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- 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

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