X-ray Diffraction Profiles Modeling Method for Layered Structures Reconstruction: Nanoclay Structural Verification

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.

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, 2, 3, 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, 6, 7, 8, 9, 10, 11, 12, 13, 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, 20, 21, 22, 23, 24, 25, 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, 31, 32, 33]. The proposed models present some technical and scientific failures. The main development was based on a formalism developed by [7, 13, 34, 35, 36, 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 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⁺_0.01 Na⁺_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:

• The positioning of the peaks (2θ°).

• The basal spacing distance of the first reflection 001, to identify the hydration state (0 W, 1 W, 2 W…) of the sample using Bragg’s law.

• FHWM values and rationality ξ [40].

• The crystallite size D_hkl in the perpendicular direction to the lattice planes [41] knowing that [8] shows that the phyllosilicates with multiscale structures begin from the nanoclay fraction to a micro agglomerate scale (> 10 μm; Figure 1). In addition, the reflection shape is checked with the aim of clearing the existence of a heterogeneous structure (possible reflection decomposition on two or more hydration states).

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:

Figure 2.

Diffracted Intensity distribution in the case of layered structure.

$$I\left(\mathrm{s}\right)=\frac{1}{{\mathrm{S}}^2}\sum _{{\mathrm{M}}_1}^{{\mathrm{M}}_{\mathrm{n}}}\mathrm{\alpha }\left(\mathrm{M}\right)\mathrm{Tr}\left\{Re\left\{\left[\mathrm{F}\right]\left[\mathrm{W}\right]\left(\left[\mathrm{I}\right]+2\sum _{\mathrm{n}}^{\mathrm{M}-1}{\left[\mathrm{Q}\right]}^{\mathrm{n}}\right)\right\}\right\},$$E1

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.

$$\overrightarrow{\mathbf{S}}\mathbf{=}\mathbf{h}\overrightarrow{{\mathbf{a}}^{\mathbf{\ast }}}\mathbf{+}\mathbf{k}\overrightarrow{{\mathbf{b}}^{\mathbf{\ast }}}\mathbf{+}\mathbf{l}\overrightarrow{{\mathbf{c}}^{\mathbf{\ast }}}$$E2

is the vector of the module reciprocal space

$$\mathbf{S}\mathbf{=}\left(2,\mathbf{sin},\mathbf{sin},\mathbf{\theta }\right)\mathbf{/}\mathbf{\lambda };\left[\mathbf{1}\right]$$E3

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

$${\mathrm{F}}_{\mathrm{ij}}{\mathrm{F}}_{\mathrm{i}}\left(\mathrm{z}\right).{\mathrm{F}}_{\mathrm{j}}^{*}\left(\mathrm{z}\right).$$E4

$$\left[\mathrm{F}\right]=\left[\begin{array}{cccc}{\mathrm{F}}_0{\mathrm{F}}_0^{\ast }& {\mathrm{F}}_0{\mathrm{F}}_1^{\ast }& ..& {\mathrm{F}}_0{\mathrm{F}}_{\mathrm{g}}^{\ast }\\ {\mathrm{F}}_1{\mathrm{F}}_0^{\ast }& {\mathrm{F}}_1{\mathrm{F}}_1^{\ast }& ..& {\mathrm{F}}_1{\mathrm{F}}_{\mathrm{g}}^{\ast }\\ .& .& ..& .\\ .& .& ..& .\\ {\mathrm{F}}_{\mathrm{g}}{\mathrm{F}}_0^{\ast }& {\mathrm{F}}_{\mathrm{g}}{\mathrm{F}}_1^{\ast }& ..& {\mathrm{F}}_{\mathrm{g}}{\mathrm{F}}_{\mathrm{g}}^{\ast }\end{array}\right].$$E5

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

$${\mathrm{F}}_{\mathrm{hk}}\left(\mathrm{z}\right)=\sum _{\mathrm{n}}{\mathrm{f}}_{\mathrm{n}}{\mathrm{e}}^{-2\mathrm{i\pi }\left(\mathrm{h}{\mathrm{x}}_{\mathrm{n}}+\mathrm{k}{\mathrm{y}}_{\mathrm{n}}+\mathrm{l}{\mathrm{z}}_{\mathrm{n}}\right)}$$E6

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, x_n and y_n are the coordinates of the atoms along the layer and z_n is their position in Å.

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

$${\sum }_{\mathrm{i}=1}^{\mathrm{g}}{\mathrm{W}}_{\mathrm{i}}=1$$E7

(W_i layer proportions of each stacking type).

$$\left[\mathrm{W}\right]=\left[\begin{array}{ccccc}{\mathrm{W}}_1& 0& .& .& 0\\ 0& {\mathrm{W}}_2& .& .& 0\\ 0& 0& .& .& 0\\ 0& 0& .& .& 0\\ 0& 0& .& .& {\mathrm{W}}_{\mathrm{g}}\end{array}\right].$$E8

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

$${\left[\mathrm{Q}\right]}^{\mathrm{n}}=\left[\begin{array}{ccccc}{\mathrm{P}}_{11}{\mathrm{\phi }}_{11}& {\mathrm{P}}_{12}{\mathrm{\phi }}_{12}& .& .& {\mathrm{P}}_{1\mathrm{g}}{\mathrm{\phi }}_{1\mathrm{g}}\\ {\mathrm{P}}_{21}{\mathrm{\phi }}_{21}& {\mathrm{P}}_{22}{\mathrm{\phi }}_{22}& .& .& {\mathrm{P}}_{2\mathrm{g}}{\mathrm{\phi }}_{2\mathrm{g}}\\ .& .& .& .& .\\ .& .& .& .& .\\ {\mathrm{P}}_{\mathrm{g}1}{\mathrm{\phi }}_{\mathrm{g}1}& {\mathrm{P}}_{\mathrm{g}2}{\mathrm{\phi }}_{\mathrm{g}2}& .& .& {\mathrm{P}}_{\mathrm{gg}}{\mathrm{\phi }}_{\mathrm{gg}}\end{array}\begin{array}{}\\ \\ \\ \\ \end{array}\right],$$E9

where P_ij is the conditional probability that a type i sheet is followed by a type j sheet. ${\mathbf{\phi }}_{\mathbf{ij}}$ is the phase shift between two waves$\left\{{\mathrm{\phi }}_{\mathrm{ij}}={\mathrm{e}}^{2\mathrm{i\pi }\overrightarrow{\mathrm{s}}\overrightarrow{{\mathrm{t}}_{\mathrm{ij}}}}\right\}$. $\overrightarrow{{\mathbf{t}}_{\mathbf{ij}}}$ is the translation between sheet i and sheet j first neighbor.

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

$${\sum }_{\mathbf{i}\mathbf{=}\mathbf{1}}^{\mathbf{g}}{\mathbf{P}}_{\mathbf{ij}}\mathbf{=}\mathbf{1}$$E10

$${\sum }_{\mathbf{j}\mathbf{=}\mathbf{1}}^{\mathbf{g}}{\mathbf{W}}_{\mathbf{j}}{\mathbf{P}}_{\mathbf{ji}}\mathbf{=}{\mathbf{W}}_{\mathbf{i}}$$E11

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).

Tendency to segregation	Tendency to regularity
$$\left\{\begin{array}{c}{\mathbf{W}}_{\mathbf{A}}\mathbf{<}{\mathbf{P}}_{\mathbf{AA}}\mathbf{\le }\mathbf{1}\\ {\mathbf{W}}_{\mathbf{B}}\mathbf{<}{\mathbf{P}}_{\mathbf{BB}}\mathbf{\le }\mathbf{1}\end{array}\right.$$E20 With total demixing for; ${\mathbf{P}}_{\mathbf{AA}}\mathbf{=}{\mathbf{P}}_{\mathbf{BB}}\mathbf{=}\mathbf{1}$	$$\left\{\begin{array}{c}{\mathbf{W}}_{\mathbf{A}}\mathbf{<}{\mathbf{P}}_{\mathbf{BA}}\mathbf{\le }\mathbf{1}\\ {\mathbf{W}}_{\mathbf{B}}\mathbf{<}{\mathbf{P}}_{\mathbf{AB}}\mathbf{\le }\mathbf{1}\end{array}\right.$$E21

Table 1.

Segregation and regularity layers stacking tendency.

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

$$\left\{\begin{array}{c}{\mathrm{W}}_{\mathrm{A}}={\mathrm{P}}_{\mathrm{AA}}={\mathrm{P}}_{\mathrm{BA}}\\ {\mathrm{W}}_{\mathrm{B}}={\mathrm{P}}_{\mathrm{BB}}={\mathrm{P}}_{\mathrm{AB}}\end{array}.\right.$$E12

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

$${\left[\mathbf{Q}\right]}^{\mathbf{n}}\mathbf{=}\left[\begin{array}{c}{\mathbf{P}}_{\mathbf{AA}}{\mathbf{\phi }}_{\mathbf{AA}}\\ {\mathbf{P}}_{\mathbf{BA}}{\mathbf{\phi }}_{\mathbf{BA}}\end{array}\begin{array}{c}{\mathbf{P}}_{\mathbf{AB}}{\mathbf{\phi }}_{\mathbf{AB}}\\ {\mathbf{P}}_{\mathbf{BB}}{\mathbf{\phi }}_{\mathbf{BB}}\end{array}\right]$$E13

$$\left[\mathbf{W}\right]\mathbf{=}\left[\begin{array}{c}{\mathbf{W}}_{\mathbf{A}}\\ \mathbf{0}\end{array}\begin{array}{c}\mathbf{0}\\ {\mathbf{W}}_{\mathbf{B}}\end{array}\right]$$E14

With

$${\mathbf{\phi }}_{\mathbf{AB}}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{i\pi }\overrightarrow{\mathbf{s}}\overrightarrow{{\mathbf{t}}_{\mathbf{AB}}}}$$E15

$\overrightarrow{{\mathbf{t}}_{\mathbf{AB}}}$ is the translation between the first neighboring layers.

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

$${\mathbf{W}}_{\mathbf{A}}\mathit{+}{\mathbf{W}}_{\mathbf{B}}\mathit{=}\mathbf{1}$$E16

$${\mathbf{P}}_{\mathbf{AA}}\mathit{+}{\mathbf{P}}_{\mathbf{AB}}\mathit{=}\mathbf{1}$$E17

$${\mathbf{P}}_{\mathbf{BA}}\mathit{+}{\mathbf{P}}_{\mathbf{BB}}\mathit{=}\mathbf{1},$$E18

$$\left\{\begin{array}{c}{\mathbf{W}}_{\mathbf{A}}\mathit{=}{\mathbf{W}}_{\mathbf{A}}{\mathbf{P}}_{\mathbf{AA}}\mathit{+}{\mathbf{W}}_{\mathbf{B}}{\mathbf{P}}_{\mathbf{BA}}\\ {\mathbf{W}}_{\mathbf{B}}\mathit{=}{\mathbf{W}}_{\mathbf{A}}{\mathbf{P}}_{\mathbf{AB}}\mathit{+}{\mathbf{W}}_{\mathbf{B}}{\mathbf{P}}_{\mathbf{BB}}\end{array}\mathit{\to }\right.{\mathbf{W}}_{\mathbf{A}}{\mathbf{P}}_{\mathbf{AB}}\mathit{=}{\mathbf{W}}_{\mathbf{B}}{\mathbf{P}}_{\mathbf{BA}}.$$E19

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

Figure 3.

(a) Junction probability diagram for two-component mixed layers (adapted from [42]). W_A and P_AA 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.

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 P_AA = W_A 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 we have; P_AB = P_BA = 0. The probability that two layers of the same type will succeed each other is equal to the unit: P_AA = P_BB = 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, P_BB = 0, based on the two relationships below:

$$\left\{\begin{array}{c}{\mathbf{P}}_{\mathbf{BA}}\mathit{=}\mathbf{1}\\ {\mathbf{P}}_{\mathbf{AB}}\mathit{=}\frac{{\mathbf{W}}_{\mathbf{B}}}{{\mathbf{W}}_{\mathbf{A}}}\end{array}\mathit{\to }\right.{\mathbf{P}}_{\mathbf{AA}}\mathit{=}\mathbf{1}\mathit{-}\frac{{\mathbf{W}}_{\mathbf{B}}}{{\mathbf{W}}_{\mathbf{A}}}\mathit{=}\frac{\mathbf{2}{\mathbf{W}}_{\mathbf{A}}\mathit{-}\mathbf{1}}{{\mathbf{W}}_{\mathbf{A}}}$$E22

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 R_WP 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.

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.

Figure 5.

Presentation of variables hydration states. The d₀₀₁ basal spacing value is the projection of the period c* on the normal to the layer.

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.

Figure 6.

Separated theoretical homogenous layer structures.

3.3.2 Water molecule abundance (nH₂O)

The water molecule amount variation has a direct impact on the increase and/or decrease in the relative diffracted intensity essentially for the higher diffraction orders. An example of this impact is reported in Figure 7. A homogeneous 1 W hydration state is considered and a discrete water molecule distribution (0.1 increasing step) is directed.

Figure 7.

Effect of varying water molecule abundance on the theoretical XRD profiles.

A remarkable reflection intensity evolution is visualized only by increasing the amount of water molecules in the IS from 0.1 to 1 (Figure 8) while maintaining the same reflection position.

Figure 8.

Relative reflection intensity evolution by increasing IS water molecule abundance.

3.3.3 Variation of the average number of layers M per stack

The variation of the average number of layers M per stack affects the theoretical profiles (Figure 9), essentially the FWHM increases dramatically when the average number of layers is reduced. Note that when the average number of layers M decreases, the width of the lines 00ℓ (with ℓ = 1, 2, … N) increases and vice versa.

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.

3.3.4 Layer abundance and the probabilities junction law (W_i and P_ij)

Variations in the relative abundances W_i and P_ij 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 W_A and P_AA values (with W_A = P_AA) 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 W_A and P_AA value.

Figure 10.

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

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 W_A and P_AA.

Figure 11.

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

Figure 12.

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

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 13.

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

Figure 14.

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

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

Figure 15.

Theoretical decomposition of the main 001 reflection.

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

$${\mathbf{R}}_{\mathbf{WP}}\mathbf{=}\sqrt{\frac{\sum {\left(\mathbf{I}{\left(\mathbf{2}{\mathbf{\theta }}_{\mathbf{i}}\right)}_{\mathbf{exp}}\mathbf{-}\mathbf{I}{\left(\mathbf{2}{\mathbf{\theta }}_{\mathbf{i}}\right)}_{\mathbf{th}\acute{\mathbf{e}}\mathbf{o}}\right)}^{\mathbf{2}}}{\sum {\left(\mathbf{I}{\left(\mathbf{2}{\mathbf{\theta }}_{\mathbf{i}}\right)}_{\mathbf{exp}}\right)}^{\mathbf{2}}}}\mathbf{\times }\mathbf{100}\%.$$E23

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.

Figure 16.

Example of intra-layer hydration heterogeneity.

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].

Sample	2θ°	d001 (Å)	FWHM (2θ°)	D (Å)	ξ (Å)	Character
SWy-Na	7.16	12.33	0.74	18.77	0.062	Homogeneous
SWy-Ba	6.83	12.93	1.15	12.08	0.397	Heterogeneous

Table 2.

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

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.

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 R_WP is very low (2.17%), which reflects the high fit quality.

Sample	d001	nH2O	WA	PAA	SM	M	C	RWP (%)
SWy-Na	10.5	0	0.80	0.85	R1-seg	12	He	2.17
	12.5	2
SWy-Ba	12.5	1	0.8	0.9	R1-seg	8	He	1.34
	15.5	1.5
		1.5

Table 3.

Structural parameters extracted from XRD modeling approach.

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; S_M, layer stacking mode; R₀, maximum order; R₁, random stacking; R_1-X, associated stacking (segregation-partial order); C, characters; Ho, homogeneous; He, heterogeneous; R_WP, confidence factor.

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 R_WP is very low (1.34%), which reflects the high fit quality.

Figure 18.

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

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 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:

1. The abundance and the junction probability law describing the layer stacking mode (W_A and P_AA).

2. The effect of fluctuations of the number of layers per crystallite M in shape and the response of the theoretical diffracted intensity.

3. The water molecule nH₂O amount effect in the improvement of the reflection intensity agreement for the higher diffraction orders (>001).

4. Role of R_WP confidence factor to validate the obtained model (≈5%).

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

Conflict of interest

The authors declare no conflict of interest.