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# lecture22 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 22 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 22 We have talked about the heat flux as 1 x y z x x k k k q fv V ω = ∑∑∑ = . The Boltzmann equation is 0 0 0 r r v F v g g v f f S m τ τ ⋅∇ + = − ⋅∇ + ⋅∇ = G G G JG G G , where 0 0 v F f m ⋅∇ = G JG for phonons, g=f-f 0 . Noticing y r g v g v y ⋅∇ G G (d<<x) and 0 0 r df dT f dT dx = G , the x direction component gives ( ) 0 0 y x df g dT g v v S x y dT dx τ τ + = − = , the solution of which is 0 exp y y g S C v τ = . One boundary condition is required to determine C. Assuming both top and bottom of the film diffusely scatter phonons, we have 0 0 0, , 0 for 0, 2 , , 0 for , 2 y f f g y d f f g π θ π θ π = = = = = = . Finally we get At y = 0, 0, 2 π θ , 0 C = -S , ( ) 0 , 1 exp cos y g y S v θ τ θ = , y d x θ v Temperature Gradient Or Electrical Field v x v y ϕ v v x v y θ v z (a) (b)

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2.57 Fall 2004 – Lecture 22 2 At y = d, , 2 π θ π , ( ) 0 C = -S exp / cos d v τ θ , ( ) 0 , 1 exp cos d y g y S v θ τ θ = .
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