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Modified Langmuir Equation for S-Shaped and Multisite Isotherm Plots Patrick G. Grant, Shawna L. Lemke, Maxene R. Dwyer, and Timothy D. Phillips* Intercollegiate Faculty of Toxicology, College of Veterinary Medicine, Texas A&M University, College Station, Texas 77843-4458 Received November 10, 1997 Standard isotherm equations do not estimate capacity (Qmax) and distribution coefficient (Kd) for complex or non-Langmuir-shaped isotherm plots. In this study, two mycotoxins, that is, aflatoxin B1 (AfB1) and cyclopiazonic acid (CPA), were mixed with kaolinite and a naturally acidic montmorillonite clay (LPHM) at 25 °C, respectively. Isotherm data gave S-type plots. The data were fitted to the models of Langmuir (LM) and multi-Langmuir (MLM); however, these models did not provide a good fit for data that displayed multisite adsorption or multiple plateaus. While a published modification of the Langmuir equation (QKLM), which defines an effective partition coefficient as a function of the surface coverage, was able to fit simple isotherm plots from all categories (H, L, S, C), it did not fit complex or multisite isotherm plots. Importantly, an equation that enables the estimation of Qmax and Kd for both S-shaped and multisite isotherm plots has not yet been reported. Since the LM, MLM, and QKLM did not provide adequate fitting of the data, several modifications of the LM were developed: shifted Langmuir, shifted squared Langmuir, shifted cubed Langmuir, shifted exponential Langmuir, exponential Langmuir, and shifted modified Langmuir. These equations were used to derive information about the adsorption of mycotoxins to clay and to gain insight into the molecular mechanism(s) and site(s) of adsorption. The objectives of this study were to present a series of modified Langmuir equations that can be used to estimate the Qmax and Kd of a specific adsorption site and to relate Qmax to available adsorption area.
Introduction A variety of hazardous chemicals (i.e., mycotoxins) are produced by common fungi and can frequently occur as unavoidable contaminants of food and feed. Many of these mycotoxins, including aflatoxin B1 (AfB1) (Figure 1) and cyclopiazonic acid (CPA) (Figure 2), have been implicated in the etiology of disease and death.1,2 Thus, safe and effective strategies to detoxify these poisons are highly desirable. One approach to this problem has been the dietary inclusion of high affinity adsorbents that can bind aflatoxins in the gastrointestinal tract and diminish their bioavailability and adverse effects.3 In previous work to delineate the molecular mechanism(s) of this action, we have studied the adsorption of AfB1 and (CPA) onto the surfaces of diverse clay minerals.4-6 One of the most effective ways of investigating the surface adsorption of mycotoxins to clay is through the use of isotherms. Multiple isotherm equations have been used to model the adsorption of compounds in aqueous solutions to solid surfaces and to provide estimates of the capacity (Qmax) and the distribution coefficient (Kd).7 On the basis of these values, suitable clays for testing in animals (as potential mycotoxin adsorbents) are delineated and investigated. However, in preliminary studies, the adsorption of CPA by a low pH montmorillonite clay * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Cole, R. J.; Cox, R. H. In Handbook of Toxic Fungal Metabolites; Academic Press: New York, 1981; pp 11-66. (2) CAST (Council for Agricultural Science and Technology) Niyo, K., Ed.; CAST: 1995; pp 1-91. (3) Phillips, T. D.; Kubena, L. F.; Harvey, R. B.; Taylor, D. R.; Heidelbaugh, N. D. Poultry Sci. 1988, 67, 243-247. (4) Sarr, A. B. Ph.D. Dissertation, 1992, pp 1-144. (5) Grant, P. G.; Phillips, T. D. J. Agric. Food Chem. 1998, 46, 599605. (6) Dwyer, M. R. Ph.D. Dissertation, 1997, 1-189. (7) Kinniburgh, D. G. Environ. Sci. Technol. 1986, 20, 895-904.
Figure 1. Structure of AfB1 (top) and the molecular model (bottom) illustrating the spacial orientation and size of the functional groups.
(LPHM) and of AfB1 onto kaolinite clay did not correlate well with the Langmuir model (LM). The lack of adequate models for these processes meant that potential mycotoxin adsorbents could not be quantitatively assessed. Consequently, this prompted a search for isotherm models that could be applied to isotherms characterized by categories other than the standard Langmuir plot (L).
S0743-7463(97)01218-3 CCC: $15.00 © 1998 American Chemical Society Published on Web 06/26/1998
Modified Langmuir Equation
Langmuir, Vol. 14, No. 15, 1998 4293
Table 1. Isotherm Equations Used To Fit Plots of Adsorption to Claysa
(
Langmuir model (LM)
q ) Qmax
multi-Langmuir model (MLM)
q ) Qmax1
modified Langmuir model with q dependent affinity (QKLM)
q ) Qmax
shifted modified Langmuir model with q dependent affinity (SQKLM)
q ) Qmax
exponential Langmuir model (ELM)
q ) Qmax
( ( ( (
q ) Qmax
shifted squared Langmuir model (SSLM)
q ) Qmax
shifted cubed Langmuir model (SCLM)
shifted exponential Langmuir model (SELM)
q ) Qmax
)
(
+ Qmax2
(Kd e-2bq Cw)
)
Kd2Cw
)
1 + Kd2Cw
1 + (Kd e-2bq Cw)
(Kd e-2bq (Cw - Cs))
+ ...
)
1 + (Kd e-2bq (Cw - Cs)) K′Cwn
)
1 + K′Cwn
(K′(Cw - Cs))
)
1 + (K′(Cw - Cs))
( ( (
q ) Qmax
Kd1Cw
1 + Kd1Cw
(
shifted Langmuir model (SLM)
)
KdCw 1 + KdCw
(K′(Cw - Cs)2)
) ) )
1 + (K′(Cw - Cs)2) (K′(Cw - Cs)3)
1 + (K′(Cw - Cs)3) (K′(Cw - Cs)n)
1 + (K′(Cw - Cs)n)
a q ) adsorbed (mol/kg), Q max ) maximum capacity (mol/kg), Kd ) distribution constant, Cw ) equilibrium concentration, Cs ) shifted concentration, b ) energy-dependent affinity parameter, n ) exponential parameter. For QKLM and SQKLM q and Cw data were transposed for fitting, while all other models were fitted with q vs Cw.
The shapes of isotherm plots have been categorized into four types of curves, labeled as H, L, C, and S, which represent different adsorption mechanisms.8-10 Standard isotherm equations do not estimate Qmax and Kd with acceptable standard error for S-shaped and multisite isotherm plots. The problem of fitting plots that are not of the L category is not new and has been addressed by Gu et al., who modified the Langmuir equation (QKLM) for application to multiple categories (H, L, S, C) of isotherm plots. However, this equation still does not fit isotherm plots with multiple sites. In response to this problem, this paper serves to introduce a series of modifications to the Langmuir model that are based on the insertion of a constant (Table 1) which represents the concentration on the x axis necessary to shift a Langmuirshaped isotherm to match part of an S- or complex-shaped L, C, or H isotherm plot. This fit also enables the estimation of Qmax for each site. To aid in the analysis of these modified equations, surface area measurements of the two clays were made. These values can provide insight into the adsorption process when coupled with information obtained from isotherm fits, in particular, the Qmax parameter. These values are used to estimate the relative surface coverage, Å2 occupied by each adsorbed ligand, and allow a prediction of ligand orientation and whether a multilayer adsorption exists. The objectives of this study were to present a series of modified Langmuir equations that can be used to (8) Giles, C. H.; MacEwan, T. H.; Nakhwa, S. N.; Smith, D. J. Chem. Soc. 1960, 3973-3993. (9) Giles, C. H.; Smith, D.; Huitson, A. J. Colloid Interface Sci. 1974, 47, 755-765. (10) Giles, C. H.; D’Silva, A. P.; Easton, I. A. J. Colloid Interface Sci. 1974, 47, 766-778.
Figure 2. Structure of CPA (top) and the molecular model (bottom) illustrating the spacial orientation and size of the functional groups.
estimate the Qmax and Kd of a specific adsorption site and to relate Qmax to available adsorption area. Materials and Methods Chemical Reagents. AfB1 was purchased from Sigma Chemical Co. (St. Louis, MO). CPA was extracted and purified from a fungal culture.6 LPHM clay was obtained from Engelhard Corp. (Cleveland, OH). Kaolinite clay (KGa-1b) was obtained from the Clay Resource Board (University of Missouri, Columbia,
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MO). All solvents were HPLC grade (Burdick and Jackson, Muskegon, MI). The water used in these studies was prepared by processing deionized water through a Milli-Quf+ system. Aflatoxin and CPA Characteristics. The cross-sectional area was estimated by drawing the structures of AfB1 and CPA with ISIS Draw 2.0 (MDL Information Systems, San Leandro, CA) and then importing them into HyperChem 4.5 (HyperCube, Ontario, Canada). The structures were energy-minimized using the semiquantitative AM1 method.11,12 The van der Waals radii of C, O, and H were set to 1.85, 1.40, and 1.20 Å, respectively.13 The structures (Figures 1 and 2) were then oriented on edge, with the dicarbonyl or keto-enol in view or planar. Structures were then transferred to the Chemplus module to print crosssectional area comparison to the carbon atom.14 Area calculations were performed by a previous method with the modification of carbon as a reference atom.15 Kaolinite and LPHM Clay Characteristics. The total surface area of kaolinite and acid montmorillonite clay was found by measuring the amount of ethylene glycol adsorbed onto the clay.16 The surface area can be calculated from the following relationship: 0.000 031 g of ethylene glycol adsorbed for each square meter of surface area.17 Isothermal Adsorption. The batch isotherm procedure consists of samples of a fixed amount of adsorbent being individually exposed to an increasing concentration of the solute.5 Along with the samples (50 mg of sorbent:5 mL of toxin solution) there were three controls consisting of 5 mL of purified water, 5 mL of stock solution without adsorbent, and 5 mL of the lowest concentration without adsorbent. The samples and controls were capped and placed on an electric shaker at 1000 rpm for 24 h in an incubator at 25 °C. After being shaken, the samples were centrifuged at 10 000 rpm for 15 min also at 25 °C. The supernatant concentration of the samples and controls was measured with a UV-vis spectrophotometer. At the highest concentration level, the supernatant was saved for analysis by high-performance liquid chromatography to check for any degradation of the parent compounds, since the adsorption calculations are dependent on a difference calculation. Derivation. When the conventional LM and multi-Langmuir model (MLM) failed to fit the mycotoxin data, the observation was made that these isotherm plots contained sections that had a shape similar to the standard Langmuir isotherm (L2) but were shifted on the x axis (equilibrium concentration). By using the mathematical principle of translation, this shift (Cs) was subtracted from the equilibrium concentration to give the following equation
q ) Qmax
(
K′ (Cw - Cs)
1 + K′(Cw - Cs)
)
(1)
which is referred to as the shifted Langmuir model (SLM). S-shaped isotherms have been characterized as describing a cooperative adsorption process. On the basis of this idea, the concentration of the shift is assumed to be the concentration necessary for the adsorption of the ligand to readily occur. The data from each multiplateau isotherm plot was divided into sets based on the delineation of each plateau (which represents a theoretical site) and was fit with this equation. This enabled the estimation of the capacity of the sites, which could not be determined from the LM, or MLM equations. The equation was only applied to sections of the curve where Cw > Cs. It was (11) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J. Am. Chem. Soc. 1985, 107, 3902-3909. (12) HyperChem: Computational Chemistry; Hypercube, Inc.: Ontario, Canada, 1996; pp 1-285. (13) Emsley, J. The Elements; Clarendon Press: Oxford, 1991; pp 1-251. (14) ChemPlus: Extensions for HyperChem; Hypercube: Ontario, Canada, 1993; pp 19-46. (15) Gray, M. J.; Mebane, R. C.; Womack, H. N.; Rybolt, T. R. J. Colloid Interface Sci. 1995, 170, 98-101. (16) Mortland, M. M.; Kemper, W. D. In Methods of Soil Analysis Part 1: Physical and Mineralogical Properties, Including Statistics of Measurement and Sampling; Black, C. A., Ed.; American Society of Agronomy: Madison, WI, 1965; pp 532-544. (17) Dyal, R. S.; Hendricks, S. B. Soil Sci. 1950, 69, 421-432.
developed to model only the plateau section of the S curve and will cause q to be negative, and therefore meaningless, at values for Cw where Cw < Cs. The SLM also resulted in a perturbation in the global Kd, which could be corrected by an alternative calculation requiring an averaging of individual Kd values. It was noted that the standard LM equation
(
)
K dC w 1 + K dC w
(2)
q (Qmax - q)Cw
(3)
q ) Qmax can be rearranged as
Kd )
This form agrees with the chemical equilibrium expression of the adsorption to the surface as expressed by
site ligand + y\z ligand-site (Qmax - q) Kd (Cw) (q)
(4)
If the SLM is rearranged as
K1′ )
q1 (Qmax 1 - q1) (Cw - Cs)
(5)
This equation is the result for the first site. This rearrangement shows that the overestimation of K′ is a result of the subtraction of Cs from Cw. Since Kd is the approximate average distribution constant for the site over a concentration range, the summation is averaged over the concentration range also. The summation used the approximation of the site capacity in the estimation of the Kd
Kd )
1
i)n
∑ (Q n i)1
qi max 1
- qi)Ci
(6)
While this modification is applicable to simple S-shaped curves, it does not fully address the problem of multiple sites, so these adsorptions were also investigated. Since multisite adsorptions can be expressed as individual chemical equilibrium expressions, then by chemical equilibrium principles, the equations can be summed.
Cw + (Qmax1 - q1) h q1
(7)
Cw + (Qmax2 - q2) h q2 2 Cw + (QmaxTot - qTot) h qTot When this summed chemical equation was arranged as an isotherm model similar to LM (eq 3), it gave an equation that may be applicable to multiple site isotherm plots
Kd )
q (Qmax - q) Cw2
(8)
Expansion upon this idea ultimately led to the formation of the exponential Langmuir model (ELM). The ELM (Table 1) is a modification of eq 8, in that the exponent of Cw is not set, but is a fitted parameter. Therefore, the exponent of ELM, acting as a variable, can be applied to a set of isotherm data and give an estimate of the number of corresponding chemical equations or adsorption sites (eq 7). This modification was also applied to the SLM to yield the shifted squared Langmuir model (SSLM), shifted cubed Langmuir model (SCLM), and the shifted exponential Langmuir model (SELM). The q-dependent Kd Langmuir model (QKLM) incorporates a Kd multiplier (Table 1) that models the coverage-dependent
Modified Langmuir Equation adsorption18 This model is applicable to S-type isotherms but did not fit either the AfB1:kaolinite or CPA:LPHM binding data very well in its original form. Therefore, our shift modification was also applied to the QKLM. Data Calculations and Curve Fitting. The UV-vis absorption data were used to calculate the amount of toxin left in solution and the amount adsorbed. The supernatant concentration of each toxin was calculated using experimentally derived values, which were equal to 21 865 and 17 357 (1/(M‚ cm)) for AfB1 and CPA in water, respectively. The data were then fit to isotherm equations to obtain estimates for the variable parameters. The isotherm equations (Table 1) were input as user defined functions. Each function had limits and beginning values or first approximations for the variable parameters. The parameter limits for Qmax were positive numbers ranging from 0 to a maximum of 5 kg/kg. The maximum represented the adsorption of an amount of ligand five times the weight of the adsorbent. The parameter limit for Kd was from 0 to 1 × 1080. The parameter range for n was from 0 to 100. Estimates for the Qmax and Kd were taken from the double logarithmic plot of the isotherm.19 The plot displayed a break or multiple breaks if there was more than one plateau in the curve. The value on the x axis where the curve breaks was an estimate of Kd-1. The value on the y axis where the curve breaks was an estimate of the Qmax. The estimate for n was 0.5. These values for the ranges and estimates were entered into the user defined functions. The QKLM equation required transposing the x and y axis data to be fit in table curve as well as the rearrangement of the equation to make Cw a function of q. The MLM has the capability to estimate the number of different types of sites on the adsorbent, the strength of binding, and the capacity of the multiple sites. User defined equations represented the linear addition of one to six Langmuir equations. These equations were fit to the data and listed by decreasing r2. The variable parameters were set values previously described. The equations that fit the data with the best r2 were then judged on the randomness of the residuals. The equation that represented the lowest number of Langmuir equations and satisfied a high r2 and had a random residual plot was selected as an estimate of the adsorption process and the number of sites. For the SLM, SSLM, SCLM, SELM, ELM, and shifted modified Langmuir model (SQKLM), the estimated Qmax was used with interpolated data points to calculate individual Kd values (eq 3), which were averaged over the concentration range of each site. This calculation was performed with MathCAD (MathSoft, Cambridge, MA). Relative surface coverage calculations used the measured surface area of the adsorbent and the Qmax of each site. The surface area (m2/g) was converted to Å2/kg and divided by the quantity of Qmax multiplied by Avogadro’s number to yield Å2/ bound molecule.
Results and Discussion Kaolinite was found to have a total surface area of 23 ( 3 m2/g by the ethylene glycol adsorption method. This surface area value is typical of the published values for the kaolinite series of clays, which range from 5 to 39 m2/g.20 The surface area of LPHM was considerably higher at 800 ( 16 m2/g, while the collapsed LPHM had a surface area of 84 ( 5 m2/g.6 Kaolinite essentially has no spacing between the platelets, while smectites such as LPHM have a considerable amount of spacing, depending on the interlamellar cation. The direct result of an interlamellar space is that the surface area of smectite clays can be considerable. In contrast, kaolinites normally average 20 m2/g due to the (18) Gu, B.; Schmitt, J.; Chen, Z.; Liang, L.; McCarthy, J. F. Environ. Sci. Technol. 1994, 28, 38-46. (19) Stumm, W.; Sigg, L.; Sulzberger, B. Chemistry of the SolidWater Interface; John Wiley & Sons: New York, 1992; pp 1-396. (20) Dixon, J. B. In Minerals in Soil Environments; Dixon, J. B., Weed, S. B. Eds.; Soil Society of America: Madison, WI, 1989; pp 468519.
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Figure 3. Isotherm plot of the adsorption of AfB1 on kaolinite. Monolayer coverages of AfB1 on kaolinite are calculated from the surface area of kaolinite relative to the cross-sectional areas of AfB1. The monolayer for the vertical orientation and horizontal orientation of AfB1 are denoted by 0 and 4, respectively.
Figure 4. Isotherm plot of the adsorption of CPA on LPHM.
absence of interlamellar space.20,21 Likewise, when the clay is collapsed, the availability of the interlamellar region is eliminated as shown by the reduced surface area of Col-LPHM. The isotherm shape of AfB1 adsorption onto kaolinite was categorized as an S4 (Figure 3). The adsorption of CPA on LPHM fits the S3 category (Figure 4). In both cases the initial adsorptions plateaued, which suggests multiple specific binding sites and the saturation of each type of site. The S isotherm has an initial slope that is shallow and the amount bound increases as a semilinear function of the concentration. The isotherm plot is concave up until an inflection point and then plateaus. This type of isotherm is observed when a molecule does not have a (21) Borchardt, G. In Minerals in Soil Environments; Dixon, J. B., Weed, S. B. Eds.; Soil Science Society of America: Madison, WI, 1989; pp 675-727.
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Table 2. Summary of Isotherm Fit Parameters for AfB1 Adsorption on the First Site of Kaolinite isotherm models (used in coverage calculations) Langmuir model (LM) multi-Langmuir model (MLM) modified Langmuir model with q dependent affinity (QKLM) shifted modified Langmuir model with q dependent affinity (SQKLM) exponential Langmuir model (ELM) n ) 9.63 shifted Langmuir model (SLM) shifted squared Langmuir model (SSLM) shifted cubed Langmuir model (SCLM) shifted exponential Langmuir model (SELM) n ) 3.856
capacity ( std error (mol/kg)
K′
Kd
b
Cs
r2
4.000 ( NC NA ( NA 0.021 ( NC
NA NA NA
206 NA 1.76 × 104
NA NA -74.2
NA NA NA
0.5776 NA 0.5168
0.021 ( NC
3.35 × 104
2.08 × 105
-61.8
3.79 × 10-6
0.8027
0.015 ( 0.001 0.029 ( 0.001 0.017 ( 0.002 0.016 ( 0.002 0.016 ( 0.005
1.15 × 1048 1.61 × 105 1.05 × 1011 2.12 × 1016 2.86 × 1020
2.55 × 105 4.53 × 104 1.57 × 105 2.58 × 105 2.48 × 105
NA NA NA NA NA
NA 7.91 × 10-6 7.49 × 10-6 6.72 × 10-6 5.49 × 10-6
0.9892 0.99997 0.9955 0.9903 0.9902
Figure 5. Isotherm data of AfB1 adsorption onto kaolinite fit by the Langmuir model (LM), the q dependent affinity modification of the Langmuir model (QKLM), and the shifted QKLM (SQKLM).
Figure 6. Isotherm data of AfB1 adsorption onto kaolinite fit by the shifted modification of the Langmuir model (SLM) for the first site and the shifted squared Langmuir model (SSLM) for the second site.
strong affinity for the surface until there is a significant amount bound, at which time, the slope increases as the affinity for the surface increases. This occurs because the solute molecule has modified the surface or has begun to bind to previously bound molecules. The plateau occurs when the saturation of that type of site has been reached. On comparison of the fit of the isotherm models for the adsorption of AfB1 to kaolinite (Table 2) it was apparent that the LM and MLM were not applicable (NA) to either the first site, second site or the whole data set because of poor correlations and reported capacities that were not possible based on the experimental conditions (Figure 5). The QKLM proposed by Gu et al., and its modification SQKLM, were able to estimate capacities, but the standard errors could not be calculated (NC) for the first site on kaolinite by table curve due to the poor correlation. In contrast, the ELM, SSLM, SCLM, and SELM had capacities that were within standard error of one another, had high values of r2, and had similar Kd values (Table 2). The capacity estimated by the SLM was chosen over ELM, SSLM, SCLM, and SELM for coverage calculations because of the smallest standard error and largest r2 value (Figure 6). Comparisons of the standard errors and r2 values of the modeling of the second site of adsorption of AfB1 to kaolinite indicated that the SSLM gave the best fit of the data (Table 3) (Figure 6). When both sites of AfB1 to kaolinite were fit simultaneously, the ELM gave the best fit (Table 4). In comparison, adsorption of CPA to LPHM was best fit by the SCLM (Table 5) (Figure 7).
On the basis of the assumptions that led to the proposal of the ELM, SSLM, SCLM and the SELM (eqs 7 and 8), it is hypothesized that in complex systems (i.e., multiple adsorption mechanisms) the exponent in ELM or SELM should be larger than that in simpler systems. It follows from this assumption that as the exponents increase in SLM, SSLM, and SCLM, the equations should model more complex systems. Furthermore, when data are fit by a series of equations with defined exponents, the level of adsorption complexity is related to the relative size of the exponent of the best fit equation. Therefore, the ELM and shifted LMs may be used similarly to the Toth isotherm model to make an estimate of the heterogeneity of the adsorption. Evidence for this argument was gathered from the evaluation of the relative surface coverage for the two adsorption sites of AfB1 on kaolinite and CPA adsorption on LPHM. As previously stated, the shifted modifications of the Langmuir equation enabled the estimation of Qmax of adsorbents with nonstandard isotherm plots. Relative surface coverage was calculated from the Qmax and the surface area measurements and was used to provide an estimate for the available surface area per bound molecule for each adsorption experiment. This value was compared to the cross-sectional areas of the probable orientations of the adsorbed molecules. The vertical orientation was defined as the binding of the dicarbonyl system for AfB1 or the keto-enol system for CPA.4,6 The horizontal orientation was defined as the positioning of molecules planar to the surface with nonplanar groups directed away
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Langmuir, Vol. 14, No. 15, 1998 4297
Table 3. Summary of Isotherm Fit Parameters for AfB1 Adsorption on the Second Site of Kaolinite isotherm models (used in coverage calculations)
capacity ( std error (mol/kg)
K′
Kd
modified Langmuir model with q dependent affinity (QKLM) shifted modified Langmuir model with q dependent affinity (SQKLM)
4.000 ( NC 0.062 ( NC 0.056 ( NC 0.158 ( 0.009
NA
0.191 ( 0.066
2.54 ×
exponential Langmuir model (ELM) n ) 13.97 shifted Langmuir model (SLM) shifted squared Langmuir model (SSLM) shifted cubed Langmuir model (SCLM) shifted exponential Langmuir model (SELM) n ) 1.803
0.133 ( 0.006 0.199 ( 0.017 0.152 ( 0.005 0.142 ( 0.005 0.155 ( 0.039
4.78 × 1066 2.47 × 105 8.82 × 1010 1.34 × 1016 7.81 × 109
Langmuir model (LM) multi-Langmuir model (MLM)
b
1.27 × 6.23 × 104 2.01 × 104 5300 103
NA NA
4.47 ×
105
NA NA
NA NA
-14.3
NA
0.5222 0.5315 0.5954 10-5
0.9936
NA 1.45 × 10-5 1.38 × 10-5 1.28 × 10-5 1.40 × 10-5
0.9898 0.9961 0.9977 0.9966 0.9977
1.46 ×
104 0.373 NA NA NA NA NA
8.80 × 105 5.68 × 104 1.53 × 105 2.50 × 105 2.54 × 105
r2
Cs
Table 4. Summary of Isotherm Fit Parameters for AfB1 Adsorption on Both Sites of Kaolinite capacity ( std error (mol/kg)
isotherm models Langmuir model (LM) multi-Langmuir model (MLM) modified Langmuir model with q dependent affinity (QKLM) shifted modified Langmuir model with q dependent affinity (SQKLM) exponential Langmuir model (ELM) n ) 12.93 shifted Langmuir model (SLM) shifted squared Langmuir model (SSLM) shifted cubed Langmuir model (SCLM) shifted exponential Langmuir model (SELM) n ) 6
b
Kd
NA NA
NA NA
0.6804 NA
-15.1 -14.1
NA 4.02 × 10-6
0.8020 0.9261
NA NA NA NA NA
NA NA 4.99 × 10-6 5.33 × 10-6 1 × 10-6
0.9924 NA 0.9416 0.9630 0.9739
0.125 ( NC NA ( NA NA ( NA 0.156 ( 0.009 0.157 ( 0.008
NA NA NA 7.20 × 105
3.41 × 104 NA NA 4.79 × 103 8.10 × 104
0.135 ( 0.005 NA ( NA 2.72 ( NC 0.26 ( 0.07 0.16 ( 0.05
5.14 × 1061 NA 1.65 × 108 2.14 × 1016 5 × 1028
2.35 × 105 NA 1.20 × 103 1.30 × 104 4.48 × 104
Cs
r2
K′
Table 5. Summary of Isotherm Fit Parameters for CPA Adsorption on LPHM isotherm models (used in coverage calculations)
capacity ( std error (mol/kg)
modified Langmuir model with q dependent affinity (QKLM) shifted modified Langmuir model with q dependent affinity (SQKLM) exponential Langmuir model (ELM) n ) 2.92 shifted Langmuir model (SLM) shifted squared Langmuir model (SSLM) shifted cubed Langmuir model (SCLM) shifted exponential Langmuir model (SELM) n ) 4
0.123 ( 0.014
K′
Kd
b
Cs
r2
NA
3.44 ×
104
-7.77
NA
0.9469
0.131 ( 0.030
5.13 × 104
6.95 × 104
-4.68
1.10 × 10-6
0.9509
0.093 ( 0.006 0.113 ( 0.017 0.101 ( 0.007 0.087 ( 0.005 0.085 ( 0.008
9.10 × 1014 2.03 × 105 1.92 × 1010 3.27 × 1015 3.0 × 1020
3.37 × 105 1.26 × 105 1.83 × 105 4.05 × 104 6.05 × 104
NA NA NA NA NA
NA 3.91 × 10-6 1.09 × 10-6 5.69 × 10-7 1.0 × 10-7
0.9783 0.9450 0.9760 0.9808 0.9750
Figure 7. Isotherm data of CPA adsorption onto LPHM fit by the shifted cubed Langmuir model (SCLM).
from the surface. The vertical and horizontal crosssectional areas were found to be 52.8 ( 0.5 and 88.3 ( 2.3 Å2 for AfB1 and 69.8 ( 1.2 and 95.1 ( 1.2 Å2 for CPA, respectively.5
The total kaolinite surface area for the first site (relative to adsorbed AfB1 molecules) was 132 Å2/molecule. This area could adsorb AfB1 in either orientation without requiring a multilayer coverage (Figure 8). For the second plateau in the isotherm plot, the coverage was 25.1 Å2/ molecule, which would be less than half the surface area required for the vertical orientation and less than a third of the surface area required for the horizontal orientation. The maximum concentration adsorbed as a monolayer was calculated by dividing the surface area of the adsorbent by the cross sectional area of the ligand. This value was located on the isotherm plot and its position compared to the corresponding plateaus or inflections. For AfB1 adsorption on kaolinite, the monolayer coverage occurred for the vertical orientation at 0.072 and 0.043 mol/kg for the horizontal orientation. These values lie within the second plateau of the isotherm plot (Figure 3). There was considerable surface area left around AfB1 at 132 Å2/molecule (Figure 8) from the adsorption to the first site. This suggests that the second plateau might be a combination of the completion of the coverage of the surface and the formation of a multilayer. It is true that the multilayer mechanism is normally a weak association and would not usually be characterized as a specific adsorption mechanism but a partition. However, examination of the isotherm plot gave further evidence for this hypothesis. The plot for the second site initially had the characteristic
4298 Langmuir, Vol. 14, No. 15, 1998
Grant et al.
Figure 8. Images representing orientations of the adsorbed molecules on areas of sorbent surface estimated by the relative surface coverage calculations. The top two figures depict AfB1 relative surface coverage (132 Å2/molecule adsorbed) for the adsorption to the first site of kaolinite in both the vertical orientation (upper right) and the horizontal orientation (upper left). The bottom figures show the relative surface coverage for CPA adsorbed on LPHM with a horizontal orientation of 1526 Å2/molecule adsorbed (lower right) and CPA adsorbed on Col-LPHM with a vertical orientation on 160 Å2/molecule adsorbed (lower left).
of an L type plot, but then changed shape and slanted upward. This again might be indicative of a combination of the two mechanisms to give the plot of the second plateau. From the relative surface coverage calculations for CPA adsorption onto LPHM, it was noted that there was considerable surface area available per bound molecule (i.e., 1536 Å2/molecule), compared to the cross-sectional area of CPA. As previously reported, collapsing the clay had insignificant effects on the adsorption but reduced the surface area from 800 to 84 m2/g.6 After collapsing the interlamellar region, the surface still had 160 Å2/ molecule left to adsorb CPA without forming a multilayer (Figure 8). This suggested that CPA is solely adsorbed on the outside of the LPHM clay at metal edge sites. This estimation confirms the previously published hypothesis.6 By using the cross sectional areas of the orientations of CPA, it was calculated that a monolayer would occur at 1.903 and 1.397 mol/kg for the vertical and horizontal
orientations on LPHM, respectively. The monolayer concentrations were also calculated for the collapsed LPHM and indicated that a monolayer would occur at 0.200 and 0.147 mol/kg for the vertical and horizontal orientations, respectively. The surface concentrations lie outside the isotherm plot, adding credence to the hypothesis that a specific site on the edge of LPHM is responsible for the adsorption of CPA. Once relative surface coverages were calculated, it was possible to further evaluate the ability of the ELM to model the adsorption experiments. Comparison between the second site of AfB1 adsorption on kaolinite and the adsorption of CPA on LPHM indicated that a correlation might exist between the size of the exponent and the complexity of the adsorption. As previously described, the adsorption of AfB1 in the second plateau is probably a combination of the completion of the surface coverage from the first plateau and the subsequent formation of a multilayer, a rather complex process. The ELM gave an
Modified Langmuir Equation
exponent of 13.97 for this adsorption experiment. For the adsorption of CPA to LPHM it was suggested that the adsorption was limited mainly to edge sites. The ELM gave an exponent of only 2.92, which suggested a less heterogeneous adsorption process, in support of prior evidence. A correlation was also observed for modeling the shifted LMs to AfB1 adsorption on kaolinite. For the first plateau it was assumed that the adsorption process was specific and homogeneous, since the total surface area exceeded the amount of area required for both orientations of AfB1 binding at the estimated capacity. The SLM, with an exponent of 1, is representative of the simplest adsorption system of the shifted LM series. This also was the best fit for this adsorption process. As previously discussed, the second plateau of the adsorption of AfB1 to kaolinite is probably a more complex process. The SCLM, a relatively more complex model due to the cubed term, was the best fit for this plateau. Modeling both plateaus with one equation increases the complexity of the system. The best fit for this system was obtained by the SELM with an exponent of 6. This result supported the correlation between exponent and system complexity. While confirmation of this trend will require further research, data from this limited set suggest the usefulness of the ELM and the shifted LM series of equations for estimating the heterogeneity of an adsorption.
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The QKLM, as proposed by Gu et al., did not adequately fit the first site, second site, or both sites combined for the adsorption of AfB1 to kaolinite. The QKLM did fit the CPA adsorption to LPHM, but not as well as the shifted modifications of the Langmuir or the exponential Langmuir modification. The QKLM fit the CPA data with a lower r2 and a standard error that was two to five times the shifted and exponential modifications of the Langmuir. It was concluded that a new set of derived equations, based on a translation along the concentration axis, or a “shift”, could be useful in fitting S-shaped data in order to apply the fitted parameters to studies of adsorption mechanisms. SAFETY. Aflatoxins and cyclopiazonic acid are hazardous chemicals and should be handled carefully. Protective clothing, gloves, fume hoods, and goggles are the first steps in preventing mycotoxin exposure. Mycotoxins in the crystalline form can become electrostatically charged and airborne. Solutions and residues on glassware can be decontaminated with a commercially available bleach solution. Acknowledgment. This study was supported by funding from the Texas Agricultural Experiment Station (H6215), USDA NRICGP # 9703230, and NIH P42-ES04197. LA971218A
