conductionXiaoshan XuThe importance of thermal conductioneEnergy consumptionetransfer through conductioneEnergy generationeThermoelectric effect Seebeckeffect eSpin SeebeckeffectHot reservoirCold rese ID: 891910
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1 Time scale of thermal conduction Xiaosh
Time scale of thermal conduction Xiaoshan Xu The importance of thermal conduction e Energy consumption: e transfer through conduction e Energy generation: e Thermoelectric effect ( Seebeck effect) e Spin Seebeck effect Hot reservoir Cold reservior Πܸ Basics of thermal conduction Heat Conductivity in Solids ( an example for irreversibility ) Re
2 member: Heat is an energy transferred fr
member: Heat is an energy transferred from one system to another because of temperature difference System ï 1 System ï 2 T 1 � T 2 Heat Q flows from ï 1 to ï 2 Example of irreversible process: Heat conduction Heat reservoir ï 1 Heat reservoir ï 2 T 1 � T 2 L T(x) x T 1 T 2 0 L * ( in the textbook T 2 �T 1 ) Heat
3 transfer per time interval through homog
transfer per time interval through homogeneous solid object: 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) T(x) x T 1 T 2 0 L Ü® â â Ý Ü¶ 2 â ܶ 1 â â ܶ ܵ Ý , Ý = â Ü
4 �ܶ Ý , Ý �Ý De
�ܶ Ý , Ý �Ý Define flux: ܵ = lim Î � â 0 ܳ � â Ý Fourierâs law Diffusion (time and space dependence of matter/energy) Ý ( Ý , Ý ) , Ü· A Ý Ý + â Ý Continuity equation: �Ý Ý , Ý �Ý = â �ܵ Ý , Ý �Ý Üµ ( Ý , Ý ) ܵ ( Ý + â Ý , Ý
5 ) � ܵ Ý + â Ý , Ý â Ü
) � ܵ Ý + â Ý , Ý â ܵ Ý , Ý â Ý = â [ Ý ( Ý , Ý + â Ý ) â Ý ( Ý , Ý ) ] � â Ý � : flux (energy flow per unit area per unit time, similar to electric current) � : energy density (energy per unit volume) How would energy change in the system? ܵ Ý + â Ý , Ý â ܵ Ý , Ý â
6 Ý = â Ý Ý , Ý + â Ý â Ý Ý
Ý = â Ý Ý , Ý + â Ý â Ý Ý , Ý â Ý Üµ Ý , Ý = â Ü �ܶ Ý , Ý �Ý �Ý Ý , Ý �Ý = â �ܵ Ý , Ý �Ý Ý ( Ý , Ý ) , Ü· A Ý Ý + â Ý Üµ ( Ý , Ý ) ܵ ( Ý + â Ý , Ý ) Flux across a boundary Energy change in a system Diffusion (time and space depende
7 nce of matter/energy) Diffusion (time an
nce of matter/energy) Diffusion (time and space dependence of matter/energy) Ý ( Ý , Ý ) , Ü· A Ý Ý + â Ý Üµ ( Ý , Ý ) ܵ ( Ý + â Ý , Ý ) � : flux (energy flow per unit area per unit time, similar to electric current) � : energy density (energy per unit volume) To study temperature distribution. Ü· = Ü· 0 + ඃ
8 5DF15;Ü· �ܶ � â
5DF15;Ü· �ܶ � â ܶ = Ü· 0 + ܯܿ � � â ܶ Ý = Ü· ( Ý ) ܸ = Ü· 0 ܸ + ܯܿ � � â ܶ ܸ = Ý 0 + � Ü¿ � � â ܶ �Ý Ý , Ý �Ý = â �ܵ Ý , Ý �Ý Ý â Ý 0 = â Ý = � Ü¿ �
9 � â ܶ �Ý Ý ,
� â ܶ �Ý Ý , Ý �Ý = � Ü¿ � � �ܶ ( Ý , Ý ) �Ý �ܵ Ý , Ý �Ý = â � Ü¿ � � �ܶ ( Ý , Ý ) �Ý Continuity equation: Diffusion (time and space dependence of matter/energy) Ý (
10 Ý , Ý ) , Ü· A Ý Ý + â Ý Üµ ( Ý
Ý , Ý ) , Ü· A Ý Ý + â Ý Üµ ( Ý , Ý ) ܵ ( Ý + â Ý , Ý ) � : flux (energy flow per unit area per unit time, similar to electric current) � : energy density (energy per unit volume) From continuity equation From Fourierâs law ܵ Ý , Ý = â Ü �ܶ Ý , Ý �Ý � ௠௫ , �
11 DC61; �௫ = â Ü �
DC61; �௫ = â Ü � 2 ௠௫ , � � ௫ 2 (2) Take � �௫ �ܶ ( Ý , Ý ) �Ý = ܦ � 2 ܶ Ý , Ý � Ý 2 �௠௫ , � �௫ = â � Ü¿ � � �௠( ௫ , �
12 ) �� (1) ܦ = Ü
) �� (1) ܦ = Ü � Ü¿ � � diffusivity Diffusion equation Application of Diffusion Equation (Decay of a hot spot) �ܶ ( Ý , Ý ) �Ý = ܦ � 2 ܶ Ý , Ý � Ý 2 Diffusion equation ܶ ( Ý , Ý ) ܶ Ý , Ý = ܶ 0 + � Ý + Ý 0 exp â Ý 2 4ܦ
13 Ý + Ý 0 ܶ Ý , 0 = ܶ 0 + �
Ý + Ý 0 ܶ Ý , 0 = ܶ 0 + � � 0 exp ( â ௫ 2 4� � 0 ) Gaussian peak. ܶ Ý One of the solution (example): Describes how the spatial distribution evolves with time. Application of Diffusion Equation (Decay of a hot spot) ܶ ( Ý , Ý ) ܶ = ܶ 0 + � ௪ exp ( â ௫ 2 ௪ 2 � ) ,
14 Ý Ý = 4ܦ ( Ý + Ý 0 ) Spatial depe
Ý Ý = 4ܦ ( Ý + Ý 0 ) Spatial dependence at Ý : ܶ Ý Ý 0 Ý 2 Ý 1 Ý 3 Spatial evolution with time: 1) Peak becomes broader 2) Total area is conserved (energy conservation ÝÜ· = Ýܶ ). �ܶ ( Ý , Ý ) �Ý = ܦ � 2 ܶ Ý , Ý � Ý 2 Scaling rule of the decay ܶ Ý , Ý = ܶ 0 + �
15 Ý + Ý 0 exp â Ý 2 4ܦ Ý + Ý 0 = Ü
Ý + Ý 0 exp â Ý 2 4ܦ Ý + Ý 0 = ܶ 0 + � Ý 0 1 + Ý Ý 0 exp â Ý 2 4ܦ Ý 0 1 + Ý Ý 0 If we use Ý 0 as the unit, and let � = � � 0 , we get a universal relation ܶ Ý , Ý = ܶ 0 + � 1 + � exp â Ý 2 4ܦ 1 + � So the decay scales with Ý 0 Scaling rule of
16 the decay Since the decay scales with
the decay Since the decay scales with Ý 0 , it scales with the � 2 . ܶ Ý , 0 = ܶ 0 + � � 0 exp ( â ௫ 2 4� � 0 ) � 2 = 2ܦ Ý 0 ܶ Ý 2� Another view using analogy to circuits Thermal conduction is like discharging a capacitor, the time constant is � = Ü
17 ´Ü¥ hot cold For thermal conduction, we
´Ü¥ hot cold For thermal conduction, we use the same formula � = Ü´Ü¥ Ü´ = Ý �� Ü¥ = Ü¿ � �ܸ = Ü¿ � � �Ý � : thermal conducitivity Ý : thickness � : area � : density Ü¿ � : specific heat So � = Ü´Ü¥ = Ý
18 �� ܿ �
�� Ü¿ � ��Ý = Ü¿ � � � � à« Ý Again, this scales with width 2 . Some hands - on data A simple experiment Less More Time scale in thin films t (ns) Ý / Ý 2 (ns/nm 2 ) For the previous water cooling experiment: � / � à« = Ø
19 35DFCE; . ��ૠ(n
35DFCE; . ��à« (ns/nm 2 ) � = 1080 s Ý = 2 cm Laser pulse Film S ubstrate 20 nm 50 nm 20 nm 50 nm 35 nm Conclusion e We discussed the scaling rule of the thermal conduction, which is proportional to the thickness^2. e A few examples (microscopic and macroscopic) are given, which can be unified using the scaling ru