A DYNAMIC MODEL OF POLYELECTROLYTE GELS

Transcription

1 A DYNAMIC MODEL OF POLYELECTROLYTE GELS M.CARME CALDERER, HAORAN CHEN, CATHERINE MICEK, AND Y. MORI Abstract. We erive a moel o the couple mechanical an electrochemical eects o polyelectrolyte gels. We assume that the gel, which is immerse in a lui omain, is an immiscible an incompressible mixture o a soli polymeric component an the lui. As the gel swells an e-swells, the gel-lui interace can move. Our moel consists o a system o partial ierential equations or mass an linear momentum balance o the polymer an lui components o the gel, the Navier- Stokes equations in the surrouning lui omain, an the Poisson-Nernst-Planck equations or the ionic concentrations on the whole omain. These are supplemente by a novel an general class o bounary conitions expressing mass an linear momentum balance across the moving gel-lui interace. Our bounary conitions inclue the permeability bounary conitions propose in earlier stuies. A salient eature o our moel is that it satisies a ree energy issipation ientity, in accorance with the secon law o thermoynamics. We also show, using bounary layer analysis, that the well-establishe Donnan conition or equilibrium arises naturally as a consequence o taking the electroneutral it in our moel. 1. Introuction. Gels are crosslinke, three imensional polymer networks that absorb solvent an swell without issolution [25, 26, 17, 44, 4]. Some gels can experience large changes in volume in response to small changes in various environmental parameters incluing temperature, ph or the ionic composition o the solvent [42]. In some gels, this change is iscontinuous with respect to changes in the environmental parameter. This is the volume phase transition, irst escribe in [43]. One interesting eature o the volume phase transition is that it exhibits hysteresis, a eature that istinguishes it rom the amiliar liqui-gas phase transition [42, 13]. These large volume changes have oun use in various artiicial evices incluing isposable iapers [37], rug elivery evices [35, 10] an chemical actuators [14, 22]. Gels aboun in nature are thought to play an important role in certain physiological systems [53, 47]. The stuy o gel swelling, an more generally o gel ynamics, is thus important rom both practical an theoretical stanpoints. In this paper, we ocus on polyelectrolyte gels; the polymer network contains ixe charge groups that issociate an eliver counterions into the solvent. Polyelectrolyte gels orm an important class o gels stuie experimentally an use in applications. Inee, the volume phase transition is most easily realize in polyelectrolyte gels [15, 42]. Most biological gels are also o this type. Many o the early theoretical stuies on gels ocuse on the static equilibrium state. A pioneering stuy on the ynamics o gels is [45], in which the authors examine the ynamics o an electrically neutral gel aroun an equilibrium swelle state. As such, this was a small eormation theory. The ynamic theory o electrically neutral gels has since been evelope by many authors [8, 51, 39, 11, 12, 28, 29]. Statics o polyelectrolyte gels are stuie in [38], which has since been extene in many irections [42, 23]. The ynamics o polyelectrolyte gels has also receive a great eal o attention, an systems o evolution equations have been propose by many authors [18, 16, 19, 31, 34, 49, 48, 52, 32, 9, 1, 23]. A stanar approach, which we aopt in this paper, is to treat polyelectrolyte gels as a two phase meium School o Mathematics, University o Minnesota, 206 Church St. SE, Minneapolis MN, Supporte by NSF-DMS School o Mathematics, University o Minnesota, 206 Church St. SE, Minneapolis MN, Department o Mathematics, Augsburg College, 2211 Riversie Ave. S, Minneapolis MN School o Mathematics, University o Minnesota, 206 Church St. SE, Minneapolis MN, Supporte by NSF-DMS , the Alre P. Sloan Founation an the McKnight Founation.. 1

2 o polymer network an lui, with the ions being treate as solute species issolve in the lui. But even within this same approach, there are various isagreements among the ierent moels propose by ierent authors. In this paper, we propose a system o partial ierential equations PDEs escribing the ynamics o a polyelectrolyte gel immerse in lui. The istinguishing eature o our moel is that its ormulation is guie by the requirement that the system, as a whole, must satisy a ree energy ientity. This energetic ramework allows or the unambiguous etermination o the orm o the ionic electroiusion equations an o the coupling between electrical eects an the mechanics o the gel. This clariies the conusion seen among ierent moels propose in the literature on the ynamics o polyelectrolyte gels. The energetic approach also allows or the ormulation o general interace conitions at the gel-lui interace; inee, in previous work, the treatment o bounary or interace conitions has been simplistic, i not an aterthought. We also examine the electroneutral it. In this it, we recover the Donnan equilibrium conition at the gel-lui interace. The paper is organize as ollows. In Section 2, we present the skeletal ramework o our moel. The gel is treate as a two-phase meium consisting o the polymer network phase an a lui phase. We write own general mass an momentum balance equations as well as the interace conitions at the gel-lui interace. In Section 3, we propose a purely mechanical moel o a neutral gel immerse in lui. We speciy the polymer stress an lui stresses an the boy orce acting between the polymer network an lui. We propose a novel class o bounary conitions at the gel-lui interace. Some previously propose bounary conitions can be seen as iting cases o the general class we present here [8, 51, 39]. We then prove a ree energy ientity satisie by this system. In Section 4, we iscuss the inclusion o ionic electroiusion. The ions satisy the electroiusion-avection equations an the electrostatic potential satisies the Poisson equation. The mechanical stress an boy orces set orth in Section 3 are now augmente by the electrical orces. We conclue the section by showing that this system too satisies a ree energy ientity. The Debye length is oten very small in polyelectrolyte gels, an thus the gel as well as the surrouning lui is nearly electroneutral except or thin bounary layers that may orm at the gel-lui interace. For gels at physiological salt concentrations, or example, the Debye length is on the orer o 1nm [40]. In Section 5, we ormulate the appropriate equations in this electroneutral it. We perorm a bounary layer analysis to euce the appropriate interace conitions in the electroneutral it an in that the van t Ho law or osmotic pressure arises naturally in this it. It is easily checke that the conition or steay state gives us the well-establishe Donnan conition, set orth in the context o polyelectrolyte gels in [38]. 2. Mass, Momentum an Energy Balance. We consier a gel that is in contact with its own lui. We moel the gel as an immiscible, incompressible mixture o two components, polymer network an solvent. The gel an the lui occupy a smooth boune region U R 3. Let U be the region where the gel is present at time t. We assume that, U =. This means that the gel is completely immerse in the lui. We enote the lui region by R t U\ Figure 2.1. At each point in, eine the volume ractions o the polymer component φ 1 an that o the solvent component φ 2. Assuming that there are neither vois nor aitional volume-occupying components in the system, we have: φ 1 + φ 2 =

3 Fig A iagram o a gel immerse in lui. The regions an R t are the gel an lui regions respectively an is the gel-lui interace. The epenent variables in are: φ 1, φ 2, v 1, v 2, p, F, c k k = 1,, N, ψ. In R t, they are: v, p, c k k = 1,, N, ψ. F is the eormation graient, to be iscusse in Section 3. c k are the ionic concentrations an ψ is the electrostatic potential, iscusse in Section 4. At the gel-lui interace, we have the variables w an q eine in 2.13 an 2.14 respectively. or any point insie. Let v i, i = 1, 2 be the velocities o the polymer an solvent components respectively. The volume ractions φ 1 an φ 2 satisy the volume transport equations: φ i t + v iφ i = 0, i = 1, The above an 2.1 implies the ollowing incompressibility conition or the gel: φ 1 v 1 + φ 2 v 2 = Let γ i, i = 1, 2 be the intrinsic mass ensities o the polymer an solvent components respectively. The mass ensity o each component is, then, given by γ i φ i, i = 1, 2. We assume that γ i, i = 1, 2 are positive constants. Multiplying 2.2 by γ i, 2.2 may also be seen as a statement o mass balance. Force balance is given as ollows: 0 = T i φ i p i + i + g i, i = 1, where T i an p i are the stress tensors an pressures in the polymer an solvent components. The term i + g i is the boy orce, which we ivie into two parts or reasons that will become clear below. Inertial eects are selom, i ever, signiicant in gel ynamics, an we have thus neglecte inertial terms. We henceorth let: p 1 = p + p, p 2 = p. 2.5 We shall reer to p 2 = p as the pressure an to p as the pressure ierence. This pressure ierence is reerre to as the capillary pressure in the theory o mixtures [7, 46]. The pressure p is etermine by the incompressibility constraint. We require that p an i satisy the ollowing conition. There exists a tensor S g such that: S g = φ 1 p The signiicance o this conition can be seen as ollows. I we take the sum o equation 2.4 in i an use 2.6, we have: 0 = T 1 + T 2 + S g pi + g 1 + g 2, 2.7 3

4 where I is the 3 3 ientity matrix. Conition 2.6 thus ensures that the total orce acting on the gel at each point can be written as the ivergence o a tensor except or the boy orces g i. To ensure local orce balance, we require g 1 + g 2 = We assume R t is ille with an incompressible lui. Let v be the velocity iel o the lui in R t. 0 = T p +, 2.9 v = 0, 2.10 where T is the stress tensor, p is the pressure an is the boy orce. We have neglecte inertial eects. We require, as in 2.6, that can be written as the ivergence o a tensor: S = In this paper, we o not consier extra boy orces in R t that cannot be written as ivergence o a tensor. We must speciy the constitutive relations or the stress tensors T i, T, the boy orces i, g i, an the pressure ierence p. We relegate this iscussion to later sections since the calculations in this section o not epen on the speciic orm o the stresses or the boy orces. Suice it to say, at this point, that the lui will be treate as viscous an the polymer phase as preominantly elastic. The boy orce may inclue a riction term an an electrostatic term. We now turn to bounary conitions. First o all, notice that is the region where the gel is present, an thus, must move with the velocity o the polymer component. Let n be the unit normal vector on pointing outwar rom into R t. Let v Γ be the normal velocity o where we take the outwar irection to be positive. We have: v 1 n = v Γ on By conservation o mass o the lui phase, we must have: v v 1 n = φ 2 v 2 v 1 n w 2.13 Let us postulate that the water velocity tangential to the surace Γ is equal on both sies o Γ: v v 1 = v 2 v 1 q 2.14 Note that q is a vector that is tangent to the membrane. It is easily checke that the bounary conitions 2.13 an 2.14 lea to conservation o lui in the whole region U. Force balance across the interace Γ is given as ollows. T + S n T 1 + T 2 + S g n + [p]n = 0, [p] = p Ωt p Rt 2.15 where Ωt an Rt shall henceorth enote evaluation on the sie an the R t sie o respectively. We also use the notation [ ] Ωt Rt to enote the jump in the enclose quantity across. In particular, we have: T + S n T 1 + T 2 + S g n q =

5 since q is tangential to the surace Γ. On the outer surace U, we let: v = At this point, we have the bounary conitions 2.14, 2.13 an 2.15 on Γ, which together give us six bounary conitions. Mass balance an total orce balance woul provie the necessary number o bounary conitions i the interior o were compose o a one-phase meium. Here, the interior o is a two-phase gel. We thus require aitional bounary conitions. This will be iscusse in subsequent sections. We conclue this Section by stating a result that will be useul later in iscussing ree energy issipation. Lemma 2.1. Suppose φ i, v i, v are smooth unctions that satisy 2.1, 2.2, 2.4, 2.9, 2.10, 2.13, 2.14, 2.15, 2.17 an suppose p, i, satisy 2.6 an We have: 0 = v 1 : T 1 + v 2 : T 2 x v : T x Ω + φ 1 p g 1 v g 2 v 2 x + v x 2.18 R t Sg S n v 1 + wπ + q Π S, Γ where Π T 1 n is the component o T 1 n parallel to the bounary an Π = n T n n T2 φ 2 n + [p] Proo. Multiply 2.4 by v i an integrate over. Likewise, multiply 2.9 by v an integrate over R t. Aing these expressions, we have 0 = v 1 T 1 + v 2 T 2 x + v T x Ω + φ 1 p g 1 v g 2 v 2 x Ω t φ 1 v 1 + φ 2 v 2 px v px R t = v 1 : T 1 + v 2 : T 2 x v : T x R t + φ 1 p g 1 v g 2 v 2 x Ω t + T 1 n φ 1 pn v 1 + T 2 n φ 2 pn v 2 T n pn v S where we use 2.3 an 2.10 in the secon equality. 5 Let us evaluate the last

6 bounary integral. Using 2.13, 2.14 an 2.15, 2.16 we have: T 1 n φ 1 pn v 1 + T 2 n φ 2 pn v 2 T n pn v =T 1 + T 2 n T n [p]n v 1 T2 + n n n T n [p] w + T 2 n T n q φ 2 T2 = S g S g n v 1 + n n n T n [p] w T 1 n q. φ 2 This yiels A Mechanical Moel. We now consier a mechanical moel o the gel immerse in lui. To escribe the elasticity o the polymer component, we consier the ollowing. Let Ω R 3 be the reerence omain o the polymer network, with coorinates X. A point X Ω is mappe to a point x by the smooth eormation map ϕ t : x = ϕ t X. 3.1 Henceorth, the small case x enotes position in an the large case X enotes position in the reerence omain Ω. Note that the velocity o the polymer phase v 1 an ϕ t are relate through: v 1 ϕ t X, t = t ϕ tx. 3.2 We let F = X ϕ t be the eormation graient, where X enotes the graient with respect to the reerence coorinate. Deine F = F ϕ 1 t so that F is the eormation graient evaluate in rather than in Ω. It is a irect consequence o 3.2 that F satisies the ollowing transport equation: F = F t t + v 1 F = v 1 F. 3.3 X=ϕ 1 t x Using the eormation graient, conition 2.2 or i = 1 can be expresse as: φ 1 ϕ t etf = φ I 3.4 where φ I is a unction eine on hat takes values between 0 an 1. This shoul be clear rom the meaning o the eormation graient, but can also be checke irectly using 3.3 an 2.2. The unction φ I is the volume raction o the polymer component in the reerence coniguration. We suppose that the stress o the gel is given by the ollowing: T 1 = T visc 1 + T elas 1, T1 visc = η 1 v 1 + v 1 T, T1 elas W elas F = φ 1 F T. F 3.5 The viscosity η 1 > 0 may be a unction o φ 1, in which case we assume this unction is smooth an boune. The unction φ 1 W elas is the elastic energy per unit volume. An example orm o W elas is given by [39]: 1 W elas = µ E F 2p I 2p + I 2p 1 etf β 1, p 1, 3.6 2p β 6

7 where is the Frobenius norm o the matrix an I is the 3 3 ientity matrix, µ E is the elastic moulus, an β is a moulus relate to polymer compressibility. The above reuces to compressible neo-hookean elasticity i we set p = 1. The pressure ierence p is given as ollows: p = W FH φ 1, W FH = k BT v s vs v p φ 1 ln φ 1 + φ 2 ln φ 2 + χφ 1 φ The unction W FH is the Flory-Huggins mixing energy where k B T is the Boltzmann constant times absolute temperature, v p an v s are the volume occupie by a single molecule o polymer an solvent respectively an χ is a parameter that escribes the interaction energy between the polymer an solvent. Since v s v p or a crosslinke polymer network, the irst term in W FH is oten taken to be 0. By substituting φ 2 = 1 φ 1 rom 2.1, we may view W FH as a unction only o φ 1. Given that W FH is a mixing energy, it oes not belong to either the solvent or the polymer phase. This is relecte in the act that the pressure ierence p is symmetric with respect to the role playe by the polymer an solvent components in the ollowing sense: p = p 1 p 2 = W FH φ 1 = W FH φ 2 = p 2 p 1, 3.8 where we use 2.5 an W FH /φ 2 above reers to the erivative o W FH when viewe as a unction o φ 2 only. We let: 1 = 2 = I we set S g = S FH g W FH φ 1 φ 1 W FH φ 1 I, 3.10 φ 1 we in that 2.6 is satisie. We point out that there is some arbitrariness in what we call the pressure ierence an what we call the stress. Inee we may prescribe T 1 an p as ollows to obtain exactly the same equations as is obtaine when using 3.5 an 3.7: T 1 = T visc 1 + T elas 1 + S FH g, p = where I is the 3 3 ientity matrix. Though mathematically equivalent, we in 3.5 an 3.7 more physically appealing since the polymer an solvent phases are treate symmetrically as ar as the Flory-Huggins energy is concerne. The boy orces g i are given by: g 1 = g ric = κv 1 v 2, g 2 = g ric Here, κ > 0 is the riction coeicient an may epen on φ 1. satisie, an we thus have local orce balance. We assume that the lui is viscous: Note that 2.8 is T 2, = η 2, v2, + v 2, T

8 The viscosity η > 0 is a constant an η 2 > 0 may be a smooth an boune unction o φ 2 or equivalently, φ 1. For the boy orce, in the lui, we let: = 0, S = Conition 2.11 is trivially satisie with the above einitions o an S. We now turn to bounary conitions. As was mentione in the previous section, 2.13, 2.14 an 2.15 provie only six bounary conitions. We nee three bounary conitions or each o the components in contact with the interace. We have three components, the polymer an solvent components o the gel an the surrouning lui. We thus nee nine bounary conitions. We now speciy the remaining three bounary conitions. They are: η w = Π, 3.15 η q = Π T 1 n, 3.16 where Π was eine in 2.19 an η an η are positive constants. We point out that Π shoul be interprete as the ierence in lui normal stresses across the gel-lui interace. Equation 3.15 thus states that lui low across the gel-lui interace is proportional to this normal lui stress ierence. I we set η = 0, there is no interacial riction or water low an the normal lui stresses balance. I we set η, w = 0 an the gel-lui interace is impermeable to lui low. Conition 3.16 is a Navier-type slip bounary conition. I η, this amounts to taking q = 0. This will give us a tangential no-slip bounary conition or the lui. The ollowing result states that the total ree energy, given as the sum o the polymer elastic energy E elas, the Flory-Huggins mixing energy E FH, ecreases through viscous or rictional issipation in the bulk I visc an on the interace J visc. Theorem 3.1. Let φ i, v i, an v, be smooth unctions that satisy 2.1, 2.2, 2.4, 2.9, 2.10, 2.13, 2.14, 2.15, 2.17, 3.15, 3.16 where the stress tensors, pressure ierence an boy orces are given by 3.5, 3.13, 3.7, 3.9, 3.12 an Then, we have ree energy issipation in the ollowing sense: t E elas + E FH = I visc J visc, E elas = φ 1 W elas Fx, E FH = W FH φ 1 x, 2 I visc = 2η i S v i 2 + κ v 1 v 2 2 x + 2η S v 2 x, i=1 R t J visc = η w 2 + η q 2 S, 3.17 where S is the symmetric part o the corresponing velocity graient an enotes the Frobenius norm o the 3 3 matrix. Proo. We can prove 3.17 by a irect application o Lemma 2.1. Substitute 3.5, 3.13, 3.7, 3.9, 3.12, 3.14, 3.15 an 3.16 into We see that 8

9 3.17 is immeiate i we can show the ollowing two ientities: φ 1 W elas Fx = v 1 : T1 elas x, 3.18 t Ω t W FH φ 1 x = S g S n v 1 S t φ 1 p + 1 v v 2 x First consier 3.18: φ 1 W elas Fx = φ I W elas F X t t Ω W elas F = φ I : F Ω F t X = W elasf φ 1 : v 1 F x F W elas F = φ 1 F T : v 1 x = v 1 : T1 elas x, F 3.20 where we use 3.4 in the irst an thir equalities an 3.3 in the thir equality. Let us now evaluate the right han sie o 3.19: S g n v 1 S + φ 1 p v 1 x Ω t W FH = W FH φ 1 v 1 ns + φ 1 Ω t WFH φ 1 = W FH v 1 ns + φ 1 t φ 1 v 1 x = t WFH φ 1 W FH x, where we integrate by parts an use 2.2 in the secon equality. x Electroiusion o Ions. We now consier the incorporation o iusing ionic species into the moel. Let c k, k = 1,, N be the concentrations o ionic species o interest an let z k be the valence o each ionic species. These electrolytes are present in the solvent as well as in the outsie lui. We eine c k as being concentrations with respect to the solvent, an not with respect to unit volume. The polymer network carries a charge ensity o ρ p φ 1 per unit volume, where ρ p is a constant. Now, eine W ion as ollows: W ion = k B T c k ln c k, 4.1 where k B T is the Boltzmann constant times absolute temperature. The quantity W ion is the entropic ree energy o ions per unit volume o solvent or lui. Much o the calculations to ollow o not epen on the above speciic orm o W ion, but this is the most commonly use orm. Using this, we eine the chemical potential µ k o the k-th ionic species as: µ k = W ion c k + qz k ψ = k B T ln c k + qz k ψ + k B T

10 where q is the elementary charge an ψ is the electrostatic potential, eine in both an R t. In, the concentrations c k satisy: t φ Dk c k 2c k + v 2 φ 2 c k = k B T µ k = D k c k + qz 4.3 kc k k B T ψ, where D k > 0, k = 1,, N are the iusion coeicients o the ions. The iusion coeicients may be unctions o φ 2, in which case we suppose that they are smooth boune unctions o φ 2. Note that the choice 4.1 or W ion leas to linear iusion. We point out here that our choice o the unction W ion resulte in ionic iusion being proportional to the graient o c k, not φ 2 c k. Recall that c k is the concentration per unit solvent volume whereas φ 2 c k is concentration per unit volume. There are moels in the literature in which ionic iusion is proportional to the graient o φ 2 c k instea o c k. Our choice stems rom the view that, since ions are issolve in water solvent, it can only iuse within the water phase. We also point out that we o not inclue the eects o ion recombination, such as the protonation an e-protonation o charge polymeric sie chains. In the lui region R t, the concentrations satisy: c k t + v c k = Dk c k k B T µ k. 4.4 In an in R t, the ions iuse an rit own the electrostatic potential graient an are avecte by the local lui low. The electrostatic potential ψ satisies the Poisson equation: ɛ ψ = { φ 1 ρ p + N qz kφ 2 c k in N qz kc k in R t 4.5 where ɛ is the ielectric constant. The ielectric constant in the gel may well be ierent rom that insie R t. We assume that ɛ may be a smooth an boune unction o φ 1 in an remains constant in R t. I we set the avective velocities to 0, equations 4.3, 4.4 an 4.5 are nothing other than the Poisson-Nernst-Planck system [41]. In many practical cases, the ielectric constant is small an it is an excellent approximation to let ɛ 0 in the above. We shall iscuss this electroneutral it in Section 5. We set the orces 1 an 2 as ollows: 1 = 1 elec = ρ p φ 1 ψ, N 2 = 2 elec = qz k φ 2 c k ψ, = elec N = qz k c k ψ. 4.6 These are the electrostatic orces acting on the polymer network an the lui. We prescribe the pressure ierence as ollows: p = p FH + p elec, p FH = W FH, p elec = 1 ɛ ψ φ 1 2 φ 1 10

11 The term p elec is known as the Helmholtz orce [30]. We point out that the einition o p elec is symmetric with respect to φ 1 an φ 2, as can be seen by an argument ientical to 3.8. Prescription 4.7 o p assumes that p can be separate into p FH, the mixing contribution, an p elec, the ielectric contribution. The interaction o ions an the charge polymeric sie chains enters only through the electrostatic potential etermine by the Poisson equation 4.5 or the electroneurality conition 5.15 i we take the electroneutral it, see Section 5. Our treatment oes not take into account interactions between the solvent ions an the charge polymeric sie chains beyon the above mean iel eects, in line with classical mean iel treatments o the statics o polyelectrolyte gels [38, 42]. Counterion conensation is such an eect, which may be signiicant especially at high ionic concentrations. Our use o W FH as the interaction energy between charge polymer an solvent is thus an approximation that may be vali only in relatively low ionic concentrations. With the ollowing einitions or S g an S, conitions 2.11 an 2.6 are satisie. S g = S FH g S = S elec + Sg elec, Sg elec = S mw, Sg, mw ɛ = Sg mw φ ɛ 1 ψ 2 I, φ 1 ψ ψ 1 2 ψ 2 I 4.8 where Sg FH was given in The tensor S mw is the stanar Maxwell stress tensor in the absence o a magnetic iel [30, 24]. Insie the gel there is an aitional term to account or the non-uniormity o the ielectric constant. We prescribe g i as in We also aopt the bounary conitions 3.15 an We must provie 4.3, 4.4 an 4.5 with bounary conitions. We require that the ionic concentrations c k the lux across be continuous: [c k ] = 0, 4.9 v v 1 c k D kc k k B T µ k n = v 2 v 1 φ 2 c k D kc k Rt k B T µ k n j k. Ωt We have name the concentration lux j k or later convenience. For the electrostatic potential ψ, we require: 4.10 [ψ] = [ɛ ψ n] = Finally, at the outer bounary o U, we require the ollowing no-lux bounary conitions or both c k an ψ, on U: v c k D kc k k B T µ k n = 0, 4.12 ɛ ψ n = It is easily checke that the above bounary conitions lea to conservation o total amount o ions. For the Poisson equation to have a solution, we must require, by the Freholm alternative, that: R t N qz k c k x + N qz k c k + ρ p φ 1 x =

12 The electrostatic potential ψ is only etermine up to an aitive constant. Given that the amount o ions an the amount o polymer integral o φ 1 are conserve, the above conition will be satisie so long as it is satisie at the initial time. We now turn to linear momentum an ree energy balance. We point out that a relate energy ientity or a ierent system was prove recently in [36]. Theorem 4.1. Let φ i, v i an v be smooth unctions satisying 2.1, 2.2, 2.4, 2.9, 2.10, 2.13, 2.14, 2.15, 2.17, 3.15, 3.16 an c k an ψ are smooth unctions satisying 4.3, 4.4, 4.5, 4.10 an Suppose the stress, pressure ierence an the boy orces are given by 3.5, 3.13, 4.7, 4.6 an Then, we have the ollowing ree energy issipation ientity: t E elas + E FH + E ion + E elec = I visc I i J visc, 1 E ion = φ 2 W ion x + W ion x, E elec = R t U 2 ɛ ψ 2 x, D k c k I i = k B T µ k 2 x, U 4.15 where E elas, E FH, I visc an J visc were eine in Proo. Substitute 3.5, into 2.18, Using Lemma 2.1 an the results o Theorem 3.1, we have: t E elas + E FH = I visc J visc + P elec, P elec = φ 1 p elec + 1 elec v elec v 2 x Ω t + elec v x Sg elec S elec n v 1 S. R t Comparing this with 4.15, we must show that: 4.16 t E ion + E elec = I i P elec First, multiply 2.2 with i = 1 by ρ p ψ an integrate in : φ1 ρ p ψ t + φ 1v 1 x = ψ ρ p φ 1 ψx ρ p φ 1 t t x + 1 elec v 1 x = 0, 4.18 where we integrate by parts an use the act that ρ p is a constant. Note that we use 4.6 to rewrite the thir integral in terms o 1 elec. Multiply 4.3 by µ k an integrate over. The let han sie gives: N µ k t φ 2c k + v 2 φ 2 c k x = + N N W ion c k t φ 2c k + v 2 φ 2 c k x qz k ψ t φ 2c k + v 2 φ 2 c k x = S 1 + S

13 To simpliy S 1, note that: W ion c k t φ 2c k + v 2 φ 2 c k = φ 2 W ion c k =φ 2 t W ion + v 2 W ion = t φ 2W ion + v 2 φ 2 W ion ck t + v 2 c k 4.20 where we use 2.2 with i = 2 in the secon an thir equalities. Thereore, we have: S 1 = t φ 2W ion + v 2 φ 2 W ion x = φ 2 W ion x + φ 2 W ion v 2 v 1 ns. t 4.21 where we integrate by parts in the secon equality. Let us now turn to S 2. Integrating by parts, we obtain: S 2 = t ψ + 2 elec v 2 x + qz k φ 2 c k x ψ ψ t qz k φ 2 c k x qz k c k φ 2 v 2 v 1 n x, 4.22 where we use 4.6 to write the thir integral in terms o elec 2. I we multiply the right han sie o 4.3 by µ k, sum in k an integrate in, we have: = N N Dk c k µ k k B T µ k x µ k D k c k k B T µ k n Collecting 4.18, 4.19, , we have: t = ψ t φ 2 W ion + ψ φ 1 ρ p + N D k c k S k B T µ k 2 x. φ 1 ρ p + qz k φ 2 c k x qz k φ 2 c k x N N j k µ k N D k c k k B T µ k 2 c k W ion c k W ion w S x 1 elec v elec v 2 x where we use 2.13 an Multiplying 4.4 by µ k, taking the sum in k an 13

14 integrating over R t, we obtain, similarly to 4.24: ψ W ion + ψ qz k c k x qz k c k x t R t R t t N N W ion = j k µ k c k W ion w S c k N D k c k k B T µ k 2 x elec v x. R t R t 4.25 Aing 4.24 an 4.25 an using the act that c k an ψ are continuous across, we have,: t E ion ψ ψ ɛ ψx + t U U t ɛ ψx 4.26 = I i P elec φ 1 v 1 p elec x Sg elec S elec n v 1 S. Now consier the two integrals on the irst line o 4.26: ψ ɛ ψx = ɛ ψ 2 x + [ ψɛ ψ ] S = ɛ ψ 2 x t U t U t n t U 4.27 where we use the continuity o ψ an 4.11 in the secon equality. Let us now turn to the secon integral in the irst line o 4.26 ψ U t ɛ ψx = ɛ ψ ψ [ U t x + ɛ ψ n = ɛ 1 ɛ t U 2 ψ 2 x + 2 t ψ 2 x + = ɛ 1 ɛ t U 2 ψ 2 x φ 1 v 1 ψ 2 x 2 φ 1 [ [ ɛ + 2 ψ 2] v 1 n + ɛ ψ ] ψ S n t = ɛ t U 2 ψ 2 x φ 1 v 1 p elec x Ω t [ ɛ + 2 ψ 2] 1 [ 2 φ ɛ 1 ψ 2 v 1 n + ɛ ψ φ 1 n ] ψ S t [ ɛ 2 ψ 2] v 1 n + ] ψ S t [ ɛ ψ n ] ψ S t 4.28 where we use 2.2 with i = 1 in the irst equality an use 4.7 in the thir equality. Substituting 4.27 an 4.27 into 4.26, we have: [ ɛ t E ion + E elec + 2 ψ 2] 1 [ 2 φ ɛ 1 ψ 2 v 1 n + ɛ ψ ] ψ S φ 1 n t = I i P elec Sg elec S elec n v 1 S

15 Using the einition o Sg, elec in 4.8, the above reuces to: [ t E ion + E elec + ɛ ψ ] ψ n t + ɛ ψ ψn v 1 S = I i P elec Let us examine the integran in the above integral: [ ɛ ψ ] ψ n t + ɛ ψ ψn v 1 = ɛ ψ [ ] ψ n t + v 1 ψ, 4.31 where we use Note that the continuity o ψ across see 4.11 implies that the jump on the right han sie must be 0 given that v 1 coincies with the velocity o. We have thus shown 4.17 an this conclues the proo. 5. Electroneutral Limit Electroneutral Moel an the Energy Ientity. To iscuss the electroneutral it, we irst non-imensionalize our system o equations. We irst consier the scalar equations 2.2, Introuce the prime imensionless variables: x = Lx, v 1,2, = V 0 v 1,2,, t = L V 0 t, D k = D 0 D k, c k = c 0 c k, ψ = k BT q ψ, ρ p = qc 0 ρ p, ɛ = ɛ ɛ, 5.1 where L is the characteristic length the size o the gel an c 0, V 0 an D 0 are the representative ionic concentration, velocity an iusion coeicient respectively. We shall prescribe V 0 in 5.9. The ielectric constant is scale with respect to ɛ, the ielectric constant o the lui. The scalar equations 2.1, 2.2, , in imensionless orm, are as ollows: φ i φ 1 + φ 2 = 1, t + v iφ i = 0, in, 5.2 φ 2 c k + φ 2 v 2 c k = Pe 1 D k c k + c k z k ψ in, 5.3 t c k t + v c k = Pe 1 D k c k + c k z k ψ in R t, 5.4 { β 2 φ 1 ρ p + N ɛ ψ = z kφ 2 c k in N z, 5.5 kc k in R t Here an in the remainer o this Section, we rop the primes rom the imensionless variables unless note otherwise. The imensionless parameters are given by: Pe = V 0 D 0 /L, β = r L, r = ɛk B T/q qc The parameter Pe is the Péclet number. The parameter β is the ratio between r, known as the Debye length, an the system size L. The Debye length is typically small compare to L, an thereore, it is o interest to consier the it β 0. This is the electroneutral it, to which we turn shortly. 15

16 The interace moves accoring to 2.12, which can be mae imensionless by scaling v Γ an v 1 with respect to V 0. The interace conitions to at are given by the ollowing imensionless orms o 4.9, 4.10 an 4.11: [ψ] = [ɛ ψ n] = [c k ] = 0, 5.7 v v 1 c k D k c k µ k n Rt = v 2 v 1 φ 2 c k D k c k µ k n Ωt, 5.8 where the chemical potential is now in imensionless orm: µ k = ln c k z k ψ. We now make imensionless the vector equations 2.4 an 2.9. Introuce the ollowing prime imensionless variables: p FH κ = κ 0 κ, η 1,2 = η η 1,2, η, = κ 0 Lη,, p = c 0k B T p, = c 0 k B T p FH, T1 elas = c 0 k B T T1 elas, Sg FH = c 0 k B T Sg FH, V 0 = c 0k B T κ 0 L, 5.9 where κ 0 is the representative magnitue o the riction coeicient. We have use the characteristic pressure c 0 k B T an κ 0 to prescribe the characteristic velocity V 0. Equations 2.4 an 2.9 now take the ollowing imensionless orm: T1 elas + ζ η 1 v 1 + v 1 T φ 1 p + p FH κv 1 v 2 1 = φ 1 ρ p ψ β 2 ɛ φ 1 ψ 2 in φ 1 ζ η 2 v 2 + v 2 T φ 2 p κv 2 v 1 = z k φ 2 c k ψ, in, 5.11 ζ v + v T p = z k c k ψ, in R t, 5.12 Expressions 3.12, 4.6 an 4.7 were use as expressions or 2, g 2, an p. The imensionless variable ζ = η /κ 0 L 2 is the ratio between the characteristic viscous an rictional orces. The interace conitions at or are given by 2.13, 2.14, 2.15, 3.15 an 3.16 in imensionless orm. Equations 2.13, 2.14 an 3.16 can be mae imensionless by rescaling the velocities w an q as well as v 1,2, are with respect to V 0 so that w = V 0 w an q = V 0 q. Equations 2.15 an 3.15 take the ollowing orm: ζ v + v T + β 2 ψ ψ 1 2 ψ 2 I n p Rt n = T1 elas + Sg FH η w =Π, + β 2 ɛ ψ ψ 1 2 Π =[p] n + ζη 1 v 1 + v 1 T + ζη 2 v 2 + v 2 T n p Ωt n ɛ ɛ φ 1 ψ 2 I n, 5.13 φ 1 ζ η 2 v 2 + v 2 T n + n ζ v + v T n φ 2 The bounary conition in the outer bounary o U are given by 2.17, 4.12 an 4.13 in imensionless orm. 16

17 In many cases o practical interest, the Debye length r is small compare to the system size. We thus consier the it β 0, while keeping the other imensionless constants ixe. Let us consier the Poisson equation 5.5. Setting β = 0, we obtain the ollowing electroneutrality conition: φ 1 ρ p + z k φ 2 c k = 0 in, z k c k = 0 in R t I we replace the Poisson equation by the above algebraic constraints, bounary conitions 4.11 or the bounary conitions or ψ in 5.7 or 4.13 can no longer be satisie. This inicates that, as β 0, a bounary layer whose thickness is o orer β or r in imensional terms evelops at the interace, within which electroneutrality is violate. This is known as the Debye layer [40]. A Debye layer oes not evelop at U given our choice o imposing no-lux bounary conitions or ψ. Thus, in eriving the equations to be satisie in the it β 0, care must be taken to capture eects arising rom the Debye layer. We reer to the resulting system as the electroneutral moel. The system beore taking this it will be reerre to as the Poisson moel. In the rest o this Section, we state the equations an bounary conitions o electroneutral moel, an establish the ree energy ientity satisie by the moel. In Section 5.2, we use matche asymptotic analysis at the Debye layer to erive the electroneutral moel in the it β 0. Let us now escribe the electroneutral moel. As state above, we replace 5.5 with the electroneutrality conitions The electrostatic potential ψ evolves so that the electroneutrality constraint is satisie everywhere at each time instant. Given the electroneutrality conition, the right han sie o 5.12, is now 0. All other bulk equations remain the same. We turn to bounary conitions. We no longer have bounary conitions or ψ 4.11 or 4.13, as iscusse above. Consier the bounary conitions or the ionic concentrations c k. We continue to require the lux conitions 5.8 at an 4.12 at U. We must, however, abanon conition 5.7, that c k be continuous across. I all the c k were continuous across, the electroneutrality conition 5.15 woul imply that φ 2 ρ p must be 0. This cannot hol in general. Instea o continuity o c k, we require continuity o the chemical potential µ k across : [µ k ] = 0, k = 1,, N This is a stanar conition impose when the electroneutral approximation is use [41]. We shall iscuss this conition in Section 5.2. Let us turn to the bounary conitions or the vector equations. Bounary conitions 2.13, 2.14 an 3.16 at remain the same, an we continue to require 2.17 at U. For bounary conition 5.13, we simply set β = 0, thereby einating stresses o electrostatic origin. The non-trivial moiication concerns the bounary conition We let: η w = Π, Π = Π π osm, [ N ] W ion π osm = c k W ion = [c k ]. c k

18 where W ion eine in 4.1 has been mae imensionless by scaling with respect to k B T. In physical imensions, π osm takes the orm: π osm = k B T [c k ] This is nothing other than the amiliar van t Ho expression or osmotic pressure. Equation 5.17 thus states that water low across the interace is riven by the mechanical orce ierence as well as the osmotic pressure ierence across. We shall erive this conition using matche asymptotics in Section 5.2. The electroneutral moel escribe above satisies the ollowing energy ientity. Theorem 5.1. Let φ i, c k an ψ be smooth unctions satisying , 5.15, with bounary conitions 5.16, 5.8, an 4.12 in imensionless orm. Let v i an v be smooth unctions satisying 5.10 with β = 0, 5.11 an For bounary conitions, we require 5.13 with β = 0 an 5.17 as well as 2.13, 2.14, 3.16 an 2.17 in imensionless orm. Then, the ollowing ientity hols: t E elas + E FH + E ion = I visc I i J visc, 5.19 where E elas, E FH, I visc, J visc, E ion an I i are the suitably non-imensionalize versions o the quantities eine in 3.17 an Proo. The proo is completely analogous to Theorem 4.1. Expressions 4.24 an 4.25 also hol in the electroneutral case. Let us now a 4.24 an We in: t E ion = π osm ws I i P elec 5.20 where we use 5.15, 5.16, an the einition o π osm in Now, Theorem 3.1 yiels: t E elas + E FH = Π w + η q 2 S I visc + P elec The integral on the right han sie o the above is not equal to J visc as eine in 3.17 since we have now aopte 5.17 instea o 3.15 as our bounary conition or w. Now, aing 5.20 an 5.21 an using 5.17, we obtain The reaer o the above proo will realize that 5.16 an 5.17 are the only conitions that will allow an energy issipation relation o the type 5.19 to hol. It may be sai that, in the it as β 0, bounary conitions 5.16 an 5.17 are orce upon us by the requirement that the iting system satisy a ree energy ientity. It is interesting that we o inee recover the classical van t Ho expression or osmotic pressure i we aopt 4.1 as our expression or W ion. We also point out that the conitions or stationary state or the above equations reuce to the Donnan conitions irst propose in [38]. In this sense, our electroneutral moel is a ynamic extension o the static calculations in [38] Matche Asymptotic Analysis. We have seen above that bounary conitions 5.16 an 5.17 arise naturally rom the requirement o ree energy issipation. The goal o this Section is to erive the iting bounary conitions 5.16 an 5.17 by way o matche asymptotic analysis. 18

19 Fig The Debye layer an the avecte coorinate system. A bounary or Debye layer whose imensionless thickness is on the orer o Oβ, β = r /L orms near the gel-lui interace. The curvilinear coorinate system y = y Γ, y 3 = y 1, y 2, y 3 avects with the surace. The y 3 irection is normal to. A rescale coorinate ξ = y 3 /β is introuce to perorm a matche asymptotic calculation. Consier a amily o solutions or the Poisson moel with the same initial conition but with ierent values o β > 0. We let the initial conition satisy the electroneutrality conition. Suppose, or suiciently small values o β, a smooth solution to this initial value problem exists or positive time. We stuy the behavior o this amily o solutions as β 0. Given the presence o the bounary layer at, we introuce a curvilinear coorinate system that conorms to an avects with Figure 5.1. Our irst step is to rewrite the imensionless Poisson moel in this coorinate system using tensor calculus see, or example, [3] or [2] or a treatment o tensor calculus in the context o continuum mechanics. Introuce a local coorinate system y Γ = y 1, y 2 on the initial surace Γ 0 so that x Γ y Γ, 0 gives the x-coorinates o the surace Γ 0. Avect this coorinate system with the polymer velocity v 1 : x Γ t = v The solution to the above equation gives a local coorinate system x Γ y Γ, t on or positive time. The coorinate y Γ may be seen as the material coorinate system o the polymer phase restricte to the gel-lui interace. Deine the signe istance unction: { y 3 istx, i x, x, t = 5.23 istx, i x R t. By taking a smaller initial coorinate patch i necessary, y = y Γ, y 3 = y 1, y 2, y 3 can be mae into a coorinate system in R 3 near x = x Γ 0, t or t 0. We enote this coorinate map by C t as ollows: x = C t y, or y N R where N is the open neighborhoo on which the coorinate map is eine. We shall nee the ollowing quantities pertaining to the y coorinate system. Deine the metric tensor g στ associate with the y coorinate system: g στ = C t y σ C t, σ, τ = 1, 2, yτ 19

20 where enotes the stanar inner prouct in R 3. We let g στ be the inverse g στ in the sense that: g σα g ατ = δ σ τ 5.26 where δτ σ is the Kronecker elta. In the above an henceorth, the summation convention is in eect or repeate Greek inices. Given 5.23, we have: g 33 = g 33 = 1, g σ3 = g 3σ = g σ3 = g σ3 = 0 or σ = 1, Deine the Christoel symbols associate with g στ as ollows: Γ α στ = 1 2 gαγ gσγ y τ + g τγ y σ g στ y γ From 5.27, it can be seen ater some calculation that: Γ 3 3σ = 0 or σ = 1, 2, Let ˆv be the velocity iel o the points with ixe y coorinate: We shall use the act that: ˆv = C t t ˆvy Γ, y 3 = 0 = v 1, 5.31 which is just a restatement o Finally, we assume the ollowing conition on g στ : g στ, g στ, y α g στ remain boune as β This conition ensures that the surace oes not become increasingly ill-behave as β 0, an ensures the presence o a bounary layer as β 0. In particular, this conition allows one to choose N in 5.24 inepenent o β. We rewrite equations in the coorinate system y, t instea o x, t. Note that expressions involving the vector variables v = v 1,2, or ˆv must be rewritten in terms o v 1, v 2, v 3, the y-components o the vector iel, eine as ollows: v = v 1 C t y 1 + v2 C t y 2 + v3 C t y Let us irst rewrite the scalar equations φ 2 c k t φ 1 + φ 2 = 1, ˆv σ D σ φ 2 c k + D σ φ 2 c k v σ 2 φ i t ˆvσ D σ φ i + D σ φ i v σ i = 0, or y 3 < = Pe 1 g στ D σ D k D τ c k + c k z k D τ ψ or y 3 < c k t ˆvσ D σ c k + D σ c k v σ = Pe 1 g στ D σ D k D τ c k + c k z k D τ ψ or y 3 > { β 2 g στ φ 1 ρ p + N D σ ɛd τ ψ = z kφ 2 c k or y 3 < 0 N z kc k or y > 0 20

21 In the above, D σ is the covariant erivative with respect to y σ. The unction v Γ is the magnitue o v Γ eine in We nee the bounary conitions 5.7: ψ y3 =0+ = ψ y 3 =0, c k y3 =0+ = c k y3 =0, ɛ ψ y 3 = ɛ ψ y y3 =0+ y3 =0 where y3 =±0 enotes the iting value as y3 = 0 is approache rom above or below. We only nee the vector equations or v 2 an v. Equations 5.11 an 5.12 are rewritten as ollows: ζg στ D σ η 2 D τ v2 α + g ατ D σ η 2 D τ v2 σ φ 2 g ασ D σ p = κv2 α v1 α + z k φ 2 c k g ασ D σ ψ, or y 3 < 0, 5.39 ζg στ D σ D τ v α + g ατ D σ D τ v σ φ 2 g ασ D σ p = z k c k g ασ D σ ψ, or y 3 > We nee bounary conition 5.14: η w = p y 3 =0 p y 3 =0+ 2ζ η 2 φ 2 v 3 2 y 3 + 2ζ v3 y We introuce an inner layer coorinate system Y = y Γ, ξ where y 3 = βξ. We aopt the ansatz that all physical quantities have an expansion in terms o β. For example: c k = c 0 k + βc 1 k +, v α 1 = v α,0 1 + βv α,1 1 +, 5.42 an likewise or other physical variables The Equilibrium Case. Suppose the system approaches a stationary state as t. We irst perorm our calculations or the stationary solutions. Our erivation here assumes the existence o solutions to the inner an outer layer equations satisying the stanar matching conitions. At the stationary state, all time erivatives are 0 an thus, the right han sie o the energy ientity in 4.15 must be 0. From the conition that I i = 0, we see that µ k = 0. Given the continuity o c k an ψ across, we have: µ k is constant throughout U This conition shoul persist in the it β 0, an we have thus erive To erive 5.17, we irst show that all velocities are ientically equal to 0. Given I visc = J visc = 0 an using the einitions o w an q in J visc, we have see 3.17,2.13 an 2.14: S v = 0 in R t, S v 2 = 0, v 1 = v 2 in, 5.44 v 2 = v on The vanishing o the symmetric graient implies that v an v 2 are velocity iels representing rigi rotation an translation. Given that v = 0 on U, we have v = 0 21

22 throughout R t. From 5.45, we see that v 2 = 0 on =, an we thus have v 2 = 0 throughout. From 5.44, we see that v 1 = v 2 = 0 in. We now examine equations 5.39 an Given that all velocities are equal to 0, the leaing orer equations are: p0 N = z k c 0 ψ 0 k, or ξ < 0 an ξ > The bounary conitions that we nee at ξ = 0 to leaing orer are see 5.38 an 5.41: ξ 0 c0 k = ξ 0+ c0 k, ξ 0 ψ0 = ξ 0+ ψ0, The matching conitions or the leaing orer terms are: ξ 0 p0 = ξ 0+ p ξ ± c0 ky Γ, ξ = y 3 0± c0 ky Γ, y 3 c 0 k,±, ξ ± ψ0 y Γ, ξ = y 3 0± ψ0 y Γ, y 3 ψ±, 0 ξ ± p0 y Γ, ξ = y 3 0± p0 y Γ, y 3 p 0 ± 5.48 Note that, given 5.43, we have: c 0 k + z kc 0 ψ 0 k = 0 or ξ < 0 an ξ > Integrating 5.46 rom ξ = to, we have: p 0 + p 0 = = p 0 ξ = N c 0 k z k c 0 ψ 0 k ξ N ξ = c 0 k, c 0 k, where we have use 5.47 an 5.48 in the irst equality, 5.46 in the secon equality, 5.49 in the thir equality an 5.55 an 5.48 in the last equality. The above expression is nothing other than 5.17 where the velocities are taken to be The Dynamic Case. The ynamic case is somewhat more involve, but the essence o the erivation remains the same as in the equilibrium case. We erive 5.16 an 5.17 at points on such that the water low w oes not vanish to leaing orer. This conition can equivalently be written as: v 3,0 1 v 3, As in the equilibrium case, we assume the existence o solutions to the inner an outer layer equations satisying the stanar matching conitions. We irst erive Explicitly write out the covariant erivatives in 5.36: c k t c ˆvσ k y σ + c kv σ y σ =Pe 1 g στ y σ + Γ σ στ c k v τ c k D k y τ + c ψ kz k y τ Γ α c k στ D k 22 y α + c ψ kz k y α. 5.52

23 We now rescale y 3 to βξ to obtain the leaing orer equation in the inner layer. Given our assumption 5.32, the terms involving the Christoel symbols Γ α στ o not make contributions to leaing orer. Using 5.27, we have, to leaing orer: 0 = c 0 D k k + z kc 0 ψ 0 k or ξ > We may perorm a similar calculation or 5.35 to obtain: 0 = c 0 D k k + z kc 0 ψ 0 k or ξ < Note that the iusion coeicient may be spatially non-constant or ξ < 0 since D k is in general a unction o φ 2. The bounary conitions that we nee at ξ = 0 to leaing orer are see 5.38: ξ 0 c0 k = ξ 0+ c0 k, The matching conitions or the leaing orer terms are: ξ 0 ψ0 = ξ 0+ ψ ξ ± c0 ky Γ, ξ, t = y 3 0± c0 ky Γ, y 3, t c 0 k,±, ξ ± ψ0 y Γ, ξ, t = y 3 0± ψ0 y Γ, y 3, t ψ±, This suggests that the ξ erivatives o c 0 k Thereore, rom 5.53 an 5.54 we have: an ψ0 shoul ten to 0 as ξ ±. c 0 k + z kc 0 ψ 0 k = 0, 5.57 where we have use the act that D k > 0 an assume that D k remains boune over the inner layer. Let µ 0 k = ln c0 k + ψ0. We have: y 3 0 µ0 ky Γ, y 3, t y 3 0+ µ0 ky Γ, y 3, t = ξ µ0 ky Γ, ξ, t ξ µ0 ky Γ, ξ, t 1 c 0 k = + z ψ 0 k ξ = 0. c 0 k 5.58 where we use the matching conitions 5.56 in the irst equality an 5.55 in the secon equality, an 5.57 in the last equality. We have thus erive We now turn to the erivation o We irst show that v 3,0 1, v3,0 2, v3,0, φ 0 i are all inepenent o ξ in the inner layer. Equation 5.40 may be written explicitly as: v ζg στ α v y σ y τ + Γα τγv γ + Γ α δ v σδ y τ + Γδ τγv γ Γ δ α στ y δ + Γα δγv γ v +ζg ατ σ v y σ y τ + Γσ τγv γ + Γ σ δ v σδ y τ + Γδ τγv γ Γ δ σ στ y δ + Γσ δγv γ ασ p φ 2 g y σ = N ασ ψ z k c k g y σ

24 Given 5.32, the terms involving Christoel symbols o not contribute to leaing orer in the inner layer. Using 5.27, we obtain: v α,0 = 0 or ξ > A similar calculation or 5.39 yiels: v α,0 2 η 2 = 0 or ξ < The matching conitions are: ξ vα,0 2 y Γ, ξ, t = y 3 0 vα,0 2 y Γ, y 3, t v α,0 2,, ξ vα,0 y Γ, ξ, t = y 3 0+ vα,0 y Γ, y 3, t v α,0, From this an the assumption that η 2 > 0 stays boune, we conclue that v α,0 2 an o not epen on ξ an that: v α,0 v α,0 2 y Γ, ξ, t = v α,0 2, y Γ, t, v α,0 y Γ, ξ, t = v α,0,+ y Γ, t Consier the equation or φ 2 in To leaing orer in the inner layer, we have, using 5.32: ˆv 3,0 φ0 2 + v3,0 2 φ0 2 = Given the einition o ˆv an 5.31, we have: ˆv 3,0 = v 3, ξ=0 1 Using this an 5.63, 5.64 becomes: v 3,0 1 + v 3,0 φ ξ=0 ξ=0 = By assumption 5.51, we conclue that φ 0 2 oes not epen on ξ. Given φ φ 0 2 = 1 rom 5.34, we see that φ 0 1 is also inepenent o ξ. Using the usual matching conitions, we thus have: φ 0 i y Γ, ξ, t = y φ0 i y Γ, y 3, t φ i, To show that v 3,0 1 oes not epen on ξ, consier the ollowing expression: 2 D σ φ i vi σ = 0, 5.68 i=1 which can be obtaine by summing the equations or φ i in 5.34 or i = 1, 2. To leaing orer, using 5.32, this equation yiels: φ0 1v 3,0 1 + φ 0 2v 3,0 2 = 0 or ξ <

25 This shows that: where C 1 oes not epen on ξ. Since φ 0 i The matching conition yiels: φ 0 1v 3,0 1 + φ 0 2v 3,0 2 = C an v3,0 2 are inepenent o ξ, so is v 3,0 1. v 3,0 1 y Γ, ξ, t = y 3 0 v3,0 1 y Γ, y 3, t v 3,0 1, y Γ, t We now examine equations 5.39 an The leaing orer equation only allowe us to show that v α,0 2 an v α,0 were constant in ξ. To obtain 5.17, we must look at the next orer in β. Let us irst consier 5.40, or equivalently Ater some calculation, using 5.32, 5.27, 5.29 an 5.62 we obtain: 2ζ v 3,1 p0 N = z k c 0 ψ k or ξ > A similar calculation using 5.39 yiels: 2ζ η 2 φ 0 2, v3,1 2 φ 0 p 0 2, = N φ 0 2, z k c 0 ψ 0 k or ξ < 0, 5.73 where 5.67 was use to rewrite φ 0 2. Using 5.57, the above two equations can be rewritten as ollows: 2ζ η 2φ 0 2, v 3,1 2 p 0 + c 0 k = 0 or ξ < 0, 5.74 φ 0 2, 2ζ v3,1 p 0 + c 0 k = 0 or ξ > 0, 5.75 where we also use the act that φ 0 2, is inepenent o ξ see 5.67 in the irst equality. At ξ = 0, we have the ollowing conition rom 5.41: η v 3,0,+ v3,0 1, = p0 ξ=0 p 0 ξ=0+ 2ζ η 2φ 0 2, φ 0 2, v 3, ζ v3, where we use the einition o w, 5.63 an 5.71 to obtain the let han sie o the above relation. We also nee the matching conition: ξ ± p0 y Γ, ξ, t = y 3 0± p0 y Γ, y 3, t p 0 ± Let us irst consier the We immeiately have: 2ζ v3,1 p 0 + c 0 k = C where C 2 oes not epen on ξ. Using 5.56 an 5.77, we have: ξ v3,1 2ζ = C 2 + p 0 + c 0 k,