Time Scale of Thermal Conduction Xiaoshan Xu the Importance of Thermal Conduction

Time scale of thermal conduction Xiaoshan Xu The importance of thermal conduction

• Energy consumption: Hot reservoir • transfer through conduction • Energy generation: Δ푉 • Thermoelectric effect (Seebeck effect) Cold reservior • Spin Seebeck effect Basics of thermal conduction Example of irreversible process: Heat conduction

Heat Conductivity in Solids (an example for irreversibility)

Remember: Heat is an energy transferred from one system to another because of temperature difference

T1 > T2

System  System 1 2

Heat Q flows from

1 to 2 Example of irreversible process: heat conductions as a non-equilibrium process: *(in the textbook T >T ) T1 > T2 2 1

Heat reservoir  Heat reservoir 1 2

T(x)

0 L x A Heat transfer per time interval through homogeneous solid object: Q K K: thermal conductivity of the rod  (T1  T2 )A where L t L A: cross-section of the rod Diffusion (time and space dependence of matter/energy)

T(x) Q K  (T1  T2 )A T1 t L

T2 퐿 → ∆푥 푇2 − 푇1 → ∆푇 0 L x Q T  K At x

Fourier’s law Define flux: 푄 휕푇 푥, 푡 푆 = lim 푆 푥, 푡 = −퐾 Δ푡→0 퐴∆푡 휕푥 Diffusion (time and space dependence of matter/energy)

A 푺: flux (energy flow per unit area per unit 푆(푥, 푡) 푢(푥, 푡), 푈 푆(푥 + ∆푥, 푡) time, similar to electric current) 풖: energy density (energy per unit volume) 푥 푥 + ∆푥

How would energy change in the system?

퐴 푆 푥 + ∆푥, 푡 − 푆 푥, 푡 ∆푡 = −[푢(푥, 푡 + ∆푡) − 푢(푥, 푡)]퐴∆푥

푆 푥 + ∆푥, 푡 − 푆 푥, 푡 푢 푥, 푡 + ∆푡 − 푢 푥, 푡 = − ∆푥 ∆푡

Continuity equation:

휕푢 푥, 푡 휕푆 푥, 푡 = − 휕푡 휕푥 Diffusion (time and space dependence of matter/energy)

A 푆(푥, 푡) 푢(푥, 푡), 푈 푆(푥 + ∆푥, 푡) Flux across a boundary 푥 푥 + ∆푥 휕푇 푥, 푡 푆 푥, 푡 = −퐾 휕푥

Energy change in a system

휕푢 푥, 푡 휕푆 푥, 푡 = − 휕푡 휕푥 Diffusion (time and space dependence of matter/energy)

휕푢 푥, 푡 휕푆 푥, 푡 To study temperature distribution. = − Continuity equation: 휕푡 휕푥 휕푈 푈 = 푈 + ∆푇 = 푈 + 푀푐푀∆푇 0 휕푇 0 푉 푉

푈(푡) 푈 푀푐푀∆푇 푢 = = 0 + 푉 = 푢 + 𝜌푐푀∆푇 푉 푉 푉 0 푉

푀 푢 − 푢0 = ∆푢 = 𝜌푐푉 ∆푇

휕푆 푥, 푡 휕푇(푥, 푡) 휕푢 푥, 푡 푀 휕푇(푥, 푡) 푀 = 𝜌푐 = −𝜌푐푉 휕푡 푉 휕푡 휕푥 휕푡 Diffusion (time and space dependence of matter/energy)

휕푆 푥,푡 휕푇(푥,푡) From continuity equation = −𝜌푐푀 (1) Diffusion equation 휕푥 푉 휕푡 휕푇(푥, 푡) 휕2푇 푥, 푡 From Fourier’s law = 퐷 2 휕 휕푡 휕푥 휕푇 푥, 푡 Take 휕푆 푥,푡 휕2푇 푥,푡 퐾 휕푥 = −퐾 (2) 푆 푥, 푡 = −퐾 휕푥 휕푥2 퐷 = 푀 diffusivity 휕푥 𝜌푐푉 Application of Diffusion Equation (Decay of a hot spot)

Diffusion equation 푇(푥, 푡)

휕푇(푥, 푡) 휕2푇 푥, 푡 = 퐷 휕푡 휕푥2 푇 Describes how the spatial distribution evolves with time.

One of the solution (example): 푥 퐴 푥2 푇 푥, 푡 = 푇0 + exp − 푡 + 푡0 4퐷 푡 + 푡0

퐴 푥2 Gaussian peak. 푇 푥, 0 = 푇0 + exp(− ) 푡0 4퐷푡0 Application of Diffusion Equation (Decay of a hot spot)

Spatial dependence at 푡: 푇(푥, 푡)

퐴 푥2 푇 = 푇 + exp(− ), 푤 푡 = 4퐷(푡 + 푡 ) 0 푤 푤2 푡 0 푇 푡0 Spatial evolution with time:

1) Peak becomes broader 푡1

푡2 2) Total area is conserved (energy conservation 푑푈 = 푑푇). 푡3

푥

휕푇(푥, 푡) 휕2푇 푥, 푡 = 퐷 휕푡 휕푥2 Scaling rule of the decay

퐴 푥2 푇 푥, 푡 = 푇0 + exp − 푡 + 푡0 4퐷 푡 + 푡0 퐴 푥2 = 푇 + exp − 0 푡 푡 푡0 1 + 4퐷푡0 1 + 푡0 푡0

푡 If we use 푡0 as the unit, and let 휏 = , we get a universal relation 푡0

퐴 푥2 푇 푥, 푡 = 푇0 + exp − 1 + 휏 4퐷 1 + 휏

So the decay scales with 푡0 Scaling rule of the decay

퐴 푥2 푇 푇 푥, 0 = 푇0 + exp(− ) 푡0 4퐷푡0

2 𝜎 = 2퐷푡0 2𝜎

푥

2 Since the decay scales with 푡0, it scales with the 𝜎 . Another view using analogy to circuits

Thermal conduction is like discharging a capacitor, the time constant is 휏 = 푅퐶 For thermal conduction, we use the same formula 휏 = 푅퐶 푑 푅 = 𝜎퐴 퐶 = 푐푀𝜌푉 = 푐푀𝜌퐴푑 𝜎: thermal conducitivity 푑: thickness 퐴: area hot cold 𝜌: density 푐푀: specific heat

So 푑 푑 푐 𝜌 휏 = 푅퐶 = 푐 𝜌퐴푑 = 푀 풅ퟐ 𝜎퐴 푀 𝜎

Again, this scales with width2. Some hands-on data A simple experiment

1.0 dT more dT less

0.8

0.6 Less

0.4

normalized

0.2

0.0

0.00 0.02 0.04 0.06 0.08 dt (day)

Time scale in thin films

Laser pulse

Film Substrate

20 nm LFO 20 nm 1.0 LFO YbFO YbFO 50 nm 1.0 50 nm BFO 0.8 35 nm

0.6

dc/c

0.5 dc/c

0.4

0.2

0.0 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0.000 0.005 tdelay t (ns) 푡/푑2 (ns/nmtdelay 2)

For the previous water cooling experiment: 흉/풅ퟐ = ퟎ. ퟎퟎퟑ (ns/nm2) 휏 = 1080s 푑 = 2 cm Conclusion

• We discussed the scaling rule of the thermal conduction, which is proportional to the thickness^2. • A few examples (microscopic and macroscopic) are given, which can be unified using the scaling rule.