Post on 03-Jan-2020
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.