1. SMT22 -Conference on Surface Modification Technologies
2008-09-23
Surface tension of liquid Ni
S. Jenkins and S.R. Kirk
Dept. of Technology, Mathematics & CS,
University West, P.O. Box 957, Trollhättan, SE 461 29, Sweden.
Project website: http://beacon.webhop.org
Production Technology Centre
Website http://www3.innovatum.se/pages/default.asp?sectionid=2503

2. The industrial importance of the surface tension of liquid metals
Key role in casting and welding process, and its dependence on temperature ->
Marangoni convection.
● Recent and considerable efforts have been directed towards the experimental
measurement of the surface tension and T dependence for molten metals.
● Vital for determining contact angle in
deposition of liquid metals, e.g. thermal
spraying
● Influences structure of deposited films

3. Diagram of a draining vessel viscometer
Experimental determination of surface tension and viscosity (see talk by S.R. Kirk).
[1] R. F Brooks, A.T Dinsdale and P.N Quested, Meas. Sci. Technol. 16 (2005) 354–362

4. The Marangoni effect
Mass transfer on, or in, a liquid layer due to surface tension differences.
● Interactions occur between the molecules of a liquid and those of any liquid or
gaseous substance which is not soluble in the liquid;
--> result in the formation of an interface
● Energy is required to change the form of this interface or surface.
● The work required to change the shape of a given surface is known as the
interfacial or surface tension.

5. Mixed results from experiments
Remarkable experimental progress achieved with the advent of levitation
processing and oscillating drop techniques [2–4].
BUT
Experiments suffers from the ambiguities in the interpretation of the resulting
frequency spectra [2].
Experimental data on liquid metal and alloy surface tension (and its temperature
dependence) is very sparse in the literature, especially above 1600oC
[2] S. Sauerland, G. Lohofer, I. Egry, J. Non-Crystalline Solids 156–158 (1993) 833–836.
[3] K.C. Mills, R.F. Brooks, Mater. Sci. Eng. A178 (1994) 77–81.
[4] J. Egry, G. Lohofer, S. Sauerland, Int. J. Thermophys. 14 (1993) 573–584.

6. Over the decades many attempts to predict surface tension of
simple liquids [5]...
The interface between laboratory experiments and theory
● Computer simulation with Monte Carlo (MC):
○ Stochastic process ; moving molecules one at a time, generates a
Boltzman-weighted chain of configuration of a given N-particle system.
○ The quantities of interest are obtained as configurational averages of the
long chain.
or
● Molecular dynamics (MD) methods
○ Repeatedly evaluate instantaneous forces on particles (atoms) e.g. using
pair-potentials, calculate acceleration, velocity and new position using
Newton's Laws, take small timestep forward
○ Adjust velocities to provide thermostatting (control temperature)
[5] J. Miyazaki, J.A. Barker, G.M. Pound, J. Chem. Phys. 64 (1976) 3364–3369.

7. Monte-Carlo (MC) and Molecular dynamics (MD) simulations
1.Surface tension can be calculated either using the mechanical expression for
the surface stress,
or
2. Surface free energy; the free energy required to create reversibly a surface in
bulk liquid is calculated directly [5].
Unfortunately ...
1. Suffers from rather high fluctuation and statistical uncertainty,
2. Introduces additional complexity into performance.

8. The importance of inter-atomic potentials- embedded atom
method (EAM) [6,7]
● EAM = Central importance for the simulation of liquid metals:
● EAM = semi-empirical potentials based on density functional theory(DFT)
DFT = energy of an arbitrary configuration of atoms is a unique function of the
electron distribution.
total energy of atomic configuration = embedding energy + short-range doubly
screened pair interaction(for the core–core repulsion).
embedding energy = energy required to ‘embed’ each atom into the local electron
density contributed by all other atoms.

9. EAM -total energy of atomic configuration (Etotal)
Etotal = ∑ Fi (ρi) + ½∑ φij (rij) (1)
i ij,i≠j
ρi = ∑ fj (rij) (2)
j≠i
Fi = embedding energy for placing an atom into that electron density (ρi)
φij = short-range pair interaction representing the core–core repulsion
rij = separation of atoms i and j
ri = total local electron density at atom i, computed as a superposition of (ρi) of the
rest of the atoms in the system
fj(rij) = atomic (ρi) of atom j due to atom i. The sums are over all atoms within a
defined cutoff radius.
MEAM = Modified EAM uses eqns. (1) and (2) with an angular dependence included in φij
[6] M.S. Daw, M.I. Baskes, Phys. Rev. Lett. 50 (1983) 1285.
[7] M.S. Daw, M.I. Baskes, Phys. Rev. B 29 (1984) 6443.

10. Surface tension via EAM/MEAM pair potentials for surfaces with
non-negligible surface curvature [8]
● Approach based on EAM/MEAM
potentials and molecular dynamics
● Melt and equilibrate approximately
spherical particle made of Ni atoms in
vacuum at fixed T
● Calculate position of centre of mass at
every timestep
● Calculate time-averaged density, force
and potential energy in radial 'shells'
r→r+∆r , as a function of r
● Find radius rs of maximum interparticle
attraction (surface minimum in F) →
surface energy Es
● Surface tension = Es /4π rs2
[8] Kirkman et. al. , Computational Materials Science 30 (2004) 126–130

11. Theoretical Surface tension calculation using work of cohesion
Surface tension can also be simulated by calculating the work of cohesion of the
liquid.
Figure 1. The process of producing two new interfaces from a liquid column[8]
[9] M. Chen, C. Yang and Z-Y. Guo, Materials Science and Engineering A292 (2000)
203–206

12. Ab-initio atomic-scale treatment of work of cohesion
Periodic unit cells:
● Identical dimensions
● Identical numbers of Ni atoms
Calculation with Quantum-
Espresso [9] ab-initio code:
● DFT, PBE/GGA exchange
correlation
● Ultrasoft pseudopotential with
non-linear core corrections
● 30 Ry basis set cutoff
● Evaluate ∆E , then surface
tension = ∆E / (2 x cross-sectional area of cell)
Future Enhancements (in progress):
● Carr-Parrinello MD treatment for relaxation at finite temperature
● Sampling over larger cell (computationally demanding)
[10] P. Giannozzi et al., http://www.quantum-espresso.org

13. Thermal spraying research- Details of computational simulations
● Molecular dynamics using EAM/MEAM LAMMPS molecular dynamics code
● Using accurate ab-initio treatment of transport properties on bulk calculations
● Multibody atomic potentials based on DFT
● Projector-augmented plane wave approach using open-source codes

14. Results: Comparison of Experiment [11] and theory for Ni
9500
9000
8500
Mass density Kg/m3
8000
7500
liquid Ni (Calculated)
Ni (Experiment)
7000
500 1000 1500 2000 2500 3000
Temperature/K
Figure 2. The variation of the mass density with temperature in Ni
[11] A.T. Dinsdale: SGTC data for pure elements, CALPHAD 15 81991) 317/425.

15. Results:Comparison of Experiment and theory for Ni
From Figure 2:
Straight line fitted (correlation = 0.999) to data from computer simulations
Periodic unit cells (1985 atoms) nPT, 1 bar pressure
● mass density at room temperature = 8908 kg/m3
● excellent agreement between theory and experiment for liquid Ni
● Melting point of Ni at 1728K corresponds to 'jump' in Figure 2.

16. Movie - MD simulation of molten Ni particle @ 1900K
● 1289 atoms (cluster is 3nm in diameter)
● 200 ps simulation run
● Carved from Ni crystal, temperature ramped from 300 K until melted
● equilibrated at 1900K

17. Future work
● Alloys: 95%Ni 5%Al
● Effect of oxidation
● inter-diffusion for metal-metal interfaces
● Temperature variation for the above
See 3D model and movie of rippling Ni particle see also beacon.webhop.org