Nanoscale interfacial heat transfer: insights from molecular dynamics S. Merabia, A. Alkurdi, T. Albaret ILM CNRS and Université Lyon 1, France K.Termentzidis, D. Lacroix LEMTA, Université Lorraine, France H. Han, F. Müller-Plathe Technische Universität Darmstadt, Germany CECAM LAMMPS workshop June 26th, 2018 Lyon
Nanoscale heat transfer Nanostructured materials T J = λ Diffusive regime Balistic regime Si @300 K : Λ>300 nm Fourier
Thermal conductivity 3
Thermal conductivity Essentially, two methods : 1. the «direct» method Apply a temperature gradient or energy flux and calculate the conductivity with Fourier law 2. equilibrium method Probe the fluctuations around equilibrium of the energy flux vector
Thermal conductivity : the direct method Steady state temperature profile Heat source Example Si/Ge LAMMPS command= Fix temp/rescale L Heat sink Fourier law : J = λ ΔT L
Thermal conductivity : the direct method Finite size effect analysis «Bulk» thermal conductivity
Thermal conductivity : direct method Advantages : - relatively easy to implement using open sources codes (LAMMPS) -may be used also to compute the thermal boundary resistance Inconvenients : -need to check if we are in the linear regime => analyze different heat fluxes -severe finite size effects! => analyze different system sizes to properly extrapolate a «bulk» conductivity
Thermal conductivity : Green-Kubo equilibrium simulations Green-Kubo formula : + 1 λ α,β= J α (t )J β (0) dt 2 0 V kbt Heat flux vector : J = d ( i E i r i ) dt J = E i v i + 1 F ij ( v i + v j ) i 2 i j In practice, run in NVE ensemble LAMMPS command = compute heat/flux
Thermal conductivity : Green-Kubo equilibrium simulations Green-Kubo formula : + 1 λ α,β= J α (t)j β (0) dt 2 0 V kbt Example amorphous Si 300 K t 1 λ xx (t )= J x (t ')J x (0) dt ' 2 0 V kbt Jx (t) Jx (0)
Thermal conductivity : Green-Kubo equilibrium simulations Green-Kubo formula : Example amorphous Si 300 K + 1 λ α,β= J α (t)j β (0) dt 2 0 V kbt t 1 λ xx (t )= J x (t ')J x (0) dt ' 2 0 V kbt t 1 λ yy (t)= J y (t ')J y (0) dt ' 2 0 V kb T Where is the plateau?
Optical modes J = E i v i + 1 F ij ( v i + v j ) i 2 i j Replaced by : J = E 0i v i + 1 F ij0 ( v i + v j ) i 2 i j Equilibrium positions λ (ω)= 1 2 0 J (t) J (0) exp(i ωt) dt V kb T Termentzidis, SM, Chantrenne IJHMT 2011 Not implemented in LAMMPS
Thermal conductivity : Green-Kubo equilibrium simulations Advantages : - less severe finite size effects - access to the full thermal conductivity tensor in a single simulation (anisotropic materials, superlattices) Inconvenients : -need to run several independent simulations (usually 10 to 20) -the plateau is sometimes difficult to identity (in principle, in a finite system the Green-Kubo formula should give a vanishing conductivity...)
Thermal anisotropy A.F. Lannord, SM, T. Albaret, D. Lacroix, K. Termentzidis, 2014
Thermal boundary resistance 14
Heat Transfer at the nanoscale Superlattices Dresselhaus, Chen, Science 2012 -Deviations from Fourier's law -Thermal boundary resistance Nanocomposites-nanowires Poulikakos, NanoLetters 2011 (dimensions < phonon mean free path)
Thermal boundary (Kapitza) resistance Molecular dynamics Si/Ge G=1/ R= C v (ω) v g (ω) t 12 (ω) d ω McGaughey, PRB 2009 Energy transmission coefficient G 1 1000 MW /m2 / K
Non-equilibrium simulations Amorphous Si/ Crystalline Si A. France Lanord et al., J. Phys. Cond. Mat. 2014 Fourier law : J = λ T z Thermal boundary resistance : R=Δ T / J
Equilibrium simulations Puech' s formula + 1 G= q(t) q(0) dt 2 0 A kbt A interfacial area Analogous to the Green-Kubo formula for the thermal conductivity Interfacial heat flux : q= i 1, j 2 F ij ( v i + v j) 2 Not computed in LAMMPS! Postprocessing! Barrat and Chiaruttini, Mol. Phys. 2003 SM and Termentzidis, PRB 2012
«Approach to equilibrium» or «Thermal relaxation» Temperature relaxation Δ T (t) exp( t / τ) G=C v /( A τ) Well adapted to imperfect interfaces! E. Lampin et al., APL 2012 S. Merabia, Termentzidis 2014 Easy to implement in LAMMPS!
Acoustic description of interfacial heat transfer Acoustic model Ar h-ar G= C v (ω)v g (ω) t 12 (cosθ1, cos θ2 ) d cos θ1 d ω 4 Z 1 Z 2 cos θ1 cos θ2 t 12 = (Z 1 cos θ1 + Z 2 cos θ2 )2 Acoustic impedances : Z i =ρm, i c i SM and K. Termentzidis, PRB 2012 24
Challenges 25
Interatomic force constants Stillinger-Weber potential Landry, phd thesis 2009 MD : G 320 MW /m2 / K Exp. : G 200 MW /m2 / K Landry, McGaughey PRB2009 26
Ab-initio Interatomic force constants Dispersion curves of Si and Ge calculated using LD with ab initio IFC up to 4th unit cell. Si Ge Necessity of going up to 4th unit cell (8th neighbour) to have a good agreement. Ab initio interatomic force constants from M. Aouissi et al Phys. Rev. B 74, 054302 (2006) 27
Theoretical Models: Ab Initio Lattice Dynamics In bulk : D α,β ( k ) e β ( k )=ω2 ( k ) e α ( k ) Zhao, H., and J. B. Freund. J. Appl. Phys. 97, 024903 (2005). D.A. Young and H.J. Maris, Phys. Rev. B 40 3685 (1989) from ab initio Medium1 Medium2 n atom / unit cell leads to: 3n equations in medium 1 3n equations in medium 2 2 x 3 x n equations at the interface in 3 dimensions. Ab initio interatomic force constants are from: 28 M. Aouissi et al Phys. Rev. B 74, 054302 (2006)
Theoretical Models: Ab Initio Lattice Dynamics from ab initio calculations At interface : L L C C R R m j u ( r j )= i Φ (r ij ) u ( r ij ) l Φ (r lj ) u ( r lj ) k Φ (r kj ) u ( r kj ) u ( r )= k A( k ) e ( k )exp(i k. r ) ω( k inc )=ω( k tra )=ω( k ref ) x x x k inc =k tra =k ref, y y y k inc =k tra =k ref C R 2 [ D ( k )+ D ( k )+ D ( k )] e ( k )=ω ( k ) e ( k ) L n atom / unit cell leads to: 3n equations in medium 1 3n equations in medium 2 2 x 3 x n equations at the interface in 3 dimensions. 29 Ab initio interatomic force constants are from: M. Aouissi et al. Phys. Rev. B 74, 054302 (2006) Zhao, H., and J. B. Freund. J. Appl. Phys. 97, 024903 (2005). D.A. Young and H.J. Maris, Phys. Rev. B 40 3685 (1989)
Phonon transmission coefficient Si/Ge interface extinction 2 MD : G 320 MW /m / K Exp. : G 200 MW /m / K transmission Critical angle 2 Heat flux Si/Ge Alkurdi, Pailhès, Merabia APL 2017
Phonon transmission coefficient Si/Ge interface extinction Difference between calculations and experiments : 2 MD : G 320 MW /m / K Exp. : G 200 MW /m2 / K -Anharmonic effects Critical angle 1 H 0= α,β Φ ij ui, α u j, β i, j, α,β 2 H '= 1 α,β, γ Χ u i, α u j,β u k, γ ijk i, j, k, α,β, γ 6 Si/Ge -Need to couple of MD Alkurdi, Pailhès, Merabia APL 2017
Challenge in LAMMPS : -build interatomic potentials deriving from ab-initio calculations -accurate description of interfacial heat transfer -thermal transmission 32
Electron-phonon couplings Two temperature model + boundary conditions h Effective conductance : 1/σ eff =1/Gh+1/(σ e +σ p ) J. Lombard, F. Detcheverry and SM, J. Phys. Cond. Mat., 2015 33
Challenge in LAMMPS : -couple MD with two-temperature model -including the cross interaction at the interface 34
Applications : How to tune interfacial heat transfer? 35
Role of the pressure Molecular dynamics simulations (OPLS force field) Temperature profile gold perfluorohexane gold conductance G=J / Δ T 39
Enhanced conductance Gold sound velocities 110 : v L (110)=3400 m / s v T (110)=1500 m /s 111 : v L (111)=3900 m / s v T (111)=1400 m/ s Three-four fold increase of the conductance with pressure H. Han, S. Merabia, F. Müller-Plathe, J. Phys. Chem. Lett., 2017 40
Vibrational analysis Vibrational DOS + D (ω) 0 C vv (t ) exp(i ω t ) d ω No participation In heat transfer C vv (t )= v i (t ) v i (0) Pressure induced DOS broadening 44
Enhanced conductance gold perfluorohexane Thermal heat flux : 45
Spectral heat flux Frequency dependent heat flux : Pure ethanol Partially wetting surface ϵnp subs =0.2 kcal mol 1 Wetting surface ϵ np subs =1.0 kcal mol 1 ω(thz) 52
Boiling simulations Significant shift of the boiling temperature due to the presence of the nanoparticles 53
Conclusion -Implement interfacial heat flux calculations -Use potentials derived from ab-initio calculations -Extend two temperature models Interfacial electron-phonon couplings 54
H. Han F. Müller-Plathe A. Alkurdi K. Termentzidis T. Albaret Thank your for your attention! 55