local expansion velocity,which causes the crack to advance fromthe bottom up.

Velocities of atoms in this snapshot cannot be directly related to the temperature.

If system is not at equilibrium it is difficult to separate different contributions tothe kinetic energy and to define temperature

Example: Acoustic emissions in a 2D simulation of fracture

Holian and RaveloPhys. Rev. B 51, 11275 (1995).

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

0 25 50 75 100Time, ps

0

0.2

0.4

0.6

0.8

1

Energy,eV Ea

th

Ee

Eac

Em

Etot = Ee + Eath + Ea

c + Em - total energy of the system

eE- heat of melting

- energy of elastic distortions and vibrations

- thermal energy of atoms

- thermal energy of electronsthaE

caE

mE

Energy redistribution in a 50 nm Ni film irradiated with a 200 fs laser pulse at an absorbed fluence of 43 mJ/cm2 (15 nm layer melts)

The energy of the collective motion of atoms can be comparable (up to ~15%) to theenergy of the thermal motion of atoms and should be taken into account in the analysisof the energy density needed to melt the material.

lase

r pul

se

50 nm

Appl. Phys. A 79, 977, 2004

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

If system is not at equilibrium it is difficult to separate different contributions tothe kinetic energy and to define temperature

Local T can be calculated by considering velocity components that are parallel to thesurface of the irradiated sample and do not have contribution from the kinetic energy ofmaterial flow in the direction normal to the surface.

Example: MD simulation of laser ablation of Al target

evolution of T and P averaged over 5 nm thick consecutive

layers located at different depths in the initial target

Wu and ZhigileiAppl. Phys. A 114, 11, 2014

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Melting in MD simulations

In an MD simulation with 3D periodic boundary conditions there are no possibilities forheterogeneous nucleation of the liquid phase and the crystal can be significantly (up to20-30%) overheated above the melting point of the material.

Solid-to-liquid transition in this case corresponds to the maximum possible overheatingof the crystal, when the initiation of the melting process takes just tens of picoseconds.

Solid to liquid transition can be identified from Visual inspection of the snapshots from the simulation Energy transformation from kinetic to potential energy (latent heat of melting) Jump in the temperature dependence of the total and potential energy Sharp increase of the diffusion coefficient Changes in the radial distribution function Jump in pressure in constant-volume simulation

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Heterogeneous melting by propagation of theliquid-crystal interface from external surfacesIn real systems and in MD simulations melting starts from freesurfaces and defects at temperatures significantly lower thanthe temperature at which an infinite perfect crystal melts.

Melting of small atomic clusters, a cross-section through the center of the cluster is shown (simulations by J. Sethna, Cornell University)

For solid/liquid/vapor interfaces, typically Solid-Vapor > Solid-Liquid + Liquid-Vapor

Therefore, no superheating is required for nucleation of a liquid layer at the surface.

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Melting temperature from liquid-crystal coexistence simulations

• A system of coexisting liquid andsolid phases is allowed to evolvewhile the energy of the system isconserved.

• If T > Tm initially, part of the solidphase melts, consuming latent heat ofmelting and reducing the temperature.

• If T > Tm initially, crystallization of apart of the system leads to thetemperature evolution toward theequilibrium melting temperature frombelow.

• Simulations can be repeated atdifferent pressures to determine thepressure dependence of the meltingtemperature.

Simulations for EAM Ni by Dmitri Ivanov(Phys. Rev. B 68, 064114, 2003)

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Thermodynamics of melting

GPaK34

PT

m

crystal – liquid coexistence simulations

m

m

m

m

m ΔV TΔH

ΔVΔS

dTdP

Can we relate these results to the Clapeyron equation?

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

0 500 1000 1500 2000

Temperature, K

11

11.5

12

12.5

13

13.5

Volumeperatom,A

3

Ni, P = 0 GPa

Vm crystal

liquid

(a)

MD simulation for EAM Ni at constant pressure with 3D periodic boundary conditions (Phys. Rev. B 68, 064114, 2003)

PTV

V1α

Volume coefficient of thermal expansion

mΔV

Thermodynamic parameters of the model material (I)

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Performing simulations at different pressures, one can find the pressure dependence of :

Thermodynamic parameters of the model material (II)

0 500 1000 1500 2000 2500

Temperature, K

10

10.5

11

11.5

12

12.5

13

Volumeperatom,A

3

-4 GPacrystal

(b)

0 GPa

12 GPa

8 GPa

4 GPa

P)α(T,α

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

0 500 1000 1500 2000

Temperature, K

-4.4

-4.3

-4.2

-4.1

-4

-3.9

-3.8

-3.7

Internalenergy

peratom,eV

Ni, P = 0 GPa

crystal

liquid

(a)

Um

Heat capacity PP THC

PVUH

latent heat of meltingmmm VPUH

m

mm T

ΔHΔS

Thermodynamic parameters of the model material (III)

MD simulation for EAM Ni at constant pressure with 3D periodic boundary conditions (Phys. Rev. B 68, 064114, 2003)

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Thermodynamic parameters of the model material (IV)

Performing simulations at different pressures,we can find the pressure dependence of Cp:

0 500 1000 1500 2000 2500

Temperature, K

-4.4

-4.3

-4.2

-4.1

-4

-3.9

-3.8

-3.7

Internalenergy

peratom,eV

-4 GPa

crystal

(b)

0 GPa12 GPa

8 GPa4 GPa

P)(T,CC PP

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

Thermodynamics of melting

-5 0 5 10Pressure, GPa

1000

1200

1400

1600

1800

2000

2200

2400

2600

Temperature,K

GPaK34

PT

m

crystal – liquid coexistence simulations

GPaK 8.464.42

ΔV TΔH

dTdP

m

m

m

Clapeyron equation:

Maximum overheating (3D PBC)

Simulations for EAM Ni (Phys. Rev. B 68, 064114, 2003)

adiabatic expansion can lead to meltingmS P

TPT

Isentropes:

GPaK 2821

CVTα

PT

PS

liquid crystal

University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei

homogeneous melting above the limit of crystal stability

heterogeneous melting

liquid - crystal coexistence(equilibrium Tm)

“classical” homogeneousnucleation

maximum overheating

Pressure (GPa)

Reduced

Temperature(T/T

m)

-5 0 5 10

0.8

1

1.2

1.4

1.6

1.8