Continuum thermodynamics of multicomponent fluids and implications for modeling electromigration of ionic species

Transcription

1 Continuum thermodynamics of multicomponent fluids and implications for modeling electromigration of ionic species Dieter Bothe Center of Smart Interfaces Technical University Darmstadt, Germany Workshop on Transport of Ionic Particles in Biological Environments Fields Institute, Toronto, July 28 - August 1, 2014 Dieter Bothe reactive multicomponent fluid systems

2 Contents 1 Motivation 2 T.I.P. Fickean vs. Maxwell-Stefan form 3 Partial Balances and Constitutive Modeling Framework 4 Closure for Non-Reactive Multicomponent Fluids 5 The Maxwell-Stefan Equations and Electromigration 6 The Nernst-Planck-Poisson System in nd

3 Motivation 1: Ion Exchange Processes Regeneration of exhausted (Cu 2+ -loaded) ion exchange pellets

4 Intro TIP Balances Nonreactive MS-Eqs NPP Regeneration of Single Pellets experiment: evolution of regeneration front radius versus time hindered diffusion inside the pellet (resin) radius evolution of single pellet can be describe with/without electrical forces hindrance factors: for Fick, 0.14 for Nernst-Planck fluxes corresponding H+ -diffusivities: m2 s 1 (Fick), m2 s 1 (Nernst-Planck)

5 Regeneration of Single Pellets Simulated concentration profiles inside the pellet: profiles: without electrical forces with electrical forces

6 CSTR Bulk Concentration Dynamics Bulk concentrations during regeneration process: evolution of the bulk concentration in the CSTR

7 Motivation 2: Mass Transfer in G/L-Systems Dissolution of CO 2 bubbles

8 Chemisorption of CO 2 in NaOH solution Simplified situation near interface: Preliminary computations: mass transfer results w/o electromigration for CO 2 in acidic solutions can be 10-20% off!

9 Chemisorption of CO 2 in NaOH solution Simulated concentration profiles:

10 DNS of Mass Transfer with Volume Effects Simulated dissolution of a CO 2 Taylor bubble

11 DNS of Conjugated Mass Transfer Simulated mass transfer at a free CO 2 bubble Dissolved air (N 2, O 2 ) from aqueous phase is transfers into the bubble

12 Motivation 3: Cross Diffusion Effects Classical experiment by Duncan and Toor 1962 on ternary diffusion initial composition: left bulb N 2 : CO 2 (1:1), right bulb N 2 : H 2 (1:1)

13 Anomalous Diffusion Typical phenomena in ternary systems

14 1 Motivation 2 T.I.P. Fickean vs. Maxwell-Stefan form 3 Partial Balances and Constitutive Modeling Framework 4 Closure for Non-Reactive Multicomponent Fluids 5 The Maxwell-Stefan Equations and Electromigration 6 The Nernst-Planck-Poisson System in nd

15 Chemically Reacting Fluid Mixture Fluid composed of N chemically reacting components A 1,..., A N N R chemical reactions between the A i : α a 1 A α a N A N β a 1 A β a N A N for a = 1,..., N R with stoichiometric coefficients α a i, βa i IN 0 Let R a = Ra f Ra b be the (molar) rate of reaction a and νi a := βi a αi a. Then N R r i = M i νi a R a with M i the molar mass of species A i a=1 is the total rate of change of mass of component A i Mass conservation in individual reactions: i M iνi a = 0 a

16 Chemically Reacting Fluid Mixture Fluid composed of N chemically reacting components A 1,..., A N N R chemical reactions between the A i : α a 1 A α a N A N β a 1 A β a N A N for a = 1,..., N R with stoichiometric coefficients α a i, βa i IN 0 Let R a = Ra f Ra b be the (molar) rate of reaction a and νi a := βi a αi a. Then N R r i = M i νi a R a with M i the molar mass of species A i a=1 is the total rate of change of mass of component A i Mass conservation in individual reactions: i M iνi a = 0 a

17 Thermodynamics of Irreversible Processes (TIP) Throughout this talk: v denotes the barycentric velocity of the mixture Classical mixture balances in T.I.P. (cf. degroot, Mazur): partial mass balances: t ϱ i + div (ϱ i v + j i ) = r i i j i = 0 t ϱ + div (ϱv) = 0 total momentum balance: t (ϱv) + div (ϱv v S) = ϱb; ϱb = i ϱ ib i internal energy balance: t (ϱe) + div (ϱev + q) = v : S + ϱπ; ϱπ = i j i b i Definition of internal energy: ϱe = ϱe tot 1 2 ϱv2

18 Thermodynamics of Irreversible Processes (TIP) Throughout this talk: v denotes the barycentric velocity of the mixture Classical mixture balances in T.I.P. (cf. degroot, Mazur): partial mass balances: t ϱ i + div (ϱ i v + j i ) = r i i j i = 0 t ϱ + div (ϱv) = 0 total momentum balance: t (ϱv) + div (ϱv v S) = ϱb; ϱb = i ϱ ib i internal energy balance: t (ϱe) + div (ϱev + q) = v : S + ϱπ; ϱπ = i j i b i Definition of internal energy: ϱe = ϱe tot 1 2 ϱv2

19 The 2 nd Law: Entropy Inequality Entropy production: ζ TIP = q 1 T + N i=1 j i ( µ i T b i ) 1 + T T Sirr : D 1 T N R a=1 R a A a S = pi + S irr, S irr : D = S : D + Π div v Notation: T denotes the (absolute) temperature µ i denotes the chemical potentials S denotes the traceless part of S D denotes the symmetric, traceless part of v Π denotes the dynamic pressure (or, irreversible pressure part) A a := i µ im i νi a are the chemical affinities.

20 The Phenomenological Equations Standard closure: fluxes linear in the (so-called) driving forces quadratic form heat flux and diffusive fluxes: q = L 00 1 T ( ) N 1 i=1 L 0i µ i µ N T 1 T (b i b N ) j i = L i0 1 T ( ) N 1 j=1 L ij µ j µ N T 1 T (b j b N ) viscous stress, dynamic pressure and chemical reaction rates: i = 1,.., N 1 S = L D, Π = l div v a l 0aA a, R a = l a0 div v b l aba b Entropy inequality: [L ij ] and [l ab ] positive semi-definite and L 0 Onsager-Casimir reciprocal relations: l 0a = l a0, [L ij ], [l ab ] symmetric

21 The Phenomenological Equations Standard closure: fluxes linear in the (so-called) driving forces quadratic form heat flux and diffusive fluxes: q = L 00 1 T ( ) N 1 i=1 L 0i µ i µ N T 1 T (b i b N ) j i = L i0 1 T ( ) N 1 j=1 L ij µ j µ N T 1 T (b j b N ) viscous stress, dynamic pressure and chemical reaction rates: i = 1,.., N 1 S = L D, Π = l div v a l 0aA a, R a = l a0 div v b l aba b Entropy inequality: [L ij ] and [l ab ] positive semi-definite and L 0 Onsager-Casimir reciprocal relations: l 0a = l a0, [L ij ], [l ab ] symmetric

22 The Phenomenological Equations Standard closure: fluxes linear in the (so-called) driving forces quadratic form heat flux and diffusive fluxes: q = L 00 1 T ( ) N 1 i=1 L 0i µ i µ N T 1 T (b i b N ) j i = L i0 1 T ( ) N 1 j=1 L ij µ j µ N T 1 T (b j b N ) viscous stress, dynamic pressure and chemical reaction rates: i = 1,.., N 1 S = L D, Π = l div v a l 0aA a, R a = l a0 div v b l aba b Entropy inequality: [L ij ] and [l ab ] positive semi-definite and L 0 Onsager-Casimir reciprocal relations: [L ij ], [l ab ] symmetric, but l 0a = l a0

23 Remarks on Classical TIP Curie s principle: driving forces couple only to fluxes of the same tensorial rank is a rigorous consequence of material frame indifference for linear constitutive relations Onsager s reciprocal relations: [L ij ] and [l ab ] are symmetric relies on microscopic theory; only derived for rates (ODE case), not for transport coefficients some couplings are anti-symmetric: Onsager-Casimir relations

24 Remarks on Classical TIP Curie s principle: driving forces couple only to fluxes of the same tensorial rank is a rigorous consequence of material frame indifference for linear constitutive relations Onsager s reciprocal relations: [L ij ] and [l ab ] are symmetric relies on microscopic theory; only derived for rates (ODE case), not for transport coefficients some couplings are anti-symmetric: Onsager-Casimir relations Some disadvantages of classical TIP / Fickean form: - the L ij show complex nonlinear dependence on the composition - the linear closure for chemical reaction rates is not appropriate - in recent applications different species can experience different BCs

25 Remarks on Classical TIP Curie s principle: driving forces couple only to fluxes of the same tensorial rank is a rigorous consequence of material frame indifference for linear constitutive relations Onsager s reciprocal relations: [L ij ] and [l ab ] are symmetric relies on microscopic theory; only derived for rates (ODE case), not for transport coefficients some couplings are anti-symmetric: Onsager-Casimir relations Some disadvantages of classical TIP / Fickean form: - the L ij show complex nonlinear dependence on the composition - the linear closure for chemical reaction rates is not appropriate - in recent applications different species can experience different BCs

26 The Maxwell-Stefan Equations Alternative approach to multicomponent diffusion: assume local balance between driving and friction forces: d i = j i f ij x i x j (v i v j ) = j i x j J i x i J j c D ij d i the thermodynamic driving forces, d i = x i RT pµmol i + φ i y i ϱrt p y i ϱrt (b i b) c = i c i total concentration, x i = c i /c molar fractions, J i = j i /M i molar mass fluxes; D ij = 1/f ij the Maxwell-Stefan diffusivities; in many cases: D ij nearly constant or affine functions of the composition Origin of the Maxwell-Stefan Equations: James Clerk Maxwell: On the dynamical theory of gases, Phil. Trans. R. Soc. 157, (1866). Josef Stefan: Über das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen, Sitzber. Akad. Wiss. Wien 63, (1871).

27 Maxwell-Stefan Equations - Criticism Problems and open issues: rigorous derivation of the Maxwell-Stefan equations, including the thermodynamic driving forces proper coupling to the mass and momentum balance extension to non-simple fluid mixtures extension to chemically reacting fluid mixtures Aim: thermodynamically consistent mathematical modeling of reacting fluid mixtures, guided by rational thermodynamics joint work with Wolfgang Dreyer (WIAS, Berlin) Preprint arxiv: v2 [physics.flu-dyn]

28 Maxwell-Stefan Equations - Derivation Four different derivations of the Maxwell-Stefan equations: I. Employing only the barycentric momentum balance: naive balance of forces ad hoc; mixes continuum balances with kinetic theory standard T.I.P. with resistance form of the closure consistency with kinetic theories only achievable via thermo-diffusive terms II. Employing partial momentum balances: diffusive approximation using time-scale separation not applicable with chemical reactions entropy invariant model reduction

29 Maxwell-Stefan Equations - Derivation Four different derivations of the Maxwell-Stefan equations: I. Employing only the barycentric momentum balance: naive balance of forces ad hoc; mixes continuum balances with kinetic theory standard T.I.P. with resistance form of the closure consistency with kinetic theories only achievable via thermo-diffusive terms II. Employing partial momentum balances: diffusive approximation using time-scale separation not applicable with chemical reactions entropy invariant model reduction

30 Maxwell-Stefan Equations - Derivation Four different derivations of the Maxwell-Stefan equations: I. Employing only the barycentric momentum balance: naive balance of forces ad hoc; mixes continuum balances with kinetic theory standard T.I.P. with resistance form of the closure consistency with kinetic theories only achievable via thermo-diffusive terms II. Employing partial momentum balances: diffusive approximation using time-scale separation not applicable with chemical reactions entropy invariant model reduction

31 1 Motivation 2 T.I.P. Fickean vs. Maxwell-Stefan form 3 Partial Balances and Constitutive Modeling Framework 4 Closure for Non-Reactive Multicomponent Fluids 5 The Maxwell-Stefan Equations and Electromigration 6 The Nernst-Planck-Poisson System in nd

32 Partial Balances of Mass, Momentum and Energy Continuum mechanical balances of the fluid components A i mass : t ϱ i + div (ϱ i v i ) = r i mom. : t (ϱ i v i ) + div (ϱ i v i v i S i ) = f i + ϱ i b i energy : t (ϱ i e i + ϱ i 2 v2 i ) + div ((ϱ i e i + ϱ i 2 v2 i )v i v i S i + q i ) = l i + ϱ i b i v i mass conservation: i r i = 0 momentum conservation: i f i = 0 energy conservation: i l i = 0 Note: power due to external forces is ϱ i b i v i, while internal forces (mechanical and chemical interactions) contribute to the l i

33 Partial Balances of Mass, Momentum and Energy Continuum mechanical balances of the fluid components A i mass : t ϱ i + div (ϱ i v i ) = r i mom. : t (ϱ i v i ) + div (ϱ i v i v i S i ) = f i + ϱ i b i energy : t (ϱ i e i + ϱ i 2 v2 i ) + div ((ϱ i e i + ϱ i 2 v2 i )v i v i S i + q i ) = l i + ϱ i b i v i mass conservation: i r i = 0 momentum conservation: i f i = 0 energy conservation: i l i = 0 Note: power due to external forces is ϱ i b i v i, while internal forces (mechanical and chemical interactions) contribute to the l i

34 Balance of internal energy Partial balance of internal energy: t (ϱ i e i ) + div (ϱ i e i v i + q i ) = v i : S i + l i v i (f i 1 2 r iv i ) ) Alternative definition of internal energy: ϱe := i ϱ ie i total internal energy = ϱ(e tot 1 2 v2 ) i P i := 1 3 tr(s i) partial pressures, P i = p i + Π i with Π i E = ϱ iu 2 i q := i (q i + (ϱ i e i + p i )u i ) mixture heat flux, p := i p i mixture internal energy balance: t (ϱe) + div (ϱev + q) = p div v + i v i : S i i u i (f i r i v i r iu i p i )

35 Balance of internal energy Partial balance of internal energy: t (ϱ i e i ) + div (ϱ i e i v i + q i ) = v i : S i + l i v i (f i 1 2 r iv i ) ) Alternative definition of internal energy: ϱe := i ϱ ie i total internal energy = ϱ(e tot 1 2 v2 ) i P i := 1 3 tr(s i) partial pressures, P i = p i + Π i with Π i E = ϱ iu 2 i q := i (q i + (ϱ i e i + p i )u i ) mixture heat flux, p := i p i mixture internal energy balance: t (ϱe) + div (ϱev + q) = p div v + i v i : S i i u i (f i r i v i r iu i p i )

36 Constitutive Modeling Variables: ϱ 1,..., ϱ N, v 1,..., v N, ϱe class-ii model requires constitutive equations for: R a, S i, f i r i v i, q Decompose the partial stresses as S i = P i I + S i = p i I + S irr i We consider non-polar fluids, hence the stresses S i are symmetric. Universal Principles: 1 material frame indifference 2 entropy principle (second law of thermodynamics) Notation: any (local) solution of the PDE-system is called thermodynamic process

37 Constitutive Modeling Variables: ϱ 1,..., ϱ N, v 1,..., v N, ϱe class-ii model requires constitutive equations for: R a, S i, f i r i v i, q Decompose the partial stresses as S i = P i I + S i = p i I + S irr i We consider non-polar fluids, hence the stresses S i are symmetric. Universal Principles: 1 material frame indifference 2 entropy principle (second law of thermodynamics) Notation: any (local) solution of the PDE-system is called thermodynamic process

38 The Entropy Principle The entropy principle comprises the following postulates: 1) There is an entropy/entropy-flux pair (ϱs, Φ) as a material dependent quantity, satisfying the principle of material frame indifference (ϱs is an objective scalar, Φ is an objective vector). 2) The pair (ϱs, Φ) satisfies the balance equation t (ϱs) + div (ϱsv + Φ) = ζ, where the entropy production ζ satisfies ζ 0 for every thermodynamic process. Equilibria are characterized by ζ = 0.

39 The Entropy Principle The entropy principle comprises the following postulates: 1) There is an entropy/entropy-flux pair (ϱs, Φ) as a material dependent quantity, satisfying the principle of material frame indifference (ϱs is an objective scalar, Φ is an objective vector). 2) The pair (ϱs, Φ) satisfies the balance equation t (ϱs) + div (ϱsv + Φ) = ζ, where the entropy production ζ satisfies ζ 0 for every thermodynamic process. Equilibria are characterized by ζ = 0.

40 The Entropy Principle 3) Every admissible entropy flux is such that the entropy production becomes a sum of binary products according to ζ = m N m P m, where N m, P m denote factors of negative, resp. positive parity. The parity of a time-dependent quantity characterizes its behavior under time reversal in the unclosed balance equations. Resulting rule: [ ] contains the time units s k with k even positive parity [ ] contains the time units s k with k odd negative parity Note: parity replaces the classical concept of flux driving force, since the latter is misleading!

41 The Entropy Principle 3) Every admissible entropy flux is such that the entropy production becomes a sum of binary products according to ζ = m N m P m, where N m, P m denote factors of negative, resp. positive parity. The parity of a time-dependent quantity characterizes its behavior under time reversal in the unclosed balance equations. Resulting rule: [ ] contains the time units s k with k even positive parity [ ] contains the time units s k with k odd negative parity Note: parity replaces the classical concept of flux driving force, since the latter is misleading!

42 The Entropy Principle 4) Each binary product in the entropy production describes a dissipative mechanism which has to be introduced in advance. Extended principle of detailed balance: N m P m 0 for every m and any thermodynamic process. Note: the specific form of the decomposition into binary products is not unique and has to be chosen as part of the modeling. Even the number of dissipative mechanisms is not fixed, but can be changed. This non-uniqueness is the basis for: introduction of cross-effects via entropy-neutral mixing improvement of classical TIP-models

43 The Entropy Principle 4) Each binary product in the entropy production describes a dissipative mechanism which has to be introduced in advance. Extended principle of detailed balance: N m P m 0 for every m and any thermodynamic process. Note: the specific form of the decomposition into binary products is not unique and has to be chosen as part of the modeling. Even the number of dissipative mechanisms is not fixed, but can be changed. This non-uniqueness is the basis for: introduction of cross-effects via entropy-neutral mixing improvement of classical TIP-models

44 Consequences of the extended principle of detailed balance the closure between the co-factors in the entropy production, ζ = m N mp m =: N, P, decouples. In a linear (in N m, P m ) constitutive theory, this enforces a block-diagonal closure. Note: N mp m refers to a single mechanism, but may itself be a sum. structure of entropy production as ζ = m N mp m is not unique In particular: mixing of different fluxes or forces is possible! ζ = A N, B P = N, A T B P = N, P N, P A 1 = B T the diagonal closure implies cross-effects with Onsager symmetry: A N := ΛB P with Λ = diag(λ i ) 0 N := B T ΛB P

45 Consequences of the extended principle of detailed balance the closure between the co-factors in the entropy production, ζ = m N mp m =: N, P, decouples. In a linear (in N m, P m ) constitutive theory, this enforces a block-diagonal closure. Note: N mp m refers to a single mechanism, but may itself be a sum. structure of entropy production as ζ = m N mp m is not unique In particular: mixing of different fluxes or forces is possible! ζ = A N, B P = N, A T B P = N, P N, P A 1 = B T the diagonal closure implies cross-effects with Onsager symmetry: A N := ΛB P with Λ = diag(λ i ) 0 N := B T ΛB P

46 Example: Entropy Inequality of T.I.P. the strengthened entropy principle can be used in T.I.P.! ζ TIP = q 1 T + N i=1 j i ( µ i T b ) i T + 1 T S : D 1 T Π div v 1 NR T a=1 R aa a Consider a coupling between volume variations and chemical reactions Parity of the factors Π, div v, R a, A a : +1, -1, -1, +1 Cross-effects via entropy neutral mixing: div v Π + N R a=1 R aa a = div v ( Π + N R a=1 l aa a ) + NR a=1 ( Ra l a div v ) A a diagonal closure (with λ > 0, [L ab ] pos. def., symmetric): div v = λ ( Π + N R a=1 l aa a ), Ra l a div v = N R b=1 L aba b

47 Example: Entropy Inequality of T.I.P. the strengthened entropy principle can be used in T.I.P.! ζ TIP = q 1 T + N i=1 j i ( µ i T b ) i T + 1 T S : D 1 T Π div v 1 NR T a=1 R aa a Consider a coupling between volume variations and chemical reactions Parity of the factors Π, div v, R a, A a : +1, -1, -1, +1 Cross-effects via entropy neutral mixing: div v Π + N R a=1 R aa a = div v ( Π + N R a=1 l aa a ) + NR a=1 ( Ra l a div v ) A a diagonal closure (with λ > 0, [L ab ] pos. def., symmetric): div v = λ ( Π + N R a=1 l aa a ), Ra l a div v = N R b=1 L aba b Hence the apparent anti-symmetry (Onsager-Casimir relations): Π = λ 1 div v N R a=1 l aa a, R a = l a div v + N R b=1 L aba b

48 Example: Entropy Inequality of T.I.P. the strengthened entropy principle can be used in T.I.P.! ζ TIP = q 1 T + N i=1 j i ( µ i T b ) i T + 1 T S : D 1 T Π div v 1 NR T a=1 R aa a Consider a coupling between volume variations and chemical reactions Parity of the factors Π, div v, R a, A a : +1, -1, -1, +1 Cross-effects via entropy neutral mixing: div v Π + N R a=1 R aa a = div v ( Π + N R a=1 l aa a ) + NR a=1 ( Ra l a div v ) A a diagonal closure (with λ > 0, [L ab ] pos. def., symmetric): div v = λ ( Π + N R a=1 l aa a ), Ra l a div v = N R b=1 L aba b Hence the apparent anti-symmetry (Onsager-Casimir relations): Π = λ 1 div v N R a=1 l aa a, R a = l a div v + N R b=1 L aba b

49 The Entropy Principle For the considered fluid mixture class we also postulate: 5) The dissipative mechanisms are: multi-component diffusion, heat conduction, chemical reaction, viscous flow. 6) The entropy density is given as ϱs = h(ϱe, ϱ 1,..., ϱ N ) with a strictly concave material function h. The absolute temperature T and chemical potentials µ i are defined as 1 ϱs := T ϱe, µ i ϱs := T ϱ i

50 The Entropy Principle For the considered fluid mixture class we also postulate: 5) The dissipative mechanisms are: multi-component diffusion, heat conduction, chemical reaction, viscous flow. 6) The entropy density is given as ϱs = h(ϱe, ϱ 1,..., ϱ N ) with a strictly concave material function h. The absolute temperature T and chemical potentials µ i are defined as 1 ϱs := T ϱe, µ i ϱs := T ϱ i

51 Entropy Principle evaluated Evaluation of the entropy principle: 1 entropy flux: Φ = q T i ϱ i u i µ i T 2 Gibbs-Duhem equation: p + ϱψ i ϱ iµ i = 0 3 restrictions to constitutive equations for dissipative mechanisms: Entropy inequality, i.e. ζ 0 with the entropy production rate ζ = 1 NR T a=1 R aa a + 1 T i Sirr i : D i + i q i 1 T ( ) i u i ϱ i µ i T + 1 T (f i r i v i r iu i p i ) (ϱ i e i + p i ) 1 T S irr i = Π i I + S i, i.e. S i = p i I + S irr i

52 Entropy Principle evaluated Evaluation of the entropy principle: 1 entropy flux: Φ = q T i ϱ i u i µ i T 2 Gibbs-Duhem equation: p + ϱψ i ϱ iµ i = 0 3 restrictions to constitutive equations for dissipative mechanisms: Entropy inequality, i.e. ζ 0 with the entropy production rate ζ = 1 NR T a=1 R aa a + 1 T i Sirr i : D i + i q i 1 T ( ) i u i ϱ i µ i T + 1 T (f i r i v i r iu i p i ) (ϱ i e i + p i ) 1 T S irr i = Π i I + S i, i.e. S i = p i I + S irr i

53 1 Motivation 2 T.I.P. Fickean vs. Maxwell-Stefan form 3 Partial Balances and Constitutive Modeling Framework 4 Closure for Non-Reactive Multicomponent Fluids 5 The Maxwell-Stefan Equations and Electromigration 6 The Nernst-Planck-Poisson System in nd

54 Nonreactive fluids without viscosity entropy production without viscosity, no chemical reactions: ζ = i u i (ϱ i µ i T + 1 T (f ) i p i ) h i 1 T + i q i 1 T with partial enthalpies h i := ϱ i e i + p i. With short-hand notation: with ζ = i u i (B i + 1 T f ) i + i q i 1 T B i := ϱ i µ i T 1 T p i h i 1 T The Gibbs-Duhem equation implies i B i = 0!

55 Nonreactive fluids without viscosity entropy production without viscosity, no chemical reactions: ζ = i u i (ϱ i µ i T + 1 T (f ) i p i ) h i 1 T + i q i 1 T with partial enthalpies h i := ϱ i e i + p i. With short-hand notation: with ζ = i u i (B i + 1 T f ) i + i q i 1 T B i := ϱ i µ i T 1 T p i h i 1 T The Gibbs-Duhem equation implies i B i = 0!

56 Exploiting the second law The interaction terms f i necessarily satisfy ( ) N i=1 u i B i + 1 T f N i 0 and i=1 B i = 0, Hence ( ) N 1 i=1 (u i u N ) B i + 1 T f i 0, N i=1 f i = 0 with build-in constraints. The standard linear Ansatz for B i + 1 T f i is B i + 1 T f i = N 1 j=1 τ ij (u j u N ) (for i = 1,..., N 1) with a positive (semi-)definite matrix [τ ij ].

57 Exploiting the second law The interaction terms f i necessarily satisfy ( ) N i=1 u i B i + 1 T f N i 0 and i=1 B i = 0, Hence ( ) N 1 i=1 (u i u N ) B i + 1 T f i 0, N i=1 f i = 0 with build-in constraints. The standard linear Ansatz for B i + 1 T f i is B i + 1 T f i = N 1 j=1 τ ij (u j u N ) (for i = 1,..., N 1) with a positive (semi-)definite matrix [τ ij ].

58 Closure for thermo-mechanical Interactions Extension to N N format (positive semi-definite): τ Nj = N 1 i=1 τ ij (i = 1,..., N 1), τ in = N 1 j=1 τ ij (j = 1,..., N) Straight forward computation: B i + 1 T f i = N j=1 τ ij (u j u N ) = N j=1 τ ij (u j u i ) Assumption of binary type interactions: (C. Truesdell) τ ij = τ ij (T, ϱ i, ϱ j ) 0 if ϱ i 0+ or ϱ j 0+ This implies symmetry of [τ ij ]! (evaluate i,j τ ij (u i u j ) = 0) τ ij = f ij ϱ i ϱ j for i j with f ij = f ji 0, f ij = f ij (ϱ i, ϱ j, T ).

59 Closure for thermo-mechanical Interactions Extension to N N format (positive semi-definite): τ Nj = N 1 i=1 τ ij (i = 1,..., N 1), τ in = N 1 j=1 τ ij (j = 1,..., N) Straight forward computation: B i + 1 T f i = N j=1 τ ij (u j u N ) = N j=1 τ ij (u j u i ) Assumption of binary type interactions: (C. Truesdell) τ ij = τ ij (T, ϱ i, ϱ j ) 0 if ϱ i 0+ or ϱ j 0+ This implies symmetry of [τ ij ]! (evaluate i,j τ ij (u i u j ) = 0) τ ij = f ij ϱ i ϱ j for i j with f ij = f ji 0, f ij = f ij (ϱ i, ϱ j, T ).

60 Momentum Balance with Thermo-mechanical Interactions partial momentum balances: with ϱ i ( t v i + v i v i ) + pi = f i + ϱ i b i f i = ϱ i T µ i T + p i + h i T 1 T T j f ijϱ i ϱ j (v i v j ) class-ii momentum balances (no viscosity, no chemical reactions): ϱ i ( t v i + v i v i ) = ϱi T µ i T + Th i 1 T T j f ijϱ i ϱ j (v i v j ) + ϱ i b i

61 Momentum Balance with Thermo-mechanical Interactions partial momentum balances: with ϱ i ( t v i + v i v i ) + pi = f i + ϱ i b i f i = ϱ i T µ i T + p i + h i T 1 T T j f ijϱ i ϱ j (v i v j ) class-ii momentum balances (no viscosity, no chemical reactions): ϱ i ( t v i + v i v i ) = ϱi T µ i T + Th i 1 T T j f ijϱ i ϱ j (v i v j ) + ϱ i b i special case of isothermal conditions: ϱ i ( t v i + v i v i ) = ϱi µ i T j f ijϱ i ϱ j (v i v j ) + ϱ i b i special case of a simple mixture: ϱψ(t, ϱ 1,..., ϱ N ) = i ϱ iψ i (T, ϱ i ) ϱ i ( t v i + v i v i ) = pi T j f ijϱ i ϱ j (v i v j ) + ϱ i b i

62 Momentum Balance with Thermo-mechanical Interactions partial momentum balances: with ϱ i ( t v i + v i v i ) + pi = f i + ϱ i b i f i = ϱ i T µ i T + p i + h i T 1 T T j f ijϱ i ϱ j (v i v j ) class-ii momentum balances (no viscosity, no chemical reactions): ϱ i ( t v i + v i v i ) = ϱi T µ i T + Th i 1 T T j f ijϱ i ϱ j (v i v j ) + ϱ i b i special case of isothermal conditions: ϱ i ( t v i + v i v i ) = ϱi µ i T j f ijϱ i ϱ j (v i v j ) + ϱ i b i special case of a simple mixture: ϱψ(t, ϱ 1,..., ϱ N ) = i ϱ iψ i (T, ϱ i ) ϱ i ( t v i + v i v i ) = pi T j f ijϱ i ϱ j (v i v j ) + ϱ i b i

63 Partial Momentum Balances due to Stefan

64 1 Motivation 2 T.I.P. Fickean vs. Maxwell-Stefan form 3 Partial Balances and Constitutive Modeling Framework 4 Closure for Non-Reactive Multicomponent Fluids 5 The Maxwell-Stefan Equations and Electromigration 6 The Nernst-Planck-Poisson System in nd

65 Scale-Reduced Model: Maxwell-Stefan Eqs Consider the difference: momentum of species i y i total momentum ϱ i ( t + v )u i + ϱ i u i v i = y i p ϱ i µ i + T (h i ϱ i µ i ) 1 T + ϱ i(b i b) T j f ijϱ i ϱ j (u i u j ) nondimensionalized form: U V C C y i( t u i + v u i + u i vi ) = y i p ϱ c0rt0 p 0 y i µ i ( h0 h i p 0 ϱ c0rt0 p 0 y i µ i ) ln T + ϱ0bl p 0 y i (b i b ) c0rt0 UL p 0 D ϱ T y i y j j Mi Mj D (u i u j ) ij some characteristic reference quantities: U diffusion velocity, V mixture velocity, τ = L/V convective time scale, C = p 0 /ϱ 0 about the speed of sound in gas mixture

66 Scale-Reduced Model: Maxwell-Stefan Eqs Consider the difference: momentum of species i y i total momentum ϱ i ( t + v )u i + ϱ i u i v i = y i p ϱ i µ i + T (h i ϱ i µ i ) 1 T + ϱ i(b i b) T j f ijϱ i ϱ j (u i u j ) nondimensionalized form: U V C C y i( t u i + v u i + u i vi ) = y i p ϱ c0rt0 p 0 y i µ i ( h0 h i p 0 ϱ c0rt0 p 0 y i µ i ) ln T + ϱ0bl p 0 y i (b i b ) c0rt0 UL p 0 D ϱ T y i y j j Mi Mj D (u i u j ) ij some characteristic reference quantities: U diffusion velocity, V mixture velocity, τ = L/V convective time scale, C = p 0 /ϱ 0 about the speed of sound in gas mixture

67 Scale-Reduced Model: Maxwell-Stefan Eqs Approximation for U C j i V C y j j i y i j j cm i M j D ij = y i RT µ i 1: generalized Maxwell-Stefan equations y i ϱrt p + ϱ iµ i h i 1 ϱr T y i ϱrt (b i b) Phenomena: molecular, pressure, thermo- (partially) & forced diffusion Chemical Eng. version of the generalized MS-eqs (isothermal case)*: j i x j J i x i J j c D ij = x i RT pµ mol i + φ i y i ϱrt p y i ϱrt (b i b) x i molar fractions, J i molar mass fluxes, µ mol i chemical potential *R. Taylor, R. Krishna: Multicomponent Mass Transfer, 1993 = M i µ i molar based

68 Scale-Reduced Model: Maxwell-Stefan Eqs Approximation for U C j i V C y j j i y i j j cm i M j D ij = y i RT µ i 1: generalized Maxwell-Stefan equations y i ϱrt p + ϱ iµ i h i 1 ϱr T y i ϱrt (b i b) Phenomena: molecular, pressure, thermo- (partially) & forced diffusion Chemical Eng. version of the generalized MS-eqs (isothermal case)*: j i x j J i x i J j c D ij = x i RT pµ mol i + φ i y i ϱrt p y i ϱrt (b i b) x i molar fractions, J i molar mass fluxes, µ mol i chemical potential *R. Taylor, R. Krishna: Multicomponent Mass Transfer, 1993 = M i µ i molar based

69 Scale-Reduced Model: Maxwell-Stefan Eqs Thermodynamic consistency? The mass fluxes j i determined by the gen. Maxwell-Stefan equations (and i j i = 0) need to satisfy: ζ TIP = ( α 1 T + i h ) iu i 1 T 1 T i j i (T µ i T b i) 0! Instead of inverting the MS-system, we use T j f ijϱ i ϱ j (u i u j ) = ϱ i T µ i T y i p h i T 1 T ϱ i(b i b) to eliminate T µ i T b i h i ϱ i T 1 T

70 Scale-Reduced Model: Maxwell-Stefan Eqs Thermodynamic consistency? The mass fluxes j i determined by the gen. Maxwell-Stefan equations (and i j i = 0) need to satisfy: ζ TIP = ( α 1 T + i h ) iu i 1 T 1 T i j i (T µ i T b i) 0! Instead of inverting the MS-system, we use T j f ijϱ i ϱ j (u i u j ) = ϱ i T µ i T y i p h i T 1 T ϱ i(b i b) to eliminate This yields: T µ i T b i h i ϱ i T 1 T ζ TIP = α ( ) 1 2 T i,j f ( ) 2 ijϱ i ϱ j ui u j 0

71 Scale-Reduced Model: Maxwell-Stefan Eqs Thermodynamic consistency? The mass fluxes j i determined by the gen. Maxwell-Stefan equations (and i j i = 0) need to satisfy: ζ TIP = ( α 1 T + i h ) iu i 1 T 1 T i j i (T µ i T b i) 0! Instead of inverting the MS-system, we use T j f ijϱ i ϱ j (u i u j ) = ϱ i T µ i T y i p h i T 1 T ϱ i(b i b) to eliminate This yields: T µ i T b i h i ϱ i T 1 T ζ TIP = α ( ) 1 2 T i,j f ( ) 2 ijϱ i ϱ j ui u j 0

72 From Maxwell-Stefan to Nernst-Planck Assume dilute solution: x 0 1, x i 1 for i = 1,..., N. Then 1 D i0 j i = c i RT p,t µ i + φ i y i ϱrt p ϱ i RT (b i b) i = 1,..., N chemical potentials: µ i (T, p, x) = g i (T, p) + RT log x i, i = 1,..., N forces on ions: b i = F M i z i φ (z i charge numbers) electrical potential: (ɛ φ) = F N k=0 z kc k neglecting pressure diffusion (& assuming c const): j i = D i ( ci + F RT (z ic i N k=0 z kc k ) φ ), i = 1,..., N.

73 From Maxwell-Stefan to Nernst-Planck Assume dilute solution: x 0 1, x i 1 for i = 1,..., N. Then 1 D i0 j i = c i RT p,t µ i + φ i y i ϱrt p ϱ i RT (b i b) i = 1,..., N chemical potentials: µ i (T, p, x) = g i (T, p) + RT log x i, i = 1,..., N forces on ions: b i = F M i z i φ (z i charge numbers) electrical potential: (ɛ φ) = F N k=0 z kc k neglecting pressure diffusion (& assuming c const): j i = D i ( ci + F RT (z ic i N k=0 z kc k ) φ ), i = 1,..., N. b 0 = 0! b i b = b i N k=0 y kb k = (1 y i )b i + N i k=1 y kb k b i, assuming again a dilute solution.

74 From Maxwell-Stefan to Nernst-Planck Assume dilute solution: x 0 1, x i 1 for i = 1,..., N. Then 1 D i0 j i = c i RT p,t µ i + φ i y i ϱrt p ϱ i RT (b i b) i = 1,..., N chemical potentials: µ i (T, p, x) = g i (T, p) + RT log x i, i = 1,..., N forces on ions: b i = F M i z i φ (z i charge numbers) electrical potential: (ɛ φ) = F N k=0 z kc k neglecting pressure diffusion (& assuming c const): j i = D i ( ci + F RT (z ic i N k=0 z kc k ) φ ), i = 1,..., N. b 0 = 0! b i b = b i N k=0 y kb k = (1 y i )b i + N i k=1 y kb k b i, assuming again a dilute solution.

75 From Maxwell-Stefan to Nernst-Planck Assume dilute solution: x 0 1, x i 1 for i = 1,..., N. Then 1 D i0 j i = c i RT p,t µ i + φ i y i ϱrt p ϱ i RT (b i b) i = 1,..., N chemical potentials: µ i (T, p, x) = g i (T, p) + RT log x i, i = 1,..., N forces on ions: b i = F M i z i φ (z i charge numbers) electrical potential: (ɛ φ) = F N k=0 z kc k neglecting pressure diffusion (& assuming c const): j i = D i ( ci + F RT z ic i φ ), i = 1,..., N. b 0 = 0! b i b = b i N k=0 y kb k = (1 y i )b i + N i k=1 y kb k b i, assuming again a dilute solution.

76 Remarks on the Nernst-Planck fluxes Shortcomings of the specialization to Nernst-Planck: If used for all constituents, Nernst-Planck (like Fickean) fluxes are inconsistent with the continuity equation If only used for the dilute components, Nernst-Planck (like Fickean) fluxes do not yield pointwise upper bounds Even for globally dilute mixtures, the diluteness assumption breaks down near interfaces (walls). The solvent concentration can actually approach zero when transversing the double layer at a wall! Pressure effects are usually not negligible (especially near walls).

77 Implications Implication for modeling transport of ions in solution: Use the full set of balance equations together with thermodynamically consistent fluxes from Maxwell-Stefan theory instead of only mass balances with Nernst-Planck fluxes! For first results from a complete and thermodynamically consistent model see the recent paper by Dreyer, Guhlke and Müller: Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys. 15, (2013). In particular, it is shown there that the complete model can be applied simultaneously inside the bulk and in the boundary layer.

78 Implications Implication for modeling transport of ions in solution: Use the full set of balance equations together with thermodynamically consistent fluxes from Maxwell-Stefan theory instead of only mass balances with Nernst-Planck fluxes! For first results from a complete and thermodynamically consistent model see the recent paper by Dreyer, Guhlke and Müller: Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys. 15, (2013). In particular, it is shown there that the complete model can be applied simultaneously inside the bulk and in the boundary layer. Final Remark: Extension to chemically reacting fluid mixtures exists.

79 Implications Implication for modeling transport of ions in solution: Use the full set of balance equations together with thermodynamically consistent fluxes from Maxwell-Stefan theory instead of only mass balances with Nernst-Planck fluxes! For first results from a complete and thermodynamically consistent model see the recent paper by Dreyer, Guhlke and Müller: Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys. 15, (2013). In particular, it is shown there that the complete model can be applied simultaneously inside the bulk and in the boundary layer. Final Remark: Extension to chemically reacting fluid mixtures exists.

80 1 Motivation 2 T.I.P. Fickean vs. Maxwell-Stefan form 3 Partial Balances and Constitutive Modeling Framework 4 Closure for Non-Reactive Multicomponent Fluids 5 The Maxwell-Stefan Equations and Electromigration 6 The Nernst-Planck-Poisson System in nd Joint work with A. Fischer, M. Pierre, G. Rolland

81 Navier-Stokes-Nernst-Planck-Poisson system (NSNPP) (NS) (NP) { (P) t v + (v )v v + p + N i=1 z ic i Φ = 0 in Ω, div v = 0 in Ω, u = 0 on Ω, v(0) = v 0 in Ω. t c i + div (c i v d i c i d i z i c i Φ) = 0 in Ω, ν c i + z i c i ν Φ = 0 on Ω, c(0) = c 0 in Ω. Φ N i=1 z ic i = σ in Ω, ν Φ + τφ = ξ on Ω. Unknowns: Data: Constants: v velocity field, p pressure, c i concentration of species i, Φ electrical potential. d i (t, x) diffusivity, σ(x) fixed charges, ξ(x) boundary datum. z i Z charge number, τ > 0 boundary capacity.

82 Energy dissipation for (NSNPP) Exploit the following Lyapunov structure: Define the functionals E and D by E(v, c, Φ) := 1 N v 2 + c i log c i + 1 Φ 2 + τ Φ 2, 2 Ω i=1 Ω 2 Ω 2 Ω N D(v, c, Φ) := v di c i + d i z i c i Φ 2 0. d i c i Ω i=1 Given a regular solution (v, c, Φ) to (NSNPP), the functional Ω V (t) = E(v(t), c(t), Φ(t)) is non-increasing in time with derivative V (t) = D(v(t), c(t), Φ(t)) 0. Analogous situation for pure (NPP) without kinetic energy term.

83 Global existence for NPP in three or higher dimensions Theorem (B., Fischer, Pierre, Rolland, Nonl. Anal. TMA 14) Let n IN, Ω IR n bounded and sufficiently smooth and d i L loc (IR +; L (Ω)), 0 < d(t ) d i (t, x) d(t ) < a.e. on Q T. c 0 L 2 (Ω) +, σ = 0, ξ L 2 ( Ω). Then there exist c L (IR + ; L 1 (Ω)) and Φ L (IR + ; H 1 (Ω)) such that (NPP) is satisfied in the following sense: For all T > 0, c i L 1 (0, T ; W 1,1 loc (Ω)), d i c i + d i z i c i Φ L 1 (Q T ) s.t., for all ψ C (Q T ) with ψ(t ) = 0, ϕ C (Ω), c i t ψ + (d i c i + d i z i c i Φ) ψ = ci 0 ψ(0), Q T Ω N Φ(t) ϕ + τφ(t)ϕ = z i c i (t)ϕ a.e. t IR +. Ω Ω Ω i=1

84 Nernst-Planck-Poisson in nd Global weak solutions in nd - Sketch of proof: A priori estimate for (NPP) Approximate (NPP) by (NPP ε ) while conserving the Lyapunov structure Global existence and uniqueness for (NPP ε ) Compactness for sequence of approximate solutions Limit as ε 0

85 Global weak solutions in nd - Sketch of proof Recall from above (without v): Integration in time yields C Q T Q T i=1 N i=1 c i + z i c i Φ 2 = c i Q T Q T i=1 V = N i=1 Ω d i c i + d i z i c i Φ 2 /d i c i N ( ci 2 ) + zi 2 c i Φ 2 + 2z i c i Φ c } i {{} 0 Integration by parts and Poisson equation give N N z i c i Φ = z i c i Φ +... = Φ Γ T Q T }{{} Γ T i=1 0!!

86 Global weak solutions in nd - Sketch of proof Recall from above (without v): Integration in time yields C Q T Q T i=1 N i=1 c i + z i c i Φ 2 = c i Q T Q T i=1 V = N i=1 Ω d i c i + d i z i c i Φ 2 /d i c i N ( ci 2 ) + zi 2 c i Φ 2 + 2z i c i Φ c } i {{} 0 Integration by parts and Poisson equation give N N z i c i Φ = z i c i Φ +... = Φ Γ T Q T }{{} Γ T i=1 0!! Boundary terms difficult to estimate only local W 1,1 -regularity, except in case n = 3, ξ L q ( Ω) for q > 2. Appr. problem with global solution and similar Lyapunov structure?

87 Global weak solutions in nd - Sketch of proof Recall from above (without v): Integration in time yields C Q T Q T i=1 N i=1 c i + z i c i Φ 2 = c i Q T Q T i=1 V = N i=1 Ω d i c i + d i z i c i Φ 2 /d i c i N ( ci 2 ) + zi 2 c i Φ 2 + 2z i c i Φ c } i {{} 0 Integration by parts and Poisson equation give N N z i c i Φ = z i c i Φ +... = Φ Γ T Q T }{{} Γ T i=1 0!! Boundary terms difficult to estimate only local W 1,1 -regularity, except in case n = 3, ξ L q ( Ω) for q > 2. Appr. problem with global solution and similar Lyapunov structure?

88 Global weak solutions in nd - Sketch of proof Approximate problem (based on an idea of Gajewski/Gröger): t c i + div( d i h(c i ) d i z i c i Φ) = 0 in Ω, ν h(c i ) + z i c i ν Φ = 0 on Ω, c i (0) = ci 0 (NPη ) in Ω, Φ N i=1 z ic i = 0 in Ω, ν Φ + τφ = ξ on Ω, where h(r) = r + ηr p for fixed, large p > 1. The solutions satisfy the same dissipation inequality, if the free energy i c i log c i is replaced by i (c i log c i + ηc p i /(p 1)). } (P)

89 Global weak solutions in nd - Sketch of proof Approximate problem (based on an idea of Gajewski/Gröger): t c i + div( d i h(c i ) d i z i c i Φ) = 0 in Ω, ν h(c i ) + z i c i ν Φ = 0 on Ω, c i (0) = ci 0 (NPη ) in Ω, Φ N i=1 z ic i = 0 in Ω, ν Φ + τφ = ξ on Ω, where h(r) = r + ηr p for fixed, large p > 1. The solutions satisfy the same dissipation inequality, if the free energy i c i log c i is replaced by i (c i log c i + ηc p i /(p 1)). appr. c η i a priori bounded in L (0, T ; L p (Ω)). appr. Φ η a priori bounded in L (0, T ; W 2,p (Ω)). appr. solution (c η i, Φη ) via fixed point argument for Φ in L (0, T ; W 1, (Ω)), using p > n. } (P)

90 Global weak solutions in nd - Sketch of proof Approximate problem (based on an idea of Gajewski/Gröger): t c i + div( d i h(c i ) d i z i c i Φ) = 0 in Ω, ν h(c i ) + z i c i ν Φ = 0 on Ω, c i (0) = ci 0 (NPη ) in Ω, Φ N i=1 z ic i = 0 in Ω, ν Φ + τφ = ξ on Ω, where h(r) = r + ηr p for fixed, large p > 1. The solutions satisfy the same dissipation inequality, if the free energy i c i log c i is replaced by i (c i log c i + ηc p i /(p 1)). appr. c η i a priori bounded in L (0, T ; L p (Ω)). appr. Φ η a priori bounded in L (0, T ; W 2,p (Ω)). appr. solution (c η i, Φη ) via fixed point argument for Φ in L (0, T ; W 1, (Ω)), using p > n. } (P)

91 Thank You for Your Attention!