**5. Modeling example**

This paragraph is intended to mention the modeling details for two different samples SWy-Na and SWy-Ba, where the CEC is fully saturated by Na+ and Ba2+ cations.

#### **5.1 SWy-Na sample**

Qualitative XRD analysis shows a homogeneous 1 W hydration state (12.33 Å; **Figure 17**). A low FWHM and ξ parameter value confirms the symmetric shape of the 001 reflection (**Table 2**).

#### **Figure 17.**

*Best agreement between theoretical and experimental XRD profiles obtained in the case of SWy-Na. \* Halite (NaCl). The sample used here is Wyoming montmorillonite (SWy-2) exchanged with sodium (SWy-Na) during the purification procedure. All the details dismaying this sample are mentioned in [26].*


*Notes: 2θ°, Bragg's angle; d001, basal distance of the 1st reflection; D, average crystalline size; FWHM, full-width at half maximum; ξ, deviation from rationality.*

**Table 2.**

*Qualitative XRD investigation in the case of the SWy-Na and SWy-Ba sample.*


*Notes: d001, interlamellar distance; nH2O, number of water molecules per half-cell; zH2O, position of the molecules along the c axis of the H2O molecule is attached to 9.6 Å for hydration states 1 W; the position of the exchangeable cations per half-cell calculated along the axis c\* is fixed à 9.6 Å for hydration states 1 W; M, average number of sheets per stack; SM, layer stacking mode; R0, maximum order; R1, random stacking; R1-X, associated stacking (segregation-partial order); C, characters; Ho, homogeneous; He, heterogeneous; RWP, confidence factor.*

#### **Table 3.**

*Structural parameters extracted from XRD modeling approach.*

The MLS used to achieve the best agreement between the calculated and experimental XRD model (**Figure 17**), shows a heterogeneous hydration character that results in the coexistence of two layers types (0 W and 1 W) with a strong dominance for the 1 W phases about 80%. This contradicts the qualitative analysis which indicates that the structure is purely homogeneous (a pure homogeneous 1 W hydration state). The structural parameters are summarized in **Table 3**. The confidence factor RWP is very low (2.17%), which reflects the high fit quality.

#### **5.2 SWy-Ba sample**

The SWy-Ba sample present a two-water hydration state 2 W [26, 46, 48]. A heterogeneous hydration behavior is the main description based on the asymmetric XRD reflection profile shape and the elevated FWHM/**ξ** parameter value (**Table 2**, **Figure 18**). The optimized theoretical model (**Figure 18**) is obtained by combining MLS with variable hydration state. The coexistence of two types of sheets (1 W and 2 W) with a major 1 W layer fraction. The existence of two types of compensating cations is probably related to the incompletion of the cation exchange process (sodium residue/excess salt). The structural parameters are summarized in **Table 3**. The confidence factor RWP is very low (1.34%), which reflects the high fit quality.

### **6. Conclusion**

This work investigates the strength of the modeling X-ray diffractograms method used to rebuild theoretically the full structure along c\* axis for the layered materials such as nanoclays. This technic is an indirect method based on the comparison

*X-ray Diffraction Profiles Modeling Method for Layered Structures Reconstruction:… DOI: http://dx.doi.org/10.5772/intechopen.107017*

**Figure 18.** *Best agreement between theoretical and experimental XRD profiles obtained in the case of SWy-Ba.*

between experimental and theoretical profiles. An intrinsic mathematical formalism was detailed in this chapter. The implemented code used to perform this operation has been described in detail by explaining the role of each input parameters such as:


A complete modeling strategy is detailed and accompanied by an application on two nanoclay specimen saturated, respectively, by Na+ and Ba2+ cations.
