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Modeling ion transport mechanisms in unsaturated porous media
Marchand, J.; Samson, E.; Snyder, K. A.; Beaudoin, J. J.
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M ode ling ion t ra nspor t m e cha nism s in
unsat urat e d porous m e dia
M a r c h a n d , J . ; S a m s o n , E . ;
S n y d e r , K . A . ; B e a u d o i n , J . J .
N R C C - 4 8 7 2 0
A version of this document is published in / Une version de ce
document se trouve dans: Encyclopedia of Surface and Colloid
Science, v. 4, June 2006, pp. 3065-3074
Modeling ion transport mechanisms in unsaturated
porous media
J. Marchand
12E. Samson
12K.A. Snyder
3J.J. Beaudoin
41CRIB Département de génie civil
Université Laval, Ste-Foy (Qc), Canada, G1K 7P4
2SIMCO Technologies inc.
1400, boul. du Parc Technologique, Québec (Qc), Canada, G1P 4R7
3 Building and Fire Research Laboratory
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
4Materials Laboratory - Institute for Research in Construction,
National Research Council, Ottawa (On), Canada, K1A 0R6
July 2, 2004
INTRODUCTION
The transport of ions in colloids, granular and consolidated porous media is important to a wide variety of environmental and engineering problems. Typical examples are the transport of contaminants in marine sediments, the containment of hazardous waste in soils, ionic exchange in biological membranes and processes related to the durability of building materials [14]. Comprehensive analyses [5, 6] and the development of new mod-eling techniques [7, 8] have largely contributed to the improvement of the knowledge of the various phenomena that control the transport of ions at the pore scale and at the
scale of the material.
This contribution is an attempt to brie y summarize the research on the mechanisms of multi-ionic species transport in reactive and unsaturated porous media. The math-ematical description of the mass conservation equations in isothermal conditions at the local (nanoscopic) scale are discussed rst. Since, in many practical cases, porous solids are often (partially) saturated with concentrated electrolytes, emphasis is placed on the behavior of non-ideal ionic solutions. Techniques to average the equations over a Rep-resentative Elementary Volume (REV) of the material are then presented. The main hypotheses underlying the application of the homogenization approach are emphasized. Problems related to the treatment of chemical and physical interaction phenomena are also reviewed. Finally, the application of the formation factor concept and the relative importance of surface (grain-boundary) diusion phenomena are discussed.
IONIC TRANSPORT AT THE PORE SCALE
The transport of ionic species in a porous solid occurs in the liquid phase of the pore space in which ions are free to move. This includes both the bulk liquid, where ions are free to move, and the physi-sorbed liquid on the pore wall surface, where ion motion is partially constrained. A schematic representation of the pore space of a material is given in Figure 1. For this speci c example, the surface of the material in contact with the pore solution is assumed to bear a negative charge. This is often the case for calcium silicate hydrates and clay minerals [9, 10]. This surface charge is usually due to a combination of the intrinsic charge of the mineral network and to surface reactions at the solid/liquid interface.
As shown in Figure 1, counterions are directly adsorbed on the surface of the solid. This layer of ions, called the Stern layer, only partially compensates the surface charge of the solid. The remaining portion of the surface charge is neutralized by an excess of positive charges is distributed in a diuse layer, forming the so-called double layer. The thickness
of the double layer is generally assumed to be small compared to the characteristics dimensions of the pore space [6]. Ions located in the free water are considered to be unaected by the surface charge of the solid.
The separation between the Stern layer and the diuse layer is called the shear plane. Ions located beyond that plane are considered to be mobile (i.e. relatively free to drift in the pore solution) [11]. In many practical cases, the transport of ions in a porous material will occur predominantly beyond the shear plane (i.e. in the electrical diuse layer and in the free water). Evidence of ionic transport at the solid particle/solution interface (i.e. in the Stern layer) has also been reported [5, 6]. The relative in uence of the latter phenomenon on the description of the transport mechanisms at the macroscopic scale will be discussed in more detail in the last section of this paper.
In many practical cases, the pore space of the solid is not fully saturated by the liquid phase. Parts of the pore space can be occupied by a gas, such as water vapor (see Figure 2). Mechanical instability at the liquid-vapor interface due to surface tension may trigger the transport of the liquid phase by capillary suction. This movement of the liquid may have a signi cant in uence on the distribution of ions throughout the material.
Numerous authors have attempted to develop ionic transport models in porous solids on the basis of more or less detailed descriptions of the microstructure of the materials. Given the intrinsic complexity of most porous systems, the eective application of these models constitutes a formidable task [12]. As emphasized by Revil [6], these models usually oversimplify the microstructure by considering pores to be formed by capillaries or parallel plates. Consequently, many of these models rely on tting parameters (determined on the basis of simpli ed assumptions which may not physically represent the system well) to fully describe the intricate nature of the microporous solid.
Other authors have elected to average the variables and equations of interest over a Rep-resentative Elementary Volume (REV). The main advantage of this approach (called the
homogenization technique) is that it does not require any detailed knowledge of the mate-rial inner structure. Furthermore, the new averaged variables appearing in the equations represent the quantities measured in practice.
The homogenization process rst requires a comprehensive description at the local (pore) scale of the transport mechanisms of the multi-ionic species system. As previously men-tioned, various phenomena can contribute to the transport of ions in solution. For in-stance, ions will move under the in uence of an electrochemical potential µ and a bulk uid velocity v. The ux ji of a given ionic species i is approximated by an independent
sum of both phenomena [13]:
ji = −Bcigradµi+ civ (1)
The quantity ci is the concentration and B is a phenomenological transport coe cient.
This formulation assumes that cross terms (a tensor transport coe cient) are either neg-ligible or nonexistent. The electrochemical potential is de ned as:
µi = µoi + RT ln(γici) + ziF ψ (2)
where µo
i is the constant standard potential, R is the ideal gas constant, T is the absolute
temperature, γi is the chemical activity coe cient, ci is the concentration of the ionic
species i, zi is the valence number, F is the Faraday constant, and ψ is the electrical
potential.
Substituting equation (2) in equation (1) gives:
ji= −BRTgradci−BRT cigrad(ln γi) − BziF cigradψ + civ (3)
The rst term on the right-hand side of equation (3) corresponds to Fick's diusion law, which relates a ux to a concentration gradient through the diusion coe cient D. Equation (3) can be modi ed by considering that:
B = D
p i
with Dp
i being the diusion coe cient of the ionic species i, also called the self-diusion
coe cient. The superscript p identi es a variable de ned at the scale of the pore. Re-placing equation (4) in equation (3) yields:
ji = −Dpigradci−Dpicigrad(ln γi) −
DpiziF
RT cigradψ + civ (5)
The rst term on the right-hand side of equation (5) has the identi able form of the Fick equation. The next term accounts for chemical activity eects, i.e. the non-ideal behavior of the solution. The third term accounts for the electrical coupling between the various ions. This coupling occurs when there are dierences among ionic species diusion coe cients. As a result, to maintain electroneutrality, a diusion potential ψ arises to slow the faster ions and accelerate the slower ones. The rst three terms on the right-hand side of equation (5) are often referred to as the extended Nernst-Planck equation.
The coe cient Dp
i appearing in equation (5) represents the self-diusion coe cient of
the ionic species. In the third term on the right had side of the equation, Dp
i is used to
represent the electrochemical mobility. The self-diusion coe cient can be related to the mobility ui, at in nite dilution, through the Einstein relationship [3]:
ui =
DipziF
RT (6)
For a given ionic species, Dp
i is assumed constant in most ionic transport models. Values
of Dp
i for the most common ionic species, in the limit of in nite dilution, can be found
in handbooks [14]. It has been argued, however, that the self-diusion concentration dependence should be inversely proportional to the solution viscosity [15]:
Dp∗i = Dpi µ ηw η
¶
(7) where ηw is the viscosity of water and η is the viscosity of the solution. Even though this
ratio changes the value of Dp∗
i by less than 15 %, even for concentrated pore solutions, the
eect on Dp∗
i is su cient to make numerical predictions of bulk binary salt diusion agree
this viscosity correction, the activity coe cient would overwhelm numerical estimates at higher concentrations.
For very dilute electrolytic solutions, the activity and electrical coupling terms in equation (5) are very small and can be neglected. However, as the ionic strength of the solution increases, these terms become more signi cant. They can be viewed as correction factors to the constant diusion coe cient Dp
i. This can be more easily viewed by applying the
following transformation to equation (5): ji = −Dpi µ 1 + d ln γi d ln ci + ziF RTci dψ dci ¶ gradci+ civ (8)
where the concentration dependence can be lumped into the transport coe cient: Dpi′ = Dpi µ 1 + d ln γi d ln ci +ziF RTci dψ dci ¶ (9) Using the Einstein equation to approximate the electrochemical mobility ui by the
diu-sion coe cient Dp
i is exact in the dilute limit, but loses accuracy at higher concentrations.
When using this approach, one has to know the concentration dependence of the mobil-ity. It has been shown [16] that the electrochemical mobility ui of ions in a solution
composed of an arbitrary mixture of ionic species can be approximated reasonably well by a relatively simple expression:
ui = u◦ i 1 + GiI 1/2 M (10) where the empirical coe cients Gi range from 0.3 to 0.8 for the ionic species typically
found in pore solutions, and the quantity IM is the molar ionic strength of the solution:
IM = 1 2 N X i z2 ici (11)
where N is the number of ionic species. Using the approximations in equations (7) and (10), one can then determine the relative accuracy of approximating the mobility by the diusion coe cient, for a particular application.
The mass conservation equation at the pore scale level is given by [12]: ∂ci
∂t +div(ji) + ri = 0 (12)
where ri is a source/sink term accounting for the chemical reactions undergone by the
species i in the aqueous phase. These reactions are called homogeneous [17] and involve only ions in the aqueous phase, like for example acid/base reactions. Heterogeneous reactions, which occur between the aqueous and the solid phase [17], can be divided into two distinct categories: dissolution/precipitation, and ionic exchange. At the pore scale, they can be handled by boundary conditions at the solid/liquid interface.
Substituting equation (5) in equation (12) one nds: ∂ci ∂t −div µ Dpigradci+ Dpicigrad(ln γi) + DipziF RT cigradψ − civ ¶ + ri = 0 (13)
This equation must be written for each ionic species present in the system.
The electrostatic potential (ψ) appearing in the previous equation can be calculated using Poisson's equation [3, 18]: ∇2ψ +F ǫ N X i=1 zici = 0 (14)
where N is the total number of ionic species and ǫ is the dielectric permittivity of the aqueous phase. Although the permittivity depends upon the solution ionic strength, it is typically approximated by it value for pure water.
The chemical activity coe cients can be calculated with various models such as Debye-Hückel or Davies, depending on the level of concentration involved. A review of such models can be found in reference [19]. All these models relate γi to the concentration of
a given ionic species and the ionic strength of the electrolytic solution. Ionic transport calculations considering both electrical coupling between the species and chemical activity eects are presented in reference [20].
A transport process that is typically ignored in many transport models is osmosis. Osmosis is a movement of solvent from a region of lower solute concentration to a region of higher
solute concentration. The driving force of this phenomenon is a gradient of water (solvent) activity. The activity of water in a solution is given by [21]:
ln aH2O = −nM φ/mH2O (15)
where n is the number of ions in the dissolved salt (n = 2 for NaCl), M is the molality of the salt, φ is the osmotic coe cient of the solution and m is the molality of water (≈ 55.6). Osmosis is also related to an osmotic pressure Π. It can be expressed as a function of the water activity aH2O. At a temperature of 300 oK, the osmotic pressure, in
units of N/m2, can be approximated using the following relationship:
Π ≈ −1.4 × 108
ln aH2O (16)
Osmotic pressure has been used to estimate concrete permeability [22, 23].
According to equation (15), the water activity for a 500 mmol/L NaCl solution is ap-proximately 0.96. The value is close to one, even at such a high concentration. This emphasizes that water activity gradients are weak for most cases of ionic transport in porous materials. Consequently, osmosis is neglected.
TRANSPORT MODEL AT THE MATERIAL SCALE
All the equations presented in the previous section were derived at the pore scale. In order to come up with a macroscopic description of the various transport processes, these equations have to be averaged over a Representative Elementary Volume (REV) of the material. In the following paragraphs, the basic concepts behind the homogenization technique are described. A comprehensive description of the technique can be found in reference [12].
As emphasized by Revil [6] and Bear [12], the dimensions of the REV should be much larger than the scales of the microscopic structure of the porous solid, and much smaller than the scale of the macroscopic phenomena. Accordingly, its size depends on the
in-trinsic properties of the material (e.g. porosity, solid matrix content, air-void content, etc.).
A REV, in which a fraction of the pore space is occupied by a gaseous phase, is shown in Figure 2. The total volume of the REV is given by Vo. The part of the volume occupied
by the liquid phase is designated by VL
o . The volumetric fraction of liquid is de ned as:
θL =
VL o
Vo (17)
When the material is fully saturated, the volumetric fraction of liquid corresponds to the total porosity φ. The gaseous phase occupies a volume VG
o . It is a mixture of air and
water vapor. It is assumed that both air and water vapor ll the entire gaseous phase volume. The volumetric fraction of gas is de ned as:
θG =
VG o
Vo (18)
In the remainder of the text, the superscripts L and G will designate the liquid and gaseous phases, respectively.
Let aγ denote the amount per unit volume of some extensive quantity A in the phase γ,
either solid, liquid or gas. The calculation of concentration or mass density can serve as examples for aγ. Two types of averages can be de ned. The volumetric phase average is
given by: aγ = 1 Vo Z Voγ aγ dV (19)
The volumetric intrinsic phase average is de ned as: aγγ = 1 Voγ Z Voγ aγ dV (20)
Using the de nition of a volumetric fraction, the volumetric phase average and the volu-metric intrinsic phase average are related by the following expression:
On the basis of these de nitions, as well as from the de nitions of the average of a time derivative, a divergence, a gradient and a product [12], equations (13) and (14) are integrated over the REV to yield the macroscopic equations de ned over the material scale. To ease the reading, uppercase symbols will be used to designate the volumetric intrinsic phase average of a given variable in the aqueous phase. It gives:
∂(θLCi) ∂t −div µ θLDigradCi+ θL DiziF RT CigradΨ + θLDiCigrad ln γi−CiV −θLc˜iv˜ L¶ + θLRi+ 1 Vo Z SLS o ji·nLS dS = 0 (22)
As can be seen, the averaging procedure introduces new parameters in the transport equations. They will be discussed in the next paragraphs.
The parameter Di is the diusion coe cient at the macroscopic level. It is related to Dpi
by the expression:
Di = τLDpi (23)
where τLis the tortuosity of the aqueous phase, a purely geometrical factor accounting for
the complexity of the porous network. For uncharged porous materials (for which surface diusion phenomena can be neglected), the diusion paths of the various ionic species present in the system all have the same tortuosity in the interconnected pore space [12]. In this case, the tortuosity can be related to the so-called formation factor (F) by the following expression:
τL =
1
Fφ (24)
For a nonconducting porous solid saturated with an ionic solution, the formation factor is the ratio of the pore solution conductivity σp to the bulk material conductivity σb [24, 25]:
F = σp
σb (25)
For granular porous materials, the formation factor can be approximated by a power relationship called Archie's law [6, 25]:
where m is the cementation exponent. For dense porous materials near a percolation threshold, the formation factor can be calculated using the following expression [6]:
F = (φ − φp)−m (27)
where φp stands for the percolated porosity. In practice, the value of the cementation
exponent m usually varies from 1.5 to 2 [6, 25]. It should be emphasized that the previous de nition of the tortuosity is only valid for saturated systems. Tortuosity values are likely to change with a reduction of the degree of saturation of the material (particularly when the degree of saturation is below 60 percent) [6]. It is also subjected to the local change in porosity if dissolution or precipitation reactions occur. A more complete discussion on the tortuosity factor can be found in references [12, 26].
The homogenization technique also generates the parameter γi in equation (22). This new
parameter corresponds to the chemical activity coe cient calculated with the averaged concentrations Ci.
The average of the divergence of the ux leads to the surface integral in equation (22). The vector nLS is an outward (to the L-phase) unit vector on the solid/aqueous phase
interface (designated as SLS). The term (ji · nLS) gives the amount of ions crossing
the solid/aqueous phase interface, as a result of dissolution/precipitation or ion exchange reactions. However, this integral might prove di cult to evaluate. It is possible to express it dierently by performing the averaging operation on the ions in the solid phase [12]. The conservation equation at the microscopic scale is:
∂cis
∂t +div(jis) = 0 (28)
where the subscript s designates the solid phase. In contrast to equation (12), it is assumed that no chemical reactions occur within the solid phase, and that all precipita-tion/dissolution phenomena are taking place at the solid/aqueous phase interface. Aver-aging equation (28) over the REV leads to:
∂(θsCis) ∂t +div(θsjis s ) + 1 Vo Z SSL o jis·nSLdS = 0 (29)
where θs is the volumetric fraction of solid phase and nSL is an outward (to the S-phase)
unit vector on the solid/aqueous phase interface (designated as SSL). The integral in
equation (29) has the same value as the one in equation (22) but with an opposite sign since the ions coming out of the aqueous phase are being bound by the solid phase. Furthermore, the ux jis is zero since there is no ionic movement in this phase. This
allows one to write:
∂(θsCis) ∂t = 1 Vo Z SLS o ji·nLS dS (30)
Substituting equation (30) in equation (22) yields the following expression for the trans-port of ionic species:
∂(θsCis) ∂t + ∂(θLCi) ∂t −div µ θLDigradCi + θL DiziF RT CigradΨ + θLDiCigrad ln γi−θLCiV −θLc˜iv˜ L¶ + θLRi = 0 (31)
Equation (31) now includes two terms to model the chemical reactions. The term Ri
accounts for the homogeneous reactions, i.e. the ones involving only the aqueous phase. These chemical reactions are concerned with the formation of complexes in solution, i.e. ions made by the combination of other ions. There is also the term ∂(θsCis)/∂t to model
the exchange of ions between the aqueous and solid phases as a result of ion-exchange or dissolution/precipitation reactions. In most cases, the problem of interest occurs over months or years. By comparison, the rates of reaction are almost always shorter than the transport time scale. For those cases when the reaction is fast compared to transport, the reactions are treated like equilibrium reactions that occur instantaneously. This is called the local equilibrium assumption. Under this assumption, chemical reactions can be modeled by algebraic relationships [27]:
Km = N Y i=1 cνmi i γ νmi j with m = 1, . . . , M (Dissolution/precipitation) (32) xj = Kj−1γj−1 N Y j=1 cνji i γ νji j with j = 1, . . . , Nx (Complexation) (33)
where M is the number of solid phases, N is the number of ions, Km is the equilibrium
constant (or solubility constant) of the solid m, Kj is the equilibrium constant of the
complex j, ciis the concentration of the ionic species i, γiis its chemical activity coe cient,
xj is the concentration of the complex j, νmi is the stoichiometric coe cient of the ith
ionic species in the mth mineral, and νji is the stoichiometric coe cient of the ith ionic
species in the jth complex.
Dierent techniques have been used to include chemical reactions in ionic transport prob-lems. In early models (see for instance Miller and Benson [28]) the equations that model the chemical reactions were solved simultaneously with the transport relationships. The current trend in reactive transport modeling is to split the transport part of the process from the chemical reactions. This method gained popularity after the publication of a paper by Yeh and Tripathi in 1989 [29] demonstrating that important CPU times could be saved by using such an uncoupled approach. Models by Grove and Wood [30], Yeh and Tripathi [31], Walter et al. [32] and Xu et al. [27] are all based on this operator splitting approach. Depending on the type of chemical reactions involved in the problem considered, dierent algorithms must be used to solve the transport equation. Discussions on this topic are found in references [33, 34].
The homogenization procedure introduces the term ˜civ˜ L
in the transport equation (22), which comes from the average of the product civ. The symbol ˜ indicates a variation
with regard to an average quantity. According to reference [12], the portion of the ux associated with this term is caused by variation of both the microscopic velocity and the concentration. It is referred to as the dispersion eect. The authors show that it can be expressed as follow:
˜ civ˜
L
= −DdispgradCi (34)
to give: ∂(θsCis) ∂t + ∂(θLCi) ∂t −div µ θLDi∗gradCi + θL DiziF RT CigradΨ + θLDiCigrad ln γi−θLCiV ¶ + θLRi = 0 (35) where D∗
i = Di+ Ddisp. This dispersion eect will be important for the cases where the
uid ow in the porous material is signi cant. The vast majority of models developed to predict the movement of ions in soils neglect the individual ionic contribution of Di
and only take into account the dispersion because pressure head gradients can lead to important groundwater ows. This is the case for references [2734]. For cases where the uid ow is weak, such as for drying, the dispersion term vanishes. This is usually the case for ionic transport models in porous construction materials such as brick or concrete. The term V stands for the volumetric average of the liquid phase ow, and can be related to a Darcy ow. For cases where the capillary forces are dominant, such as in drying or capillary absorption situations, the transport of the uid phase can be modeled with Richards' equation. The development of this equation within the mathematical framework of the homogenization technique was presented by Withaker [35]. Richards' model, for variably saturated water ow in soils [27] can be written as:
³ φ∂Sw ∂h + SwSs ´∂h ∂t −div¡V¢ + w = 0 (36) V = KrKgrad(h + z) (37)
where h is the pressure head, z is the elevation, w is a uid source/sink term, φ is the porosity, K is the saturated hydraulic conductivity, Kr is the relative conductivity, Ss is
the speci c elastic storage coe cient, and Swis the water saturation. The latter parameter
is de ned as Sw = θL/φ. The relative conductivity parameter is a function of the pressure
head h; for fully saturated media, it is equal to one. The term w can be used to account for water uptake by roots in the soil.
A similar equation is used to model water movement in porous construction materials [36] like bricks, mortars and concrete. The terms involving gravity are neglected, since they
are much less important than capillary eects in the very ne pores of these materials. The source sink/term w could be used to model water uptake by the hydration of cement, for instance. However, this eect is most of the time neglected.
∂θL
∂t −div(V) = 0 (38)
V= −DθgradθL (39)
where Dθ is moisture diusivity, accounting for water under both liquid and gaseous form.
More complex models have been developed to predict water content pro les in porous media. Models by Selih et al. [37] and Coussy et al. [38] separately take into account the contribution of vapor and liquid water, as well as the interactions between both phases, to the overall water ow. However, these models include many parameters for which there is still no reliable measurement method. Accordingly, Richards' approach is still the most commonly used.
Finally, for multiionic models where the electrical coupling between the ions is considered, one must proceed to the averaging of Poisson's equation. This gives:
div (θLτLgradΨ) + θL F ǫ N X i=1 ziCi = 0 (40)
FUNDAMENTAL ASSUMPTIONS AT THE BASIS OF THE HOMOGENIZATION PROCEDURE
It should be emphasized that the validity of equations (31) and (40) rests on a series of classical assumptions. For instance, the derivation of the equation is based on the hypothesis that the porous material is in mechanical equilibrium and kept in isothermal conditions. The porous solid is an open system that can exchange ions, liquids and gases with its environment.
The validity of equations (31) and (40) also rests on a series of simplifying assumptions. It is hypothesized that the electrical charges at the surface of the solid do not have
any signi cant in uence on the transport mechanisms at the local scale. As previously mentioned, it is well known that the concentration pro les within a pore are disturbed as a result of the double layer eect [39]. The derivation of equation (31) is thus based on the hypothesis that these perturbations do not extend far from the interface and have only negligible eects on the transport mechanisms at a macroscopic scale. Similar assumptions are made for the diusion potential Ψ. The eect of the surface charges on its distribution across a pore is neglected.
In a very comprehensive investigation of ionic transport mechanisms in porous media, Revil [5, 6] demonstrated that the charged nature of the surface of most porous materials can have a signi cant in uence on the transport of ions at the nanoscopic scale. For instance, his results indicate that surface transport phenomena in the Stern Layer tend to have a signi cant in uence on the eective tortuosity of the system. Furthermore, this eect appears to be dierent for cations than for anions. Revil has proposed a very elegant model that takes into account the in uence of surface phenomena on macroscopic transport mechanisms. Unfortunately, the model is only valid for granular porous media saturated with a binary symmetric 1:1 electrolyte (such as NaCl).
Revil's study indicates that grain-boundary eects can be neglected for systems saturated with a concentrated electrolytic solutions [5, 6]. This is very fortunate because many porous systems are, in practice, saturated with relatively concentrated solutions. This conclusion is in good agreement with the conclusion of other studies [1, 2].
CONCLUSION
The mechanisms of ionic diusion in unsaturated porous media have been the subject of much attention over the past decade. Research done over this period has clearly allowed a better description of the various transport processes at the pore scale. Signi cant eort has also been made towards a better description of macroscopic transport mechanisms of multi-ionic systems.
More research is needed to reliably describe the in uence of grain-boundary transport phenomena on the transport mechanisms in multi-ionic systems. Research is also needed to understand the in uence of temperature variations on the behaviors of these systems.
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Toward the center of the pore
Surface of the material
Stern layer Diffuse layer Free water
Solid phase Aqueous phase
Shear plane
Figure 1: Distribution of ions in a typical pore. The ions adsobed in the Stern layer are considered to be part of the solid phase. The ions in the free water as well as in the diuse layer are free to move and participate in the diusion process.
Solid Solid Solid Solid Liquid Gas Vo Vo Vo L G
