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Thermal boundary conditions for molecular dynamics simulations Simon P.A. Gill and Kenny Jolley Dept. of Engineering University of Leicester, UK A pragmatic approach to

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Overview 1.Imposing a steady state temperature gradient 2.Remote boundary conditions by coarse-graining

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1.0 Imposing a steady state temperature gradient Consider a 1D LJ chain of 100 atoms subjected to a different temperature at each end temperature position, x T1T1 TNTN Heat flux Thermal conductivity Temperature gradient Classical model (Ficks law)

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1.1 Thermostats Nosé-Hoover –a global deterministic thermostat –enforces average temperature only Langevin –a local stochastic thermostat –enforces temperature on each atom –no feedback from actual temperature x T(x) ? T1T1 T2T2 x T1T1 T2T2

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1.2 Results for 1D LJ chain

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Kapitza effect – boundary conductivity different from bulk conductivity Langevin cannot control temperature in a 1D chain away from equilibrium

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1.3 Boundary effect Green-Kubo (assuming local equilibrium) Steady state –constant heat flux at all points Reduced k at boundary for NH k nearly zero at boundary for Langevin Reduce boundary effect using Memory Kernels?

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1.4 Cooling of Ubiquitin (NAMD) Cooling of 1000 atom LJ chain NH : Unphysically large temp gradient maintained at boundary

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1.5 Thermostatting far from equilibrium Concept – use feedback control loop to regulate temp in centre of chain by thermostating ends T1T1 TNTN { { Controlled region Thermostat at T 1c to maintain T 1 Thermostat at T Nc to maintain T N boundary zone

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1.6 Feedback Control of 1D chain Algorithm { Thermostat at T 1c Adjust T 1c to maintain T 1 here

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1.7 Divergence of k The thermal conductivity of a body with a momentum-conserving potential scales with the system size N as 3D molecular chain exhibits convergent conductivity Transverse and longitudinal waves in higher dimensions in 1D in 2D in 3D.

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1.8 Feedback control results 3D rod (8x8x100) Nosé-Hoover thermostat

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1.9 Feedback control results Langevin thermostat Langevin can control temperature in 3D but less effective than Nosé-Hoover

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1.10 Feedback control results Stadium damping thermostat

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1.11 Feedback control results Temperature distributions

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2.0 Remote boundary conditions by coarse-graining Some problems are not well represented by : –periodic boundary conditions, particularly where there are long range interactions, e.g. elastic fields in solids –standard ensembles, especially in cases where we are doing work on the system, e.g. NVE, NVT, NPT Work dissipated as heat. Only N is constant

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2.2 Concurrent modelling approach Do not model dynamics in continuum region Dynamic atomistic region DOF : positions of atoms, q momenta of atoms, p Punch Elastostatic continuum region DOF : positions of nodes, q temperatures at nodes, T

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2.3 Stadium damping Diffuse thermostatting interface Proposed by B.L. Holian & R. Ravelo (1995) Constant temperature simulation embedded in elastostatic FE (Qu, Shastry, Curtin, Miller 2005) Shown to produce canonical ensemble Simple solution to phonon reflection problem MD Damping zone Stadium damping

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2.4 Steady State Concurrent model { { unthermostatted MD region Thermostat at T 1c to maintain continuum FD region Thermostat at T Nc to maintain

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2.5 Steady state FD/MD simulation Blue line is MD boundary zone Red line is FD/MD result Blue line is full FD solution Red line is FD at ends (1-20 and ) and MD in middle Nosé-Hoover Stadium damping

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2.6 Transient Concurrent model { { unthermostatted MD region Thermostat at T 1c to maintain continuum FD region Thermostat at T Nc to maintain Heat must be conserved on average

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2.6 Transient FD/MD simulation Blue line is full FD solution Red line is FD at ends (1-20 and ) and MD in middle

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3.0 Conclusions Kapitza boundary effect –conductivity near an interface (real or artificial) is less than bulk value –can obtain desired temperature by feedback control of thermostatted boundary zone. Thermostats for non-isothermal boundary conditions –deterministic NH thermostat minimizes Kapitza effect –although Langevin naturally thermostats each particle individually.. –..global NH thermostat determines average temp but not distribution –responsiveness of global NH thermostat depends on no. of thermostatted atoms (important for transient b.c.s) FD/MD coupling –Stadium damping effectively removes spurious phonon reflection from atomistic/coarse-grained interface –simple matching conditions ensure coupling between continuum and atomistic regions

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