This document summarizes work using molecular dynamics simulations to calculate the viscosity of liquid nickel. The researchers used a modified embedded atom method potential to simulate liquid nickel across a range of temperatures. Preliminary results for the viscosity fell within the range of available experimental data. Future work involves further testing and developing optimized potentials for nickel alloys and calculating other physically relevant parameters for larger scale simulations.

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

2. Viscosity of liquid metals important for manufacturing processes
Great influence on many metallurgical manufacturing processes:
● Fluid flow in a vessel
● Metallic glass formation
● Thermal spraying - splat formation
Experimental data for viscosity of liquid metallic elements is sparse,
● Available data set[1] for the shear viscosity of liquid Ni yields estimates
spanning some 60% around mean value.
● Data coverage for alloys even worse than for pure metals
Require development of reliable, universal models for predicting liquid metal
viscosities[2].
[1] T. IIda and R.I.L. Guthrie, The Physical Properties of Liquid Metals Oxford Science, UK, 147 (1988)
[2] T. Iida, R. Guthrie, M. Isac and N. Tripathi, Metallurgical and Materials Transactions B, 37B, 411 (2006)

3. Open-source codes for molecular simulation
● Plenty to choose from, well-developed with active user communities
● Now mostly well parallelized
○ Some with multiple levels of parallelization, e.g. domain decomposition, k-
point decomposition ..
○ Designed to be straightforward to extend to new problem classes
LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator)
(http://lammps.sandia.gov)
● Solid state (metals, insulators and semiconductors), liquid & gas phases, soft
matter (biomolecules, polymers), coarse-grained and mesoscopic problems.
● Potentials: two-body, many-body, granular (inc. peridynamics, SPH), long-
range electrostatics, colloidal, hydrodynamic lubrication
Quantum-Espresso (http://www.quantum-espresso.org)
● Ab initio quantum mechanics, DFT, choice of exchange-correlation functionals
● Structural optimization, pressure response
● Phonon, dielectric, optical properties
● Car-Parrinello / Born-Oppenheimer molecular dynamics (with 'ensemble DFT'
for metals), isothermal & isoenthalpic dynamics

4. Standard theory of viscosity of liquid metals
Viscosity at melting point by Andrade [3]:
(Tm) = Ac (MTm) / Vm3/2
Vm = molar volume, M = average atomic weight, Ac=1.8x10-7 (J K-1 mol1/3 )1/2
Temperature dependence : Arrhenius form
 = A exp [ B/ (T - T0, )]
● Fits some elements well (not so good for alloys, esp. glass-forming) , but
essentially still a fitted curve
● More complex fitting formulae have also been employed
[3] E.N.C. Andrade, Nature, 1931

5. Methods for viscosity simulations
● Want ability to calculate viscosity, and its variation with temperature
○ Access to experimentally difficult temperatures
○ Straightforward extension to alloys
○ Ideally, get insight into dynamics and structure of the liquid
● Ideal candidate - molecular dynamics methods (MD)
○ Integrating Newton's equations of motion for atoms
○ Physics encapsulated in the interaction potential
○ Well-tested algorithms for time integration of Newton's equations
○ Thermostat and barostat algorithms well understood
○ Interaction potentials can be derived from quantum-mechanically accurate
ab-initio (DFT etc.) calculations
○ Can uncover structural phenomena and inform higher-level
(macroscopic/mesoscopic) simulations.
● Equilibrium methods
○ Use pressure and momentum fluctuations, or periodic perturbations
○ Drawback: slow convergence, must extrapolate to k=0

6. Non-equilibrium MD methods for calculating viscosity
In regime of linear response [4], equation
relating momentum transfer to velocity
gradient is:
jz(px) = -η(∂νx/∂z) (1)
Viscosity relates
● Flux in z-direction of momentum along x
● x-velocity gradient in z-direction
Conventional NEMD method
● Impose known velocity gradient
● Measure momentum flux
○ (or off-diagonal component of stress
tensor)
[4] Hansen J-P and McDonald I R 1986 Theory of Simple Liquids (San Diego, CA: Academic)

7. Conventional NEMD
Replace small natural fluctuations with large artificial ones - better convergence [5]
Method: continuously deform simulation box from orthogonality to induce velocity
gradient, compute momentum flux
Requires careful consideration of the equations of motion
● SLLOD algorithm with Nose-Hoover thermostat [6]
Drawbacks: lose computational advantages of orthogonal cell, need to thermostat
velocities differently
[5] Evans D J and Morris G P 1990 Statistical Mechanics of Non-Equilibrium Liquids
(London: Academic) Todd B D Comput. Phys. Commun. 142 14, 2001.
[6] Tuckerman et. al., J Chem Phys, 106, 5615 (1997).

8. Alternative: reverse NEMD
Impose momentum flux, observe velocity gradient (aka Norton Ensemble methods)
Advantages: good if flux difficult to define microscopically,slowly converging [7]
Unphysical ('Maxwell's demon')
● Divide cell into equal slabs
● Compute <vx> of atoms in each slab
● Every N timesteps, find
○ Set A: atoms with biggest +ve vx in
slab 1
○ Set B: atoms with biggest -ve vx in
slab z=Lz/2
○ Swap velocities in sets A and B
○ Keep track of total momentum
transferred Px
Conserves momentum and energy
After time t, momentum flux = jz(px) = Px /(2tA)
[7] F. Muller-Plathe, Phys. Rev. E. 59(5), 4894 (1999)

9. Testing MD potentials for liquid metals
Choosing and checking an MD
potential for liquid metals
● Use in (current!) literature
● Expected phase behaviour,
e.g. via radial distribution
function g(r)
● MEAM (Modified Embedded
Atom Method) potential [8]
chosen for MD simulations
○ Replicates expected phase
behaviour
○ Multibody, based on density
functional theory (see talk by
S.Jenkins)
[8] M.I. Baskes, Phys. Rev. B. 46(5), 2727 (1992)

10. Results
Protocol:
● 1985 atoms, 300ps NVE ensemble melting phase from source FCC structure
● 300 ps equillibration under Nose-Hoover thermostat + barostat, 1 bar pressure,
in NPT ensemble
● 5 ns viscosity sampling using Muller-Plathe algorithm
Results in range 4 - 5 mPas-1 across liquid range of temperatures
● Already well within range of available experimental data
● Exchange frequency needs tuning to improve velocity gradient
● Larger simulations (more atoms) for better velocity gradient
○ Large parallel calculations currently running on SHARCNET computing facility
via research partners.

11. Conclusions and future work
MEAM potentials are a good candidate for MD work in liquid metals, yielding
physically reasonable predictions across a wide range of temperatures.
Future work:
● Further tests of MEAM potentials
● Construction of optimised MEAM potentials for alloys (e.g. Ni5%Al)
● Prediction of further physically relevant parameters for mesoscopic / continuum
simulations
○ e.g. heat capacity
● Reparameterization of MEAM potentials via ab-initio Car-Parrinello MD