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Published online 2 December 2005
Published in Soil Sci Soc Am J 70:14-23 (2006)
DOI: 10.2136/sssaj2005.0044
© 2005 Soil Science Society of America
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Soil Physics and Soil Chemistry

Percolation Treatment of Charge Transfer in Humidified Smectite Clays

Allen G. Hunta,*, Sally D. Logsdonb and David A. Lairdb

a Dep. of Physics and Dep. of Geology, Wright State Univ., Dayton, OH 45435
b National Soil Tilth Lab., 2150 Pammel Dr., Ames, IA 50011

* Corresponding author (allen.hunt{at}wright.edu)


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 Appendix
 REFERENCES
 
Bulk electrical conductivity of soil is generally assumed to be dominated by the electrical conductivity of the soil solution, with perhaps a small contribution from surface charges associated with soil solids. Soils high in smectites often exhibit high electrical conductivity due to water associated with the clays. These charges are either associated with the exchangeable cations or with proton migration in the clay-associated water. The purpose of this study was to use percolation principles to elucidate mechanisms of charge transfer in humidified smectites. The theory predictions were compared with electrical conductivity spectra measured for mono-ionic humidified smectites. The measured spectra were consistent with a percolation model of proton hopping in the maximum interlayer spacing beyond a threshold water content, plus an offset term for some samples. The threshold water content for percolation in the maximum interlayer spacing was 0.07 m3 m–3, which was assumed the same for all basal spacings. The spectra should converge at the phonon frequency (vph), but for some samples at the higher humidity levels, the spectra were offset. The samples with this offset were determined by examining the spectra for the same mono-ionic smectite at different humidity levels. The mean offset level was 0.015 m3 m–3 and was likely related to charge transfer on external surfaces. The phonon frequency fitted across all samples was 3.54 x 108 Hz, lower than the 1012 Hz often assumed. These results are generally consistent with the idea that water associated with smectite clays generated the observed electrical conductivity.

Abbreviations: 2D, two-dimensional • 3D, three-dimensional • ac, alter-nating current • dc, direct current


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 Appendix
 REFERENCES
 
BULK ELECTRICAL CONDUCTIVITY is often described in relation to soil water content. Field measurements of the direct current (dc) electrical conductivity are typically related to water content through two terms, the first term representing the contribution of water in the pore space, and the second term taken to represent a surface, or solid, contribution (Cremers and Laudelout, 1965; Rhoades et al., 1976). Rhoades et al. (1976) give the bulk electrical conductivity ({sigma}a) as

[1]
in which {sigma}w is electrical conductivity in the liquid phase, {theta}w is the volumetric water fraction, tr is the transmission coefficient (= a + b{theta}w, in which a and b are fitted), and {sigma}s is the electrical conductivity associated with the soil surfaces.

Reevaluation of the contribution from the water in the pore space to the electrical conductivity indicates that the analytical form is a nonlinear power (Hunt, 2004). Although the solid contribution is often neglected, at low water contents the bulk electrical conductivity of the porous media is dominated by the electrical conductivity associated with the solids (Letey and Klute, 1960; Cremers et al., 1966), especially water associated with clay surfaces.

High bulk electrical conductivity is observed for non-saline soils high in 2:1 clay minerals (Saarenketo, 1998; Logsdon, 2000). The source of this charge is often attributed to the exchangeable cations (Oster and Low, 1963) or to proton transfer from dissociated interlayer water (Fripiat et al., 1965; Calvet, 1975). A more thorough analysis of this charge transfer contribution is needed, particularly since the process of electrical conduction is related to other transport processes in these systems. For clay-associated water, Low (1979) summarizes earlier work that showed lower density but higher heat capacity, viscosity, expandability, specific entropy, and decreased ion mobility (Kemper et al., 1964) compared with bulk water. The dc electrical conductivity shows strong increases with water content for humidified smectites (Logsdon and Laird, 2004a). The Na-saturated clays had the highest electrical conductivity because the Na ion polarizes the hydrated water less than the divalent cations. The K-saturated clays had the lowest electrical conductivity because of limited water in the interlayers. Otay samples (highest charge density) had the lowest electrical conductivity because of less water in the interlayers. Hectorite samples (lowest charge density) had the largest electrical conductivity because the interlayer water molecules were less influenced by the local electrical fields (Logsdon and Laird, 2004a). These trends are consistent with a proton migration hypothesis, and inconsistent with exchangeable cation migration as a dominant mechanism for electrical conductivity. Complex interactions between water and clay mineral surfaces influence dielectric properties of near surface water as clays swell in response to changes in relative humidity (Laird, 1996).

Spectra of the complex alternating current (ac) electrical conductivity showed a strong frequency-dependence for electrical conductivity of hydrated clay minerals (Logsdon and Laird, 2004a, 2004b). Over the range of experimentally accessible frequencies (300 KHz to 1 GHz), the electrical conductivity of four reference smectites increased as much as two orders of magnitude. Bidadi et al. (1988) observed similar frequency-dependent conductivity for humidified Na- and Li- saturated smectites. The strong-frequency-dependence demonstrates that the contribution of solids and surfaces to electrical conductivity is substantial for clays and probably also for soils. By contrast, electrical conductivity in homogeneous conductors is independent of frequency, which substantiates the heterogeneous nature of humidified smectites.

Percolation theory has been used to model the electrical conductivity of heterogeneous systems. In this study we use percolation theory at the molecular level to find the principle limitation to charge transport, and hence the dc conductivity. We assume this limitation is due to Coulomb energy barriers encountered by hopping charges, which we believe to be protons. We will treat this limitation as overcoming an energy barrier. The purpose of this study was to use percolation principles to elucidate mechanisms of charge transfer in humidified smectites.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 Appendix
 REFERENCES
 
The measured electrical conductivity spectra are from Logsdon and Laird (2004a). Briefly, the spectra were measured from mono-ionic smectites that had been equilibrated at four humidity levels. The ions were Ca, Mg, Na, and K, the smectite clays were SPV bentonite, Otay, hectorite, and IMV bentonite, and the humidity levels were 99, 85, 74, and 56%, which resulted in a range of water contents. The surface charge densities were 1.20, 1.28, 2.11, and 1.7 µmolc g–1 for the hectorite, SPV bentonite, Otay, and IMV bentonite clays; whereas the percentage of tetrahedral charge was 0, 8, 26, and 20 for these clays (Logsdon and Laird, 2004a).

There were duplicate samples for each ion–smectite–humidity level combination, which were averaged for comparison with the theory. The spectra were measured in a truncated coaxial cell using a vector network analyzer for frequencies from 300 KHz to 3 GHz, although the data were not good beyond 1 GHz. The inner conductor of the truncated coaxial cell was 5 mm long and 4 mm in diameter, and the outer conductor was 10.6 mm long and 13.2 mm inner diameter. Basal spacings of the clays were determined by X-ray diffraction for each of the ion-smectite-humidity level combinations.

For each sample, the volume fraction of interlayer water within each basal spacing was calculated as described in Logsdon and Laird (2004a), and external water was calculated as total water volume fraction minus interlayer water. If any of the interlayers were not completely filled with water, this would result in an overestimate of water volume for that basal spacing, and an underestimate of external water volume fraction.

Theory
Throughout the theory section, electrical conductivity will be simply called "conductivity," and angular frequency will just be called "frequency" as the terms are used in the physics literature. We propose to evaluate the contribution to the electrical conductivity from mobile protons within the context of the theory of hopping conduction. The term "hopping" means that charges that are located on specific sites (in the case here, protons on water molecules) most of the time jump to another site in a much shorter time period. While the typical time taken to jump to another site is essentially zero, the time a charge spends "waiting" to jump is an exponential function of the energy barrier, E, between the sites, that is, {tau} = {nu}ph–1exp(E/kT). Here the quantity {nu}ph is a vibrational, or "attempt" frequency and kT is the product of the Boltzmann constant and the temperature. The exponential dependence on energy E, ultimately derives from the probability that the energy to transport the particle over the barrier can be absorbed by the particle from thermal fluctuations in the surroundings. This "waiting" time may also be loosely referred to as a relaxation time, or a hopping time. In disordered systems, such energy barriers can vary widely from place to place and the total time required to transport charges through the material (related to an effective velocity, or current) depends on all the waiting times along the particular path followed. For dc conduction in macroscopic natural systems there is usually enough time, enough individual charges, and sufficient local heterogeneity that the dominant transport paths can be identified as those with the "least resistance," or with smallest transport times, that is, smallest activation energies.

Over the last 30 yr considerable literature has developed over the theory of hopping conduction in disordered systems. Although this theory is relatively mature for "non-interacting" systems, in cases where the hopping motions of the individual charge carriers are strongly correlated, controversy remains (Jonscher, 1977; Funke, 1991). The present discussion will utilize the material given in Hunt (2001a). First consider the possible charge pathways through the humidified smectite clay (Fig. 1 ). Assume that proton hopping is the mechanism with which water transfers charge. Calvet (1975) assumed that proton hopping was associated with "fill" water, that is, water not as closely associated with the exchangeable cations, since the water hydrating the cations is strongly oriented in response to the local electrical fields of the cation and surface charge sites on the smectites. On the other hand, the "fill" water is associated with more hydrophobic nanosites located between charge sites (Laird and Sawhney, 2002). Hydrophobic surfaces result in more rigid, less fluid water (Derjaguin and Churaev, 1986; Vogler, 1998), which would tend to push the charges closer to the cations and charge sites on the clays. For dc conduction, protons must hop along pathways, which connect from one end of the system to the other. Thus, when considering dc electrical conductivity, the highest unavoidable activation energy barrier is of interest (Dyre and Schroeder, 2000) because the total time required for charge transport is dominated by the slowest jumps. This particular effect arises from the exponential dependence of the waiting time on the energy.



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  		Fig. 1. Diagram of possible pathways for proton transfer (a) along interlayer face through fill water away from interlayer cation, or (b) in two-dimensions passing close to cation.

 
One way to accommodate chemical equilibrium and electrical conduction in pure water is for individual proton hops to be highly correlated, such that as one proton vacates a given location, a second proton moves into position vacated by the first. If correlated proton hopping is a conduction mechanism for pure water, then the mechanism can be logically extended to water associated with smectites. Smectites have spatially variable density along the interior clay surfaces, which reduces the mobility of the protons in the water adjacent to these surfaces. We express this effect in terms of Coulombic energy barriers. We thus have a problem of correlated hopping in a relatively slowly varying Coulombic potential (since the typical charge separation is larger than the separation of the water molecules). But the most important process on such a path is the one, which is slowest (i.e., has the highest energy barrier, Dyre, 1991), while the most important paths are those, for which the effective hopping rate is fastest (for which the largest energy barrier is as small as possible). This idea of a limiting barrier height on the most conducting path is in the spirit of percolation theory.

Analysis here will reveal that the height of the dominant energy barrier is on the order of thermal fluctuations for systems with more than four water layers, but increases to values of about 4kT for single water layers. This increase produces a diminution of the conductivity of about exp(–4kT/kT) = exp(–4) (ca. 1/50), approximately the range of variation of the electrical conductivity observed in this study. This energy scale will be seen to be that of a Coulombic energy (q1q2/4{varepsilon}0{varepsilon}wr) for which the length scale, r, is the thickness of the surface water film, q1 is the charge of a proton, q2 that of a typical cation, {varepsilon}0 is the permittivity of free space, and {varepsilon}w the dielectric constant of water.

Start with the activation energy of the dc conductivity. Assume that the negative charge on the basal surfaces of 2:1 phyllosilicates as located in basal oxygen atoms that are proximal to sites of isomorphic substitution. The counter positive charges are associated with the exchangeable cations in the interlayers. Although the cations are highly mobile, at any given moment they will tend to be located as close to the negative surface charge sites as possible and as far from each other as possible. Thus the spatial distribution of the interlayer cations is determined by the distribution of negative surface charge. The charge of the cations is qe. These positive charge sites associated with the interlayer cations are separated within plane geometry by a typical distance that we will call l (See Fig. 1). The potential at a site in the water due to one of these cations is not a "naked" potential, but is reduced through the dielectric properties of the water itself. The energy of interaction of a charge e (the protonic charge) and a single cation charge, qe, at a separation, r, is then,

[2]

Consider a problem in two dimensions (2D) with no significant water thickness (Fig. 1a). A hopping charge must, on average, be brought within a distance, l/2, of a cation of charge qe to find a path through the system. In three dimensions (3D) (represented in cross-sections in Fig. 1a and 1b), with water thickness, wr0, (with w the number of water layers and r0 the thickness of a water layer) this distance becomes d = [(l/2)2+w2r02]1/2. However, for smectites dominated by octahedral charges at high water contents, the distance may become d = [(l/2)2+1/4(w2r02)]1/2, especially for divalent cations. We ignore this modification, which would result in activation energy increasing as water content increases rather than decreasing as water content increases. If, on average, a proton starts at zero energy (arbitrary starting position under conditions of charge neutrality), its total energy has increased by an amount,

[3]
by the time it makes its closest approach to a counterion. At this point it becomes important to consider how l depends on the concentration of cation charges, N. In a 3D system, l {propto} N–1/3, it is possible that a 2D result, l {propto} N–1/2, would be more appropriate (in platy systems), but it turns out not to matter which choice is made. We are ignoring orientation of the clay plates in relation to the external field (which introduces a numerical factor around 1/3, which is appropriate for random orientations). Now,

[4]

This Eac was our first estimate of the actual value of the activation energy for the dc conductivity due to the Coulombic repulsion of the counterions. Comparison with experimental data presented here (not shown), which show a strong dependence of charge mobility with water layer thickness, proves that Eq. [4] is incorrect (R2 = 0.0). A Taylor series expansion (in the small quantity, N2/3r02w2) of Eq. [4] shows immediately that the calculated Eac is almost independent of r0w for r0w l. For successful comparison with experiment it will be necessary to modify Eq. [4] to drop the first term in the square root. Thus, if the proposed mechanism of transport is correct, it is not possible for the hopping protons to avoid the counterions in the plate parallel direction, only in the perpendicular direction. Then we have,

[5]
with w as number of water layers of r0 = 0.25 nm. (See Fig. 1.) The last expression was written in anticipation of comparison with experiments conducted at T = 298 K. Although, as noted, using {varepsilon}w = 80 may underestimate Eac, consider the case for w = 1. Then it is highly probable that the proposed mechanism (of highly correlated hopping motions) during any individual hop would not produce an actual change in the number of protons on the water molecule nearest the counterion. One proton would simply replace another at a given location. But the conduction process would require a proton to jump between that site and a neighboring site. This means that the nearest distance of approach (for the purpose of calculating a barrier height) would be somewhat larger than wr0 and the energy somewhat smaller. For rough estimation of this effect consider that the highest energy that the proton experiences (counting the Coulombic attraction to the water molecules) is likely to occur at about half the water molecule spacing. At such a distance, the Coulombic effects due to the counterion will be reduced by a factor (depending on orientation) somewhere between (4/5)1/2 and 2/3, (from 11 to 33%). Such a numerical uncertainty is on the same order of magnitude as what would arise from using a dielectric constant of, say, 50 rather than 80, which would increase the Coulomb interaction strength by 38%. As a consequence we ignore these complications and use the numerical factor of Eq. [5].

The pre-exponential for the conductivity was estimated from the perspective of a random resistor network (disordered medium). In such a disordered medium, the optimal paths for current flow avoid large resistances as much as possible, but the majority of the resistance is concentrated in relatively widely spaced bottlenecks (Bernabe and Bruderer, 1998; Hunt 2001a, 2001b). The calculation of the conductivity can be reduced to the determination of the bottleneck resistance values on the dominant current-carrying paths, the frequency of occurrence of such paths (per unit cross-sectional area) and the separation of the dominant resistances on such paths (Friedman and Pollak, 1981; Hunt, 2001a, 2001b). In such a network, in 3D, the dc conductivity is given by the following equation (Friedman and Pollak, 1981; Hunt, 2001a, 2001b):

[6]
where (e2/kT)Rc = [{nu}phexp(–Eac/kT)]–1, l0 is the linear separation of critical (bottleneck) resistances on a critical path, and L is the linear separation of such paths, making L–2 the number of current-carrying paths per given cross-sectional area. In d dimensions L–2 is replaced by L–(d–1). Right at critical percolation L->{infty} (Stauffer, 1979), but when critical path analysis (Friedman and Pollak, 1981) is used to develop {sigma}, the bottleneck resistance value is slightly larger than the critical value, and the value of L is more nearly a molecular separation. Thus L can be taken to be a numerical constant (Hunt, [2001b] found values between 5 and 15) times r0. In our problem, however, these considerations do not strictly apply. Use of L2 is related to the dimensionality of the optimization procedure and would be replaced by L if the optimization were performed in 2D. This is consistent with structural constraints for the 3D in the clay, for example, a distance between proton carrying paths of (4 + w)r0, where 4 + w is the thickness (in units of water molecule size) of a simple clay sheet. It is also possible that in the plate parallel direction (Fig. 1) the path separation is structurally controlled, and is approximately equal to l. The largest resistance values, however, will be separated by l0 = l, when protons come into the vicinity of a counterion. Thus the length scales in the pre-exponential are all multiples of r0 with numerical values greater than 1. Since two such numerical constants are in the denominator, but only one in the numerator, the combined numerical constant is likely to be less than 1. Altogether we have

[7]

The factor, exp[–Eac/(kT)], is given in Eq. [5].

Alternating Current Conductivity
Calculation of the ac conductivity of hopping systems is based on the fundamental result from the fluctuation-dissipation theorem (Reif, 1965) that at an angular frequency {omega} (= 2{pi}v), all transitions with relaxation times {tau} such that {tau} {approx} {omega}–1 contribute importantly to the real part of the conductivity, while all those with {tau} ≤ {omega}–1contribute importantly to the imaginary part of the conductivity. This concept is generally consistent with the argument in Sposito and Prost (1982) that reorientations of dipoles (if the reorientation takes place over a very short time compared with the time the dipole remains in a given configuration this motion is also "hopping" transport) with {tau} ≤ {omega}–1 can contribute to the polarization at frequency {omega} because the current is the time derivative of the polarization. In other words, precisely those reorientations which could just barely proceed within the time that the applied field points in a given direction then take place in phase with the field, while those which occur much more rapidly have a phase which actually leads that of the field. The relaxation time of a transition is directly related to the activation energy of the transition,

[8]

To calculate or estimate the ac conductivity one needs information regarding the spectrum of relaxation times and therefore activation energies. The lowest energy barrier encountered by protons (midway between counter ions, i.e., l/2) is likely nearly zero, but one can make the following estimation. Take the derivative with respect to l/2 of the above potential (Eq. [3]) for a single cation and multiply by r0, a typical water molecule separation. The result is,

[9]

It is likely not necessary to approximate this result better than to let w2r02N2/3 << 1. Then we have,

[10]

This smaller activation energy corresponds to a smaller time and a higher frequency, {omega}m, which is given by,

[11]
with {Delta}E given above. The ac conductivity at this frequency is given by,

[12]

Here nac is the concentration of contributing charges. This density will be determined by comparison with experimental data. In general, the fact that the time of the hops contributing to the real part of the conductivity is proportional to the inverse of the frequency means that the conductivity must increase roughly linearly with the frequency. If hopping processes of all waiting times were equally common as well as equally long, then s = 1, but generally s < 1 since hopping processes with larger energies and longer times tend to be more common. So assume that {sigma}({omega}){propto}{omega}s, with s a power somewhat less than one. How does one find s? First note that the present argument for finding s cannot be applied for energies E > Eac, since by construction the hopping charges can avoid such energy barriers. Then,

[13]

Note that {omega}c{equiv}{nu}phexp(–Eac/kT). Using the above values for the frequencies and noting that {sigma}({omega}c) = {sigma}dc (very nearly), one finds for s,

[14]

This result is approximately valid between the two frequencies {omega}c and {omega}m. A simple approximation to Eq. [14], which can be used between the frequencies {omega}c and {nu}ph is,

[15]

Such approximations have been used before (MacDonald, 1987). It has often (Nakajima, 1972; Namikawa, 1975; Dyre, 1991; Dyre and Schroeder, 2000) been shown that the hopping contribution to the dc dielectric constant, {Delta}{varepsilon}, and the dc conductivity are related by,

[16]

Here B is the Barton-Nakajima-Namikawa (Nakajima, 1972; Namikawa, 1975) numerical coefficient. Thus it is possible using Eq. [7] and Eq. [19] below to make a first estimation of {Delta}{varepsilon},

[17]

Consider the ac response from the perspective in which the applied frequency is gradually lowered. As the time frame for response of the hopping charges continues to be increased, eventually at a frequency, {omega}c there is sufficient time, tc = {omega}c–1 for these charges to follow the paths defined by the dc conductivity. This physical argument, interpreted within the framework of percolation theory, leads to the analytical approximation,

[18]
as long as the approximation of a power-law form for the ac conductivity is valid. The important physical result associated with Eq. [18] is

[19]
where Eac is the same activation energy as the dc conductivity.

In the limit of {omega}->{nu}ph (on account of Eq. [8]) the only charges, which can move in phase with the applied electric field are those with activation energies of magnitude kT or less. If the distribution of activation energies, E, is slowly varying in the limit E->0, and the width of the distribution of activation energies has a scale given by Eac, then the fraction of excitations with excitation energy ≤ kT is proportional to kT/Eac. Such a condition would imply that at frequencies approaching the phonon frequency,

[20]

Note that Eq. [20] gives for small w a conductivity proportionality to the moisture content, but for w > 4 {sigma}({nu}ph) gradually saturates. Also, Eq. [20] is independent of T and only weakly dependent on the water content for w on the order of 4 (or larger).

Analysis
The first step was to fit each measured {sigma}({omega}) mean curve to Eq. [18], and tabulate {sigma}dc, {omega}c, and s as fit parameters. Then we developed the protocol for predicting {sigma}dc. We calculated the charge separation l from the layer charge density for the clay (Logsdon and Laird, 2004a), and we adjusted cation charge by subtracting tetrahedral charge. For predicted results to be consistent with measured quantities, it was necessary to assume that for dc electrical conductivity, the protons pass close to the cation/smectite charges giving the largest energy barrier as the alternate formulation (Eq. [4]) that the protons avoid these charges could not be confirmed. Also interlayer charges would be transported only in interlayers with the largest basal spacing if water content in these interlayers was beyond a threshold, {theta}t. For simplicity the threshold was assumed to be the same for all combinations of mineral and counterion. This allowed a cutoff basal spacing for charge transport associated with each cation-smectite-humidity level and calculation of Eac. Note that use of Eq. [7] contains the phonon frequency, {nu}ph, which is not known before hand. The data for ac conduction can be used to extract {nu}ph, which can then be used in the calculation of {sigma}0 (as described next).

At or near the phonon frequency, hopping motion in phase with the electric field begins to be suppressed, because even with zero energy barriers, (Eq. [7]) it is not possible to excite hopping transitions over shorter time periods than the phonon frequency. Thus the real part of the electrical conductivity begins to level off at a phonon frequency. Above the phonon frequency, the frequency-dependence of the electrical conductivity begins to be influenced strongly by atomic and molecular polarization effects rather than by hopping motions. Such effects introduce different dependences of the electrical conductivity on frequency. The phonon frequency was first approximated to be ≤1010 Hz by visual inspection of the intersections in the {sigma}({omega}) curves. We also approximated Eq. [14] with Eq. [15], 1 – s {propto} kT/Eac. Then we graphed [–log({omega}c/{nu}ph)]–1 vs. 1 – s for arbitrary values of {nu}ph, and optimized the value of R2 with respect to the choice of {nu}ph. Because of the uncertainties in Eq. [7] for l/[Lr0(4 + w)], we compared the predicted conductivity without any prefactor, as a function of the measured conductivity minus the 0.015 S m–1 offset, forcing regression through the origin. The inverse of this slope should be the best-fit prefactor. For 2D, the L should not be squared, but L2 matches the data well. This would suggest a 3D relation inconsistent with the percolation ideas developed in Eq. [5] and [7]. An alternate approach would be to consider mean water thickness instead. The wr0 would be replaced with a mean value in Eq. [5] and Eq. [7]. Both approaches were calculated using L2 = 152 (obtained by using L = l). The percolation approach resulted in some potassium samples that had no threshold water thickness in the interlayer; for these samples, the predicted {sigma}dc was set to 0. The threshold percolation water content (obtained by optimization) of 0.07 m3 m–3, and the offset of 0.015 S m–1 were other uncertainties. The offset was determined through an analysis described in the next section.

The ac hopping conduction is known to show a weak (power-law) temperature dependence, in contrast to the exponential dc value (Friedman and Pollak, 1987). Similarly we showed that the ac conductivity at the phonon frequency (Eq. [20]) has a much weaker dependence on the moisture content than does the dc conductivity (Eq. [7] and [5]). Thus, the values of the ac conductivity with different water contents should tend to converge at the phonon frequency. In some cases, the higher water content values of {sigma}({nu}ph) were larger by the same value as the dc conductivity, rather than converging. This offset indicated a contribution to {sigma}, which was frequency independent. Once these offset curves were identified, the mean offset ({sigma}o) was used for all of them and added to the {sigma}dc predicted from Eq. [7].

Once those offset cases were identified, it became possible to analyze the correlation between the high frequency ac conductivity and the dc conductivity. We regressed [{sigma}({nu}ph) – {sigma}dc] as a function of ln({sigma}dc – {sigma}o), in which {nu}ph is the ac conductivity at 2.22e9 Hz. Because of the activated behavior of {sigma}dc, the described operation on the dc conductivity should yield kT/Eac.

A diagnostic of critical importance was to compare {sigma}dc with {omega}c. If these two quantities are proportional to each other, that is, have the same activation energy; it clearly indicates that the same processes control the ac conductivity as control the dc conductivity (Hunt, 2001a; Dyre and Schroeder, 2000). This result would also clearly indicate the relevance of percolation theory to both the ac and the dc conduction. We graphed {sigma}dc – {sigma}o as a function of {omega}c to show that the activation energies were the same for both processes. We compared the predicted and measured {sigma}dc using Eq. [7] with Eq. [5] for Eac, and the value obtained above for {nu}ph. We also compared predicted values using mean interlayer water thickness rather than cutoff interlayer water thickness. Finally we tested the proportion in Eq. [20]. The 95% confidence intervals were calculated for the regression equations (based on mean), and intercepts and slopes were tested to see if they were significantly different from zero (p = 0.05).


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 Appendix
 REFERENCES
 
Typical ac conductivity results are shown in Fig. 2 . These fits were not unique; as described above, an additional term assumed to relate to conduction on external surfaces could be included. In some cases charge transport should occur through other pathways than the cut-off interlayer, most likely through external water. Addition of such a constant term raises the ac conductivity over the entire frequency range, producing what could be called a "parallel offset." It was only through comparison with other spectra that it was possible to estimate which water contents allowed this parallel offset. This was apparent from inspection of the frequency spectra, which should converge at high frequencies across the water content levels for a given cation-smectite combination. For instance, Ca-hectorite showed offsets for the three highest water content levels. The predicted curves diverge from the measured curves at frequencies higher than the phonon frequency.



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  		Fig. 2. Measured and predicted electrical conductivity spectra at the four humidity levels for Ca-hectorite showing the offset for the highest three humidity levels.

 
Optimization of the comparison of the predicted and observed dc conductivity values led to the choice {theta}t = 0.07. This value appears to us to be reasonable. Consider that the water contents of any given layer range between 0 and 0.216 (except one value of 0.256). Thus we have a critical volume fraction on the order of 30% (i.e., 0.07/0.216). Critical thresholds for percolation in 2D tend to be near 50%, while critical thresholds for percolation in 3D can be as small as 2 to 3%, with a well-known early citation for continuum percolation (Scher and Zallen, 1970) giving 15% and bond percolation on a simple cubic lattice yielding 25%.

It is important to note that hydrated clay minerals are disordered. The usual approximation of a single "phonon" frequency (from the zero wavenumber limit of the phonon dispersion relations) is not valid, and there is a range of phonon frequencies. The appropriate means to calculate this theoretical spectrum is still controversial, so the important point is the caution on assuming a single value. Nevertheless, the analysis of the electrical conductivity is greatly simplified by the assumption that the external electric field couples with the charges through optical phonons with a given frequency. Thus by visual examination of the figures one can restrict the phonon frequency already to within the range 109 to 1010 Hz, rather than 1012 Hz (indicated by Sposito and Prost, 1982). From graphs of [–log({sigma}c/{nu}ph)]–1 vs. 1 – s for arbitrary values of {nu}ph, we optimized the value of R2 with respect to the choice of {nu}ph (Fig. 3 ). Although the optimization did not lead to an impressive value of R2 (only 0.268) there was a clear maximum at the value of {nu}ph = 2.22 x 109 Hz, which was within the range of values estimated visually. The correlation was significant, and both slope and intercept were significantly different from zero. Thus this value was chosen for the purpose of comparing the observed {sigma}dc with its predicted value.



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  		Fig. 3. Relation between the exponent s and the activation energy.

 
The mean contribution offset to dc electrical conductivity was 0.015 S m–1. This was also apparent from a sequential plot of the electrical conductivity at the phonon frequency, in which the last 18 values were offset 0.015 S m–1 above the line (not shown). These were generally the same as those indicated from the frequency plots; however, there were two of the 18 samples indicated from the frequency spectra plots but not the sequential phonon plots, and two indicated from the sequential plots but not the frequency plots. The term of {sigma}0 = 0.015 S m–1 was then added to the 18 samples identified through comparing the electrical conductivity at the phonon frequency with its value in the dc limit.

The comparison of Fig. 4 showed that, in contrast to the dc conductivity, which was proportional to {sigma}0ex-p[–Eac/kT], the ac conductivity at phonon frequencies was proportional to {sigma}0kT/Eac. The correlation was significant (i.e., the slope was significantly different from zero), but the intercept was not significantly different from zero, as expected. Thus only a fraction, approximately kT/Eac of the protons, actually contributed to the ac conductivity near the phonon frequencies, implying at most a weak dependence of the distribution of proton barrier heights with energy in the limit of zero barrier height.



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  		Fig. 4. The dc electrical conductivity as a function of the characteristic angular frequency, {omega}c.

 
The comparison of {sigma}dc with {sigma}c (Fig. 5 ) confirmed that the same processes control the ac conductivity and the dc conductivity. The 95% confidence intervals were 0.208 to 0.445 (for 0.3237) and 0.72 to 1.35 (for 1.04), and the correlation was significant. This result demonstrated a clear indication of the relevance of percolation theory to both the ac and the dc conduction (Hunt, 2001a). Note that this comparison (Fig. 5) was made only after subtracting 0.015 from the dc conductivity of those samples, which showed the parallel conducting mechanism that was frequency independent.



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  		Fig. 5. The electrical conductivity at the phonon frequency minus the dc electrical conductivity as a function of the dc electrical conductivity minus the offset.

 
The slope of the predicted (no prefactor) as a function of the measured dc electrical conductivity minus the offset was around 30.3 (Fig. 6a ), which would indicate a prefactor around 0.033. This prefactor would be close to that obtained by using L–2 in the pre-exponential of Eq. [7] with L = 15. Both intercept and slope were significantly different from zero. Comparison of the theoretical calculations of the electrical conductivity with experiment (Fig. 6b) demonstrated an r2 {approx} 0.54. Though 0.54 is not that much greater than a comparison of the dc conductivity with a simple linear function in the water content (r2{approx}0.47), such a simplified model would predict no ac conductivity at all (because of the assumed homogeneous nature of the water), and moreover had a large, and unphysical, negative intercept. The slope was closer to one and the intercept closer to zero for the percolation threshold case than for the mean water case. For the mean water case, both intercept and slope were significantly different from zero, but for the threshold case, only the slope was significantly different from zero. This again points to the superiority of the threshold case since the intercept is not expected to be different from zero. The regression was improved somewhat by omitting the zero values (Fig. 6c), and the slope was significantly different from zero, but the intercept was not significantly different from zero. The use of L2 in Eq. [7] would suggest a 3D component to charge transfer. Also including 1/(1 + w) did not improve the fit (not shown). These factors do not support the percolation threshold case. The proton transfer could have only a small 3D nature, following a path more like 1b than 1a. Nevertheless, uncertainties in the threshold value of 0.07 m3 m–3 and in the 0.015 S m–1 offset did not allow conclusive evidence for or against the percolation approach.



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  		Fig. 6. Predicted dc electrical conductivity as a function of measured: (a) predicted with no prefactor vs. measured minus offset (0.015 S m–1) forced through origin; (b) predicted percolation basal spacing with offset including samples with zero spacing (solid) or mean water content (dashed) as a function of measured; (c) predicted basal spacing not including samples with zero spacing.

 
The proportionality for Eq. [20] (Fig. 7 ) was only weak, and both intercepts and slopes were significantly different from zero. With some variation, any transport property is affected by a similar idea that transport is more difficult when protons approach the cations under diminishing water content. For example, neutral water molecules are attracted to the counterions, and the stronger this attraction is (with increasing proximity) the harder it will be to induce their motion past the obstacle. The range of predicted (and observed) values for the dc conductivity (one and a half orders of magnitude) is generally consistent with what is observed in other transport properties as well, lending support to the argument that Coulombic repulsion from the counterions is relevant to all transport properties of hydrated clay minerals. The magnitude of the prefactor associated with predicted activation energy was roughly in accord with experiment. This rough correspondence would not have been possible without independent verification from the ac conductivity that the value of the phonon frequency was <10 GHz rather than the usually assumed value more than three orders of magnitude larger. Such a small value of a vibrational frequency implied a relatively low stiffness of the medium, perhaps not surprising for hydrated clay minerals.



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  		Fig. 7. Relation between 4{pi}{varepsilon}0{varepsilon}phlwvph/[qr0(4 + w)] and {sigma}ph as described in Eq. [20].

 
We performed analyses to determine if there was a recognizable threshold in the external water content to produce a contribution, {sigma}0, in the electrical conductivity. If a set of criteria exists, by which one can predict in advance; which samples will have such a contribution, we were unable to discover it. There was certainly a tendency for these samples to have higher external water content, but this tendency was not pronounced, nor could we define consistently the conditions when it was violated. It is possible that the determination of the external water content, by subtraction of calculated internal water contents in given layers from the total water content, was not accurate enough to make such an analysis productive at this time. But we anticipate that this criterion should also depend on such parameters as counterion strength and the fractions of charge in the octahedral and tetrahedral layers, respectively, that is, mineralogy. For these reasons we expect that development of a predictive criterion is not feasible without considerably more detailed knowledge and higher experimental precision.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
Appendix
 REFERENCES
 
The results of the present study of the behavior of the frequency-dependent conductivity of hydrated clay minerals are:

These results support the idea that protons in the water associated with smectite clays generated the observed electrical conductivity.


   Appendix
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 Appendix
REFERENCES
 
List of Symbols

B
Barton–Nakajima–Namikawa coefficient (varies with material and temperature)

d
distance between proton and interlayer cation

e
protonic charge = 1.602 x 10–19 C

Eac
total increase in energy of proton when passing near interlayer cation, or activation energy due to Coulomb repulsion of countercations

k
Boltzman's constant = 1.38 x 10–23 J/°K

l
distance between charges on silicate sheet

lc
linear separation of critical (bottleneck) resistances on critical paths

L
linear separation of critical paths

nac
concentration of contributing charges

N
concentration of cation charges

qe
single charge on silicate sheet or of cation

r
distance between protonic charge and single charge on silicate sheet

r0
thickness of water layer (assume 0.25 nm)

Rc
critical resistance

s
power of frequency-dependence, < 1

t
minimum time as which charges follow paths defined by dc electrical conductivity

T
Temperature (assume 298°K)

tr
the transmission coefficient = a + b{theta}t

w
number of water layers in smectite interlayer

{varepsilon}w
relative dielectric constant of water

{varepsilon}0
permittivity of free space = 8.854 x 10–12 C N–1 m–2

{theta}w
volumetric water content

{theta}t
threshold water content for a basal spacing (0.07 m3 m–3)

{sigma}a
bulk electrical conductivity

{sigma}w
electrical conductivity in the liquid phase

{sigma}s
the electrical conductivity associated with the soil surfaces

{sigma}o
offset electrical conductivity

{sigma}dc
dc electrical conductivity

{tau}
relaxation time

vph
phonon frequency

{omega}
angular relaxation frequency

{omega}m
higher angular frequency with smaller activation energy

Received for publication February 7, 2005.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 Appendix
 REFERENCES
 





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