Accuracy of Dubinin-Astakhov and Dubinin-Radushkevich Adsorption Isotherm Models in Evaluating Micropore Volume of Bentonite

Accuracy of Dubinin-Astakhov (DA) and Dubinin-Radushkevich (DR) isotherm models were examined for evaluating micropore volume of several natural and pillared bentonite samples with type I and II isotherms. Afterwards, the obtained results were compared with the micropore volume calculated by Horvath–Kawazoe (HK) model. The results showed that the accuracy of DA and DR models for this purpose depends on isotherm types and range of relative pressure (P/P0) in which the models are applied. It can be generally concluded that DA and DR can be used to calculate micropore volume of bentonite instead of HK which imposes more complicated calculations.


Introduction
Clay can be used in several applications due to its physical and chemical properties, such as large specific surface area, cation exchange capacity and adsorptive affinity for inorganic and organic ions from aqueous solution and water.Among the clays, bentonite is considered as a main candidate in the removal of pollutants such as lead and other heavy metal ions.Bentonite consists mostly of microcrystalline particles of montmorillonite which belongs to 2:1 clay minerals meaning that it has two tetrahedral sheets sandwiching a central octahedral sheet [1][2][3][4].Since the existence of huge deposits of bentonite, there is a great potential for its utilization in removing various dilute pollutants and adsorption technology including the removal of heavy metal ions [5][6][7][8], phenols [9][10][11], organic molecules [12,13], polymers [14,15], pesticides [16], radio nuclides [17], dyes [18][19][20].To apply bentonite in adsorption, the textural properties including surface area, pore size, micropore volume and mesopore volume must be determined.The textural properties of porous materils are very often determined by gas (N 2 ) adsorption isotherm data [8,21,22].
The empirical form of an adsorption isotherm was recognized as early as 1926 by Freundlich [23], and was later derived theoretically from the Langmuir equation [24] extended to heterogeneous surfaces considered to be a composite surface, composed of many homogeneous patches [25].By adopting the Langmuir mechanism, but introducing a number of simplifying assumptions, the Brunauer-Emmet-Teller (BET) equation was subsequently derived for multilayer adsorption [26].It is unquestionable that together with the concept of multilayer adsorption (leading to the BET equation) the theory of volume filling of micropores is one of the most stimulating concepts occupying the principal position in adsorption science [27].Developed formerly, and based on the Weibull distribution of adsorption potential, the proposed equation by M. M. Dubinin [28] was considered to be a semi-empirical one.The fundamental relations are the Dubinin-Astakhov and Dubinin-Radushkevich equations [28,29].The surface area, mesopore volume and micropore volume of such porous materials can be evaluated by means of BET, BJH [30] and both HK [31] and density functional theory (DFT) [32] methods, respectively.According to literature, the HK and DFT methods are accurate ones; however they're somewhat complicated and time-consuming [33,34].
In this paper DA and DR isotherm models were applied to calculate the micropore volume of some bentonite samples and the results were compared with the results obtained by HK model.To calculate micropore volume by the methods, the data collected through N 2 adsorption-desorption isotherms was used.Several authors have applied the methods based on the HK model to determine the micropore size distributions and micropore volume of bentonite [21,22], [35][36][37][38][39][40].Therefore, this model is considered as a valid one.However, as mentioned above, this model is complicated and time-consuming.In short, the aim of this paper is to study whether DA and DR isotherm models, which are simpler than HK, are accurate to assess micropore volume of bentonite or not.

Theoretical approach 2.1 Dubinin-Astakhov and Dubinin-Radushkevich equations
Dubinin-Astakhov and Dubinin-Radushkevich equations are given, respectively, in the following forms: In Eqs.(1) and (2), w represents the volume of adsorbate filling the micropores, in the unit mass of adsorbent (cm 3 /g), at temperature T and P/P 0 , w 0 is the maximum volume of adsorbent per adsorbed mass, the micropore volume (cm 3 /g), B is a parameter characterizing the microporous structure, β is the affinity coefficient of the characteristic curves, E 0 is the characteristic energy of adsorption, n is an equation parameter and A is the differential molar work of adsorption i.e., the differential Gibbs energy of adsorption, ∆G, defined by It must be noted that the DA equation is valid at the micropore filling mechanism, over a range of P/P 0 varying from 1E-7 to 0.02, and the DR equation is valid at both micropore filling and sub-monolayer formation mechanisms, over a range of P/ P 0 varying from 1E-7 to 0.02 and 0.01 to 0.1, respectively.

The Horvath-Kawazoe (HK) model
Everett and Powl [41] extended the work of Sams et al. [42] on the interaction energy of one adsorbate molecule with a single infinite-layer plane of adsorbent molecules.They indicated that the potential energy of interaction, ɛ, between one adsorbate molecule and two parallel infinite layer planes whose nuclei are separated by a distance, x, can be expressed via the Lennard-Jones (12:6) potential.Later, Horvath and Kawazoe concluded that for a slit-like shaped pore filled with adsorbate molecules, ɛ, can be arranged as follows: Where d 0 being the arithmetic mean diameter of the adsorbate molecule and the adsorbent atom.In Eq. ( 1), z is the internuclear distance between an adsorbate molecule and the plane of surface atoms of the adsorbent (one plate), N a is the number of oxide ions per unit area of surface, and N A is the number of adsorbate molecules per unit area of adsorbent surface.A a and A A are the dispersion constants, which are given by the Kirkwood-Muller equations [43] as follows: Where m is the mass of an electron, c is the speed of light, α a and λ a are the polarizability and magnetic susceptibility of an adsorbent atom, respectively, and similarly α A and λ A are the polarizability and magnetic susceptibility of an adsorbate molecule.The sum N a A a + N A A A is the so-called interaction parameter, which summarizes the physicochemical characteristics of the adsorbate-adsorbent system.Horvath and Kawazoe indicated the free energy change upon adsorption to the net potential energy of interaction between the layers.Thus, the following expression is derived for a slit-like pore geometry: Where P 0 is the saturation vapour pressure, R is the universal gas constant, and N Av is Avogadro number.
Everett and Powl indicated that the potential energy of interaction, ɛ, between one adsorbate molecule and the inside wall of a cylindrical pore of radius x can be expressed via the Lennard-Jones potential.Thus, for a cylindrical pore composed of a single layer of oxide ions and filled with adsorbate molecules, ɛ, can be arranged as follows: (1) Where r is the distance of the nucleus of a gas molecule from the central axis of the cylinder, and the expansion coefficients, α k and β k , are given by And Where α 0 = β 0 = 1.Using the mentioned results, Saito and Foley [44] obtained the following equation for cylindrical pores by means of a method similar to that of Horvath and Kawazoe: On deriving this equation, the authors deemed that adsorption only takes place on the inside wall of the cylinder for micropores, and approximated interactions upon adsorption by the interactions between the gas molecules and the oxide ions of the adsorbent.Taking into account the experimental results that can be attained from adsorption of probe molecules, adsorbed volume and P/P 0 at equilibrium conditions, the pore size distribution of a given material can be derived from Eqs. (8) or (12).In both cases, the microporous volume is calculated as the adsorbed volume corresponding to pores with size up to 20 Å [45].It must be mentioned that the HK method is valid at the micropore filling mechanism.

Result and discussion
In this work, the nitrogen adsorption-desorption isotherm experimental data [22,23] was used to calculate micropore volume.The data is concerning a natural bentonite, S 1 , some intercalated and pillared bentonite samples, S 2 , S 3 , S 4 and S 5 , and some pillared bentonite samples, S 6 , S 7 , S 8 , S 9 , S 10 , S 11 and S 13 .Figure 1 illustrates the nitrogen adsorption-desorption isotherm data of the mentioned samples.
As can be seen in Fig. 1A, the adsorption isotherm of the natural clay, S1, is of type II in the Brunauer, Deming and Teller (BDDT) classification [43].The adsorption isotherms at low P/ P 0 of the intercalated and pillared samples, S 2 , S 3 , S 4 and S 5 , are of type I in the same classification with an important uptake at low P/P 0 , characteristic of highly microporous materials.The Fig. 1 Nitrogen adsorption-desorption isotherms of the samples under study [22,23] Table 1 Textural properties of the samples under study.

Sample
S BET (m 2 /g) V meso (cm 3 /g) V total (cm 3 /g) V meso / V total V HK (cm Abbreviations: S BET , BET specific surface area; V meso , mesopore volume assessed by BJH method; V total , total pore volume at P/ P 0 = 0.99; V HK , micropore volume derived from Korvath-Kawazoe (HK) model.
adsorption isotherm of the other samples is of type II in the mentioned classification (Figs.1B and C) with a lower uptake at low P/P 0 than type I classification.The observed narrow hysteresis behaviour showed that these samples had mesopores with parallel layers.According to the types of isotherms and the above mentioned remarks it can be concluded that S 2 , S 3 , S 4 and S 5 samples contain more micropores volume than other samples.Table 1 summarizes the textural properties of the samples.
As can be seen in Table 1, the fractions of mesopore volume, V meso /V total , in S 2 , S 3 , S 4 and S 5 samples are lower than 0.4.However, these fractions in other samples are above 0.6.Thus, in good agreement with the above mentioned remarks about type I and II isotherms, it can be said that S 2 , S 3 , S 4 and S 5 samples contain more micropore volumes than other samples.As mentioned earlier, the DA equation is valid over a range of P/P 0 varying from 1E-7 to 0.02, and the DR equation is valid over ranges of P/P 0 , from 1E-7 to 0.02 and 0.01 to 0.1 individually.Thus, the adsorption isotherms were fitted to the DA and DR equations over these ranges.Table 2 presents the obtained results, the micropore volume, w 0 , by fitting the adsorption isotherm data of the samples to both the DA and DR equations.According to the correlation coefficients, R 2 , it can be deduced that the adsorption isotherms have properly correlated to the both isotherm models.
In this work, the obtained results through fitting Nitrogen adsorption isotherm data to DA and DR isotherm models were compared with the calculated micropore volume by HK method.Tables 3 and 4 show the results.
Table 3 shows the results concerning the samples with the isotherms of type I.As can be seen in Table 3, the error differences, ∆, relevant to the obtained results by fitting isotherms to DA model are very low and, also, less than the error difference calculated in other cases.
Table 4 indicates the results regarding the samples with the isotherms of type II.As can be seen in Table 4, the error differences, ∆, relevant to the obtained results by fitting isotherms to DR 1 model are very low and, also, less than the error difference calculated in other cases.According to the both tables, as can be predicted, it was concluded that DR 2 , fitting isotherm data to DR model at sub-monolayer formation mechanism, isn't proper to evaluate micropore volumes of the both types of samples, the samples with isotherms of type I and type II.It is likely due to that in set out of this mechanism and after the knee, in the curve isotherms, micropores cease to contribute to the adsorption process.
In DA equation, Eq. ( 1), n is an exponent usually varying between 1 and 3. Exponent n depends on the width of the distribution curve of adsorption energy (of pores with different sizes).As be said about activated carbon [46], values of n equal or above 3 represent molecular sieve carbons with narrow pores, while values less than 2 belong to less microporous carbons with wider pore size distribution.The value of 2 (original DR equation) shows an intermediate condition among different cases.Therefore, based on the above mentioned remarks and Tables 3 and 4, it can be said when a sample contains high micropore volume then its n, in DA equation, is higher than 2. Thus, DR   equation with n equal 2 can't properly evaluate micropore volume of the sample.However, in intermediate condition, about our samples with fraction of micropore volume around 0.4 and less, the DR equation can properly assessed the micropore volume of sample over the range of micropore filling mechanism.

Conclusions
In this paper, micropore volume of several bentonite samples with various textural properties was calculated by DA and DR isotherm models and compared with the micropore volume obtained by HK method.The results showed that DA model is accurate for the sample with high micropore volume and with type I isotherms.In contrast, DR model is valid for the sample having mediocre micropore volume and with type II isotherms.Finally, DR model is not precise for evaluating micropore volume of the all samples over the range of sub-monolayer mechanism (0.01˂ P/P 0 ˂0.1).

Table 2
Micropoe volume calculated by DA and DR.DA , correlation coefficient in fitting the isotherm data to DA equation; R 2 DR1 , correlation coefficient in fitting the isotherm data to DR equation in the range 0.0001˂ P/ P 0 ˂0.01;R 2 DR2 , correlation coefficient in fitting the isotherm data to DR equation in the range 0.01˂ P/ P 0 ˂0.1.

Table 3
Comparison of the micropore volume obtained by DA and DR isotherm models by HK method for the samples having type I isotherms.

Table 4
Comparison of the micropore volume obtained by DA and DR isotherm models by HK method for the samples having type II isotherms.