free energy calculations

8/11/2019 Free energy calculations

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Department of Applied Physics, Chalmers, Gteborg

Martin Petisme, Paul Erhart, 2014-02-09

Free energy calculations

In this exercise you will use the embedded atom method (EAM) potential for aluminum developed in the

first task. In principle you can use your own version but for simplyfing debugging and comarison we

rather have you use the potential file provided on the homepage. You will perform molecular dynamics(MD) simulations to test the potential and carry out free energy calculations.

1 Free energy calculation

1.1 ermodynamic integration,-integration

We can couple our system the one we seek the free energy of with potential energy U1to a reference

system with potential energyU0, for which the free energy is known. is will enable us to find the free

energyFof our system.

We begin by constructing the potential energy of the coupled system

U() =U0+ (U1 U0), 0 1, (1)

which interpolates between the reference system and the system of interest as runs from 0 to 1. e

canonical partition function (NVTensamble) of the coupled system is

Z=

dq3Ndp3N exp [U()/kBT] . (2)

en we can write the derivative of the free energy of the coupled system as an ensemble average

F

NVT

= kBT

ln Z=

kBT

Z

Z

=

dq3N dp3N U

exp [U/kBT]

dxdpexp [U/kBT] =

U

. (3)

We can thus obtain the free energy difference between the reference system and the system of interest by

integration

F1= F0+

10

d

U

=F0+

10

dU1 U0. (4)

By performinge.g., a molecular dynamics (MD) simulation with the potential energy Ugiven by Eq. (1)

we can sample the quantity U1 U0 for different values ofand then evaluate numerically the integralin Eq. (4).

1.2 e Einstein crystal

In the Einstein crystal all atoms are aached to their equilibrium laice positions with springs with a

spring constant. e potential energy for such a system with Natoms is

UES({qi}) =

2

3Ni=1

qi q

(0)i

2. (5)

To obtain converged results with the least effort it is advisable to keep the integrand in Eq. (4) smooth

and the integral small. is can be accomplished by choosing such that it yields a similar mean square

displacement as the system of interest. is can be realized by employing the relation

= 2kT/

r2. (6)

Additional issues can be encountered in the limit 0, i.e. when sampling U1U0from an ensemblethat is dominated by the Einstein crystal. Firstly, in this limit the coupling between different modes is very

small, which can slow down equilibration and require longer simulation times to converge

U1

U0

.

Secondly, one might observes a steeping of the integrand. is effect is due to the lack of interatomic

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repulsion in the Einstein crystal. If this problem is encountered it may be advisable to choose somewhat

stiffer springs than what is obtained from Eq. (6).

For numerical reasons, one uses the fixed center of mass constraint. In this case the Helmholtz energy

of the system of interest is given by [1]

F

NkT=

3

2ln

k2T2m

2

1

NkT

10

dUref U 3

2Nln

2kT

3

2Nln N+

1

Nln

N

V, (7)

whereNis the number of atoms, Vthe system volume,Tthe temperature, m the atom mass, Plancks

constant, andkBoltzmanns constant.

2 Melting point calculation

In the co-existence method, a system is set up so that a solidliquid interface is obtained at a temperature

T. en an NPTsimulation is performed, and if the system is below the melting point, the crystalline

phase will grow and the system will solidify. In the other case the system will melt. is method allowone to determine upper and lower bounds on the melting temperature.

A more accurate value can be obtained using NPHsimulations (constant enthalpy), seee.g. Mendelev

et al. [2]. If the average kinetic energy and thus the average temperature in the system deviates strongly

from the actual melting temperature the system will still either melt or solidify. If the average kinetic

energy, which is determined by the initial conditions of the MD simulation, is, however, sufficiently close

to the melting temperature the system will establish an equilibrium between solid and liquid and the

average temperature will then yield the melting point.

Figure 1: A snapshot from a co-existence melting point simulation.

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Task

1. Use to compute the laice thermal expansion up to 900 K. Compare with experimental

results from literature.

2. Use the co-existence method with theNPHensamble to determine the melting point with a proper

error estimatefor your Al potential.

An example script that you may find useful is provided. 1

You should visualize the dump files of the atomic coordinates in order to ensure that solid and

liquid are indeed in equilibrium. To this end it is strongly to use the very powerful visualization

and analysis soware .2

3. Determine the entropy of fusion, i.e. the increase in entropy upon melting. Use calculations

using the NPTensamble for two bulk systems at the melting point. Also determine the volume

change upon melting. Find experimental values and compare.

4. Using the configurations obtained in the previous task, calculate the coordination numbers for thesolid and the melt. e (first) coordination numbern1 is defined through the radial distribution

functiong(r)as

n1= 4

r1r0

r2g(r) dr (8)

wherer0 is where g(r)is first non-zero, r1 is at the first minimum ofg(r), and is the density ofparticlesN/V. To obtaing(r)it is possible to use or the compute command rdf.3

5. e partition function and free energy for a classical harmonic oscillator are given by

Z=kBT/, and F= kBTln(/kT), (9)

respectively. Derive these expressions.

e corresponding quantum expressions are

Z=exp(/2kBT)/[1 exp(/kT)] and F= kTln[2 sinh(/2kBT)]. (10)

Show that the quantum expressions reduce to the classical ones in the appropriate limit.

6. Use the quasi-harmonic approximation to compute the Gibbs energy for your fied potential using

up to the calculated melting point of your potential. Usease.thermochemistry.CrystalThermo.4

By default, the quantum harmonic oscillator expressions are used. By providing the keyword

classical=Trueusing the modified thermochemistry.py5 one can instead use the classical ex-

pression, which is the relevant case for comparison with the-integration method.

7. Use the-integration method described in the text to compute the Gibbs energy for temperatures

up to the calculated melting point of your potential. An example script that you may find

useful is provided.6 Use a reasonable number ofvalues, on the order of 10. How do you choose

your Einstein crystal spring constant?

e following code snippet may be useful to read in data from the log.lammpsfile.7

1https://db.tt/WRK7rAK82http://www.ovito.org3http://lammps.sandia.gov/doc/compute_rdf.html4https://wiki.fysik.dtu.dk/ase/ase/thermochemistry.html5Available for download at https://db.tt/omQYSQbh. To use this module copy the file into your working directory.

en add the line from thermochemistry import CrystalThermo to your python script and invoke the function asCrystalThermo(...).

6https://db.tt/4HYAcVDg7https://db.tt/I2rwqWY1

3
https://db.tt/I2rwqWY1https://db.tt/4HYAcVDghttps://db.tt/omQYSQbhhttps://wiki.fysik.dtu.dk/ase/ase/thermochemistry.htmlhttp://lammps.sandia.gov/doc/compute_rdf.htmlhttp://www.ovito.org/https://db.tt/WRK7rAK8


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Compare your results with the 8 expression[3]

G=

7976.15+137.093038T 24.3671976Tln T

1.884662 103T2 0.877664 106 T3 +74092 T1, 298.15< T< 700,

11276.24+223.048446T 38.5844296Tln T

+18.531982 103T2 5.764227 106 T3 +74092 T1, 700< T< 933.473.

(11)

In the above expressionGis given in J mol1. Compare also with the results from the previous task

and comment on your results.

References

[1] J.M. Polson, E. Trizac, S. Pronk, and D. Frenkel,Finite-size corrections to the free energies of crystalline

solids, J. Chem. Phys.112, 5339, (2000)

[2] M.I. Mendelev et al., Development of New Interatomic Potentials Appropriate for Crystalline and Liquid

Iron, Phil. Mag. A83, 3977 (2003)

[3] A.T. Dinsdale,SGTE data for pure elements, Calphad15, 317 (1991)

http://resource.npl.co.uk/mtdata/SGTEelementdata.pdf

8CALculation of PHAse Diagrams, see hp://en.wikipedia.org/wiki/CALPHAD

4
http://resource.npl.co.uk/mtdata/SGTEelementdata.pdf

free energy calculations

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