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Soil Science Society of America Journal 66:1430-1438 (2002)
© 2002 Soil Science Society of America

DIVISION S-1—SOIL PHYSICS

Effective Diffusion Coefficients of Soil Aggregates with Surface Skins

J. Maximilian Köhne*,a,c, Horst. H. Gerkeb and Sigrid Köhnea,c

a Texas A&M University, Dep. of Biological & Agricultural Engineering 201 Scoates Hall, 2117 TAMU, College Station, TX 77843-2117
b Centre for Agricultural Landscape and Land Use Research, Dep. of Soil Landscape Research, Eberswalder Str. 84, D-15374 Müncheberg, Germany
c 1912 Vinewood, Bryan, TX 77802

* Corresponding author (mkoehne{at}cora.tamu.edu)


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 CALCULATIONS
 DIFFUSION EXPERIMENT
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Dual-permeability model simulations and sensitivity studies indicate that preferential solute leaching in structured soil is intensified by a limited solute mass transfer between preferential flow paths and matrix. It is currently unknown how much aggregate skins may affect solute mass transfer. The objective of this study was to evaluate the effect of a skin layer on the effective diffusion coefficient, De, of water saturated soil aggregates. In a diffusion experiment, the concentration decrease of Br- and Cl- in a solution being in contact with soil aggregates was measured. Aggregates with intact and removed surface skins were studied. Two different theoretical approaches were compared: (i) fitting a modified analytical solution of Crank for diffusion out of a solution of limited volume into a plane sheet to obtain De for intact and for skin-removed aggregates; and (ii) calibrating a numerical solution for diffusion through a two-layer system to estimate De of the skin layer and the interior part, separately. The De-values of the skins differed for Br- and Cl-. Using the first approach, De was six (Br-) and 15 (Cl-) times smaller for the intact than for the skin-removed aggregates. The second approach resulted in effective diffusion coefficients of the skin layer being 30 (Br-) and 45 (Cl-) times smaller than those of the interior parts. Aggregate skins may reduce diffusive anion transfer between interaggregate region and soil matrix, an effect which may significantly influence preferential solute transport in structured soil.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
CALCULATIONS
 DIFFUSION EXPERIMENT
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
SOIL AGGREGATION leads to porous structures that may act as preferential flow paths. In structured soil, the liquid phase solution can be conceptually partitioned into two fractions, one located in the intra- (or matrix) and the other in the interaggregate (macropore or fracture) pore system. While the matrix porosity provides most of the soils' storage volume for water and solutes, the interaggregate porous network especially near saturation provides highly conductive pathways in which preferential flow may occur. The preferential displacement of water and dissolved ions is marked by physical nonequilibrium conditions as given by matric potential and solute concentration differences between the two flow regions. For instance, significant matric potential differences between intra- and interaggregate locations were measured, for example, using microtensiometers (e.g., Türk et al., 1991), and concentration differences were detected by using mini-solution sampling devices in an aggregated forest soil (Wilcke and Kaupenjohann, 1998). Local potential and concentration gradients between intra- and interaggregate regions initiate a water and solute mass transfer to restore equilibrium. Assuming typical, relatively small aggregate sizes (i.e., of at most some centimeters) as distances between flow paths, nonequilibrium between the transport regions of mobile-immobile and dual-permeability models can be maintained only if water and solute transfer is slow as compared with the transport within the regions, such that transport is preferential (e.g., van Genuchten and Wierenga, 1976; Rao et al., 1980; Jarvis et al., 1991; Gerke and van Genuchten, 1993a, 1993b; Gwo et al., 1996). The solute mass transfer may be advective (i.e., water flux bound) and diffusive (water flux independent). While for highly transient flow conditions, advection often dominates the overall mass transfer, for less transient to steady state flow conditions, diffusive mass transfer may dominate the equilibration of local solute concentrations (e.g., Gwo et al., 1996). A major difficulty involved in the practical application of dual-permeability models is the estimation of parameters of interregion mass transfer. Gerke and Köhne (2002) found that surface skins may substantially reduce the hydraulic conductivity of soil aggregates, thus limiting advective solute mass transfer. This study investigates the effect of aggregate surface skins on the solute-diffusion coefficient, thus on the diffusive part of mass transfer.

The first-order solute transfer terms in the models quoted above contain time-independent diffusive mass transfer coefficients that may be further subdivided at least into an effective diffusion coefficient, a diffusion path length factor, and a geometry factor (Gerke and van Genuchten, 1996). The diffusion path length, referred to as "the equivalent aggregate radius", could be derived from a mobile-immobile mass transfer term and an analytical solution of the diffusion equation to lump diffusion time scales and other variables like, for example, water content, flow rates, and adsorption (Cote et al., 2000).

When, among other parameters, the diffusive mass transfer coefficient was adjusted using the dual-permeability model MACRO (Jarvis et al., 1991) to describe Cl- concentrations in tile-drain outflow of tracer experiments, diffusion path lengths of 6 cm (Andreu et al., 1994) and time-dependent values between 6 cm and effectively infinite (2000 cm) (Villholt and Jensen, 1998) were calibrated. If in a mechanistic sense, the diffusion path length is regarded to be comparable with an effective equivalent aggregate half width, a value much larger than an observed average aggregate radius or half-distance between fractures is difficult to explain. On the other hand, an effective diffusion coefficient in the transfer term of 100-fold smaller than in the soil matrix was necessary to describe Br- transport in a tile-drained structured field soil using the dual-permeability model of Gerke and van Genuchten (1993a) (Köhne 1999). These results might be explained by two subsequent rate-limited processes, either like diffusion through a viscous water film on aggregate surfaces and diffusion into aggregates (e.g., Wallach and Parlange, 1998), or, what may be applicable to many aggregated soils, by the presence of a coating or dense surface layer covering the aggregates. Surface skins are often fine textured and the resulting small pores may reduce diffusion by anion exclusion (Nye and Staunton, 1994), or the layered structure of coatings may increase tortuosity (Olesen et al., 1996). Hence, it could be expected that diffusive mass transfer into the aggregates may be reduced by coatings on the matrix–fracture interface.

Clayey and silty coatings are a widespread phenomenon particularly in illuvial or argillic horizons of Alfisols and Ultisols. With respect to pedogenesis, mineralogical and physicochemical properties, coatings have been intensively studied (e.g., Cabrera-Martinez et al., 1989; Feijtel et al., 1989; Sanborn and Lavkulich, 1989; Habecker et al., 1990; Hiller et al., 1993; Ajmone-Marsan et al., 1994; Jongmans et al., 1994, 1998). For brevity, the expression skin is in the following used for all kinds of surface-altered layers on peds.

Few experimental studies have investigated effects of skins on solute mass transfer. Chen et al. (1997) inferred such clay skin effects from the disappearance of Sr and Br- from a solution percolating through a soil column with artificial macropores coated by clay. The clay surfaces reduced the transport rate into and out of soil aggregates (Chen et al., 1997). Clay films that coat pore surfaces in structured soil may affect ion diffusion and root growth (Gerber et al. 1974). Other studies focussed on diffusion into aggregates without skins. Aggregate-diffusion measurements have been performed with sieved aggregates (e.g., Pinner and Nye, 1982; Addiscott et al., 1983) where skins, if originally present, were mechanically destroyed. Addiscott (1982) detected an effect of anion exclusion on the diffusion of Br- into sieved silty-clayey aggregates. Rao et al. (1980) calculated the effective-diffusion coefficient for diffusion data of Cl- and tritiated water in porous spheres by using an analytical solution (Crank 1975) of Fick's second diffusion law. Addiscott (1982) numerically simulated the data of diffusion in porous spheres of Rao et al. (1980) and data of diffusion in cubes. Addiscott et al. (1983) used Addiscott's (1982) model to simulate anion diffusion in silty-clayey sieved-soil aggregates of different sizes assuming spherical geometry, to analyze the effect of the diffusion length on the impedance factor. In diffusion experiments with porous ceramic spheres (Al-Sibai et al., 1997) and porous silica beads (Lafolie et al., 1997), effective-diffusion coefficients were derived for use in the transfer term of the mobile-immobile convection-dispersion equation to simulate the respective column transport data. Furthermore, mass transfer parameters were obtained by numerical inversion of dual- permeability transport equations using data of steady-state solute displacement experiments (Schwartz et al., 2000). Because of the uncertainties involved in the parameter estimation using inverse methods, Schwartz et al. (2000) recommended an independent determination of diffusive mass transfer parameters using soil peds. Results could be analyzed and compared with estimates obtained by inverse methods.

The objectives of this study were to investigate if soil aggregate skins affect diffusion of conservative (i.e., assumed as nonreactive) salt anions into water saturated soil aggregates, to estimate the skins' effect on the overall diffusion coefficient of soil aggregates and to estimate the effective diffusion coefficient of the skins themselves. For comparison, diffusion was investigated for water saturated aggregates with intact skins and where the skins had been removed. Two different anions, Br- and Cl-, were used to detect possible deviations from ideal diffusion behavior.


   CALCULATIONS
TOP
 ABSTRACT
 INTRODUCTION
 CALCULATIONS
DIFFUSION EXPERIMENT
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Macroscopically, one-dimensional diffusion of a dilute conservative solute in a saturated porous medium can be described with Fick's second law (Eq. [1]) (e.g., Atkins, 1990)

[1]
where c (M L-3) is the concentration in the soil aggregates' solution at time t (T) in a distance x (L), and where the effective diffusion coefficient, De (L2 T-1), is defined as (Eq. [2])

[2]
with the molecular diffusion coefficient of a solute, D{ell} (L2 T-1), and the impedance factor F (-) (e.g., Collis-George and Geering, 1993). The impedance, F, is the ratio of molecular to effective diffusion coefficient (Eq. [2]) and depends on water content and tortuosity of diffusion paths. Here, F is optimized as the specific parameter quantifying the porous medium diffusion properties. For skin-removed aggregates, F is the impedance factor of the aggregate interior. For aggregates with skin, F is an effective impedance factor of the two-layer porous medium. In the numerical approach, Eq. [1] is solved for both layers with different values of impedance factor, F, for the aggregate interior and Fskin for the skin layer. The parameter De in Eq. [1] encompasses porous medium, chemical, and solvent properties; De is derived from Eq. [2] using results of F. Here, values of D{ell} = 1.797 cm2 d-1 for Br- and D{ell} = 1.754 cm2 d-1 for Cl- were used (Atkins, 1990).

Analytical Fitting of F
Crank (1975) described the analytical solution of Eq. [1] for one-dimensional diffusion out of a well-stirred solution of limited normalized volume v (L) that occupies the space 0 <= x <= v into an infinite porous medium of plane sheet geometry with normalized volume l (L). The plane sheet is at x = v in contact with the solution and occupies the space v <= x <= v + l (Fig. 1) . This analytical solution (Crank, 1975) was here modified for the description of diffusion into a number of water saturated soil aggregates of average thickness l. The free solution depth v was calculated as solution volume V (L3) normalized with the contact area, A (L2), between solution and aggregates, such that v = V/A.



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  		Fig. 1. Geometry of the imaginary diffusion experiment of Crank (1975).

 
As initial condition for this diffusion problem it was assumed that all solute was in the free solution (Eq. [3a], i.e., aggregates were solute-free (Eq. [3b]).


[3a, 3b]
where c is the concentration (M L-3) of the solute in the well-stirred solution at time t, with the initial value of c0, and ca0 is the zero initial aggregate (a) concentration. The mass conservation law states that the solute mass leaving the solution is identical to that entering the soil at x = v, which leads to the boundary condition defining the diffusive flux through the soil–solution interface (Eq. [4a]). Assuming that no solute left the other side of the aggregates, a zero gradient boundary condition was used at x = v + l (Eq. [4b]).


[4a, 4b]
with the aggregates' saturated water content, {theta}s, [L3 L-3]. Here, the solute concentration at the surface of the aggregates (at x = v) was assumed to be the same as in the solution (i.e., c[t] = ca[t,v]).

The solution of Crank (1975) describing the mass of solute that diffused in the sheet at time t as a fraction of the mass in the sheet at infinite time (i.e., at concentration equilibrium) was here expressed as the mass, M (M), of solute in the free solution at time t > 0 normalized with its corresponding initial value, M0, at time t = 0. Moreover, the effect of solute sampling on the shift of the equilibrium mass distribution between aggregates and free solution and, hence, on the diffusion rate was considered. In modified form, the analytical solution of Crank (1975) is then (see appendix for the derivation):

[5]
where k (-) is the ratio of the aggregates' liquid volume to free solution volume at time t, k = l{theta}sv-1, and {alpha}i are the nonzero positive solutions of tan({alpha}i) = -{alpha}ik-1, Msamp is the cumulative mass at time t > 0 removed from the free solution by sampling. The subscript "end" in Msampend and kend defines the respective quantity at the end of the experiment. The free solution normalized volume v diminished slightly from the initial value, v0, owing to solute sampling—therefore v and k were used as step functions of time. In the summation term of Eq. [5], only the first five terms were considered, which gave accurate results even for early times. The theoretical equilibrium mass distribution in the free solution is

[6]

The matrix diffusion impedance, F, was estimated using a nonlinear optimization scheme (Press et al., 1986) by minimizing the sum of squared differences between Eq. [5] and the normalized measured Br- and Cl- mass in free solution (M/M0) being in contact with intact or skin-removed surfaces of aggregates.

Numerical Calibration of F andFskin
For diffusion through a laminate comprising two layers, analytical solutions exist (e.g., Barrer, 1968), however, they are mathematically cumbersome and contain a multitude of parameters (Crank, 1975). Here instead, the diffusion impedance, Fskin, in the skin layer was calibrated using a fully implicit numerical solution of Eq. [1] utilizing the finite element code HYDRUS (Vogel et al., 1996).

The initial concentration in the aggregates was zero for Cl- (compare Eq. [3b]). For Cl-, the initial concentration was equal to the measured Cl- concentration prior to the experiment. A first-type boundary condition was used at x = v assuming the transient solute concentration to be continuous across the soil–solution interface, that is, c(t) = ca(t,v) at x = v. The first type boundary condition of a simulation was derived from the respective measured concentrations in the free solution, values between data points were linearly interpolated. The boundary condition at x = v + l is of second-type and represents a zero solute flux (compare Eq. [4b]). Using HYDRUS, the cumulative solute mass uptake of the aggregates, Ma, was calculated for different times as

[7]
where the initial mass term, {theta}slc0, was always zero for Br-. As for the analytical fitting approach, here also the effect of sampling on the equilibrium mass distribution was taken into account such that the fitted diffusion impedance was not biased by the sampling effect: the normalized mass in the free solution, M/M0, was derived from the mass balance (Eq. [8])

[8]

The mass balance (Eq. [8]) states that the solute mass in the solution, M, is at any time t equal to the initial (total) mass M0 minus the mass Ma diffused into the aggregates minus the mass Msamp removed by solute sampling.

The numerical approach included two steps. First, the diffusion of Br- and Cl- into skin-removed aggregates was calibrated to yield the diffusion impedance of the aggregate sample interior. Second, Fskin of the skin was calibrated assuming a two-layer-profile using the data of Br- and Cl- diffusion into intact aggregates. Here, F of the interior (upper) layer was fixed at the value found for the skin-removed aggregate samples.

To compare the goodness-of-fit between the analytical fitting and the numerical calibration approaches, the normalized residual sum of squares, r2, was calculated as

[9]
where the yi represent the observations with an arithmetic mean of , and yi denote the model calculations.


   DIFFUSION EXPERIMENT
TOP
 ABSTRACT
 INTRODUCTION
 CALCULATIONS
 DIFFUSION EXPERIMENT
RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Aggregate Sample Preparation
Samples were taken from the tile-drained agricultural field-site Bokhorst near the city of Kiel, Northern Germany, where preferential flow had been investigated based on conservative and reactive tracer experiments (Wichtmann et al., 1998; Lennartz et al., 1999). The soil was heterogeneous according to its glacial till origin, the subsoil varying at places within single meter distances between clay-loam with angular aggregates and rather structureless silty sand (Lennartz et al., 1999). Soil types were ranging from Stagnic Calcaric Regosols to Stagnic Luvisols according to the Food and Agricultural Organization classification scheme (i.e., Calcaric or Eutric Epiaquent and Argillic Epiaqualf of the U.S. system).

A soil block of 10 dm3 was taken at 35- to 50-cm depth from the subsoil horizon of the Calcaric Regosol. From this field-moist soil block, 42 stable angular aggregates of about 20-mm edge length were carefully separated by hand. These primary clods were cut into cubic slabs with two different modes of preparation for the flat side that was a matrix–fracture interface in the soil. For 21 of the soil aggregates, this former matrix–fracture interface was left intact. For the second group of 21 aggregates, the skin layer was removed from the surface. A layer of approximately 0.5 mm was carefully separated by hand using a scalpel. The procedure and principal of sample preparation is exlained in greater detail in Gerke and Köhne (2002). The average thickness of about 0.5 mm of the slices was determined from microscopic observations. As clay skin layers have been reported to be thinner (e.g., Sullivan and Koppi, 1991), presumably all parts of the clayey skins were completely removed. A visual inspection using a scanning electron microscope showed that sliding and pressing of the scalpel blade created local patterns of rupture cracks alternating with more smeared areas. The intact surfaces showed a layered and rather smooth surface, aside from very few cylindrical pores and planar cracks. After preparation, the two groups of 21 aggregates each had about the same mean thickness of 7 to 8 mm (Table 1).


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  		Table 1. Selected properties of two soil aggregate samples and of the diffusion experiment: A, contact area between the respective sample and the solution; l, mean aggregate thickness of a sample; {theta}s, saturated water content; {rho}b, bulk density; , time-averaged normalized solution volume; , time-averaged volume ratio of aggregate solution to free solution; mean, arithmetic average; SD, standard deviation; co, initial concentration (Br-, Cl-).

 
Soil texture and organic C content was determined from 50 g of the aggregates' interior and the skin, respectively, for other aggregates separated from the same soil block (see Table 1 in Gerke and Köhne, 2002). The results indicate a higher organic C content and a slightly higher fine silt and clay content of the skin compared with the matrix. Prior to the diffusion experiment, the soil aggregates were capillary saturated with distilled water by putting them on a cheese-cloth placed on a water saturated silt bed for 12 h.

Experimental Procedure
Anion diffusion was measured at constant room-temperature of 20°C using a simple vessel containing 295 cm3 of distilled water (Fig. 2) . With respect to the experimental setup, the aggregates' skins and the skin-removed sides, respectively, are denoted lower sides, which are in contact with the free solution (x = v); the upper sides are not in contact with the solution (x = v + l) (see Fig. 1 and 2). A mesh with a sample of 21 water-saturated aggregates was placed into the diffusion vessel such that the lower sides of the aggregates were just in contact with the water. The vessel was closed with a lid and gentle stirring was commenced by means of a magnetic stirrer. After 15 h of chemical equilibration between distilled water and ionic compounds possibly present in the soil aggregates' solution, a sample was taken from the water to determine the background concentration. Immediately after sampling, a 5-mL solution with a concentration of 1000 mg L-1 Br- (KBr) and 1000 mg L-1 Cl- (NaCl) were applied within 10 s by applying portions of 1 mL with a pipette through each of the five holes in the lid, which were then closed again (Fig. 2). The initial concentration in the free solution was theoretically c0 = 5 mg/300 mL = 16.7 mg L-1, practically it was 15.6 mg L-1 for Br- and 17.9 mg L-1 for Cl- for intact aggregates and 16.6 mg L-1 for Br- and 19.5 mg L-1 for Cl- for skin-removed aggregates (Table 1). The aggregates originally contained some Cl- that diffused into the water resulting in a Cl- concentration of 3 mg L-1 which gave a slightly increased initial Cl- concentration of the solution. The time of application defined the start of diffusion at t = 0. To monitor the concentration c during the experiment, samples of 0.5 mL of solution were taken from the center at initially relatively close and later increasing time intervals until the end at about 13 h. During the first 15 min after application, an additional sample was taken from a side of the vessel to verify that a homogeneous concentration distribution in the free solution was realized. Bromide and Cl- concentrations were determined by using an ion-chromatography system with a conductivity detector (GAT Wescan, Kontron Instruments, Exchingen, Germany). The solid phase was a 100-mm-long cartridge (Methrom, Super-Sep 6.1009.000, Deutsche Methrom GmbH & Co. KG, Filderstadt, Germany) together with a 20-mm guard-cartridge (Bischoff chromatography, part No. 6302137, Metrohm AG, Herisau, Switzerland) placed in an oven at 40°C. A 2.5 mM phtalic acid eluant (pH 4.2) at a flow rate of 1.5 mL min-1 served as the mobile phase. The eluant was degassed via an automated degasser (Degasys) when pumped from its reservoir to the cartridge. A sample volume of 100 µL was injected via an automatic sampler and the detection limit was 0.2 mg L-1 for Br- and Cl-, respectively. Standard solutions of 0.5 to 25 mg L-1 were used for calibration. Concentrations were calculated from integrated peaks. Aggregate heights, volumes, and densities were determined as described in Gerke and Köhne (2002). Table 1 shows selected physical properties of the aggregates together with initial concentrations, time-averaged normalized solution volume, = T-1{int}vdt, and time-averaged volume ratio of free solution to aggregate solution, = T-1{int}kdt.



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  		Fig. 2. Schematic drawing (top view and cross-section view) of the apparatus for measurement of salt tracer diffusion out of a well-stirred solution into water-saturated soil aggregates. The numbers beside the plugs indicate the sequence of tracer application through the respective holes in the lid.

 

   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 CALCULATIONS
 DIFFUSION EXPERIMENT
 RESULTS
DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Data of Br- and Cl- are shown in Fig. 3 as mass fractions of their initial mass in the free solution, M/M0, over time. In the solution in contact with intact aggregates M/M0 decreased more slowly than in the solution with skin-removed aggregates.



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  		Fig. 3. Normalized anion mass in the solution during diffusion into 21 soil aggregates without (removed) and with skin (intact), and results of fitting F in an analytical diffusion equation (Eq. [5]) to data of aggregates without (dashed line) and with (solid line) surface. (a) Br-, (b) Cl-.

 
The Analytical Fitting of F.

The analytical fitting of F (Eq. [5]) yielded a close approximation of data for intact and skin-removed aggregates (Fig. 3) as was indicated by a coefficient of determination of r2 {approx} 0.99 (Table 2). The theoretical equilibrium (Eq. [6]) between aggregates and free solution was neither attained for the intact nor for the skin-removed aggregates. However, at the end of the experiment, Mend/M0 of the free solution in contact with skin-removed aggregates was 0.862 (0.871) for Br- (Cl-) and thus closer to the theoretical equilibrium than respective data for the intact aggregates which was 0.913 (0.926) for Br- (Cl-) . For diffusion into a soil sample at constant water content, F is a property of the porous medium and should theoretically be equal for Br- and Cl-. This was found for the skin-removed sample, where F = 0.48 (Table 2). For intact aggregates, the fitted impedance values were much smaller. For Cl- (F = 0.0324), the impedance was only half as large as for Br-, F = 0.0764 (Table 2). For skin-removed aggregates F was found to be six (Br-) to 15 times (Cl-) as large as an effective F for aggregates with surface layer. The effective diffusion coefficients, De, calculated with Eq. [2], show the same (Table 2).


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  		Table 2. Parameters obtained by fitting different models to measured data. F, impedance factor; r2, coefficient of determination; numerical, numerical solution using HYDRUS.

 
The Numerical Calibration of F and Fskin.

The numerical calibration of F and Fskin using HYDRUS (Vogel et al., 1996) was applied to data of skin-removed and intact aggregates. For skin-removed aggregates, the observed Br-– and Cl-–data could be numerically predicted (Fig. 4) assuming F = 0.48, which was the same result of fitting Eq. [5]. Numerical simulation and the fitted analytical solution (Eq. [5]) gave almost the same result (Fig. 4). Consequently, no further calibration was attempted. Additionally, Fig. 4 shows a numerical calculation with F = 0.48 not corrected for solute mass loss by sampling; the resulting curve deviates markedly from the data.



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  		Fig. 4. Temporal change of normalized anion mass M/M0 in the stirred free solution versus time during the diffusion experiment with surface skins-removed aggregates (data), numerical forward simulation (num.-samp) compared with analytical fitting (Eq. [5]) results, and numerical forward simulation not corrected (num.) for sampling, (a) Br-, (b) Cl-.

 
Next, Fskin was calibrated using HYDRUS and assuming a two-layer system consisting of an upper layer (interior) and a lower layer (skin). The thickness of the skin was fixed at a value of L = 0.5 mm given by the experimental procedure of removing the surface. Only the diffusion impedance of the skin was calibrated, whereas for the interior the value of F = 0.48 as obtained for the skin-removed aggregates was used as an input parameter. The numerical calibration yielded a value of Fskin = 0.017 for Br- and Fskin = 0.011 for Cl- (Table 2). The good agreement between data and numerical simulation as shown in Fig. 5 is indicated by r2 > 0.99 (Table 2). A ratio F/Fskin = 45 for Cl- and of F/Fskin = 30 for Br- could be deduced.



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  		Fig. 5. Temporal change of normalized anion mass M/M0 in the free solution during the diffusion experiment with intact aggregates (data), numerical calibration corrected (num.-samp) and not corrected (num.) for sampling, respectively, (a) Br-, (b) Cl-.

 
The sampling effect is relatively larger for the slower diffusion into the intact aggregates, such that the early decrease of M/M0 for intact aggregates is mainly caused by withdrawing solute mass by sampling (Fig. 5). First, in the early stage the concentration in the solution is highest such that more solute is removed per sample than later, second, in the beginning two samples were taken during each sampling, and third, sampling was initially more frequent. The cumulative effect of sampling during the experiment duration can be estimated with the help of Table 3, where the mass components removed from the solution by diffusion and sampling are listed. For the skin-removed aggregates, 22% of the mass that left the solution was withdrawn by sampling. For intact aggregates the sampled mass fraction amounted to 42 to 52% (Table 3, last column).


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  		Table 3. Anion mass distribution at the end of the diffusion experiment. Ma + Msampend - total mass removed from the solution by diffusion (Ma) and sampling (Msampend), - fraction of sampled mass to total mass withdrawn from the solution.

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 CALCULATIONS
 DIFFUSION EXPERIMENT
 RESULTS
 DISCUSSION
CONCLUSIONS
 REFERENCES
 
Applied to measurements of diffusion into water-saturated soil aggregates, the analytical fitting approach based on Crank (1975) yielded a six- to 15-fold higher F for skin-removed than for intact aggregates. A comparable ratio was obtained for the effective-hydraulic conductivities at a pressure head of -3 cm for aggregates with and without skin (Gerke and Köhne, 2002). When the skin was regarded as an explicit layer with individual diffusion properties, as it was done in the numerical calibration approach assuming a two-layer porous medium, the ratio F/Fskin was 30 to 45, that is, several times larger. However, the value for Fskin depends on the thickness of the skin layer that was here fixed at 0.5 mm. For a thinner skin layer the ratio F/Fskin would increase. Sullivan and Koppi (1991), for instance, reported values of the skin layer thickness for a desert loam of 0.2 mm for clay and 0.3 mm for silt coatings, while composite coatings had a thickness of up to 1 mm.

Comparable reductions in the interior/skin ratio were estimated for hydraulic conductivities of aggregates close to water saturation (Gerke and Köhne, 2002). These congruent results for diffusion impedance and hydraulic resistance might reflect that hydraulic resistance or hydraulic conductivity and effective-diffusion coefficient (of soil aggregates) are related via the same tortuous pathways for water flow and ion diffusion. The tortuosity coefficient in a permeability model by Millington and Quirk (1961) was used by other authors to calculate the effective diffusion coefficient (e.g., Vogel et al., 1996).

The smaller the volume ratio between free solution and aggregate moisture, the higher will be the accuracy of the experimental diffusion method, since the measured concentration differences caused by diffusion out of the free solution will be accordingly larger. On the other hand, a lower volume of free solution leads to a relatively larger mass removal by sampling, which leads to a less sensitive diffusion measurement even if mass removal is considered in the calculation of diffusion parameters. As a compromise, the volume ratio between free stirred solution to aggregate moisture was here about 7:1. Although the accuracy of this experimental diffusion device may be limited to some extent, the results of this study provide a first data set of average diffusion impedance or diffusion coefficient values for a group of soil aggregates. This could be deduced from the fact that the experimental data could be described with diffusion models, giving parameters for F of nonsorbing anions for the skin-removed aggregates comparable with values reported for water-saturated soil (e.g., Pinner and Nye, 1982; Darrah, 1991; Collis-George and Geering, 1993; Al-Sibai et al., 1997).

The fact that for the intact aggregates, F was larger for Br- than for Cl-, could be possibly explained by reactive behaviour of Br-. Both Br- and Cl- have been observed to be affected by anion exclusion (e.g., Addiscott, 1982; Melamed et al., 1994; Thomas and Swoboda, 1970). Adsorption of Br-, but not of Cl- to ferrihydrite has been reported at pH values below the zero charge point of ferrihydrite (e.g., Brooks et al., 1998). However, Wang and Yu (1989) found adsorption of Cl- to Fe2O3. Furthermore, the identity of the balancing cation may affect the anion-diffusion coefficient (e.g., Seaman et al., 1995). We have no explicit information about the reactive behavior of Br- and Cl- in the soil under study, however, there might be a different reactivity of the two anions because of special constituents of the aggregate skins.

The results of this paper are limited to water saturated aggregates. The diffusion rate declines if the water content of the soil decreases (e.g., Pinner and Nye, 1982; So and Nye, 1989), however, the effect of desaturation is complicated. All the more, a combined effect of fracture coatings and variable water saturation on diffusion is an issue that still needs to be addressed.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 CALCULATIONS
 DIFFUSION EXPERIMENT
 RESULTS
 DISCUSSION
 CONCLUSIONS
REFERENCES
 
An experimental diffusion device was developed for the determination of the diffusion impedance or the effective diffusion coefficient of a group of soil aggregates. Comparing the experimentally measured diffusive Cl- and Br- uptake of intact and skin-removed soil aggregates demonstrated that surface skins may reduce the effective anion-diffusion coefficient of soil aggregates. The method adapted from Crank (1975) gave overall impedance factors for intact aggregates that implicitly accounted for the effect of a skin. The numerical method allowed the calibration of an impedance factor for the skin as an explicit layer with individual diffusion properties. The effect of removing solute samples on the measured concentrations had to be considered for the calculation of diffusion parameters. The methods presented here may help to estimate parameters for use in solute mass transfer terms describing the solute exchange between preferential flow region and soil matrix in dual-permeability models. The determination of the diffusion impedance of aggregates, as shown here, can be applied to stable, water saturated aggregates. Only for the skin but not for the aggregates interior, Br- and Cl- had different effective-diffusion coefficients which could not be explained. Therefore further studies may address the diffusion properties of aggregates and skins for different ions, and also for unsaturated conditions.

APPENDIX

The analytical solution (Eq. [5]) for conservative solute diffusion out of a solution of limited volume from which samples are taken into a group of aggregates is derived based on the analytical solution given by Crank (1975) (in the notation of this paper):

[A.1]
where Ma (Ma{alpha}) (M) is the mass of solute in the aggregates' (Crank: plane sheet's) solution at time t > 0 (t = {infty}).

The following mass balance is required

[A.2a]
where c{infty} is the final solution equilibrium concentration. The mass balance Eq. [A.2a] relates the initial mass in the solution, M0 = v0c0, to the solute mass finally distributed between aggregates, Ma{infty} = {theta}lc{infty}, and solution, vendc{infty}, and the mass Msampend taken in all n solute samples, Msampend = {sum}ni=1vsampici, where vsampi is the normalized volume of a sample i with concentration ci. A shorter notation of Eq. [A.2a] in terms of masses is then

[A.2b]

The Eq. [A.2a] can be rearranged to solve it for c{infty}, from which follows that the final solute mass in the aggregates can be expressed as

[A.3]
which, expanded with ({theta}l)-1, gives

[A.4]

The mass, Ma, diffused into the aggregates at time t > 0, is

[A.5]
where Msamp = {sum}ji=1vsampici is the solute mass in j (j <= n) removed samples. When Ma{infty} and Ma in Eq. [A.1] are substituted by the expressions given by Eq. [A.4] and [A.5], the Eq. [A.6] results

[A.6]

Rearranging Eq. [A.6] yields the solution Eq. [5].


[5]


   ACKNOWLEDGMENTS
 
This study was financially supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn, (contract Wi 671/9-1). We thank Prof. Dr. h.c.P. Widmoser and Prof. Dr. B. Lennartz for helpful support during the project.

Received for publication July 9, 2001.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 CALCULATIONS
 DIFFUSION EXPERIMENT
 RESULTS
 DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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