Multicomponent, Multiphase Thermodynamics of Swelling Porous Media with Electroquasistatics: II. Constitutive Theory

Transcription

1 Multicomponent, Multiphae Thermodynamic of Swelling Porou Media with Electroquaitatic: II. Contitutive Theory Lynn Schreyer Bennethum John H. Cuhman Augut 8, 2001 Abtract In Part I macrocopic field equation of ma, linear and angular momentum, energy, and the quaitatic form of Maxwell equation for a multiphae, multicomponent medium were derived. Here we exploit the entropy inequality to obtain retriction on contitutive relation at the macrocale for a 2-phae, multiple-contituent, polarizable mixture of fluid and olid. Specific emphai i placed on charged porou media in the preence of electrolyte. The governing equation for the tre tenor of each phae, flow of the fluid through a deforming medium, and diffuion of contituent through uch a medium are derived. The reult have application in welling clay (mectite, biopolymer, biological membrane, puled electrophorei, chromotography, drug delivery, and other welling ytem. Key word: porou media, mixture theory, electrodynamic, welling, contitutive equation 1 Introduction We continue our invetigation into the form of the governing equation for a multiple-component, multiple-phae, polarizable, welling porou medium with charged particle ubect to an electric field. In Part I of thi erie we derived the macrocopic field equation in which it i the total electric field which affect the pecie conervation of momentum and energy. In thi paper, we exploit the entropy inequality in the ene of Coleman and Noll 14 in order to obtain retriction on contitutive equation in term of macrocopic variable. Thi approach differ from claical averaging 29, 30, 31, 35, 36 or Univerity of Colorado at Denver, Center for Computational Mathematic, Campu Box 170, P.O. Box , Denver, CO bennethum@math.cudenver.edu. To whom correpondence hould be addreed. Center for Applied Math, Math Science Building, Purdue Univerity, W. Lafayette, IN

2 Swelling Porou Media with Electroquaitatic 2 homogenization 10, 16, 32 where both the field equation and microcopic contitutive equation are upcaled to obtain macrocale equation in term of microcale geometry and microcale contitutive coefficient. The advantage of homogenization and averaging i that if one know the microcale contitutive equation and geometry, then one can obtain to firt-order (and econd-order... contitutive equation (including the coefficient at the macrocale in term of microcopic variable. The HMT approach aume only the variable upon which contitutive variable may depend, and produce retriction of the form of contitutive equation in term of macrocopic variable. HMT doe not aume any microcale contitutive relation nor any pecific microcale geometry, which in uch complex media a conidered here are not atifactorily known. Thi point i of ignificant import for natural media uch a geophyical environ or polymeric ytem, which are never well-characterized geometrically at the microcale. For implicity we conider only a liquid-olid ytem. The mixture i charge neutral, although neither the phae nor pecie face thi requirement individually. We aume that interfacial propertie uch a exce ma denity, free charge on interface, and interface current are negligible; although the preent theory can be extended to explicitly incorporate thee effect 19. However, it hould be noted that for many practical material where the ratio of urface area to volume of the olid phae remain contant, interfacial characteritic can be accounted for in thi theory. Thi i becaue in the cae where urface to volume ratio i contant, charge per unit ma of the olid phae i directly proportional to charge per unit urface area. Likewie the theory preented here alo applie to material in which the charge of the olid itelf change. Thi theory would not apply to material in which the charge of the olid phae and of the interface vary independently. So, for example, the theory applie to clay mineral, which ha a charge aociated only with it urface, and to polymer which ha a charge aociated with the polymer but not with it urface if a urface can be defined. In deriving retriction on the form of the contitutive equation, we follow 17, 18 and view fluxe a contitutive. In earlier work, Eringen 18 conidered contituent electric field throughout for a non-welling porou medium; Huyghe and Janen 23 conidered a ingle electric field in a deformable two-phae porou medium with no exchange term in the conervation equation; and Gu et al. 20 conidered a charged welling medium with no electric field. 2 Contitutive Aumption and the Entropy Inequality The full entropy inequality which i exploited in ubequent ection i preented in Appendix B. The baic notation i found in Part I o only new term are defined herein. We aume the medium conit of a liquid phae (denoted by = l, and a olid phae, (denoted by =, and that the medium i macro-

3 Swelling Porou Media with Electroquaitatic 3 copically neutrally charged, however charge may move between contituent and phae. We aume that entropy generation mut be non-negative for the total body, i.e. ρ Λ = ε ρ Λ 0. (1 Further we aume a form of local equilibrium wherein there i one temperature for all contituent and all phae, i.e. T (x, t = T (x, t for all contituent and all phae. Thi effectively tate that the rate of heat tranfer between contituent i much fater than the time cale of interet to the problem. To couple the entropy and energy equation, it i neceary to relate the fluxe and ource of entropy to the fluxe and ource of heat. We aume the procee are imple in the ene of 17. In thi ene everal poible relation are admiable. Among thee are: φ = q T b = h T φ = q b = h T T ε φ = ε q T ε b = (2 (3 ε h T. (4 The expreion in (4 have been ued in 18, and the relation in (2 have been ued in 1, 2, 4, 22. More general expreion which implify to the above are ued in extended thermodynamic 24, 26. The quetion i whether the procee governing the behavior of the contituent themelve, the individual phae, or the bulk material are imple. The aumption of any one of thee doe not imply any other due to microcale/macrocale relationhip between heat fluxe and heat ource 5. Thi problem i complicated further becaue the macrocale definition of the heat flux depend on how one incorporate microcale fluctuation (ee Part I, 5. The difference manifet themelve explicitly in the contitutive relation obtained for diffuive fluxe and the chemical potential. For example, if the relation in (4 are ued then it can be hown that the chemical potential of two different pecie at equilibrium mut be equal, which i inconitent with Gibbian thermotatic 13. To the author knowledge, the relation in (2 do not reult in any phyical inconitencie in near-equilibrium procee, and thee are the relation ued herein. Define the Helmholtz free energy for the pecie and the internal Helmholtz free energy for the bulk phae a A = e T η A = C A (5

4 Swelling Porou Media with Electroquaitatic 4 and introduce a modified Helmholtz free energy a à = e T η 1 ρ E T P à = C Ã, (6 where C i the ma fraction of contituent in phae given by C = ρ /ρ. The purpoe of introducing the modified Helmholtz potential i to reduce the amount of manipulation required to obtain the entropy inequality, a either P or E T mut be contitutive (dependent variable. With thi notation, eliminating φ, b, Q β, Q, Φ β, and η from the entropy balance and re-writing it in term of bulk-phae variable one obtain ε ρ T Λ = ε ρ ( D à + η D T + ε J E T Dt Dt ε t + ε P E T I + ε ρ (à I + v, v, : v ε t + ε P E T I : v, ε 1 2 ε oe T E + E T P + ρ A ε T T { q + ρ v, (à v, v, t v, } ε D ρ 1 Dt ρ E T P + A +v l, ε l ρ l T l + ε l E T P l ε oe T E l ε l ε E T P ( A l (ε l ρ l + 1 ρ l E T P l (ε l ρ l v, ε ρ (î T β ε ρ à ε q e E T β + 1 ρ E T P (ε ρ ε E T P ε oe T E ε ε ρ ê 1 β ρ E T P + + à à (v, (v, 2 β ε ρ r 1 2 (v, 2 à 1 ρ E T P 0, (7

5 Swelling Porou Media with Electroquaitatic 5 where a comma in the upercript denote difference (e.g. v, = v v, a uperimpoed dot denote the material time derivative with repect to the olid phae (e.g. ε = ε / t+v ε, I i the identity matrix, and the contraction operator A : B i, in indicial notation, A i B i. We enforce many of the balance law weakly uing the Lagrange Multiplier approach 25. The equation and their aociated Lagrange multiplier are lited below. Lagrange Equation from 5 Lagrange Equation from 5 Mult. Mult. λ ρ Continuity Eqn, (42 λ D Gau Law (49 Λ E Faraday Law (55 λ q e Conerv. of Charge (70 D Λ Dt (εl qe l + ε qe = 0 Ampère law i derivable from the conervation of charge and Gau law and o i not enforced directly. The expreion correponding to the Lagrange multiplier Λ enforce charge neutrality locally. Thi retriction alone implie that the total charge could vary in pace. However we have in mind that the mixture i charge neutral initially everywhere o that it i aumed the time cale at which imbalance may occur i mall compared to the time cale involved with other procee. The unknown in thi ytem include: ε l, ρ, v, T, E, q e, (8 ê β, r, t, t, T β, T β, î, (9 A, q, η, (10 P, d, d β, σ, σ β, J, q, Ẑ β. (11 The variable in the firt row, (8, are the primary unknown. The remaining variable, (9 11, are conidered contitutive and are a function of contitutive independent variable. In order to cloe the ytem, one additional equation i needed, which correpond to the unknown ε l. Thi i known a the cloure problem, and it arie from the homogenization of the microcopic geometry. To cloe the ytem we follow 1, 9, 12 and view the time rate of change of the volume fraction Dε l /Dt a a contitutive variable. The choice of contitutive independent variable i made baed on knowledge of the ytem being modeled. Here we aume the fluid may behave a a Newtonian fluid and the olid a an elatic olid, hence we include the rate of deformation tenor, d l, and the train tenor, E. Since the olid phae may be diconnected, the macrocale train tenor i not the average of the microcale train, but i defined in term of the deformation gradient F = o x, E = 1 2 (F T F I, where o denote differentiation with repect to the macrocopic material particle. Thu the train tenor i a meaure of the geometry of the olid phae. Further, we are particularly intereted in modeling material in which the olid and fluid phae have electro-chemical interaction, o that

6 Swelling Porou Media with Electroquaitatic 6 the behavior of the liquid phae may trongly depend upon it proximity to the olid phae. Thu we incorporate the volume fraction, ε l, a an independent variable. The independent variable which are ued to define the contitutive variable include: ε l, T, ρ, v l,, v,, E, E T, z, ε l, T, ρ, d l, ω, v l,l, E, E T, = 1,..., N, = l, (12 where becaue the liquid phae may be polarizable and may depend trongly on the geometry of the olid phae 5, we have alo included the vorticity tenor, ω l = ( v ( v T. Note that we have incorporated the total electric field a an independent variable, a oppoed to the electric field of each contituent or each phae. Thi i becaue it i aumed that all contitutive variable are meaured with repect to the total electric field. The charge of a pecie, z, ha unit of charge of per unit ma of. Thi reult in term uch a A z which i evaluated holding all other independent variable in (12, uch a volume fraction and denitie, fixed. Thi allow for diaociation of ion, or change in the charge denity of the olid phae, either through the change in urface charge denity or through charge denity of the bulk phae, but not both (ee the introduction. To implify the reult we relax the Principle of Equipreence 34 and aume that the modified Helmholtz potential energie à are a function of a ubet of the above contitutive independent variable: à l = Ãl( ε l, T, ρ l, v l,, v l,l, E, E T, z l, T, ρ l, d l, ω l, v l,l (13 à = Ã( ε l, T, ρ, v,, E, E T, z, T, ρ, ω. (14 We note that including the additional independent variable doe not change the reult if one modifie the definition of preure, chemical potential, etc. ee e.g. 4. The entropy inequality i now expanded in the traditional manner 9, 17, 21 and i preented in Appendix B. 3 Non-Equilibrium Contitutive Retriction The following variable are neither contitutive nor independent, D ρ D z,, T, Ė T, d, v,, T Dt Dt, (15 ( D v, ρ,, v l,, ḋ l, ω l, v l,l, (16 Dt E, E, (17

7 Swelling Porou Media with Electroquaitatic 7 where = 1,..., N for all variable not containing v,, ince N ρ v, = 0. Thu for example, v, i indexed from = 1,...N 1 in order to keep the lit of variable functionally independent. Since thee term appear linearly in the entropy inequality, their coefficient mut be zero. Thi reult in the following retriction (correponding directly with the term in (15: λ ρ λ q e =l, = ρ Ã ρ + Λ = ρ ρ Ã Ã z 1 ρ E T P z Λ, = 1,..., N (18 (19 ε ρ (η + Ã T = 0 (20 ε P = ε t ym = ε ρ Ã E T (21 ε ρ (λ ρ + Ã + ε t e + ε l t l ε P E T I ε ρ v, v, (22 t ρ ρ t N = (P E N T ρ ρ P N E N T I ρ (λ ρ λ N ρ I (23 ε l ρ l Ãl T + ε ρ Ã = 0, (24 T where we have defined the effective tre tenor and hydration tre tenor a t e = ρ F Ã E ( F T t l = ρ l F Ã l E ( F T, (25 repectively. The retriction obtained from the coefficient of the variable lited in (16 indicate that the modified Helmholtz free energie are not a function of v,, ρ, v l,, d l, ω, v l,l, and the retriction correponding to the variable lited in (17 require that the Lagrange multiplier, λ q e, Λ E, and λ D mut all be identically zero. Equation (18 and (19 define the Lagrange multiplier λ ρ and λ q e, repectively. Equation (20 tate that η and T are dual variable with repect to the modified Helmholtz potential, and thi i in agreement with 2, 18. Equation (24 tate that the modified Helmholtz potential of the entire ytem i independent of T. The definition of the effective tre tenor indicate that the effective tre tenor meaure a change in energy of the olid phae with repect to the train in the olid phae. The train in the olid phae i purely a geometrical quantity (intuitively, one take the olid phae, mear it out to obtain a continuou

8 Swelling Porou Media with Electroquaitatic 8 medium at the macrocale, and meaure it deformation - ee the definition of E following (11. If the olid phae i connected, the train repreent an average of the microcale train, and the effective tre tenor i non-negligible if train occur. For a diconnected olid phae, the energy of the olid phae i negligible for large movement of particle, and hence the effective tre tenor i negligible. However in the cae where the medium well (high interaction between liquid and olid phae, and where the olid phae i diconnected, the hydration tre tenor meaure the tre the liquid phae upport when the olid phae i heared. Thi can occur at low moiture content, ee, e.g. 15, o that in thi cae although the effective tre tenor i negligible, the hydration tre tenor i not. Mot welling porou media have a tructure uch that both the olid and liquid phae can upport hear, in which cae neither the hydration nor the effective tre tenor can be neglected. See 8 for a more detailed dicuion on thi topic. Equation (21 implie that it i the polarization of the entire medium which i dual to the electric field in thi repreentation. Thi i in contrat with 18 in which it i hown that the polarization of each component i dual to the contituent electric field. The diparity reult from our choice of independent variable - we ue E T and Eringen ue E a independent variable. We define the preure thermodynamically by p = = ρ (λ ρ + à + P E T + q e Λ (26 ( ρ ρ à ρ (27 To ee how thi variable correpond to what i claically thought of a preure, we re-write (22, the expreion for the ymmetric part of the olid-phae tre tenor, a t ym = (p q eλi + t e + εl ε tl ρ v, v, (28 o that by comparion 17, we ee that p qeλ repreent what i traditionally thought of a preure. Charge and the electric field enter into thi expreion through the term containing the Lagrange multiplier enforcing charge neutrality, qeλ, and through the definition of à in the remaining term. The role played by the effective and hydration tre tenor are dicued in 7, 8, 27, 28. We remark that if the olid phae i conidered incompreible then the material time derivative of ρ i zero, ρ i not conidered an independent variable, and λ ρ i the Lagrange multiplier which enforce the remaining part of the conervation of ma for contituent. In that cae we obtain the ame reult, except that p i a primary unknown which mut be olved for directly.

9 Swelling Porou Media with Electroquaitatic 9 We define the chemical potential a the change of the total Helmholtz potential with repect to the ma of pecie (keeping the total volume fixed 13. Intuitively it i the calar quantity repreenting the chemical energy required to inert that pecie into the mixture. Formally dividing thi thermodynamic definition by the volume of the REV, we have µ = (ε ρ Ã (ε ρ = Ã + ρ Ã ρ. (29 The electro-chemical potential, incorporate the chemical and electrical energy required to inert the particular pecie into the mixture. In thi formulation thi i: µ = µ + z ( Λ ρ ρ Ã ε z ε,ρ,.... (30 The additional term are generically termed the electrical potential, and claically they are written a z eψ where z i the charge number of, e i the charge on an electron, and ψ i the electrical potential. The Lagrange multiplier, Λ, ha unit of energy/charge, and i ometime referred to a the treaming potential 33. The firt part of the electrical potential ha been derived before 23, 20, and i the energy aociated with inerting a charged ion into a ytem when the ytem want to remain charge neutral. The econd term i novel, and repreent the energy of changing the charge of a particular pecie if, for example, an electron were added to a molecule. Interchanging between the different independent variable z and z, require the molar weight, M ma per mole, and Faraday contant, F c (96, 000 C/mole: Ã z = M F c Ã. z With (29 a the definition of the chemical potential and by uing (23 we can determine the relationhip between the chemical potential and the partial tre tenor. By eliminating the Lagrange multiplier and uing the definition of µ, (23 become t ρ ρ t N = ρ (µ µ N + ρ N (Ã Ã N +ρ (z z N Λ I. (31 Summing thee equation on from 1 to N and uing ρ µ = ρ Ã + p t (32 = (p q eλi + t e + εl ε tl + t a, (33

10 Swelling Porou Media with Electroquaitatic 10 where a indicate the anti-ymmetric portion of the tre tenor, t a = t t ym, one obtain an expreion for µ N, which when ubtituted back into e- quation (31 yield µ I = Ã I 1 ρ t + 1 ρ (t e + εl ε tl + t a + z Λ = 1,..., N. (34 The firt two term on the right-hand-ide form the claical chemical potential (ee Bowen, 11, although thee term do not produce a calar. In 7, 9 it wa hown that incorporating the tre tenor yield an appropriate definition of chemical potential when no electric field or charge exit. Note that changing the effective tre or hydration tre in a porou medium reult in a change in the chemical potential, which in turn produce a different balance of pecie within phae. The lat term incorporate charge neutrality and ugget that changing the location of charge change the chemical potential a well. Taking advantage of thee relation and implifying give u the diipative portion of the entropy inequality in Appendix A. 4 Near-Equilibrium Contitutive Retriction Equilibrium i defined to occur when the following variable, defined generically a x a, are zero: d l, ε l, v l,, v l,l, v,, = l, = 1,..., N 1 (35 ε l ρ l ê l, ω, ε l ρ l Ẑ l, = l, = 1,..., N. (36 Uing a dimenionality argument we can how that thee variable are functionally independent. Hence we have ( ε ρ T x Λ D 2 a = 0, (x a x b ( ε ρ T Λ D 0, (37 e e =l, =l, where ubcript e denote equilibrium. Note that we have not incorporated r and q into the above et. Thi i becaue without incorporating pecific chemical reaction, incorrect reult are obtained. So for example, conidering x a = d l, we have at equilibrium t l ym = (p l q l eλi. (38 To obtain reult which hold near equilibrium, we expand linearly about equilibrium. For example ( t l ym + (p l qeλi l + ρ l v l,l v l,l ( t l ym + (p l qeλi l eq neq + f 1 : d l + f 2 ε l +... (39

11 Swelling Porou Media with Electroquaitatic 11 where f 1 i a fourth order tenor and f 2 i a calar. Thee linearization coefficient are function of all independent variable which are not in the lit (35 or (36. In thi manner one can obtain cro effect, e.g. 18, and nonlinear term, e.g. 6. With the exception of v l, and v,, we chooe to linearize only about the one variable which produce a quadratic term in the entropy inequality, e.g. for the liquid phae tre tenor: t l ym (p l qeλi l + ν : d l ρ l v l,l v l,l (40 where ν i a fourth-order tenor. Note that imilar to (28, it i the term p l qeλ l which repreent normal force per unit area. We remark here that in order to obtain equilibrium reult the variable lited in (35 and (36 mut be functionally independent. However near equilibrium, one can linearize about any independent variable which i zero at equilibrium. We define the welling preure a π = ε ρ Ãl ε. (41 ρ,t,... which repreent the change in energy of phae with repect to the relative quantity of phae in the ytem. If the liquid and olid phae are noninteracting (non-welling, then the energy of either phae would not change if the volume fraction were changed (keeping the denitie fixed and thi quantity would be zero. A more detailed dicuion of p and π i forthcoming in another manucript. Uing thee definition and linearizing about ε l, we obtain µ l ε l = p l + π l (p + π ε oe T (E l E, (42 where µ l i the linearization coefficient and i not to be confued with the chemical potential. Thu if there are no effect of the electric field and p l +π l > p +π the volume fraction will change o a to increae the amount of the liquid phae. The lat term involve the difference in a portion of the electro-tre tenor t E = DE 1 2 ε oe EI (ee the conervation of linear momentum equation in Part I. Thu if the contribution of the electric field of one phae i greater than the other, then the equilibrium volume fraction will be affected, but the electrical potential doe not effect the equilibrium volume fraction. Linearizing about ω = v l d l yield the traditional reult ε t a = Q : ω, (43 where Q i a econd-order tenor which may be a function of all independent variable not necearily zero at equilibrium, ω i the anti-ymmetric part of the gradient of the velocity of phae, and t a i the anti-ymmetric part of the

12 Swelling Porou Media with Electroquaitatic 12 tre tenor. Thi expreion tate that the tre tenor are in general not ymmetric, which wa known from the conervation of angular momentum (ee Part I 5. Cro effect can alo be obtained 18. Linearizing about the conervation of charge exchange term, Ẑl, yield the following near-equilibrium reult ε l ρ l G l Ẑ l = ρ à ρ z ρl Ãl ρ l z l, (44 where G l i the linearization contant and where we aumed there are no cro effect. Thu a tranfer of ion of one pecie between phae occur only if there i an imbalance in the part of the electric-potential involving ion diaociation. Adorption relation are obtained by linearizing about the rate at which ma i tranfered from the olid phae to the liquid phae, ε l ρ l ê l. At equilibrium we obtain or ( µ l µ = (z z l z l ρ l l Ã Λ + ρ l z l z ρ ρ à z, (45 µ l = µ (46 which i why we defined the electro-chemical potential a we did. Thi i the boundary condition between phae, and in a more general framework (ee e.g. 3, 4 thi would produce the boundary condition for an omotic experiment in which the mixture on one ide contain ion. To obtain a near-equilibrium reult governing phae tranition, we linearize about ê l β. Neglecting quadratic term of relative velocitie yield ρ l K l ê l = µ µ l + ρ l z G l Ẑ l (47 where, by (44, the lat term on the right-hand-ide i zero at equilibrium. Thu the larger the difference in the electro-chemical potential, the fater the phae tranition occur. The relationhip between the chemical potential and the partial tre tenor of the liquid phae i obtained exactly a in the previou ection for the olid phae, except that the liquid phae reult hold only at equilibrium: µ l I = Ãl I 1 ρ l tl + z l ΛI. (48 5 Bulk-Phae Flow and Diffuion The equation which govern momentum balance in porou media are known a generalized Darcy equation, after Darcy, who in 1856 empirically derived

13 Swelling Porou Media with Electroquaitatic 13 the rather imple relationhip that flux i proportional to the gradient in fluid preure: ε l v l, = K p l + ε l ρ l g, (49 where K i the conductivity of the material. It i generally thought to be valid for low-moving vicou fluid through a homogeneou granular media. We would like to determine the generalization of thi law for the welling charged porou media conidered here. To begin with, we obtain a near equilibrium expreion for T l by linearizing the coefficient of v l, about v l, and the diffuive velocitie, v l,l in order to capture the effect of ion hydration (ee e.g. (39. Thi expreion may then be ubtituted into the conervation of momentum equation. Neglecting inertial effect, the Brinkman correction term, v l, a well a the term involving ω l (ee (43, and uing (38 to eliminate t l we obtain: K v l, = ε l p l + ε l ρ l (g l + g l I + π l ε l + ε l qee l T +ε l qe Λ l ε l ρ l Ãl E : ( E T ε l ρ l à l ( E T T E T ε l ρ l Ãl ( T ( 2 T r l v l,l, (50 where E Al : ( E T = E Al E i,k in indicial notation. The linearization coefficient K and r l i are econd-order tenor which may be a function of all independent variable not equal to zero at equilibrium, including ε l. The firt 2 term on the right-hand ide recover the tandard Darcy equation (49, except that p l i not the claical preure of normal force per unit area, but a thermodynamic definition of preure, ee equation (38. The third term involving g I, i due to fluctuation in the electric field, ee it definition in Appendix A of 5. The term not involving the electric field have been derived before 4 and thee reult are dicued in detail in 8. They indicate that flow can be driven by a gradient in the volume fraction if the medium well (π l 0 and alo by gradient in hear train, the latter of which may be appropriate for welling media with low water content. Thee term account for the chemical/hydration force between the olid and liquid phae. The term involving ( 2 T i likely negligible - firt order thermal effect are obtained by conidering cro term. In addition to thee term, we have the Lorentz force (ε l qee l T, the Kelvin force (uing equation (21 ε l P l E T, and a term enforcing charge neutrality, ε l qe Λ. l The lat term involving ummation over pecie i a cro-term put in pecifically to capture the hydrating effect of ion. Each charged particle urround itelf with water molecule, thereby impeding or enhancing bulk phae flow when the diffuive velocity of the charged pecie are non-negligible. Thi term i negligible for non-hydrating pecie. Alternatively we can re-write Darcy law in term of the bulk phae chemical potential, a thi i the formulation often ued in application 20, 23. Define

14 Swelling Porou Media with Electroquaitatic 14 the bulk phae potential a the Gibb potential: G l = C l µ l = Ãl + ρ l Ãl ρ l = Ãl + pl ρ l. (51 Further we wih to re-write the electrical forcing term in term of the Maxwell tre tenor: t l M = D l E T 1 2 ε oe l E T I, (52 Auming that the electric field are defined uch that the exchange term in Gau law i zero ( (ε l D l ε l q l e = 0 we have ε l q l ee T + ε l P l E T = (ε l D l E T 1 2 ε oe l E T I ε oe l E T ε l 1 2 ε oε l (E l E T E l E T. (53 Further, auming ε l ρ l g l I 1 2 ε oε l (E l E T E l E T = N 1 2 ε oε l < E E > l < E E > l (ee Part I, Appendice B and C, i negligible, we have K v l, = ε l ρ l G l + ε l (ρ l 2 Ãl ρ l Cl + ε l q l e Λ + ε l ρ l Ãl z l zl ε l E T P l + ε l ρ l g l + (ε l t l M + ε l ρ l Ãl T T ε oe T E l ε l r l v l,l, (54 which i comparable to what i derived in 20, 23. The firt term i denoted a the mechanochemical force ince it doe not incorporate the electrical potential, the econd term i due to omotic effect, the third and fourth term due to the electrical potential. The next two term on the econd line account for the Kelvin force and gravity, repectively, and the following term tate that the Maxwell tre tenor effect flow. Thermal effect appear explicitly in thi form, and ince à T < 0 (ee (20, we ee that flow goe from hot to cold region. The lat term on the econd line magnifie the effect due to gradient in volume fraction if electric field are non-negligible. The lat term i due to hydration and wa dicued above. Diffuion in a ingle-phae mixture i governed by Fick law, which tate diffuive velocity i proportional to the gradient of the chemical potential. Here we derive a novel form of Fick law. Begin with the coefficient of v l,l in the reidual entropy inequality which, when et to zero, give at equilibrium: ε ρ (î + T β ε ρ (î N + T N β = ε ρ (à à N

15 Swelling Porou Media with Electroquaitatic 15 ε E T (P ρ ρ P N + 1 N 2 ε oe T (E ρ ρ E N ε N ( +ρ Ã ρ Ã ( ρ (ε ρ + ε ρ Ã N z z ρ Ã ρ N z z N N Λ ( ε ρ (z z N ε (qe ( ρ ε t N = 1,..., N 1. ρ N ρ ρ q N e N Summing (55 over from 1 to N and making ue of the equilibrium relationhip ε l ρ l T l = ε l ρ l T l = ε l ρ l Ãl + E T ρ l Ãl ρ l (εl ρ l + ε l ρ l Ãl z l zl +ε l ρ l Ãl T T εl E T P l ε oe T E l ε l Λ (ε l q l e (56 we obtain (57 for the cae = N. Subtituting thi reult back into (55 and again making ue of (29 and (48 we obtain the equilibrium reult: ε l ρ l (îl + T l = ε l ρ l µ l (ε l t l + ε l q l e Λ +ε l ρ l Ãl T T εl E T P l + ε l ρ l (z l z l E T ε oe T E l ε l (57 Next we linearize the coefficient of v l,l about equilibrium by expanding the coefficient in the original entropy inequality given in Appendix B in term of both v l,l and v l, o that Onager principal i till atified. Thi allow u to obtain non-relative reult ince = 1,..., N. Here we make ue of the fact that Ãl and the primary independent variable, lited in (8, are the ame at equilibrium and near-equilibrium. That thi hold for Ãl i utified by Ãl not being a function of any of the variable which define equilibrium. Uing thi reult to eliminate ε l ρ l (îl + T l in the conervation of momentum equation, neglecting the inertial term, and approximating µ l uing equation (48, we obtain R l v l,l = ε l ρ l µ l + ε l ρ l (g l g l I + εl ρ l Ãl z l zl + ε l q l e Λ ε l ρ l z l E T + ε l ρ l Ãl T T kl v l, (58 (55

16 Swelling Porou Media with Electroquaitatic 16 where R l and k l are linearization contant and mut be uch that K, r l, R l and k l atify the Onager relationhip. Note that thi i imilar in form to the bulk flow equation (54. The Lorentz term, ε l ρ l z l E T, i actually a relative Lorentz term, and the form i a conequence of the bulk phae velocity, v l, being a ma-averaged velocity. To analyze thi further, aume that body force, g l i gravity, g, temperature gradient are negligible, the body force due to electric field fluctuation i negligible g l I = 0, and we enforce charge neutrality with the Lagrange multiplier, Λ, and that the charge number on pecie i contant ( z l = 0. In thi cae (58 implifie to R l v l,l = ε l ρ l µ l + ε l q l e Λ + ε l ρ l g ε l C l q l ee T k l v l, (59 The firt two term (on the right-hand-ide tate that the driving force are the chemical potential and the treaming potential. The Lorentz term involve the bulk-phae force weighted by the ma fraction, ut a the gravitational force i weighted by the ma denity. Thi term i new and hould be evaluated carefully, although imilar bulk term have been derived before, 2. The lat term in (59 i the reult of linearizing the diipative term about both v l, and v l,l and account for hydrating effect. 6 Dicuion We exploited the entropy inequality to obtain retriction on the form of contitutive relation for welling porou media compoed of a poibly polarizable olid and liquid phae, with charge and an electric field. Thi ha application in welling clay oil, biopolymer, biological membrane, puled electrophorei, chromotography, drug delivery, and other welling ytem. We did o under the philoophy that it i the total electric field which contribute to the force and work term in the conervation of momentum and energy, and that it i only the total electric field which i meaurable. Thi produced an additional forcing term involving the gradient of the volume fraction, and an additional body force which i a reult of fluctuation in the electric field, appearing in the macrocale conervation of momentum equation. The new body force term appear wherever gravity appear, and the extra term involving the gradient of the volume fraction, 1 2 ε oe T E l ε l, affect only the bulk phae flow when thi equation i written in term of the Maxwell tre tenor of the liquid phae, ee (54. Thee term are a conequence of auming that the primitive form of the momentum and energy equation i the form written in term of the Lorentz force and Kelvin force, intead of the Maxwell tre tenor (ee Part I. The Lagrange multiplier which enforce charge neutrality i hown to correpond with the electrical potential and i een to affect the macrocopic preure (ee e.g. (28 and (40, o that claical preure i not the thermodynamically defined preure, p, but i p q e Λ. Thu care mut be taken in interpreting the preure term given in the generalized Darcy law, (50.

17 Swelling Porou Media with Electroquaitatic 17 It wa aumed that ion-diaociation can occur, and thi i repreented by term which contain the partial of the Helmholtz free energy denity with repect to the charge per unit ma. Thi term can alo be related to the charge per molecule, ee the dicuion following (30. Thi term doe not affect preure, but doe manifet itelf in bulk phae flow and diffuion. The boundary condition between phae i a natural by-product of thi formulation, and it i hown that the electric field doe not influence the boundary condition, but that it i the balance of the electro-chemical potential which determine equilibrium, ee (46. Polarization enter into it only through the definition of Ã, ee (6. It wa alo hown that the rate at which the medium well i determined by the difference in the thermodynamic and welling preure, (p l + π l (p + π and the difference of a portion of the electro-tre tenor, ee equation (42. Acknowledgment. JHC would like to acknowledge the ARO / Terretrial Science and Mathematic diviion for upport under grant DAAG Reference 1 S. Achanta, J. H. Cuhman, and M. R. Oko. On multicomponent, multiphae thermomechanic with interface. International Journal of Engineering Science, 32(11: , R. Benach and I. Müller. Thermodynamic and the decription of magnetizable dielectric mixture of fluid. Archive for Rational Mechanic and Analyi, 53(4: , L. S. Bennethum and J. H. Cuhman. Multicale, hybrid mixture theory for welling ytem - I: Balance law. International Journal of Engineering Science, 34(2: , L. S. Bennethum and J. H. Cuhman. Multicale, hybrid mixture theory for welling ytem - II: Contitutive theory. International Journal of Engineering Science, 34(2: , L. S. Bennethum and J. H. Cuhman. Multicomponent, multiphae thermodynamic of welling porou media with electroquaitatic: I. macrocale field equation. Tranport in Porou Media, to appear, L. S. Bennethum and T. Giorgi. Generalized forchheimer law for twophae flow baed on hybrid mixture theory. Tranport in Porou Media, 26(3: , L. S. Bennethum, M. A. Murad, and J. H. Cuhman. Clarifying mixture theory and the macrocale chemical potential for porou media. International Journal of Engineering Science, 34(14: , 1996.

18 Swelling Porou Media with Electroquaitatic 18 8 L. S. Bennethum, M. A. Murad, and J. H. Cuhman. Modified darcy law, terzaghi effective tre principle and fick law for welling clay oil. Computer and Geotechnic, 20(3/4: , L. S. Bennethum, M. A. Murad, and J. H. Cuhman. Macrocale thermodynamic and the chemical potential for welling porou media. Tranport in Porou Media, 39(2: , A. Benouan, J.L. Lion, and G. Papanicolau. Aymptotic analyi of periodic tructure. North-Holland, Amterdam, R. M. Bowen. Theory of mixture. In A. C. Eringen, editor, Continuum Phyic. Academic Pre, Inc., New York, R. M. Bowen. Compreible porou media model by ue of the theory of mixture. International Journal of Engineering Science, 20: , H. B. Callen. Thermodynamic and an Introduction to Thermotatitic. John Wiley and Son, New York, B. D. Coleman and W. Noll. The thermodynamic of elatic material with heat conduction and vicoity. Archive for Rational Mechanic and Analyi, 13: , J. H. Cuhman. Molecular-cale lubrication. Nature, 347(6290: , J. Dougla, Jr. and T. Arbogat. Dual poroity model for flow in naturally fractured reervoir. In J. H. Cuhman, editor, Dynamic of Fluid in Hierarchical Porou Media, page Academic Pre, New York, A. C. Eringen. Mechanic of Continua. John Wiley and Son, New York, A. C. Eringen. A mixture theory of electromagnetim and uperconductivity. International Journal of Engineering Science, 36(5,6: , W. G. Gray and S. M. Haanizadeh. Unaturated flow theory including interfacial phenomena. Water Reource Reearch, 27: , W. Y. Gu, W. M. Lai, and V. C. Mow. Tranport of multi-electrolyte in charged hydrated biological oft tiue. Tranport in Porou Media, 34: , S. M. Haanizadeh and W. G. Gray. General conervation equation for multiphae ytem: 2. Ma, momenta, energy, and entropy equation. Advance in Water Reource, 2: , 1979.

19 Swelling Porou Media with Electroquaitatic S. M. Haanizadeh and W. G. Gray. General conervation equation for multiphae ytem: 3. Contitutive theory for porou media. Advance in Water Reource, 3:25 40, J. M. Huyghe and J. D. Janen. Thermo-chemo-electro-mechanical formulation of aturated charged porou olid. Tranport in Porou Media, 34: , D. Jou, J. Caa-Vázquez, and G. Lebon. Extended Irreverible Thermodynamic. Springer-Verlag, New York, I-Shih Liu. Method of lagrange multiplier for exploitation of the entropy principle. Archive for Rational Mechanic and Analyi, 46: , I. Müller and T. Ruggeri. Extended Thermodynamic. Springer-Verlag, New York, M. A. Murad, L. S. Bennethum, and J. H. Cuhman. A multi-cale theory of welling porou media: I. Application to one-dimenional conolidation. Tranport in Porou Media, 19:93 122, M. A. Murad and J. H. Cuhman. Multicale flow and deformation in hydrophilic welling porou media. International Journal of Engineering Science, 34(3: , O. A. Plumb and S. Whitaker. Diffuion, adorption and diperion in porou media: Small-cale averaging and local volume averaging. In J. H. Cuhman, editor, Dynamic of Fluid in Hierarchical Porou Media, page Academic Pre, New York, J. A. del Rio and S. Whitaker. Maxwell equation in two-phae ytem I: Local electrodynamic equilibrium. Tranport in Porou Media, 39: , J. A. del Rio and S. Whitaker. Maxwell equation in two-phae ytem II: Two-equation model. Tranport in Porou Media, 39: , E. Sanchez-Palencia. Non-homegeneou media and vibration theory. In Lecture Note in Phyic. Springer-Verlag, New York, V. Saidhar and E. Rucketein. Electrolyte omoi through capillarie. Journal of Colloid and Interface Science, 82(2: , C. Truedell and W. Noll. The Non-Linear Field Theorie of Mechanic. Handbuch der Phyik III/3. Springer-Verlag, S. Whitaker. Diffuion and diperion in porou media. AIChEJ, 13: , S. Whitaker. Advance in theory of fluid motion in porou media. Indutrial and Engineering Chemitry, 61(12:14 28, 1969.

20 Swelling Porou Media with Electroquaitatic 20 Appendix A. Diipative Entropy Inequality The diipative portion of the entropy inequality i ( ε ρ T Λ = D d l : ε l t l ym + + ε l ε l ρ l Ãl ε l ε l ρ l v l,l v l,l + ε l p l ε l qeλ l +v l, ε l ρ l Ãl + N 1 + ε ρ à ε l + p l p ε oe T (E l E ρ l Ãl ρ l (εl ρ l + ε l ρ l Ãl zl l z +ε l ρ l Ãl T T εl E T P l ε oe T E l ε l Λ (ε l qe l ε l ρ l l T v l,l : ε l t l ρl (q l e N 1 =l, =l, ρl ρ l ql N N e v, ΛI ρ l N tl N + ρ l ρ l ( l à ρ l à l ρ l N ρ l (Ãl Ãl N I ε ρ (î + T β + ε ρ (î N + T N β ( ε ρ (A A N ε E T (P ρ ρ P N N ( ε oe T (E ρ ρ E N ε + ρ à N ρ à ρ (ε ρ N ( ( +ε ρ à z z ρ à ρ N z z N Λ (ε ρ z (ε ρ z N N ( ε (qe ρ ρ e E T ε t N ε T T ρ T =l, ρ N q N q + (η + à T ρ N ( ρ v, (à (v, 2 t v, v, ε l ρ l ê l à l + à ρ l Ãl ρ l + ρ à ρ + ρ à z ρ z (zl

21 Swelling Porou Media with Electroquaitatic 21 +(z l z Λ v, v, 1 2 vl,l v l,l 1 2 vl, v l, =l, =l, ε ρ r ρ Ã ρ + z Λ 1 2 (v, 2 ω : ε t a + ε J E T + =l, ε l ρ l Ẑ l ρ ρ =l, =l, Ã z ρl Ãl ρ l z l (ε J ρ Ã q e ε q ρ Ã Λ qe Λ 0, (60 where ubcripted ym and a mean the ymmetric and anti-ymmetric part of the tenor, repectively.

22 Swelling Porou Media with Electroquaitatic 22 Appendix B. Entropy Inequality The entropy inequality in it entirety i: ε ρ T Λ = =l, =l, ε D ρ Dt ρ à ρ + E T ε l ρ l Ãl ε ρ à ε l P l ε P E T E T + T ε l ρ l η l ε l ρ l Ãl T ε ρ η ε ρ à T +d l : ε l t l ym + ε l P l E T I + +d : ω : =l, ε t ym + ε P E T I + =l, ε l ρ l F Ãl ε t a =l, + λ ρ + à + 1 ρ E T P + z Λ ε l ρ l (λ l ρ I + Ãl I + v l,l v l,l ε ρ (λ ρ I + à I + v, v, E ( F T ε ρ F à N 1 D v, Dt + T ε l ρ l Ãl T ε ρ à T D ρ ε ρ à Dt ( ρ E ( F T ε ρ à v, N + ε l (ρ l λ l ρ ρ λ ρ + ρ l A l ρ A ε oe T (E l E +E T (P l P + Λ(qe l qe ε l ρ l Ãl ε l ε ρ à ε l + v l, ε l ρ l Ãl v l, + ḋl : ε l ρ l Ãl d l + ẇ l : ε l ρ l Ãl w l + v l,l : ε l ρ l Ãl +v l, ε l ρ l Ãl ε l εl ( v l,l ε l ρ l Ãl ρ l ρl ε l ρ l à l E : ( E T ε l ρ l à ( E T T E T l

23 Swelling Porou Media with Electroquaitatic 23 + ε l ρ l Ãl z l zl ε l E T P l ε l ρ l T l ε oe T E l ε l (Ãl + 1 ρ l E T P l (ε l ρ l + λ l ρ (ε l ρ l + ε l ρ l λ l qe z l ε l ρ l Ãl N 1 v l, ( vl, T ε l ρ l Ãl v l,l ( vl,l T ε l ρ l à T 2 T ε l ρ l Ãl v l,l : 2 v l,l ε l ρ l Ãl ( ρ l 2 ρ l ε l ρ l Ãl d l : ( d l T ε l ρ l à l w l : ( wl T ( v, T : ε t + ε P E T I + ε ρ λ ρ I =l, =l, ε T T T ρ ( =l, q + η + à T v, ( ρ v, (à (v, 2 t v, v, ε ρ î ε ρ T β β ε ρ à + 1 ρ E T P (ε ρ ε E T P ε oe T E ε +λ ρ (ε ρ + ε ρ λ q e z ε qe E T ε J E T =l, =l, β ε ρ ê β à 1 2 (v, 2 =l, ε ρ r 1 à D =l, ε λ D à 1 ρ E T P 1 2 (v, 2 λ ρ ρ E T P λ ρ 1 2 (v, 2 l

24 Swelling Porou Media with Electroquaitatic 24 =l, =l, =l, =l, =l, D z ε ρ Ã Dt z E Λ E ε λ D + ε ρ λ q e + ε ρ Λ D ε ε qe ε d ε d β β Λ E ε E ε σ ε σ β β λ q e (ε J ε q ε ρ Ẑ β 0, β where ubcripted ym and a mean the ymmetric and anti-ymmetric part of the tenor, repectively.