Fitting a little-known isotherm equation to S-shaped adsorption equilibrium data

A research topic of current interest concerns the measurement and modeling of adsorption equilibrium isotherms of water contaminants. The bulk of measured equilibrium isotherms exhibit the convex-upward or favorable curve shape, which can be described by the two-parameter Langmuir and Freundlich isotherms. An increasing number of recent studies have however reported S-shaped or sigmoidal equilibrium isotherms. Modeling such equilibrium data requires the use of isotherm equations with a highly flexible functional form. This work introduces a little-known isotherm useful for tracking the trajectory of S-shaped equilibrium data. This relatively simple three-parameter isotherm, first proposed by Krishnamurti in 1951, has theoretical merit because it is based on a co-operative adsorption mechanism. It is shown that previously published S-shaped equilibrium data of water contaminants can be represented by the Krishnamurti isotherm. Specific examples discussed are (1) ammonium ion adsorption by the clay mineral sepiolite, (2) uptake of the antibiotic ciprofloxacin by a magnetic nanosorbent, and (3) fluoride ion removal by a layered double hydroxide adsorbent. Further, it is found that a modified form of the Krishnamurti isotherm is superior to its original counterpart in tracing sigmoidal equilibrium data. The original and modified Krishnamurti isotherms can be practically very useful to describe S-shaped equilibrium data and for adsorptive process modeling.


Introduction
Adsorption is an efficient technology for trace contaminant removal from large volumes of aqueous wastes. Mathematical simulations of various adsorption processes of practical interest (fixed bed, expanded bed, fluidized bed, moving bed, and simulated moving bed) require precise knowledge of adsorption equilibria and kinetics.
Although the equilibrium characteristics of organic contaminant adsorption may be estimated using predictive correlations based on specific compound and adsorbent properties [1,2], such predictive methods are limited to certain classes of organic contaminants and adsorbents. In general, equilibrium isotherms of various water contaminants are invariably obtained from laboratory experiments.
It is well known that experimental equilibrium isotherms are capable of exhibiting different curve shapes. Adsorbent heterogeneity appears to play a significant role in establishing the adsorption isotherm shapes [3,4]. Although the major classes of isotherm shapes according to the Brunauer-Deming-Deming-Teller (BDDT) classification have been observed in aqueous adsorption systems [5], the vast majority of measured equilibrium data of water contaminants have been found to exhibit 2 the convex-upward or favorable isotherm shape. This particular isotherm shape can be easily represented by the two-parameter Langmuir and Freundlich isotherms. To fit equilibrium data of this kind, it is rarely necessary to use more intricate isotherms, particularly those involving three or more adjustable parameters.
In recent years, an increasing number of studies have however reported Sshaped or sigmoidal equilibrium data. Such isotherms are concave to the horizontal x-axis at low liquid-phase concentrations and convex at higher liquid-phase concentrations (type V isotherm shape according to the BDDT classification). Some recent examples include the works of Soares et al. [6], Jiang et al. [7], Mitrogiannis et al. [8], and Gago et al. [9]. Soares et al. [6] reported the uptake of the antibiotic ciprofloxacin by a silica-based magnetic nanosorbent. The equilibrium data set displaying a sigmoidal profile was reasonably well fitted by the Dubinin-Radushkevich isotherm. In the work of Jiang et al. [7], equilibrium experiments produced S-shaped isotherms for the adsorption of 2,4,6-trichlorophenol on highsilica zeolites. Mathematical simulations were used to explore the mechanism responsible for the sigmoidal adsorption trend, but no attempts were made to fit macroscopic isotherms to the S-shaped data. Mitrogiannis et al. [8] found that the uptake of phosphate by a zeolite adsorbent was characterized by an S-shaped adsorption profile, which was well tracked by the Zhu-Gu isotherm. Gago et al. [9] used the Sips isotherm to represent the sigmoidal curve exhibited by the adsorption of methylene blue on a cellulose-based adsorbent. It is of note that some of the isotherms used to describe aqueous sigmoidal equilibrium data are semi-empirical in nature. Their principal utility is to offer a practical way to track the sigmoidal shape of experimental isotherms, without implying the inherent validity of their mechanistic assumptions. By contrast, rigorous theoretical models are available for a priori prediction of gas-phase sigmoidal equilibrium isotherms [10].
Mathematical simulations of adsorption processes such as fixed bed and simulated moving bed adsorbers have shown that S-shaped isotherms exerted a significant impact on the process dynamics [11][12][13]. Since isotherm models serve as critical input parameters to process models, there is a need to develop reliable isotherm models capable of representing different types of sigmoidal equilibrium data. To correlate S-shaped equilibrium data to a significant degree of precision, isotherm equations with an appropriate functional form are needed. Although a number of macroscopic isotherms capable of fitting sigmoidal equilibrium data are available [14,15], there is a shortage of simple mechanistic isotherm models. This communication attempts to address this gap by exploring the ability of a simple isotherm proposed by Krishnamurti [16] to fit S-shaped equilibrium data. The Krishnamurti isotherm, which is based on a co-operative adsorption mechanism, seems to have attracted no attention thus far. According to Google Scholar, the work of Krishnamurti [16], published in 1951, has not been cited in the open literature. In this work, three sets of previously published S-shaped equilibrium data of water contaminants are used to test the little-known Krishnamurti isotherm. Further, this paper reports the modeling results obtained with a modified form of the Krishnamurti isotherm.

2. Theory
The main assumption of the adsorption model proposed by Krishnamurti [16] is that the uptake of an adsorbate by an adsorbent follows a co-operative adsorption mechanism where previously adsorbed molecules facilitate the adsorption of more molecules in neighboring positions. Under such conditions, the rate of increase in the number of adsorbed molecules (n) with increase in the concentration of the molecules in the solution (c) will depend upon the number of molecules already adsorbed and also upon the concentration of the molecules still available for adsorption in the solution. Mathematically this adsorption model can be expressed as where n  the number of adsorbed molecules; c  the solution phase concentration; k  a constant; and N  the total number of molecules originally present in the solution.
Integrating the preceding equation gives where k1 and k2 are constants.
Dividing both sides of this last equation by the mass of adsorbent yields where q  the adsorbed phase concentration and N0  the total number of molecules per unit mass of adsorbent. The three parameters to be fitted are N0, k1, and k2. An

Nonlinear least-squares regression
Nonlinear least-squares regression was applied to the models tested in this work to estimate their free parameters. Model fit was assessed using average relative error (ARE) and residual root mean square error (RRMSE). The dimensionless ARE is a relative measure of overall fit, whereas the RRMSE [18], which has the same units as q, is an absolute measure of overall fit. The two statistical metrics are given by Eqs. (4) and (5).

Results and discussion
To our knowledge, a paper by the originator of the isotherm defined by Eq. (3) appears to be the only work where the Krishnamurti isotherm was used to fit S-shaped equilibrium data [19]. As such, there is a need to conduct a comprehensive testing of Eq. (3). In this work, three sets of previously published equilibrium data of water contaminants are used to test the Krishnamurti isotherm: (1) ammonium ion adsorption by sepiolite [20], (2) ciprofloxacin removal by a magnetic nanosorbent [6], and (3) fluoride ion adsorption by a layered double hydroxide adsorbent [21]. Balci [20] reported several sets of equilibrium data on the uptake of ammonium ion by sepiolite, a naturally occurring clay mineral. Most data sets exhibited Sshaped curves, one of which is shown in Fig. 1A. Balci [20] found that the S-shaped data sets were well represented by the Toth and Langmuir-Freundlich (Sips) isotherms.

Ammonium ion adsorption by sepiolite
The mathematical forms of these two isotherms are quite similarboth are power law  Soares et al. [6] synthesized three different types of magnetic nanoparticles which were used to remove the antibiotic ciprofloxacin from aqueous solutions. Of interest to the present study are the equilibrium characteristics of the three adsorption systems. Two of the three adsorption systems were found to exhibit favorable convex-upward curve shapes, which were well fitted by the Langmuir isotherm.
The remaining one exhibited a sigmoidal curve, as shown in Fig. 2A. The S-shaped curve was attributed to the presence of cooperative binding in the adsorption system.
The authors fitted the Langmuir, Freundlich, and Dubinin-Radushkevich isotherms to the Fig. 2A data. As expected, the data were poorly described by the Langmuir and 6 Freundlich isotherms. By contrast, the Dubinin-Radushkevich isotherm fit was satisfactory. Here, an adequate fit with the Krishnamurti isotherm has also been obtained, as may be seen in Fig. 2A The preceding equation predicts a non-zero q value at c = 0. The Krishnamurti isotherm therefore has the undesirable property of not being constrained to go through the origin. This data set reveals the deficiencies of the Krishnamurti isotherm as a modeling tool for S-shaped equilibrium data. The next section discusses how the Krishnamurti isotherm may be modified in order to improve its data fitting ability.

Modified Krishnamurti isotherm
To reduce its lack-of-fit error, the Krishnamurti isotherm defined by Eq. (3) is modified via a logarithmic transformation of the variable c. This method of logarithmic transformation was previously successfully used to enhance the ability of a mathematical function to fit fixed bed breakthrough curves [22]. Here we show that the Krishnamurti isotherm can be transformed in the same spirit as the fixed bed model. Because c is a dimensioned quantity, it must first be made dimensionless before the logarithmic transformation can be applied to the Krishnamurti isotherm.
The specific steps involved are described below.
Step 1: Assign the variable c an appropriate set of units, e.g., mg/L.

8
Step 2: Define a new variable c * , which is numerically equal to 1 mg/L.
Step 5: We can now take the logarithm of (c/c * ), which is a dimensionless quantity.
Step 6: Eq. (9) can be simplified and expressed as   Step 7: Since c * = 1 mg/L, for convenience, it may be omitted from Eq. (10). Nevertheless, it can track q values that correspond to very small c values, as may be seen in Fig. 3A.
The performance of an isotherm model is often assessed according to the sole criterion of overall fit. One may adopt a broader approach by evaluating the statistical significance of fitted parameters. A 95% confidence interval is commonly used to assess uncertainty in parameter estimates. For the modified and original Krishnamurti isotherm fits shown in Fig. 3A suggesting that the fitted k2 values are not statistically significant. Given that the width of a confidence interval is influenced by the number of degrees of freedom in the estimation process, a feasible way to reduce the width is to use more data points. Fig. 3A shows that only eight data points were collected for the fluoride adsorption system, with a cluster of five appearing in the low concentration range.
It seems that more data points might be able to reduce the standard error (and thus the confidence interval) of k2.
For comparison, the Sips isotherm was fitted to the Fig. 3A  concentrations. In addition, Sips [23] points out that the maximum value that the exponent k1 in Eq. (12) can assume is unity. It is well known that the Sips isotherm reduces to the Langmuir isotherm when k1 is set to unity. Since the case of k1 > 1 was not considered by Sips, one can conclude that the Sips isotherm lacks theoretical merit when it is applied to S-shaped equilibrium data.
The superiority of the modified Krishnamurti isotherm versus its original counterpart is largely due to the fact that the former has a floating inflection point while the latter has a fixed one. The original isotherm is mathematically analogous to the well-known logistic equation of population growth as well as the Bohart-Adams, Thomas, and Yoon-Nelson models of fixed bed adsorption [22]. The inflection point of a sigmoidal curve predicted by the logistic equation is invariant, and is located at the curve's mid-point. Accordingly, the inflection point of an Sshaped curve predicted by the original isotherm is always located at q/N0 = 0.5. This means that the original isotherm produces sigmoidal curves that are always symmetric, that is, the convex and concave curves on either side of the inflection point have the same curvature. The data fitting ability of the original isotherm is thus somewhat limited since it is confined to fitting highly symmetric S-shaped curves. By contrast, the inflection point of the modified isotherm varies with the parameter k1, and is given by the following expression: With a floating inflection point, the modified isotherm can fit both symmetric and asymmetric S-shaped curves to a significant degree of precision. Like the Sips isotherm, the modified Krishnamurti isotherm has no theoretical merit and should thus be treated as a convenient empirical representation of S-shaped equilibrium data.

Conclusions
This study tested the ability of the Krishnamurti isotherm to fit S-shaped equilibrium data of water contaminants. The Krishnamurti isotherm was found to track closely the sigmoidal data trends of ammonium ion adsorption by sepiolite and ciprofloxacin adsorption by a nanosorbent. Thus, its practical value is evident.
However, the isotherm was found less accurate in representing the S-shaped equilibrium data of fluoride adsorption by a layered double hydroxide adsorbent. In this particular case, a modified form of the Krishnamurti isotherm was found to outperform its original counterpart. Both the original and modified Krishnamurti isotherms are useful additions to the existing pool of isotherm equations capable of fitting S-shaped equilibrium data. Further testing using observed equilibrium data of other water contaminants will be required to confirm that they can indeed track different types of sigmoidal equilibrium data.

Funding
This work was not in any way directly or indirectly supported, funded, or sponsored by any organization or entity.

Declaration of competing interest
The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.