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Biological fluid mechanics at the micro‐ and nanoscale Lecture 7: Atomistic Modelling Classical Molecular Dynamics Simulations of Driven Systems Anne Tanguy University of Lyon (France)

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Atomistic Modelling: Classical Molecular Dynamics Simulations of Driven Systems. I.Description II.The example of Wetting III.The example of Shear Deformation

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Classical Molecular Dynamics Simulations consists in solving the Newton’s equations for an assembly of particles interacting through an empirical potentiaL; In the Microcanonical Ensemble (Isolated system): Total energy E=cst In the Canonical Ensemble: Temperature T=cst with if no external force Different possible thermostats: Rescaling of velocities, Langevin-Andersen, Nosé-Hoover… more or less compatible with ensemble averages of statistical mechanics.

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Equations of motion: the example of Verlet’s algorithm. Adapt the equations of motion, to the chosen Thermostat for cst T.

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Langevin Thermostat: Random force (t) Friction force – .v(t) with =cste.2 k B T. (t-t’) Andersen Thermostat: prob. of collision t, Maxwell-Boltzman velocity distr. Nosé-Hoover Thermostat: Rescaling of velocities: Berendsen Thermostat:with Heat transfer. Coupling to a heat bath. after substracted the Center of Mass velocity, or the Average Velocity along Layers () 1/2 Thermostats:

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Examples of Empirical Interactions: The Lennard-Jones Potential: 2-body interactions cf. van der Waals Length scales ij ≈ 10 Å Masses m i ≈ kg Energy ij ≈ 1 eV ≈ J ≈ k B T m Time scale or Time step t = 0.01 ≈ s 10 6 MD steps ≈ s = 10 ns or 10 6 x10 -4 =100% shear strain in quasi-static simulations N=10 6 particles, Box size L=100 ≈ 0.1 m for a mass density =1. 3.N.N neig ≈10 8 operations at each « time » step.

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The Stillinger-Weber Potential: For « Silicon » Si, with 3-body interactions Stillinger-Weber Potential F. Stillinger and T. A. Weber, Phys. Rev. B 31 (1985) Melting T Vibration modes Structure Factor The BKS Potential: For Silica SiO 2, with long range effective Coulombian Interactions B.W.H. Van Beest, G.J. Kramer and R.A. Van Santen, Phys. Rev. Lett. 64 (1990) Ewald Summation of the long-range interactions, or Additional Screening (Kerrache 2005, Carré 2008) 2-body interactions (Cauchy Model) 3-body interactions

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Example: Melting of a Stillinger-Weber glass, from T=0 to T=2.

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Microscopic determination of different physical quantities: -Density profile, pair distribution function -Velocity profile -Diffusion constant -Stress tensor (Irwin-Kirkwood, Goldenberg-Goldhirsch) -Shear viscosity(Kubo)

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II. The example of Wetting

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Surface Tension: coexistence beween the liquid and the gas at a given V. (L. Joly, 2009)

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The Molecular Theory of Capillarity: Intermolecular potential energy u(r). Total force of attraction per unit area: Work done to separate the surfaces: (I. Israelachvili, J.S.Rowlinson and B.Widom) Surface Tension: h (Hautman and Klein, 1991)

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III. The example of Shear Deformation

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Boundary conditions:

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Quasi-static shear at T=0. Fixed walls Or biperiodic boundary conditions (Lees-Edwards) Example: quasi-static deformation of a solid material at T=0°K At each step, apply a small strain ≈ on the boundary, And Relax the system to a local minimum of the Total Potential Energy V({ri}). Dissipation is assumed to be total during . Quasi-Static Limit

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uxux LyLy Rheological behaviour: Stress-Strain curve in the quasi-static regime

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uxux LyLy X y Local Dynamics: Global and Fluctuating Motion of Particles

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uxux LyLy Local Dynamics: Global and Fluctuating Motion of Particles Transition from Driven to Diffusive motion due to Plasticity, at zero temperature. cage effect (driven motion) Diffusive y _ max n ~ xy pp Tanguy et al. (2006)

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Driving at Finite Temperature: The relative importance of Driving and of Temperature must be chosen carefully.

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Low Temperature Simulations: Athermal Limit Typical Relative displacement due to the external strain larger than Typical vibration of the atom due to thermal activation >>

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Convergence to the quasi-static behaviour, in the athermal limit: At T=10 -8 (rescaling of the transverse velocity v y et each step) M. Tsamados (2010)

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T= Tg =0.435 Rescaling of transverse velocities in parallel layers Effect of aging at finite T

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Non-uniform Temperature Profile at Large Shear Rate Time needed to dissipate heat created by applied shear across the whole system Heat creation rate due to plastic deformation Time needed to generate k B T,

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Visco-Plastic Behaviour: Flow due to an external force (cf. Poiseuille flow) F. Varnik (2008) Non uniform T

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