Transport de chaleur. Samy Merabia ILM CNRS and Université Lyon 1. Ecole d'été «GDR Modmat» July 20-24th, Istres

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1 Transport de chaleur Samy Merabia ILM CNRS and Université Lyon 1 Ecole d'été «GDR Modmat» July 0-4th, Istres

2 Outline Introduction - motivations Vibrations Thermal conductivity Phonon spectroscopy Phonon scattering at interfaces

3 Nanoscale heat transfer 1950's 000's

4 Materials for thermoelectricity Electricity The ideal thermoelectric material is a good electronic conductor, but a poor thermal conductor

5 Thermoelectricity and thermal conductivity Figure of merit S σ Z= λ ph +λ e S Coefficient Seebeck σ Electronic conductivity λ Thermal conductivity

6 Thermoelectricity and thermal conductivity Figure of merit S σ Z= λ ph +λ e

7 Nanoscale heat transfer Nanostructured materials T J = λ K : Λ>300 nm Fourier

8 Fourier law Heat flux Th Tc L 0 Fourier law : T h T c T q= λ =λ x L ( ) x

9 Fourier law in anisotropic materials Superlattices isotropic anisotropic T J = λ λ J α = λ α,β β T λ IP λ IP λ CP λ IP In-plane conductivity λ CP Cross-plane conductivity The flux is not always parallel to the temperature gradient!

10 Thermal conductivity Y. Touloukian, Thermophysical properties of matter

11 Thermal conductivity Y. Touloukian, Thermophysical properties of matter

12 Energy carriers Crystalline phases Metals : electrons Less extent phonons Semi-conductors : mainly phonons Other excitations : magnons etc not discussed Disordered phases Phonons and diffusons

13 Electronic thermal conductivity Thermal conductivity λ=λ e +λ ph phononic electronic λe Wiedemann Franz law =L σt L0 = Lorenz number π k B 3 e 8 L0 = V K

14 Lorenz number 8 L0= V K Tritt, thermal conductivity

15 What is a phonon? Vibrations

16 Vibrations : assumptions -adiabatic assumption -harmonic approximation -quasi-classical treatment

17 Dynamics of a monoatomic chain C Equations of motion : m Solutions : dt =C (un+1 +u n 1 u n ) u n (t)=q k exp(i(kna ω(k)t)) Dispersion relation : 4C ω(k )= m d un 1/ ( ) ka sin ( ) k= π p ( p integer) N

18 Dynamics of a monoatomic chain Dispersion relation 1/ 4C ω(k )= m ( ) sin ka ( ) Phase velocity 1/ 4C v ϕ (k)= m ( ) ka sin /k ( ) Group velocity C v g (k)= m 1/ () a cos ka ( )

19 Dynamics of a monoatomic chain Dispersion relation 1/ 4C ω(k )= m ka sin ( ) ( ) Phase velocity 1 / 4C v ϕ (k )= m ka /k ( ) ( ) sin Group velocity C v g (k)= m 1 / ( ) Vibrational density of states g(ω)= ka a cos dn 1 1 d ω v g (ω) (ω max ω )1 / ( )

20 Dynamics of a monoatomic chain Dispersion relation 1/ 4C ω(k )= m ka sin ( ) ( ) Phase velocity 1 / 4C v ϕ (k )= m ka /k ( ) ( ) sin Group velocity C v g (k)= m 1 / ( ) Vibrational density of states Acoustic limit : g(ω)= ka a cos dn 1 1 d ω v g (ω) (ω max ω )1 / C k 0 v ϕ (k) v g ( k) v s k v s= m 1/ () a ( )

21 Dynamics of a diatomic chain C c Equations of motion : M m d U n dt d un dt = C (U n u n ) c (U n un 1 ) = c (un U n+1) C (un U n ) Solutions : U n (t)=q k exp(i(kna ω(k)t )) 1 u n (t)=qk exp(i(k (n+ )a ω(k )t)) Dispersion relation : π p k= ( p integer) N (C+c) (( M +m) (C+c) 16MmCc sin (ka/))1/ ω (k )=( M +m) ± Mm Mm

22 Dynamics of a diatomic chain Case C=c ( ( M +m) C Mm 1/ ) C m 1/ ( ) C M 1/ ( )

23 Optical vs acoustic modes

24 Three dimensional dispersion curve Harmonic Hamiltonian : H =H αβ α β Φ ij u i u j i, j α,β x, y, z Harmonic force constants α Equations of motion : mi Solution : d ui dt α i u = = j Φαij,β uβj Q k mi e α ( k)exp(i( k r i ω t)) Polarisation vector D α,β ( k) eβ ( k)=ω ( k)e α ( k) Dynamic matrix : 1 α,β αβ Dij = Φij mi m j αβ D α β ( k)= j D ij exp(i k ( r j r i))

25 Dynamic matrix and dispersion relations Dynamic matrix : D α,β ij 1 αβ = Φij mi m j αβ D α β ( k)= j D ij exp(i k ( r j r i)) D α,β ( k)eβ ( k)=ω ( k)e α ( k) α,β β,α α,β ij α,β ji Properties : D ij =D ji D =D Dα β ( k )=D*β α ( k ) Dαβ ( k )=D*αβ ( k ) αβ i, j D ij =0 D α β ( k = 0 )=0 (Taylor) (inversion symmetry) (acoustic rule)

26 Phonon dispersion in gold Transverse modes Displacement perpendicular to k Longitudinal mode Displacement parallel to k A. Alkurdi, C. Adessi et al.

27 Dispersion relations for complex crystals Eigenmodes : u ( b)= α i Q k mi eα ( k, b)exp(i( k r i ( b) ωt )) vector defining the position in the primitive cell Eigenvalues : b ω ( k )e α ( k, b)= b ' D αβ ( k ; b, b ')eβ ( k, b ') ω s ( k ); eα, s ( k, b) Three dimensional crystal : index s Total = 3*p branches with p = number of atoms in the primitive cell

28 Phonon dispersion in Si Landry, phd thesis 009

29 Phonon dispersion in graphene D material g(ω)= ω v ϕ (ω) v g (ω) Bending modes dominant at low q E. Pop, V. Varshney and A.K. Roy, MRS Bull., 37 (01) 173

30 Molecular dynamics Molecular dynamics : m d r i Φ = r i dt Interatomic potential : Φ= i< j V ( r ij )+ i< j<k V 3 ( r i, r j, r k ) Stilinger-Weber potential (Si) : + thermal bath

31 Dispersion from molecular dynamics We use the property : 1 Φ α,β ( k )=k B T G α, β ( k ) Harmonic force constants F=-Hu u Φα,β ( k )= j Φα,β (i, j) exp(i k ( r i r j )) cc G α,β ( k )= u α ( k)u β ( k) cc cc G α,β ( k )= r α ( k) rβ ( k) r α ( k) rβ ( k ) Atomic displacement Green's function Equipartition : H= kbt u 1 D α,β ( k)= Φα, β ( k) m u α ( k)= i u α, i exp(i k r i)

32 FCC Cu Implemented in LAMMPS «fix phonon«command

33 Vibrational density of states Definition Identity 1 g(ω)= V k δ(ω ωk ) + g(ω) 0 C vv (t )exp(i ω t )d ω C vv (t)= v i (t) v i (0) See Dove «Introduction to lattice dynamics»

34 Silicon crystalline and amorphous A. France-Lanord et al., J.Phys. Cond. Mat., 014

35 Thermal conductivity

36 Thermal conductivity Essentially, two methods : 1. the «direct» method Apply a temperature gradient or energy flux and calculate the conductivity with Fourier law. equilibrium method Probe the fluctuations around equilibrium of the energy flux vector

37 1. Thermal conductivity : the direct method Steady state temperature profile Heat source Example Si/Ge L Heat sink Fourier law : J = λ ΔT L

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

39 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

40 Thermal conductivity : Green-Kubo equilibrium simulations Green-Kubo formula : + 1 λ α,β= J α (t)j β (0) dt 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 i j In practice, run in NVE ensemble

41 Thermal conductivity : Green-Kubo equilibrium simulations Green-Kubo formula : + 1 λ α,β= J α (t)j β (0) dt 0 V kbt Example amorphous Si 300 K t 1 λ xx (t )= J x (t ')J x (0) dt ' 0 V kbt Jx (t) Jx (0)

42 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 0 V kbt t 1 λ xx (t )= J x (t ')J x (0) dt ' 0 V kbt t 1 λ yy (t)= J y (t ')J y (0) dt ' 0 V kb T Where is the plateau?

43 Optical modes λ (ω)= 1 0 J (t)j (0) exp(i ω t) dt V kb T Termentzidis, SM, Chantrenne IJHMT 011

44 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 0) -the plateau is sometimes difficult to identity (in principle, in a finite system the Green-Kubo formula should give a vanishing conductivity...)

45 Thermal anisotropy K A.F. Lannord, SM, T. Albaret, D. Lacroix, K. Termentzidis, 014

46 Thermal conductivity 1 k c v v g Λ k 3 Kinetic formula : λ= Phonon lifetime τ(ω) Mean Free path Λ(ω)=v g τ(ω) v g λ (ω) Λ(ω)

47 Thermal conductivity Local phonon occupation number : n n( k, s, r,t) n= n Boltzmann transport equation : n + v g t ( t ) coll (n n eq ) n Single relaxation time approximation : + v g n= t τ( k, s) Thermal conductivity : ρ ω λ= 0 g(ω)c v (ω)v g (ω) τ(ω)d ω 3 max λ= Mean free path 1 k c v v g Λ k 3 Λ(ω)=v g τ(ω)

48 Relaxation times : phonon-phonon scattering Three phonons Umklapp processes : τ 1 (ω,t )= A ω T exp( B /T ) k B T ω Klemens formula τ (ω,t )=γ Mv ω D 1 P.G. Klemens, «Solid State Physics» Academic Press New-York 1958

49 Other relaxation mechanisms v τ = Boundary scattering b L L

50 Other relaxation mechanisms L v τ = Boundary scattering b L m mi m ω 4 Point defects τ =V 3 m 4πv 1 m ( ) mi

51 Other relaxation mechanisms L v τ = Boundary scattering b L m mi m 1 Point defects τm =V m ( ) 4 ω 4 π v3 1 τ =N b ω d Dislocations d number of dislocation lines per unit area mi

52 Other relaxation mechanisms L v τ = Boundary scattering b L m mi m ω 4 Point defects τ =V 3 m 4πv 1 m 1 ( ) mi Dislocations τ d =N d b ω number of dislocation lines per unit area v Grain boundaries τ g d d

53 Thermal conductivity Matthiessen's rule = + τ τ 3 ph τ b + τ m + τ d + τ g k ' k k ' '

54 Phonon-phonon scattering :Perturbation theory Ef Hamiltonian : H =H 0 + H ' Pif = π i H ' f ρ(e i )δ (E f E i ) h Ei E. Fermi Initial state Final state Initial state Final state

55 Thermal conductivity : perturbation Hamiltonian Reference Hamiltonian : (harmonic crystal) operator a + 1 α,β H 0 = i, j, α,β Φij ui, α u j, β 1 H 0 = k, s ℏ ω( k, s) a + ( k, s) a ( k, s)+ ( operator a )

56 Thermal conductivity : perturbation Hamiltonian Reference Hamiltonian : (harmonic crystal) Perturbation Hamiltonian : 1 α,β H 0 = i, j, α,β Φij ui, α u j, β 1 H 0 = k, s ℏ ω( k, s) a + ( k, s) a ( k, s)+ ( H '= ) 1 α,β, γ Χ u i, α u j,β u k, γ ijk i, j, k, α,β, γ 6 H ' = k, s, k ', s ', k ' ', s' ' δg, k + k ' + k ' ' V ( k, s, k ', s ', k ' ', s ' ') ( a ( k, s) a ( k, s))( a ( k ', s ') a ( k ', s '))( a ( k ' ', s ' ') a ( k ' ', s ' ')) : G Reciprocal lattice vector

57 Momentum (non) conservation Normal processes k= k ' + k k ' k k ' + k ' ' k ' '

58 Momentum (non) conservation Normal processes k= k ' + k k ' k k ' + k ' ' k ' ' Umklapp processes k= k ' + k ' ' + G k ' k G k ' + k ' ' k ' '

59 Three phonon processes a ( k) a ( k ') a ( k ' ') k ' ' Simultaneous creation of three phonons k k ' a + ( k ) a + ( k ') a ( k ' ' ) k ' Destruction of two phonons and creation of one phonon k ' ' k a + ( k ) a ( k ') a ( k ' ') Destruction of one phonon and creation of two phonons k k ' ' k ' a ( k) a ( k ' ) a ( k ' ' ) Simultaneous destruction of three phonons k ' k k ' '

60 Three phonon processes a + ( k ) a + ( k ') a ( k ' ' ) k ' Destruction of two phonons and creation of one phonon k ' ' k a + ( k ) a ( k ') a ( k ' ') Destruction of one phonon and creation of two phonons k k ' ' k ' Only these processses do not violate energy conservation!!

61 Single relaxation time Class 1 events : (n' ' eq +1) 1 G = k ', k ' ' γ n' eq δ(ω+ω' ω ' ') (neq +1) τ( k)

62 Single relaxation time Class 1 events : (n' ' eq +1) 1 G = k ', k ' ' γ n' eq δ(ω+ω' ω ' ') (neq +1) τ( k) kbt High temperature regime : n ' eq ℏ ω' kbt n eq +1 ℏ ω' 1 G =k B T k ', k ' ' γ ω δ(ω+ω ' ω' ') ω'ω'' τ( k) Difficult to estimate because Dispersion : ω ' ω ' ( k ' ) Crystal anisotropy : ω ' ω ' ( k ' ) Polarisation : k ' ( k ', s)

63 Some references Isotropic crystal, no polarisation P.G. Klemens, «Solid State Physics» Academic Press New-York 1958 J. Callaway, «Model for lattice thermal conductivity at low temperatures», Phys. Rev Effect of the crystal anisotropy C. Herring, «Role of low-energy phonons in thermal conduction», Phys. Rev Effect of the normal processes : P. Carruthers, «Theory of thermal conductivity of solids at low temperatures» Rev. Mod. Phys Effect of the polarisation M.G. Holland, «Analysis of lattice thermal conductivity», Phys. Rev 'Exact' model for a model crystal A.A. Maradudin and A.E. Fein, «Scattering of neutrons by an anharmonic crystal», Phys. Rev., 196

64 Phonon spectroscopy

65 Phonon lifetimes Thermal conductivity : (single relaxation time approximation) λ= k, ν c v ( k, ν) v g ( k, ν)λ ( k, ν) Phonon mean free path : Λ( k, ν)= v g ( k, ν) τ ( k, ν) Importance to calculate the phonon lifetimes : -understand the physical mechanisms at play in Umklapp and/or normal process -use the phonon lifetimes in a Monte-Carlo simulation based on the solution of Boltzmann's equation : k, ν) f f ( f ( k, ν) eq + v g f ( k, ν)= t τ( k, ν) allows large scale simulations-> nanowires, superlattices...

66 Phonon lifetimes : some elements of history First computation of phonon lifetimes in MD goes back to 1986! Ladd, Moran and Hoover, Phys. Rev. B, 34 (1986) 5058 Small system sizes => limited range of k values, but this idea was there

67 Phonon lifetimes : some elements of history First computation of phonon lifetimes in MD goes back to 1986! Ladd, Moran and Hoover, Phys. Rev. B, 34 (1986) 5058 Small system sizes => limited range of k values, but this idea was there Until almost nothing...! McGaughey, Kaviany, Phys. Rev. B 69 (004) Since then, considerable amount of work to describe : -silicon (Henry and Chen, J. Comp. Theo. Nanores., 008) -alloys (Larkin and Mcgaughey, J. App. Phys. 114 (013) 03507) -amorphous solids (He, Donadio, Galli, App. Phys. Lett., 98 (011) polymers (Henry and Chen, Phys. Rev. B 79 (009) )

68 Phonon lifetimes How to put it in practice? 1. Determine the harmonic frequencies and eigenvectors : ω ( k) e ( k)=d( k) e ( k) dynamic matrix. Run MD simulations and compute the normal coordinates : (1/) m 0 Q( k,t)= i ( i ) exp(i k r i ) e i ( k ) u i (t) N 1 dq dq 3. Calculate the kinetic and potential energies : T ( k,t )= (k ) ( k ) dt dt 1 U ( k,t )= ω ( k )Q ( k )Q( k ) k, t) E ( k,0) E ( E( k, t)=t ( k,t)+ U ( k,t) τ( k)= dt E ( k,t ) But this route doesnot give access to the anharmonic frequencies ω a ( k)!! 0

69 Phonon lifetimes How to extract the phonon lifetime of mode k? T ( k,t)t ( k,0) =cos (ω a ( k )t)exp( t / τ( k)) T ( k,t) Example LJ Argon, 10 K E ( k,t) E( k,0) =exp( t / τ( k )) E ( k,t ) k, t) E ( k,0) E ( τ( k)= dt E ( k,t) 0 Turney, Landry, McGaughey, Amon, Phys. Rev. B, 009

70 Phonon lifetimes Alternative computation : the frequency domain decomposition 3. Calculate the kinetic and potential energies : Expectation value : Parseval theorem : 1 dq dq T ( k,t )= (k ) ( k ) dt dt 1 τ dq dq T ( k, t) =lim τ ( k,t) ( k,t )dt 0 τ0 dt dt 0 0 k, ω) T ( k, ω) =ω Q( τ 1 k,t)exp( i ω t) dt k, ω)=lim τ Q( Q( π Spectral energy density 1/ π τ( k) T ( k, ω) =C ( k ) (ωa ( k ) ω) + (1/4 π τ( k ))

71 Phonon lifetimes Alternative computation : the frequency domain decomposition Spectral energy density 1/ π τ( k) T ( k, ω) =C ( k ) (ωa ( k ) ω) + (1/4 π τ( k )) Example LJ Argon, 10 K Turney, Landry, McGaughey, Amon, Phys. Rev. B, 009

72 Phonon lifetimes Example LJ Argon, 10 K Turney, Landry, McGaughey, Amon, Phys. Rev. B, 009

73 Phonon lifetimes Thermal conductivity λ= k cv ( k ) v g ( k ) τ ( k ) Example LJ Argon, 10 K Towards large scale simulations : Boltzmann transport equation : (n neq ) n + v g n= t τ( k, s)

74 Accumulation function : phonon spectroscopy Crystal Si ω acc (ω)= 0 c v (ω) vg (ω) Λ (ω) d ω Regner et al., Nat. Comm. 013

75 Amorphous materials Amorphous Si λ (ω) v g Λ(ω) phonons phonons Λ>λ He, Donadio and Galli, APL 011

76 Amorphous materials Only localized modes phonons Larkin and McGaughey, PRB 014

77 Phonon scattering at interfaces

78 Heat Transfer at the nanoscale Superlattices Dresselhaus, Chen, Science 01 -Deviations from Fourier's law -Thermal boundary resistance Nanocomposites-nanowires Poulikakos, NanoLetters 011 (dimensions < phonon mean free path)

79 Superlattices as high ZT materials

80 Crystalline/amorphous superlattices Bulk crystalline K λ=148w / K / m Bulk amorphous K λ=0.68 W / K /m 6% amorphised Silicon => 0 fold drop in the thermal conductivity A. France Lanord et al., J. Phys. Cond. Mat. 014

81 Thermal boundary (Kapitza) resistance Thermal boundary conductance G=q/ Δ T G=1/ R= C v (ω) v g (ω) t 1 (ω)d ω Energy transmission coefficient G MW /m / K Equivalent thickness : l=λ /G= nm Hopkins 013

82 Acoustic mismatch model (AMM) W.A. Little (1959) θ Z > Z 1 Phonons treated as plane waves propagating in continuous media acoustic impendance Z i=ρi c i Z1 θ1 θ1 Reflection/refraction at the interface obey laws analogous to Snell's laws sin θ1 sin θ = c1 c 4 Z 1 Z cos θ1 cos θ Transmission coefficient : t 1= (Z cos θ + Z cos θ ) 1 1

83 Diffuse mismatch model (DMM) (Swartz 1989) All the phonons are diffusively scattered by the interface The transmission coefficient is given by detailed balance c g (ω) α1 (ω)= c g (ω)+ c1 g1 (ω) The conductance is related to the overlap between the density of states characterizing the two media

84 Factors affecting the conductance : (a non exhaustive list) «Bonding strength» between the two solids Lattice parameters mismatch Interfacial roughness Defects at the interface (vacancies) Intermixing Electron-phonon coupling

85 Non-equilibrium simulations Amorphous Si/ Crystalline Si Fourier law : J = λ T z Thermal boundary resistance : R=Δ T / J

86 Equilibrium simulations Puech formula + 1 G= q(t) q(0) dt 0 A kbt A interfacial area Analogous to the Green-Kubo formula for the thermal conductivity Interfacial flux : ( v i + v j) q= i 1, j F ij Barrat and Chiaruttini, Mol. Phys. 003 SM and Termentzidis, PRB 01

87 Equilibrium simulations Equivalent formula 1 dc EE G= A k B T dt ( ) t=0 C EE (t)= δ E (t)δ E(0) C EE (t) C EE (0)exp( t / τ) 1 C EE (0) G= τ Ak B T SM and Termentzidis, PRB 01

88 Finite size effects equilibrium non-equilibrium

89 «Approach to equilibrium» Temperature relaxation Finite size effects Δ T (t) exp( t / τ) G=C v /( A τ) E. Lampin et al., APL 01

90 Comparison with the models Thermal boundary resistance G=1/ R= g(ω)c v (ω) v g (ω)t 1 (ω)d ω AMM DMM A. France Lanord et al., J. Phys. Cond. Mat., Z 1 Z cos θ1 cos θ t 1= (Z 1 cos θ1 + Z cos θ) c g (ω) t 1 (ω)= c g (ω)+c 1 g1 (ω)

91 Phonon transmission coefficient crystalline Si/heavy Si interface First paper : Shelling,Keblinski APL 00 Transmission coefficient

92 Phonon transmission coefficient Au/ Si interface extinction Critical angle

93 Electron-phonon couplings Two temperature model + boundary conditions Measurements : Hopkins et al. JAP 009 Theory : Sergeev, PRB 1998 Mahan, PRB

94 Two temperature model σ e =0 Two temperature model Te ce =k e xx T e G(T e T p ) t T p cp =k p xx T p +G(T e T p ) t Ts cs =k s xx T s t Boundary conditions x=h J = k s x T s =σ e (T e T s )+σ p (T p T s) = k e x T e k p x T p J. Lombard et al. J. Phys. Cond. Mat. 015

95 Two temperature model 3 ( ( )) ul h γ ul σe τ 1+ k B e ph ut J. Lombard et al. J. Phys. Cond. Mat. 015

96 Heat transport at solid/liquid interfaces

97 Heat transfer at solid/liquid interfaces Wen, Int. J. Hyperthermia, 009

98 Molecular dynamics results 300 K 0 nw 100 K 600 nw SM et al., PNAS K 400 nw 1000 nw Gold Np in octane

99 Gold nanoparticles in liquid octane

100 Gold nanoparticles in liquid octane

101 Nanobubbles Eau 300 K Vapeur Nanoparticule (NP) Hashimoto et al., J. of Photochemistry and Photobiology C: Photochemistry Reviews (01)

102 Biomedical applications Diagnostic and tumor therapy Lapotko et al., ACS Nano (010 ) Lapotko, Cancer (011)

103 Questions Threshold energy to drive nanobubble generation? Cancer cell What does control nanobubbles size? => hydrodynamic models (phase field) Cancer cell

104 Fluid modeling Ø Conservation equations : mass, momentum and energy Ø Capillary effects : density functional theory f f δ α,β +w α ρ β ρ Pressure tensor : Π αβ = ρ ρ ( )

105 Nanoparticle modeling LASER Liquid/nanoparticle interface : interfacial conductance Vapor/nanoparticle interface : ballistic thermal transport

106 Threshold for explosive boiling 0 ps r(nm) r(nm) 118 ps Water (73K) Hot NP Cold NP Rb Fs pulse r (nm) Minimal laser fluence to observe nanobubbles Nanobubbles No nanobubble s J.Lombard, T. Biben, SM, PRL (014) r (nm) ρ

107 Maximal size of the nanobubbles Ballistic flux J =αρs k 3B 3/ (T p T 3/ l ) m Experimental results : A. Siems et al., New J. Phys. 011

108 Thank you for your attention!