Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt Professor and Chair Engineering Thermodynamics Delft University of Technology, The Netherlands t.j.h.vlugt@tudelft.nl, http://homepage.tudelft.nl/v9k6y August 13, 2012 Fluid Phase Equilibria, 2011, 301, 110-117. Chem. Phys. Lett., 2011, 504, 199-201. J. Phys. Chem B, 2011, 115, 8506-8517. J. Phys. Chem B, 2011, 115, 10911-10918. J. Phys. Chem B, 2011, 115, 12921-12929. Ind. Eng. Chem. Res., 2011, 50, 4776-4782. Ind. Eng. Chem. Res., 2011, 50, 10350-10358.
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [1] Collaborators • Xin Liu (RWTH Aachen University, Delft) • Sondre K. Schnell (Delft) • Andr´ e Bardow (RWTH Aachen University, Delft) • Signe Kjelstrup (NTNU, Delft) • Dick Bedeaux (NTNU, Delft) • Jean-Marc Simon (Dijon)
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [2] (Empirical) Fick formulation for multicomponent diffusion n − 1 � J i = − c t D ik ∇ x k k =1 • n components, ( n − 1) 2 Fick diffusivities • � n i =1 J i = 0 (molar reference frame) • D ij � = D ji for n > 2 • D ij strongly dependent on the mole fractions x 1 · · · x n • D ij can become negative for n > 2 • multicomponent D ij unrelated to binary counterpart • R. Krishna and J.A. Wesselingh, Chem. Eng. Sci., 1997, 52, 861-911.
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [3] Maxwell-Stefan formulation at constant T, p n n x i ∇ µ i x i x j ( u j − u i ) x i J j − x j J i � � d i = = = RT D ij c t ¯ D ij ¯ j =1 ,j � = i j =1 ,j � = i J i = u i c i = u i x i c t n � x i ∇ µ i = 0 i =1 • Gradient in chemical potential as driving force d i • n components, n ( n − 1) / 2 MS diffusivities, ¯ D ij > 0 • ¯ D ij = ¯ D ji (Onsager’s reciprocal relations) • ¯ D ij is less composition dependent than Fick diffusivities • Possibility to predict ¯ D ij using theory...
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [4] Diffusion coefficients depend on concentration! (a) 10 C2mimCl-H2O H 2 O-C 2 mimCl C4mimCl-H2O H 2 O-C 4 mimCl C8mimCl-H2O H 2 O-C 8 mimCl -1 ) 1 2 s -9 m Đ IL /(10 0.1 0.01 0.0 0.2 0.4 0.6 0.8 1.0 x CnmimCl
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [5] Vignes equation in ternary systems x j ( ¯ x i ( ¯ x k x j → 1 D x i → 1 D x k → 1 D ij ≈ ( ¯ ) D ) ) ¯ ij ij ij • Recommended in chemical engineering • Is it any good? D x k → 1 • What about ¯ ? Models for this in literature: ij � x j → 1 D x k → 1 D x i → 1 WK (1990) = D ¯ ¯ ¯ ij ij ij � D x k → 1 D x k → 1 D x k → 1 KT (1991) = ¯ ¯ ¯ ij ik jk xj xi � � xi + xj � � xi + xj D x k → 1 D x k → 1 D x k → 1 VKB (2005) = ¯ ¯ ¯ ij ik jk x j x i D x k → 1 D x k → 1 D x k → 1 DKB (2005) = + ¯ x i + x j ¯ x i + x j ¯ ij ik jk x j → 1 D x k → 1 D x k → 1 D x k → 1 D x i → 1 ) 1 / 4 RS (2007) = ( ¯ D ¯ ¯ ¯ ¯ ij ik jk ij ij
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [6] (Open?) Questions • Is the quality of Vignes increased/decreased by a particular model choice for D x k → 1 ? ¯ ij D x k → 1 • Which of the predictive models for ¯ is best for a certain system? ij (if any...) D x k → 1 • VKB and DKB suggest that ¯ does not exist, i.e. its value depends on ij the ratio x i /x j . Is this correct? � xj → 1 xk → 1 xi → 1 WK (1990) D = D D ¯ ¯ ¯ ij ij ij � xk → 1 xk → 1 xk → 1 KT (1991) D = D D ¯ ¯ ¯ ij ik jk xj xi � � xi + xj � � xi + xj xk → 1 xk → 1 xk → 1 VKB (2005) D = D D ¯ ¯ ¯ ij ik jk x j x i xk → 1 xk → 1 xk → 1 DKB (2005) D = D + D ¯ x i + x j ¯ x i + x j ¯ ij ik jk xj → 1 xk → 1 xk → 1 xk → 1 xi → 1 ) 1 / 4 RS (2007) D = ( ¯ D D D D ¯ ¯ ¯ ¯ ij ij ij ik jk
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [7] Obtaining Maxwell-Stefan diffusivities from MD � N i 1 1 � × Λ ij = 6 N lim ( r l,i ( t + m · ∆ t ) − r l,i ( t )) m · ∆ t m →∞ l =1 N j � � ( r k,j ( t + m · ∆ t ) − r k,j ( t )) k =1 � N i N j � ∞ � 1 � � = dt v l,i (0) · v k,j ( t ) 3 N 0 l =1 k =1 Note: Λ ij = Λ ji (Onsager) Krishna & Van Baten, Ind. Eng. Chem. Res., 2005, 44, 6939-6947.
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [8] Obtaining ternary Maxwell-Stefan diffusivities from MD A + ( x 1 + x 3) B C D 12 = ¯ x 1 D − Λ11Λ23 x 2 − Λ12 x 3Λ21 + Λ11 x 3Λ22 − Λ11 x 3 x 2Λ22 + Λ11Λ23 x 2 A = 2 − Λ11 x 3 x 2Λ32 +Λ11Λ33 x 2 2 − Λ13 x 1Λ22 − x 1 x 3Λ11Λ22 + Λ13Λ22 x 2 1 − x 1Λ31 x 3Λ22 + Λ33 x 2 1Λ22 +Λ12Λ23 x 1 + Λ12 x 3 x 2Λ21 + Λ13 x 1 x 2Λ22 + Λ12 x 3Λ31 x 2 + Λ13 x 2Λ21 − Λ13 x 2 2Λ21 − Λ13 x 2 2Λ31 + x 1Λ12 x 3Λ21 − x 2 1Λ12Λ23 + x 1Λ32 x 3Λ21 − x 2 1Λ32Λ23 + Λ13 x 1 x 2Λ32 + x 1Λ11Λ23 x 2 + x 1Λ31Λ23 x 2 − Λ12 x 1Λ23 x 2 − Λ12 x 1Λ33 x 2 − x 1Λ13 x 2Λ21 − x 1Λ33 x 2Λ21 B = Λ12 x 3 − Λ13 x 2 − x 1 x 3Λ12 + x 1 x 2Λ23 − x 1 x 3Λ32 + x 1 x 2Λ33 − Λ11Λ23 x 2 − Λ12Λ21 x 3 + x 3Λ11Λ22 − x 2 x 3Λ11Λ22 + Λ11Λ23 x 2 C = 2 − x 2 x 3Λ11Λ32 +Λ11Λ33 x 2 2 − Λ13Λ22 x 1 − x 1 x 3Λ11Λ22 + x 2 1Λ13Λ22 − x 1 x 3Λ31Λ22 + x 2 1Λ22Λ33 + x 1Λ12Λ23 + x 2 x 3Λ12Λ21 + x 1 x 2Λ13Λ22 + x 2 x 3Λ12Λ31 + x 2Λ13Λ21 − x 2 2Λ13Λ21 − x 2 2Λ13Λ31 + x 1 x 3Λ12Λ21 − x 2 1Λ12Λ23 + x 1 x 3Λ32Λ21 − x 2 1Λ32Λ23 + x 1 x 2Λ13Λ32 + x 1 x 2Λ11Λ23 + x 1 x 2Λ31Λ23 − x 1 x 2Λ12Λ23 − x 1 x 2Λ12Λ33 − x 1 x 2Λ13Λ21 − x 1 x 2Λ33Λ21 Λ22 x 3 − Λ23 x 2 − Λ22 x 3 x 2 + Λ23 x 3 2 − x 2Λ12 x 3 + Λ13 x 2 2 − x 2Λ32 x 3 + Λ33 x 2 D = 2
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [9] Limit when x k → 1 (1) � Ni � ∞ Ni � 1 � � Λ ii = dt v l,i (0) · v g,i ( t ) 3 N 0 g =1 l =1 � ∞ N i ≈ dt � v i, 1 (0) · v i, 1 ( t ) � 3 N 0 = x i C ii � Nk Nk � ∞ � 1 � � Λ kk = dt v l,k (0) · v g,k ( t ) 3 N 0 l =1 g =1 � Nk � Nk Nk � ∞ � ∞ � � 1 1 � � � = dt v l,k (0) · v l,k ( t ) + dt v l,k (0) · v g,k ( t ) 3 N 3 N 0 0 l =1 l =1 g =1 ,g � = l x k C kk + x 2 k NC ⋆ ≈ kk � Ni Nj � ∞ � 1 � � Λ ij,j � = i = dt v l,i (0) · v k,j ( t ) 3 N 0 l =1 k =1 � ∞ N i N j ≈ dt � v 1 ,i (0) · v 1 ,j ( t ) � 3 N 0 = Nx i x j C ij
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [10] Limit when x k → 1 (2) Set a = x j /x i and x k = 1 − x i − x j , fill in the equations for Λ ii , Λ jj , Λ kk , Λ ij , Λ ik , Λ jk and take the limit x k → 1 Final result (after a lot of math):
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [11] Limit when x k → 1 (2) Set a = x j /x i and x k = 1 − x i − x j , fill in the equations for Λ ii , Λ jj , Λ kk , Λ ij , Λ ik , Λ jk and take the limit x k → 1 Final result (after a lot of math): D x k → 1 i, self · D x k → 1 D x k → 1 j, self = ¯ ij D x k → 1 k, self + C x N ( C ij − C ik − C jk + C ⋆ C x = kk ) which is independent of x j /x i !! (Note: C x converges to a finite value when N → ∞ ). When cross-correlation are neglected: D x k → 1 i, self · D x k → 1 D x k → 1 j, self ≈ ¯ ij D x k → 1 k, self
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [12] Ternary system of WCA particles that only differ in mass D x 3 → 1 MS diffusivity ¯ 12 incl. C x C x = 0 WK KT VKB DKB RS Prediction 1.401 1.441 1.296 0.952 0.952 0.952 1.111 MD b 1.411 1.411 1.411 1.411 1.411 1.411 1.411 AD a 1% 2% 8% 32% 32% 32% 21% Prediction 0.310 0.315 0.390 0.248 0.248 0.248 0.311 MD c 0.318 0.318 0.318 0.318 0.318 0.318 0.318 AD a 2% 1% 23% 22% 22% 22% 2% Prediction 3.344 3.288 1.296 0.682 0.682 0.683 0.940 MD d 3.348 3.348 3.348 3.348 3.348 3.348 3.348 AD a 0% 2% 61% 80% 80% 80% 72% Prediction 0.172 0.161 0.389 0.101 0.101 0.101 0.198 MD e 0.172 0.172 0.172 0.172 0.172 0.172 0.172 AD a 0% 7% 126% 42% 42% 42% 15% a absolute difference normalized with corresponding value from MD simulations b ρ = 0.2; M 1 = 1; M 2 = 1.5; M 3 = 5; x 1 /x 2 = 1; x 3 = 0.95; T = 2 c ρ = 0.5; M 1 = 1; M 2 = 1.5; M 3 = 5; x 1 /x 2 = 1; x 3 = 0.95; T = 2 d ρ = 0.2; M 1 = 1; M 2 = 1.5; M 3 = 100; x 1 /x 2 = 1; x 3 = 0.95; T = 2 e ρ = 0.5; M 1 = 1; M 2 = 1.5; M 3 = 100; x 1 /x 2 = 1; x 3 = 0.95; T = 2
Predicting Fick- and Maxwell-Stefan diffusivities in liquids Thijs J.H. Vlugt [13] Increasing the mass of the solvent ( M 3 ) 1000 ρ = 0.1 rho = 0.1 ρ = 0.2 rho = 0.2 ρ = 0.3 rho = 0.3 ρ = 0.4 rho = 0.4 100 ρ = 0.5 rho = 0.5 Đ 12 10 1 0.1 1 10 100 1000 10000 M 3 xk → 1 xk → 1 D i, self · D xk → 1 j, self D ≈ ¯ ij xk → 1 D k, self
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