Lecture 4

Heat transfer at small scales

Linear laws between fluxes and forces proposed by NET are not valid: -In ’far-from-equilibrium situations’ For transport processes, the linear response regime holds even for large gradients For activated processes, linearity occurs only very close to equilibrium -At small scales: anomalous heat transfer

1 2( )q dJ G T T= − −

Thermal rectification: Resistance to the heat flow depends on the sign of the T-grad. Asymmetry in the phonon distribution.

Thermal grandient as a driving force

• Water polarization under thermal gradients: F. Bresme, A. Lervik, el al., PRL, 101, 020602 (2008).

• Thermomolecular orientation of nonpolar fluids: F. Römer et al. PRL, 108, 105901 (2008)

• Nanoscale mass conveyors: Regan et al., Nature, 428, 924 (2004)

• Traping of DNA by thermophoretic depletion: D. Braun , A. Libchaber, PRL, 89, 188103 (2002)

• To drive the flow of fluids in nanotubes, nanopores and microfluidoic devices

D. Braun , A. Libchaber, PRL, 89, 188103 (2002)

Very large thermal gradients: typically range from 10^5 K/m on the mesoscale to 10^8 K/m

on the nanoscale

A thermal gradient may induce actuation in DWCN

DWNT

Actuated by the temperature difference the outer NT moves along and/or around the inner tube

Chiral numbers: (n,m)

50-100 m/s ; 1K/nm; thermal force: pico Newtons

0.142cca nm=

0.142cca nm=

1 2hC na ma= +

out inL L<<

Quan-Wen Hou et al, Nanotechnology, 20, 495503 (2009)

Experiments and simulations

A. Barreiro et al., Science, 320, 775 (2008)

0.1I mA

AFM

Energy barrier

Mechanism: in the presence of a thermal gradient, there will be a current of phonons which transfer momentum to the mobile nanotube. - Reverse of the heat dissipation that occurs in friction

Translational motion

Joule heating

Molecular dynamics

Thermal driving force T zz

TFz

κ ∂=

∂

3

2 3

1010 10in

T KL nm∆

−

/ 40 /TF m sζ 4TF pN

3

2 3

100.01 0.1

10 10

inL nmI mA

T K−

∆ −

Observed in experiments 211.5 10 /zz J Kκ −⋅

Observed in simulations

Interwall potential

Structures depend on the chirality Darker color: lower potential energy

Model Two degrees of freedom:

: ( ): ( )

axial z tazimutal tθ

Forces: friction, van der Waals, thermal gradient, random

12

1.5

1 2 10 /frictionf pN

Kg sζ −− ⋅

2.1 4.2vdWF pN−

( ) ( ) ( ) ( )1( ) ( ) ( )

vdW Rz zz z z

vdW R

mx t x t F T F t

m t t F F tr θ θ

ζ κ

θ ζθ

= − + + ∇ +

= − + +

( ) (0) 2 ( )

( ) (0) 2 ( )

R Rz z

R Rrot

F t F kT tF t F kT tθ θ

ζδ

ζ δ

< >=

< >=

Equilibrium photon gas

4eqJ Tσ=

Energy conservation # microstates

Planck distribution

Stefan-Boltzmann

1

exp 1N

kT

ω ω=

−

TdS=dE

Planck’s theory is not valid when length scales are comparable to the wave length of thermal radiation

Enhancement of the heat flux in the near-field

Heat Transfer coefficient

Hu et al., Appl. Phys. Lett. 92, 133106 (2008)

Thermal conductance

Domingues et al., Phys. Rev. Lett. 94, 085901 (2005)

Molecular dynamics

Ther

mal

con

duct

ance

Fluctuation-dissipation regime

FDT

Fluctuating electrodynamics

Domingues et al., PRL, 94, 085901 (2005)

Valid when thermalization is very fast

Mesoscopic thermodynamics

( , )x pΓ

( , )n tΓ ?nt

∂

∂

Photons, phonons, electrons,…:

Flux of particles, Heat flux, Energy flux, etc. Second law: entropy

dQ T s nδ µδ= = −

n Jt

∂= −∆

∂

1( )s n J Jt T t T T

µ µ µ∂ ∂= − = ∆ − ∆

∂ ∂

2 1( )LJT

µ µ= − −

2 1

( )

kT kT

kT kT

LJ kLe eT

J l e e

µ µ

µ µ

µ−

= − ∆ ≈ − ∆

= − −

Linear

Nonlinear (Arrhenius)

Nanosystems: fluctuations are important One has to adopt a probabilistic description

kinetics

-D. Reguera, J.M. Rubi and J.M. Vilar, J. Phys. Chem. B (2005) -J.M. Rubi, Scientific American, Nov., 40 (2008)

entropy production

ii) Mesoscopic conductors (Landauer):

iii) Photons

iv) Thermal emission, adsorption (Langmuir), electrokinetic phenomena (Butler-Volmer): Activated processes.

0 01( ) ( )n L n n

t Tµ µ

τ∂

= − − ≈ − −∂

i) Boltzmann (BGK): 0

0

1

0

ln nkTn

LTn

µ µ

τ −

= +

=

[ ]( ) ( ) ( ) ( , ) ( , )st L R R LJ D D n T n Tω ω ω ω ω= − −

( )1( , )

exp / 1n T

kTω

ω=

−

3 4 43 3 ( ) ( )

4st p st R LJ d d J T Tc

ω ω ω σπ

= Ω = −∫

LT RT

ELECTROMAGNETIC WAVES CONFINED IN A VERY NARROW GAP

Spectrum of electromagnetic energy for different detection distances above surface of silicon carbide normalized by its maximum value in far field at T= 300 K. (J.-J. Greffet and C. Henkel, 2007)

18

When the distance diminishes, thermal radiation becomes a highly directional beam of coherent radiation. The narrow gap acts as a resonant cavity

Electromagnetic energy density above a plane interface separating glass (amorphous, optical phonons poorly defined) at T = 300 K from vacuum at T = 0 K

Photon Kinetics

2 1*

*

( )

( ) ( )

J n n

t n n d

τ

τ τ

= −

= Γ∫

) ;

)

;

T

T

R

i d black body

ii dx p h

h Kcp E h hd d

λ

λ

ω

>>

≤ ∆ ∆ ≥

∆ ≥ ∆ ≥ ≡

pp xxpx

xp

D D nJ DD x

∂= − ∂

A. Perez, L. Lapas, J.M. Rubi, PRL, 103, 048301 (2009)

Heat flux results from the contribution of Different modes

Input of the theory

Elastic:

Inelastic:

Lognormal DOS

-Statistical model which can describe collective motions. -Stems from the existence of a hierarchy of relaxations mechanisms in the material -The energy consists of a large number of contributions; central limit theorem

Y. V. Denisov, A.P. Rylev, JEPT Lett. 52, 411 (1990) 0: ,Fitting parameters ω σ

Heat transfer coefficient

21

A. Perez, L. Lapas, J.M. Rubi, PRL, 103, 048301 (2009)

FF

NF

30

0

H TH T

A. Perez-Madrid, L. Lapas, J.M. Rubi, Plos One, 8, e58770 (2013)

Summary

• At small scales heat transfer shows anomalous behaviour, not explained by the NET laws. New phenomena: Near-field thermodynamics, thermal rectification, ….

• The linear response regime may extend to the case of large gradients.