I have used NPT simulation to find out the melting point of silver. I have used EAM potential as force-field. My result is too near to experimental value.

1. Course: Computational condensed matter
2nd
Assignment
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Q. Find the melting point of silver by using Monte-carlo (MC) or Molecular dynamics (MD)
simulation. In the simulation use the potential obtained by DFT calculation in 1st
assignment.
Ans.
I have used MD simulation to find melting point. Under NPT ensemble simulation I have measured
many quantities that characterizes the liquid-solid transition (melting). e.g; density, internal
energy,enthalpy, specific heat(Cp), radial-distribution function (g(r)) etc. and also took snapshot of
atomic positions of atoms at different temperatures. I will describe these results one-by-one later. I
have used EAM (Embedded atomic model) potential of silver as force-field(1)
. My results show that
melting point of silver (2048 atoms) is (1350-1400) K and experimental value is 1281K(2)
. This
assignment is described into three sections: (1) simulation details , (2)results , and (3) analysis and
discussion.
(1) Simulation details:
The crystal structure of crystalline silver is face-centred cubic (fcc) with lattice constant (a)4.084 Å.
I have taken an 8a×8a×8a size box and generated 2048 lattice points where silver atoms are sitting
in and on the box in an FCC fashion. Then I started MD simulation. I injected Gaussian random
velocities to the atoms at a desired temperature. As in MD simulation temperature is determined by
velocities of the particles [equipartition theorem of energy (E=1/2 Kb*T)] so here the given
temperature will fluctuate. We minimize the fluctuation in temperature by applying some damping
force and this method is called “thermostatting” (fixing temperature). I have used Nose-Hover
thermostat throughout my simulation. In NPT ensemble we have to keep constant pressure too and
the fixing of pressure is called “barostatting”. The main aim of my work is to determine melting
point of silver. Melting (and boiling too) point is defined as the temperature at which solid phase of
a matter changes to liquid phase at 1 atmospheric (atm) pressure. So I have fixed the pressure of the
system at 1 atm. First I tried to choose suitable parameters for thermostatting and barostatting such
that temperature ,pressure ,energy, density etc. should equilibrate around chosen value. I took
integration time-step equal to 0.001 picosecond and ran 100000 MD steps. Then I collected the
desired quantities in a temperature range 800K-2000K for fixed pressure (1 atm) on 81000 MD
steps (from 20000-100000).

4. (2) Results:
Thermodynamic quantities showing abrupt change in their value at melting point.
This plot has been generated after collecting data after each 100 MD steps.

5. All the following thermodynamic quantities plot has been generated by collecting data after each
1000 MD steps. We see a sharp change in these quantities at melting temperature.
Density of silver at room temperature is 10.49 gm/cc. We see that with
increase in temperature the density decreases (due to volume expansion)
and it is in agreement with ρs
> ρl
. ρs
is the density of solid and ρl
of liquid.

7. Structural quantities showing difference between pre -melting and post-melting temperature. e.g;
g(r) and structures. The below figures are structures (atomic position details) at different
temperatures. The first figure is the perfect fcc crystal structure of silver at absolute temperature
(0K).

8. These three given pictures show the snapshot of atomic positions at temperatures 1000K, 1200K
and 1350K respectively(in the given order).
T = 1000K
T = 1200K
T = 1350K

9. This figure is the snapshot of atomic positions of bulk silver (2048 atoms) at 1400K. If we compare
this with the above snapshots (taken at T<1400K) then we found that it is in liquid phase. Here
there is no any order in arrangement (as we see in above images). Till 1350 K temperature the
system is maintaining structure and in 1350K-1400K range melting happened and atoms became
free and moving randomly(not as random as gas molecules). After melting the system is dense-fluid
and we see very few peaks in g(r) and becomes constant as r increases. In this force field (EAM
potential) the cut-off distance(rc) is 5.85 Å . In crystalline form g(r) shows sharp peaks. In perfect
crystal these peaks are δ-function peaks but at finite temperature the broadening of peaks happen
due to anharmonic effect (defects, electron-phonon coupling etc.).
T = 1400K

10. (3) analysis and discussion: We can say that the melting point of silver is in temperature
range 1350-1400 K by looking the snapshots of atomic positions taken at different temperatures but
NPT simulation shows it should be around 1430K. The experimental value is 1281K. I visualized
the atomic positions in VMD (visual molecular dynamics) software. This difference in melting point
may be due to not an exact known potential for silver. Embedded Atomic Model (EAM) potential is
an approximation describing the interaction energy between two atoms of metal. EAM is related to
the 2nd
moment approximation to tight-binding theory. This potential is not universal and applicable
only for some metals. In previous assignment (assignment 1) I tried to find out the interaction

11. potential form for bulk silver but couldn’t guess suitable parameter to plot this. That’s why I used
EAM potential.
References:
(1)The embedded-atom method: a review of theory and applications (Murray S. Daw et.al)
(2)An embedded-atom potential for the Cu–Ag system: P L Williams 1 , Y Mishin 1 and J C
Hamilton 2 doi:10.1088/0965-0393/14/5/002
(3)Simulation of molecular dynamics of silver subcritical nuclei and crystal clusters during
solidification : JIAN ZengYun et.al doi: 10.1007/s11431-010-4171-5
(4) [BOOK]
Intermolecular and surface forces : Jacob N. Israelachvili
(5)[BOOK]
Computer simulation of Liquids : Allen & Tildesley
(6)[BOOK]Understanding molecular simulation : Frenkel & Smith