Soil Science Society of America Journal 66:1218-1224 (2002)
© 2002 Soil Science Society of America
DIVISION S-2—"PARTICLE INTERACTIONS IN COLLOIDAL SYSTEMS"
A pH-Dependence Implicit Formulation of Cation- and Anion-exchange Capacities of Variable-charge Soils
Hidetaka Katou*
Water Quality and Solute Dynamics Group, National Institute for Agro-Environmental Sciences, Kannondai 3-1-3, Tsukuba, Ibaraki, 305-8604 Japan
* Corresponding author (katouh{at}niaes.affrc.go.jp)
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ABSTRACT
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Cation- and anion-exchange capacities (CEC and AEC) of variable-charge soils are often expressed as a function of pH and the electrolyte concentration. A difficulty following this formulation is that changes in pH with time and space must be computed precisely if ion transport is to be predicted. In this paper, an alternative formulation of CEC and AEC is presented in which pH does not appear explicitly. The formulation is based on the notion that, unless H+ or OH- is added, changes in the total cation adsorption, Qcat (= CEC), and the total anion adsorption, Qan (= AEC), in response to a change in the total electrolyte concentration, C, are nearly equivalent. This means that the net surface charge remains virtually constant and leads to an equation that relates d[H+]/dC to partial derivatives of Qcat and Qan with respect to [H+] and C. Using this relation, we show that, if the exchange capacities are expressed by Qcat = kcat[H+]-acatCbcat and Qan = kan[H+]-aanCban, the change in Qcat (or Qan) with C in a soil saturated with monovalent ions is described by an ordinary differential equation which does not contain pH as a variable. The Qcat–C relations predicted by solving the equation closely agreed with those by the conventional method in which pH-dependent adsorption, mass balance, and electroneutrality equations were solved simultaneously. The advantage of this approach is that once the adsorption parameters are known, the transport equation for H+ need not be solved in predicting transport of electrolyte ions in soil.
Abbreviations: AEC, anion-exchange capacities • C, total electrolyte concentration • CEC, cation-exchange capacity • Qan (= AEC), total anion adsorption • Qcat (= CEC), total cation adsorption
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INTRODUCTION
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VARIABLE-CHARGE SOILS have surface electrical charge which varies in response to changes in the ionic composition of the aqueous solution phase (Parfitt, 1980; Bolan et al., 1999). In soils containing considerable amounts of allophane, imogolite, or other oxide and hydroxide minerals, the net surface charge may be positive, negative, or zero with the cation and anion adsorption dependent on pH and the ionic strength of the bulk solution (van Raij and Peech, 1972; Ilton Morais et al., 1976; Okamura and Wada, 1983; Charlet and Sposito, 1989). For predicting the velocity of electrolyte ions moving through these soils, the effects of adsorption by the variable-charge surface need to be properly taken into account (Wong et al., 1990; Bellini et al., 1996; Katou et al., 1996, 2001).
The CEC and AEC of variable-charge soils, assumed by several workers (Wong et al., 1990; Bellini et al., 1996) to be linearly related to the retardation of electrolyte ions relative to water, have often been expressed as a function of pH and the electrolyte concentration. For example, Okamura and Wada (1983) found that CEC and AEC of the Andisols and Ultisols they measured using NH+4 and Cl- as index ions were described by regression equations of the type
 | [1] |
 | [2] |
where C is the ion concentration in the aqueous solution phase, and a, b, c, a', b', and c' are empirical constants. Ishiguro et al. (1992) conducted steady state CaCl2–SrBr2 miscible-displacement experiments in an Andisol at constant pH and electrolyte concentrations, and observed that the delays in Sr2+ and Br- breakthrough were consistent with the CEC and AEC of the soil as determined at the same pH and electrolyte concentration.
The condition of constant pH and electrolyte concentration is, however, rarely met in the field. A more realistic situation is that the electrolyte concentration is variable, and pH is uncontrolled. Unless pH is at the point of zero salt effect (PZSE), pH of a variable-charge soil changes in response to a change in the ionic strength of the bulk solution (Sposito, 1984, p. 81–88). Under such conditions, prediction of changes in CEC and AEC requires that concurrent change of pH be predicted as well. For a homoionic soil, changes in the pH and cation and anion adsorptions caused by addition or removal of a neutral salt may be predicted by solving simultaneously the following adsorption, mass balance, and the electroneutrality equations (Wada, 1985):
 | [3] |
 | [4] |
 | [5] |
 | [6] |
 | [7] |
where Qcat and Qan are the amounts of exchangeable cation and anion adsorbed per unit mass of soil (molc kg-1) and equal to CEC and AEC, respectively. Here, Ccat and Can are the electrolyte cation and anion concentrations in the liquid phase (molc m-3), Mcat and Man are the cation and anion contents per unit mass of soil (molc kg-1), 
is the volumetric water content (m3 m-3), 
is the bulk density of soil (kg m-3), and f and g are functions of some arbitrary form.
A difficulty encountered when this approach is taken in predicting the transport of electrolyte ions under varying ionic strength conditions, is that changes in pH with time and distance must be computed precisely. This is because adsorption of electrolyte ions by a variable-charge soil is very sensitive to a small change in the H+ concentration, and requires that the transport equation for H+ be solved in addition to those for the electrolyte ions. Moreover, calculation of dQcat/dCcat, dQan/dCan, and hence the retardation factor R [= 1 + (/)dQi/dCi for the ionic species i] (van Genuchten and Cleary, 1982), is not straightforward and results in much computational work.
In this paper, an alternative formulation of CEC and AEC is presented in which pH does not appear explicitly. The formulation is based on the notion that, unless H+ or OH- is added, the net surface charge of soil, qH, remains essentially constant. This enables the change in the H+ concentration in response to a change in the electrolyte concentration, C, to be related to partial derivatives of CEC and AEC with respect to [H+] and C. Using this relation, we show that if the exchange capacities are expressed by Eq. [1] and [2], the change in CEC (or AEC) with C in a homoionic soil is described by an ordinary differential equation which does not contain pH as a variable. Changes in CEC upon addition of a neutral salt to soil are predicted by solving this equation. These predictions are then compared with those obtained by the conventional method in which CEC and AEC are expressed as an explicit function of pH and the electrolyte concentration. The advantage of the pH-dependence implicit formulation is that once the empirical adsorption parameters are known, the transport equation for H+ need not be solved in predicting the transport of electrolyte ions in soil.
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THEORY
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Suppose a neutral salt is added to or removed from a variable-charge soil saturated with the monovalent cation and anion species making up the salt. Changes in the cation and anion adsorption in response to the change in the electrolyte concentration are given by
 | [8] |
and
 | [9] |
The second term of the right-hand side of Eq. [8] and [9] represents the change in the ion adsorption due to the pH change induced by the addition or removal of the neutral salt. Here we assume that, unless H+ or OH- is added to the soil, changes in the cation adsorption and the anion adsorption are nearly equivalent so that
 | [10] |
This assumption is based on the notion that any increase of Qcat in excess of the increase of Qan, for example, tends to increase H+ concentration in the solution, but the released H+ will be almost entirely consumed in the protonation of possible anion adsorption sites such that the net surface charge, qH (= Qan - Qcat), remains essentially constant. Wada (1984) postulated such a stoichiometric transfer of H+ from the surface SiOH group to the surface AlOH group as a mechanism of the apparent salt absorption observed in Andisols.
Substituting Eq. [8] and [9] into Eq. [10] and assuming Ccat >> [H+] and Can >> [OH-], such that Ccat 
Can C, we obtain the following equation as a close approximation to describe the change in the H+ concentration in response to a change in the electrolyte concentration:
 | [11] |
When the dependence of CEC and AEC on pH and the electrolyte concentration is described by Eq. [1] and [2], respectively, we may write
 | [12] |
 | [13] |
where kcat, acat, bcat, aan, ban, and kan are empirical constants, and the activity coefficient for H+ is assumed to be unity. Partial derivatives appearing in Eq. [11] are then given by
 | [14] |
 | [15] |
 | [16] |
 | [17] |
Using Eq. [14] through [17], and substituting Eq. [11] into Eq. [8] and [9] yields, respectively,
 | [18] |
and
 | [19] |
where the relation qH = Qan – Qcat has been used. Equations [18] and [19] are identical, and describe the changes in Qcat (= CEC) and Qan (= AEC) with the electrolyte concentration when neither H+ nor OH- is added to the soil. Note that qH has been taken as constant and that pH does not appear explicitly in these equations. By solving Eq. [18] and [19] subject to an appropriate initial condition, Qcat and Qan may be obtained as a function of C without knowing the concurrent change in pH.
Illustrative Examples
Adsorption isotherms pertinent to the adsorption of electrolyte ions, from a neutral salt solution moving through a variable-charge soil, should describe the Qcat - Ccat and Qan - Can relations when the solution pH varies in response to changes in the electrolyte concentration. We predict cation adsorption isotherms following addition of a neutral salt by solving Eq. [18] for soils having contrasting surface charge characteristics and at different initial pH. The predicted cation adsorption isotherms are compared with those obtained by simultaneously solving Eq. [5] through [7], [12], and [13] in which CEC and AEC are expressed as an explicit function of pH and the electrolyte concentration. Two example soils are taken from published data of Okamura and Wada (1983) who measured CEC and AEC at different pH and NH4Cl concentrations after saturating the soils with NH+4 and Cl-.
Soil Predominated by Allophane and Imogolite
The "905" soil studied by Okamura and Wada (1983) was taken from the B horizon of an Andisol and contained allophane and imogolite as the predominant clay minerals. The soil had a total C content of 13 g kg-1 and a clay content of 0.14 to 0.44 kg kg-1. The Si/Al molar ratio of the clay minerals was 0.5. The net surface charge was positive or negative, depending on pH of the equilibrated solution. They reported the coefficients in Eq. [1] and [2] to be a = 0.307, b = 0.253, c = -0.717, a' = -0.204, b' = 0.195, and c' = 2.36 when CEC and AEC were expressed in cmolc kg-1 and C in molc L-1. For use in Eq. [12] and [13], we find acat = 0.307, bcat = 0.253, kcat = 103a-3b+c-2 = 0.00279, aan = -0.204, ban = 0.195, and kan = 103a'-3b'+c'-2 = 0.146. Solid lines in Fig. 1
represent CEC (= Qcat) and AEC (= Qan) of the soil as a function of pH, as calculated from Eq. [12] and [13] for different electrolyte concentrations.
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Fig. 1. Changes in pH, cation adsorption, Qcat, and anion adsorption, Qan, upon addition of a neutral salt to "905" soil, obtained by solving Eq. [5] through [7], [12], and [13] simultaneously. The soil was initially at Ccat = 0.1 molc m-3 with initial pH = 4, 5, 6, 7, or 8. Solid lines represent Qcat (= CEC) and Qan (= AEC) as a function of pH when the electrolyte concentration, C, is kept constant.
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Symbols in Fig. 1 show the changes in pH, the cation adsorption, and the anion adsorption upon addition of a neutral salt to the "905" soil, as predicted by solving the pH-dependence explicit equations (Eq. [5]–[7], [12], and [13]) simultaneously. In the calculation, the soil was assumed to be initially at Ccat = 0.1 molc m-3 with initial pH of 4, 5, 6, 7, or 8. The volumetric water content and the bulk density were arbitrarily taken as = 0.6 m3 m-3 and = 800 kg m-3, respectively. For simplicity, the soil was assumed to be saturated with the monovalent cation and anion species making up the salt, so that ion exchange need not be considered. The values of Can, Qcat, Qan, Mcat, and Man prior to the salt addition were calculated from Eq. [5] through [7], [12], and [13], taking [H+][OH-] = 10-8 mol2c m-6. Table 1 lists the initial conditions thus obtained. To simulate the addition of neutral salt, Mcat and Man were both increased by 0.0025 molc kg-1 and the values of Ccat, Can, Qcat, Qan, and [H+] after reequilibration were computed by solving Eq. [5]–[7], [12], and [13] simultaneously. Then, Mcat and Man were further increased by 0.0025 molc kg-1 and the calculation repeated until the cumulative amount of the salt added (i.e., the increase in Mcat and Man from the initial values) reached 0.1 molc kg-1. We see that as the electrolyte concentration increases as a result of salt addition, not only the cation and anion adsorption increase but also the solution pH changes. The results question the use of CEC and AEC measured at a constant pH in predicting the retardation of ion transport in variable-charge soils.
Figure 2
depicts the changes in the net surface charge, qH (= Qan - Qcat), with an increase in the electrolyte cation concentration, Ccat (Can), as obtained from the calculation described above. Irrespective of initial pH, increases in the cation adsorption and the anion adsorption are nearly equivalent, so that the net surface charge remains essentially constant. These results corroborate the assumption of dQcat/dCcat dQan/dCan that was made in deriving Eq. [18] and [19]. Unless H+ or OH- is added, Qcat and Qan in a variable-charge soil do not necessarily traverse over the entire surface bounded by the CEC vs. pH and AEC vs. pH curves at the highest electrolyte concentration (e.g., C = 50 mmolc L-1 in Fig. 1). Instead, they vary only along those lines, represented by sequences of the symbols in Fig. 1, in which qH is kept constant. It is the changes in Qcat and Qan with the electrolyte concentration along these lines that the pH-dependence implicit Eq. [18] and [19] seek to describe.
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Fig. 2. Changes in the net surface charge, qH (= Qan - Qcat), with the electrolyte cation concentration, Ccat, caused by addition of a neutral salt to "905" soil. pHin = initial pH when Ccat = 0.1 molc m-3.
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In Fig. 3
, the cation adsorption isotherms obtained from numerical solution of Eq. [18], subject to the initial conditions listed in Table 1, are compared with those obtained by solving Eq. [5] through [7], [12], and [13] as described above and plotting Qcat against Ccat. In solving Eq. [18], C was taken to be equal to Ccat. The isotherms obtained from Eq. [18] are in excellent agreement with those based on the solution of the pH-dependence explicit equations, as long as the condition Ccat >> [H+] is fulfilled. Where this condition is not fully met, as in the italicized initial conditions in Table 1 and the isotherms represented by dashed lines in Fig. 3, the agreement is somewhat poorer because the approximation Ccat Can C made in the derivation of Eq. [11] becomes less accurate. In such cases, Eq. [18] is no longer a good approximation of the change of Qcat with Ccat and the errors produced when Ccat is small tend to accumulate even after Ccat is increased and the condition Ccat >> [H+] has been met. It should be remembered that in solving Eq. [18], the initial value of C, and hence Qcat, may be chosen arbitrarily. Thus, where the initial condition did not satisfy Ccat >> [H+] (i.e., Ccat = 0.1 molc m-3 with initial pH = 4 and 5), an alternative initial condition for use in solving Eq. [18] was sought by solving Eq. [5] through [7], [12], and [13] at an arbitrarily specified Ccat for Mcat, [H+], Can, Qcat, and Qan while keeping (Man - Mcat) constant. The initial conditions chosen to satisfy Ccat > 10 [H+] are also listed in Table 1. Note that given qH as a constant, C (taken equal to Ccat) and Qcat are the only variables appearing in the calculation. The isotherms based on the solution of Eq. [18] subject to these initial conditions closely agreed with those obtained from the pH-dependence explicit equations (Fig. 3).
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Fig. 3. Comparison of the cation adsorption isotherms in "905" soil, obtained by solving the pH-dependence implicit Eq. [18] (solid lines) and those by simultaneously solving Eq. [5] through [7], [12], and [13] in which CEC and AEC are expressed as a function of pH and the electrolyte concentration (symbols). Dashed lines represent the isotherms obtained from Eq. [18] when the condition Ccat >> [H+] is not always fulfilled. qH = net surface charge of soil (molc kg-1).
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Soil Containing Layer Silicate Minerals
Another soil we examined is the "H-1" soil studied by Okamura and Wada (1983). The soil was taken from the B horizon of an Ultisol, and contained kaolinite and vermiculite–chlorite intergrade as dominant clay minerals. The soil had a total C content of 6 g kg-1 and a clay content of 0.32 kg kg-1. In contrast to the "905" soil, the net surface charge was always negative in the range pH 4 to 8. The coefficients in Eq. [1] and [2] were a = 0.094, b = 0.078, c = 0.471, a' = -0.253, b' = 0.136, and c' = 1.81, so we find acat = 0.094, bcat = 0.078, kcat = 103a-3b+c-2 = 0.0330, aan = -0.253, ban = 0.136, and kan = 103a'-3b'+c'-2 = 0.0440 in Eq. [12] and [13]. Solid lines in Fig. 4
represent CEC and AEC of the soil as a function of pH, at constant electrolyte concentrations.
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Fig. 4. Changes in pH, cation adsorption, Qcat, and anion adsorption, Qan, upon addition of a neutral salt to "H-1" soil, obtained by solving Eq. [5] through [7], [12], and [13]. The soil was initially at Ccat = 0.1 molc m-3 with initial pH = 4, 5, 6, 7, or 8. Solid lines represent Qcat (= CEC) and Qan (= AEC) as a function of pH when the electrolyte concentration, C, is kept constant.
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Symbols in Fig. 4 depict the changes in pH, Qcat, and Qan upon addition of a neutral salt to the "H-1" soil, as predicted by solving Eq. [5] through [7], [12], and [13] simultaneously. The soil was assumed to be saturated with monovalent cation and anion species, and initially at Ccat = 0.1 molc m-3 with initial pHs of 4, 5, 6, 7, or 8. Considering the less porous nature of the soil, the volumetric water content and the bulk density were taken as = 0.48 m3 m-3 and = 1200 kg m-3, respectively. The initial conditions are listed in Table 2. The values of Ccat, Can, Qcat, Qan, and [H+] after addition of different amounts of neutral salt were calculated in the same way as already described for the "905" soil. We observe in Fig. 4 that the addition of the neutral salt tends to decrease the pH in this soil dominated by layer silicate minerals. Nevertheless, both Qcat (= CEC) and Qan (= AEC) increase, owing to the increase in the electrolyte concentration. Figure 5
presents the changes in the net surface charge qH (= Qan – Qcat) with Ccat obtained from this calculation. Despite the significant differences in the surface charge characteristics from the "905" soil, we see that the increases in Qcat and Qan in response to a change in the electrolyte concentration are nearly equivalent, so that the net surface charge remains virtually constant in this soil as well.
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Fig. 5. Changes in the net surface charge, qH (= Qan - Qcat), with the electrolyte cation concentration, Ccat, caused by addition of a neutral salt to "H-1" soil. pHin = initial pH when Ccat = 0.1 molc m-3.
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Figure 6
compares the cation adsorption isotherms obtained from the numerical solution of Eq. [18], subject to the initial conditions listed in Table 2, with those based on the simultaneous solution of Eq. [5] through [7], [12], and [13]. The isotherms from Eq. [18] closely agreed with those obtained by solving the pH-dependence equations, though the agreement was slightly poorer for the case Ccat = [H+] = 0.1 molc m-3. For the latter case, the agreement was improved by choosing an alternative initial condition so as to satisfy Ccat > 10 [H+].
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Fig. 6. Comparison of the cation adsorption isotherms in "H-1" soil, obtained by solving Eq. [18] (solid lines) and those by solving Eq. [5] through [7], [12], and [13] simultaneously (symbols). Dashed lines represent the isotherms obtained from Eq. [18] when the condition Ccat >> [H+] is not always fulfilled. qH = net surface charge of soil (molc kg-1).
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DISCUSSION
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In comparison with the well-known dependence of CEC and AEC of variable-charge soils on pH and ionic strength of the bulk solution, less attention has been paid to the changes in pH caused by changes in the solution concentration. This lacuna has been the focus of the current analysis.
Unless H+ or OH- is added, an increase in the solution concentration induces a change in pH of a variable-charge soil in such a way that there are nearly equivalent increases in CEC and AEC (Fig. 1 and 4). The resultant increases in cation and anion adsorption are responsible for the phenomenon known as apparent salt sorption (Thomas, 1960; Imai and Okajima, 1980; Wada, 1984; Pearce and Sumner, 1997). If the soil is saturated with indifferent electrolyte ions, such addition or removal of a neutral salt causes negligible change in the net surface charge, and CEC and AEC of the soil can be regarded as a function of the electrolyte concentration only. The pH-dependence implicit Eq. [18] and [19] derived in the present study describe changes in CEC and AEC with the electrolyte concentration under such conditions. When C Ccat Can is a good approximation (i.e., Ccat >> [H+] in the examples soils), as we assumed in deriving Eq. [11], the cation adsorption isotherms predicted by solving Eq. [18] are in excellent agreement with those obtained by solving Eq. [5] through [7], [12], and [13] in which CEC and AEC are expressed as an explicit function of pH and the electrolyte concentration. The advantage of the pH-dependence implicit formulation over the conventional method is that once the empirical adsorption parameters in Eq. [12] and [13] are known, changes in cation and anion adsorption in response to a change in the electrolyte concentration can be calculated without knowing the concurrent change in pH, so that the transport equation for H+ need not be solved in predicting the transport of electrolyte ions in soil.
It is interesting to note that the parameter / does not appear in Eq. [18], but the isotherms obtained from Eq. [18] closely agree with those by the conventional method in which / appears in the mass balance equations. This stems from the fact that the amount of H+ or OH-, adsorbed on the solid phase in a given volume of bulk soil is much greater than that of H+ present in the aqueous solution phase. Unless H+ or OH- is added, any increase or decrease in the excess H+ concentration (= [H+] - [OH-]) in the liquid phase must be compensated with a corresponding decrease, or increase, in the net surface charge of the soil (i.e., the net adsorption of H+ onto the surface). However, in the "905" soil at pH 5, for example, the excess H+ in the liquid phase per unit mass of soil [= (/)([H+] - [OH-])] amounts to 7.5 x 10-6 molc kg-1, which is negligibly small when compared with the densities of positively and negatively charged sites (i.e., AEC and CEC) in the soil. Thus, for a realistic value of /, changes in [H+] in the liquid phase are expected to have negligible effects on the H+ balance, and the net surface charge of the soil. The lack of the term / in Eq. [18] and [19] is equivalent to the neglect of such effects.
Variable-charge soils possessing positive charge typically contain appreciable amounts of adsorbed SO2-4 (Wong et al., 1990; Kamewada, 1994). Unless the electrolyte concentration has been thoroughly lowered and weakly reactive ions desorbed by leaching with water, ions having different valence and affinity to the surface may coexist in the soils. Katou et al. (1996) suggested that adsorption of monovalent anions during invasion of a salt solution into Andisols is largely due to increase in AEC caused by the increase in the ionic strength, and that desorption of native SO2-4 in exchange for the monovalent anions proceeds only to a limited extent. They also argued that during absorption of a neutral salt solution, the net surface charge is kept nearly constant, so that the adsorption of monovalent anions can be described as a function of the liquid-phase anion concentrations only. The results obtained in the present study demonstrate that under certain simplified conditions, where the soil is saturated with monovalent cation and anion species, the net surface charge is indeed kept constant and ion adsorption can be described as a function of the electrolyte concentration only (Eq. [18] and [19]). Extension of the pH-dependence implicit formulation to soils having mixed adsorbed ion composition, in which SO2-4 is the dominant adsorbed anion species or where the desorption and hydrolysis of native Al3+ upon addition of a neutral salt may create additional positive surface charge (Seaman et al., 1995), will be the subject of future study.
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ACKNOWLEDGMENTS
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I gratefully acknowledge valuable comments by Dr. B.E. Clothier of HortResearch, New Zealand on the manuscript. This work was partly supported by a fellowship under the OECD Co-operative Research Programme: Biological Resource Management for Sustainable Agricultural Systems.
Received for publication June 20, 2001.
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REFERENCES
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TOP
ABSTRACT
INTRODUCTION
