Mechano-chemical effects in weakly charged porous media.

Advances in Colloid and Interface Science 222 (2015) 779–801

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Advances in Colloid and Interface Science journal homepage: www.elsevier.com/locate/cis

Historical perspective

Mechano-chemical effects in weakly charged porous media Emiliy K. Zholkovskij a,⁎, Andriy E. Yaroshchuk b, Volodymyr I. Koval'chuk a, Mykola P. Bondarenko a a b

Institute of Bio-Colloid Chemistry of Ukrainian Academy of Sciences, Vernadskogo, 42, 03142 Kiev, Ukraine ICREA &Department d'Enginyeria Química (EQ), Universitat Politècnica de Catalunya, Av. Diagonal, 647, Edifici H, 4a planta, 08028 Barcelona, Spain

a r t i c l e

i n f o

Available online 2 October 2014 Keywords: Osmosis Pressure driven separation Reflection coefficient Conductivity formation factor Darcy coefficient Standard electrokinetic model

a b s t r a c t The paper is concerned with mechano-chemical effects, namely, osmosis and pressure‐driven separation of ions that can be observed when a charged porous medium is placed between two electrolyte solutions. The study is focused on porous systems with low equilibrium interfacial potentials (about 30 mV or lower). At such low potentials, osmosis and pressure‐driven separation of ions noticeably manifest themselves provided that the ions in the electrolyte solutions have different diffusion coefficients. The analysis is conducted by combining the irreversible thermodynamic approach and the linearized (in terms of the normalized equilibrium interfacial potential) version of the Standard Electrokinetic Model. Osmosis and the pressure‐driven separation of ions are considered for an arbitrary mixed electrolyte solution and various porous space geometries. It is shown that the effects under consideration are proportional to a geometrical factor which, for all the considered geometries of porous space, can be expressed as a function of porosity and the Λ- parameter of porous medium normalized by the Debye length. For all the studied geometries, this function turns out to be weakly dependent on both the porosity and the geometry type. The latter allows for a rough evaluation of the geometrical factor from experimental data on electric conductivity and hydraulic permeability without previous knowledge of the porous space geometry. The obtained results are used to illustrate how the composition of electrolyte solution affects the mechanochemical effects. For various examples of electrolyte solution compositions, the obtained results are capable of describing positive, negative and anomalous osmosis, positive and negative rejection of binary electrolytes, and pressure‐driven separation of binary electrolyte mixtures. © 2014 Elsevier B.V. All rights reserved.

Contents

1. 2.

3.

4.

5.

6.

Introduction: mechano-chemical effects in charged porous media . . . . . . . . . . . . . . . . Irreversible thermodynamic consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Basic equation set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Relationship between electroosmotic mobility and reflection coefficients . . . . . . . . . 2.3. Zero-current mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reflection coefficients for weakly charged porous systems . . . . . . . . . . . . . . . . . . . 3.1. Derivation of relationship between the reflection coefficient and interfacial potential . . . 3.2. Obtaining the geometrical factor, s . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Apparent reflection coefficient and rejection . . . . . . . . . . . . . . . . . . . . . . Role of electrolyte composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Dependencies of apparent reflection coefficient on the ion charges and diffusion coefficients 4.2. Dependencies of rejections on the ion charges and diffusion coefficients . . . . . . . . . Role of porous space geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Important limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Various geometries of porous space . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. The Λ-parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Comparison of predictions from various geometrical models of porous space . . . . . . . Examples of mechano-chemical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⁎ Corresponding author. E-mail address: [email protected] (E.K. Zholkovskij).

http://dx.doi.org/10.1016/j.cis.2014.09.006 0001-8686/© 2014 Elsevier B.V. All rights reserved.

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E.K. Zholkovskij et al. / Advances in Colloid and Interface Science 222 (2015) 779–801

6.1. Positive, negative and anomalous osmotic pressure . . . . . . . . . . . . . . . . . . . . . . 6.2. Pressure driven separation of ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Applicability of linear approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Applicability of SEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Normalized electroosmotic mobility for fibers packed in parallel to external electric field strength References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction: mechano-chemical effects in charged porous media The present paper deals with a remarkable and practically important property of charged porous media: being placed between two electrolyte solutions, a porous medium behaves as a mechano-chemical transducer. For the first time, such behavior was observed in 1748 by JeanAntoine Nollet who was an Abbot of Grand Convent of the Carthusians in Paris [1]. However, systematic experimental and theoretical studies of relevant effects were started more than 100 years later by Traube [2], Pfeffer [3] and Van't Hoff [4]. In the beginning of the 20th century these classical studies were extended by many researchers, in particular, by Bartell [5], Loeb [6] and Teorell [7,8]. In all the studies cited above, the authors observed Osmosis, i.e., a solution volume flow through a porous medium placed between two solutions having different solute concentrations and being maintained at equal hydrostatic pressures. Also, when the solution volume flow was blocked, the authors observed the pressure difference between the compartments, osmotic pressure (OP). The effect opposite to Osmosis was experimentally discovered in the 1950's in the studies of McKelvey et al. [9,10] who observed the salt concentration changes produced in the adjacent solutions while imposing a pressure-driven flow through a membrane separating the compartments. This discovery initiated thousands of publications and widely employed technologies of water desalination [11]. As shown by the many authors (see review [12], for example), the aforementioned mechano-chemical effects can be addressed using the so-called standard electrokinetic model (SEM). According to the SEM, the solid–liquid interface of a porous material bears electric charges originating from dissociation of alkaline or acidic interfacial groups, or preferential adsorption of cations or anions. The interfacial charges repulse or attract the ions that have the same (coions) or opposite (counterions) signs of charge, respectively. Due to the thermal motion of the attracted counterions and the repulsed coions, an electric diffuse space charge opposite to the interfacial charge is formed within the solution in the vicinity of the interface. Thus, in the thermodynamic equilibrium state, the interfacial region contains Electric Double Layer (EDL) formed by the interfacial charge and the oppositely charged diffuse part which exists in the solution region adjacent to the interface. Within the frameworks of the SEM, the mechano-chemical effects are explained in terms of excess hydrostatic pressure gradient, which is formed when the porous system is placed between two solutions with different electrolyte compositions, and the electric fields generated in the presence of concentration difference and/or hydrodynamic flow through the charged porous system. While using the SEM, a quantitative analysis of the mechanochemical effects is conducted by solving a complex boundary value problem within the porous space. This problem includes the Poisson– Boltzmann equation describing the distribution of electric potential in the thermodynamic equilibrium state, equations expressing the continuity of individual ionic fluxes and local hydrodynamic velocity and a version of the Stokes equations which accounts for the bulk electric force acting on the EDL space charge. At the solid–liquid interface, these equations are subject to boundary conditions setting a

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792 794 796 798 798 799 800 800 800

given value of interfacial potential in the equilibrium state, ζ, zero normal fluxes of the ions and zero velocity. Another group of boundary conditions is imposed to interrelate the mechanical and chemical system responses and external perturbations. For over five decades, a number of theoretical studies were intended to address the mechanochemical-effects by using various versions of the SEM, as reported in the literature. To our knowledge, for the first time, a quantitative model analysis of Osmosis was provided by Schloegl [13,14] on the basis of the Fine Pore Model (FPM) proposed by Schmidt and Schwarz [15] to address porous media where the EDL thickness is much larger than the distance between the pore walls. Being a particular case of the SEM, the FPM describes situations when the equilibrium distribution of both the ion concentrations and the space charge density within the porous space are nearly uniform. It can easily be shown that the FPM is a modified version of the Mayers–Sievers [16] and Teorell [17] model. The modification amounts to introducing the terms which, in addition to the electro-diffusion, describe the convective transport of ions. While using the FPM, Schloegl [13,14] considered small applied thermodynamic forces when the system responses are linear functions of external perturbations. For the nonlinear case, the analysis of Schloegl was generalized by Hoffer and Kedem [18]. The studies [13,14,18] dealt with binary electrolytes. In the later publications of various authors [12,19–29], the FPM was employed for analyzing mechano-chemical effects for systems containing mixed electrolytes. For sufficiently wide pores (for example, with the pore radii more than 10 nm at the solute concentration higher than 0.001 M) one can expect a non-uniform distribution of both the ion concentrations and the space charge density within the pores. To consider such systems quantitatively, one should assume certain geometry of the porous space. The simplest assumption is that the porous medium can be represented as a set of straight capillaries, the Straight Capillary Model (SCM). The first analysis of Osmosis using the SCM was conducted by Derjaguin et al. [30], and Dukhin and Derjaguin [31] who addressed the flow of solution due to a symmetric binary electrolyte electrolyte concentration gradient. The authors considered the opposite limiting case with respect the FPM: the EDL thickness was assumed to be much smaller than the distance between the pore walls. In the original papers this type of Osmosis was referred to as the Capillary Osmosis. A number of later theoretical studies dealing with the SCM have been concerned with addressing Osmosis and/or the concentration changes under pressure driven flow conditions for arbitrary ratio of the Debye length to the pore dimension [12,32–46]. In these papers, the authors mostly dealt with binary electrolyte solution and circular capillaries except for refs. [12,46], where a slit capillary is additionally considered, and refs [12,43,46] analyzing the systems with mixed electrolyte solutions. For other than the straight-capillary geometries, the osmotic and separation effects were theoretically studied by analyzing systems of spherical particles or cylindrical fibers [31,47–61]. Some of these studies dealt with Diffusiophoresis-motion of solid particles due to ion concentration gradients. By reversing the sign of Diffusiophoresis velocity, one obtains the Osmosis velocity through a bed which is made up by


such particles. Accordingly, the results obtained for Diffusiophoresis of single particle in binary [30,46–49] and mixed [50] electrolyte solutions can be rearranged for addressing asymptotic behavior of Osmosis through a porous system whose porosity approaches to unity. In many studies dealing with mechano-chemical effects the authors used the Cell Model approach for describing the systems with binary electrolyte solutions [51–62]. When all the ions have the same diffusion coefficients, all the aforementioned theoretical studies predict an expected asymptotic behavior at low interfacial potentials: the leading term in the expansions by powers of interfacial potential is of the second order. When the diffusion coefficients are different, the expansion additionally contains a linear term. At sufficiently low interfacial potentials and sufficiently large difference between the diffusion coefficients, the linear term makes a dominant contribution. In particular, the expression for the osmotic velocity derived for binary symmetric electrolyte in refs. [30,31] consists of two terms: linear and nonlinear. The linear term is zero for equal diffusion coefficients and the nonlinear term is independent of the diffusion coefficients. When the ion diffusion coefficients are essentially different, even for moderate interfacial potentials of about 30mV, the linear term approximates the osmotic velocity with error of about 10%. Clearly, at lower potentials, the error is smaller. Notably, the cases of strongly different ionic diffusion coefficients are rather typical than exceptional. Widely used solutions with acidic and alkaline pH's as well as solutions containing big organic ions belong to this class. As well, the interfacial potentials of order of 30mV or lower have often been reported. Thus, theoretical analysis of mechanochemical effects in the linear approximation in interfacial potential is relevant to practically important systems. As it follows from the above survey, mathematical descriptions of mechano-chemical effects observed in charged porous systems have been conducted for two groups of problems, only: (i) binary electrolytes and several examples of porous space geometries and (ii) mixed electrolytes and two simplest models of porous space geometry (SCM and FPM). The present study is intended to address, the mechanochemical effects for a variety of pore geometries and for electrolyte mixture of general type. Such a generality is achieved due to the linear approximation in interfacial potential. The paper is organized as follows. In Section 2, we conduct an irreversible thermodynamics analysis to define convenient sets of kinetic coefficients that characterize the porous system and quantify Osmosis and pressure driven separation of ions. In Section 3, by using the linear approximation, we derive some general expressions interrelating the chosen kinetic coefficients and porous system geometry, interfacial potential, electrolyte composition. The roles of electrolyte composition and porous system geometry are discussed in Sections 4 and 5, respectively. Section 6 is concerned with examples of various mechanochemical effects and with the analysis of linear approximation applicability. Extended concussions are present in Section 7. 2. Irreversible thermodynamic consideration We consider a layer of porous medium having thickness h and being in contact with two compartments containing electrolyte solutions. External (with respect to the layer) convective and diffusion kinetics of ion transport is assumed to be infinitely rapid. For such a regime, within each of the compartments, the ionic concentrations, Ck′ and Ck″, are uniform and time-independent (Fig. 1). While dealing with the linear irreversible thermodynamic case, we consider slight deviations of physical quantities from their values in the thermodynamic equilibrium state. Consequently, we assume that the deviation, ΔC k ¼ C k 0 −C k 00 , from ¨the equilibrium value, Ck, which is the same for both compartments, is small, i.e., 0

00

C k =C k ¼ C k =C k þ OðΔC k =C k Þ ¼ 1 þ OðΔC k =C k Þ

ð1Þ

781

P

h

Ck

IS

P

Jk

Eext

Eext

Ck

IS

P

U

S

- Electrode area

Fig. 1. Two compartments separated by porous layer.

Due to the electroneutrality condition: X k

C k zk ¼

X

0

C k zk ¼

k

X

″

C k zk ¼

k

X

ΔC k zk ¼ 0

ð2Þ

k

where zk is the kth ion valence The mechano-chemical effects discussed in Section 1 are observed in porous systems that simultaneously behave as electromechanical (electrokinetic) and electrochemical transducers. The observed complex coupling between the mechanical, chemical and electrical processes motivated the use of the Linear Irreversible Thermodynamics approach developed by Onsager [63,64] for interrelating the mechano-electrochemical external perturbations and system responses. A rather general version of such irreversible-thermodynamics treatment was proposed by Staverman [65] who expressed the individual transmembrane fluxes of multi-ionic electrolyte solution components (solvent molecules and ions) as linear combination of applied differences of electric potential, hydrostatic pressure and solution component chemical potentials. Using a single set of kinetic coefficients (individual ionic electrical and mechanical transport numbers, mechanical permeability and electric conductivity), Staverman addressed major mechano-chemical, electrochemical, and electromechanical effects, namely, OP, Diffusion Electric Potential, Electroosmosis, etc. 2.1. Basic equation set In ref. [12], a reduced version of the Staverman [65] equation set was developed for a sufficiently diluted multi-ionic electrolyte solution. This equation set can be represented in the form X λ U¼ ΔP− C k σ k Δμ k h k

J k ¼ UC k ð1−σ k Þ þ

!

tk g 1X ch Δμ k þ l Δμ h n≠k kn n hðzk F Þ2

ð3Þ

ð4Þ

where F is the Faraday constant, ΔP = P′ − P″ is the pressure difference between the compartments (Fig. 1), U and Jk are, respectively, the solution volume flow and the kth ion flux through the porous system per unit area. The volume flow, U, coincides with the solution local velocity relative to solid phase sufficiently far from the porous system.

782


mobility, χ, through the kinetic coefficient featuring in Eqs. (3) and (4) or Table 1. The electroosmotic mobility is defined as

In Eqs. (3) and (4), the electrochemical potential difference Δμk = μk′ − μ″k, takes the form ch

Δμ k ¼ Δμ k þ zk FΔΦ

ð5Þ

χ¼

where ΔΦ = Φ′ − Φ″ is the electric potential difference between the compartments, and Δμch k is the chemical potentials difference which is a function of all the ion concentrations, Cm. When the compartments contain ideal electrolyte solutions, Δμch k takes the form ch

Δμ k ¼ RTΔð ln C k Þ ¼ RT

ΔC k ΔC k 1þO Ck Ck

U Eext ΔC k ¼ 0 ΔP ¼ 0

ð9Þ

i.e., χ, yields the volume flow per unit area and per unit external electric field strength, Eext, (Fig. 1). Combining Eqs. (3)–(7) and (9) one can represent the electroosmotic mobility in this form λF

ð6Þ

X

gþλ F R is the gas constant and T is the absolute temperature. The definitions of kinetic coefficients, g, λ, σk, tk and lkn = lnk are collected in Table 1. All the definitions given in Table 1 can be obtained by specifying Eqs. (3) and (4) according to the respective conditions given in the last column of Table 1 and taking into account that the electric current density, I, is expressed through the ion fluxes as

C k σ k zk

k

χ ¼ −g S

X

ð10Þ

!2 C k σ k zk

k

where gS is the electric conductivity of solutions. In linear approximation, gS is considered to be nearly equal for both the adjacent compartments. N F X 2 C z D RT k¼1 k k k 2

I¼F

N X

gS ¼

ð7Þ

J k zk

ð11Þ

k¼1

In Eq. (11), Dk is the kth ion diffusion coefficient in the solution. Thus, Eq. (10) interrelates the electroosmotic mobility, χ, with the porous system electric conductivity, g, hydraulic permeability, λ, and set of reflection coefficients σk. This important equation will be used in Section 4 for analyzing the dependency of reflection coefficients on the porous system morphology and various system parameters.

Note that interdiffusion permeability, lkn, is defined under conditions of zero difference of all the electrochemical potentials except for the nth one, Δμk ≠ n = 0. Accordingly, by combining the latter equality with Eqs. (2), (5) and (6), one obtains the equalities presented in the last two cells of the last row of Table 1 where we introduced the dimensionless quantity, ξn, ξn ¼

2.3. Zero-current mode

C n z2n N X 2 C m zm

ð8Þ

Now, using Eqs. (3) and (4) we will derive a version of irreversible thermodynamic equation set intended for addressing Osmosis and pressure driven separation that are observed at zero electric current, I = 0. Using Eq. (7), the zero-current condition is written as

m¼1

Thus, irreversible-thermodynamics equation set (3)–(4) describes the ion fluxes and volume flow through the porous system layer per unit area. Using this equation set one can address all the electrokinetic (Electroosmosis, Streaming Potential, etc.), electrochemical (Concentration Potential, Electrodialysis, etc.) and mechano-chemical phenomena (osmosis, electrolyte separation etc) phenomena in terms of the kinetic coefficients featuring in Eqs. (3) and (4) and defined in Table 1.

N X

J k zk ¼ 0

ð12Þ

k¼1

2.2. Relationship between electroosmotic mobility and reflection coefficients

Combining Eqs. (2), (4), (5) and (12) we eliminate the electric potential difference, ΔΦ. Consequently, by using Eqs. (3),(5) and (6), one obtains expression for the OP, ΔΠ, which is the pressure difference at zero volume flux across the porous medium, U = 0 (OP)

The major objective of the present section is to derive relationships to be employed for addressing Osmosis and pressure driven separation in weakly charged porous media by using equation set (3)–(4). However, prior to that, we will obtain an expression for the electroosmotic

ΔΠ ¼ ðΔP Þ I ¼ 0 ¼ U¼0

N X

ch

C k σ k Δμ k ¼ RT

k¼1

N X

σ k ΔC k

ð13Þ

k¼1

Table 1 Definitions of the kinetic coefficients. Coefficient

Notation

Definition

Conditions

1

Electric conductivity

g

I g ¼ ΔΦ=h

ΔCk = 0; U = 0

2

Hydraulic permeability

λ

U ΔP=h

ΔCk = 0; ΔΦ = 0

3

Reflection coefficient of the kth ion

σk

Jk σ k ¼ 1− UC k

ΔCn = 0; ΔΦ = 0

5

Hittorf transport number of the kth ion

tk

ΔCn = 0; U = 0

5

Interdiffusion coefficients

lkn = lnk

F J k zk I Jk Δμ n≠k =h

Jk ¼ ξn zn FΔΦ=h

ΔC k≠n zk FΔΦ C k≠n ¼ RT ΔC n zn FΔΦ 1−ξn C n ¼ RT ξn

U¼0


and the volume flux at zero pressure difference (velocity of Osmosis) N N λ X λ X ch ðU Þ I ¼ 0 ¼ − C k σ k Δμ k ¼ −RT σ ΔC h k¼1 h k¼1 k k ΔP ¼ 0

ð14Þ

In Eqs. (13) and (14), we introduced the coefficients σk∗ and λ∗ that are related to the kinetic coefficients featuring in Eqs. (3) and (4), as

σ k ¼ σ k−

N tk X C z σ zk C k m¼1 m m m

λ

λ ¼ 1þ

F

N X

ð15Þ

ð16Þ

!2

C m zm σ m

λ=g

m¼1

Thus, Eqs. (13) and (14) describe, respectively, the pressure difference (at zero volume flow) and the volume flow (at zero pressure difference) produced due to the difference in compositions of two electrolyte solutions adjacent to the porous layer under consideration. The opposite effect amounts to changes in ion concentrations in the adjacent compartments due to the volume flux through the porous system. We consider the situation when the solution is driven by a pressure difference from the left hand side compartment through the porous system toward the right hand side compartment (Fig. 1). In this case, the total number of kth ion remains constant, − d(V′Ck′)/dt = d(V″Ck″)/ dt = SJk, where S is the porous layer cross-section area; V′ and V″ are the solution volumes. Consequently, by assuming conservation of solution volume, − d(V′)/dt = d(V″)/dt = SU, and by using Eqs. (4) and (12) one obtains ″

V S

0 ″ dC k V ¼ dt I¼0 S

″

dC k dt

!

¼ −UC k σ k þ I¼0

X

ch

lkn Δμ n

ð17Þ

n≠k

⁎ = lnk ⁎ are expressed in where the apparent interdiffusion coefficients lkn terms of the kinetic coefficients defined in Table 1, as

lkn ¼ lnk ¼ lkn þ

gt k t n F 2 zn zk

ð18Þ

Thus, under conditions of zero electric current, Eq. (17) describes the time evolution of concentrations in the solutions due to the volume flow, U, through the porous system. Let us now consider the situation when, in the initial moment of time, the kth ion concentrations in both compartments are equal, Δμch k = RTΔCk/Ck = 0. Applying a pressure difference between the compartments gives rise to a volume flow and, according to Eq. (17), to changes of ion concentrations in both compartments. In the first moment, the concentration of ions with σk∗ N 0 should increase with time in the feed (higher pressure) and decrease in permeate (lower pressure). Oppositely, for ions with σk∗ b 0 the concentration should decrease in the feed and increase in permeate. The concentration differences, ΔCk, increase by magnitude with time and give rise to the ion diffusion and interdiffusion transport. As a result, each of the ion concentration differences, ΔCk, approaches a steady state value, (ΔCk)St, which corresponds to the case when dCk′ /dt = dCk′ /dt = 0. The rejection for the kth ion, rk, can be expressed through the respective steady state concentration difference, (ΔCk)St, as rk ¼

ðΔC k ÞSt Ck

ð19Þ

For obtaining the kth-ion rejection, rk, one should set dCk′/dt = dCk″/ dt = 0 on the left hand side of Eq. (17) and solve the obtained equation

783

set with respect to Δμch k = RTΔCk/Ck. As shown in ref. [12], an explicit analytical solution of this equation set can be obtained for the cases when ⁎ = lnk ⁎ ≈ gtktn/F 2znzk. The latter assumplkn b b gtktn/F2znzk and, thus, lkn tion is a good approximation for various models of ion rejection, in particular, for the SEM. The respective result obtained in [12] is rk ¼

N F 2 Uh X ξ σ z ξ σ z C n zn k k n − n n k RTg n¼1 tk tn

ð20Þ

Thus, Eqs. (13), (14) and (20) describe mechano-chemical-effects in terms of phenomenological coefficients defined in Table 1. An important role of the reflection coefficient, σk, in coupling between chemical and mechanical processes is clear from the structure of Eqs. (13)–(16) and (20). For an uncharged porous system, where σk = 0, all the effects described by these equations disappear. In this paper, we take into account only the linear terms in the Taylor expansions of corresponding parameters by powers of ζe ¼ ζ F=RT where ζ is the equilibrium surface potential defined with reference to the bulk of the solutions adjacent to the porous medium. In such an approximation, the mechano-chemical effects can be described by Eqs. (13)–(16) and (20) simplified in the following manner. Since σ k ¼ O ζe , for all the parameters represented in these equations except for the reflection coefficients, σk, one can substitute their values attributed to an uncharged porous system having the same geometry of porous space. As for the reflection coefficients, one should substitute to Eqs. (13)–(16) and (20) the linear terms in their expansion by powers of ζe. Thus, within the frameworks of the SEM, obtaining the reflection coefficient by using the linear approximation in terms of the normalized surface potential is a key step in addressing the mechano-chemical effects in weakly charged porous media. Predicting the reflection coefficients from the SEM is associated with solving a complex boundary value problems linked to given geometry of the porous space. However, the linear term in the Taylor expansion of the reflection coefficient, as a function of surface potential, can be computed by using a number of results already reported in the literature dealing with calculation of other kinetic coefficients for various porous space geometries. The analysis demonstrating such a possibility is given next. 3. Reflection coefficients for weakly charged porous systems Below, it is shown that, within the cope of SEM, the following relationship is valid. 2 σ k ¼ zk ζes þ O ζe

ð21Þ

where s is a dimensionless factor which is common for all the ions and does not depend on ζ. Eq. (21) gives an important relationship between the kth ion individual reflection coefficient and interfacial potential of a weakly charged porous medium. 3.1. Derivation of relationship between the reflection coefficient and interfacial potential Within the frameworks of SEM, the solid phase of the porous system is considered as a perfect dielectric. In the thermodynamic equilibrium state, the solid/liquid interface is assumed to bear a surface electric charge and, thus, an interfacial electric potential, ζ. The liquid phase is assumed to be an ideal electrolyte solution which is described as a continuous medium by using equations of macroscopic electrostatics, hydrodynamics and the Nernst–Plank equations describing individual fluxes of ions (Fig. 2).

784


The Debye parameter (inverse Debye screening length), κ is given by the following relationship

B

A

U

F2

Jk n

P

2

κ ¼

k

ð25Þ

εRT

D C ! 2 ! ð0Þ ! j k ¼ u C k 1−zk ζeψ − k k ∇ μ k þ O ζe RT

Ck

ix

2

zk C k

where ε is the dielectric permittivity of the electrolyte solution. Expression for the kth ion flux given by Eq. (23) can be rewritten as

P

Ck

X

x h

Structure of the second term on the right hand side of Eq. (26) is explained through the fact that, under conditions ΔCn = 0; ΔΦ = 0 presented in Eq. (22), there is no electric field and concentration gradients within the uncharged porous space: a hydrodynamic flow through an uncharged porous system does not produce concentration changes and electric field, itself. Accordingly, the zero order term is ! e i.e., ! absent in the expansion of ∇ μ by powers of ζ, ∇ μ ¼ O ζe . Therek

Fig. 2. Porous system composed by two phases.

To demonstrate that relationship (21) is valid, we consider the definition of reflection coefficient in the last two cells of the last raw of Table 1. Taking into account that the measured values of volume flow ! and the kth ion fluxes are obtained by averaging the local velocity, u , ! and kth ion flux, j k , over the porous system volume, the definition of the reflection coefficient, σk, (Table 1) can be rewritten, as 8! hD! E D!E i9 > < i x RT = 2 ∇ μ k þ zk ζe u ψð0Þ LD E L σk ¼ þ O ζe ! ! > > : ; ΔΦ ¼ 0 ix u L ΔC n ¼ 0

ð27Þ

! Obviously, when ∇ μ k ¼ 0 everywhere in the porous space, relationship (21) directly follows from Eq. (27). In particular, under the conditions presented in Eq. (27), ΔCn = 0; ΔΦ = 0, the local ! electrochemical potential gradient is zero, ∇ μ k ¼ 0, for the pressure driven flow through a sufficiently long straight capillary. However, for ! other porous space geometries, generally, ∇ μ k ≠0 . Below, we will show that relationship (21) is also valid for the general geometry in ! spite of the fact that ∇ μ k ≠0. The electrochemical potential distribution within the liquid phase, ! μ k r , which should be substituted into Eq. (27), is obtained as a solution of a boundary value problem formulated for the solution region bounded by the solid–liquid interface and two hypothetic planes A and B that are considered as external boundaries of the porous system (Fig. 2). The planes A and B are chosen to be parallel to the porous medium layer and separated by the equal distances, δ from it. While choosing the planes A and B, the following inequalities are assumed to be satisfied: κδ N N 1 and b b b δ b b h where b is the largest length scale parameter characterizing the internal geometry of porous medium. Such a choice of the planes A and B enables one to set at these planes the solution-compartments values of pressure, velocity, ion concentrations and electric potential. The boundary value problem formulated for the above described solution region contains governing equations expressing the steady state conservation law for each of the ions ! ! ∇ jk¼0

ð28Þ


785

By combining Eqs. (26) and (28), after some transformation one obtains

Eq. (11), solving the obtained simple algebraic equation with respect 2 to s and omitting the small terms of order of O ζe , we arrive at the fol-

2 z ! ! ð0Þ 2 ∇ μ k ¼ −RT ζe k u ∇ ψ þ O ζe Dk

lowing result

ð29Þ

Governing equation (29) is subject to boundary condition imposing zero values of normal flux at the interface and differences between the concentrations and electric potential at the planes A and B. ! ! n ∇ μk ¼ 0

A

at the interface

B

μ k −μ k ¼ 0

ð30Þ

ð31Þ

! where n is a unit normal vector and μkA,B are the electrochemical potentials at the planes, A and B, respectively (Fig. 2) Further, one can introduce a set of unknowns, Μk, which turn out to be common for all ions, Μk = Μ. The corresponding substitution takes the form z μ k ¼ RT ζe k Μ Dk

2 !ð0Þ ! ð0Þ ∇ Μ ¼ − u ∇ψ ! ! n ∇Μ ¼ 0 at the interface A B Μ −Μ ¼ 0

ð0Þ

hence, can be ignored while dealing with linear approximation in terms of ζe. Thus, we have proved the validity of relationship (21). Generally, for any specific geometry of the porous space, the kth ion reflection coefficient can be determined by solving consequently boundary value problems (24) and (34) specified for the assumed geometry. The obtained distributions of ψ and Μ should be substituted into Eq. (33) for obtaining the factor s which is finally substituted into Eq. (21). 3.2. Obtaining the geometrical factor, s Now, we will demonstrate that one can avoid solving the above mentioned complex problems by expressing the factor s through kinetic coefficients predicted earlier by others who considered various geometries of porous space. While deriving the respective expression, we will use Eq. (21) and irreversible thermodynamic expression of Eq. (11) derived in Section 2.2 to interrelate the electroosmotic mobility, χ, and the reflection coefficients, σk. Consequently, substituting Eq. (21) into

¼ g S =f

ðaÞ λ

ð0Þ

¼ K=η ðbÞ χ

ð1Þ

¼

εζ e χ η

ðcÞ

ð36Þ

where f is referred to as the conductivity formation factor; K is the Darcy e is the coefficient, η is the electrolyte solution viscosity coefficient; χ electroosmotic mobility normalized by the Smoluchowski value corresponding to the limiting case of the vanishingly thin Debye length, κ−1. Making use of substitutions (36), Eq. (35) is rewritten as

s¼

!ð0Þ In Eqs. (33) and (34), u is the local velocity of the purely pressure ! !ð0Þ driven flow. Clearly, u differs from u which contains a contribution originating from the bulk force due to the electric field and the concentration gradient acting the equilibrium space charge of EDL. However, 2 the respective terms bring contributions of order of O ζe and,

ð35Þ

It is convenient to rewrite Eq. (35) by introducing the following substitutions

ð33Þ

ð34Þ

ð Þ

Thus, Eq. (35) yields the factor s represented in relationship (21) which describes the kth ion reflection coefficient at low interfacial potentials. It is clear from Eq. (35) that the factor s is common for all the ions and is expressed through parameters of a porous medium having the same geometry of porous space as the medium of interest, namely, the electric conductivity and the hydraulic permeability, g(0) and λ(0), of the uncharged system, and the linear term, χ(1), in the Taylor expansion of electroosmotic mobility χ, by powers of ζe.

ð32Þ

where the function Μ is determined as a solution of boundary value problem which is obtained by substituting Eq. (32) into Eqs. (29)–(31)

ð Þ

g0 χ1 X 2 g S F ζeλð0Þ C k zk k

g

Substituting Eq. (32) into Eq. (27) we arrive at relationship (21). Consequently, the factor s, which is common for all the ions in Eq. (21), is expressed as ! hD! E D! ð0Þ Ei i x ∇Μ þ u ψ s¼ ! D!E ix u

s¼−

e χ f κ2K

ð37Þ

Thus, the geometrical factor, s, is expressed through the normalized linear term in the expansion of the electroosmotic mobility by powers of e the conductivity formation factor, f, and the Darcy surface potential, χ, coefficient, K. Both the formation factor and the Darcy coefficient are attributed to the uncharged system with the same internal geometry. The above three coefficients can be measured independently. To interrelate the geometrical factor, s, and the porous space geometry, one can use a considerable number of already published theoretie obtained for porous media cal predictions of coefficients, f, K and χ having various geometries of porous space. From Maxwell's times, the electric conductivity of two phase system has been studied by using various assumptions about the system architecture [71–80]. From these publications, one can take the formation factor f. The Darcy coefficient, K, can be taken from the papers on the creeping flow through the systems of solid particles or capillaries [81–97]. e can be obtained by norThe third coefficient featuring in Eq. (37), χ, malizing, according to Eq. (36c), the electroosmotic mobility predicted in the linear approximation in terms of ζe. For an individual spherical particle and cylindrical fiber, the corresponding expressions have been deduced in the classical paper of Henry [98]. These results can be used for addressing porous systems having porosity close to unity. In refs. [99,100], the authors addressed electroosmotic flow in slit and cylindrical capillaries. A number of studies analyzed Electroosmosis by assuming that the solid phase of the porous medium is made up by particles having spherical, cylindrical or other shapes [101–110]. In the latter references, the particles formed a regular array or were randomly distributed within the continuous liquid phase. A detailed discussion on how the geometrical factor, and thus the reflection coefficients depend on the porous space geometry will be presented in Section 5.

786


3.3. Apparent reflection coefficient and rejection Now, using relationship (21) we determine the kth ion apparent reflection coefficient, σk∗, and rejection, rk, given by Eqs. (15) and (20), respectively. Clearly, for an uncharged porous system the Hittorf transport numbers coincide with those attributed to the adjacent electrolyte solutions. Therefore the transport numbers attributed to a charged porous medium can be represented in the form D C z2 t k ¼ X k k k 2 þ O ζe Dn C n zn

ð38Þ

force that occurs when the EDL space charge is acted by the electric field of Concentration Potential (CP). Due to the linearity of equation set (3)–(4), the CP, ðΔΦÞ I ¼ 0 , can S

U¼0 be represented as a sum of contributions of individual ions. Using Eqs. (2)–(4), (25) and (38) one obtains ðΔΦÞ I ¼ 0 ¼ S U¼0

Consequently, by combining Eqs. (15), (21) and (38), we obtain 2 D σ k ¼ szk ζe 1− k þ O ζe D

ð39Þ

As well, combining Eqs. (20), (21) and (38) yields rk 2 rk D ¼ szk ζe 1− k þ O ζe Pek D

ð40Þ

where D¼

X n

ξn Dn

ðaÞ D ¼

1 N X

ðbÞ

Pek ¼ Uh=Dk

ðcÞ

ð41Þ

ξk =Dk

k¼1

Recall that the parameter ξk is given by Eq. (8). Thus, Eqs. (37) and (39)–(41) yield the major result of the present study: the kth ion apparent reflection coefficient, σk∗ and rejection, rk, attributed to a weakly charged porous system are represented through the normalized surface potential, ζe, the geometrical factor, s, which is common for all the ions, the kth ion valence, zk and the diffusion coefficient normalized by a certain version of the mean ionic diffusion coefficients in the solution. These mean diffusion coefficients take different forms given by Eqs. (41a), and (41b), D and D, for the apparent reflection coefficients and the rejections, respectively. It should be noted that, at given Debye parameter, κ, the linear term in the expansion of the electroosmotic mobility by powers of the normalized surface potential, χ(1) does not depend on the electrolyte composition. Hence, at a given value of κ, the apparent reflection coefficients, σk∗, and the rejections, rk, are influenced by the electrolyte composition due to the presence other quintiles than s in Eqs. (37) and (39)–(41). 4. Role of electrolyte composition 4.1. Dependencies of apparent reflection coefficient on the ion charges and diffusion coefficients While discussing the behavior of kth ion apparent reflection coefficient, σk∗, we will distinguish between “fast” and “slow” ions whose diffusions coefficients satisfy the inequalities Dk ND and Dk bD, respectively, where D is given by Eq. (41a). According to Eq. (39), the apparent reflection coefficient, σ ∗, of “fast” counterions, z ζeb0, and “slow” coions, k

k

zk ζe N0, is always positive, σk∗ N 0. Oppositely, for the “fast” coions and the “slow” counterions, σk∗ b 0. The above formulated rule can qualitatively be explained by considering the thought experiments linked to Eq. (13) which describes the OP (the pressure difference which should be applied in the presence of ion concentration differences to maintain zero volume flux at zero electric current). This pressure difference opposes the driving electric

ΔΦk

ð42Þ

k

where the kth ion contribution, ΔΦk, is expressed as

n

σk∗

X

ΔΦk ¼

Fzk Dk 1− ΔC k εκ 2 D

ð43Þ

As it follows from Eq. (43), the kth ion contribution into CP, ΔΦk, excerpts on this ion a force, ΔΦFzk/h, which is directed toward a higher concentration compartment when the kth ion is “fast” (Dk ND). Similarly, the force is directed toward a slower concentration compartment when the kth is “slow” (Dk bD) (Fig. 3). These components of CP electric field, ΔΦk, exert the electric force on the EDL space charge. This force coincides by direction with the force acting on counterions. In the view of above discussion, we conclude that the concentration differences of “fast” counterion or “slow” coion drive the liquid toward the compartment with a higher concentration of these ions. Similarly, the concentration differences of “fast” coions or “slow” counterions drive the liquid toward lower concentrations of these ions (Fig. 3). For maintaining zero volume flow in the presence of the above discussed electric force of the CP origin, it is required to apply the positive pressure difference (the higher pressure is in the compartment with a higher concentration), in the presence of concentration difference of the “fast” counterions or “slow” coions. Similarly, the negative pressure difference (the higher pressure is in the compartment with lower concentration) should be applied to block volume flow when concentration differences of “fast” coions and “slow” counterion are imposed. Thus, while inspecting Eq. (13), we conclude that, for the “fast” counterions and “slow” coions, the apparent reflection coefficients should be positive, σk∗ N 0, whereas for the “fast” coions and “slow” counterions, σk∗ b 0. In the case of binary electrolyte, the apparent reflection coefficients attributed to individual ions are equal σ ¼ σ 1 ¼ σ 2 ¼ sjz1 z2 jζe

¼ sjz1 z2 jζe

D1 −D2 signðz2 Þ jz1 jD1 þ jz2 jD2

D2 −D1 signðz1 Þ jz1 jD1 þ jz2 jD2

ð44Þ

The latter equality in Eq. (44) is valid since sign(z1) = − sign(z2). The common reflection coefficient, σ ∗, is referred to as the electrolyte reflection coefficient. The binary electrolyte ions having higher and lower diffusion coefficients are, respectively, the “fast” and the “slow” ions. Accordingly, when the coion has a higher diffusion coefficient the electrolyte reflection coefficient is negative, σ ∗ b 0. When the counterion has a higher diffusion coefficient, σ ∗ N 0. Inspecting Eq. (44) yields jσ j≤sz1;2 ζe. It will be shown in the next section that s ≤ 1. As well, within the frameworks of the low potential approximation z1;2 ζeb1. Thus, for weakly charged porous system the absolute value of binary electrolyte reflection coefficient is always less than unity, |σ ∗| b 1. While considering electrolyte mixture of the general type, one

should make some important remarks regarding the factor 1−Dk =D in Eq. (39). For “slow” ions, the following inequality is always valid: 0b

1−Dk =D b1. As for the “fast” ions, the factor 1−Dk =D is always negative and can have the absolute value which noticeably exceeds unity.


Force acting “fast” counterion

787

Force acting “slow” coion

Force acting “slow” counterion

Force acting “fast” coion

“fast” counterion and “slow” coion Contribution into force acting space charge: “slow” counterion and “fast” coion

Ck x Fig. 3. The CP electric forces acting individual ions and space charge.

The latter situation is impossible for binary electrolytes but can be observed when the kth ion has the highest diffusion coefficient and, simultaneously, is represented in the mixture in such small amounts that its presence does not affect the mean diffusion coefficient given by Eq. (41a). The simplest example of that is the NaCl solution containing trace amounts of HCL. For such a solution one can easily estimate

that 1−DH =D ≈−5 where DH is the diffusion coefficient of the hydrox

yl ions. According to Eq. (39), such a high value of factor 1−Dk =D can lead to the apparent reflection coefficients with absolute value higher than unity, |σk∗| N 1, even for z ζeb1. Remarkably, when such a “fast” 1;2

ion is a counterion the apparent coefficient is positive, i.e., σ∗k N 1. Hence, the respective value of reflection coefficient is higher than that in the case of a membrane completely impermeable for this ion. The expected experimental consequences of this result will be discussed in Section 6. 4.2. Dependencies of rejections on the ion charges and diffusion coefficients Let us now consider the kth ion rejection given by Eq. (40). Note that the right hand side of Eq. (40) has nearly the same form as the expression for the kth ion apparent reflection coefficient given by Eq. (39) with the only difference: on the right hand side of Eq. (40), D is represented instead of D in Eq. (39). It is clear from Eqs. (41a and b) that D and D do differ. Because of such a similarity of the right hand sides of (39) and (40), all the above conclusions regarding the sign of σk∗ retain their validity with respect to the sign of rk. The only reservation is that the terms “fast” and “slow” ions should be redefined with reference to D. Now, the kth ion is “fast” or “slow” if, respectively, Dk ND or Dk b D. It can easily be shown that D ≥D. Therefore, the signs of the rejection and the reflection coefficient always coincide when either Dk ND or Dk b D. An example of that is a binary electrolyte solution. Specifying Eq. (40) for the case of binary electrolyte solution yields expression for the common rejection, r: r ¼ r 1 ¼ r 2 ¼ Pe σ

ð45Þ

where the electrolyte apparent reflection coefficient, σ ∗, is given by Eq. (42) and Pe ¼ Uh=D. Thus, Eq. (45) shows that, for binary electrolyte

solutions, the signs of the apparent reflection coefficient and rejection always coincide. For the kth ion, whose diffusion coefficient, Dk, satisfies inequality DbDk b D, the reflection coefficients and the rejection have opposite signs. As stated in Section 2.3, the apparent reflection coefficient, σ∗k, and rejection rk, describe different stages of the pressure driven process. The value of σk∗ defines the rate of concentration changes within the compartments in the beginning of the pressure driven processes when the kth ion concentration attributed to different compartments are assumed to be equal. The positive (negative) value of the apparent reflection coefficient corresponds to an increase (decrease) with time of the kth ion concentration in the feed and, thus, decrease (increase) in the permeate compartment. As for the rejection, rk, according to Eq. (19), it defines the kth ion concentration difference in the end of a transition period when a steady state regime, dCk′/dt = dCk′/dt = 0, is already reached. Hence, for ions having different signs of σk∗ and rk, the signs of the concentration changes, which are finally reached in each of the compartments, are opposite to the direction of the concentration changes in the first moment of the process. The latter means that, in each of the compartments, the concentrations of the kth ion, whose diffusion coefficient satisfies the inequality D b Dk b D, changes with time nonmonotonously. Example of the above discussed behavior will be considered in Section 6.2. In summary, for weakly charged porous media, each of the kinetic coefficients, σk∗ and rk, which describes the mechano-chemical effects can take either positive or negative value. The signs of σ∗k and rk, depend on whether a given ion is counterion or coion and on relationships between the ion diffusion coefficients attributed to the electrolyte solution.

5. Role of porous space geometry In expressions (39) and (40), which yield the apparent reflection coefficient and rejection, respectively, the dimensionless factor s given by Eq. (37) is defined by the porous space geometry and thus depends on the length scale parameters characterizing this geometry. Additionally, s depends on the Debye length, κ−1. It is expected that the geometrical factor s depends on the ratios of the geometrical parameters to Debye length, strongly. To demonstrate that, we consider two limiting cases

788


that correspond to the Debye length mach shorter and much longer than, respectively, the shortest and longest length scale parameters characterizing the porous system geometry.

Hence, the electroosmotic velocity corresponding to the limit κ → 0, ðU κ→0 Þ ΔP ¼ 0 takes the form

5.1. Important limiting cases

ðU κ→0 Þ ΔP ¼ 0 ¼ λ ρΔΦ ΔC k ¼ 0

ΔC k ¼ 0

ð0Þ

For the vanishingly short Debye length, κ → ∞, the distributions of velocity and electric current density are similar under conditions of measuring electroosmosis [31]. Consequently, the electroosmotic mobility, χ, approaches the Smoluchowski limiting value, εζ/η. Hence, eκ→∞ →1. Thus, for vanishingly short Debye according to Eq. (36a), χ length, the geometrical factor, s, given by Eq. (37) takes the following asymptotic form sκ→∞ ¼

1 f κ2K

Combining Eqs. (9), (36a), (36c) and (48) yields e ¼ κ2K f χ

sκ→0 ¼ 1

ð50Þ

Thus, while the Debye length is becoming infinitely large, the geometrical factor, s, approaches unity. The asymptotic expressions given by Eqs. (46) and (50) describe the geometrical factor, s, for arbitrary geometry of the porous space. Asymptotic result (50) does not depend on geometry at all. In the expression given by Eq. (46), the geometry manifests itself through the Darcy coefficient, K, and the conductivity formation factor, f, and, for given values of these coefficients, yields the same prediction for all the geometries.

Let us now consider an opposite limiting case of the infinitely large Debye length, κ → 0. This case corresponds to the FPM [12–15] discussed in Section 1. For such a case, the unique solution of boundary value problem (24) yields ψ = 1 everywhere in the porous space. The latter corresponds to the uniform distribution of both the equilibrium electric potential and space charge density, ρ, within the porous space. Within the frameworks of the linear approximation in terms of ζe , such a spatially uniform equilibrium charge density, ρκ → 0, can be written as 2

ð49Þ

Substituting Eq. (47) into the expression for geometrical factor given by Eq. (37) yields

ð46Þ

ρκ→0 →εκ ζ

ð48Þ

5.2. Various geometries of porous space In order to analyze how the geometrical factor, s, depends on the porous space geometry for finite values of the Debye parameter we will apply Eq. (37) for five models of the porous system, namely, (i and ii) weakly curved slit (i) and circular (ii) capillaries; (iii) packed beds of spherical particles; (iv and v) fibrous porous system with fibers parallel (iv) or perpendicular (v) to the axis x shown in Fig. (2)). In Table 2, we collected earlier derived relationships that express each of the coefficients represented on the right hand side of Eq. (37), namely, the Darcy coefficient, K, the conductivity formation factor, f, and the nore through the structure parameters malized electroosmotic mobility, χ, employed in the above listed five models of the porous medium. The coefficients presented in Table 2 for the cases of the packed spherical particles or cylindrical fibers have earlier been derived by different authors by using the spherical or cylindrical cell model approach. Within the frameworks of such an approach a representative cell of a porous system is considered to be a sphere or cylinder with, respectively, a spherical particle or cylindrical fiber in the center (Fig. 4). The cell

ð47Þ

While analyzing electroosmotic flow, which is determined under condition of zero ion concentration and pressure differences between the solutions separated by the porous medium, see Eq. (9), one should solve the hydrodynamic equation set consisting of the Stokes equation with local volumetric electric force and the continuity equation for the local liquid velocity. When the space charge is distributed uni! formly such an electric force can be expressed as a gradient, −ρ ∇ Ψ ¼ ! − ∇ ðρΨÞ. Consequently, the total local force acting the liquid becomes ! [− ∇ ðp þ ρΨÞ], where p is the local pressure. At zero applied pressure difference and in the presence of electric potential difference, ΔΦ, the above discussed force creates a flow having exactly the same structure as the pressure driven flow which would be generated through an uncharged porous system with the same geometry of the phases by an applied pressures difference equal to ρΔΦ.

Table 2 Coefficients for different geometries. χe

f

K

Slit Capillary, τ-tortuosity, 2 h-capillary width Circular capillary, τ -tortuosity, a-cross-section radius Packed bed of spherical particles, a-particle radius

τ ϕ

ϕh 3τ

1−

τ ϕ

a2 ϕ 8τ

2I1 ðκaÞ 1− κaI [100] 0 ðκaÞ

3−ϕ ϕ

ϕ ϕ2 1=3 a2 1−ð1−ϕÞ − 3 − 9

Fibers perpendicular to axis x

2−ϕ ϕ

2

5

1−ϕ

1 ϕ

[99]

Λ pffiffiffiffiffiffiffiffi h 8=3

2h

a

[91]

a2 −ϕðϕþ2Þ−2 ln ð1−ϕÞ 1−ϕ 16

[90]

3 1 3 þ 5 r 2r [101,103] o ð0Þ

dψ 3 6 þ 1−ϕ dr 10 1−10r − r 5 dr 2 − 3ϕ

a2 −ϕðϕþ2Þ−2 ln ð1−ϕÞ 1−ϕ 8

[89]

ð1−ϕÞ

∫

1−

a 2

2a

1

ð0Þ

Þϕ ψb −2 1þϕ 1 þ ð1−ϕ 2 ϕ ð1−ϕÞ−1=2

∫

1

Þ 1− 2ð1−ϕ ϕ

ð1−ϕÞ−1=2

∫

1

ψð0Þ dr r5

−1=2

ð1−ϕÞ ψð0Þ dr− 1−ϕ ∫ ϕ r3 1

ð1−ϕÞ−1=2

∫

1

Λ0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

( −1=3

þ ϕ1 Fibers parallel to axis x

tanh ðκhÞ κh

a

ϕ ϕ 1=3 2 1−ð1−ϕÞ − 3 − 9 ð1−ϕÞϕ 5

ð3−ϕÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −ϕðϕþ2Þ−2 ln ð1−ϕÞ ð2−ϕÞ 2ð1−ϕÞϕ

2a ϕð1−ϕ=3Þ 1−ϕ 3

aϕ 1−ϕ=2 1−ϕ

[104]

rψð0Þ dr

ψð0Þ xdx (Appendix)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a

−ϕðϕþ2Þ−2 ln ð1−ϕÞ ð1−ϕÞϕ

ϕ a 1−ϕ


b a

Fig. 4. Common sketch for spherical and cylindrical cells: a — radius of spherical particle or cylindrical fiber; b — radius of the 3D spherical or 2D cylindrical cells.

radius, b, is chosen such that the ratio of the cell volume free from the particle or fiber to the total cell volume coincides with the porosity, ϕ, i.e., ( ϕ¼

3

1−ða=bÞ 2 1−ða=bÞ

for spherical cell for cylindrical cell

ð51Þ

where a is the radius of spherical particles and cylindrical fibers, respectively. The boundary value problems, which are originally formulated for porous space, are reformulated and solved within the space surrounding the particle in the cell, and the interfacial boundary conditions are set at the particle surfaces. While obtaining the macroscopic kinetic coefficients, averaging over the porous system volume is replaced by averaging over the spherical or cylindrical cell volumes. The cell model approach has a long history. Details regarding the hydrodynamic and electrokinetic cell models can be found in recently published reviews [97,110] In the column (Table 2) containing expressions for the formation factor, f, we collected simple results following from the relationships widely used in the literature. Expressions for capillaries contain the tortuosity factor, τ ≥ 1, which accounts for a larger length of the curved capillary compared to a straight one for which the formation factor is 1/ϕ where ϕ is the porosity, i.e., the ratio of the porous space to total system volume. The latter expression for f is valid for the fibrous porous systems when the fibers are packed in parallel to axis x (Fig. 2) since such a configuration is a version of the straight capillary system. For the formation factor attributed to the packed spheres and fibers (perpendicular to the axis x), we presented expression following from respective versions (3D and 2D, respectively) of the Maxwell relationship. Note that these expressions can be deduced by using spherical or cylindrical cell model approach [97,110]. In the column containing expressions for the Darcy coefficient, K, (Table 2), the results presented for curved capillaries follow from the respective versions of the Poiseuille formula modified by the inverse tortuosity factor, 1/τ, which accounts for the increased capillary length. The expressions presented for packed spherical particles and fibers have been obtained in the literature using the above outlined cell model approach. In the respective publications, within the cell part free fro the particle, the authors solved the Stokes and continuity equations subject to the boundary conditions of zero velocity at the particle surface. One of the outer boundary conditions is obtained by the equating the ! volume flow, U , and the velocity averaged over the cell volume. Another

789

boundary condition sets zero total force applied to the cell. The ! ! Kuwabara boundary condition [87], ∇ u ¼ 0 , is additionally set at the cell outer boundary. Note that for the case of fibers parallel to velocity the Kuwabara and Happel boundary conditions coincide. Remarkably, the expressions for K obtained for fibers parallel and perpendicular to axis x differ by factor ½, only/. e In the column containing the normalized electroosmotic mobility, χ, we collected the results obtained on the basis of SEM for various geometries of the porous space, earlier. Note that each of the results obtained in refs. [99] and [100] is related to a single capillary and yield the electroosmotic mobility as the cross-sectional mean velocity, 〈ux〉 per unit applied electric field strength inside the capillary, E. However, realizing that U = ϕ〈ux〉 and Eext = Eϕ (the equalities reflect continuity of the volume flow and electric current, respectively), we arrive at the conclusion that the electroosmotic mobility, χ, given by definition (9) coincides with the expressions deduced in refs. [99] and [100]. Using Eq. (36c), such expressions lead to expressions for e given in the respective cells the normalized electroosmotic mobility χ, of Table 2. e for the packed spheres and fiExpressions in Table 2 that describe χ bers contain the function ψ(0)(r) under integrals. This function is the zero perturbation term in the expansion of the solution of the Poisson–Boltzmann problem given by Eq. (24) by powers of ζe. In the cases of spherical and cylindrical cell models the boundary condition at infinity (the last boundary condition in Eq. (24)) is replaced by condition dψ(0)/dr = 0 which is set at the cell outer boundary (Fig. 4). In this boundary condition, r is the radial coordinate in the spherical and 2D polar coordinate systems, respectively. Forms of functions ψ(0)(r) have been obtained from Levine and Neale [100] and Ohshima [103] 8 a sinh ½κ ðb−rÞ−κb cosh ½κ ðb−r Þ > > sperical cell < r sinh ½κ ðb−aÞ−κb cosh ½κ ðb−aÞ ð0Þ ψ ðrÞ ¼ > > K 1 ðκbÞI0 ðκr Þ þ I1 ðκbÞK 0 ðκr Þ : cylindrical cell K 1 ðκbÞI0 ðκaÞ þ I1 ðκbÞK 0 ðκaÞ

ð52Þ

where a is the radius of spherical particles and cylindrical fibers, respectively, and the cell radii. b, are obtained from Eq. (51). In the case of the spherical cell model, the expression presented e was derived by Levine and Neale [101] for binary elecin Table 2 for χ trolyte solution. This expression was rederived by Ohshima [103] for mixed electrolyte solutions. The expression for the cylindrical cell model when the fibers are perpendicular to axis x was obtained by Ohshima [104]. As shown in ref. [111], the electroosmotic mobility predicted in refs. [101,103,104] corresponds to the definition given in the present paper by Eq. (9) containing the external electric field strength, Eext, in the denominator. Thus, by normalizing this quantities according e represented in Eq. (37) that to Eq. (36c), we obtain the parameter χ yields the geometrical factor under consideration. The latter cell in the column with normalized electroosmotic mobility contains expression e for the fibers packed in parallel to the axis x. Surprisingly, describing χ we did not find in literature an expression addressing electroosmotic mobility for this relatively simple case. Therefore, in the Appendix A, we present the straightforward derivation of such an expression. 5.3. The Λ-parameter e from reBy substituting into Eq. (37) the expressions for K, f and χ spective rows of Table 2 one obtains the geometrical factor, s, for each of the discussed geometries. The geometrical factor obtained in such a manner turns out to be a function of two dimensionless parameters, namely, the porosity, ϕ and the Debye parameter κ timed by a single length scale parameter characterizing each of the porous space geometries under consideration (radii of capillaries, particles and fibers, half of the slit channel width). In order to compare the predictions from various geometrical models in common terms, instead of the above

790


mentioned various length scale parameters, we will use a common length scale parameter, Λ, which was introduced in ref. [112–114], as 2

Λ ¼8K f

ð53Þ

Convenience of employing the Λ-parameter is that it is expressed through two phenomenological parameters, K and f, that, for a given porous system, can be measured independently. Thus, the Λ-parameter can experimentally be determined for a porous systems whose morphology is not known, preliminary. In the respective column of Table 2, we collected the expressions for Λ derived from various models. These expressions were obtained with the help of Eq. (53) by combining the results presented in the columns for K and f. Remarkably, for the weekly curved capillary model, at a given cross-section dimension, Λ is independent of both the porosity, ϕ, and the tortuosity, τ, and is interrelated with the pore radius, a, or the half of the slit channel width, h, in an unique manner. For the models of packed spheres and fibers, the parameter Λ turns out to be an increasing function of the porosity, ϕ. These dependencies are illustrated by the solid curves of Fig. 5. We present the results for ϕ N 0.6 where the Kuwabara cell model yields reasonable predictions [96]. Within the displayed range, the strongest dependencies, Λ(ϕ), are observed for sufficiently high porosities, 0.9 b ϕ b 0.99 where the Λ-parameter increases in a few times for all the models. At lower porosities, 0.6 b ϕ b 0.9, the Λ-parameter changes slower and increases, for packed spherical particles, by factor of about 2 and, for both the models of packed fibers, about 1.5. At ϕ = 0.6, the Λ-parameter takes value close to the fiber cross-section radius, a, for both the cylindrical cell models, and, a value of about 0.6a for the spherical cell model (a is the particle radius, in the latter case). The dashed curves are presented in Fig. 5 to display the behavior of another parameter, Λ0 as a function of ϕ. The parameter Λ0 was suggested in ref. [111] as a good approximation of Λ. According to ref. [111] Λ0 is determined from the relationship Z

2

E dV Λ0 ¼ 2

porous space

Z

ð54Þ

2

medium with the same geometry of the porous space. For obtaining Λ0, one should solve the Laplace equation written for the local potential Φ within the porous space. After satisfying the boundary condition ! ! n ∇ Φ ¼ 0, which set at the particle surface, and condition imposing ! ! externally applied voltage, one should determine E ¼ − ∇ Φ. The obtained magnitude E should be substituted into Eq. (54). For the capillary models, including the case of fibers packed along the axis x, the above outlined scheme yields uniform electric ! field E . Consequently, using Eq. (54) one obtains that, for such systems, the parameter Λ0 coincides with the hydrodynamic radii (ratio of the porous space volume to the interface area) timed by coefficient 2. Accordingly, for capillaries having circular cross-section the parameters Λ0 and Λ exactly coincide, Λ = Λ0 = a (Table 2). For slit capillaries, Λ0 pffiffiffiffiffiffiffiffi and Λ differ, Λ ¼ 8=3h≈0:82Λ 0 . The dished curve in Fig. 4 where plotted for packed spheres and fibers by using the respective expressions given in the last column of Table. 2. While obtaining Λ0 for the packed spheres and for the fibers packed in perpendicular to axis x, we used the above described scheme and assumptions that are specific for the Cell Model approach. Accordingly, the potential Φ was assumed to have the same angular symmetry as that in the case of a single particle (Φ ∼ cos(θ), where θ is the polar angle either in the spherical or 2D polar coordinate systems). Also, while using Eq. (54), the integrations over the porous space volume and the interface are replaced by integrations over the cell part filled by the liquid and the particle surface, respectively. For fibers packed in parallel of axis x (Fig. 2), as mentioned above, the parameter Λ0 coincides with hydrodynamic radius which is calculated as the ratio of the area of the cross-section, π(b2 − a2), (annulus bound by the external circular boundary of the 2D cell and the internal circle at the fiber surface) to the length of internal circle, 2πa (Fig. 4). As it is clear from Fig. 5, within the displayed range of porosities, for each of the geometry types Λ0(ϕ) N Λ(ϕ). At ϕ = 0.6, all the models yield the closest values of Λ0(ϕ) and Λ(ϕ). In particular, for the spherical cell model, Λ ≈ 0.75Λ0; for the cylindrical cell model with fibers packed in perpendicular to the axis x, Λ ≈ 0.9Λ0; and for the cylindrical cell model with fibers packed in parallel to the axis x, Λ ≈ 0.66Λ0 With increasing porosity, the difference between Λ0(ϕ) and Λ(ϕ) becomes stronger, and, at ϕ N 0.98, these two parameters differ more than in two times for all the models.

E dS 5.4. Comparison of predictions from various geometrical models of porous space

interface

where E is the magnitude of the local electric field strength which occurs under conditions of measuring conductivity of an uncharged porous

3

10

2 1 3 2

/a;

0

/a

1

In Table 2, for each of the porous space geometries the normalized e is represented as a function of the two electroosmotic mobility χ dimensionless parameters, namely, the porosity, ϕ, a length scale parameter (capillary radius or width, spherical particle or cylindrical fiber radii) normalized by the Debye length, κ−1. Such a length scale parameter can be expressed through the Λ-parameter and, for the e can be represented packed spheres and fibers, the porosity, ϕ. Hence, χ as a function of the normalized Λ-parameter, κΛ, and the porosity, e¼χ eðκΛ; ϕÞ . Consequently, using Eq. (53), one can rewrite i.e., χ Eq. (37), as

1

sðκΛ; ϕÞ ¼ 1- Spherical Cell

8 eðκΛ; ϕÞ χ ðκΛ Þ2

ð55Þ

2- Cylindrical Cell (fibres perpendicular to x) 3- Cylindrical Cell (fibres parallel to x)

0.1 0.6

0.7

0.8

0.9

1.0

Fig. 5. Length scale parameters Λ and Λ0 as a functions of the porosity ϕ for different cell models; a — radii of particles or fibers.

The curves plotted in Figs. 6 and 7 display the dependency of the geometrical factor, s, on the parameters, κΛ and ϕ for various models of porous space geometry. Prior to discussing the behavior of curves, we recall asymptotic relationships (46) and (50) that yield s(0, ϕ) = 1 and s(κΛ, ϕ) → 8/(κΛ)2 when κΛ → ∞. Since, at given values of κΛ, both the above asymptotic expressions are independent of the porosity, ϕ, one can expect that dependency of s on ϕ and on the type of porous


a

1

=1

791

1

8 / (

2

)

3

0.99

0.1

10

= 0.5

Cylindrical cell model (perpendicular to x)

0.01

s

s

Circular pores

Cylindrical cell model (parallel to x)

0.1 slit capillary

Spherical cell model

1E-3

circular capillary cylindrical cell model (fibres parallel to x )

100 0.6

0.7

0.8

0.9

1.0

Fig. 6. Dependency of the geometrical factor, s, on the porosity, ϕ, at given κ Λ for different geometries of the porous space.

space geometry is not strong. The latter is confirmed by the behavior illustrated in Fig. 6. At ϕ ≤ 0.6, all the curves, s(ϕ), plotted in Fig. 6 for various κΛ deviate from respective straight lines attributed to the circular capillaries less than 10%. For κΛ = 1, all the curves nearly merge with each other. For κΛ = 3 and κΛ = 10, there are small but noticeable changes in s with increase in ϕ and small differences between the predictions from various models. At κΛ = 100, the curves nearly merge except for the cases of very high porosities, 0.9 b ϕ b 0.999, where different models lead to noticeably different results. As it is clear from Table 2, for the curved capillary models, the geometrical factor is independent on the porosity. Accordingly, in Fig. 6, we see the horizontal straight lines built for the circular capillary. Remarkably, for the fibers packed in parallel to the axis x, the dependencies s(ϕ) nearly coincide with those for the circular capillary with a few percent deviations. Such a result could be expected since the latter model is somewhat similar to the capillary model The strongest dependency, s(ϕ), is observed for fibers packed in perpendicular to the axis x. For such geometry, the geometrical factor decreases with increasing porosity. Within the displayed range, 0.5 b ϕ b 0.999, the observed maximum decrease is about 25%, for κΛ = 3; 35% for κΛ = 10; and 40% for κΛ = 100. For the spherical cell model, we observe a weaker decrease of s. Within the same range of porosities, the maximum deviation of the corresponding curve from the straight line plotted for a circular capillary is less than 10%, for κΛ = 3; about 20%, for κΛ = 10 and 25%, for κΛ = 100. Note that, at κΛ = 100, all the curves nearly merge except for the range of rather high porosities, ϕ N 0.9. According to Eq. (52), the above discussed behavior is defined by the dependency of the electroosmotic mobility on the porosity at a given value of κΛ. As shown in refs. [103,104], at given values of κa (a is the particle or fiber radius), the electroosmotic mobility increases with increasing porosity. Remarkably, at given κΛ, we have weakly expressed opposite trend: both the electroosmotic mobility, χ, and thus the factor s decrease while increasing the porosity. The above mentioned behavior is explained through the dependency of the Λ-parameter on the porosity (Fig. 5): for a given type of packing of particles or fibers, the Λ-parameter normalized by the particle or fiber radius, Λ/a (a is the radius of either sphere or fiber), is an increasing function of the porosity. At a given value of κΛ, an increase in Λ/a reveals about a decrease in .the parameter κa. As shown in refs. [103,104], at constant ϕ and ζ, a decrease in κa results in the decrease of electroosmotic mobility. For the fibers packed in perpendicular to the axis x and

0.1

b

1

10

1

8/(

2

)

0.1 Sperical Cell

s

1E-4 0.5

Cylindrical Cell (fibres parallel to x)

0.01

Cylindrical Cell (fibres perpendicular to x)

1E-3 0.1

1

10

100

Fig. 7. Dependency of the geometrical factor, s, on κ Λ at given porosity, ϕ, for different capillary models.

for the packed spheres the latter trend turns out to be slightly stronger than that defined by the increase in χ with increasing ϕ. Therefore, at constant κΛ, we observe a decrease in the geometrical factor, s, which is proportional to χ. The curves plotted in Figs. 7 a and b display the dependencies s(κΛ) for various porosities, ϕ, and the porous space geometry models. All the curves confirm the above discussed asymptotic behavior at κΛ → 0 and κΛ → ∞. In Fig7a, we observe that, for various types of the capillary model, the curves nearly merge. In the case of fibers packed in parallel to the axis x there is a small difference, only, between the results corresponding to ϕ = 0.5 and ϕ = 0.99. All the curves noticeably deviate from the limiting value s = 1 when κΛ N 1 and practically merge with the straight line displaying the asymptotic dependency 8/(κΛ)2 at κΛ ≥ 20. For each of the two models describing the packed spherical particles or the fibers packed in perpendicular to the axis x, the curves plotted for ϕ = 0.5 and ϕ = 0.99 do not merge with each other within the displayed range, κΛ ≤ 100 (Fig. 7b). For the latter two models, at κΛ N 30, the curves plotted for ϕ = 0.5, become coinciding with the straight line describing the asymptotic dependency 8/(κΛ)2. When ϕ = 0.99, both the curves still noticeably (by factor of about 2 or less) deviate from the asymptote 8/(κΛ)2. In the latter case, the curves merge with the asymptote out of the displayed range when values of s become too small for providing measurable values of the apparent reflection coefficients, σk∗.

792


6. Examples of mechano-chemical effects In previous section we interrelated between the parameters responsible for mechano-chemical effects in weakly charged porous systems and the porous system morphology. Now, we consider examples of addressing the mechano-chemical effects by using the parameters under consideration. Such an analysis is conducted to understand how physicochemical properties and geometrical structure of porous system affect experimentally measured quantities. 6.1. Positive, negative and anomalous osmotic pressure For analyzing behavior of OP in ideal mixed electrolyte solutions, we combine Eqs. (1), (6), (13) and (39) and obtain the following expression N X zk Dk ΔΠ ¼ −RT ζes ΔC k k¼1 D

ð56Þ

Thus, Eq. (56) yields expression for OP in the case of a weakly charged porous system separating two mixed electrolyte solutions. We will confine ourselves by analyzing the case of a mixture of N − 1 binary electrolytes with a single common the N-th ion. By using, the electroneutrality condition of Eq. (2) one can eliminate ΔCN ΔC N ¼ −

−1 1 NX z ΔC zN k¼1 k k

ð57Þ

By combining Eqs. (56) and (57) the notation introduced by Eqs. (57), (56) can be rewritten in the terms of electrolyte concentration differences, Δck, as N −1 X

ΔΠ ¼ −RT ζes

zk ΔC k

Dk −DN

k¼1

D

ð58Þ

Thus, Eq. (58) yields expression for the OP when a weakly charged porous layer separates mixtures of N − 1 binary electrolytes with the common Nth ion. For a single binary electrolyte, N = 2, Eq. (58) can be represented, as ΔΠ ¼ ΔΠ0 σ

ð59Þ

where σ∗ is the binary electrolyte apparent reflection coefficient given by Eq. (44) and ΔΠ0 is the Van't Hoff pressure difference describing OP for a membrane absolutely impermeable for solute [4]. This quantity is given by the following relationship ΔΠ0 ¼ RT ðjz1 j þ jz2 jÞΔc ¼ RTωΔcs

ð60Þ

where Δc = ΔC1/|z2| = ΔC2/|z1|; the Van't Hoff factor, ω = (|z1| + |z2|)/ν, yields total number of ions, which is stoichiometricallydefined by the chemical reaction of complete dissociation of strong binary electrolyte under consideration; Δcs = νΔc is the binary electrolyte concentration. In Eq. (60), ν is the maximum integer common factor of numbers |z1| and |z2|. Below, we will consider only examples where ν = 1. Accordingly, for all these examples, ω = |z1| + |z2| and Δcs = Δc. As discussed in Section 4.1 while analyzing Eq. (44), σ∗ can be negative and positive Hence, the OP can be positive and negative, as well, i.e., a higher pressure can occur in the compartments with a higher (positive osmosis) or lower (negative osmosis) electrolyte concentration. While recalling the rule formulated in Section 4.1 regarding the sign of σ∗, one can reformulate this rule with respect to osmosis: when the counterion have a higher diffusion coefficient one observes positive osmosis, when the coion has a higher diffusion coefficient one deals with negative osmosis. Also, according to the discussion below Eq. (44), in the linear approximation in terms of ζe yields |σ∗| b 1. Consequently, in the presence of a single binary electrolyte, the magnitude of OP is always lower than the Van't Hoff value, ΔΠ0 given by Eq. (60). The prediction of OP given by Eqs. (59) and (60) is valid for sufficiently small concentration differences because the coefficient σ∗ depends on electrolyte concentration, c, in the equilibrium solution, σ ∗ = σ ∗(c). While replacing the finite differences in Eqs. (59) and (60), Δc and ΔΠ, by the respective differentials and integrating the obtained simple ordinary differential equation, one obtains 0

Zc ΔΠ ¼ RT ðjz1 j þ jz2 jÞ σ ðcÞdc

ð61Þ

c″

Thus, Eq. (61) yields expression for the OP produced due to the difference between the binary electrolyte concentrations in the electrolyte solutions adjacent to the porous layer, c′ and c″. Hence, Eq. (59) is a generalization of the result given by Eqs. (59) and (60). According to Eq. (44), for weakly charged porous media, the dependency σ∗(c) is defined by the concentration dependencies of both s and ζ. It is convenient to change variable of integration and integrate over the Debye parameter, κ. By combining Eqs. (25) and (57) and specifying the result for the case of binary electrolyte, one obtains

κ¼F

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jz1 z2 jðjz1 j þ jz2 jÞc εRT

ð62Þ


793

Consequently, after substituting Eq. (44) into Eq. (62) and changing the variable of integration with the help of Eq. (62), the OP is expressed as ΔΠ ¼ 2ε

0 2 Zκ RT D2 −D1 signðz1 Þ ζes κdκ F jz1 jD1 þ jz2 jD2

κ

ð63Þ

″

where the integration limits, κ′ and κ″, are obtained by substituting the electrolyte concentrations in the adjacent solutions, c′ and c″ into Eq. (62). To make rough estimations, we assume that ζ(c) = const and, thus, ζ(κ) = const, as well. By representing the geometrical factor s, in the form given by Eq. (55) and conducting some transformations, we arrive at the following result ΔΠ ¼ 16ε

RT FΛ

2

ζe

h i

D2 −D1 0 ″ signðz1 Þ W κ Λ; ϕ −W κ Λ; ϕ jz1 jD1 þ jz2 jD2

ð64Þ

where the function Wζ(w) is given by the following integral Zw W ðw; ϕÞ ¼ 0

eðκΛ; ϕÞ χ dðκΛ Þ κΛ

ð65Þ

The behavior of function W(w) is displayed by curves in Fig. 8, for two types of porous space geometry, namely, the curved capillary (solid curve) and cylindrical cell (dotted curve) models. The curved capillary model was considered for the case of slit cross-section, only, because, as it follows from Fig. 7a, the curves displaying the dependency s(κΛ) for various capillary cross-sections nearly merge with each other and are nearly independent of the porosity. The curves plotted for cylindrical cell model describe the case of fibers packed in perpendicular to the axis x. In this case, the geometrical factor, s, take the lowest value at given values of both κΛ and ϕ (Fig. 7b). We do not display the predictions from the spherical cell model since they are expected to be intermediate between the abovementioned two models that give rather close results: for ϕ ≤ 0.4 the prediction using the capillary and cylindrical cell models nearly coincide and, for ϕ = 0.9, the deviations of results from each other is less than 30%. Let us now evaluate the OPs that are achievable by using weakly charged porous media. In Table 3, we collected the values of OP computed for various 1:1 electrolytes by using Eq. (64) and the data displayed by the curves of Fig. 8. We consider the case of c″ b b c′ = 10−2M and porous system with ζe ¼ 0:5. Examples are presented for three values of the Λ-parameter: 30 nm, 100 nm and 300 nm. While using Eq. (65) we assumed that RT/F = 3 ⋅ 10−2V, ε = 7 ⋅ 10−10F/m. According to the data given in Table 3, when Λ ≤ 30 nm, one can observe easily measurable values of the OPs, more than 1 kPa, for all the examples of electrolytes Note that such a pressure corresponds to the height of liquid in the measuring capillary of about 10cm. At Λ = 100 nm, the pressure difference close to 1 kPa is expected for bases and acids. For Λ ≥ 300 nm the OP in weakly charged porous systems requires is less than 200Pa that requires special equipment for measuring. Now, we consider mixed electrolyte solutions. Our discussion will be focused on the case discussed in Section 4.2 where the apparent reflection coefficient attributed to the so-called “fast” ions was shown to take value more than unity in some mixed electrolyte solution. Below, by using examples of ternary electrolyte solutions containing such a “fast” ion, we will see how the presence of “fast” ion is expected to manifest itself in the measured value of OP. Thus, we will analyze the OP which occurs when a weakly charged porous system separates two electrolyte solutions being mixtures of two 1:1 electrolytes, namely, an acid and salt having a common anion. For such an analysis, we combine Eqs. (8), (41a) and (56) and assume that N = 3, c1 = cA; c2 = cS, z1 = zH = 1, z2 = 1, z3 = − 1, D1 = DH, D2 = D+, D3 = D−. Consequently, by considering a concentration difference of acid, only, i.e., Δc1 = ΔcA and Δc2 = ΔcS = 0, we arrive at the following expression for the OP ΔΠ=ΔΠ 0 ¼ ζes

ðDH −D− Þð1 þ cA =cS Þ ðD− þ DH ÞcA =cS þ Dþ þ D−

ð66Þ

where ΔΠ0 = 2RTΔcA is the Van't Hoff OPs difference for 1:1 electrolyte solution (see Eq. (60)). Remarkably, under certain conditions, Eq. (66) predicts OPs higher than that defined by the Vant's Hoff value ΔΠ0 = 2RTΔcA. Earlier, this interesting effect was carefully studied in refs. [12,115–117] for 10

= 0.4

w2 / 16

W

1

= 0.9

0.1

Curved capillary model 0.01

Cylindrical cell model

1E-3 0.1

1

10

100

w Fig. 8. Function W (w, ϕ) for the curved capillary (slit cross section) and cylindrical cell (fibers perpendicular to axis x) models.

794


Table 3 Values of osmotic pressure for different electrolytes for c″ b b c′ = 10−2M and ζe ¼ 0:5. Λ

30nm

100nm

300nm

κ′Λ Electrolyte Organic acid

10 |ΔΠ|(kPa) 6

33 |ΔΠ|(kPa) 1

100 |ΔΠ|(kPa) 0.17

DHþ ≈9:3 10−9 m2 =s Danion = 0.5 ⋅ 10−9m2/s HF

4.8

0.8

0.13

4.3

0.7

0.12

1.3

0.2

0.04

4

0.66

DHþ ≈9:3 10−9 m2 =s, D F‐ ≈1:5 10−9 m2 =s HCl DHþ ≈9:3 10−9 m2 =s DCl‐ ¼ 2 10−9 m2 =s NaCl DCl‐ ¼ 2 10−9 m2 =s DNaþ ¼ 1:3 10−9 m2 =s NaOH

0.11

DNaþ ¼ 1:3 10−9 m2 =s DOH− ¼ 5:3 10−9 m2 =s

the simplest geometries of porous space. In Fig. 9, for various ternary mixtures, we presented the curves displaying the dependency of OP normalized by the Vant's Hoff value on the ratio of the acid to salt concentration, cA/cS. The ion diffusion coefficients employed in the calculations were taken from ref. [118] and collected in Table 4. While obtaining the curves in Fig. 9, it was assumed that ζes ¼ 0:5. Earlier, the possibility to observe pressure difference higher than the Vant's Hoff value was shown in ref. [12]. By using the examples of electrolyte mixtures given in Fig. 9, one can see how the possibility to observe this effect depends on both the porous medium properties (interfacial potential and porous space geometry) and the electrolyte composition. In all the displayed curves, one observes the expected behavior: with increasing the ratio CA/CS the normalized OP decreases. Among the examples of Fig. 9, the highest normalized pressure difference is observed for mixture of an organic acid and lithium organic salt. When CA/CS ≈ 0.1, the OP exceeds the Van't Hoff value by factor of about two. Let us assume that CS = 10−3N and ΔCA ≈ CA ≈ 10−4N. The latter assumption violates inequality ΔCA/CA b b 1 which is a condition of using linear relationship (58). However the linearity holds since, for CA/CS b b 1, variations of CA nearly do not affect the coefficients of superposition on the right hand side of Eq. (58). By using the above data, one can evaluate the expected value of OP as ΔΠ ≈ 2RTΔcA ≈ 1kPa. Such pressure differences are easily measurable. On the basis of similar evaluation, a lower but also easily measurable value of OP, ΔΠ ≈ 0.6kPa, is expected for the mixture HF and LiF for which the Van't Hoff value is exceeded by factor of about 1.2 at CA/CS ≈ 0.1. Two other examples require lower concentrations of acids and much lower values of OP, difference 1 − 10Pa whose measurements require special experimental technique. The curves of Fig. 9 were plotted for ζes ¼ 0:5. Accordingly, for a given value of interfacial potential one can obtain the respective value of geometrical factor s. For 10−3N, κ−1 = 10nm. Using these data, one can determine the Λ-parameter with the help of Figs. 7 a or b. In particular, while assuming ζe ¼ 2 or ζe ¼ 1, the above outlined scheme yields Λ ≈ 50nm or Λ ≈ 30nm, respectively. 6.2. Pressure driven separation of ions Now, similarly to Section 6.1, we consider a mixture of N − 1 binary electrolytes having one, the Nth, common ion. One can easily demonstrate that rk ≠ N = ΔCk ≠ N/Ck ≠ N = Δck/ck, where ck is the kth binary elecrtrolyte concentration. Hence, while introducing the electrolyte rejection as Δck/ck,

3 Organic Acid +Organic Lithium Salt HF+LiF HCl+NaCl HCl+KCl

/ 2 CA R T

2

1

0 1E-3

0.01

0.1

1

10

100

CA / CS Fig. 9. Osmotic pressure difference produced by acid concentration difference in mixed electrolyte solution as function of ratio of the acid to salt concentration.


795

Table 4 Diffusion coefficients of ions employed in calculations of osmotic pressures and rejections in various ternary electrolyte solutions (Figs. 9 and 10).

Diffusion Coeffcient (10−9m2/s) Notation

H+

Na+

K+

Li+

Cl‐

F‐

Organic anion or cation

9

1.3

2

1

2

1.5

0.5

DH

D+

D+

D−

D−

D−

D− or D+

we arrive at the conclusion that such an electrolyte rejection coincides with the rejection which is attributed to the respective kth ion, k ≠ N, and thus one can use Eq. (40) for evaluating the electrolyte separation effect. For evaluating the rejection in the particular case of binary electrolyte solution, it is convenient to use Eqs. (44) and (45). When the ions have strongly different diffusion coefficients, for example in the case of an organic acid, the fraction in Eq. (44) containing the combination of diffusion coefficients approaches unity, by magnitude. Thus, by using Eqs. (44) and (45) for the case of 1:1 electrolyte and assuming that Pe ≃ 1, ζe≃1, we obtain that r ≃ s. The latter means that noticeable, r ≃ 0.1, concentration changes can be observed even for κΛ ≃ 10 (see Figs. 7 a and b). For 10−3N solution such a result is expected for the porous system characterized by Λ ≃ 100nm. Indeed, while using narrower pours one can achieve a stronger effect. The above estimations do not change substantially if we consider inorganic acid, instead of organic one, or organic salt. In the case of inorganic salt, the multiplayer depending on diffusion coefficients in Eq. (44) is considerably less than unity. For example, in the case of NaCl, it takes value of about 0.2. By using the above discussed evaluation scheme employing Figs. 7 a and b and the same parameters, Pe ≃ 1 and ζe≃1, we conclude that, for 10−3N solution, a noticeable concentration changes are achieved for a porous system with Λ ≃ 20nm. Importantly, using narrower pours in the latter example can increase the rejection not more than in two times [119]. Now, we consider the next example: a ternary electrolyte solution being a mixture of two binary electrolytes having a common ion. According to the discussion of Section 4.2, the structure of multiplayer ð1−Dk =D Þ in Eq. (40) allows one to obtain opposite signs of rejection for the “fast” and “slow” ions. In the case of mixture of two electrolytes with a common ion, such a property can be employed for separating the electrolytes when their non-common ions have different diffusion coefficients. To illustrate the aforementioned possibility, we will use Eqs. (8), (40) and (41b) for the mixture of HCl with a corresponding inorganic (NaCl) or organic (organic cation + Cl) salt. By assuming that, in Eqs. (8), (40) and (41b), N = 3 and taking account the discussion given in the first paragraph of present Section 6.2, we identify the concentrations of H+ ions, C1, as the acid concentration, C1 = cA, and the concentration of another cation, C2, as the salt concentration, C2 = cS. Consequently, we consider the rejections of H+-ion, r1 and another cation r2, as the rejections of acid and salt, respectively: r1 = rA and r2 = rS. Using the electroneutrality condition given by Eq. (2) for the system under consideration yields C3 = cA + cS. Accordingly, combining Eqs. (8), (40) and (41b) with above stated specifications enables us to represent the normalized salt and acid rejections, rA,S/PeCl (where PeCl = Uh/D−.) as functions of the ratio of the acid to salt concentration, cA/cS and the ion diffusion coefficients. ð1 þ DH =D− ÞcA =cS þ 1 þ DH =Dþ rA D ¼ sζe − 1− PeCl DH 2ð1

þ cA =cS Þ 1 þ Dþ =DH cA =cS þ 1 þ Dþ =D− rS D− e ¼ sζ 1− PeCl Dþ 2ð1 þ cA =cS Þ

ð67Þ

The curves of Fig. 10 where plotted with the help of Eq. (67) to display the dependency of normalized salt and acid rejections on the ratio cA/cS for two mixtures of salt and acid. While plotting the graphs of Fig. 10 we assumed that ζes ¼ 0:5. The diffusion coefficients of respective ions were taken from Table 4. It is clearly seen from the plots that the rejections attributed to the salt and acid have opposite signs that leads to a noticeable separation of salt and acid during the pressure driven process. Now, we present an example of system mentioned in Section 4.2 where we demonstrated an interesting possibility for some ions to have opposite signs of their apparent reflection coefficient, σ∗k, and rejection, rk. According to the discussion of Section 4.2, such a situation takes place for the kth ion provided that its diffusion coefficient, Dk, satisfies the inequality DbDk bD. Let us consider an organic acid solution additionally containing a trace

Fig. 10. Salt and acid mixture. Dependencies of salt and acid rejections on the ratio of the salt to acid concentration.

796


amount of HCl. We assume strong dissociation of organic acid with release of large univalent organic anion. By using Eqs. (8), (41a) and (41b), we determine for such a system the parameters D and D represented, respectively, in Eqs. (39) and (40) that yield the apparent reflection coefficient, σ∗k and the rejection, rk. Such a calculation yields D ≈ 4:8 10−9 m2 =s and D ≈ 0:95 10−9 m2 =s. For obtaining the latter results we substituted the values of diffusion coefficients of the Cl‐-, H+-ions and the organic anion from Table 4. Importantly, the chloride ion diffusion coefficient turns out to be higher than D and lower than D, DbDCl bD, as it is necessary for obtaining opposite signs of σ∗Cl and the rejection, rCl (Section 4.2) Let us discuss the dynamic behavior of the above described system after instant applying a constant volume flow through the porous medium. For certainty, we assume that ζ N 0. In the beginning of the process, the concentration changes of Cl‐-ions are defined by their apparent reflection ch coefficient, σk∗, (see Eq. (17) where Δμch n = ΔμCl = 0, in the initial moment). By using Eq. (39) for ζ N 0, zk = − 1, Dk ¼ DCl bD, one obtains that ∗ ∗ ′ σk = σCl b 0. Hence, the concentration, Ck = CCl′, decreases in the feed compartment and increases in the permeate compartment where Ck′ = CCl′. Thus, in the beginning of process the Cl‐-ion concentration difference is negative, ΔCcl = Ccl′ − C″cl b 0, and increases by its magnitude. Since DbDCl , in the end of the process (when the steady state regime is established), ΔCcl N 0 The latter is clear from Eqs. (19) and (40). We see that, in the beginning of process, ΔCcl becomes negative and increases by magnitude, but turns out to be positive in the end of process. Thus, the concentration of Cl‐-ions in both the solutions separated by the porous system and the concentration difference change non-monotonously with time. Such behavior can be understood while taking into account that, in the beginning of the process, the Cl‐-ions are “slow” torn out to be counterions, and, their concentration changes in adjacent compartments are defined by their excess convective transfer from the feed to permeate compartment. The migration in opposite direction driven by the Streaming Potential electric field is weak. The reverse of concentration changes occurs due to the CP electric field which is formed during the process. The above mentioned CP electric field is produced be the concentration changes of the organic acid. Recall that, in the mixture under consideration, the Cl‐-ions are assumed to be presented in trace amounts. The organic acid concentration changes in the compartments occur according to the regularities discussed in Section 4.2 for the case of binary electrolytes. Consequently, as it follows from Eq. (45), which gives a relationship between the binary electrolyte reflection coefficient, σ∗, and rejection, r, these parameters have the same sign. At ζ N 0, σ∗ is negative since the “fast” H+-ion, is a coion (see the discussion of Section 4.1) After beginning of the pressure driven process, during a transition period, the organic acid concentration difference between the compartments is formed with a higher concentration in the permeate compartment. This concentration difference gives rise to the CP electric field which drives the slow organic coion toward the feed compartment where the acid concentration is lower. Simultaneously, this field transports the Cl‐-ions toward the feed compartment. When the increasing acid concentration difference becomes sufficiently high, the migration of the Cl‐-ions becomes sufficiently strong to reverse signs of concentration changes within the compartments. Finally, the reached steady state change of Cl‐-ions concentration have a sign opposite to the sign of initial changes of its concentration. 6.3. Applicability of linear approximation While conducting the above estimations, we used rather high values of interfacial potential, ζe≃1, that allowed predicting measurable values of OP. Meanwhile, usually, the first order term in the Taylor series expansion gives a good approximation for sufficiently small value of parameter of expansion, i.e., there should be ζebb1, in the present case. However, in some electrokinetic studies, the linearization yields quite accurate results for much higher ζe. In particular, the electrophoretic mobility obtained for the Smoluchowski [120,121] and Debye [122] limiting cases (κa → ∞ and κa → 0, respectively, a is the particle radius), turns out to be proportional to ζe for its arbitrary value. In the case of electrophoretic mobility, for 1 b κa b 10, univalent ions and equal diffusion coefficients, the linear approximation yields more or less reliable description for ζe ≤2 [31,68,70]. The validity range of linear approximation is not affected noticeably while varying diffusion coefficients of ions within reasonable ranges (see the expansions of Eqs. (34) and (35) from ref. [123]). Clearly, higher valences of ions spoil the linear approximation accuracy. There some reasons to expect that the error due to the linearization will be more substantial for the case of the apparent reflection coefficients, σ∗k, that define the mechano-chemical effects under consideration. In particular, on the right hand side of Eq. (15), there are ion transport numbers, tk, e In the linear theory, this dependency does not manifest itself because the linear term in the expansion of σ∗ by powers of ζe contains that depend on ζ. k the zero order term of the similar expansion of tk, only. This zero order term is defined by the transport numbers of ions in the solution that are ine As it is clear from the analysis given in Section 4, the relationships between the transport numbers t define values and signs of σ∗ . dependent of ζ. k k These relationships can be substantially different from those between the ion transport numbers attributed to the solution. Thus, one can expect that, in the case of mechano-chemical effects, there are stronger limitations for using the linear analysis than that for classical electrokinetic phenomena. Consequently, we face a problem of determining the range of validity of the employed linear approximation. Indeed, a rigorous analysis of such problem should be conducted by determining the next terms in the expansion or by using numerical calculations. However, some conclusions can be obtained from a relatively simple analysis of error which is produced while using the linear approximation in addressing the system within the frameworks of FPM [12–15]. This model deals with the strongest deviation of ion transport numbers inside the porous system from their values in the solution. The linearized version of the FPM expression for σ∗k is easily obtained for Eq. (39) where, according to Eq. (50) one should set s = 1. The complete FPM expression for σ∗k is derived by substituting the following expression for the kth ion reflection coefficient σk and migration transport number, tk, into Eq. (15)

σ k ¼ 1−exp −zk ζe

ðaÞ;

tk ¼

ξk Dk exp −ζezk N X

ξn Dn exp −ζezn

ðbÞ

ð68Þ

n¼1

Following ref. [12], the expressions given by Eq. (68) are easily obtained while using the respective definition presented in Table 3 and realizing that, within the frameworks of FPM, the equilibrium values of the electric potential and kth ion concentration are uniform within the porous space and take the values ζ and C exp −z ζe , respectively. k

k


797

Fig. 11. Complete and linearized and versions of FPM: Comparison between predictions of apparent reflection coefficients attributed to (a) coions and counterions (b) of a ternary electrolyte solution (parameters in Table 5).

Let us consider the result of such a substitution as applied to a ternary electrolyte solution. Consequently, while taking into account the electroneutrality condition given by Eq. (2), one obtains for σ∗1

σ1 ¼

9 8 h i h i C z 2 2 > > > exp −z1 ζe −exp −z3 ζe þ exp −z2 ζe −exp −z3 ζe = < > C 1 z1 e 1−z1 1−exp −z1 ζ > D D D C z 2 2> > > ; : z1 exp −ζez1 −z3 3 exp −ζez3 þ 2 z2 exp −ζez2 −z3 3 exp −z3 ζe D1 D1 D1 C 1 z1

ð69Þ

Thus, Eq. (69) describes the dependency of the 1st ion apparent reflection coefficient, σ∗1 on interfacial potential. As it is clear from Eq. (69), at given valences, zk, the value of σ∗1 also depends on the concentration and diffusion coefficient ratios, C2/C1, C3/C1, D2/D1 and D3/D1. According to Eq. (69), the solid curves in Figs. 11 a and b display the behavior of apparent reflection coefficients attributed to the 1st ion of a ternary electrolyte solution, σ∗1, as a function of the normalized interfacial potential magnitude ζe. The dashed straight lines in the same graphs display the linear dependencies corresponding to the limiting case given by Eqs. (39) and (50). The curves were plotted for two possible cases, namely, the ternary electrolyte contains two coions (Fig. 11a) and counterions (Fig. 10b). The curves in Fig. 11a display two types of dependencies typical for coion reflection coefficients. These curves were plotted for ternary electrolyte containing two univalent coions, 1st and 2nd, and one univalent counterion, 3rd. Other parameters in Eq. (69) are assumed to take values: D3 = D2 and C1 = C2 = C3/2. The curve with minimum was built for D2/D1 = 1/5. The monotonous, curve was obtained for D2/D1 = 5. When σ∗1 is attributed to counterion, we face a wider variety of dependency types, as it seen from Fig. 11b where the curves were plotted be using Eq. (69) for ternary electrolyte whose specific parameters are collected in Table 5. The graphs presented in Figs. 11 a and b allows us to make the following conclusions: 1) The linear approximation gives correct sign of apparent reflection coefficient for the “fast” (Dk N D) counterions and the “slow” (Dk bD) coions. At ζe ≤1:5, the correct sign is predicted for all the examples. Thus, for the abovementioned cases, the rule formulated in Section 4.1 concerning the correlation between the value of Dk and sign of σ∗1 is valid. 2) Quantitatively, for all the examples, the linear approximation gives a reasonable accuracy (the error less than 10%) at ζe ≤0:5. For two cases associated with counterions (curves 1 and 6 of Fig. 11b), the approximation works good within a wider range, ζe ≤1.

798


Table 5 Parameters employed for plotting curves of Fig. 11b. Curve number

D1/D2

D1/D3

|z1| counterion

|z2| counterion

|z|3 coion

C2/C1

1 2 3 4 5 6

2 0.5 0.5 3.33 0.5 0.5

5 5 0.2 0.1 5 2

1 1 1 1 1 1

1 1 1 3 1 1

1 1 1 1 1 1

1 0 0 10−5 0.1 1

3) Except for one example (curve 1 of Fig. 11b), the linear approximation overpredicts the absolute value of the apparent refection coefficient at sufficiently high potentials. In some cases (lower curve of Fig. 11a, curves 3–5 of Fig. 11b), at sufficiently high potentials, the behavior of interfacial function σ 1 ζe , differs from that defined by the linear approximation, substantially It should be noted, that the above conclusions are linked to the FPM for which we expect the strongest limitations for using the linear approximation 6.4. Applicability of SEM Very often, authors who studied electrokinetic phenomena arrived at a conclusion that SEM requires modifications (see review papers [124,125]). Such a conclusion looks reasonable while taking into account that SEM deals with a smooth interface and vanishingly thin interfacial region. At the same time, usually, interfaces are characterized by roughness whose scale appreciably exceeds molecular dimensions. Also, in many cases, interfaces are covered by adsorbed macromolecules that make up an extended adsorption layer. Even for some interfaces that are expected to be smooth, the electrokinetic experiments reveals existence of a hydrodynamically stagnant layer adjacent to the interface. Above, we mentioned only complications existing at low interfacial potentials that are considered in the present paper. For this case, the most popular modification of SEM amounts to introducing a stagnant layer between the flowing electrolyte solution and the interface. The external boundary of such a stagnant layer is referred to as the slip surface since the hydrodynamic equations are solved in the solution outside the volume bound by this surface. Within the stagnant zone between the slip surface and the interface, the ionic transfer is addressed by using transport coefficients that differ from those attributed to the solution [124,125]. For low equilibrium potentials, the results obtained from CEM modified in the above discussed manner can noticeably deviate from the predictions of present study when the stagnant layer thickness, δst, is of order of Λ-parameter. For δst b b Λ, the deviations are not expected to be strong provided that ζ is understood as the equilibrium potential attributed to the slip surface, not to the actual interface. Importantly, there can be situations when ζeb1, but the actual interfacial potential is not small. In such a case, the conductance through the stagnant layer can bring a noticeable contribution into total conductivity. Clearly, this conductivity is not taken into account by using the approach employed in the present paper. 7. Conclusions When a weakly charged porous medium is placed between two electrolyte solutions, one can observe mechano-chemical Effects: Osmosis and pressure driven separation of ions, provided that the ions have different diffusion coefficients inside the solution. Osmosis occurs due to the electric driving force which acts on the equilibrium space charge of the diffuse part of EDL within the porous space adjacent to the solid–liquid interface. The electric field exerting such a force is produced in the presence of concentration gradients of ions having different diffusion coefficients. The separation effect originates from interplay between the positive (negative) excess convective fluxes of counterions (coions) and their migration fluxes in the electric field of Streaming Potential which maintains zero electric current through the porous medium. When the ion diffusion coefficients are equal, the migration fluxes exactly compensate for the excess convective flux of each of the ions. However, when diffusion coefficients differ, the effect of separation manifests itself. It is shown that, within the frameworks of linear approximation (in e for addressing both the terms of the normalized interfacial potential, ζ), OP and pressure driven separation of ions it is sufficient to know a set of reflection coefficients, σk, attributed to each of the ions from electrolyte solution and characterizing the porous medium. When all the reflection coefficients are known, one can determine two sets of parameters: the apparent reflection coefficients, σ∗k, and the rejections, rk. When the ion concentration differences between the solutions separated by porous medium, ΔCk, are known, one can predict the OP by substituting the obtained σ∗k into Eq. (14). Also, having obtained all the rejections, rk, one can use Eq. (19) to determine the steady state concentration differences (ΔCk)St that are produced between the adjacent solutions in the presence of volume flow through the porous system.

The obtained expression for σk is given by Eq. (21) which demonstrates that the reflection coefficient is proportional to coefficient, s, common for all the ions. This coefficient depends on the porous space geometry and, therefore, is referred to as the geometrical factor. By using the derived irreversible thermodynamic relationship given by Eq. (10), the geometrical factor, s, is expressed through the electroose the conductivmotic mobility normalized by its Smoluchowski limit, χ, ity formation factor, f, the Darcy coefficient, K and the Debye parameter of electrolyte solution, κ, Eq. (37). The above quantities can be measured independently, and, by combining them according to Eq. (37), one can e f, and K obtain the geometrical factor s. Obtaining the coefficients χ, for various geometries of the porous space was in the focus of a tremendous number of earlier studies. Consequently, using results of these studies enables one to predict the geometrical factor, s, for all the considered geometries. The obtained expression for σk , Eq. (21), allowed us to deduce Eqs. (39) and (40) that yield expressions for apparent reflection coefficients, σ∗k, and the rejections, rk, respectively. These expressions have convenient form for analyzing how σ∗k and r k depend on the kth ion valence and diffusion coefficient and are affected by the solution composition. By analyzing Eqs. (39) and (40), we formulated a rule which allows one to determine the sign of both σ∗k and rk: The apparent reflection coefficient, σ∗k, and rejection of the “fast” counterions and the “slow” coions are always positive, whereas for the “fast” coions and the “slow” counterions, they are negative. To understand whether the kth coion is “fast” or “slow”, one should inspect whether its diffusion coefficient is, respectively, higher or lower than a mean diffusion coefficient. Remarkably, these mean diffusion coefficients represented in the expressions for σ∗k and rk, D and D (Eqs. (39) and (40)), differ and are given by Eqs. (41a) or (41b), respectively. As D≥D , the signs of the rejection and the apparent


reflection coefficient of the kth ion always coincide when either Dk ND or Dk bD. In particular, these signs always coincide for binary electrolyte, as follows from Eq. (45). When DbDk bD, the kth ion have different signs of σ∗k and rk. In the latter case, during the transition period of establishing a steady state concentration difference, one can expect a nonmonotonous change of the kth ion concentrations in the adjacent solutions with time. Analysis of Eq. (39) also confirmed the results earlier reported in literature that, for binary electrolyte, the apparent reflection coefficient is always less than unity by absolute value whereas, in mixed electrolytes, the “fast” ions can noticeably be more than unity. The latter case comprises a seemingly paradoxical situation of positive reflection coefficients more than unity, i.e. more than that for an ion whose transfer through porous system is completely blocked. The role of porous space geometry was analyzed by comparing between the predictions of geometrical factor, s, from various geometrical models of porous space, namely, the weakly curved capillary model (circular and slit cross-sections), randomly packed spheres and cylinders (along and perpendicular axis x, Fig. 2). The expressions for s were obtained by substituting into Eq. (37) the expressions for the nore the conductivity formation factor, malized electroosmotic mobility, χ, f, and the Darcy coefficient K that were earlier derived by different authors. All these expressions are collected in Table 2. For all the considered geometries, it is possible to represent the geometrical factor, as a function of the dimensionless parameter, κΛ, where Λ is a length scale parameter of porous systems which is widely used in literature and defined by Eq. (53), and the porosity, ϕ. The expressions for Λ are also collected in Table 2. The plots illustrating the behavior of function s(κΛ, ϕ) (Figs. 6, 7 a and b) reveals the several important properties of this dependency. At given values of the parameter, κΛ, the porosity, ϕ, the geometrical factor, s, takes close values for all the porous space geometries. For various versions of the capillary model, s, is independent (or nearly independent, for fibers packed in parallel to axis x) on the porosity. Accordingly, the function s(κΛ) and is displayed by nearly coinciding curves that are common for all the versions of the capillary model. Another situation takes place for the packed spheres and the fibers packed in perpendicular to the axis x. For these geometries, the curves describing the dependency s(κΛ, ϕ) deviate from the curve s(κΛ) describing the geometrical factor for capillary systems. Within the porosity range 0.5 b ϕ b 0.99, this deviation is not very strong. The strongest deviation occurs in the case of the fibers packed in perpendicular to the axis x at ϕ = 0.99. In the latter case, s(κΛ, 0.99) ≤ s(κΛ), and the deviation of s(κΛ, ϕ) from s(κΛ) is less than 50%. The above properties define the practical importance of Fig. 7a and b. By measuring the porous system conductivity and hydraulic permeability, one can determine the coefficients f and K, and, with the help of Eq. (53), the Λ-parameter. Consequently, for a given value of the Debye parameter, one can use Figs. 7a and b for rough estimating s. We considered several examples of addressing the OP and the pressure driven separation of ions in the cases of binary and ternary electrolyte solutions. The considered examples confirmed the reported earlier possibilities to observe positive and negative OP and electrolyte rejection. For binary electrolyte, we derived Eq. (63) which describes the OP when the electrolyte concentration difference is not small. While assuming that ζ is independent on concentration in equilibrium solution, we arrived at Eq. (65) containing a function W(w, ϕ) which depends on the porosity and the type of porous pace geometry. This function is given by Eq. (66) and, for several types of geometry and porosities, is displayed by curves in Fig. 8. The OP computed for various binary electrolyte solutions and values of Λ-parameters are presented in Table 3. Another example was the OP in a mixture of salt and acid having a common anion. Analysis was conducted for the case when there is a concentration difference of acid, only. It was shown that, at sufficiently

799

low concentration of acid, one can observe the OP whose magnitude is higher than the Van't Hoff value (anomalous OP) given by Eq. (60). From Fig. 9, it is clear that this effect is more profound for mixtures containing organic ion as a common one. We also considered an example of separating mixture of salt and acid. Curves in Fig. 10 show that this effect is stronger when the salt contains organic cation. Also, we presented a specific example of a system containing an ion whose diffusion coefficient satisfies the inequality Db Dk bD and discussed non-monotonous changes of this ion concentration in adjacent solutions with time. Finally, the applicability of the linear approximation in terms of ζe was discussed by using examples obtained with the help of FPM, for which the general expressions for the apparent reflection coefficients have been reported in the literature and the linear approximation yields result given by Eqs. (21) and (50). List of Symbols Latin letters a radius of a spherical particle or fiber b radius of the cell Ck concentration of the kth ion in equilibrium solution ck concentration of the kth binary electrolyte in a mixture cA acid concentration cS salt concentration Dk the kth ion diffusion coefficient D+ and D− diffusion coefficients of cations and anions D and D mean ionic diffusion coefficients given by Eqs. (41a and b) ! E ext electric field strength in the solution adjacent to the porous layer F Faraday constant f = gS/g(0) conductivity formation factor g porous medium conductivity g(0) conductivity of uncharged porous medium gS conductivity of adjacent solutions h thickness of porous layer I electric current density ! ix unity vector along axis x Jk the kth ion flux through porous layer ! jk the kth ion local flux K Darcy coefficient lkn interdiffusion permeability l∗kn apparent interdiffusion permeability ! n unit vector normal to solid–liquid interface P local pressure; R gas constant rk the kth ion rejection S area T absolute temperature tk the kth ion transport number tSk the kth ion transport number in solution U volume flow through the porous system per unit area ! u local velocity of liquid; Ueo electroosmotic velocity V volume x Cartesian axis normal to the porous medium solution interfaces (Fig. 2) zk valence of the kth ion

Greek letters δ thickness of the gap between the disperse system layer and planes A and B (Fig. 2) δst thickness of the stagnant layer between the interface and the slip surface χ electroosmotic mobility

800


ξk ¼ C k z2k =∑ Z 2k C k dimensionless coefficient n

η ε κ Λ

viscosity dielectric permittivity Debye parameter phenomenological lambda parameter of porous system given by Eq. (53) Λ0 theoretical lambda parameter of porous system given by Eq. (54) λ hydraulic permeability λ∗ apparent hydraulic permeability μk the kth ion electrochemical potential μ ch the kth ion chemical potential k ν maximum common factor of valences of binary electrolyte ions ΔΠ osmotic pressure ΔΠ0 the Van't Hoff limit of osmotic pressure ρ space charge density σk the kth ion reflection coefficient σ∗k the kth ion apparent reflection coefficient τ tortuosity ω Van't's Hoff factor Φ electric potential; ϕ porosity Ψ equilibrium electric potential ψ = Ψ/ζ normalized local equilibrium potential ζ electric potential at the interface in equilibrium state

Abbreviations CO Concentration Potential EDL Electric Double Layer FPR Fine Pore Model OP osmotic pressure P–B-Poisson Boltzmann (equation, problem) SEM standard electrokinetic model SCM Straight Capillary Model

Acknowledgments Financial supports from EC within the frameworks of the Seventh Framework Program (project acronym “CoTraPhen”, grant agreement number: PIRSES-GA-2010-269135) is gratefully acknowledged. Appendix A. Normalized electroosmotic mobility for fibers packed in parallel to external electric field strength For addressing electrically driven flow in the system of fibers packed in parallel to the externally applied field, we consider circularly symmetrical version of the Stokes equation η

1d du 1d dψ r ¼ εζ r E int r dr dr r dr dr

ðA1Þ

where r is the radial coordinate in the 2D polar coordinate system ! ! whose origin coincides with the cell center; E int ¼ E int i x is the magnitude of the uniform electric field strength existing in the pore; the whole expression multiplied by Eint on the right hand side is the local electric charge density (taken with opposite sin). It is easy to integrate Eq. (A1) twice while using boundary conditions at the particle surface (u = 0 and ψ = 1) and at the cell outer boundary (du/dr = 0 and dψ/dr = 0). The first of the latter two boundary conditions is the Kuwabara condition which coincides with the Happel condition in the case. The second of the conditions reflects

total electroneutrality of the cell. Result of the integration leads to the following expression u¼−

εζ ð1−ψÞE int η

ðA2Þ

Taking into account that Eq. (2) yields velocity distribution within the liquid phase and N within the solid phase the velocity is zero, one can average the velocity over the cell volume 0 1 Zb εζ 2π hui ¼ − E int @ϕ− 2 ψrdr A η πb a

where a and b are the fiber and cell radii that satisfy equality (a/b)2 = 1 − ϕ. Finally, using relationships (9) and (36) and taking unto account that due to the continuity of electric current, Eintϕ = Eext/ we arrive at the following equality

e ¼ 1− χ

2ð1−ϕÞ ϕ

−1=2 ð1−ϕ ZÞ

ψxdx

ðA3Þ

1

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