Environ. Sci. Technol. 2002, 36, 1312-1319
Coupling Effects during Steady-State Solute Diffusion through a Semipermeable Clay Membrane MICHAEL A. MALUSIS† AND CHARLES D. SHACKELFORD* Geoenvironmental Engineering, Department of Civil Engineering, Colorado State University, Fort Collins, Colorado 80523-1372
Two separate coupling effects are evaluated with respect to steady-state potassium chloride (KCl) diffusion through a bentonite-based geosynthetic clay liner (GCL) that behaves as a semipermeable membrane. Both of the coupling effects are correlated with measured chemico-osmotic efficiency coefficients, ω, that range from 0.14 to 0.63 for the GCL. The first coupling effect is an explicit (theoretical) salt-sieving effect expressed as a coupled effective salt diffusion coefficient, Dω*, that is lower than the true (uncoupled) effective salt diffusion coefficient, Ds*, because of the observed membrane behavior. However, the maximum difference between Dω* and Ds* based on measured chloride concentrations is relatively small (i.e., ) 10%), and the difference decreases with decreasing ω (i.e., Dω* f Ds* as ω f 0). The second coupling effect is implicit (empirical) and is characterized by the measurement of concentration-dependent effective salt diffusion coefficients that results in an ∼300% decrease in Ds* as ω increases from 0.14 to 0.63. The decrease in Ds* resulting from implicit coupling is attributed to solute exclusion described in terms of a restrictive tortuosity factor.
Introduction Geosynthetic clay liners (GCLs) are factory manufactured rolls consisting of thin layers (∼5-10 mm) of bentonite clay (∼5.0 kg/m2) sandwiched between two geotextiles or bonded to a geomembrane. The structural integrity of a GCL is maintained by stitching or needle-punching through the geotextiles and bentonite or by adding an adhesive compound to the bentonite to bind the bentonite to the geotextile or geomembrane. The primary differences among the GCLs currently available pertain to the mineralogy (e.g., montmorillonite content), type (e.g., sodium vs calcium montmorillonite) and form (e.g., powder vs granular) of the bentonite used in the GCL, the type of geotextile (e.g., woven vs nonwoven geotextiles), and the nature of the structural integrity (1). GCLs commonly are used as components in engineered liners for hydraulic containment applications, such as landfills, surface impoundments, and waste piles (2). The preferential use of GCLs in containment applications relative to other alternatives, such as compacted clay liners, stems primarily from economic benefits and the relatively low * Corresponding author phone: (970) 491-5051; fax: (970) 4913584; e-mail:
[email protected]. † Present address: GeoTrans, Inc., 9101 Harlan St., Suite 210, Westminster, CO 80030. 1312
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hydraulic conductivities to water (e5 × 10-11 m/s) typically measured for these materials (3, 4). Except in the case of a GCL that contains a geomembrane, the low hydraulic conductivity and, therefore, the hydraulic performance of GCLs are attributed to the bentonite component of the GCL. While the use of GCLs in geoenvironmental containment applications has been based primarily on favorable hydraulic performance and economic benefits, other factors may be important in terms of the overall performance of GCLs in containment applications. For example, the potential for breakthrough of contaminants through GCLs in less than 1 day as a result of solute (liquid-phase) diffusion has been reported (5), and the premature arrival of a salt (2.0 N NaCl) front through a needle-punched GCL during permeation at low flow rates has been attributed to the dominance of diffusive transport (6). Also, partly on the basis of the measured high rate of chloride diffusion through a needlepunched GCL, a lining containment system comprised of a geomembrane overlying a GCL was found to be essentially equivalent to a lining containment system consisting of geomembrane overlying a compacted clay liner (7). Thus, diffusion apparently is an important, if not dominant, process in determining the rate of aqueous miscible contaminant migration through GCLs. Membrane behavior, or the ability of clay soils to impede the passage of solutes, also could affect contaminant migration through GCLs. Restricted passage of charged solutes (ions) through the pores of a clay soil is attributed to electrostatic repulsion of the ions by electric fields associated with the diffuse double layers of adjacent clay particles (811). Nonelectrolyte solutes (uncharged species), such as aqueous miscible organic compounds, also may be restricted from migrating through clay soils when the size of the organic molecule is greater than pore size of the clay soil (12). In addition to restricting the passage of solutes, membrane behavior also results in chemico-osmotic flow, or the movement of liquid in response to a solute concentration gradient (13). The ability of sodium bentonite, in particular, to exhibit membrane behavior in the presence of common electrolytes (e.g., NaCl) has been illustrated extensively (9, 11, 14-16). For example, the chemico-osmotic membrane efficiency of compacted sodium bentonite pastes has been measured in the presence of NaCl and CaCl2 solutions over a range of concentrations, resulting in observed membrane behavior that varied as a function of the soil porosity, electrolyte concentration, and ion valence (14). Also, reduced salt migration rates through saturated bentonite pastes due to the permi-selective properties of the bentonite have been reported (16). More recently, a reduced diffusive solute flux of naphthalene, an aqueous miscible organic compound (nonelectrolyte), through a montmorillonite-based shale has been attributed to exclusion of the naphthalene from the pore space (17). Similar solute exclusion effects for a variety of inorganic solutes have been reported in studies involving the use of compacted sodium bentonites being considered for use as containment barriers for nuclear waste repositories (18-21). Although the results of all of these studies indicate that montmorillonite-rich soils, such as bentonite, may exhibit membrane behavior (i.e., solute exclusion) and that this behavior can affect solute diffusion, no correlation between membrane behavior and solute diffusion has been shown. The existence of such a correlation may have important implications with respect to the use of bentonite-based soil barriers in waste containment and in situ remediation 10.1021/es011130q CCC: $22.00
2002 American Chemical Society Published on Web 02/09/2002
applications (e.g., GCLs, compacted sand-bentonite mixtures, soil-bentonite cutoff walls), because the primary purpose of such barriers is to restrict the migration of aqueous miscible contaminants. On the basis of the aforementioned considerations, the hypothesis of this research is that the diffusive flux of solutes through semipermeable membranes is coupled to the efficiency of the membranes such that the diffusive solute flux becomes increasingly more restrictive as the membrane efficiency approaches that of an ideal semipermeable membrane. This hypothesis is evaluated through the simultaneous measurement of chemico-osmotic efficiency coefficients and effective diffusion coefficients during steadystate diffusion of potassium chloride (KCl) through a GCL over a range in source concentrations for which the GCL behaves as a semipermeable membrane. The results of these measurements are also evaluated on the basis of coupled flux transport theory that explicitly accounts for the effects of membrane-related coupling on solute diffusion.
Materials and Experimental Methods Materials. The GCL tested in this study is a needle-punched GCL containing granular sodium bentonite (see Supporting Information). The liquids used in this study consist of processed tap water (PTW) and solutions of PTW and potassium chloride (KCl) (certified ACS, Fisher Scientific, Fair Lawn, NJ) dissolved in PTW at measured KCl concentrations ranging from 0.0039 to 0.047 M (290-3500 mg/L). PTW (pH ) 6.93; EC at 25 °C ) 0.32 mS/m) consists of tap water passed through three Barnstead ion exchange columns placed in series. The chloride (Cl-) concentrations in the KCl solutions were measured using ion chromatography (IC), whereas the potassium (K+) concentrations were measured using inductively coupled plasma-optical emissions spectroscopy (ICPOES). The measured pH of the KCl solutions ranged from 6.68 to 6.91, and the measured EC at 25 °C for the KCl solutions ranged from 58.7 to 682 mS/m. Testing Apparatus and Procedures. The testing apparatus and procedures used in this study are described in detail by Malusis et al. (22). In essence, a test specimen (GCL) is confined within a rigid acrylic cylinder between a top piston and base pedestal, and a source solution at an initial KCl concentration, Cot (>0), is circulated through a porous stone located adjacent to the top of the specimen, while PTW is circulated through a porous stone located adjacent to the bottom of the specimen. Because (a) the circulation rates are constant and equivalent, (b) the test specimen is confined and saturated, and (c) the system is closed, there is no volume change within the system (∆V ) 0) and, therefore, no solution flow (q ) 0) during the test. The resulting differences in solute (Cl- and K+) concentrations between the top and bottom boundaries of the specimen result in the generation of an induced pressure difference across the specimen that is measured with a differential pressure transducer, as well as solute diffusion from the higher concentration boundary (top) to the lower concentration boundary (bottom). As a result of diffusion, the solute concentration in the circulation outflow from the base pedestal, Cb, is greater than the solute concentration in the circulation inflow from the base pedestal, Cob (i.e., Cb > Cob) (see Figure S2, Supporting Information). Therefore, the measured difference between these concentrations can be used as a basis for calculating the diffusive mass flux exiting the specimen. Further details are given in ref 22. Specimen Preparation. Circular specimens of the GCL with nominal diameters of 71.1 mm were cut from a larger GCL sheet and placed on the base pedestal inside the testing cell. The cylinder then was filled with PTW to submerge the specimen, and the top piston was lowered into the cylinder to compress the GCL to the desired thickness. After comple-
tion of compression, the top piston was locked in place to prevent volume expansion of the specimen due to swelling of the bentonite. Each specimen was permeated under backpressure with PTW before testing to saturate the specimen, to remove excess soluble salts, and to measure the initial hydraulic conductivity. After permeation, PTW was circulated at the top and bottom boundaries of the specimen for approximately 5 days to establish a steady baseline differential pressure. The diffusion tests then were initiated by circulating a KCl solution in the top piston (i.e., Cot > 0) while continuing circulation of PTW in the base pedestal. Thus, in this study, the initial concentration of solute (KCl) in the base pedestal was maintained at zero (i.e., Cob ) 0). At the end of diffusion testing, the GCL specimen was permeated with the source solution until steady-state hydraulic conductivity was achieved. After permeation, the cell was disassembled, and the water content of the specimen was measured to determine the final degree of saturation of the specimen. Determination of Chemico-Osmotic Efficiency. Under no-flow (q ) 0) conditions, as imposed in this study, the chemico-osmotic liquid flux that occurs in a soil membrane in response to the applied concentration difference is balanced by an equal hydraulic liquid flux in the opposite direction, resulting in the development of an induced pressure gradient. At steady-state induced pressure under no-flow conditions, the chemico-osmotic efficiency coefficient, ω, that reflects the existence of membrane behavior, is defined as follows (22, 23):
ω)
|
∆P ∆π q)0
(1)
where ∆P (0), during the diffusion tests are shown in Figure 1. Before the introduction of KCl, PTW was circulated at both specimen boundaries (i.e., Cot ) Cob ) 0) for 5 days to establish a baseline differential pressure, -∆Po. Measured values of -∆Po ranged between 0.62 and 4.0 kPa in the four tests. Introduction of KCl into the top piston after the initial 5 days resulted in steady-state values of induced differential pressure, -∆Pss, ranging from 14.1 to 32.0 kPa. The difference between -∆Pss and -∆Po represents the “effective” induced differential pressure, -∆Pe (i.e., -∆Pe ) -∆Pss - (-∆Po)), that is due solely to the membrane behavior of the GCL (22). Values of -∆Pe in the four tests were 11.5, 19.7, 27.8, and 28.0 1314
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FIGURE 1. Differential pressures induced across GCL specimens during steady-state diffusion tests.
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FIGURE 2. Chemico-osmotic efficiency coefficients versus source potassium chloride concentrations for GCL specimens. kPa, respectively, as shown in Figure 1. The effective induced differential pressure values were used in eq 1 (i.e., ∆P ) ∆Pe) to calculate ω (22). The differences in chemico-osmotic pressure, -∆π, across the GCL specimens due to the applied concentration differences of 0.0039, 0.0087, 0.020, and 0.047 M were calculated as 18.4, 39.9, 86.4, and 201 kPa, respectively. These values of -∆π were corrected to account for slight changes in the boundary concentrations due to diffusion, as described by Malusis et al. (22). Values of ω based on eq 1 corresponding to source KCl concentrations (Cot) of 0.0039, 0.0087, 0.020, and 0.047 M are 0.63, 0.49, 0.32, and 0.14, respectively, as shown in Figure 2. On the basis of these results, the GCL exhibited membrane behavior in the presence of KCl solutions during the diffusion tests. The decrease in ω with increasing salt concentration is consistent with the results of previous studies (14, 15, 25) and is attributed to compression of the diffuse double layers of water and ions surrounding the clay particles caused by the higher ion concentrations in the pore space (10). The trend in the data shown in Figure 2, when extrapolated to concentrations beyond the highest source concentration used in this study (i.e., Cot ) 0.047 M), indicates that the GCL is expected to exhibit virtually no membrane effect at concentrations beyond approximately 0.10 M KCl (i.e., ω f 0 as Cot f 0.10 M KCl). Diffusion Results. Measured exit concentrations of chloride and potassium from the base pedestal (i.e., Cb) are presented versus time in Figure 3. Essentially constant values of Cb are obtained in each test as steady-state diffusion is achieved. Greater values of Cb at steady state correspond to
FIGURE 3. Measured chloride and potassium exit concentrations (Cb) as a function of time. greater values of the source concentration, Cot, because a greater Cot results in a greater rate of diffusion through the soil. The results also indicate that steady-state diffusion is achieved more rapidly for chloride than for potassium, likely because of the exchange of potassium (K+) for the exchangeable cations of the bentonite, principally sodium (Na+). The resulting plots of Qt versus time for chloride and potassium based on the measured concentrations in Figure 3 are shown in Figure 4. The slopes, ∆Qt/∆t, based on bestfit linear regression of the steady-state data range from 7.84 × 10-7 to 3.06 × 10-5 g/m2/s for chloride and from 5.38 × 10-7 to 2.87 × 10-5 g/m2/s for potassium. The increase in ∆Qt/∆t with increasing source concentration, Cot, reflects an increase in diffusive flux through the soil. The x intercept of the linear fit through the steady-state data represents the time lag, TL, that provides a relative measure of the attenuation (e.g., adsorption) of the solute. For example, TL for chloride ranged from 37.8 to 51.3 h, whereas TL for potassium ranged from 340 to 738 h. Effective Diffusion Coefficients. Values of the coupled effective salt diffusion coefficient, Dω*, and the true effective salt diffusion coefficient, Ds*, based on the chloride and potassium data in Figure 4 are summarized in Table 1. As previously indicated, Dω* and Ds* should be the same for both chloride and potassium at steady state, in accordance with the requirement for electroneutrality. As indicated in Table 1, Dω* and Ds* based on the chloride data are practically equivalent to Dω* and Ds* based on the potassium data in three of the four tests. In the test with the lowest source concentration (Cot ) 0.0039 M), Dω* and Ds* based on the potassium data are slightly lower than Dω* and Ds* based on the chloride data. This particular test had the longest duration (48 days), and steady-state conditions with respect to diffusion of the potassium were approached but may not have been completely established (Figure 3). However, steady-state transport of chloride was achieved in all four tests (Figure 3). Therefore, Dω* and the associated Ds* values for chloride likely are more representative of the actual effective salt diffusion coefficients for KCl because the
FIGURE 4. Diffusion test results for chloride and potassium in terms of cumulative mass per unit area (Qt) of test specimen versus time. chloride concentrations essentially remained unchanged after the chloride initially reached steady-state diffusion, indicating that differences in the number or types of codiffusing cations (e.g., Na+) had very little, if any, measurable effect on the diffusion of the chloride. The results in Table 1 also indicate that Dω* < Ds* for all tests in this study are due to the observed membrane behavior for the GCL, as expected. This relationship between Dω* and Ds* is illustrated in Figure 5a. In essence, Dω* represents the value that would be obtained by analyzing the test data using Fick’s law that inherently assumes ω ) 0. Therefore, unless there is explicit evidence to indicate that the soil does not exhibit membrane behavior (i.e., ω ) 0), effective salt diffusion coefficients previously reported for sodium bentonites or other montmorillonite-dominated soils based on Fick’s law may actually represent coupled effective diffusion coefficients (i.e., Dω*) rather than true effective diffusion coefficients (i.e., Ds*). In this study, the error associated with using Fick’s law to evaluate the true coefficient Ds* for GCL, illustrated in Figure 5b, increases with increasing chemico-osmotic efficiency. However, for practical purposes, the maximum error in all cases is relatively small (i.e., 10% based on Cl- data, and 16% based on K+ data) and decreases with decreasing ω or increasing source concentration, Cot.
Discussion Dependence of Ds* and Dω* on ω. The more significant aspect of the influence of membrane behavior on the diffusion results in Table 1 is the apparent dependence of Ds* and Dω* on the source KCl concentration, Cot. For example, as shown in Figure 6a, the Ds* based on the chloride data decreases by approximately 300% (i.e., (2.38 × 10-10 m2/s)/(7.86 × 10-11 VOL. 36, NO. 6, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 1. Test Results specimen propertiesa
diffusion resultsb
test no.
n
kh (m/s)
Sf (%)
ω
Cot (M)
solute
1
0.80
1.65 × 10-11
97.0
0.63
0.0039
2
0.79
1.33 × 10-11
99.4
0.49
0.0087
3
0.79
2.06 × 10-11
97.5
0.32
0.020
4
0.78
1.48 × 10-11
96.3
0.14
0.047
ClK+ ClK+ ClK+ ClK+
∆Qt/∆t (g/m2/s) 7.84 × 10-7 5.38 × 10-7 2.82 × 10-6 2.43 × 10-6 1.20 × 10-5 1.18 × 10-5 3.06 × 10-5 2.87 × 10-5
Dω* (m2/s)
Ds* (m2/s)
7.05 × 10-11 4.39 × 10-11 1.16 × 10-10 9.07 × 10-11 2.14 × 10-10 1.91 × 10-10 2.34 × 10-10 1.99 × 10-10
7.86 × 10-11 5.20 × 10-11 1.25 × 10-10 9.96 × 10-11 2.24 × 10-10 2.01 × 10-10 2.38 × 10-10 2.03 × 10-10
a n ) specimen porosity; k ) hydraulic conductivity based on permeation with source KCl solution; S ) final degree of saturation; ω ) h f chemico-osmotic efficiency coefficient at steady-state induced pressure. b Cot ) source KCl concentration; ∆Qt/∆t ) slope of steady-state diffusion test data; Dω* ) coupled effective salt-diffusion coefficient; Ds* ) true effective salt-diffusion coefficient.
FIGURE 5. Coupled versus true effective salt diffusion coefficients: (a) comparison of measured values; (b) error resulting from explicit coupling versus chemico-osmotic efficiency coefficient. m2/s) × 100%) as Cot decreases from 0.047 to 0.0039 M. Similar results are shown for Dω*. These decreases in Ds* and Dω* with decreasing source KCl concentration are related to the observed increase in chemico-osmotic efficiency of the GCL with decreasing source KCL concentration, as shown in Figure 2. For example, the measured values of Ds* and Dω* based on the four different source concentrations, Cot, are plotted versus the corresponding measured values of ω in Figure 6b. As expected, these results illustrate that Ds* and Dω* decrease as the chemico-osmotic efficiency of the GCL increases. Because the relationship between Dω* (or Ds*) and ω shown in Figure 6b is not explicitly included in the governing theory for diffusive solute transport through membranes, the effect of ω on Dω* (or Ds*) is referred to, herein, as an implicit coupling effect. While this implicit coupling effect has not been previously shown, the relationship between Dω* (or Ds*) and ω illustrated in Figure 6b reflects expected behavior on the basis of our current understanding of the mechanisms associated with solute restriction in clay mem1316
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FIGURE 6. True (Ds*) and coupled (Dω*) effective salt diffusion coefficients versus (a) potassium chloride source concentration and (b) chemico-osmotic efficiency coefficient (i.e., implicit coupling). branes. For example, the degree of solute restriction is greatest when the double layers of adjacent clay particles overlap in the pore space, leaving no “free” solution for solute transport (8). In this case, the membrane is considered an “ideal” membrane such that ω ) 1. Because, by definition, no solute transport can occur into or through an “ideal” membrane, Jd ) 0; thus, Dω* must be zero on the basis of eq 2. However, the pores in most clay soils that exhibit chemico-osmotic membrane behavior vary over a range of sizes such that not all of the pores are restrictive. In such cases, ω typically falls within the range 0 < ω < 1, and clay soils are referred to as “nonideal” semipermeable membranes such that Dω* > 0. The higher solute (ion) concentrations in the pore space associated with an increase in the source concentration, Cot, causes contraction of the diffuse double layers that results in a decrease in chemico-osmotic efficiency and a corresponding increase in Dω* as more pores become available for solute transport. If the solute concentration is sufficiently high such that the diffuse double layers are compressed to the extent that ω ) 0, then Dω* equals Ds* and Dω* approaches
a matrix tortuosity factor, τm (0 e τm e 1), representing the tortuous nature of the actual diffusive pathways through the porous medium due only to the geometry of the interconnected pores (i.e., not including diffuse double layer effects) and a generalized restrictive tortuosity factor, τr (0 e τr e 1), or N
τa ) τmτr ) τm
∏τ ) τ i
m(τ1
‚ τ2 ‚ ‚ ‚ τN)
(5)
i)1
FIGURE 7. Implicit (empirical) coupling effect in terms of (a) the apparent tortuosity factor and (b) the restrictive tortuosity factor or effective porosity ratio. a maximum value (i.e., for a given state of stress or porosity). The estimated maximum value of Dω* ()Ds*) at ω ) 0 in this study is ∼2.4 × 10-10 m2/s on the basis of extrapolation of the measured trends shown in Figure 6b. Membrane Behavior and Solute Restriction. Apparent Tortuosity. The apparent tortuosity factor, τa, reflects the effect of the porous medium on the relative rate of solute diffusion that would occur in absence of the porous medium and, as defined by Shackelford and Daniel (27), is calculated as the effective diffusion coefficient divided by the free-solution (aqueous) diffusion coefficient for a given solute and medium. Therefore, an increase in Dω* also represents an increase in τa, as shown in Figure 7a. In general, τa ranges from zero, for the case where there are no interconnected pores in the porous medium, to unity, for the case where there is no porous medium (i.e., 0 e τa e 1). As a result, higher values of τa represent a less tortuous pathway for solute migration. The values of τa shown in Figure 7a were computed by dividing the measured Dω* and Ds* values by the salt diffusion coefficient for KCl in free (aqueous) solution, Dso, ) 19.93 × 10-10 m2/s at 25 °C (26, 27). Values of τa based on Dω* are similar to the values of τa based on Ds* because there is little difference between Dω* and Ds* in this study for a given source concentration (see Figure 6). The measured correlation between τa and ω based on the experimental results extends only within the range 0.14 < ω < 0.63. However, because, τa must approach zero as ω approaches unity (i.e., τa f 0 as ω f 1), the expected general trend for higher values of ω can be approximated on the basis of extrapolation of the data (dashed line) as shown in Figure 7a. Conversely, τa is expected to approach a maximum value when the chemico-osmotic efficiency is zero (i.e., ω ) 0). The estimated maximum value τa at ω ) 0 is ∼0.12 on the basis of extrapolation of the trend in the measured data shown in Figure 6a. Although the threshold source concentration corresponding to ω ) 0 is not known exactly for the results in this study, extrapolation of the semilogarithmic regression of the data in Figure 2 suggests that ω would be zero when the source concentration is approximately 0.1 M KCl. However, the relationships between ω and Cot and τa and ω given in Figures 2 and 7, respectively, likely are unique for the GCL and testing conditions used in this study. Restrictive Tortuosity Factor. On the basis of Shackelford and Daniel (27), τa may be defined further as the product of
where τr ) the product of N other factors (τi) that contribute to the apparent tortuosity by acting to reduce or restrict the diffusive flux of solutes through the porous medium. For example, Kemper et al. (35) included a factor, R (τr ) R), to account for the reduction in diffusive mass transport due to the increased viscosity of the water adjacent to the clay mineral surfaces (i.e., the adsorbed water) relative to that of the bulk water. Kemper and van Schaik (36) included both R and γ (i.e., τr ) Rγ), where γ is a factor to account for anion exclusion resulting from the existence of membrane behavior for electrolytes. Porter et al. (37) lumped these two effects into a single factor, also represented by γ (i.e., τr ) γ), presumably because of the difficulty associated with distinguishing between the effects of viscosity and anion exclusion. Finally, in the case of the exclusion of aqueousphase organic compounds (nonelectrolytes) from soil pores, τr ) δ where δ is referred to as a “constrictivity factor” (12). Whereas the matrix tortuosity factor accounts for the solute diffusion pathways not being parallel with the direction of the macroscopic concentration gradient, the constrictivity factor accounts for the variation in the cross section of the pathway (38). At ω ) 0, the diffuse double layers surrounding the clay particles likely are sufficiently compressed that both ion exclusion and viscosity effects are negligible, particularly for the relatively high porosities (0.78 e n e 0.80) associated with the GCL specimens in this study. On the basis of this assertion, τr ) 1 and the matrix tortuosity factor, τm, is represented by the maximum value of τa, as shown in Figure 7b or
τm ) τa,max ) τa|ω)0
(6)
The resulting expression for the restrictive tortuosity factor, τr, is obtained by substituting eq 6 into eq 5 to yield
τr )
τa τa ) τm τa,max
(7)
Because the range in porosities for the GCLs tested in the study is narrow, τm likely is approximately 0.12 for all specimens for the purposes of this discussion. The resulting values of τr, based on eq 7 and τm ) 0.12, are plotted versus the chemico-osmotic efficiency coefficient, ω, in Figure 7b. The results indicate that τr decreases (i.e., the migration pathway is more restrictive) as ω increases. This decrease in τr can be attributed to increased solute restriction or increased viscosity effects associated with greater thickness of diffuse double layers surrounding the clay particles (i.e., because no distinction can be made between these two individual effects). Comparison of Results. The potential influence of solute concentration on measured values of Dω* ()D*) has been examined in previous experimental studies on bentonites and GCLs. For example, Dω* values ranging from 2.9 × 10-12 m2/s to 3.2 × 10-11 m2/s were measured for cesium (Cs+) diffusion in Avonlea bentonite (i.e., 80% sodium montmorillonite) subjected to source concentrations (Co) of 3.8 × 10-8 M and 3.8 × 10-6 M (39). Although the higher values VOL. 36, NO. 6, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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diffusive solute flux based on a single value of Dω* likely will yield conservative estimates as long as the Dω* value used in the analysis corresponds to a concentration level sufficiently high such that membrane behavior is negligible.
Acknowledgments
FIGURE 8. Comparison of effective salt diffusion coefficients from this study versus ref 40. of D* in this range were obtained using the higher of the two Co values, the effect of source concentration was reported to be insignificant on the basis of a statistical analysis of the data (39). The influence of source concentration on Dω* for NaCl under constant volume conditions in granular sodium bentonite extracted from a GCL (Bentofix B4000) has also been investigated by conducting a four-stage diffusion test on a single specimen (n ) 0.71) (40). The source NaCl concentrations in the four stages were 0.08 M (4.6 g/L), 0.16 M (9.1 g/L), 0.60 M (35.1 g/L), and 2.0 M (114.3 g/L). The resulting Dω* values, based on the average of the values reported individually for Na+ and Cl- in ref 40, are plotted versus the source concentration, Co, in Figure 8. The Dω* values obtained in the present study for KCl also are shown in Figure 8 for comparison. The results in Figure 8 indicate that a similar trend of increasing Dω* with increasing Co was obtained relative to the results of the present study. Although chemico-osmotic efficiency coefficients were not measured in ref 40, chemico-osmotic flow was reported to be sufficiently negligible to conclude that membrane behavior probably was not significant for the range of NaCl concentrations used (i.e., Co g 0.08 M). This conclusion also is supported by the results reported in Figure 2 of the present study, although the relationship between ω and Co for the granular bentonite used in ref 40 may not be the same as the relationship shown in Figure 2 , in part because of the different porosity of the specimens (n ) 0.78-0.80 vs n ) 0.71), different salts used in the tests (KCl vs NaCl), and the potentially different properties of the granular bentonites in the two GCLs (Bentomat vs Bentofix B4000). Despite these differences, the results shown in Figure 8 suggest that there is general agreement between the test results previously reported in ref 40 and those reported in the present study. Practical Significance of Results. The influence of diffusion on the transport of miscible contaminants through clay barriers has been recognized to be an important, if not dominant, transport mechanism in geoenvironmental containment applications (41). In these applications, steadystate diffusive flux of a solute typically is assumed to occur in accordance with Fick’s first law, and the effective diffusion coefficient, Dω*, is assumed to be constant regardless of the solute concentration. However, the results of this study suggest that prediction of steady-state diffusive flux through a clay barrier material that exhibits membrane behavior may be accurate only when the analysis is based on the value of Dω* corresponding to the concentration level of the contaminants. The source concentration of contaminants contained by an actual soil barrier may not be known a priori. Also, performance of multiple diffusion tests for different values of Co to obtain a trend of Dω* versus Co, such as that shown in Figure 6a for an actual soil barrier, typically is not practical from an economic standpoint. However, prediction of 1318
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Financial support for this study, which is part of a joint research effort between Colorado State University and the Colorado School of Mines, was provided by the U.S. National Science Foundation (NSF), Arlington, VA, under Grant No. CMS-9634649. The assistance of Professor Harold (Hal) W. Olsen of the Colorado School of Mines is appreciated. The opinions expressed in this paper are solely those of the writers and are not necessarily consistent with the policies or opinions of the NSF.
Supporting Information Available Information regarding the geosynthetic clay liner (GCL) and testing procedures used in this study as well as the theoretical basis for eqs 2 and 3. This material is available free of charge via the Internet at http://pubs.acs.org.
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Received for review July 11, 2001. Revised manuscript received November 29, 2001. Accepted December 28, 2001. ES011130Q
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