Nanoscale interfacial heat transfer: insights from molecular dynamics

Transcription

1 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

2 Nanoscale heat transfer Nanostructured materials T J = λ Diffusive regime Balistic regime K : Λ>300 nm Fourier

3 Thermal conductivity 3

4 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

5 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

6 Thermal conductivity : the direct method Finite size effect analysis «Bulk» thermal conductivity

7 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

8 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

9 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)

10 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?

11 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)= J (t) J (0) exp(i ωt) dt V kb T Termentzidis, SM, Chantrenne IJHMT 2011 Not implemented in LAMMPS

12 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...)

13 Thermal anisotropy A.F. Lannord, SM, T. Albaret, D. Lacroix, K. Termentzidis, 2014

14 Thermal boundary resistance 14

15 Heat Transfer at the nanoscale Superlattices Dresselhaus, Chen, Science Deviations from Fourier's law -Thermal boundary resistance Nanocomposites-nanowires Poulikakos, NanoLetters 2011 (dimensions < phonon mean free path)

16 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 MW /m2 / K

17 Non-equilibrium simulations Amorphous Si/ Crystalline Si A. France Lanord et al., J. Phys. Cond. Mat Fourier law : J = λ T z Thermal boundary resistance : R=Δ T / J

18 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 SM and Termentzidis, PRB 2012

19 «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!

20 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

21 Challenges 25

22 Interatomic force constants Stillinger-Weber potential Landry, phd thesis 2009 MD : G 320 MW /m2 / K Exp. : G 200 MW /m2 / K Landry, McGaughey PRB

23 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, (2006) 27

24 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, (2005). D.A. Young and H.J. Maris, Phys. Rev. B (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, (2006)

25 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, (2006) Zhao, H., and J. B. Freund. J. Appl. Phys. 97, (2005). D.A. Young and H.J. Maris, Phys. Rev. B (1989)

26 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

27 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

28 Challenge in LAMMPS : -build interatomic potentials deriving from ab-initio calculations -accurate description of interfacial heat transfer -thermal transmission 32

29 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.,

30 Challenge in LAMMPS : -couple MD with two-temperature model -including the cross interaction at the interface 34

31 Applications : How to tune interfacial heat transfer? 35

32 Role of the pressure Molecular dynamics simulations (OPLS force field) Temperature profile gold perfluorohexane gold conductance G=J / Δ T 39

33 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.,

34 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

35 Enhanced conductance gold perfluorohexane Thermal heat flux : 45

36 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

37 Boiling simulations Significant shift of the boiling temperature due to the presence of the nanoparticles 53

38 Conclusion -Implement interfacial heat flux calculations -Use potentials derived from ab-initio calculations -Extend two temperature models Interfacial electron-phonon couplings 54

39 H. Han F. Müller-Plathe A. Alkurdi K. Termentzidis T. Albaret Thank your for your attention! 55