Heat Transport Across a Small Gap: Transition from Radiation to - PowerPoint PPT Presentation

Heat Transport Across a Small Gap: Transition from Radiation to Conductance Bair V. Budaev and David B. Bogy Computer Mechanics Laboratory, UC Berkeley CML Sponsors Meeting 2014 Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 1 / 12

A typical Problem Q Find K ( H ) = for the structure: T A − T B Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 2 / 12

A typical Problem Q Find K ( H ) = for the structure: T A − T B Textbooks: Only EM radiation carries heat in vacuum Radiative heat transport is H -independent Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 2 / 12

A typical Problem Q Find K ( H ) = for the structure: T A − T B Textbooks: Only EM radiation carries heat in vacuum Radiative heat transport is H -independent Common Sense: As gap collapses ( H → 0 ), heat transport SHOULD increase Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 2 / 12

Physics behind heat transport across vacuum gap ◮ Heat flows because electric charges interact through electric fields Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 3 / 12

Physics behind heat transport across vacuum gap ◮ Heat flows because electric charges interact through electric fields ◮ The field of a moving charge has three terms: � � � � � � � � � d 2 E = − q r 3 + r r d r + 1 r � , r 3 c 2 dt 2 4 πǫ c dt r � �� � � �� � Conduction terms Radiation term ◮ Conduction terms ∼ 1 / r 2 ( � r is a retarded vector) � � � � � Because d 2 r = � a v ˙ r ◮ Radiation term ∼ 1 / r r − 2 � r 2 + . . . dt 2 r Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 3 / 12

Physics behind heat transport across vacuum gap ◮ Heat flows because electric charges interact through electric fields ◮ The field of a moving charge has three terms: � � � � � � � � � d 2 E = − q r 3 + r r d r + 1 r � , r 3 c 2 dt 2 4 πǫ c dt r � �� � � �� � Conduction terms Radiation term ◮ Conduction terms ∼ 1 / r 2 ( � r is a retarded vector) Dominates the “near-field”. Provides “heat conduction by phonons” � � � � � Because d 2 r = � a v ˙ r ◮ Radiation term ∼ 1 / r r − 2 � r 2 + . . . dt 2 r Dominates the “far-field”. Provides “heat radiation by photons” Both mechanisms work at all distances, both work in vacuum Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 3 / 12

A model for radiative transport Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 4 / 12

A model for radiative transport Reflection coefficient � H , H ≪ wavelength R ∼ const , otherwise Energy transport � 1 / H 2 , K ∼ 1 H ≪ wavelength R 2 ∼ const , otherwise For narrow gaps, radiative transport is not constant ! Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 4 / 12

Models for conductance through vacuum In solids heat is carried by lattice vibrations (phonons) Lattices can be modeled by mass-spring chains: One body: Isolated bodies: Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 5 / 12

Models for conductance through vacuum In solids heat is carried by lattice vibrations (phonons) Lattices can be modeled by mass-spring chains: One body: Isolated bodies: Interacting bodies: Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 5 / 12

Models for conductance through vacuum In solids heat is carried by lattice vibrations (phonons) Lattices can be modeled by mass-spring chains: One body: Isolated bodies: Interacting bodies: Continuum model ( p = pressure): Reflection coefficient R ∼ H ǫ , ǫ ≥ 4 (at least ∼ 1 / H 8 ) Energy transport K ∼ 1 / H 2 ǫ , Phonons contribute to heat transport across NARROW gaps Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 5 / 12

First Conclusions ◮ Both radiation and conduction carry heat through vacuum Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 6 / 12

First Conclusions ◮ Both radiation and conduction carry heat through vacuum ◮ Both mechanisms are described by similar mathematics: ◮ wave equations ◮ interface conditions ◮ reflection coefficients Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 6 / 12

First Conclusions ◮ Both radiation and conduction carry heat through vacuum ◮ Both mechanisms are described by similar mathematics: ◮ wave equations ◮ interface conditions ◮ reflection coefficients ◮ Both radiation and conduction across narrow gaps should be studied by similar methods ◮ The classical theory may not be used, because: ◮ predicts H -independence of radiative transport ◮ does not admit conductance through vacuum Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 6 / 12

Classical theory of radiation breaks down because: It assumes that: ◮ Only radiation carries heat across vacuum gap ◮ Each body radiates as if there are no other bodies ◮ Each body radiates as if it is in equilibrium Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 7 / 12

Classical theory of radiation breaks down because: It assumes that: ◮ Only radiation carries heat across vacuum gap ◮ Each body radiates as if there are no other bodies ◮ Each body radiates as if it is in equilibrium Classical scheme is inconsistent: ◮ It does not comply with conservation of energy ◮ For a vanishing gap the flux remains finite, instead of diverging These inconsistencies are not negligible in nanoscale Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 7 / 12

Self-consistent approach New scheme: connects two temperatures with the heat flux Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 8 / 12

Self-consistent approach New scheme: connects two temperatures with the heat flux New problem: How to get � Q A → B ( T , � Q ) ? ◮ In the classical theory, � Q A → B ( T ) is computed using Planck’s spectrum of thermal radiation in equilibrium . ◮ We generalize Planck’s spectrum to systems with a heat flux ! Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 8 / 12

Spectrum of thermal radiation with a heat flux � � ω P ( ω, θ, T , Q ) = P 1 − Q cos θ/ c E , T T – temperature P ( ω, T ) – equilibrium spectrum ω – frequency θ – angle between the flux and the wave vector Q – heat flux E – energy density Annalen der Phyzik, 2011, v. 523, no. 10, pp. 791–804 Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 9 / 12

Spectrum of thermal radiation with a heat flux � � ω P ( ω, θ, T , Q ) = P 1 − Q cos θ/ c E , T T – temperature P ( ω, T ) – equilibrium spectrum ω – frequency θ – angle between the flux and the wave vector Q – heat flux E – energy density Explanation: ◮ Q is the flux in the reference frame R Annalen der Phyzik, 2011, v. 523, no. 10, pp. 791–804 Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 9 / 12

Spectrum of thermal radiation with a heat flux � � ω P ( ω, θ, T , Q ) = P 1 − Q cos θ/ c E , T T – temperature P ( ω, T ) – equilibrium spectrum ω – frequency θ – angle between the flux and the wave vector Q – heat flux E – energy density Explanation: ◮ Q is the flux in the reference frame R ◮ No flux in the frame M with v = Q / E Annalen der Phyzik, 2011, v. 523, no. 10, pp. 791–804 Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 9 / 12

Spectrum of thermal radiation with a heat flux � � ω P ( ω, θ, T , Q ) = P 1 − Q cos θ/ c E , T T – temperature P ( ω, T ) – equilibrium spectrum ω – frequency θ – angle between the flux and the wave vector Q – heat flux E – energy density Explanation: ◮ Q is the flux in the reference frame R ◮ No flux in the frame M with v = Q / E ◮ In M radiation has Planck’s spectrum ◮ Spectra in R and M are related by Doppler shift Annalen der Phyzik, 2011, v. 523, no. 10, pp. 791–804 Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 9 / 12

The self-consistent scheme in action: • Get an equation for T A , T B , Q � ∞ � π/ 2 1 � � 2 − R 2 ⊥ ( θ ) − R 2 Q = � ( θ ) 2 π 2 c 2 0 0 × {P ( ω, θ, T A , Q ) − P ( ω, θ, T B , − Q ) } ω 2 d ω d θ 4 Heat transport coefficient (W/K/m 2 ) 10 Two half − spaces with c=0.45c 0 • Fix T A and T B 3 • Solve for Q 10 Adjusted to a half − space and a sphere with c=0.45c 0 • Compute Q K = 2 T B − T A 10 Experiments with a plate 1 and a sphere from silica 10 Classical limit adjusted to a half − space and a sphere 0 10 1 2 3 4 10 10 10 10 Width of the vacuum gap (nm) Budaev, Bogy (CML, UC Berkeley) Transition from Heat Radiation to Conduction January, 2014 10 / 12