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
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 36, NO. 6, 2002
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) ) the measured differential pressure induced
across the specimen as a result of prohibiting chemicoosmotic flux of solution and ∆π (<0) ) the theoretical
chemico-osmotic pressure difference across an “ideal”
semipermeable membrane (i.e., ω ) 1) subjected to an
applied difference in solute (KCl) concentration that is
calculated on the basis of the boundary solute concentrations
in accordance with the van’t Hoff expression (22, 24). The
generation of an induced pressure gradient across finegrained soil specimens under q ) 0 conditions in a closed
system previously has been shown experimentally (11, 22,
25).
Determination of Effective Diffusion Coefficients. For
the special condition of no net liquid flux imposed in this
study, the general expressions for the diffusive flux (Jd) of the
anion (subscript a) and the cation (subscript c) for a binary
salt solution (e.g., KCl) at steady state can be written in a
form analagous to Fick’s first law for diffusion in soil as follows
(see Theoretical Development, Supporting Information):
∆Ca
∆Cc
; Jdc ) -nDω*
∆x
∆x
Jda ) -nDω*
(2)
where n ) specimen porosity, C ) molar concentration, x
) direction of transport, and Dω* ) a coupled effective salt
diffusion coefficient that can be expressed as follows:
Dω* ) Ds* -
ω2(C
ha + C
h c)khRT
nFwg
(3)
where Ds* ) the true (uncoupled) effective salt diffusion
coefficient given by the Nernst-Einstein expression (26)
modified for the effect of tortuosity as described in ref 27,
C
h a and C
h c ) arithmetic mean molar concentrations across
the membrane (10, 28, 29), kh ) the hydraulic conductivity
of the specimen, R ) the universal gas constant (8.314 J mol-1
VOL. 36, NO. 6, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
1313
K-1), T ) the absolute temperature (K), Fw ) the density of
water (assuming dilute solutions), and g ) acceleration due
to gravity. The expression for Dω* given by eq 3 is the same
as that derived by Groenevelt et al. (30) but with different
notation and using a different approach.
The second term in eq 3 represents the explicit diffusive
coupling term that arises due to the soil membrane behavior.
This explicit diffusive coupling term has been called salt
sieving (30, 31), hyperfiltration (32), or streaming current
(13). If the soil exhibits no membrane behavior, then ω ) 0
and eqs 2 reduce to the traditional form for Fick’s first law
for diffusion in soil (27). Equations 2 and 3 form the theoretical
basis for evaluating the effect of explicit (theoretical) coupling
on steady-state solute diffusion through the GCL used in this
study.
On the basis of eq 3, Dω* will always be less than Ds* when
ω > 0 because of the salt sieving associated with the
membrane behavior of the soil. Thus, when a clay soil exhibits
membrane behavior, the diffusive fluxes given by eqs 2 will
be less than the fluxes given by Fick’s first law in which Ds*
is used rather than Dω*. However, when membrane behavior
is not present, ω ) 0 and Dω* ) Ds* in accordance with eq
3.
The determination of the coupled effective salt diffusion
coefficient, Dω*, in this study is based on the steady-state
approach in which the measured solute concentrations from
the collected circulation outflow are converted to cumulative
solute mass per unit area of the specimen, Qt, and plotted
versus elapsed time (22, 33, 34). Under constant boundary
conditions, the diffusive solute flux will eventually approach
a steady-state condition at which Dω* can be determined
using the following expression (22):
( )
Dω* ) -
∆Qt
L
∆t nwA∆C
(4)
where ∆Qt/∆t ) the diffusive solute flux through the specimen
at steady state, L ) thickness of specimen, wA ) the atomic
weight of the solute, and ∆C ) the molar concentration
difference of the solute across the specimen. The true effective
salt diffusion coefficient, Ds*, then can be determined by
substituting Dω* obtained from eq 4 into eq 3 (i.e., provided
that all of the other parameters in the second term on the
right-hand side of eq 3 also are measured simultaneously).
Testing Program. Four diffusion tests were conducted in
this study. In each test, the thickness, L, of the GCL specimen
was controlled at 10 mm, resulting in a narrow range of
measured specimen porosities (0.78 e n e 0.80). A different
source KCl concentration, Cot, was used in each test (i.e., Cot
) 0.0039, 0.0087, 0.020, and 0.047 M) to evaluate the influence
of solute concentration on measured values of Dω* and Ds*.
Values of Cb were measured using ICP-OES for K+ and IC
for Cl-.
Results
Chemico-Osmotic Efficiency Coefficients. The measured
differential pressures induced across the GCL specimens,
-∆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
9
FIGURE 1. Differential pressures induced across GCL specimens
during steady-state diffusion tests.
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 36, NO. 6, 2002
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
9
1315
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
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 36, NO. 6, 2002
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
9
1317
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
9
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 36, NO. 6, 2002
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.
Literature Cited
(1) Shackelford, C. D.; Benson, C. H.; Katsumi, T.; Edil, T. B.; Lin,
L. Geotext. Geomembr. 2000, 18, 133-161.
(2) Gleason, M. H.; Daniel, D. E.; Eykholt, G. R. J. Geotech.
Geoenviron. Eng. 1997, 123, 438-445.
(3) Daniel, D. E. In Geotechnical Practice for Waste Disposal; Daniel,
D. E., Ed.; Chapman and Hall: New York, 1993; Chapter 7, pp
137-163.
(4) Koerner, R. M. Designing with Geosynthetics, 3rd ed.; Prentice
Hall: Englewood Cliffs, NJ, 1994.
(5) Rowe, R. K.; Petrov, R. P.; Lake, C. In Proceedings, 6th
International Landfill Symposium, Sardinia ‘97; Cagliari, Italy,
1997; Vol 3, pp 301-310.
(6) Petrov, R. J.; Rowe, R. K.; Quigley, R. M. Geotech. Test. J. 1997,
20, 49-62.
(7) Rowe, R. K.; Lake, C.; von Maubeuge, K.; Stewart, D. In
Proceedings, GeoEnvironment ‘97; Bouazza, Kodikara, and
Parker, Eds.; Balkema: Rotterdam, The Netherlands, 1997; pp
295-300.
(8) Marine, I. W.; Fritz, S. J. Water Resour. Res. 1981, 17, 73-82.
(9) Fritz, S. J.; Marine, I. W. Geochim. Cosmochim. Acta 1983, 47,
1515-1522.
(10) Fritz, S. J. Clays Clay Miner. 1986, 34 (2), 214-223.
(11) Keijzer, Th. J. S.; Kleingeld, P. J.; Loch, J. P. G. In Geoenvironmental
Engineering, Contaminated Ground: Fate of Pollutants and
Remediation; Yong, R. N., Thomas, H. R., Eds.; Thomas Telford
Publishers: London, U.K., 1997; pp 199-204.
(12) Grathwohl, P. Diffusion in Natural Porous Media, Contaminant
Transport, Sorption/Desorption and Dissolution Kinetics; Kluwer
Academic Publishers: Norwell, MA, 1998.
(13) Mitchell, J. K. Fundamentals of Soil Behavior, 2nd ed.; John
Wiley and Sons: New York, 1993.
(14) Kemper, W. D.; Rollins, J. B. Soil Sci. Soc. Am. Proc. 1966, 30,
529-534.
(15) Kemper, W. D.; Quirk, J. P. Soil Sci. Soc. Am. Proc. 1972, 36,
426-433.
(16) Elrick, D. E.; Smiles, D. E.; Baumgartner, N.; Groenevelt, P. H.
Soil Sci. Soc. Am. J. 1976, 40, 490-491.
(17) Sawatsky, N.; Feng, Y.; Dudas, M. J. J. Contam. Hydrol. 1997, 27,
25-41.
(18) Muurinen, A. Eng. Geol. 1990, 28, 359-367.
(19) Oscarson, D. W.; Hume, H. B.; Sawatsky, N. G.; Cheung, S. C.
H. Soil Sci. Soc. Am. J. 1992, 56, 1400-1406.
(20) Oscarson, D. W. Clays Clay Miner. 1994, 42, 534-543.
(21) Oscarson, D. W.; Hume, H. B. Transp. Porous Media 1994, 14,
73-84.
(22) Malusis, M. A.; Shackelford, C. D.; Olsen, H. W. Geotech. Test.
J. 2001, 24, 229-242.
(23) Van Oort, E.; Hale, A. H.; Mody, F. K.; Roy, S. SPE Drill. Compl.
1996, 11, 137-146.
(24) Olsen, H. W.; Yearsley, E. N.; Nelson, K. R. In Transportation
Research Record; No. 1288; Transportation Research Board:
Washington, DC, 1990; pp 15-22.
(25) Olsen, H. W. Soil Sci. Soc. Am. Proc. 1969, 33, 338-344.
(26) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.;
Butterworth Scientific Publications: London, U.K., 1959.
(27) Shackelford, C. D.; Daniel, D. E. J. Geotech. Eng. 1991, 117, 467484.
(28) Katchalsky, A.; Curran, P. F. Nonequilibrium Thermodynamics
in Biophysics; Harvard University Press: Cambridge, MA, 1965.
(29) Whitworth, T. M.; Fritz, S. J. Appl. Geochem. 1994, 9, 533-546.
(30) Groenevelt, P. H.; Elrick, D. E.; Laryea, K. B. Soil Sci. Soc. Am.
J. 1980, 44, 1168-1173.
(31) McKelvey, J. G.; Milne, I. H. Clays Clay Miner. 1962, 9, 248-259.
(32) de Marsily, G. Quantitative Hydrogeology; Academic Press:
Orlando, FL, 1986.
(33) Jessberger, H. L.; Onnich, K. In Proceedings, XIIIth International
Conference on Soil Mechanics and Foundation Engineering;
New Delhi, India, 1994; 1547-1552.
(34) Shackelford, C. D. J. Contam. Hydrol. 1991, 7, 177-217.
(35) Kemper, W. D.; Maasland, D. E. L.; Porter, L. K. Soil Sci. Soc. Am.
Proc. 1964, 28, 164-167.
(36) Kemper, W. D.; van Schaik, J. C. Soil Sci. Soc. Am. Proc. 1966,
30, 534-540.
(37) Porter, L. K.; Kemper, W. D.; Jackson, R. D.; Stewart, B. A. Soil
Sci. Soc. Am. Proc. 1960, 24, 460-463.
(38) Van Brackel, J.; Heertjes, P. M. Int. J. Heat Mass Transfer 1974,
17, 1093-1103.
(39) Cho, W. J.; Oscarson, D. W.; Gray, M. N.; Cheung, S. C. H.
Radiochim. Acta 1993, 60, 159-163.
(40) Lake, C.; Rowe, R. K. Geotext. Geomembr. 2000, 18, 103-131.
(41) Shackelford, C. D. Geotech. News 1988, 6, 24-27.
Received for review July 11, 2001. Revised manuscript received November 29, 2001. Accepted December 28, 2001.
ES011130Q
VOL. 36, NO. 6, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
9
1319