Time scale of thermal conduction

Xiaoshan Xu

The importance of thermal conduction

• Energy consumption:• transfer through conduction

• Energy generation:• Thermoelectric effect (Seebeck effect)

• Spin Seebeck effect

Hot reservoir

Cold reservior

Δ𝑉

Basics of thermal conduction

Heat Conductivity in Solids (an example for irreversibility)

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

System 1System 2

T1 > T2

Heat Q flows from

1 to 2

Example of irreversible process: Heat conduction

Heat reservoir 1Heat reservoir 2

T1 > T2

L

T(x)

x

T1

T2

0 L

*(in the textbook T2>T1)

Heat transfer per time interval through homogeneous solid object:

A)TT(L

K

t

Q21

where

K: thermal conductivity of the rod

A: cross-section of the rod

A

L

Example of irreversible process: heat conductions as a non-equilibrium process:

Diffusion (time and space dependence of matter/energy)

A)TT(L

K

t

Q21

T(x)

x

T1

T2

0 L

x

TK

tA

Q

𝐿 → ∆𝑥𝑇2 − 𝑇1 → ∆𝑇

𝑆 𝑥, 𝑡 = −𝐾𝜕𝑇 𝑥, 𝑡

𝜕𝑥

Define flux:

𝑆 = limΔ𝑡→0

𝑄

𝐴∆𝑡

Fourier’s law

Diffusion (time and space dependence of matter/energy)

𝑢(𝑥, 𝑡), 𝑈

A

𝑥 𝑥 + ∆𝑥

Continuity equation:

𝜕𝑢 𝑥, 𝑡

𝜕𝑡= −

𝜕𝑆 𝑥, 𝑡

𝜕𝑥

𝑆(𝑥, 𝑡) 𝑆(𝑥 + ∆𝑥, 𝑡)

𝐴 𝑆 𝑥 + ∆𝑥, 𝑡 − 𝑆 𝑥, 𝑡 ∆𝑡 = −[𝑢(𝑥, 𝑡 + ∆𝑡) − 𝑢(𝑥, 𝑡)]𝐴∆𝑥

𝑺: flux (energy flow per unit area per unit time, similar to electric current)𝒖: energy density (energy per unit volume)

How would energy change in the system?

𝑆 𝑥 + ∆𝑥, 𝑡 − 𝑆 𝑥, 𝑡

∆𝑥= −

𝑢 𝑥, 𝑡 + ∆𝑡 − 𝑢 𝑥, 𝑡

∆𝑡

𝑆 𝑥, 𝑡 = −𝐾𝜕𝑇 𝑥, 𝑡

𝜕𝑥

𝜕𝑢 𝑥, 𝑡

𝜕𝑡= −

𝜕𝑆 𝑥, 𝑡

𝜕𝑥

𝑢(𝑥, 𝑡), 𝑈

A

𝑥 𝑥 + ∆𝑥

𝑆(𝑥, 𝑡) 𝑆(𝑥 + ∆𝑥, 𝑡) Flux across a boundary

Energy change in a system

Diffusion (time and space dependence of matter/energy)

Diffusion (time and space dependence of matter/energy)

𝑢(𝑥, 𝑡), 𝑈

A

𝑥 𝑥 + ∆𝑥

𝑆(𝑥, 𝑡) 𝑆(𝑥 + ∆𝑥, 𝑡)

𝑺: flux (energy flow per unit area per unit time, similar to electric current)𝒖: energy density (energy per unit volume)

To study temperature distribution.

𝑈 = 𝑈0 +𝜕𝑈

𝜕𝑇𝑉

∆𝑇 = 𝑈0 +𝑀𝑐𝑉𝑀∆𝑇

𝑢 =𝑈(𝑡)

𝑉=𝑈0𝑉+𝑀𝑐𝑉

𝑀∆𝑇

𝑉= 𝑢0 + 𝜌𝑐𝑉

𝑀∆𝑇

𝜕𝑢 𝑥, 𝑡

𝜕𝑡= −

𝜕𝑆 𝑥, 𝑡

𝜕𝑥

𝑢 − 𝑢0 = ∆𝑢 = 𝜌𝑐𝑉𝑀∆𝑇

𝜕𝑢 𝑥, 𝑡

𝜕𝑡= 𝜌𝑐𝑉

𝑀𝜕𝑇(𝑥, 𝑡)

𝜕𝑡

𝜕𝑆 𝑥, 𝑡

𝜕𝑥= −𝜌𝑐𝑉

𝑀𝜕𝑇(𝑥, 𝑡)

𝜕𝑡

Continuity equation:

Diffusion (time and space dependence of matter/energy)

𝑢(𝑥, 𝑡), 𝑈

A

𝑥 𝑥 + ∆𝑥

𝑆(𝑥, 𝑡) 𝑆(𝑥 + ∆𝑥, 𝑡)

𝑺: flux (energy flow per unit area per unit time, similar to electric current)𝒖: energy density (energy per unit volume)

From continuity equation

From Fourier’s law

𝑆 𝑥, 𝑡 = −𝐾𝜕𝑇 𝑥, 𝑡

𝜕𝑥

𝜕𝑆 𝑥,𝑡

𝜕𝑥= −𝐾

𝜕2𝑇 𝑥,𝑡

𝜕𝑥2(2)

Take 𝜕

𝜕𝑥

𝜕𝑇(𝑥, 𝑡)

𝜕𝑡= 𝐷

𝜕2𝑇 𝑥, 𝑡

𝜕𝑥2

𝜕𝑆 𝑥,𝑡

𝜕𝑥= −𝜌𝑐𝑉

𝑀 𝜕𝑇(𝑥,𝑡)

𝜕𝑡(1)

𝐷 =𝐾

𝜌𝑐𝑉𝑀 diffusivity

Diffusion equation

Application of Diffusion Equation (Decay of a hot spot)

𝜕𝑇(𝑥, 𝑡)

𝜕𝑡= 𝐷

𝜕2𝑇 𝑥, 𝑡

𝜕𝑥2

Diffusion equation 𝑇(𝑥, 𝑡)

𝑇 𝑥, 𝑡 = 𝑇0 +𝐴

𝑡 + 𝑡0exp −

𝑥2

4𝐷 𝑡 + 𝑡0

𝑇 𝑥, 0 = 𝑇0 +𝐴

𝑡0exp(−

𝑥2

4𝐷𝑡0)Gaussian peak.

𝑇

𝑥One of the solution (example):

Describes how the spatial distribution evolves with time.

Application of Diffusion Equation (Decay of a hot spot)

𝑇(𝑥, 𝑡)

𝑇 = 𝑇0 +𝐴

𝑤exp(−

𝑥2

𝑤2 𝑡), 𝑤 𝑡 = 4𝐷(𝑡 + 𝑡0)

Spatial dependence at 𝑡:

𝑇

𝑥

𝑡0

𝑡2

𝑡1

𝑡3

Spatial evolution with time:

1) Peak becomes broader

2) Total area is conserved (energy conservation 𝑑𝑈 = 𝑑𝑇).

𝜕𝑇(𝑥, 𝑡)

𝜕𝑡= 𝐷

𝜕2𝑇 𝑥, 𝑡

𝜕𝑥2

Scaling rule of the decay

𝑇 𝑥, 𝑡 = 𝑇0 +𝐴

𝑡 + 𝑡0exp −

𝑥2

4𝐷 𝑡 + 𝑡0

= 𝑇0 +𝐴

𝑡0 1 +𝑡𝑡0

exp −𝑥2

4𝐷𝑡0 1 +𝑡𝑡0

If we use 𝑡0 as the unit, and let 𝜏 =𝑡

𝑡0, we get a universal relation

𝑇 𝑥, 𝑡 = 𝑇0 +𝐴

1 + 𝜏exp −

𝑥2

4𝐷 1 + 𝜏

So the decay scales with 𝑡0

Scaling rule of the decay

Since the decay scales with 𝑡0, it scales with the 𝜎2.

𝑇 𝑥, 0 = 𝑇0 +𝐴

𝑡0exp(−

𝑥2

4𝐷𝑡0)

𝜎2 = 2𝐷𝑡0

𝑇

𝑥

2𝜎

Another view using analogy to circuitsThermal conduction is like discharging a capacitor, the time constant is

𝜏 = 𝑅𝐶

hot cold

For thermal conduction, we use the same formula𝜏 = 𝑅𝐶

𝑅 =𝑑

𝜎𝐴𝐶 = 𝑐𝑀𝜌𝑉 = 𝑐𝑀𝜌𝐴𝑑

𝜎: thermal conducitivity𝑑: thickness𝐴: area𝜌: density𝑐𝑀: specific heat

So

𝜏 = 𝑅𝐶 =𝑑

𝜎𝐴𝑐𝑀𝜌𝐴𝑑 =

𝑐𝑀𝜌

𝜎𝒅𝟐

𝑑

Again, this scales with width2.

Some hands-on data

A simple experiment

0.00 0.02 0.04 0.06 0.08

0.0

0.2

0.4

0.6

0.8

1.0

dT

norm

aliz

ed

dt (day)

dT more

dT less

Less

More

Time scale in thin films

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 320.0

0.5

1.0

dc/c

tdelay

LFO

YbFO

0.000 0.005

0.2

0.4

0.6

0.8

1.0

dc/c

tdelay

LFO

YbFO

BFO

t (ns) 𝑡/𝑑2 (ns/nm2)

For the previous water cooling experiment: 𝝉/𝒅𝟐 = 𝟎. 𝟎𝟎𝟑 (ns/nm2)𝜏 = 1080s𝑑 = 2 cm

Lase

rp

uls

e

FilmSubstrate

20 nm

50 nm

20 nm

50 nm

35 nm

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.