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The effect of membrane potential on the development of chemical osmotic pressure in compacted clay
Journal of Colloid and Interface Science 297 (2006) 329–340 www.elsevier.com/locate/jcis
The effect of membrane potential on the development of chemical osmotic pressure in compacted clay S. Bader a,∗ , K. Heister b a Environmental Hydrogeology Group, Faculty of Geosciences, Department of Earth Sciences, Utrecht University, PO Box 80021, 3508TA, Utrecht,
The Netherlands b Geochemistry Group, Faculty of Geosciences, Department of Earth Sciences, Utrecht University, PO Box 80021, 3508TA, Utrecht, The Netherlands
Received 4 August 2005; accepted 11 October 2005 Available online 10 November 2005
Abstract When clay soils are subjected to salt concentration gradients, various interrelated processes come into play. It is known that chemical osmosis induces a water flow and that a membrane potential difference develops that counteracts diffusive flow of solutes and osmotic flow of water. In this paper, we present the results of experiments on the influence of membrane potential on chemical osmotic flow and diffusion of solutes and we show how we are able to derive the membrane potential value from theory. Moreover, the simultaneous development of water pressure, salt concentration and membrane potential difference are simulated using a model for combined chemico-electroosmosis in clays. A new method for short-circuiting the clay sample is employed to assess the influence of electrical effects on flow of water and transport of solutes. 2005 Elsevier Inc. All rights reserved. Keywords: Chemical osmosis; Membrane potential; Modeling; Clay membranes; Electroosmosis
1. Introduction Although generally not taken into account during modeling of groundwater flow, coupled effects such as chemical and electroosmosis may be quite significant in clayey soils in, e.g., coastal areas where salt concentration gradients are abundant. Quite a body of literature exists on experimental evidence for chemical osmosis in laboratory situations [1–3], but only recently substantial proof has been obtained for this effect occurring in situ [4,5]. In this paper, we report on experiments involving measuring membrane potential and its influence on chemical osmotic pressure evolution. Each experiment was performed in short-circuited and non-short-circuited conditions to assess the influence of electrical effects. Details of these experiments and, in particular, the method of short-circuiting are given in [6]. The theory of Revil [7] is used to give an interpretation of the observed value of the membrane potential.
* Corresponding author.
E-mail address: [email protected] (S. Bader). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.10.036
A model for simulating transient behavior of osmotic pressure and salt concentration was presented in [8,9]. This model is used in this study in a similar manner to compare experimental and numerical results. However, we extend the model with electrical effects, and investigate subsequently the behavior of the model with and without these effects. 2. Chemical osmosis and membrane potential: an introduction As water flow driven by gradients of temperature and electrical potential is called thermoosmosis and electroosmosis, respectively [10], we define osmosis in the context of soil to be non-hydraulically driven water flow. Chemical osmosis is then the water flow driven by a salt concentration gradient. As clay consists of negatively charged platelets that restrict movement of ions passing through, it can be considered as a semi-permeable membrane, and when a compacted clay is subjected to a salt concentration gradient, chemical osmosis may drive water from low to high salt concentration. Furthermore, an electrical potential gradient may induce water flow as well, as moving ions drag along water [10]. This phenomenon is called
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electroosmosis. Mathematically, chemical and electroosmosis are seen as extensions of Darcy’s law, k q = − ∇p + λ∇cf − ke ∇V , µ
(1)
i=+,−
[m2 ]
where q [m/s] is specific discharge, k is intrinsic permeability, µ [kg/(m s)] is fluid viscosity, p [Pa] is pressure, cf [kg/m3 ] is salt concentration, ke [m2 /(V s)] is electroosmotic permeability, V [V] is electrical potential, and the socalled chemical osmotic mobility λ is given by λ = σ νRT
We start with the electrical current density I [A/m2 ], which is expressed as I= (3) jdi zi F,
k , µ
(2)
where σ [–] is the reflection coefficient that describes the degree of restriction on solutes, or reflection by the membrane, and ranges between 0 and 1. Furthermore, ν [–] is the dissociation constant, R [J/(mol K)] is the gas constant and T [K] is the temperature. The membrane potential is the electrical potential difference that compensates the separation of charge due to different velocities of cations and anions in a free electrolyte solution or in a porous medium (partly) saturated with an electrolyte solution. Different terminologies for this effect are en vogue. For example, in [11], Bolt states that a potential difference caused by the restrictive behavior of the membrane on the ionic mobilities is called the membrane potential whereas the sum effect of liquid and membrane on the ions is called the diffusion potential. In [12], however, the membrane potential is the sum of the so-called concentration potential (the potential difference in a porous medium between two solutions) and the Donnan potential (between the solution and the membrane). We follow the approach of Revil [7], where membrane potential is the sum effect, and the part caused by the membrane is called the exclusion potential. Schematically, the Bolt and Revil definitions are given by Bolt [11]:
Vmembrane + Vliquid = Vdiffusion ,
Revil [7]:
Vexclusion + Vliquid = Vmembrane .
Although the Bolt [11] approach may be somewhat more logical, in most papers, the term membrane potential specifies the sum effect. We apply the combined set of equations for chemicoelectroosmosis to assess the influence of membrane potential on chemical osmotic flow. A number of experiments were performed quite similarly to those reported in [1]. Details on these experiments can be found in [6]. In a laboratory, a clay sample was subjected to a salt concentration gradient and pressure and concentration were simultaneously measured. However, the clay was equipped with electrodes to measure the electrical potential as well, and an experimental method was found to short-circuit the clay, in order to turn on and off the electrical effects. 3. Derivation of an expression for membrane potential In this section we derive an expression for the membrane potential.
where jdi [mol/(m2 s)] are the ionic fluxes in a free solution, zi [–] are the ionic valences and F [C/mol] is Faraday’s constant. For diffusion of separate ions of concentration ci [mol/m3 ], assuming an ideal solution, we write (see [7,13]) ui ci jdi = − (4) ∇µi , F µi = µ0i + RT ln ci + zi F V , (5) where ui [m2 /(V s)] is ionic mobility and µ0i is a reference chemical potential such that ∇µ0i = 0. Substituting (5) in (4) yields ui RT ∇ci − ui ci zi ∇V . (6) F For the electrical current density in a 1:1 electrolyte this implies jdi = −
I = −(u+ − u− )RT ∇cf − (u+ + u− )cf F ∇V .
(7)
When electroneutrality is enforced, the concentrations of anions and cations are equal, cf = c+ = c− , as are the ionic fluxes: jds = jd+ = jd− . We define u = u+ − u− as the effective solute mobility and σf [S/m] = (u+ + u− )F cf as the electrical conductivity of the electrolyte to get to the following expression for the electrical current density: I = −uRT ∇cf − σf ∇V .
(8)
If we assume the mobility and the electrical conductivity to be constant, we obtain a simple expression to quantify membrane potential: I = 0 → ∇V = −
uRT ∇cf . σf
(9)
This expression is often found in literature [13,14] to describe the membrane potential. Here, most notably, a constant electrical conductivity is assumed. However, especially for larger salt concentration gradients, the concentration dependence of the electrical conductivity becomes increasingly significant [9]. For solutions, such as NaCl, it is known that a Cl− ion moves faster than a Na+ ion, or u− > u+ , hence u < 0 and the potential gradient is of the same sign as the salt concentration gradient. However, when for example in a clay, the movement of anions is more restricted than the movement of cations, viz. in the case of anion exclusion, the mobility of cations is higher than the mobility of anions and the potential gradient and the salt concentration gradient have different signs. We may follow the approach of [7] and adopt the expression for electrical conductivity; then ∇V = −
(u+ − u− )RT RT ∇cf = − (t+ − t− )∇ ln cf , (u+ + u− )F cf F
(10)
where ti are Hittorf transport numbers that specify relative mobility of the ionic constituents: ui ti = (11) . i=+,− ui
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For a porous medium it is shown in [7] that these equations are valid if we substitute for σf , ui , ti macroscopic parameters. In [7] it is shown that we can write ∇V = − Ti =
RT (T+ − T− )∇ ln cf , F
σi , σf
(12) (13)
where σi [S/m] is the ionic contribution to the bulk electrical conductivity σe = σ+ + σ− and Ti [–] are the macroscopic Hittorf numbers. This expression can be decomposed in a liquid junction potential and a exclusion potential: RT 2RT (2t+ − 1)∇ ln cf − (T+ − t+ )∇ ln cf . (14) F F In [7] it is shown qualitatively how in the absence of a porous medium, the macroscopic parameter Ti reduces to ti . Alternatively, we may consider the presence of the porous medium to be only accounted for by the formation factor (as in [15]), i.e., σf σe = (15) . F0 ∇V = −
The formation factor is denoted by F0 , and ∇V = −
RT F0 (2t+ − 1)∇ ln cf . F
(16)
331
as d Df Js nsc = − γ ∇cf , F0
(20)
where F0 [–] is the formation factor, Df is free solution diffusivity and γ = T+ /t+ is a factor indicating the influence of the charged medium on diffusion. By definition, in a free solution, γ → 1 and F0 → 1. More details on this can be found in [9]. When we consider the short-circuited situation, we disregard all electrical contributions. The result is the following: the anionic and cationic diffusion coefficients, concentrations and fluxes are no longer equal. This implies that a successful shortcut separates the charges in the solution. The liquid-junction potential, generated by this separation of charge, is immediately compensated by the potential generated by the virtual shortcut. In the experiments considered, the Cl− sensitive electrodes only measure the amount of anions in the solution, so instead of the salt flux we have to use the expression for the anion flux to interpret the results of the experiments. Accordingly, the electrolyte anion flux is, with D− the microscopic anionic diffusivity,
Jd−
sc
=−
D− ∇c− , F0
(21)
as the diffusion of anions is assumed not to be influenced by the charge of the porous medium, but only by its geometry F0 . 6. Experiments
4. Membrane potential and chemical osmosis We can extend the derived expression for the electrical potential difference with the streaming potential [10]: ∇V = −
ke RT ∇p − (T+ − T− )∇ ln cf . σe F
(17)
Substituting (17) in (1), we find RT k 1 ke2 qnsc = − − (T+ − T− ) ∇p + λ + ke ∇cf . µ σe F cf (18) The transition from the expression with electrical effects to the one without electrical effects is formally done by shortcircuiting. In the experiments, the system is short-circuited to eliminate any electrical potentials. The flux equations therefore come in two flavors: short-circuited (sc) (or shorted), i.e., without electrical effects, and non-short-circuited (nsc) (or nonshorted), i.e., inducing electrical effects. The short-circuited Darcy flux is simply given by k qsc = − ∇p + λ∇cf , µ
(19)
i.e., all electrical terms are absent. 5. Membrane potential and diffusion If we consistently follow the theoretical reasoning of Revil, as given in [7], we can derive an expression for the effective diffusivity as in [7]. The non-short-circuited solute flux is written
A number of experiments were carried out to investigate how the membrane potential influences the buildup of osmotic pressure. We show here the results of experiments on bentonite clay for an initial concentration difference of 0.01–0.1 M NaCl and 0.01–0.05 M NaCl for natural, heteroionic clay and a c = 0.01–0.1 M NaCl experiment for homoionic clay. In Table 1 it is shown how the different experiments are classified. 7. Experimental setup To contain the clay, a rigid-wall permeameter was used, consisting of plastics for electrical insulation with an inner diameter of 50 mm. Within this mould, the clay sample is situated between two porous stones and nylon filters, where it is saturated with a fresh water solution before the experiment is started such that distortion of the clay membrane is minimal. Compaction of the clay during the experiment is ensured by putting a mechanical load on the piston of the sample mould. Table 1 Designation of experiments Designation
Concentration difference (NaCl)
Type of clay
Shorted?
be1 nsc be1 sc be05 nsc be05 sc hbe1 nsc hbe1 sc
0.1–0.01 M 0.1–0.01 M 0.05–0.01 M 0.05–0.01 M 0.1–0.01 M 0.1–0.01 M
bentonite bentonite bentonite bentonite homoionic bentonite homoionic bentonite
no yes no yes no yes
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Fig. 1. Photograph of the experimental setup.
Via the porous stones, the clay is in contact with two fluid reservoirs that are equipped with double junction Ag/AgCl reference electrodes and Cl-sensitive electrodes to measure the electrical potential and the Cl concentration, and with a calibrated standpipe to quantify the water flow. Ag/AgCl electrodes are used to measure the electrical potential because their potential is determined by the activity of Cl ions, so they are reversible to chloride. The outer liquid junction consists of KNO3 which minimizes diffusion potentials at the interface of the electrodes. Moreover, polarization effects, which occur at metal electrodes, are eliminated. To prevent concentration polarization and to homogenize the solutions, the reservoirs are continuously circulated with a peristaltic pump. The setup is shown in Fig. 1. To gain insight in the effect of the potential build-up on the water flow across the clay sample, it should be possible to eliminate this potential by short-circuiting the electrodes. However, physical shortening is not viable because conducting current through the reference electrodes produces unpredictable additional potentials. Therefore, an instrument was designed, here referred to as “virtual shortcut,” which brings the potential across the sample to zero with a negative feedback current. The virtual shortcut measures the potential with the Ag/AgCl electrodes and applies a corresponding current by a feedback amplifier on noble metal electrodes, which are also inserted in both fluid reservoirs. Such a four-electrode arrangement is common in electrochemistry. Interferences and deterioration of the reference electrodes are avoided because the electrical current is conducted through the metal electrodes, which are chemically inert. The problem of polarization potentials at the metal electrodes can be ignored since this is also compensated by the electrical potential applied on them. A detailed description of the experimental setup and procedure is given in [6].
up more or less instantaneously, and then declined, mainly because of diffusion of salt into the clay. The membrane potential, as calculated in the next section, refers to the initial potential difference. Also, because of the salt concentration difference, chemical osmosis induced a water flow through the clay from low to high salt concentration, inducing a relatively slow (by storativity) buildup of pressure. As seen in comparable studies [8], the pressure declined after a maximum, also because of diffusion of solute through the clay. Two experiments were performed for each case, with different clay samples prepared form the same batch of sample material. We assume these samples to be identical. Every experiment was performed with and without shortcut. Without shortcut, which is actually the default situation for clays, an electrical potential is present throughout the clay: a sum of membrane and streaming potential. This potential induces electroosmotic flow, or in particular: a concentration gradient shifts the excess of cations, thereby creating a potential that induces electroosmotic flow which counteracts flow of water due to chemical osmosis. 8. Measuring the membrane potential The membrane potential cannot be measured directly [12]. Usually the total electromotive force (EMF) of a system is measured, as displayed in Fig. 2. The EMF is evaluated as the sum of the electrode potentials, the Donnan potentials and the membrane potential. The electrode potential is the electrical potential difference between the electrode and the solution. The
7.1. Experimental results In all experiments, the following occurred: the applied concentration difference induced a membrane potential that built
Fig. 2. Overview of electrical potentials.
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electrode potential Eel is defined as
Now, in [7], an expression is derived for T+ :
RT ln ai , (22) F where E0 is the standard electrode potential and ai is the activity of ion i. When calculating the EMF of the system, the two E0 ’s cancel out. The potential between two phases in contact is called the phase-boundary potential [12]. When a solution is brought in contact with a membrane this is called the Donnan potential. In a negatively charged membrane, the concentration of cations is larger than the concentration of anions, and the concentration of cations is larger in the membrane than in the solution. The cations want to diffuse out of the membrane, but this is prevented by the Donnan potential to attain electroneutrality. The Donnan potential is defined by
1 1 − t+
=1+ T+ F0 ξ + 12 (t+ − ξ ) 1 − tξ+ + 1 −
Eel = E0 +
RT a¯ i ln , EDon = − F ai
(23)
where a¯ i [mol/m3 ] is the activity of ion i in the membrane. As the membrane potential is related to the activities of the ions in the membranes, the formula for the Donnan potential “translates” the values for the activities in the membranes to the values of the activities in the solutions. As in [12], we get for a 1:1 electrolyte and a cation-exchanger: 1 2 − EDon Emem = Ediff + EDon (24) a¯ 1+ a¯ 2+ a¯ 2+ RT + 2 t¯− d ln a¯ f + ln − ln ln =− F a¯ 1+ a1+ a2+ (25) RT a2 =− (26) + 2 t¯− d ln af ln F a1 RT a2 = (27) − 2 t¯+ d ln af . ln F a1
Sub- and superscripts 1, 2 refer to different sides of the sample and af is salt activity in the free solution. In the last lines the definition t¯+ = 1 − t¯− was used and it was assumed that the activity of the salt is equal to the activity of the ions. Introducing the notation of [7] for the macroscopic transference number, i.e., T+ instead of t¯+ , where L is the membrane thickness, RT Emem = − (28) (2T+ − 1) d ln af . F When we assume T+ (L) = T2+ and T+ (0) = T1+ to be constant, we find RT (2T2+ − 1) ln a2 − (2T1+ − 1) ln a1 F RT a1 2RT + ln (T1+ ln a1 − T2+ ln a2 ). =− F a2 F
Emem = −
(29) (30)
The total EMF, i.e., the measured potential, now consists of the electrode potential and the membrane potentials; as in [7], we write RT a2 ln Etot = Eel1 + Emem + Eel2 = Emem − (31) F a1 2RT = (32) (T1+ ln a1 − T2+ ln a2 ). F
ξ 2 t+
+
4F0 ξ t+
. (33)
Na+
is t+ = 0.38. The paThe microscopic Hittorf number for rameter ξ is defined as the ratio of surface conductivity σs and fluid conductivity σf : ξ=
σs . σf
(34)
The surface conductivity is defined in [7] as σs =
2 n βs Qv , 31−n
(35)
where Qv [C/m3 ] is the excess of charge, n [–] is porosity and βs [m2 /(V s)] is the mobility of the counterions at the surface of the solid particles. This definition however, is based on the assumption of low porosity, whereas we deal with high porosity. The approximation for high porosity [16] is σs =
βs Qv . 2 − 2n/3
(36)
The excess of charge is calculated using Qv = ρs
1−n F C, n
(37)
where ρs [kg/m3 ] is the rock density and C is the cation exchange capacity, expressed in terms of molc /kg ∼ meq/g. 9. Membrane potential value As an example, we review the results of experiment be1 sc in detail. Here, the overburden pressure applied was 1 bar and the applied concentration difference was roughly 0.1–0.01 M. The precise activities are listed in Table 2. With these values, the total potential is Etot =
2RT (T1+ ln 11.6 − T2+ ln 94.1). F
(38)
Table 2 Some a priori measured and calculated parameters Parameter
be1 (n)sc
be05 (n)sc
Low concentration [mol/m3 ]
12.2 105 11.6 94.1 53 0.80 298 2650 [1] 3.4 × 107 0.118 −32 −34
13.77 55 13.1 50.4 53 0.80 298 2650 3.4 × 107 0.118 −23 −23
High concentration [mol/m3 ] Low activity [mol/m3 ] High activity [mol/m3 ] Cation exchange capacity C [cmolc /kg] Porosity n [–] Temperature T [K] Rock density ρs [kg/m3 ] Excess of charge Qv [C m3 ] Surface conductivity σs [S/m] Membrane potential: measured [mV] Membrane potential: calculated [mV]
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The corresponding theoretical liquid-junction potential Elp in the absence of a membrane would be Elp =
2RT t+ (ln 11.6 − ln 94.1) = −41 mV. F
(39)
The excess of charge and the surface conductivity are in Table 2. The value of the mobility of the counterions at the surface of the solid particles βs is based on experiments on montmorillonite [7]. Another parameter considered is the formation factor F0 . It is customary to relate this parameter to the porosity via Archie’s law [17], F0 = n−m ,
(40)
where m is the so-called cementation exponent, usually assumed to be 2. If we assume constant porosity and formation factor, we get F0 = 1.56. Using all these values and definitions, the macroscopic Hittorf numbers depend only on the concentrations in the reservoirs. Using formula (33), we find T+1 = 0.54, T+2 = 0.44, and consequently Etot = −34 mV.
(41)
As the experimental result is Etot = −32 ± 3 mV, we have quite a satisfactory agreement between the experimental result and our theory. Now, the actual membrane potential is RT a1 ln F a2 = −32 mV + 55 mV = +23 mV.
Emem = Etot −
(42)
Hence, the actual membrane potential is positive, i.e., of the same sign as the concentration gradient. This is in accordance with flow data of non-shorted tests and short-circuited tests, that strongly indicate a counterflow of water hindering chemical osmotic transport. This result is, however, not in accordance with the interpretation as given in [7]. Here, a theoretical range is given for the membrane potential, which amounts for the current experiment to Etot ∈ [−110, −41], where Etot = −41 mV corresponds to the case of “pure” liquid junction potential and Etot = −110 mV to a “strong” influence of the membrane on the potential. However, in Fig. 3, it is shown how the membrane potential, using the values of the activities from the experiment considered in this study, depends on the formation factor and on the parameter α = ξ af . This figure shows that, in the range of α and F0 that we consider, the membrane potential stays clearly above the −41 mV value. Only for rather large values of F0 and α, the membrane potential is smaller than −41 mV, as predicted in [7]. Apparently, the interpretation of membrane potential in this respect has to be adapted. Most likely, the large difference in transference numbers for the fresh and salt sides causes this “anomaly” in the membrane potential. Or, the influence of cation exchange may be different than as proposed in [7].
Fig. 3. Membrane potential values for varying α and F0 .
10. Modeling procedure 10.1. Equations The simultaneous evolution of pressure, membrane potential and concentration was modeled using equations from [8]. The model equations are the following (see [8]), assuming incompressible fluid, no chemical reactions, no temperature effects, low Peclet number. Fluid mass balance: ∂n + ∇ · q = 0. (43) ∂t Salt mass balance: ∂ncf + ∇ · (cf q) + ∇ · Jds = 0. (44) ∂t Flux equations: k 1 k2 qnsc = − − e ∇p + λ + ke RT (T+ − T− ) ∇cf , µ σe cf (45) k qsc = − ∇p + λ∇cf , (46) µ d Df Js nsc = − (47) γ ∇cf , F0 d D− J− sc = − (48) ∇c− . F0 Constitutive equation: n = 1 − (1 − n0 )e−αp ,
(49)
where n0 is a reference porosity and α [1/Pa] the compressibility of the porous medium. 10.2. Model domain In Fig. 4 the general modeling domain for all membrane potential experiments is presented. The parameters not specified are given in the tables corresponding to the respective experiments.
S. Bader, K. Heister / Journal of Colloid and Interface Science 297 (2006) 329–340
Parameter
Value in clay
Value in porous stone
n (–) k (m2 ) D (m2 /s) α (1/Pa) σ (–) Length (mm)
0.5 2.34 × 10−18 D 1 × 10−9 σ a = 2.9
n 1 × 10−13 1.5 × 10−9 4.6 × 10−10 0 b − a = 7.2
335
Fig. 4. Modeling domain membrane potential experiment and relevant parameters.
Fig. 5. Pressure development for the shorted and non-shorted case; comparison of experimental and numerical results.
10.3. Initial and boundary conditions Initially, the pressure and electrical potential are zero in the whole domain, and the concentration is c0 (high value) in the porous stone at the left-hand side and ca (low value) in the clay and the porous stone at the right-hand side. The following boundary conditions apply: ∂p = 0, ∂x
∂c (50) = 0, at x = −L, ∂x ∂c p = 0, (51) = 0, at x = b. ∂x We imposed p ≡ 0 in the left-hand side reservoir, because the pressure in this reservoir is atmospheric. The solutions in the reservoir are well stirred, such that the measured concentrations should correspond to the concentrations at the interface between the reservoirs and the porous stones. 11. Modeling results on bentonite: high c 11.1. Pressure development In this section, we present the experimental and modeling results of experiment be1 (n)sc, which was performed on the clay sample subjected to a concentration gradient of about
90 mol/m3 . For numerical modeling, we used the scripted finite element builder and numerical solver FlexPDE [18]. In Fig. 5, the pressure buildup in the salt water reservoir is depicted. Obviously, in the shorted case, water flow is not hindered by electroosmosis, and pressure reaches a higher point than in the non-shorted case. The graph breaks down at some point, for the shorted as well as the non-shorted case. This is probably due to temporary cracks in the clay, through which water can preferentially flow in the direction of the fresh-water reservoir. It is interesting to observe, that in the non-shorted case, the pressure development continues after the degeneration in about the same way as before. 11.2. Membrane potential development In Fig. 6 the dissipation of membrane potential is shown. The graph for the numerical and experimental results compare rather well, especially when the numerical graph is shifted to correspond with the initial membrane potential value, as is represented by the middle line. The dissipation of membrane potential occurs somewhat faster in the experiment than in the simulation. This can be explained by a streaming potential development, that becomes significant for higher pressures. However, the streaming poten-
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Fig. 6. Membrane potential development for the non-shorted case; comparison of experimental and numerical results.
Fig. 7. Concentration development for the non-shorted case; comparison of experimental and numerical results.
tial can be approximated by ke (52) p ≈ 7 µV, σe which is obviously much too small to contribute to the total potential. Other effects should therefore be considered to be responsible for the dissipation of the potential. Vsp ≈ −
11.3. Concentration development In Fig. 7 the development of the concentration in the salt water reservoir is shown for the non-shorted setup. Here, we corrected for the volume increase due to the inflow of water.
Obviously, after t = 180,000 s, something has happened during the experiment that “degenerates” the diffusion. Using the shorted diffusion coefficient, we find for the shorted case the concentration evolution as depicted in Fig. 8. In both cases, the agreement between numerical and experimental results is excellent, aside from the degeneration and the scattering of the concentration data in the shorted experiment. 11.4. Coefficients During the experiments, the permeability was not measured directly, but was assumed to be equal to the permeability found
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337
Fig. 8. Concentration development for the shorted case; comparison of experimental and numerical results.
in conductivity tests on similar samples of the same clay, under equal overburden pressures [19]. Therefore, its value is assumed to be k = 2.3 × 10−18 m2 . From the results of the short-circuited experiment we can derive the value of the reflection coefficient. This can be calculated using van ’t Hoff’s law, resulting in σ = 0.024 ± 0.005. Van ’t Hoff’s law applies at the equilibrium flow conditions, and henceforth the concentration difference used in the expression for van ’t Hoff’s law was evaluated at the time of flow equilibrium. Obviously, it is smaller than the initial salt concentration difference. It is difficult to assess precisely the moment of flow equilibrium, resulting in the relatively large error. The value of the electroosmotic permeability is inferred from the difference between the water flux in the shorted and the nonshorted experiment. The difference corresponds to the counterflow of water induced by electroosmosis, and equals the electroosmotic permeability multiplied by the membrane potential gradient. This calculation, using the results of this experiment, yields a value of ke = 1.2 × 10−9 m2 /(V s), which is on the low side of the spectrum of values ke ∈ (0.5, 5) × 10−9 m2 /(V s), as reported by Bolt [11] for instance. As this value is directly correlated to the value of the reflection coefficient, which is apparently quite small as well, this is not a surprising result. The diffusion coefficient for the non-shorted case is deduced from flow measurements and equals Dnsc = 2.7 × 10−10 m2 /s. The diffusion coefficient has to be corrected for cation exchange, which may be represented by a retardation term R [14]:
experiment is about equal, viz. D− (54) = 2.6 × 10−10 m2 /s. RF0 Finally, the storativity was chosen to fit pressure evolution. This value was subsequently used in all simulations, as it reflects a property of the clay which is not likely to be dependent on the salt concentration or cation occupation. D=
12. Modeling results on bentonite: low c
(53)
A similar experiment was performed with a smaller concentration gradient, to assess the consistency of the results. Most coefficients were equal to the ones in the previous experiment, except for the diffusion coefficient, which, based on solute flux data, was seen to be slightly larger, i.e., D = 4 × 10−10 m2 /s. The reflection coefficient, which is expected to be larger for lower concentrations, equals σ = 0.03 ± 0.005 in this setup. Actually, if we would calculate the reflection coefficient based on the values of the diffuse double layer thickness inferred from the σ in the high c experiment, we would get a higher value, if we assume the concentration difference to be constant in the clay and equal to the mean of the concentrations in the reservoirs. However, as salt diffuses into the clay, the reflection coefficient becomes quite position dependent, and prediction of σ becomes very difficult. The membrane potential was calculated using the same theory as before. The concentrations, as measured in the nonshorted experiment, are 0.055 and 0.014 M for the salt and the fresh reservoir, which translates to activities of 0.051 and 0.013 M, respectively, yielding
This retardation parameter is inferred from the experimentally derived diffusion coefficient. It is subsequently used for the shorted case. Hence, the diffusion coefficient for the shorted
2RT (55) (T1+ ln 13.1 − T2+ ln 51.4). F The corresponding Hittorf transport numbers are T1+ = 0.52 and T2+ = 0.45, and therefore the calculated total membrane
D=
Df γ . RF0
Etot =
338
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Fig. 9. Membrane potential development for the non-shorted case of experiment be05 (n)sc; comparison of experimental and numerical results.
Fig. 10. Pressure development for the non-shorted and the shorted case of experiment be05 (n)sc; comparison of experimental and numerical results.
potential turns out to be −23 mV. This is in excellent agreement with the experimental result Etot = −23 ± 3 mV. The development of the membrane potential is displayed in Fig. 9. The numerical result was shifted to a lower initial membrane potential value to obtain a better comparison of the evolution. It is remarkable to observe that in the experiment, the development is apparently more linear than in the simulations. The pressure development was simulated and the results are displayed in Fig. 10. Unfortunately, the shorted experiment suffered a fate comparable to what happened in the previous experiment, but now in the initial stage: the pressure drops unexpectedly, or evolution of pressure is somehow slowed down.
However, from the results of the simulations, we can observe what most likely would have happened without this mishap. Finally, in Fig. 11 the evolution of concentration in the fresh water reservoir is shown. 13. Results with homoionic clay In a second type of experiment, the results of be1 (n)sc where shown to be reproducible. The experiment was performed with a homoionic clay to assess the influence of cation occupation on the clay platelets. In Table 3 the differences in coefficients between the two experiments are shown. The only differences are
S. Bader, K. Heister / Journal of Colloid and Interface Science 297 (2006) 329–340
339
Fig. 11. Concentration development in the fresh water reservoir for the shorted and non-shorted cases of experiment be05 (n)sc; comparison of experimental and numerical results. Table 3 Comparison between parameters of the experiment with heteroionic clay and the one with homoionic clay Parameter
hbe1 (n)sc
be1 (n)sc
Low concentration [mol/m3 ] High concentration [mol/m3 ] Low activity [mol/m3 ] High activity [mol/m3 ] Cation exchange capacity C [meq/g] Porosity n [–] Rock density ρs [kg/m3 ] Excess of charge Qv [C m3 ] Surface conductivity σχ [S/m] Membrane potential: measured [mV] Membrane potential: calculated [mV] Reflection coefficient σ [–]
12.9 90.4 12.6 81.4 53 0.80 2650 5.1 × 107 0.178 −33 −32 0.025
12.2 105 11.6 94.1 53 0.80 2650 3.4 × 107 0.118 −32 −34 0.024
the initial concentrations, the cation occupation and hence the surface conductivity. From the table we see that the homoionicity of the clay does not influence the values of membrane potential and reflection coefficient much: the initial concentration in the salt water reservoir is somewhat higher in the homoionic experiment compared to the heteroionic experiment. This should imply a somewhat smaller reflection coefficient, such as is observed in Table 3. The initial membrane potential difference is nearly equal, and differences could be the result of different cation occupation. However, all values lie within the margin of error. From Fig. 12 we observe that the evolution of pressure is rather similar compared to the experiment with the heteroionic clay, although the non-shorted pressure “degenerates” in the homoionic experiment. An interesting deviation
Fig. 12. Comparison of pressure evolution for experiment be1 (n)sc and experiment hbe1 (n)sc.
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Fig. 13. Comparison of membrane potential evolution for be1 (n)sc and experiment hbe1 (n)sc.
from the first experiment can be seen in Fig. 13: the (unknown) effect responsible for the divergence of numerical and experimental results in Fig. 6, is seen to be even stronger here. Again, we could attribute this to streaming potential, although this effect is probably too small.
the development of membrane potential difference for numerical and experimental results, other than by streaming potential? And, finally, does the cation occupation on the clay perhaps influence this development, and if so, in what way? References
14. Conclusions We have adapted the model of Revil [7] and validated it with results from laboratory experiments on bentonite clay. The model was shown to be quite suitable to predict membrane potential values. Also, it is shown how the model equations are perfectly able to simulate the transient development of pressure, concentration and membrane potential for shorted as well as for non-shorted conditions. As expected, when turning off the electrical terms in the equations, the numerical results agree with the experimental findings for the shorted experiment. The model was shown to be applicable for homo- as well as heteroionic clays. However, there does not seem to be much difference between the results of the experiments on the two types of clay. As the cation occupation is the significant difference between the homoionic and heteroionic clay, this is apparently not of great influence on processes such as membrane potential development and chemical osmosis. Some questions, however, remain unanswered: how can we come to an interpretation of the result for membrane potential that can agree with the interpretation of Revil? How can we explain the divergence of
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
T.J.S. Keijzer, P.J. Kleingeld, J.P.G. Loch, Eng. Geol. 53 (1999) 151–159. J.D. Sherwood, B. Craster, J. Colloid Interface Sci. 230 (2000) 349–358. S.J. Fritz, I.W. Marine, Geochim. Cosmochim. Acta 47 (1983) 1515–1522. C.E. Neuzil, Nature 403 (2000) 182–184. A.M. Garavito, H. Kooi, C. Neuzil, submitted for publication. K. Heister, P.J. Kleingeld, J.P.G. Loch, J. Colloid Interface Sci., in press. A. Revil, J. Colloid Interface Sci. 212 (1999) 503–522. S. Bader, H. Kooi, Adv. Water Resources 28 (2005) 203–214. S. Bader, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands, 2005. J.K. Mitchell, Fundamentals of Soil Behavior, Wiley, New York, 1976. G.H. Bolt (Ed.), Soil Chemistry B. Physical-Chemical Models, Elsevier, Amsterdam, 1982. F. Helfferich, Ion Exchange, McGraw–Hill, New York, 1962. A.T. Yeung, J. Non-Equilibrium Thermodyn. 15 (1990) 247–267. M.A. Malusis, C.D. Shackelford, J. Geotech. Geoenviron. Eng. 128 (2002) 97–106. J. Bear, Dynamics of Fluids in Porous Media, Dover, New York, 1972. P.N. Sen, Physica A 157 (1989) 446–453. G.E. Archie, AIME 146 (1942) 54–67. FlexPDE version 2.20e1, Scripted finite element model builder and numerical solver, http://www.pdesolutions.com. K. Heister, P.J. Kleingeld, T.J.S. Keijzer, J.P.G. Loch, Eng. Geol. 77 (2005) 295–303.
The effect of membrane potential on the development of chemical osmotic pressure in compacted clay S. Bader a,∗ , K. Heister b a Environmental Hydrogeology Group, Faculty of Geosciences, Department of Earth Sciences, Utrecht University, PO Box 80021, 3508TA, Utrecht,
The Netherlands b Geochemistry Group, Faculty of Geosciences, Department of Earth Sciences, Utrecht University, PO Box 80021, 3508TA, Utrecht, The Netherlands
Received 4 August 2005; accepted 11 October 2005 Available online 10 November 2005
Abstract When clay soils are subjected to salt concentration gradients, various interrelated processes come into play. It is known that chemical osmosis induces a water flow and that a membrane potential difference develops that counteracts diffusive flow of solutes and osmotic flow of water. In this paper, we present the results of experiments on the influence of membrane potential on chemical osmotic flow and diffusion of solutes and we show how we are able to derive the membrane potential value from theory. Moreover, the simultaneous development of water pressure, salt concentration and membrane potential difference are simulated using a model for combined chemico-electroosmosis in clays. A new method for short-circuiting the clay sample is employed to assess the influence of electrical effects on flow of water and transport of solutes. 2005 Elsevier Inc. All rights reserved. Keywords: Chemical osmosis; Membrane potential; Modeling; Clay membranes; Electroosmosis
1. Introduction Although generally not taken into account during modeling of groundwater flow, coupled effects such as chemical and electroosmosis may be quite significant in clayey soils in, e.g., coastal areas where salt concentration gradients are abundant. Quite a body of literature exists on experimental evidence for chemical osmosis in laboratory situations [1–3], but only recently substantial proof has been obtained for this effect occurring in situ [4,5]. In this paper, we report on experiments involving measuring membrane potential and its influence on chemical osmotic pressure evolution. Each experiment was performed in short-circuited and non-short-circuited conditions to assess the influence of electrical effects. Details of these experiments and, in particular, the method of short-circuiting are given in [6]. The theory of Revil [7] is used to give an interpretation of the observed value of the membrane potential.
* Corresponding author.
E-mail address: [email protected] (S. Bader). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.10.036
A model for simulating transient behavior of osmotic pressure and salt concentration was presented in [8,9]. This model is used in this study in a similar manner to compare experimental and numerical results. However, we extend the model with electrical effects, and investigate subsequently the behavior of the model with and without these effects. 2. Chemical osmosis and membrane potential: an introduction As water flow driven by gradients of temperature and electrical potential is called thermoosmosis and electroosmosis, respectively [10], we define osmosis in the context of soil to be non-hydraulically driven water flow. Chemical osmosis is then the water flow driven by a salt concentration gradient. As clay consists of negatively charged platelets that restrict movement of ions passing through, it can be considered as a semi-permeable membrane, and when a compacted clay is subjected to a salt concentration gradient, chemical osmosis may drive water from low to high salt concentration. Furthermore, an electrical potential gradient may induce water flow as well, as moving ions drag along water [10]. This phenomenon is called
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electroosmosis. Mathematically, chemical and electroosmosis are seen as extensions of Darcy’s law, k q = − ∇p + λ∇cf − ke ∇V , µ
(1)
i=+,−
[m2 ]
where q [m/s] is specific discharge, k is intrinsic permeability, µ [kg/(m s)] is fluid viscosity, p [Pa] is pressure, cf [kg/m3 ] is salt concentration, ke [m2 /(V s)] is electroosmotic permeability, V [V] is electrical potential, and the socalled chemical osmotic mobility λ is given by λ = σ νRT
We start with the electrical current density I [A/m2 ], which is expressed as I= (3) jdi zi F,
k , µ
(2)
where σ [–] is the reflection coefficient that describes the degree of restriction on solutes, or reflection by the membrane, and ranges between 0 and 1. Furthermore, ν [–] is the dissociation constant, R [J/(mol K)] is the gas constant and T [K] is the temperature. The membrane potential is the electrical potential difference that compensates the separation of charge due to different velocities of cations and anions in a free electrolyte solution or in a porous medium (partly) saturated with an electrolyte solution. Different terminologies for this effect are en vogue. For example, in [11], Bolt states that a potential difference caused by the restrictive behavior of the membrane on the ionic mobilities is called the membrane potential whereas the sum effect of liquid and membrane on the ions is called the diffusion potential. In [12], however, the membrane potential is the sum of the so-called concentration potential (the potential difference in a porous medium between two solutions) and the Donnan potential (between the solution and the membrane). We follow the approach of Revil [7], where membrane potential is the sum effect, and the part caused by the membrane is called the exclusion potential. Schematically, the Bolt and Revil definitions are given by Bolt [11]:
Vmembrane + Vliquid = Vdiffusion ,
Revil [7]:
Vexclusion + Vliquid = Vmembrane .
Although the Bolt [11] approach may be somewhat more logical, in most papers, the term membrane potential specifies the sum effect. We apply the combined set of equations for chemicoelectroosmosis to assess the influence of membrane potential on chemical osmotic flow. A number of experiments were performed quite similarly to those reported in [1]. Details on these experiments can be found in [6]. In a laboratory, a clay sample was subjected to a salt concentration gradient and pressure and concentration were simultaneously measured. However, the clay was equipped with electrodes to measure the electrical potential as well, and an experimental method was found to short-circuit the clay, in order to turn on and off the electrical effects. 3. Derivation of an expression for membrane potential In this section we derive an expression for the membrane potential.
where jdi [mol/(m2 s)] are the ionic fluxes in a free solution, zi [–] are the ionic valences and F [C/mol] is Faraday’s constant. For diffusion of separate ions of concentration ci [mol/m3 ], assuming an ideal solution, we write (see [7,13]) ui ci jdi = − (4) ∇µi , F µi = µ0i + RT ln ci + zi F V , (5) where ui [m2 /(V s)] is ionic mobility and µ0i is a reference chemical potential such that ∇µ0i = 0. Substituting (5) in (4) yields ui RT ∇ci − ui ci zi ∇V . (6) F For the electrical current density in a 1:1 electrolyte this implies jdi = −
I = −(u+ − u− )RT ∇cf − (u+ + u− )cf F ∇V .
(7)
When electroneutrality is enforced, the concentrations of anions and cations are equal, cf = c+ = c− , as are the ionic fluxes: jds = jd+ = jd− . We define u = u+ − u− as the effective solute mobility and σf [S/m] = (u+ + u− )F cf as the electrical conductivity of the electrolyte to get to the following expression for the electrical current density: I = −uRT ∇cf − σf ∇V .
(8)
If we assume the mobility and the electrical conductivity to be constant, we obtain a simple expression to quantify membrane potential: I = 0 → ∇V = −
uRT ∇cf . σf
(9)
This expression is often found in literature [13,14] to describe the membrane potential. Here, most notably, a constant electrical conductivity is assumed. However, especially for larger salt concentration gradients, the concentration dependence of the electrical conductivity becomes increasingly significant [9]. For solutions, such as NaCl, it is known that a Cl− ion moves faster than a Na+ ion, or u− > u+ , hence u < 0 and the potential gradient is of the same sign as the salt concentration gradient. However, when for example in a clay, the movement of anions is more restricted than the movement of cations, viz. in the case of anion exclusion, the mobility of cations is higher than the mobility of anions and the potential gradient and the salt concentration gradient have different signs. We may follow the approach of [7] and adopt the expression for electrical conductivity; then ∇V = −
(u+ − u− )RT RT ∇cf = − (t+ − t− )∇ ln cf , (u+ + u− )F cf F
(10)
where ti are Hittorf transport numbers that specify relative mobility of the ionic constituents: ui ti = (11) . i=+,− ui
S. Bader, K. Heister / Journal of Colloid and Interface Science 297 (2006) 329–340
For a porous medium it is shown in [7] that these equations are valid if we substitute for σf , ui , ti macroscopic parameters. In [7] it is shown that we can write ∇V = − Ti =
RT (T+ − T− )∇ ln cf , F
σi , σf
(12) (13)
where σi [S/m] is the ionic contribution to the bulk electrical conductivity σe = σ+ + σ− and Ti [–] are the macroscopic Hittorf numbers. This expression can be decomposed in a liquid junction potential and a exclusion potential: RT 2RT (2t+ − 1)∇ ln cf − (T+ − t+ )∇ ln cf . (14) F F In [7] it is shown qualitatively how in the absence of a porous medium, the macroscopic parameter Ti reduces to ti . Alternatively, we may consider the presence of the porous medium to be only accounted for by the formation factor (as in [15]), i.e., σf σe = (15) . F0 ∇V = −
The formation factor is denoted by F0 , and ∇V = −
RT F0 (2t+ − 1)∇ ln cf . F
(16)
331
as d Df Js nsc = − γ ∇cf , F0
(20)
where F0 [–] is the formation factor, Df is free solution diffusivity and γ = T+ /t+ is a factor indicating the influence of the charged medium on diffusion. By definition, in a free solution, γ → 1 and F0 → 1. More details on this can be found in [9]. When we consider the short-circuited situation, we disregard all electrical contributions. The result is the following: the anionic and cationic diffusion coefficients, concentrations and fluxes are no longer equal. This implies that a successful shortcut separates the charges in the solution. The liquid-junction potential, generated by this separation of charge, is immediately compensated by the potential generated by the virtual shortcut. In the experiments considered, the Cl− sensitive electrodes only measure the amount of anions in the solution, so instead of the salt flux we have to use the expression for the anion flux to interpret the results of the experiments. Accordingly, the electrolyte anion flux is, with D− the microscopic anionic diffusivity,
Jd−
sc
=−
D− ∇c− , F0
(21)
as the diffusion of anions is assumed not to be influenced by the charge of the porous medium, but only by its geometry F0 . 6. Experiments
4. Membrane potential and chemical osmosis We can extend the derived expression for the electrical potential difference with the streaming potential [10]: ∇V = −
ke RT ∇p − (T+ − T− )∇ ln cf . σe F
(17)
Substituting (17) in (1), we find RT k 1 ke2 qnsc = − − (T+ − T− ) ∇p + λ + ke ∇cf . µ σe F cf (18) The transition from the expression with electrical effects to the one without electrical effects is formally done by shortcircuiting. In the experiments, the system is short-circuited to eliminate any electrical potentials. The flux equations therefore come in two flavors: short-circuited (sc) (or shorted), i.e., without electrical effects, and non-short-circuited (nsc) (or nonshorted), i.e., inducing electrical effects. The short-circuited Darcy flux is simply given by k qsc = − ∇p + λ∇cf , µ
(19)
i.e., all electrical terms are absent. 5. Membrane potential and diffusion If we consistently follow the theoretical reasoning of Revil, as given in [7], we can derive an expression for the effective diffusivity as in [7]. The non-short-circuited solute flux is written
A number of experiments were carried out to investigate how the membrane potential influences the buildup of osmotic pressure. We show here the results of experiments on bentonite clay for an initial concentration difference of 0.01–0.1 M NaCl and 0.01–0.05 M NaCl for natural, heteroionic clay and a c = 0.01–0.1 M NaCl experiment for homoionic clay. In Table 1 it is shown how the different experiments are classified. 7. Experimental setup To contain the clay, a rigid-wall permeameter was used, consisting of plastics for electrical insulation with an inner diameter of 50 mm. Within this mould, the clay sample is situated between two porous stones and nylon filters, where it is saturated with a fresh water solution before the experiment is started such that distortion of the clay membrane is minimal. Compaction of the clay during the experiment is ensured by putting a mechanical load on the piston of the sample mould. Table 1 Designation of experiments Designation
Concentration difference (NaCl)
Type of clay
Shorted?
be1 nsc be1 sc be05 nsc be05 sc hbe1 nsc hbe1 sc
0.1–0.01 M 0.1–0.01 M 0.05–0.01 M 0.05–0.01 M 0.1–0.01 M 0.1–0.01 M
bentonite bentonite bentonite bentonite homoionic bentonite homoionic bentonite
no yes no yes no yes
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Fig. 1. Photograph of the experimental setup.
Via the porous stones, the clay is in contact with two fluid reservoirs that are equipped with double junction Ag/AgCl reference electrodes and Cl-sensitive electrodes to measure the electrical potential and the Cl concentration, and with a calibrated standpipe to quantify the water flow. Ag/AgCl electrodes are used to measure the electrical potential because their potential is determined by the activity of Cl ions, so they are reversible to chloride. The outer liquid junction consists of KNO3 which minimizes diffusion potentials at the interface of the electrodes. Moreover, polarization effects, which occur at metal electrodes, are eliminated. To prevent concentration polarization and to homogenize the solutions, the reservoirs are continuously circulated with a peristaltic pump. The setup is shown in Fig. 1. To gain insight in the effect of the potential build-up on the water flow across the clay sample, it should be possible to eliminate this potential by short-circuiting the electrodes. However, physical shortening is not viable because conducting current through the reference electrodes produces unpredictable additional potentials. Therefore, an instrument was designed, here referred to as “virtual shortcut,” which brings the potential across the sample to zero with a negative feedback current. The virtual shortcut measures the potential with the Ag/AgCl electrodes and applies a corresponding current by a feedback amplifier on noble metal electrodes, which are also inserted in both fluid reservoirs. Such a four-electrode arrangement is common in electrochemistry. Interferences and deterioration of the reference electrodes are avoided because the electrical current is conducted through the metal electrodes, which are chemically inert. The problem of polarization potentials at the metal electrodes can be ignored since this is also compensated by the electrical potential applied on them. A detailed description of the experimental setup and procedure is given in [6].
up more or less instantaneously, and then declined, mainly because of diffusion of salt into the clay. The membrane potential, as calculated in the next section, refers to the initial potential difference. Also, because of the salt concentration difference, chemical osmosis induced a water flow through the clay from low to high salt concentration, inducing a relatively slow (by storativity) buildup of pressure. As seen in comparable studies [8], the pressure declined after a maximum, also because of diffusion of solute through the clay. Two experiments were performed for each case, with different clay samples prepared form the same batch of sample material. We assume these samples to be identical. Every experiment was performed with and without shortcut. Without shortcut, which is actually the default situation for clays, an electrical potential is present throughout the clay: a sum of membrane and streaming potential. This potential induces electroosmotic flow, or in particular: a concentration gradient shifts the excess of cations, thereby creating a potential that induces electroosmotic flow which counteracts flow of water due to chemical osmosis. 8. Measuring the membrane potential The membrane potential cannot be measured directly [12]. Usually the total electromotive force (EMF) of a system is measured, as displayed in Fig. 2. The EMF is evaluated as the sum of the electrode potentials, the Donnan potentials and the membrane potential. The electrode potential is the electrical potential difference between the electrode and the solution. The
7.1. Experimental results In all experiments, the following occurred: the applied concentration difference induced a membrane potential that built
Fig. 2. Overview of electrical potentials.
S. Bader, K. Heister / Journal of Colloid and Interface Science 297 (2006) 329–340
333
electrode potential Eel is defined as
Now, in [7], an expression is derived for T+ :
RT ln ai , (22) F where E0 is the standard electrode potential and ai is the activity of ion i. When calculating the EMF of the system, the two E0 ’s cancel out. The potential between two phases in contact is called the phase-boundary potential [12]. When a solution is brought in contact with a membrane this is called the Donnan potential. In a negatively charged membrane, the concentration of cations is larger than the concentration of anions, and the concentration of cations is larger in the membrane than in the solution. The cations want to diffuse out of the membrane, but this is prevented by the Donnan potential to attain electroneutrality. The Donnan potential is defined by
1 1 − t+
=1+ T+ F0 ξ + 12 (t+ − ξ ) 1 − tξ+ + 1 −
Eel = E0 +
RT a¯ i ln , EDon = − F ai
(23)
where a¯ i [mol/m3 ] is the activity of ion i in the membrane. As the membrane potential is related to the activities of the ions in the membranes, the formula for the Donnan potential “translates” the values for the activities in the membranes to the values of the activities in the solutions. As in [12], we get for a 1:1 electrolyte and a cation-exchanger: 1 2 − EDon Emem = Ediff + EDon (24) a¯ 1+ a¯ 2+ a¯ 2+ RT + 2 t¯− d ln a¯ f + ln − ln ln =− F a¯ 1+ a1+ a2+ (25) RT a2 =− (26) + 2 t¯− d ln af ln F a1 RT a2 = (27) − 2 t¯+ d ln af . ln F a1
Sub- and superscripts 1, 2 refer to different sides of the sample and af is salt activity in the free solution. In the last lines the definition t¯+ = 1 − t¯− was used and it was assumed that the activity of the salt is equal to the activity of the ions. Introducing the notation of [7] for the macroscopic transference number, i.e., T+ instead of t¯+ , where L is the membrane thickness, RT Emem = − (28) (2T+ − 1) d ln af . F When we assume T+ (L) = T2+ and T+ (0) = T1+ to be constant, we find RT (2T2+ − 1) ln a2 − (2T1+ − 1) ln a1 F RT a1 2RT + ln (T1+ ln a1 − T2+ ln a2 ). =− F a2 F
Emem = −
(29) (30)
The total EMF, i.e., the measured potential, now consists of the electrode potential and the membrane potentials; as in [7], we write RT a2 ln Etot = Eel1 + Emem + Eel2 = Emem − (31) F a1 2RT = (32) (T1+ ln a1 − T2+ ln a2 ). F
ξ 2 t+
+
4F0 ξ t+
. (33)
Na+
is t+ = 0.38. The paThe microscopic Hittorf number for rameter ξ is defined as the ratio of surface conductivity σs and fluid conductivity σf : ξ=
σs . σf
(34)
The surface conductivity is defined in [7] as σs =
2 n βs Qv , 31−n
(35)
where Qv [C/m3 ] is the excess of charge, n [–] is porosity and βs [m2 /(V s)] is the mobility of the counterions at the surface of the solid particles. This definition however, is based on the assumption of low porosity, whereas we deal with high porosity. The approximation for high porosity [16] is σs =
βs Qv . 2 − 2n/3
(36)
The excess of charge is calculated using Qv = ρs
1−n F C, n
(37)
where ρs [kg/m3 ] is the rock density and C is the cation exchange capacity, expressed in terms of molc /kg ∼ meq/g. 9. Membrane potential value As an example, we review the results of experiment be1 sc in detail. Here, the overburden pressure applied was 1 bar and the applied concentration difference was roughly 0.1–0.01 M. The precise activities are listed in Table 2. With these values, the total potential is Etot =
2RT (T1+ ln 11.6 − T2+ ln 94.1). F
(38)
Table 2 Some a priori measured and calculated parameters Parameter
be1 (n)sc
be05 (n)sc
Low concentration [mol/m3 ]
12.2 105 11.6 94.1 53 0.80 298 2650 [1] 3.4 × 107 0.118 −32 −34
13.77 55 13.1 50.4 53 0.80 298 2650 3.4 × 107 0.118 −23 −23
High concentration [mol/m3 ] Low activity [mol/m3 ] High activity [mol/m3 ] Cation exchange capacity C [cmolc /kg] Porosity n [–] Temperature T [K] Rock density ρs [kg/m3 ] Excess of charge Qv [C m3 ] Surface conductivity σs [S/m] Membrane potential: measured [mV] Membrane potential: calculated [mV]
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The corresponding theoretical liquid-junction potential Elp in the absence of a membrane would be Elp =
2RT t+ (ln 11.6 − ln 94.1) = −41 mV. F
(39)
The excess of charge and the surface conductivity are in Table 2. The value of the mobility of the counterions at the surface of the solid particles βs is based on experiments on montmorillonite [7]. Another parameter considered is the formation factor F0 . It is customary to relate this parameter to the porosity via Archie’s law [17], F0 = n−m ,
(40)
where m is the so-called cementation exponent, usually assumed to be 2. If we assume constant porosity and formation factor, we get F0 = 1.56. Using all these values and definitions, the macroscopic Hittorf numbers depend only on the concentrations in the reservoirs. Using formula (33), we find T+1 = 0.54, T+2 = 0.44, and consequently Etot = −34 mV.
(41)
As the experimental result is Etot = −32 ± 3 mV, we have quite a satisfactory agreement between the experimental result and our theory. Now, the actual membrane potential is RT a1 ln F a2 = −32 mV + 55 mV = +23 mV.
Emem = Etot −
(42)
Hence, the actual membrane potential is positive, i.e., of the same sign as the concentration gradient. This is in accordance with flow data of non-shorted tests and short-circuited tests, that strongly indicate a counterflow of water hindering chemical osmotic transport. This result is, however, not in accordance with the interpretation as given in [7]. Here, a theoretical range is given for the membrane potential, which amounts for the current experiment to Etot ∈ [−110, −41], where Etot = −41 mV corresponds to the case of “pure” liquid junction potential and Etot = −110 mV to a “strong” influence of the membrane on the potential. However, in Fig. 3, it is shown how the membrane potential, using the values of the activities from the experiment considered in this study, depends on the formation factor and on the parameter α = ξ af . This figure shows that, in the range of α and F0 that we consider, the membrane potential stays clearly above the −41 mV value. Only for rather large values of F0 and α, the membrane potential is smaller than −41 mV, as predicted in [7]. Apparently, the interpretation of membrane potential in this respect has to be adapted. Most likely, the large difference in transference numbers for the fresh and salt sides causes this “anomaly” in the membrane potential. Or, the influence of cation exchange may be different than as proposed in [7].
Fig. 3. Membrane potential values for varying α and F0 .
10. Modeling procedure 10.1. Equations The simultaneous evolution of pressure, membrane potential and concentration was modeled using equations from [8]. The model equations are the following (see [8]), assuming incompressible fluid, no chemical reactions, no temperature effects, low Peclet number. Fluid mass balance: ∂n + ∇ · q = 0. (43) ∂t Salt mass balance: ∂ncf + ∇ · (cf q) + ∇ · Jds = 0. (44) ∂t Flux equations: k 1 k2 qnsc = − − e ∇p + λ + ke RT (T+ − T− ) ∇cf , µ σe cf (45) k qsc = − ∇p + λ∇cf , (46) µ d Df Js nsc = − (47) γ ∇cf , F0 d D− J− sc = − (48) ∇c− . F0 Constitutive equation: n = 1 − (1 − n0 )e−αp ,
(49)
where n0 is a reference porosity and α [1/Pa] the compressibility of the porous medium. 10.2. Model domain In Fig. 4 the general modeling domain for all membrane potential experiments is presented. The parameters not specified are given in the tables corresponding to the respective experiments.
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Parameter
Value in clay
Value in porous stone
n (–) k (m2 ) D (m2 /s) α (1/Pa) σ (–) Length (mm)
0.5 2.34 × 10−18 D 1 × 10−9 σ a = 2.9
n 1 × 10−13 1.5 × 10−9 4.6 × 10−10 0 b − a = 7.2
335
Fig. 4. Modeling domain membrane potential experiment and relevant parameters.
Fig. 5. Pressure development for the shorted and non-shorted case; comparison of experimental and numerical results.
10.3. Initial and boundary conditions Initially, the pressure and electrical potential are zero in the whole domain, and the concentration is c0 (high value) in the porous stone at the left-hand side and ca (low value) in the clay and the porous stone at the right-hand side. The following boundary conditions apply: ∂p = 0, ∂x
∂c (50) = 0, at x = −L, ∂x ∂c p = 0, (51) = 0, at x = b. ∂x We imposed p ≡ 0 in the left-hand side reservoir, because the pressure in this reservoir is atmospheric. The solutions in the reservoir are well stirred, such that the measured concentrations should correspond to the concentrations at the interface between the reservoirs and the porous stones. 11. Modeling results on bentonite: high c 11.1. Pressure development In this section, we present the experimental and modeling results of experiment be1 (n)sc, which was performed on the clay sample subjected to a concentration gradient of about
90 mol/m3 . For numerical modeling, we used the scripted finite element builder and numerical solver FlexPDE [18]. In Fig. 5, the pressure buildup in the salt water reservoir is depicted. Obviously, in the shorted case, water flow is not hindered by electroosmosis, and pressure reaches a higher point than in the non-shorted case. The graph breaks down at some point, for the shorted as well as the non-shorted case. This is probably due to temporary cracks in the clay, through which water can preferentially flow in the direction of the fresh-water reservoir. It is interesting to observe, that in the non-shorted case, the pressure development continues after the degeneration in about the same way as before. 11.2. Membrane potential development In Fig. 6 the dissipation of membrane potential is shown. The graph for the numerical and experimental results compare rather well, especially when the numerical graph is shifted to correspond with the initial membrane potential value, as is represented by the middle line. The dissipation of membrane potential occurs somewhat faster in the experiment than in the simulation. This can be explained by a streaming potential development, that becomes significant for higher pressures. However, the streaming poten-
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Fig. 6. Membrane potential development for the non-shorted case; comparison of experimental and numerical results.
Fig. 7. Concentration development for the non-shorted case; comparison of experimental and numerical results.
tial can be approximated by ke (52) p ≈ 7 µV, σe which is obviously much too small to contribute to the total potential. Other effects should therefore be considered to be responsible for the dissipation of the potential. Vsp ≈ −
11.3. Concentration development In Fig. 7 the development of the concentration in the salt water reservoir is shown for the non-shorted setup. Here, we corrected for the volume increase due to the inflow of water.
Obviously, after t = 180,000 s, something has happened during the experiment that “degenerates” the diffusion. Using the shorted diffusion coefficient, we find for the shorted case the concentration evolution as depicted in Fig. 8. In both cases, the agreement between numerical and experimental results is excellent, aside from the degeneration and the scattering of the concentration data in the shorted experiment. 11.4. Coefficients During the experiments, the permeability was not measured directly, but was assumed to be equal to the permeability found
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337
Fig. 8. Concentration development for the shorted case; comparison of experimental and numerical results.
in conductivity tests on similar samples of the same clay, under equal overburden pressures [19]. Therefore, its value is assumed to be k = 2.3 × 10−18 m2 . From the results of the short-circuited experiment we can derive the value of the reflection coefficient. This can be calculated using van ’t Hoff’s law, resulting in σ = 0.024 ± 0.005. Van ’t Hoff’s law applies at the equilibrium flow conditions, and henceforth the concentration difference used in the expression for van ’t Hoff’s law was evaluated at the time of flow equilibrium. Obviously, it is smaller than the initial salt concentration difference. It is difficult to assess precisely the moment of flow equilibrium, resulting in the relatively large error. The value of the electroosmotic permeability is inferred from the difference between the water flux in the shorted and the nonshorted experiment. The difference corresponds to the counterflow of water induced by electroosmosis, and equals the electroosmotic permeability multiplied by the membrane potential gradient. This calculation, using the results of this experiment, yields a value of ke = 1.2 × 10−9 m2 /(V s), which is on the low side of the spectrum of values ke ∈ (0.5, 5) × 10−9 m2 /(V s), as reported by Bolt [11] for instance. As this value is directly correlated to the value of the reflection coefficient, which is apparently quite small as well, this is not a surprising result. The diffusion coefficient for the non-shorted case is deduced from flow measurements and equals Dnsc = 2.7 × 10−10 m2 /s. The diffusion coefficient has to be corrected for cation exchange, which may be represented by a retardation term R [14]:
experiment is about equal, viz. D− (54) = 2.6 × 10−10 m2 /s. RF0 Finally, the storativity was chosen to fit pressure evolution. This value was subsequently used in all simulations, as it reflects a property of the clay which is not likely to be dependent on the salt concentration or cation occupation. D=
12. Modeling results on bentonite: low c
(53)
A similar experiment was performed with a smaller concentration gradient, to assess the consistency of the results. Most coefficients were equal to the ones in the previous experiment, except for the diffusion coefficient, which, based on solute flux data, was seen to be slightly larger, i.e., D = 4 × 10−10 m2 /s. The reflection coefficient, which is expected to be larger for lower concentrations, equals σ = 0.03 ± 0.005 in this setup. Actually, if we would calculate the reflection coefficient based on the values of the diffuse double layer thickness inferred from the σ in the high c experiment, we would get a higher value, if we assume the concentration difference to be constant in the clay and equal to the mean of the concentrations in the reservoirs. However, as salt diffuses into the clay, the reflection coefficient becomes quite position dependent, and prediction of σ becomes very difficult. The membrane potential was calculated using the same theory as before. The concentrations, as measured in the nonshorted experiment, are 0.055 and 0.014 M for the salt and the fresh reservoir, which translates to activities of 0.051 and 0.013 M, respectively, yielding
This retardation parameter is inferred from the experimentally derived diffusion coefficient. It is subsequently used for the shorted case. Hence, the diffusion coefficient for the shorted
2RT (55) (T1+ ln 13.1 − T2+ ln 51.4). F The corresponding Hittorf transport numbers are T1+ = 0.52 and T2+ = 0.45, and therefore the calculated total membrane
D=
Df γ . RF0
Etot =
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Fig. 9. Membrane potential development for the non-shorted case of experiment be05 (n)sc; comparison of experimental and numerical results.
Fig. 10. Pressure development for the non-shorted and the shorted case of experiment be05 (n)sc; comparison of experimental and numerical results.
potential turns out to be −23 mV. This is in excellent agreement with the experimental result Etot = −23 ± 3 mV. The development of the membrane potential is displayed in Fig. 9. The numerical result was shifted to a lower initial membrane potential value to obtain a better comparison of the evolution. It is remarkable to observe that in the experiment, the development is apparently more linear than in the simulations. The pressure development was simulated and the results are displayed in Fig. 10. Unfortunately, the shorted experiment suffered a fate comparable to what happened in the previous experiment, but now in the initial stage: the pressure drops unexpectedly, or evolution of pressure is somehow slowed down.
However, from the results of the simulations, we can observe what most likely would have happened without this mishap. Finally, in Fig. 11 the evolution of concentration in the fresh water reservoir is shown. 13. Results with homoionic clay In a second type of experiment, the results of be1 (n)sc where shown to be reproducible. The experiment was performed with a homoionic clay to assess the influence of cation occupation on the clay platelets. In Table 3 the differences in coefficients between the two experiments are shown. The only differences are
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339
Fig. 11. Concentration development in the fresh water reservoir for the shorted and non-shorted cases of experiment be05 (n)sc; comparison of experimental and numerical results. Table 3 Comparison between parameters of the experiment with heteroionic clay and the one with homoionic clay Parameter
hbe1 (n)sc
be1 (n)sc
Low concentration [mol/m3 ] High concentration [mol/m3 ] Low activity [mol/m3 ] High activity [mol/m3 ] Cation exchange capacity C [meq/g] Porosity n [–] Rock density ρs [kg/m3 ] Excess of charge Qv [C m3 ] Surface conductivity σχ [S/m] Membrane potential: measured [mV] Membrane potential: calculated [mV] Reflection coefficient σ [–]
12.9 90.4 12.6 81.4 53 0.80 2650 5.1 × 107 0.178 −33 −32 0.025
12.2 105 11.6 94.1 53 0.80 2650 3.4 × 107 0.118 −32 −34 0.024
the initial concentrations, the cation occupation and hence the surface conductivity. From the table we see that the homoionicity of the clay does not influence the values of membrane potential and reflection coefficient much: the initial concentration in the salt water reservoir is somewhat higher in the homoionic experiment compared to the heteroionic experiment. This should imply a somewhat smaller reflection coefficient, such as is observed in Table 3. The initial membrane potential difference is nearly equal, and differences could be the result of different cation occupation. However, all values lie within the margin of error. From Fig. 12 we observe that the evolution of pressure is rather similar compared to the experiment with the heteroionic clay, although the non-shorted pressure “degenerates” in the homoionic experiment. An interesting deviation
Fig. 12. Comparison of pressure evolution for experiment be1 (n)sc and experiment hbe1 (n)sc.
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Fig. 13. Comparison of membrane potential evolution for be1 (n)sc and experiment hbe1 (n)sc.
from the first experiment can be seen in Fig. 13: the (unknown) effect responsible for the divergence of numerical and experimental results in Fig. 6, is seen to be even stronger here. Again, we could attribute this to streaming potential, although this effect is probably too small.
the development of membrane potential difference for numerical and experimental results, other than by streaming potential? And, finally, does the cation occupation on the clay perhaps influence this development, and if so, in what way? References
14. Conclusions We have adapted the model of Revil [7] and validated it with results from laboratory experiments on bentonite clay. The model was shown to be quite suitable to predict membrane potential values. Also, it is shown how the model equations are perfectly able to simulate the transient development of pressure, concentration and membrane potential for shorted as well as for non-shorted conditions. As expected, when turning off the electrical terms in the equations, the numerical results agree with the experimental findings for the shorted experiment. The model was shown to be applicable for homo- as well as heteroionic clays. However, there does not seem to be much difference between the results of the experiments on the two types of clay. As the cation occupation is the significant difference between the homoionic and heteroionic clay, this is apparently not of great influence on processes such as membrane potential development and chemical osmosis. Some questions, however, remain unanswered: how can we come to an interpretation of the result for membrane potential that can agree with the interpretation of Revil? How can we explain the divergence of
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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