Dynamic Van der Waals Theory

Transcription

1 Dynamic Van der Waal heory A diffue interface model for two-phae hydrodynamic involving the liquid-ga tranition in non-uniform temperature [A. Onuki, PRL (005) & PRE (007)] Hydrodynamic equation for liquid-ga flow in the bulk region [well developed] + Boundary condition at fluid-olid interface [J. Chem. Phy. (010) and ongoing]

2 Helmholtz free energy denity: f ( n, ) = nk ln( 3 ) ( ) B λthn 1 ln 1 vn 0 εvn 0 Internal energy denity, entropy per molecule, and preure e n, = 3 nk / εv n ( ) B 0 3 ( ) λ ( ) n, = kbln thn/ 1 v0n + 5 kb / p n, = nk/ 1 vn εvn ( ) ( ) B 0 0 Gradient contribution to the internal energy denity and entropy denity K ( n ) eˆ = e+ n ˆ (, ) ( ) C n S = n n e n Free energy minimization lead to the equilibrium tructure of a diffue liquid-ga interface. van der Waal Elaticity in one-component liquid-ga ytem, manifeted through a reverible tre tenor Π, which i aniotropic.

3 Balance equation for particle number, momentum, and energy n + ( nv) = 0 the continuity equation ( ρv) + ( ρvv) = M the momentum equation M Π+ σ i the total tre tenor. reverible irreverible (vicou) eˆ + ( eˆ v) = Π : v + σ : v q for the internal energy denity Ue of tandard thermodynamic relation Sˆ ( ˆ ) ˆ S Sv = J + : ( ˆ f σ v q + Π + pi + M n n) : v the balance equation for the denity of entropy ( ) ˆ S f M n n/ t+ n + / J v q the total entropy flux

4 he rate of entropy production in the bulk region σ = σ : v q + Π + pi+ M n n : v ( ) ( /3) q = λ heat flux ( ˆ ) he denity inhomogeneity doe not contribute to entropy production. Π = M n n pˆ I the reverible tre M nm ˆ n M p p n n n n Mn n and = + (, ) = ( ) + ( ) σ = 1 1 σ : v q he poitive definitene of M n K n C n can be enured by the contitutive relation σ = η v+ v + ζ η I v vicou tre

5 v γ = w γ v = v w lip τ τ τ A chematic illutration. of the fluxe into the urface region bounded by the cloed curve C Σ No adorption at the fluid-olid interface. he area denitie of urface energy and urface entropy are flunction of n, the boundary value of fluid denity. Surface tre tenor and urface heat flux are preent, but no urface vicoity.

6 he general boundary (jump) condition in differential form (a pecial cae of the extended Kotchine theorem) φ + ( ) ˆ τ φ τ = τ + + π with γˆ J γˆ J Jw v J γ J ( ) Here the prime denote the urface quantitie whoe dimenion are different from the correponding bulk quantitie. Surface entropy and urface energy S daσ ( n) E = dae n = ( ) Helmholtz free energy per unit area f ( n, ) = e ( n) σ ( n) 1 1 f dσ = de dn n A Gibb-type equation γ f = f the fluid-olid interfacial tenion Surface tre tenor i tangential and ymmetric: M γ fτ = f τ τ I γγ ˆˆ

7 Balance equation at the fluid-olid interface Force balance Energy balance e f ˆ τ γ M+ F = 0 f ˆ τ γ M τ + Fτ = 0 γˆ M γˆ + Fγ = 0 hree force by the interface, fluid, and wall lip + e v = f v q + γˆ q γˆ M v w f ( ) ( ) τ τ τ τ τ τ τ τ v = v w lip τ τ τ Firt law of thermodynamic applied at the fluid-olid interface Entropy balance from σ 1 e 1 f n = n tangential lip velocity

8 Entropy balance σ q ( ) ( ˆ ˆ S ˆ ˆ S + σ v = + ) + σ γ J γ J τ τ τ w urf ( ) ˆ ˆ S γ J γˆ q + Mn γ n / ˆ ˆ S γ J γˆ q / w w w entropy fluxe σ lip 1 q ˆ τ γ q Fτ vτ Ln w urf w entropy production γ ( / ) L= M n+ f n i equal to zero in equilibrium. Interfacial contitutive relation 1 1 q = λ τ χf τ κ γˆ qw = w lip χ βvτ = β τ Fτ α n = L

9 Balance equation (conervation law) Contitutive relation } Hydrodynamic equation From the bulk region to the interface Balance equation Contitutive relation } Hydrodynamic boundary condition A continuum hydrodynamic model formed by differential equation and boundary condition.

10 Hydrodynamic Boundary Condition Denity relaxation n f α + vτ τn = M γn n Impermeability Velocity lip γˆ v = γˆ w= 0 lip f χ βv = η v + M n n σ β τ γ τ + γ + n τ τ emperature lip Heat fluxe 1 1 κλw γw = w e λ γ + λw γw = + τ ( σ vτ ) + vτ τ f 1 χ τ ( λ τ) Fτ Fτ τ ( χfτ ) Fτ τ β F v σ ( / ) = η M n n f τ γ τ γ τ τ f = + f n n τ τ τ

11 A limiting cae Xu and Qian, J. Chem. Phy. 133, (010) No cro coupling: χ 0 Contant temperature at the fluid-olid interface: κ 0 and fat relaxation toward thermal equilibrium in the olid = w = cont. Denity relaxation n f α + vτ τn = M γn n Impermeability Velocity lip γˆ v = γˆ w= 0 βv = η v f + M n+ n lip τ γ τ γ τ n Dirichlet temperature condition = w = cont

12 =0.875c =0.99c