**Abstract**

The nanoclay properties find a large environmental application domain as depolluant, ion exchanger, natural geological barrier for industrial and radioactive waste confinement, clay-based nanocomposite for drug delivery, and more. Layered materials, such as nanoclay, present rather complex structures whose classical characterization requires a complementarity between several analysis methods to decipher the effects of interstratification (and its cause) on the intrinsic functional properties. The appearance of defects related to the layers stacking mode, which differ in their thickness and/or their internal structure are directly related to the reactivity of the mineral's surface. During the last decades, and with the development of computer codes, the modeling of X-ray diffraction profiles has proven to be an important tool that allows detailed structural reconstruction. The quantitative XRD analysis, which consists of the comparison of experimental (00l) reflections with the calculated ones deduced from structural models, allowed us to determine the optimal structural parameters describing interlamellar space (IS) configuration, hydration state, cation exchange capacity (CEC), layer stacking mode, and theoretical mixed-layer structure (MLS) distribution. This chapter will review the state of the art of this theoretical approach as a basic technique for the study of nanoclays. The basic mathematical formalism, the parameters affecting the theoretical models, and the modeling strategy steps will be detailed in concrete examples.

**Keywords:** nanoclay, layered materials, modeling of X-ray diffraction profiles, layer thickness, layer stacking mod, modeling strategy

### **1. Introduction**

Hydration properties of nanoclays are controlled by several factors such as the type of the interlayer cation and the amount and the layer charge location (created by isomorphic substitutions in octahedral or tetrahedral sites). The nanoclay swelling process is controlled by the balance between the repulsive force owing to the layer interactions and the attractive forces between exchangeable cations and the negatively

charged surface of siloxane layers [1–4]. XRD is the basic analysis method for the structural characterization of clay minerals. The structural characterization of nanoclays, and given their nanoscopic properties, through a simple XRD qualitative analysis does not provide clear and convincing answers if we really want to take advantage of these intrinsic properties. Indeed, the qualitative XRD investigation based on the basal spacing *d*00l from the first order (001) Bragg reflections, the full-width at half maximum (FWHM), the profile geometry (i.e., symmetric or asymmetric reflections) description, and the deviation rationality parameter (ξ), respectively (reference), remains insufficient for a precise description. The information remains obsolete and incomplete on what is really happening at the level of the IS configuration and its content as well as the layers distribution within the crystallites. Also, analysis of XRD effects from defective nanoclay materials cannot be reached adequately with conventional XRD methods such as single-crystal diffraction and/or Rietveld structure refinement because of this reduced periodicity. This has driven the advancement of specific algorithms for the calculation of diffraction intensity occurring from defective structures.

The development of specific modeling XRD techniques based on the theoretical approach is imposed. Pioneering authors [5–14] studied the IS configuration focusing on the atomic positions of exchangeable cations and the associated H2O molecules of Na-montmorillonite samples by assuming a homogeneous hydration state and neglecting the coexistence of different hydration states in a sample. The heterogeneous layer charge distribution is investigated by [9] who showed that the insertion of the IS water molecule is accompanied by a progressive expansion of the basal spacing value which is done by the discrete hydration state going from the dehydrated (0 W, *d*00l = 10 Å) to the strongly hydrated ones (4 W, *d*00l ≈ 21 Å) passing by the 1 W, 2 W, and 3 W hydration states [9, 15]. Among the problems solved by XRD modeling approach is the coexistence of several hydration states within the same layer and the coexistence of different types of layers within the same crystallite.

Indeed, [16] demonstrated that the presence of different exchangeable cations in different interlayers is lead to the presence of segregated domains. These domains attributed to the demixed state are described in the works of [17]. The modeling XRD approach correlated to the adsorption–desorption measurements are used by [18] to determine the proportion of the different layer types coexisting along the isotherms. More recently, [19–26] used this approach to fit reflection positions and 00l profiles over a large angular range focusing study of the smectite hydration behavior and the selective ion exchange process.

They demonstrate the existence of mixed-layer structure (MLS) and a specific response to mechanical and geochemical disturbances has been resolved. Authors [27–28] showed the great possibility of using diffraction techniques to root out structural information from poor 3D crystal periodicity. They also show that a reliable characterization of the structural and chemical heterogeneities of the layered structures mainly depends on an optimized and reliable interpretation of the diffraction data [29]. Several proposed methods allowing the theoretical calculation of the diffracted intensity for the non-periodic interstratified structures have been proposed in the literature [30–33]. The proposed models present some technical and scientific failures. The main development was based on a formalism developed by [7, 13, 34–37] to describe the diffracted 00l and hkl intensities by a set of crystals containing different layer types.

The XRD modeling method is used to quantify the MLS, the layer hydration state, CEC fluctuations, optimum IS configuration, crystallite size/distribution, average

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

layer number per crystallite, and structural heterogeneities [29]. Despite the various works carried out that rely on the modeling of nanoclay structure, to our knowledge, no study has described the details of modeling and strategy execution. In some cases, the modeling method is briefly described without details [21, 25–26, 29, 37]. Hence there are always shortcomings to discover. This work focuses on a detailed description of the diffractogram modeling strategy of nanoclay structures based on the comparison of experimental reflections of 00l with those calculated, which makes it possible to reconstruct a theoretical model describing the layers stacking at the crystallite along the c\*.

### **2. Materials and methods**

#### **2.1 Baseline material**

A standard dioctahedral smectite SWy-2 extracted from the cretaceous formations of Wyoming (USA) and provided by the clay mineral repository is selected for the present study [38]. The structural formula per half-cell is given by [39]:

(Si4+3.96, Al3+ 0.04) (Al3+ 1.53, Fe3+ 0.18, Fe2+ 0.045, Mg2+ 0.26, Ti4+0.01) O10 (OH)2 (Ca2+ 0.07, K<sup>+</sup> 0.01 Na<sup>+</sup> 0.2)

This bentonite exhibits a low octahedral charge and extremely limited tetrahedral substitutions. The clay cation exchange capacity (CEC) is 101 meq/100 g [39]. Pretreatment of the starting material consists of preparing Na-rich montmorillonite suspension (SWy-Na), is realized following a classic protocol detailed by [25]. A cation exchange process is carried out for Barium cation (Ba2+) in order to obtain the second reference sample SWy-Ba. The experimental protocol established consists of applying a mechanical shake for 48 h, followed by centrifugation at 4000 rpm. This step is repeated five times to ensure process achievement. After recovery of the solid fraction, a series of washes with distilled water will take place to remove excess salt from chloride ions. To collect XRD diffraction data, oriented samples were prepared by placing the obtained suspensions on a glass slide at air dry for 24 h.

#### **2.2 Qualitative analysis**

Qualitative XRD analysis is an essential task for the identification of a given diffractogram. It allows us to simplify the modeling approach. The following parameters are determined:


**Figure 1.** *Scheme of a clay particle at different scales.*

#### **2.3 Mathematical formalism: modeling (001) reflections**

Theoretical diffracted intensity of lamellar structures generally based on powder diagrams is reported in **Figure 2**. Considering the case of a powder formed by crystallites having a layer with large lateral extension, the diffracted XRD intensity is expressed by:

$$I(\mathbf{s}) = \frac{1}{\mathbf{S}^2} \sum\_{\mathbf{M}\_1}^{\mathbf{M}\_n} \mathbf{a} \left( \mathbf{M} \right) \operatorname{Tr} \left\{ \operatorname{Re} \left\{ \left[ \mathbf{F} \middle| \left[ \mathbf{W} \right] \left( \left[ \mathbf{I} \right] + 2 \sum\_{\mathbf{n}}^{\mathbf{M} - 1} \left[ \mathbf{Q} \right]^{\mathbf{n}} \right) \right\} \right\} \right\}, \tag{1}$$

where **Tr** is the trace of the matrix; **Re** is the real part of the expression; **M** is the number of layers per stack; **α(M)** is the weight distribution in thicknesses of the layer, M varies from M1 to M2.

$$\overrightarrow{\mathbf{S}} = \overrightarrow{\mathbf{h}\,\mathbf{a}^\*} + \overrightarrow{\mathbf{k}\,\mathbf{b}^\*} + \overrightarrow{\mathbf{l}\,\mathbf{c}^\*} \tag{2}$$

is the vector of the module reciprocal space

$$\mathbf{S} = (2\sin\sin\mathbf{\theta})/\lambda; [\mathbf{1}] \tag{3}$$

**Figure 2.** *Diffracted Intensity distribution in the case of layered structure.*

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

is the identity matrix.

**Matrix [F]:** Matrix of structural factors, each element (ij) corresponds to the passage from a layer of type *i* to a neighboring layer *j*, this is expressed as

$$\mathbf{F}\_{\text{ij}} \mathbf{F}\_{\text{i}}(\mathbf{z}).\mathbf{F}\_{\text{j}}^{\*}(\mathbf{z}).\tag{4}$$

$$\begin{bmatrix} \mathbf{F} \end{bmatrix} = \begin{bmatrix} \mathbf{F\_0F\_0^\*} & \mathbf{F\_0F\_1^\*} & \dots & \mathbf{F\_0F\_g^\*} \\\\ \mathbf{F\_1F\_0^\*} & \mathbf{F\_1F\_1^\*} & \dots & \mathbf{F\_1F\_g^\*} \\\\ \cdot & \cdot & \dots & \cdot \\\\ \cdot & \cdot & \dots & \cdot \\\\ \mathbf{F\_gF\_0^\*} & \mathbf{F\_gF\_1^\*} & \dots & \mathbf{F\_gF\_g^\*} \end{bmatrix} \cdot \tag{5}$$

For a given pair of Miller hk indices, the structure factor is calculated using the following expression:

$$\mathbf{F\_{hk}(z)} = \sum\_{\mathbf{n}} \mathbf{f\_n} \mathbf{e^{-2i\pi \left(\mathbf{hx\_n} + \mathbf{ky\_n} + \mathbf{kz\_n}\right)}} \tag{6}$$

While the individual structural factors F1, F2 ... are obtained from the same relation using the same 2D network rotated from angle 2п/v, 4п/v, ... compared wih the fixed repository. Also, xn and yn are the coordinates of the atoms along the layer and zn is their position in Å.

Matrix [W]: It is a diagonal matrix of order g, avec

$$\sum\_{i=1}^{g} \mathbf{W}\_i = \mathbf{1} \tag{7}$$

(Wi layer proportions of each stacking type).

$$[\mathbf{W}] = \begin{bmatrix} \mathbf{W}\_1 & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & \mathbf{W}\_2 & \dots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \dots & \mathbf{0} \end{bmatrix} . \tag{8}$$

Matrix [Q]<sup>n</sup> : It characterizes the interference between diffracted waves by adjacent layers.

$$\begin{bmatrix} \mathbf{Q} \end{bmatrix}^{\mathrm{n}} = \begin{bmatrix} \mathbf{P}\_{11}\mathbf{q}\_{11} & \mathbf{P}\_{12}\mathbf{q}\_{12} & \cdot & \cdot & \mathbf{P}\_{1\mathrm{g}}\mathbf{q}\_{1\mathrm{g}} \\ \mathbf{P}\_{21}\mathbf{q}\_{21} & \mathbf{P}\_{22}\mathbf{q}\_{22} & \cdot & \cdot & \mathbf{P}\_{2\mathrm{g}}\mathbf{q}\_{2\mathrm{g}} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \mathbf{P}\_{\mathfrak{g}1}\mathbf{q}\_{\mathfrak{g}1} & \mathbf{P}\_{\mathfrak{g}2}\mathbf{q}\_{\mathfrak{g}2} & \cdot & \cdot & \mathbf{P}\_{\mathfrak{g}\mathfrak{g}}\mathbf{q}\_{\mathfrak{g}\mathfrak{g}} \end{bmatrix}, \tag{9}$$

where **Pij** is the conditional probability that a type i sheet is followed by a type j sheet. **<sup>φ</sup>ij** is the phase shift between two waves <sup>φ</sup>ij <sup>¼</sup> e2i<sup>π</sup> <sup>s</sup> ! tij � � ! . **tij** ! is the translation between sheet i and sheet j first neighbor.

The relationships between probabilities and abundances are given by [7, 13, 37]:

$$\sum\_{\mathbf{i}=\mathbf{1}}^{\mathbf{g}} \mathbf{P}\_{\mathbf{i}\parallel} = \mathbf{1} \tag{10}$$

$$\sum\_{\mathbf{j=1}}^{\mathbf{g}} \mathbf{W}\_{\mathbf{j}} \mathbf{P}\_{\mathbf{j}\mathbf{i}} = \mathbf{W}\_{\mathbf{i}} \tag{11}$$

The theoretical XRD profile allows us to determine [37]: (i) the layers succession law within the stack; (ii) the number of different types of layers; (iii) relative layers type abundances; (iv) the average number of layers per stack; and (v) the weight distribution in thickness of the stacks.

The quasi-homogeneous model that assumes Markovian statistics is tremendously utilized [34, 37]. For illustration purposes, one may consider the case of mixed layers containing two types (A and B) and different sets of junction probability parameters. It is an interaction with the first neighbors since the translation between layers depends only on the nature of the previous layer or the next layer. In this type of interstratification, two main trends appear (**Table 1**).

Case of the random system that corresponds to aleatory layers succession within the stack:

$$\begin{cases} \mathbf{W\_A} = \mathbf{P\_{AA}} = \mathbf{P\_{BA}} \\ \mathbf{W\_B} = \mathbf{P\_{BB}} = \mathbf{P\_{AB}} \end{cases} \tag{12}$$

For all cases, the matrices [Q] and [W] have the following forms:

$$\begin{bmatrix} \mathbf{Q} \end{bmatrix}^{\mathbf{n}} = \begin{bmatrix} \mathbf{P\_{AA}}\boldsymbol{\uprho}\_{\mathbf{AA}} & \mathbf{P\_{AB}}\boldsymbol{\uprho}\_{\mathbf{AB}} \\\\ \mathbf{P\_{BA}}\boldsymbol{\uprho}\_{\mathbf{BA}} & \mathbf{P\_{BB}}\boldsymbol{\uprho}\_{\mathbf{BB}} \end{bmatrix} \tag{13}$$

$$\mathbf{[W]} = \begin{bmatrix} \mathbf{W\_A} \mathbf{0} \\ \mathbf{0} & \mathbf{W\_B} \end{bmatrix} \tag{14}$$

With

$$
\mathfrak{q}\_{\rm AB} = \mathbf{e}^{2i\pi \overline{s} \overleftarrow{\mathbf{t}\_{\rm AB}}} \tag{15}
$$

**tAB** �! is the translation between the first neighboring layers.

Relative abundances and junction conditional probabilities are linked by the following expressions:

$$\mathbf{W\_A} + \mathbf{W\_B} = \mathbf{1} \tag{16}$$

$$\mathbf{P\_{AA}} + \mathbf{P\_{AB}} = \mathbf{1} \tag{17}$$

$$\mathbf{P\_{BA}} + \mathbf{P\_{BB}} = \mathbf{1},\tag{18}$$

$$\begin{cases} \mathbf{W\_A} = \mathbf{W\_A}\mathbf{P\_{AA}} + \mathbf{W\_B}\mathbf{P\_{BA}} \\ \mathbf{W\_B} = \mathbf{W\_A}\mathbf{P\_{AB}} + \mathbf{W\_B}\mathbf{P\_{BB}} \end{cases} \rightarrow \mathbf{W\_A}\mathbf{P\_{AB}} = \mathbf{W\_B}\mathbf{P\_{BA}}.\tag{19}$$

All possible types of layer stacks are characterized in (**Figure 3**).

In the case of a random stack, no stacking sequence is prohibited, so the probability of a layer appearing in a sequence depends only on its abundance. Thus PAA = WA and this type of stacking is characterized in **Figure 3** by an increasing linear function. In the opposite case, where the succession of the two layers with different nature is prohibited *X-ray Diffraction Profiles Modeling Method for Layered Structures Reconstruction:… DOI: http://dx.doi.org/10.5772/intechopen.107017*


**Table 1.**

*Segregation and regularity layers stacking tendency.*

#### **Figure 3.**

*(a) Junction probability diagram for two-component mixed layers (adapted from [42]). WA and PAA represent the relative proportion of A layers and the probability of finding an A layer after an A layer, respectively. Specific junction probabilities correspond to random interstratification. Possible stacking sequences corresponding to the different cases are schematized. Between the probability of occurrence of a layer B following a layer A as a function of the proportion of A layer [42]. (b) Type of stack deduced according to the parameters reported on the diagram: series (x) Regular, (y) random, and (z) segregated.*

we have; PAB = PBA = 0. The probability that two layers of the same type will succeed each other is equal to the unit: PAA = PBB = 1. Then it is no longer a question of interstratification because the two types of layers no longer coexist at the heart of the same crystal but of total segregation or total separation. The maximum order is defined in the case of a prohibited succession between two minority layers. For example, if the type B layer is a minority, PBB = 0, based on the two relationships below:

$$\begin{cases} \mathbf{P\_{BA}} = \mathbf{1} \\ \mathbf{P\_{AB}} = \frac{\mathbf{W\_B}}{\mathbf{W\_A}} \to \mathbf{P\_{AA}} = \mathbf{1} - \frac{\mathbf{W\_B}}{\mathbf{W\_A}} = \frac{2\mathbf{W\_A} - \mathbf{1}}{\mathbf{W\_A}} \end{cases} \tag{22}$$

### **3. Implemented code**

#### **3.1 Code description**

The modeling program is a "Fortran" code developed and improved by several authors [7, 8, 10] based on the mathematical formalism detailed by [37]. It makes it possible to create theoretical X-ray diffractograms in the case of lamellar structures. The quality control agreement between the two profiles is carried out separately using a calculated RWP and/or Rp confidence factor [21, 29].

#### **3.2 Code architecture**

The program is essentially divided into three parts, respectively, (i) XRD parameters (experimental conditions and technical parameters related to the diffractometer), (ii) layered materials specification, and (iii) intrinsic structural properties (layers distribution, average number of layer per crystallite, and layer distribution function). The basic architecture of the executable file is reported in **Figure 4**.

**Figure 4.** *Basic architecture of the executable file.*

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

## **3.3 Input parameters affecting theoretical X-rays diffractograms: case of nanoclay**

### *3.3.1 Hydration state*

The nanoclay hydration state is simply defined by the amount of water inserted into the IS which induces an increase in the layer thickness [43]. A discrete simple hydration state is detailed in **Figure 5**.

The theoretical profile for each hydration state is reported on **Figure 6**. A logical translation toward low 2θ values is accompanied by an increase in the layer thickness input (according to Bragg Brentano's law). The proposed structure in this case is composed of two types of layers with a major contribution (100%) of the first layer, neglecting the second.
