The empirical value of the statistical thickness t is expressed in the form of the equation of Harkins et al. [1] [7] [8] [9] [10] , valid for as long as (P/P₀) < 0.8:

† ( A ˙ ) = 13.99 0.034 − log ( P P 0 ) (1)

The Magee proposes a calculation of the statistical thickness t for carbon black valid for 0.2 < (P/P₀) < 0.5 [1] [6] [11] :

t ( A ˙ ) = 0.88 ( P P 0 ) 2 + 6.45 ( P P 0 ) + 2.98 (2)

Halsey’s statistical thickness is represented by the following equation [11] [12] :

t ( A ˙ ) = 3.54 ( − 5 In P P 0 ) 1 3 (3)

Jaroniec’s statistical thickness is given by the following equation [8] :

t ( A ˙ ) = ( 60.65 0.03071 − log P P 0 ) 0.3968 (4)

The external specific surface σ e x t is obtained by determining the slope (standard linear regression) from the graph of the quantity of nitrogen adsorbed per gram of sample (Qa) as a function of the statistical thickness t. The value of σ e x t is calculated using equation Halsey statistical thickness is represented by the following equation [11] Halsey statistical thickness is represented by the following equation [1] [13] :

σ e x t = P × 15.47 (5)

With

P, the value of the slope of the curve va = f (t);

15.47: a constant related to the conversion of the volume of nitrogen and the units in m²/g.

If the statistical thickness is given in nm, a conversion factor is added, giving:

σ e x t = P × 1.547 (6)

The difference between the specific surface area value determined by the BET method and the external specific surface area obtained by the t method is used to determine the microporous surface area according to equation [1] [13] :

σ B E T − σ e x t = σ m i c r o p o r e s (7)

Four criteria are used to compare and select models: the adjusted R² criterion, the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Chi-square. The best model is the one with the lowest AIC, the lowest BIC, the highest R²adj and the lowest Chi-square [14] . The values of the criteria were obtained using Origin pro software.

Nitrogen adsorption experiments on clay concretes were carried out using a Micro Active for ASAP 2460 Version 2.01 apparatus.

Table 1 gives the geotechnical characteristics of the clay soil.

The sugar cane molasses used comes from the “Société Agricole de Raffinage Industriel du Sucre (SARIS-Congo)”, a sugar industry organized in the town of Nkayi, Republic of Congo. The molasses used has the following characteristics: the Brix value is 82.85%, which represents a sugar content of 82.85%. The corresponding polarity is 29.07% and the purity 35.09% [2] .

Table 2 shows the composition of stabilized and non-stabilized clay concretes.

The experimental ratio of nitrogen absorption on clayey concrete, in particular the relative pressure (P/P₀) and the quantity of nitrogen adsorbed, is given in Table 3. The statistical thickness values according to the models obtained using: formula 1 for the Harkins and Jura model, formula 2 for the Carbon Black model, formula 3 for the Halsey model, formula 4 for the Jaroniec et al. model, are also presented in Table 3.

The shape of the curves in Figure 1 reveals capillary condensation in the pores that begins as the slope starts to increase, which justifies the presence of condensed water molecules in the mesopores, between the particles and grains in the clay concrete, which is obvious because clay concrete is water-based with an initial water content of 63% (Table 2).

Figure 2 shows the linear interpolation of the curves for the amount of nitrogen adsorbed per gram of sample.

Linear interpolation of the curves showing the quantities of nitrogen adsorbed as a function of statistical thickness (Figure 2), enabled us to obtain linear regression lines of the form Q_ads = Pt + Q₀ (y = Q_ads, x = t (statistical thickness)). The slope P is used to calculate the external surface of the material (formula 6). The ordinate at the origin Q₀ is used to obtain the volume of the micropores. However, in this study we are only interested in the surface area of the micropores. Table 4 gives the values of the model selection criteria clay concrete.

The Hasley model has the highest R a d j 2 value, the lowest χ 2 value, the lowest BIC value and the lowest AIC value (Table 4), indicating that it is the best performing of the four models, followed by the carbon black model, then the Jaroniec model and finally the Harkins and Jura model.

Table 5 gives the ratio of nitrogen absorption on clay concrete stabilized at 8%, i.e. the relative pressure (P/P₀) and the corresponding quantity of adsorbed nitrogen. It also gives the statistical thickness values calculated according to the Carbon Black model, the Harkins and Jura model, the Halsey model and the Jaroniec model.

The shape of the curves (Figure 3) reveals multilayer adsorption on a surface with low porosity, which can be explained by the occupation of the pores by sugarcane molasses. We observe a straightening of the curves (linearization) compared with the curves obtained for non-stabilized clay concrete. This corresponds to the absence of capillary condensation in the mesopores between the particles and the grains. In fact, the presence of molasses, mainly made up of sucrose, glucose and fructose [2] , which are molecules rich in hydroxyl groups (OH) in the mesopores of the material, favours the establishment of hydrogen bonds with the water molecules. The latter are then integrated into the molasses macromolecules in the material’s mesopores. The linearization of these curves will become clearer as the molasses content of the clay concrete increases.

The linear interpolation of the curves for the quantity of nitrogen adsorbed per gram of sample (Figure 3) is given in Figure 4.

Table 6 gives the values of the model selection criteria for clay concrete stabilized with 8% molasses.

The Harkins and Jura model has the highest R²_adj value, the lowest χ² value, the lowest BIC value and the lowest AIC value (Table 6), indicating that it is the best performing of the four models, followed by the Jaroniec model, then the Halsey model and finally the Carbon black model.

Table 7 gives the experimental ratio of nitrogen absorption on clay concrete stabilized at 12%: the relative pressure (P/P₀) and the quantity of nitrogen adsorbed. The statistical thickness values according to the models are calculated using formulae 1, 2, 3 and 4 (page 3).

The shape of the curves (Figure 5) shows a linearization with respect to the curve at 0% and 8%, which corresponds to multilayer adsorption on a surface with low porosity, which can be justified by the occupation of the pores by sugarcane molasses.

The linear interpolation of the curves for the quantity of nitrogen adsorbed per gram of sample (Figure 5) is shown in Figure 6.

Table 8 gives the values of the model selection criteria for 12% stabilized concrete.

The Harkins and Jura model has the highest R²adj value, the lowest χ² value, the lowest BIC value and the lowest AIC value (Table 8), indicating that it is the best performing of the four models, followed by the Jaroniec model, then the Halsey model and finally the Carbon black model.

The experimental ratio of nitrogen absorption on clay concrete stabilized at 16% and the statistical thickness values calculated using the model formulae (page 3) are shown in Table 9.

The curves in Figure 7 are more linear than at 8% and 12% and at 0%, which corresponds to multilayer adsorption on a surface with low porosity, which may be justified by the prior occupation of the pores by sugarcane molasses. The adjusted correlation coefficients obtained for the Harkins and Jura model: 0.99836 (Table 6) for clay concrete stabilized at 8%, 0.99825 (Table 8) for clay concrete stabilized at 12%, 0.99936 (Table 10) for clay concrete stabilized at 16%, show an increase in linearization (Figure 8) with molasses content, which can be explained by the absence of condensation in the pores, as the molasses molecules occupy these pores.

The linear interpolation of the curves for the quantity of nitrogen adsorbed per gram of sample is shown in Figure 8.

Table 10 gives the values of the model selection criteria for 16% stabilized concrete.

The Harkins and Jura model has the highest R²_adj value, the lowest χ² value, the lowest BIC value and the lowest AIC value (Table 10), indicating that it is the best performing of the four models, followed by the Jaroniec model, then the Halsey model and finally the Carbon black model.

Table 11 gives the values of slope and external specific surface as a function of the statistical thickness models.

Table 12 gives the BET specific surface area and micropore surface area values for stabilized and non-stabilized concretes according to the models used.

The results obtained show that the Halsey model performs best for non-stabilized clay soil. However, the external specific surface area values obtained by this model are higher than the BET specific surface area values, so it is not possible to obtain the micropore surface area by this model. Thus, the Carbon Black model becomes the most relevant for non-stabilized clayey concrete, followed by the Jaroniec model, and finally by the Harkins and Jura model.

The results obtained for stabilized clay concretes show that the Harkins and Jura model is the most relevant, followed by the Jaroniec model and, finally, the Carbon Black model. The Halsey model does not provide micropore surfaces for the same reason as mentioned above.

This change compared to unsterilized clayey concrete leads us to believe that stabilization using molasses brings about modifications to the external surface of clayey concrete by occupying the pores [4] . Ngouallat Mfoutou (2020) reports that 53.10⁸ molasses molecules occupy the accessible surface of clay concrete [3] .

The change in the shape of the curve from the curved form for clay concrete without molasses to the more linear form for stabilized clay concretes also expresses the change in texture of clay concrete from a porous material to a less and less porous material.

According to Tchemmou and Gherbi (2018), the t-plot method can be used to determine the external surface of microporous materials of the zeolite type [7] . For Magee (1995) cited Moulin (2018) [1] , the Carbon black model is designed for the specific case of carbon blacks. Nevertheless, the results obtained show that the Carbon black model is also relevant for the case of clay concretes. Ngouallat (2022) used the Carbon Black statistical thickness model to obtain the textural properties of clay concretes stabilized with molasses [2] .

According to Jeffrey Kevin (2008), the Carbon black statistical thickness model can also be used to determine the external surface of cetyltrimethyl ammonium bromide [3] .

The carbon black and Halsey equations (Eq. 2 and 3) are applicable to carbon black and related materials [15] .

According to Yijing Zheng (2008), the t-plot method (Halsey model, Harkin and Jura model, Carbon black model) used for the physical characterization of common adsorbents may not be applicable to carbon nanotubes [5] .

The Harkins and Jura equation has been developed for well-selected alumina samples; however, it is applicable to other materials, including graphitized carbon blacks [17] .

The most commonly applied thickness curves are those of Harkin-Jura and Halsey (Webb and Orr, 1997) [6] . Utpalendu Kuila and Manika Prasad (2011), have used Halsey thickness curves to estimate micropore volume and “open area”, mesopore area, macropore area and external area (i.e. total area excluding micropore area) in natural clay minerals and shales [18] .