*3.3.4 Layer abundance and the probabilities junction law (Wi and Pij)*

Variations in the relative abundances Wi and Pij probabilities of layers stacking make it possible to visualize essentially three types of stacks: (i) random, (ii) regular, and (iii) segregated. Two-layer types were considered (i.e. 0 W (anhydrate) and 1 W (hydrated).

A **random distribution** refers to the chaotic stacking mode between two different types of layers (**Figure 10**). Note that the simultaneous increase of the WA and PAA values (with WA = PAA) provoke a 001 reflection shift while keeping the same relative intensity. For the second reflection 002 (14° (2θ)), we also visualize displacement toward the low angles accompanied by a slight increase in diffracted intensity. The intensity of the 4th reflection order (position at 28° (2θ)) present a fluctuation versus WA and PAA value.

The **regular distribution** presents a well-ordered stacking mode between two different layers types (**Figures 11** and **12**). The theoretical reflections of this distribution exhibit the same behavior for random configuration by increasing the values of WA and PAA.

The **segregated distribution** involved an intermediate state between the regular and the random distribution. It is a mixture at short range between regular and chaotic layer stacking mode. The graphical effect of this distribution type on the theoretical diffracted intensity is reported in **Figures 13** and **14**.

#### **Figure 9.**

*Effect of the variation of the average number of layers M per stack on the theoretical profiles shape in the case of total segeregation.*

**Figure 10.** *Theoretical profiles of the variation of relative weights and probabilities (random configuration).*

**Figure 11.**

*Theoretical profiles of the change in relative abundance and junction probabilities law in the case of partial order stacking trend.*

## **4. Modeling strategy**

Before starting the modeling, several technical parameters such as improvement of the diffraction experimental data acquisition, chemical formula (extracted from literature or other analytical technics), experimental diffractions conditions, layer composition, atomic and onic scattering factors [44], and atoms coordinates must be checked and controlled in order to minimize the input variables thereafter. The

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

**Figure 12.**

*Theoretical profiles of the change in relative abundance and junction probabilities law in the case of maximum order stacking.*

#### **Figure 13.**

*Theoretical profiles change when varying relative abundance and junction probabilities law in the case of partial segregation trend.*

fitting strategy consists of reproducing the experimental XRD pattern using a main homogeneous structure.

If necessary, additional contributions to the diffracted intensity are introduced to account for improve agreement between calculated and experimental patterns (i.e., if we have more one main structure, a MLS can be introduced). Indeed, the main 001 reflection can be decomposed into several theoretical weighted phases (**Figure 15**). The presence of two MLSs does not imply that two populations of particles are physically present in the sample [21, 45, 46]. Therefore, layers with the same hydration state present in the different MLSs contributing to the diffracted intensity are

#### **Figure 14.**

*Theoretical profiles change when varying relative abundance and junction probabilities law in the case of total segregation.*

**Figure 15.** *Theoretical decomposition of the main 001 reflection.*

assumed to have identical properties (chemical composition, layer thickness, and z coordinates of atoms).

Agreement between theoretical and experimental XRD profile is evaluated by the calculation of the RWP trust factor based on the expression quoted [47] which must be around 5%.

$$\mathbf{R\_{WP}} = \sqrt{\frac{\sum \left( \mathbf{I(2\theta\_i)\_{exp}} - \mathbf{I(2\theta\_i)\_{tibeo}} \right)^2}{\sum \left( \mathbf{I(2\theta\_i)\_{exp}} \right)^2}} \times \mathbf{100\%.} \tag{23}$$

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

**Figure 16.** *Example of intra-layer hydration heterogeneity.*

Similarly, there is another alternative that involves combining two or more hydration states in the same layer (interstratification intra-layer) by varying the percentage of their abundance (**Figure 16**). This physically results in fluctuations in the layer thickness as a function of the (0kl) surface which affects the properties of the clay particle for lateral extension (a and b ∞). Although this method is easier than theoretical decomposition, it is not recommended because the width of the lateral extension cannot be determined qualitatively.
