S-shaped adsorption isotherms modeled by the Frumkin–Fowler–Guggenheim and Hill–de Boer equations

In this study, we used the Frumkin–Fowler–Guggenheim (FFG) and Hill–de Boer (HDB) isotherm models to describe previously published S-shaped isotherms involving water contaminant adsorption, specifically auramine-O and ortho-nitrophenol, onto solid materials. Although the two models differ in their mathematical formulations, they share three key parameters: the monolayer adsorption capacity (qm), the equilibrium constant representing adsorbate−adsorbent interactions, and the equilibrium constant representing adsorbate–adsorbate interactions. Upon comparing the data fitting capabilities of the FFG and HDB models, we found minimal differences. However, substantial discrepancies emerged in the fitted qm values. The FFG model yielded qm values that aligned with the plateaus evident in the auramine-O and ortho-nitrophenol isotherms, whereas the HDB model did not. This inconsistency in the HDB model was also observed when analyzing a hyperbolic tetracycline isotherm. In addition, we delved into the impact of the adsorbate−adsorbent and adsorbate–adsorbate interaction constants on simulated isotherm curves. Our findings revealed that the adsorbate−adsorbent interaction constant influenced the steepness of the simulated curves, while the adsorbate–adsorbate interaction constant determined the nature of isotherm curve generated by the two models (S-shaped or hyperbolic).

Extended author information available on the last page of the article Both models account for lateral interactions among adsorbed molecules within a monolayer on a homogeneous surface.The primary distinction lies in the assumption regarding adsorption behavior.The FFG model assumes localized adsorption, whereas the HDB model considers a mobile monolayer.Formulated for aqueous phase adsorption, the FFG and HDB models are expressed by Eqs. ( 1) and (2), respectively.In these equations, c represents the equilibrium aqueous phase concentration, q represents the corresponding adsorbed phase concentration, q m denotes the monolayer adsorption capacity, and k 1 through k 4 are constants.The parameters k 1 and k 3 characterize adsorbate−adsorbent interactions, while k 2 and k 4 represent lateral adsorbate-adsorbate interactions.Equations ( 1) and (2) demonstrate the mathematical similarity between the FFG and HDB models, with the latter incorporating an additional concentration ratio within the exponential term to account for mobile adsorption.
In cases where lateral interactions are absent (k 2 or k 4 = 0), the FFG model simplifies to Eq. ( 3), which corresponds to the well-known Langmuir model [5].Conversely, the HDB model transforms into the Volmer model [6], as shown in Eq. (4).Equation (5) gives an alternative representation of the Volmer model, which is explicit in q, where W(x) denotes the Lambert W function with x = k 3 c [7].
In aqueous phase adsorption studies, Eqs. ( 1) and ( 2) are rarely used to correlate isotherm data.Instead, their linear forms, denoted by Eqs. ( 6) and (7), respectively, are generally preferred.Equation (6) or (7) suggests that plotting the left-hand side against q/q m would result in a straight line.However, generating such linear plots requires prior knowledge of q m . (1) Many studies have used Eq. ( 6) or (7) for isotherm data analysis without disclosing the method used to estimate q m .Examples of recent studies following this approach include those conducted by Ramadoss and Subramaniam [8], Wu et al. [9], Ali et al. [10], Mathangi et al. [11], Hamadeen and Elkhatib [12], and Elkhatib et al. [13].In the case of the FFG model, various methods have been used to estimate q m [14].However, the reported values of q m have been found to be highly questionable [15].Treating q m in both the FFG and HDB models as a known parameter lacks scientific justification.Its primary purpose is to facilitate the use of linear regression for estimating the remaining unknown parameters k 1 -k 4 in these two models.
Another issue related to the application of the FFG and HDB models pertains to fitting hyperbolic or type I isotherm data.Such simple curve shapes generally pose minimal challenges for the FFG and HDB models.In fact, simpler models like the Langmuir [5] and Freundlich [16] isotherms provide adequate representations for this type of data.Apart from interpreting hyperbolic isotherms, the FFG and HDB models are also capable of capturing S-shaped or type V isotherms [17].In the recent studies investigating water contaminant adsorption [18][19][20], the FFG and HDB models were evaluated using S-shaped isotherm data.However, a comprehensive comparison between these two models has not been reported.The objective of the present study was not only to critically assess and compare the FFG and HDB models in relation to S-shaped isotherm data, but also to explore the impact of model parameters on simulated isotherm curves.

Data fitting method
The FFG and HDB models defined by Eqs. ( 1) and (2) were fitted to published experimental data by minimizing the sum of squared residuals (SSR) as described in Eq. ( 8), where q i,obs represents the experimental value of q, q i,fit denotes the value of q calculated from the chosen isotherm model, and m is the number of data points.Engauge Digitizer (https:// marku mmitc hell.github.io/ engau ge-digit izer/) was used to extract data points from the original sources.
It should be noted that Eqs. ( 1) and (2) are implicit in q, indicating that there is no closed algebraic expression of q in terms of the model parameters and c.To obtain q i,fit from Eq. ( 1) or (2), a numerical approach using Newton's method was integrated into a nonlinear least-squares fitting procedure implemented in Mathematica.The specific Mathematica commands employed were FindRoot and Nonlinear-ModelFit.The uncertainties associated with all estimated parameter values were determined with a confidence level of 95%.A comprehensive report detailing parameter estimation using Mathematica can be found in the existing literature [21].Alternatively, a nonlinear least-squares method called orthogonal distance regression can be used for fitting the loading-implicit FFG and HDB models to experimental data.This data fitting procedure, available in OriginPro, minimizes the residual errors in both c and q.

Verifying the data fitting method
In the field of aqueous phase adsorption, the FFG model is often used in a linearized form with two unknown parameters (k 1 and k 2 ) for correlating hyperbolic adsorption isotherms.This modeling approach has been discussed in some detail in the work of Chu and Tan [15].The HDB model is handled in a similar manner as demonstrated in previous work [8][9][10][11][12][13][14].However, only a limited number of studies have focused on extracting the three unknown parameters of the FFG model (q m , k 1 , and k 2 ) or the HDB model (q m , k 3 , and k 4 ) from S-shaped isotherm data.The studies conducted by Fideles et al. [18] and Teodoro et al. [19] are noteworthy in that the FFG and HDB models were fitted to S-shaped isotherm data by means of nonlinear regression.
Panels a and b of Fig. 1 present the S-shaped isotherm illustrating the adsorption of auramine-O dye on a modified cellulose adsorbent at pH 7, as extracted from the study by Teodoro et al. [19].In their study, Teodoro et al. [19] fitted the FFG and HDB models to the experimental isotherm using a nonlinear regression procedure implemented in MATLAB.The parameter estimates reported by Teodoro et al. [19] are reproduced in Table 1.In our study, we used a nonlinear least-squares regression procedure coded in Mathematica to fit the FFG and HDB models to the auramine-O adsorption data.Panels a and b of Fig. 1 display the fits generated by the two models.Our parameter estimates presented in Table 1 show excellent agreement with those reported in the original study, thereby validating the accuracy of our data fitting procedure.Although panels a and b of Fig. 1 demonstrate that the modeling capabilities of the FFG and HDB models were similar, there was significant disparity in the resulting q m values (indicated by the dashed lines in panels a and b of Fig. 1), as observed also by Teodoro et al. [19].Notably, the fitted value of q m in the FFG model (q m = 5.44 mmol g -1 ) corresponded well with the experimental plateau.In contrast, the fitted value of q m in the HDB model (q m = 7.39 mmol g -1 ) substantially exceeded the experimental plateau.Consequently, it can be concluded that q m in the FFG model holds physical significance while the same term in the HDB model does not.
From a modeling perspective, the relatively close agreement between the calculated isotherm and the experimental data indicates that either of the two models can be used as a suitable mathematical representation for the equilibrium behavior of auramine-O.Panels c and d of Fig. 1 present the residual plots for the FFG and HDB model fits, respectively.In both cases, the presence of relatively large residuals in the low concentration range suggests that the convex portion of the experimental isotherm was not accurately captured by the two models.

Data fitting using linear and nonlinear regression methods
As mentioned, the linearized versions of the FFG and HDB models are commonly used to describe hyperbolic isotherm data.This linearization approach has also been extended to S-shaped isotherm data.For instance, in the study conducted by Benosmane et al. [20], linear forms of the FFG and HDB models were fitted to the S-shaped isotherms representing the adsorption of paracetamol and niflumic acid on a hyper-crosslinked cellulosic adsorbent.However, it is worth reiterating that the process of plotting the linearized FFG and HDB models, as defined by Eqs. ( 6) and ( 7), requires prior knowledge of q m .The methodology used for estimating q m prior to performing the linear fits reported in the study by Benosmane et al. [20] remains unclear.
In a separate investigation conducted by Koubaissy et al. [22], the S-shaped isotherm for ortho-nitrophenol adsorption on a zeolite adsorbent (referred to as FAU in the original study) was analyzed using the linearized form of the FFG model described in Eq. (9).In this equation, K and w are constants, R is the gas constant, and T is the absolute temperature.The reported parameter estimates for K and w were 0.0134 dm 3 mg -1 and −4.2 kJ mol -1 , respectively.Although the specific method used to estimate q m was not mentioned, Koubaissy et al. [22] provided a value of 240 mg g -1 for q m .This q m value was likely obtained through a rough estimation of the plateau observed in the experimental isotherm.With the known value of q m , it becomes feasible to generate a linear plot using Eq. ( 9).
To validate the reported values of K and w, we applied Eq. ( 9) to the ortho-nitrophenol adsorption data using a linear regression procedure.The resulting linear fit, presented in panel a of Fig. 2, exhibited a satisfactory fit quality with an R 2 value of 0.993.However, the value of K, obtained from the intercept of the linear fit in panel a of Fig. 2, was estimated to be 0.009 dm 3 mg -1 , which differed from the reported value of K. Similarly, the value of w, obtained from the slope of the linear fit in panel a of Fig. 2, was −5.1 kJ mol -1 (at T = 298.15K), demonstrating a discrepancy with the reported value of w.It is noteworthy that the reported value of w (−4.2 kJ mol -1 ) closely resembled the slope of the linear fit (−4.12), which corresponds to 2w/RT as described by Eq. (9).
Upon comparing Eqs. ( 6) and ( 9), it becomes evident that K is equivalent to k 1 and 2w/RT corresponds to −k 2 .As a result, the linear fit conducted in this study on the orthonitrophenol adsorption data resulted in estimated values of k 1 = 0.009 dm 3 mg -1 and k 2 = 4.12, as indicated in Table 2.These estimates of k 1 and k 2 , along with the assumed value of q m (240 mg g -1 ), were used in Eq. ( 1) to generate the q versus c plot depicted in panel b of Fig. 2. The FFG prediction displayed in panel b of Fig. 2 exhibited a reasonably good fit with an R 2 score of 0.973, considering that the k 1 and k 2 estimates were derived from linear regression.
Nevertheless, as illustrated in the analysis of the auramine-O isotherm, the FFG model can also be fitted to experimental data using nonlinear regression to estimate all three unknown parameters.Figure 3 showcases the nonlinear fit of the FFG model to the ortho-nitrophenol adsorption data, while Table 2 presents the fitted parameters.The FFG model fit presented in Fig. 3   In the original study [22], the HDB model was not applied to the ortho-nitrophenol adsorption data.However, in our analysis, we applied the linearized HDB model given by Eq. ( 7) to the ortho-nitrophenol adsorption data, assuming a q m value of 240 mg g -1 , as depicted in Fig. 4. Notably, the linearized HDB model exhibited poor performance with an R 2 value of 0.646.The fitted values for k 3 and k 4 are presented in Table 3, where it is observed that their uncertainties were either comparable to or much larger than the fitted values.
Figure 5 shows the nonlinear fit of the HDB model to the ortho-nitrophenol adsorption data.The fitted values for k 3 and k 4 derived from this nonlinear regression analysis significantly deviated from those obtained via the linear regression analysis, as demonstrated in Table 3.In addition, Fig. 2 a Fitting Eq. ( 9) to ortho-nitrophenol adsorption data [22] by linear regression.b Isotherm curve calculated from the FFG model using k 1 and k 2 estimates obtained by linear regression and the assumed value of q m Table 2 Parameter estimates obtained by linear and nonlinear fits of the FFG model to ortho-nitrophenol adsorption data [22] Parameter Linear regression Nonlinear regression q m /mg g -1 -248.1 ± 18.3 k 1 /dm 3 mg -1 0.009 ± 0.002 0.012 ± 0.002 k 2 4.12 ± 0.33 3.54 ± 0.39 Fig. 3 Fitting the nonlinear FFG model to ortho-nitrophenol adsorption data [22] Fig. 4 Fitting Eq. ( 7) to ortho-nitrophenol adsorption data [22] by linear regression Table 3 Parameter estimates obtained by linear and nonlinear fits of the HDB model to ortho-nitrophenol adsorption data [22] Parameter Linear regression Nonlinear regression q m /mg g -1 -333.7 ± 37.6 k 3 /dm 3 mg -1 0.0003 ± 0.003 0.007 ± 0.002 k 4 22.7 ± 19.3 6.21 ± 0.90 the fitted value of q m obtained from the nonlinear regression analysis (q m = 333.7 mg g -1 ) was noticeably larger than the assumed value of q m used in the linear regression procedure (q m = 240 mg g -1 ).With an R 2 score of 0.954, the HDB model fit presented in Fig. 5 was inferior to the FFG model fit shown in Fig. 3 (R 2 = 0.995).As observed in Figs. 3 and  5, the q m value obtained from the HDB model fit was substantially larger than that produced by the FFG model fit, consistent with the anomaly initially observed in Fig. 1.

Fitting hyperbolic isotherm data
To further investigate the issue regarding the erroneous estimation of q m values during data correlation, the HDB model was applied to a typical hyperbolic isotherm extracted from the study conducted by Shen et al. [23], involving tetracycline adsorption on carboxymethyl starch-modified magnetic bentonite.Figure 6 displays the hyperbolic isotherm data alongside the HDB model fit, calculated using the parameter estimates provided in Table 4.The fitted value of q m , shown in Fig. 6, exceeded the plateau of the experimental isotherm.This observation suggests that regardless of the type of isotherm data to which it is applied, the HDB model will yield an inaccurate q m value.Consequently, it can be inferred that the HDB model suffers from an inherent deficiency in its mathematical formulation.
The tetracycline isotherm was also fitted to the FFG model, as demonstrated in Fig. 7.The fitted curve was generated using the parameter estimates provided in Table 4.The R 2 values in Table 4 indicate that both the FFG and HDB models exhibited good agreement with the tetracycline isotherm.Upon comparing Figs. 6 and 7, it becomes apparent that the FFG model fit closely resembled the HDB model fit, with the exception that the former's q m value aligned more closely with the plateau of the experimental isotherm.This finding was consistent with the results obtained from fitting the FFG model to the S-shaped auramine-O isotherm.As a result, the FFG model was highly effective in representing both the S-shaped and hyperbolic isotherm data.However, it is important to note that the uncertainties associated with k 2 Fig. 5 Fitting the nonlinear HDB model to ortho-nitrophenol adsorption data [22] Fig. 6 Fitting the nonlinear HDB model to tetracycline adsorption data [23] Table 4 Parameter estimates obtained by nonlinear fits of the HDB and FFG models to tetracycline adsorption data [23] HDB parameter Value FFG parameter Value q m /mg g -1 218.6 ± 28.4 q m /mg g -1 159.8 ± 16.3 k 3 /dm 3 mg -1 0.021 ± 0.005 k 1 /dm 3 mg -1 0.031 ± 0.008 Fig. 7 Fitting the nonlinear FFG model to tetracycline adsorption data [23] and k 4 in the FFG and HDB models, respectively, exceeded their fitted values, as shown in Table 4.

Effects of model parameters
There are three adjustable parameters that influence the shapes of isotherm curves predicted by the FFG and HDB models.The impact of q m can be intuitively understood without resorting to modeling, as it primarily governs the isotherm plateau.To put it simply, when both models predict a higher q m value, it corresponds to a higher level of the predicted isotherm plateau.In this study, the focus was placed on investigating the effects of k 1 and k 2 in the FFG model, as well as k 3 and k 4 in the HDB model.These parameters elucidate the interactions between the adsorbate and adsorbent (k 1 and k 3 ) and the lateral interactions between adsorbate molecules (k 2 and k 4 ), as mentioned earlier.A comprehensive discussion on the effects of these parameters on adsorption at the solid-gas interface can be found in a book published approximately three decades ago [24].Figure 8 presents a series of simulated curves generated by the FFG model while varying the values of k 1 and k 2 .The isotherm curves illustrated in panel a of Fig. 8 were obtained by varying k 1 from 1 to 4 dm 3 mmol -1 , while keeping q m and k 2 constant (q m = 5 mmol g -1 ; k 2 = 3).Negative values of k 1 were not selected since, by definition, k 1 is a positive constant.As observed in panel a of Fig. 8, an increase in the value of k 1 led to progressively steeper S-shaped isotherm curves.Even with k 1 exceeding 4 dm 3 mmol -1 , the resulting isotherm curves remained S-shaped (not shown).It can be concluded that k 1 alone did not determine the type of isotherm curve generated by the FFG model, be it S-shaped or hyperbolic.When k 2 is zero, the FFG model reverts to the Langmuir model.Consequently, k 1 assumes the role of the equilibrium constant in the Langmuir model.It is recognized that hyperbolic curves predicted by the Langmuir model become steeper or more favorable as the value of the equilibrium constant increases.Hence, in both the FFG and Langmuir models, the parameter k 1 serves the same purpose.
Panel b of Fig. 8 displays a set of simulated curves generated by the FFG model by varying k 2 from −1 to 4, while maintaining q m constant at 5 mmol g -1 and k 1 constant at 2 dm 3 mmol -1 .The parameter k 2 represents the nature of the adsorbate-adsorbate interaction, where a positive value indicates an attractive interaction and a negative value indicates a repulsive interaction between adsorbed molecules.In the case of Eq. ( 9), the parameter w is negative for an attractive interaction and positive for a repulsive interaction [25].Upon observing panel b of Fig. 8, it becomes evident that the FFG model generated hyperbolic isotherm curves when k 2 = 0 or −1.As the value of k 2 increased, the FFG model produced S-shaped isotherm curves with an inflection point.Therefore, the parameter k 2 determined the type of isotherm curve generated by the FFG model (hyperbolic or sigmoid).
The results obtained when simulating the effects of k 3 and k 4 on isotherm curves generated by the HDB model are presented in Fig. 9.A quick comparison of Figs. 8 and  9 indicates that the effects of k 3 and k 4 were similar to those of k 1 and k 2 , respectively.To analyze the impact of k 3 , its values were varied from 1 to 4 dm 3 mmol -1 .Using these values, along with q m = 5 mmol g -1 and k 4 = 4, the HDB model generated the isotherm curves shown in panel a of Fig. 9. Consistent with the effect of k 1 demonstrated in panel a of Fig. 8, panel a of Fig. 9 shows that an increase in k 3 resulted in sharper S-shaped isotherm curves.To investigate the influence of k 4 , its values were varied from −2 to 6 while maintaining q m at 5 mmol g -1 and k 3 at 1 dm 3 mmol -1 .The results of these simulations are presented in panel b of Fig. 9.It is evident that k 4 exerted a major influence on the type of isotherm curve generated by the HDB model, aligning with the previous observation

Conclusions
In this study, we assessed the performance of the FFG and HDB models in fitting previously published S-shaped isotherms involving water contaminant adsorption onto solid materials.Since both models are implicit in q, a nonconventional least-squares fitting algorithm was used in the data fitting process.The results obtained in this study indicated that there was minimal discrepancy in the modeling capabilities of the FFG and HDB models.Nevertheless, the fitted q m values in the HDB model exhibited considerable deviation from the plateaus observed in the experimental isotherms, raising concerns about the physical significance of q m within the model.To unequivocally ascertain the physical meaning of q m in the HDB model, further testing using experimental data from other water contaminants will be essential.In contrast, the fitted values of q m in the FFG model aligned well with the plateaus of the experimental isotherms, making the FFG model a recommended choice for modeling sigmoid isotherm data.In addition, we explored the influence of model parameters on simulated curves generated by the two models.The adsorbate−adsorbent interaction constant in both the FFG and HDB models influenced the steepness of the simulated isotherm curves, while the adsorbate-adsorbate interaction constant governed the type of isotherm curve generated by both models (S-shaped or hyperbolic).

Fig. 1
Fig. 1 Fitting the FFG (a) and HDB (b) models to auramine-O adsorption data [19].Residual plots for the FFG (c) and HDB (d) model fits higher accuracy as compared to the calculated FFG curve shown in panel b of Fig. 2 (R 2 = 0.973).

Table 1
Parameter estimates obtained by nonlinear fits of the FFG and HDB models to auramine-O adsorption data