Mapping Molecular Dynamics to Mesoscopic Models: Challenges at Interfaces

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1 Mapping Molecular Dynamics to Mesoscopic Models: Challenges at Interfaces Mark Robbins, Johns Hopkins University Shiyi Chen, Shengfeng Cheng, Colin Denniston, Weinan E, Jin Liu, Xiaobo Nie, Peter Thompson Development and Analysis of Multiscale Methods University of Minnesota, Novermber 6, 008 Funding: National Science Foundation Nanomotor Challenge When composition, strain rate, stress or other property changes rapidly compared to correlation length or mean free path, hard to describe with coarse-grained equations. Show examples for fluid/solid and fluid/fluid interfaces, and flows where continuum equations predict singularities Funding: National Science Foundation Nanomotor 1

2 Atomic Interactions Fix Continuum Behavior Continuum models need: Boundary conditions (BC) + Constitutive Relations velocity or stress stress vs. strain (rate) slip, friction, adhesion viscous, elastic, plastic Traditionally assume simple form, fit to experiment: Assume no-slip for fluids, Simple analytic functions simple friction and Linear response gives adhesion laws for solids. viscosity or elastic modulii Molecular simulations determine BC s and constitutive relations. Can answer: Do usual assumptions for BC and constitutive laws work? Down to what scale do continuum equations apply? Is there new mesoscopic behavior between atomic and bulk? Issue: When is it better to find relations on the fly vs. fitting simple relations? Simulations: Generic behavior simple potential Spherical molecules, Lennard-Jones (LJ) interactions: V LJ =4ε ij [(a ij /r) 1 -(a ij /r) 6 ] for r<r c (usually r c = 1/6 a) Add unbreakable FENE bonds to make n-mers Solid walls Usually flat surface of fcc crystal (111) Usually rigid or atoms held to sites by springs Characteristic LJ energy and length depend on species w solid wall, 1 and fluid atoms Vary ε 1, a 1, make fluids miscible or immiscible Vary ε Iw, a Iw, change wetting, flow BC Units: average a (0.3nm), ε, m, and τ=a(m/ε) 1/ (3ps)

3 Continuum Eqs: Newtonian, steady, low Re, Ma 0 Conservation of each species i: ( ρ ) i uiα ρ i -mass density, u iα velocity Conservation of momentum: α σ βα = 0 σ αβ =stress tensor, σ αα /3=-p, p-pressure Constitutive relation for stress: σ x z =ηb η b - viscosity, u=mass-averaged velocity α = Need BC s: u iz =0, σ zz =const u x (0) = slip velocity or length L s? z u ix (0)-u x (0) = surf. diffusion? σ (0)-σ (wall) = surf. stress? L s wall at z=0 Want to apply to nearly molecular scale flows Have fit both sharp interface and mesoscopic BC s u x z u x x Single Fluid Structure Near walls layering & in-plane order Average over layers to generate mesoscopic model and compare with continuum Choice of wall position z=0 is not unique at molecular level. Choose center of first layer of solid atoms Midpoint between solid and fluid layer gives same physics, different numbers - local ρ - layer ave. Wall σ z 3

4 Single Fluid Slip Bulk flow extrapolates (lines) to u(0) u wall (shaded bars) u s u(0)-u wall Define local η from local v η z v x (z)=σ (or finite difference) Then find Navier slip condition w u s = dz z ( u v ) = L sσ / η b b b L s dz ( η b / η 1) w L s > 0 slip, L s < 0 stick L s /a typically ~- 0a can be much bigger for polymers (Thompson & Robbins PRA41, 6830 (1990)) u s u s monomers 16-mers 16-mers exponential decay Wall Induces layering Density vs. height z between walls Deviations from no-slip BC Viscosity bulk within ~ σ Thompson & Robbins, Phys. Rev. A41, 6830 (1990) 4

In-plane Order Controls L S ρ(r) Layer Layer 1 S(q) Fluid atoms cluster between wall atoms Atoms in next fluid layer between atoms of first layer Results for all interaction parameters, densities,

5 In-plane Order Controls L S ρ(r) Layer Layer 1 S(q) Fluid atoms cluster between wall atoms Atoms in next fluid layer between atoms of first layer Results for all interaction parameters, densities, collapse when plotted against S max =S(G) Quantify with S(q)= ρ(q) Thompson & Robbins PRA41, 6830 (1990)) Boundary Conditions For Mixtures? What average velocity should we apply the boundary condition (& const. relation) to? Mass average u= (ρ 1 u 1 + ρ u )/ ρ Number average? Volume average? Koplik and Banavar, PRL 80, 515 ( 98). v 1 =v at wall, both vanish. Ganesan & Brenner, PRL 8, 1333 ( 99); PRE 61, 6879 ( 00). If true, not a useful B.C. for macroscopic equations: D=0. Denniston and Robbins, PRL 87, ( 01), JCP 15, 1410 ( 06) Diffusion leads to v 1 v, wetting drives flows, no unique choice of average velocity for BC s Unexpected contributions to slip velocity 5

Diffusion in Concentration Gradient Maxwell Demon s Wall z y x Lennard-Jones interactions Diffusion bulk on same length as viscosity bulk Slip D higher at wall slip J A

6 Diffusion in Concentration Gradient Maxwell Demon s Wall z y x Lennard-Jones interactions Diffusion bulk on same length as viscosity bulk Slip D higher at wall slip J A =ρ A (u A -u)=-ρd c A stick Simulation times ~ μs. 1 rich Wetting + Conc. Gradient Couette or Poisseuille-like flow between static walls rich z x u s u s u s ~ m/s z/a z/a 6

7 Flux of Individual Species Large diffusive flux J i =ρ i (v i -v) v 1 Const in center since x ρ i, D const. Drops near walls since x ρ i, D drop asymmetric wetting symmetric wetting v v 1 v Generalized Slip Boundary Condition Sharp interface model for slip can only depend on continuum values at wall: σb, x γ and x c 1 γ=surface stress, c 1 =concentration of species 1 L b L f1 f s m us = σ + xγ + D xc 1 ηb ηb c1 c L L s m dz( η / η 1) b dz( η / η)( σ b σ (0)) / γ Usual Navier condition is limiting case x 0 f i depend on relative momentum transfer of two components last term vanishes for some average u BUT which depends on atomic interactions, mass, For any other average can have net flow with no drag x 7

8 Marangoni Surface Stress Surface stress x γ near wall since σ xx =σ yy σ zz =-p b γ dz( σ xx + p) w Force balance gives shear stress BC b wall Δ σ σ ( w) σ = xγ -p Δσ a 3 /ε asymmetric wetting Stress BC obeyed in simulations Gradient in σ xx in fluid drives interfacial flow x γ a 3 /ε L M dz( η / η)( σ σ (0)) / γ b x Test of Slip Boundary Condition Measured slip agrees with calculated u L = L + f γ + s b m 1 s σ x D xc 1 ηb ηb c1 c f Diffusion term =0 for identical atoms Constant L s ~ -0.6a, stick Constant L m ~ 0.7a measured slip calculated L s term x γ term Denniston & Robbins, JCP 15, 1410 ( 06) 8

u Flow without drag neutral wetting L L f s b m 1 s = σ + xγ + D xc 1 ηb ηb c1 c P>0 =0 P=0 f 0 if m 1 > m, couple to wall differently, etc. P<0 Mass flux due to diffusion with no pressure gradient.

9 u Flow without drag neutral wetting L L f s b m 1 s = σ + xγ + D xc 1 ηb ηb c1 c P>0 =0 P=0 f 0 if m 1 > m, couple to wall differently, etc. P<0 Mass flux due to diffusion with no pressure gradient. Adding pressure gradient Extra Poiseuille flow with constant L s Denniston & Robbins, PRL 87, ( 01); JCP 15, 1410 ( 06) Optimize Parameters for Nanomotor Efficiency of interfacial stress transfer b dz( η / η 1)( σ σ (0)) / γ b x v x a/τ Velocities ~m/s Δσ a 3 /ε Surface stress ~MPa z/a x γ a 3 /ε 9

10 Summary for Solid-Fluid Interfaces Atomistic flows fit by generalized sharp interface boundary conditions down to scales ~ a few nm Stress balance b wall Δ σ σ ( w) σ = xγ Slip boundary condition L b L f1 f s m us = σ + xγ + D c x 1 ηb ηb c1 c Navier Surface stress Diffusive Can choose average velocity to make last term vanish, but not always mass, number or volume average Considered design and optimization of nanomotors and pumps: u s ~ m/s, stresses ~ MPa Difficult to find on fly because diffusive flux small What about Fluid Interfaces? Mesoscopic Model for Mixtures Conservation of each species i: ( ) α ρ i uiα = 0 ρ i -mass density, u iα velocity Conservation of momentum: Navier-Stokes Equations Constitutive relation for stress: 1 σ αβ = p0δ αβ + η ( ) αuβ + βuα +??? η - viscosity, u=mass-averaged velocity??? stress from variations in free energy F with mass-averaged density ρ and conc. φ= ρ A - ρ B Diffusive flux from gradient in chemical potential μ tφ + α ( φuα ) = Γ μ Lattice-Boltzmann method efficient dynamics that guarantees conservation, but need F and μ 10

11 Phase Field Model for Mixtures If correlation length ξ» σ coarse-grained F as functional of ρ, φ Usually write simplest form with proper symmetry 1 { ψ ( ρ, φ, T ) + ( ) + ( ) + ( } 1 1 φ K ρ ρ Kφ φ Kρφ ρ F = ) Typically ignore K ρφ (odd in φ), assume K ρρ >0 Often fix ρ, expand ψ as quartic polynomial in φ Goal: Obtain ψ and K s from MD None of usual assumptions is good Denniston & Robbins, Physical Review E69, (004) Model Lennard-Jones truncated to give pure repulsion Temperature mostly irrelevant Interaction energy between unlike species greater by factor ε* drives segregation coexistence region 11

12 Finding K From Linear Response Apply perturbation: δμ ρ = μ ρθ sinqx, δμ φ =-μ φθ sinqx Measure resulting ρ q, φ q Lρρ Lρφ ρq μρq = Lρφ Lφφ φq uφq L ρρ = ψ/ ρ + K ρρ q, Find L s linear in q to π/q σ, but K ρρ,k ρφ <0 Usual to add q 4 terms if K ρρ <0 but expect K ρρ 0 as q since cost of forcing atoms onto arbitrarily fine lattice 0 Fit F using only: Fitting F 1) Phase boundaries of coexistence region ) Linear response near coexistence 1

13 Test of Model No information about the interface is used in the fit, but the model reproduces interfacial width and tension γ lines fit symbols -MD Liquid-liquid Interface Small dip in ρ at interface big change in free energy and surface tension 13

Fit near coexistence line reproduces Laplace pressure Δp=γ/R Molecular Dynamics Lattice Boltzmann R 1.

08 sêr Laplace pressure (sharp interface) prediction Using models like those of Chen, Jasnow and Vinals: an unphysical

Removal of truncation error removes driving, but mode is still present and is easily driven in contact line motion.

14 Fit near coexistence line reproduces Laplace pressure Δp=γ/R Molecular Dynamics Lattice Boltzmann R 1.5 Δp Denniston & Robbins, Phys. Rev. E 69, (003). a/r Dp ÿ1 0 0 s t êm sêr Laplace pressure (sharp interface) prediction Using models like those of Chen, Jasnow and Vinals: an unphysical mode driven by numerical truncation error. Removal of truncation error removes driving, but mode is still present and is easily driven in contact line motion. Δ = γ μ φ R = Δp Recall, isotropic part of pressure was: ( ρμ Φ ) P0 = kbt ψ ρ μφ In a system taking all parameters from molecular dynamics simulation the unphysical mode is gone. Spurious velocities from Numerical truncation errors ~

Lattice Boltzmann simulation of contact line motion Concentration and velocity field (same velocity field at top and

Continuum solution gives MD boundary condition Innermost MD gives continuum boundary condition Middle Two solutions

displacements Streamlines in L~0.3mm channel with moving top wall.

15 Lattice Boltzmann simulation of contact line motion Concentration and velocity field (same velocity field at top and bottom) Linking Atomistic and Continuum Regions Three overlap regions where solve both continuum and MD Outermost Continuum solution gives MD boundary condition Innermost MD gives continuum boundary condition Middle Two solutions equilibrate independently Fluids: Apply boundary conditions to velocities Solids: Apply boundary conditions to displacements Streamlines in L~0.3mm channel with moving top wall. Atomistic solution in small area (green) removes continuum singularity Model contact region atomistically, elastic deformations with finite-elements, constrain deformations in overlap region 15

Hybrid Algorithm Applied to Fluids Continuum: Incompressible Navier-Stokes (Projection method) Atomistic: Molecular dynamics of Lennard-Jones atoms, no-slip Potential: U(r) =4ε((σ/r) 1 -(σ/r) 6 ] ;

16 Hybrid Algorithm Applied to Fluids Continuum: Incompressible Navier-Stokes (Projection method) Atomistic: Molecular dynamics of Lennard-Jones atoms, no-slip Potential: U(r) =4ε((σ/r) 1 -(σ/r) 6 ] ; Units ε, σ N 1 J MD Continuum u J = vi N Continuum MD 1 N NJ J i= 1 J i= 1 v = u ( t) i J NJ 1 DuJ mx && i = Fi Fk + m, NJ k= 1 Dt LJ Fi = Vik. x Potential confines particles at y 3 Insert/remove number of particles equal to net flux k F y Particle Confinement and Mass Flux External force F for y y y3 ( y y) = αp0σ 1 ( y y ) /( y 3 y ) Continuum Maintain a mass flux by introducing particles n(x,t) near y = y mn ( x, t) = Aρu y ( x, y, t) Δt MD Overlap y 3 y y 1 Langevin thermostat for y < y < y3 : LJ V ( rij ) m&& yi = mγy& i + ζ i x j i i ζ ( t) ζ ( t ) = δ δ ( t t ) k TΓ i j ij B MD Werder et al. J. Comp. Phys. 05, 373 (005) claim artifacts, but use different implementation y 0 16

17 Schematic of simulation Moving Wall Continuum Dynamic Couette Flow U Hybrid solution (symbols) tracks full continuum (lines) as a function of time after motion starts Overlap MD Still Wall Δt FE =40Δt MD X. B. Nie, S. Y. Chen, W. N. E and M. O. Robbins, J. Fluid Mech Singular Cavity Flow Continuum approach: Navier-Stokes + no-slip boundary condition (bc) Usually phenomenological no-slip bc has little effect at large scales r Corner flow Molecular scale influences macroscopic forces No-slip boundary condition is discontinuous at corners a, b Stress diverges as 1/r Log divergence in total force on wall y x Only need atomic information near corners Use hybrid method that treats bulk with continuum Navier-Stokes equations, corners with MD 17

18 Coupling in Overlap Region MD Navier Stokes Mean atomic velocity gives boundary condition to NS eqs. Continuum MD 1) Average tangential MD velocity in shadowed bins forced to NS value: Fi 1 && xi = m mn N J Du Fj + Dt J,Fj = x J j= 1 k ) Normal MD velocity constrained by matching mass flux at boundary V Have tested: Agrees with pure MD calculations. Independent of continuum grid 1, 3 and 6σ and specific set of constrained velocities (within MD noise) X.B. Nie, S.Y. Chen and M. R. Robbins, Physics of Fluids 004. LJ jk NS and Hybrid Velocities Near Corners hybrid Effect like slip BC on scale S S is larger of ~σ and U/0.1σ/τ discreteness shear-thinning Hard to use effective Navier BC: spatially varying, nonlinear 18

Treating Large Range of Length Scales Problem: Size of atomistic region independent of system size L BUT time to equilibrate NS flow field grows with L. Initial approach limited to L~0.1μm.

19 Treating Large Range of Length Scales Problem: Size of atomistic region independent of system size L BUT time to equilibrate NS flow field grows with L. Initial approach limited to L~0.1μm. Solution: Multigrid and time approach Integrate to steady state at each scale with optimum time step. Iterate between scales till self-consistent (~10 times). Result: Size limited only by onset of non-steady, turbulent flow Show results for 0.3mm cavities. > 10 orders of magnitude faster than fully atomistic ~ 0 minutes per iteration Use average over 16 MD representations to accelerate Schematic of Local Refinement M Coarse ->Fine: Prolongation. Fine->Coarse: Restriction. M Flow at each scale reaches steady state at its own characteristic time 19

20 Multiscale Solution for Re=6400 (U=0.068σ/τ) 0.1mm Ten grid levels, largest 56x56, others 64x64, smallest mesh 0.95σ Dashed lines: the regions expanded in successive plots. Final plot MD region Stokes equations bottom corners self-similar under mag. by ~16 (red arrows) This scaling is cut off by atomic structure. Computational time saving more than over fully atomistic. Stress along the moving wall Three regions contribute to force F: Atomistic, Stokes, high Re Re=ρUL/μ Shear stress on wall Re=6400 U=0.7 σ/τ Re= U=0.7 σ/τ Breakdown of Stokes for r<s atomistic or r>r I μ/ρu inertial Little change for r < R I as increase Re by increasing L Large r contribution gives change in F for fixed U, atomic props. 0

21 Total Force on the Moving Wall Re - only parameter in continuum theory Find strong variation with U at fixed Re, atomic model Re=400 U (σ/τ) F μu f Stokes f Re = fs + fstokes + f Re r<s S<r<R I R I <r f S = 4.3 8π = ln( R / ) I S π = Re f S given by assumption that stress saturates at S S= Ut LJ ; f Re is phenomenological fit Summary for Multiscale Fluid Flow We have developed a multiscale hybrid method that can simulate a macro-length scale flow while still resolving the atomistic structure in a small region. Treats mass and heat flux The ability to resolve the stress on all scales enables the first calculation of the drag force on the moving wall in cavity flow. The force depends on three dimensionless numbers: Re=ρUL/μ, I=Uτ/σ and R m =S/r I =ρus/μ Recently extended to time extrapolation X. Nie, S. Chen and M. O. Robbins, Physics of Fluids 16, (004). X. B. Nie, S. Y. Chen, W. N. E and M. O. Robbins, J. Fluid Mech. 500, (004). X. Nie, M. O. Robbins and S. Chen, Phys. Rev. Lett. 96, (006). B. Luan, S. Hyun, J. F. Molinari, N. Bernstein, and Mark O. Robbins, Phys. Rev. E74, (006). J. Liu, S. Chen, X. Nie and M. O. Robbins, J. Comp. Phys. 7, (007). 1

22 Time Extrapolation Scheme J. Liu, S. Chen, X. Nie and M. O. Robbins, Commun. Comp. Phys. 4,179 (008). Couette flow driven by oscillating wall Δx=Δy=15.6σ, Δt FD =10τ, Δt 1 =Δt =0.5 τ Δx=Δy=31.3σ, Δt FD =50τ, Δt 1 =Δt =.5 τ

23 See Delay If No Time Extrapolation 3

Modeling the combined effect of surface roughness and shear rate on slip flow of simple fluids

Modeling the combined effect of surface roughness and shear rate on slip flow of simple fluids Anoosheh Niavarani and Nikolai Priezjev www.egr.msu.edu/~niavaran November 2009 A. Niavarani and N.V. Priezjev,