Applications of X-ray Powder Diffraction Microstructural Analysis in Applied Clay Mineralogy

Abstract

Clay minerals and sheet silicates are the main constituents of lutites and clays. These materials are relevant in earth science research as well as in economic geology because of the great variety of applications, based on their particular features at different levels of aggregation in mineral assemblages and on the microstructural and structural characteristics of the mineral constituents frequently characterized by micro- and nanocrystalline appearance. Thus, X-ray diffraction is a main tool for fundamental and applied research of these materials. The present review concerns their microstructural research from powder X-ray diffraction data.

1. Introduction about Clays and Applied Clay Mineralogy

Sheet silicates and clay minerals are the main constituents of lutites and clays, which account for more than 80% of the earth crust according to different estimates [1]. Mineral abundances for >24,000 samples of 21 lithological units representing the brittle continental and oceanic crust were provided [2], and among clay mineral types, illite was the most abundant (7.7 wt.%). A book [3] is dedicated to this mineral.

Ball clays, bentonite, common clays and shales, fullers earth, kaolin, palygorskite and sepiolite are considered as the main clay commodities [4].

Clay is used as a rock term and also as a particle size term in mechanical analysis (with variable upper size limit in the range 1–4 µm) in sedimentary rocks and soils, etc. The main related terms were explained in the book of Grim [5]. The different main contributions to the knowledge of clays in sediments and rocks including an annotated bibliography can be found in [6]. In his book, Grim included the main concepts about structure, classification, main physical and chemical properties and behaviour, as well as origin and occurrence of clay minerals.

The joint nomenclature committees of the Association Internationale pour l’Etude des Argiles (AIPEA) and the Clay Minerals Society (CMS) gave definitions of clays and clay minerals and particular terms [7].

Updated information on structure properties, applications, analytical techniques for clay minerals and history of clay science is provided in [8].

There is a great number of applications of clay minerals grouped in chapters of another classic book of Grim [9]: ceramic and cement products, foundry and moulding sands, engineering properties, recovery and discovery of petroleum, refining and preparation of organic materials (including catalysis for petroleum cracking), decolourization and a variety of miscellaneous uses related with adhesives, aluminium ores, waste disposal, clarification, coatings, de-inking, desiccants, absorbents, emulsifiers, suspending and stabilizing agents, foods, greases, ink, leather, medicine and pharmaceutical, soaps and cleaning, paints, paper, pelletizing ores, fluxes, fuels, etc.

The applications of the main industrial clays (kaolins, bentonites, palygorskite and sepiolite) and common clays, as well as chapters concerning geology, and major deposits, testing methods and exploration, mining and processing were included in a handbook [10].

The fundamentals and applications of clay colloids were the aim of a classical book [11] and of a chapter [12]. Conventional uses of clays based on their inertness and stability or their reactivity and catalytic activity were reviewed in the chapter of Harvey and Lagaly [13].

Clay mineral–organic reactions have a great number of application: identification of kaolinites and 2:1 clay minerals, modification of the surface character of clay particles and of colloidal behaviour of clay dispersions, suitable adsorbents in pollution control, enhancement of properties for mineral–dye interactions or for clay mineral–hybrid film formation, as well as in clay–polymer nanocomposites, and intercalation of clay minerals in different types of guest compounds, among others [14].

The contribution of Ruiz-Hitzky and Van Meerbeek [15] is an extended summary of the literature about clay nanocomposite dealing with very relevant features concerning clay mineral–polymer interactions and structures. Different considered items are (i) methods of preparation, (ii) clay–polymer compatibilization, (iii) properties and applications.

The different structures proposed for clay–polymer materials are (i) polymer–clay micro-composite (clay is predominantly present as large size aggregates, with dimensions larger than 1 μm); (ii) polymer–clay nanocomposites: (a) intercalated (when the polymer chains are inserted into the interlayer space of the clay mineral); (b) delaminated or exfoliated (when the silicate layers are no longer close enough to interact with each other). According to these authors, a well-accepted definition of nanocomposites is that the dispersed particles have at least one dimension in the nanometer range (nanofillers) and can be classified according to the number of the nanodimensions of the filler.

Thus, in the case of phyllosilicates like smectites integrated into nanocomposites, the clay–polymer nanocomposite could be considered as ‘one-nanodimensional’ (1D) because the clay filler has one dimension at the nanometer scale, although the clay filler is two-dimensional (2D) in the microscopic sense.

The different methods of preparation of nanoclay–polymer composites in thermoplastic and thermosetting polymers with different resins and their different uses are described in a review focused on sepiolite and palygorsite [16]. Particular topics, as nanostructured hybrids (e g nanocarbonaceus materials), carriers of active particles, and metal or ceramic–clay composites, are also considered in this paper.

The book of Gurses [17] includes a chapter showing uses of polymer–clay nanocomposites in different fields: food packing, biomedical application, wastewater pre-treatment, electrical/electronics, optoelectronics, automobile, and others.

There is a great number of papers and books about the use of clays in polymer composites mainly concerning introduction of kaolin, bentonite and fibrous clay minerals (sepiolite, and palygorskite or attapulgite, and other sheet silicates as mica and vermiculite). Frequent topics of research are the influence of the silicate dosage on the microstructure of the polymer, usually described from X-ray powder diffraction data in terms of crystalline and amorphous regions of the polymer, and the influence of the sheet silicates on properties of the composites.

Fundamental and applied research on clay minerals covers a variety of fields from climate and environment to nanotechnologies, as shown in different conferences [18].

The scientific production about clays is immense. A recent Scopus search of articles with the word ‘clay’ in the title gives 60,002 titles, and filtering by subject areas Earth Science, Engineering, Materials Science, Environmental Science and Chemistry yielded 23,902, 20,874, 15,421, 11,789 and 10,592 results, respectively. That same search filtered by document type (articles, conference papers, book chapters and reviews) produced 51,475, 10,490, 1371 and 1056 results, respectively.

There are chapters about clay mineralogy in the mentioned books [3,5,6,8,9] and also in other books about clays in rocks, sediments or soils, e.g., [19,20,21,22]; in addition, there are reference books on clay mineralogy, e.g., [23,24,25,26,27,28,29].

For more than 60 years, microstructural aspects at a sub-microscopic level have been studied mainly via X-ray powder diffraction microstructural analysis, and X-ray texture analysis. Different techniques have been subsequently used to imaging materials from microscopic to atomic level imaging: conventional transmission electron microscopy (TEM), scanning electron microscopy (SEM), emission electron microscopy (EEM), high-resolution transmission electron microscopy (HRTEM), low-energy electron microscopy (LEEM), X-ray micro-tomography (micro-CT, high-resolution micro-CT, etc.), scan tunnelling microscopy (STM) and atomic force microscopy (AFM), among others.

Those methods of microstructure analysis contribute to better explaining the properties of raw materials and products in different fields (earth sciences, engineering geology, economic geology, materials sciences, etc.). The understanding of microstructures frequently contributes to the interpretation of conditions of formation and behaviour of mineral assemblages.

X-ray diffraction line broadening has been used to extract microstructural and structural information of clay minerals since the initial paper of Brindley [30].

Clay minerals and sheet silicates are frequently relevant constituents in rocks, sediments and soils, and their microstructural characteristics can be potentially useful in the study of their formation and growth conditions, as well as in the exploration and exploitation of their deposits, or in the interpretation of their properties, processing conditions and properties of the products obtained with them.

The aim of the present paper is to show general principles of microstructural analysis in X-ray powder diffraction and to provide a wide compilation of bibliographical references (with thematic classification and chronological ordering) where parameters of line profile analysis were used for research of clays and sheet silicates in different fields.

2. X-ray Powder Diffraction Microstructural Analysis Fundamentals and Methods

2.1. Prelimnary Comments on Crystalline Microstructure from X-ray Powder Diffraction Data

Polycrystalline materials contain imperfections that modify the intensity distribution of a Bragg reflection. This departure from an ideal structure, generally known as microstructure, can profoundly influence the physical, mechanical and chemical properties of materials. These imperfections mainly refer to the existence of crystallites and lattice deformations and reflect the crystalline microstructure that can be analyzed from the X-ray Diffraction (XRD) data. We will examine this concept in more detail.

As far as the XRD is concerned, the crystal appears as a mosaic of little crystallites slightly disoriented among themselves. The notion of crystallite is equivalent to that of coherent diffraction domain. In a real crystal, the perfect geometric periodicity does not extend to the entire volume of the crystal. In reality, only inside small volumes—crystallites or coherent diffraction domains—can it be considered that there is a perfect order. These crystallites are arranged very close to each other, separated by more or less crystalline material, and not in a perfect parallelism but with a slight disorientation of these mosaic pieces or adjacent sub-grains of only a few seconds or minutes of arc.

As a result of that disorientation, the Bragg law ($n\lambda =2d sen \theta$) is not satisfied for each of them at same value of $\theta$ which causes a broadening of the XRD peaks (mainly when the crystallites have a very small size, usually <1000 Å in the case of clay minerals).

On the other hand, the existence of lattice strains, or distortions, inside the domains, also causes a broadening of the diffraction peaks and affects the shape of the profiles. These distortions consist of a random displacement of unit cells or groups of unit cells from their ideal positions. There are also other imperfections, such as irregularities in the stacking sequence of the atomic planes in the crystal lattice (stacking faults).

A given reflection hkl informs only about the microstructure in the direction perpendicular to the hkl planes [31]. Therefore, if the material to be examined is anisotropic (in the sense that the broadening of its profiles depends on the reflection), an analysis of several reflections is required for a more complete microstructural characterization. The first approach by Scherrer [32], to extract the crystallite size, using a single hkl diffraction peak, was

$$<D>=\frac{K\lambda }{ {\beta }_f\cos \theta }$$

(1)

where $<D>$ = mean crystallite size in the normal direction to the hkl planes, $\lambda$ = wavelength of radiation (Å), β_f = integral breadth of diffraction peak (expressed in radians of 2θ) with correction of the instrumental broadening, $\theta$ = Bragg angle of the considered reflection, $K$ = a constant, of value depending on (a) shape of the crystals, (b) size distribution and (c) the hkl indices of the diffraction planes. Normally $K=1$ is taken and then apparent crystallite size is considered. The value of β_f is obtained from a deconvolution procedure that is applied to the experimental integral breadth and which is discussed below.

Scherrer’s equation does not allow distinguishing the broadening due to the size of the crystallites from that caused by other microstructural factors (micro-strains, stacking faults, etc.). It is, therefore, an approximate procedure for estimating the average size of the crystallites. Note that, in general, two types of strain are considered in powders: macro-strain (macroscopic homogeneous strain affecting all grains equally) and micro-strain (non-homogenous strain field at scale of individual crystallites, which can significantly vary from grain to grain). Macro-strain affects mainly peak positions, while micro-strain results in peak width changes [33].

The found relationship between broadening of the diffraction peaks with lattice micro-strain [34] led to the development of different microstructural analysis methods.

The size parameter determined in these methods is generally denoted by the symbol $<D>$ and represents the average D-length value of a column of unit cells, measured in the direction perpendicular to the diffraction plane, averaged in each diffraction domain. Usually, a distinction is made between $<Dv>$ and $<Ds>$ (volume weighted or surface weighted) both being characteristic parameters of the distribution of $D$ in the sample [35]: $<Ds>$ is the average height of the columns of cells normal to the diffracting planes of the crystallites determined via the method of Warren and Averbach [36,37]. $<Dv>$ represents the mean value of the apparent thicknesses of the domains in the direction normal to the plane of reflection, averaged over the total volume of the diffracting sample. The apparent thickness is considered to be the length of the longest column and it is the parameter that is determined in the simplified methods of microstructural analysis. For a given sample, the values of $<Dv>$ should, in principle, be greater than those of $<Ds>$.

The shape of the diffraction domains was investigated by several authors considering the crystallite sizes calculated for various crystallographic directions, and ideal models were proposed for different cases. Wilson [35] used a spherical model, relating the diameter $\left(D\right)$ of a sphere with the value of $<Dv>$ for any value of hkl. Other papers [38,39,40,41,42,43] considered cases with spherical and cylindrical models as well as prismatic models [41,42,43].

The different meaning of the concept of crystallite versus that of grain or particle should be noted. The use of other characterization techniques (transmission electron microscopy, BET specific surface area, etc.) can—although not necessarily—lead to particle dimensions different from those that can be calculated from XRD measurements. Thus, the meaning of the term particle differs depending on the used technique.

The comparison of the results obtained by direct observation, by means of TEM or SEM at high magnification and high resolution, with those offered via the methods of powder X-ray diffraction microstructural analysis appears frequently in the bibliography. In general, it is verified that the measurements obtained via XRD are often lower than those via microscopic techniques. This may be due, in the case of materials made of isolated crystallites, to the fact that the XRD measurements represent average values, and in other cases to the existence of particles subdivided into crystallites.

2.2. Powder X-ray Diffraction Microstructural Analysis

Before the description of usual methods of X-ray diffraction microstructural analysis (line profile analysis) of clay minerals, some preliminary comments are needed about the information contained in X-ray diffraction profiles and the previous management of that information.

The diffraction condition for a family of planes hkl is expressed by Bragg’s law:

$$2 d_{hkl} sin \theta =\lambda$$

(2)

X-ray diffraction by a family of planes separated by a distance $d_{hkl}$ is produced and the previous equation is fulfilled. Three conditions must be verified: (1) dimensions of the crystal considered infinite related to the distance between two adjacent diffraction centres; (2) perfect three-dimensional order of the structure of the crystal; (3) monochromatic X-ray source: negligible dimensions of the examined sample and a point-like detector. If all three conditions were fulfilled, the intensity of the diffracted beam would produce a Dirac delta distribution. Otherwise, failure of any of the three conditions will cause a broadening of the diffraction peak.

The existence of crystallites, understood as domains of coherent diffraction, of very small dimensions—less than 1 µm in most cases—and of micro-deformations implies that the first two conditions are not met. As for condition (3), whatever the X-ray monochromatization and collimation system used, no X-ray beam incident on a crystal is perfectly monochromatic, but it consists of a small range of wavelengths and consequently broadening of the diffraction peaks occurs:

$$d\lambda = 2 d_{hkl} cos \theta d\theta$$

(3)

$$d\theta = \frac{d\lambda }{2 d_{hkl} cos \theta }$$

(4)

Thus, related to $d\lambda$ and around $\theta$, there is a small interval $d\theta$, for which the Bragg equation is verified, and the total energy diffracted by a crystal is not concentrated in a maximum, but distributed around $\theta$, producing a broadening of the peak profile.

In addition, other factors related to instrumental conditions of the diffractometer also cause broadening of the diffraction peaks. In summary, the experimental profile of an XRD peak is determined by three components: (a) the intrinsic profile of the material, caused by its microstructural characteristics; (b) the profile caused by the spectral distribution of the radiation emitted by the anode of the X-ray tube; (c) the profile caused by the geometric aberrations of the diffractometer.

Both contributions of components (b) and (c) are usually included in an instrumental function denoted as g(x), where $x=2\theta$. In the same way, the function including the intrinsic contribution of the material, that is, the real profile of the sample is called f(x), while the observed or experimental profile is represented by the function h(x). In mathematical terms, these three functions are related by a convolution product [35,44]:

$$h\left(x\right)=g\left(x\right) \ast f\left(x\right)$$

(5)

where the symbol $\ast$ represents the convolution operation.

The function g(x) is expressed [45] as the convolution product of the contribution of spectral dispersion (W) with that that originated in instrumental aberrations (G). The function h(x) that defines the experimental profile of a diffraction peak is then:

$$h\left(x\right)=(W\ast G \ast f(x\left)\right)+background$$

(6)

2.3. Fundamental Parameters of Diffraction Profiles

The main aim of diffraction measurements is to accurately determine the positions of the peaks and their intensities, as well as their broadening and shape. This information is included in the basic measurement parameters used in the description of the diffraction profiles: position, area and dispersion parameters.

2.3.1. Position and Area Parameters

The two most commonly used position parameters are the angle 2θ_p, corresponding to maximal intensity, and the centre of gravity (or centroid), expressed as:

$$〈2\theta 〉=\frac{\int \left(2\theta \right)I\left(2\theta \right)d\left(2\theta \right)}{\int I\left(2\theta \right)d\left(2\theta \right)}$$

(7)

The area parameter is represented by the integrated intensity of the peak, Iint, representing the area that delimits the diffraction profile above background line.

$$I_{int}=\int I\left(2\theta \right)d\left(2\theta \right)$$

(8)

I_p is the maximum intensity of the peak.

2.3.2. Dispersion Parameters

These parameters characterize the broadening of the diffraction peaks and are therefore used in the detection of microstructural effects. The most used are the width at half height and the full width.

The width at half height—introduced by Scherrer—is determined by directly measuring the width of the profile at the position corresponding to half the height of the peak (estimated relative to the baseline). It is usually symbolized by FWHM (for Full Width at Half Maximum), although the notations 2ω, H or Γ are also used to refer to this parameter.

The integral breadth—introduced by Laue—is denoted by β, and is defined as

$$\beta =\frac{1}{I_p}\int I\left(2\theta \right)d\left(2\theta \right)$$

(9)

where I_p is the maximum intensity of the peak.

The value of β for a peak is equivalent to the width of a rectangle with the same area and height as the diffraction profile. There are other dispersion parameters—like the variance of the profile [35] much less used.

2.3.3. Shape of the Diffraction Profiles

Some procedures of microstructural analysis such as the Williamson and Hall method [46] and that of the Voigt function method [47] base their equations on assumptions about the shape of the profiles.

Thus, the XRD profiles were initially described by the Cauchy (or Lorentz) function and the Gauss function (Figure 1). The symmetric curves generated by these two functions are significantly adjusted to XRD profiles, and thus have been widely used [48].

As stated above, the experimental X-ray diffraction profiles (x) are the result of the convolution of real profiles f(x) with the instruments g(x). The shape of each of these contributions must be considered separately.

Considering the real profiles f(x), the functions of Cauchy and Gauss represent the characteristic broadenings produced by extreme microstructural effects. According to Langford [47]—among other authors—Cauchy’s function properly describes an f(x) profile when its broadening is due to the effect of crystallite size. If broadening is caused by strain, the profile f(x) fits a Gauss curve. If both effects concur, the diffraction profile will define a curve composition of both functions. Therefore, the functions of Gauss and Cauchy are known as base functions in the field of profile-fitting procedures.

It is also possible to separate two contributions in the shape of instrumental profiles g(x), The broadening effects due to the dispersion of λ contribute to the instrumental profile with a Cauchy-type curve, while the geometric effects present a Gaussian contribution.

The experience in X-ray powder diffraction shows that at low angles (2θ < 90°) there is a good fitting to Gaussian functions, since geometric effects predominate over X radiation. At higher angles, the profile g(x) is Lorentzian because the effect of the dispersion of λ predominates over other instrumental and geometric factors.

To obtain the real profile f(x) from the experimental h(x), it is necessary to know the shape of the instrumental profile g(x). Once g(x) is known, through deconvolution it is possible to isolate f(x), from which the microstructural information is obtained.

The instrumental profile, g(x), is obtained in practice by using well-crystallized standard samples, which present a negligible broadening due to the microstructural characteristics. The materials that can be used for this purpose must meet a series of requirements, such as the following: (1) large crystallite size, greater than 1 μm; according to the kinematic theory of diffraction, for diffraction domains of this size, the width of the diffraction maxima is much smaller (on the order of thousandths of a degree in 2θ) than the instrumental width; (2) wide angular distribution of intense peaks; (3) high crystallinity intense, non-overlapping peaks; (4) homogeneity and chemical purity; (5) readily available, non-toxic, stable; (6) free of strains and micro-strains. NIST standards for XRD are available at https://www.nist.gov/programs-projects/powder-diffraction-srms (accessed on 28 January 2024)

Some usual standards are SiO₂, Si, Al₂O₃, Y₂O₃, BaF₂, LaB₆ and ZnO. A perfectly crystallized sample of the same chemical composition as the product to be analyzed can also be used as standard.

As said before, the profile g(x) is determined by the spectral distribution of the radiation used (W) and by the instrumental contributions (G). The effect that the spectral distribution has on the profile logically depends on the degree of monochromatization of the radiation. If non-monochromatic K_α1 radiation is used, the presence of the K_α2 component produces an asymmetric broadening of the profiles.

The following instrumental contributions can be basically considered [49]: (1) Width of the X-ray source: understood as the dimensions of the projection of the linear focus on the plane normal to the absorption coefficient axis of rotation of the goniometer. This factor produces a symmetrical broadening of the profile that is symmetrical and independent of the angle. Its effects can be reduced by using narrower divergence slits. (2). Flat sample: as the sample is flat, the Bragg–Brentano condition is not fulfilled, causing an asymmetric widening of the profiles, mainly at low angles. (3) Axial divergence of the incident beam: the primary X-ray beam has considerable lateral divergence in the direction of the axis of rotation of the goniometer. This divergence tends to broaden the beam in the focusing plane, producing substantial asymmetry in the profile, particularly for extreme angular values (<20° 2θ and >160° 2θ). This effect can be corrected by using Soller slits. (4) Penetration of X-rays into the sample: as the absorption coefficient, (μ) of a sample decreases, the X-ray beam penetrates deeper, so that the effective diffraction surface moves away from the focusing circle. This factor introduces an asymmetry in the profiles for materials with low coefficients of absorption. (5) Width of the receiving slit: its opening produces a broadening symmetric to and independent of the diffraction angle.

The width and shape parameters of the profile g(x) vary with 2θ. This variation is indicative of instrumental resolution, and is usually described using the equation of Caglioti [50]:

$${\left(FWHM\right)}^2=U {tg}^2\theta +V tg\theta +W$$

(10)

where U, V and W are characteristic parameters of the used instrumental device and of the radiation source.

The analysis of the real profile of a diffraction peak cannot be performed without previous separation of the microstructural and instrumental contributions, allowing the production of profile f(x) from h(x) and g(x) by means of an operation of separation known as deconvolution, which is part of the stages of all microstructural analyses.

Detailed description of the aberrations comprising the geometric component of the instrumental profile are provided in [51]. There are procedures to calculate instrumental line broadening like the fundamental parameter approach [52,53], by quantification of the different instrumental aberrations, and ray tracing without making any a priori assumptions [54].

2.3.4. Some Considerations Prior to the Microstructural Analysis

A complete microstructural analysis consists of three stages: (a) sample preparation and XRD data collection, (b) correction of experimental profiles, (c) calculation of microstructural parameters.

The experimental conditions used to obtain the XRD patterns are of great importance, and mainly concern the preparation of the sample and conditions of XRD data collection [55].

The conditions of data collection must allow obtaining a pattern of quality, and that implies precision in the position of both the peaks and the intensities. The latter are fundamental since they define the diffraction profile. If, as usual, a diffractometer working in discontinuous mode is used, the intensity measurement precision is improved by increasing the number of accumulated counts in each step and/or the number of steps measured in the considered angular range; extensive details can be found in [56].

As a practical rule to achieve an adequate level of precision, it can be considered that the measurement conditions (mainly the time per step) must be adjusted in such a way that the maximum of the peak exceeds, as far as possible, 10⁴ counts [57].

The corrections that must be applied to the profiles, both to h(x) and g(x), are as follows: (1) elimination of K_α2, (2) Lorentz polarization correction and (3) elimination of the background. Deconvolution would also be a profile correction, although applied only to h(x).

The contribution of K_α2 to the experimental diffraction pattern can be avoided by using a monochromator. When this is not possible, a fairly common circumstance, then it is necessary to work with the characteristic K_α(1+2) doublet. Under these conditions, the elimination of the broadening produced by K_α2 can be achieved by computational procedures [58].

The use of profiling techniques offers another way to eliminate the K_α2 contributions. In this case, the studied profile is fitted to two peaks, one originating in K_α1 (which appears at a lower θ) and the other originating in K_α2. The correction of factors dependent on θ, such as the Lorentz polarization factor, must also be considered. The broadening produced by this factor is relatively small for values of 2θ < 30°, being instead more important for reflections corresponding to high 2θ values [59].

The correct estimation of the background of the profile is decisive in the quality of the data obtained. Generally, the bottom line is eliminated assuming a linear variation of it with 2θ. Overestimation of the background can lead to truncation of the profile, causing important errors. In the Fourier analysis of profiles, one of these errors is the so-called ‘hook effect’ [60]. Profile-fitting methods largely obviate this problem, since the analysis is done on the modelled profile, that is, on the analytical function that replaces the experimental profile.

2.3.5. Deconvolution Procedures

There are several deconvolution procedures, including the most rigorous methods that use Fourier transforms [61,62], the Ergun iterative method [63], the LWL method [64] and methods that use the integral breadth β of the diffraction profiles.

An analytical peak-shape function is explicitly included in all Rietveld refinements, a function that as accurately as possible represents the observed peak shapes. Although earlier Rietveld refinement programs used Gaussian or Cauchy profile functions, more recent programs include other functions such as pseudo-Voigt (combination of Gaussian and Cauchy) or Pearson VII functions; descriptions of different profile functions were described [65]. These profile functions have commonly been combined with an angle-dependent peak breadth function, like that of Caglioti [50] relating peak breadth to diffraction angle. The basic assumptions made in line profile decomposition methods and the type of line profile methods and the type of obtained size and strain data are summarized in [66] as shown in [67,68].

Simplified Deconvolution Procedures

• Method of Williamson and Hall [46]

This is a simplified method initially introduced as a means of separating the effects of size and strain on peak broadening. In the original version of the method, the contributions of size and deformation were assumed to be represented by Cauchy functions; in subsequent applications, a modified version is used assuming Cauchy and Gauss profile functions both for the investigated material and for the standard used to estimate the broadening.

In the first case, the experimental profile is obtained by association of two Cauchy curves; therefore, the integral breadths of the peaks are additive, verifying:

$${\beta }_h={\beta }_f+{\beta }_g \mathrm{Cauchy}\text{-}\mathrm{Cauchy}$$

(11)

In a similar way, for the second case, the squares of the integral breadths are additive, and thus:

$${\beta }^2={{\beta }_f}^2+{{\beta }_g}^2 \mathrm{Gauss}\text{-}\mathrm{Gauss}$$

(12)

These two last expressions are the basis of the deconvolution procedure prior to the microstructural analysis, allowing one to obtain the values of β_f (corresponding to the real profile of the material).

When the broadening is only a result of size effects, the expression of β_f is

$${\beta }_f=\frac{K\lambda }{<D>\cos \theta }$$

(13)

If the broadening is attributed exclusively to micro-strains, we have [34]:

$${\beta }_f=4 \mathit{e}\tan \theta$$

(14)

where e is the strain parameter

In the general case where β_f can be affected by both effects, and assuming that they can be associated with either Cauchy profiles or Gauss profiles, we can write:

$${\beta }_{fC}=\frac{K\lambda }{<D>\cos \theta }+4 \mathit{e}\tan \theta$$

(15)

Cauchy–Cauchy

$${{\beta }_{fC}}^2={\left(\frac{K\lambda }{<D>\cos \theta }\right)}^2+{4 \mathit{e}\tan \theta }^2$$

(16)

Gauss–Gauss

Subscripts C and G refer to Cauchy and Gauss, respectively. Other equations based on the association of Cauchy and Gaussian profiles can be used, showing a better approximation, but with the disadvantage of more complex expressions [48,69].

When these last two expressions are used, the integral breadths β can be substituted by their respective FWHM values [39].

The Williamson and Hall method must be considered as an approximate procedure for the estimation of microstructural parameters. If several orders of the same reflection are used to obtain the Williamson and Hall diagram, the results obtained refer to the crystallographic direction considered. If different reflections can be used, the obtained values of <D> and e represent the average over several crystallographic directions.

2.

The method of Voigt function [47]

This method is based on the fact that the profile of a diffraction peak can be described in a first approximation by the convolution of a Cauchy (Lorentz) curve and a Gaussian curve. The two main microstructural contributions to the broadening of the diffraction peaks, crystallite size and micro-strain, produce two different profiles: a Lorentzian-type intensity distribution, from crystallite sizes, and a Gaussian distribution from micro-strains [47,70].

Thus, the K_α1 pure profile f(x), resulting from these two contributions, can be described as the convolution product.

$$f\left(x\right)=f_{G}G\left(x\right) \ast f_{C}C\left(x\right) \mathrm{Voigt} \mathrm{function}$$

(17)

On the other hand, it has been found experimentally that the instrumental function g(x) can also be adequately described as a Voigt function, with a Gaussian-type profile predominating at low angles (2θ < 90°) and a Cauchy profile for higher angles:

$$g\left(x\right)=g_G\left(x\right) \ast g_C\left(x\right)$$

(18)

Thus, the function h(x) of the experimental profiles can be written as:

$$h\left(x\right)=\left[f_G\right(x) \ast f_C(x\left)\right] \ast \left[g_G\right(x) \ast g_G(x\left)\right]$$

(19)

Since the convolution operation ∗ is commutative and associative

$$h\left(x\right)=\left[f_G\right(x) \ast g_G(x\left)\right] \ast \left[f_C\right(x)\ast g_C(x\left)\right]$$

(20)

Or in a more abbreviated expression:

$$h=h_G\ast h_C (h_G=f_G\ast g_G; h_C=f_C\ast g_C)$$

(21)

The Voigt function h(x) has a variable form, since its profile changes according to the proportion of Gaussian and Cauchy contributions. To uniquely describe the profile of a Voigt function, the so-called ‘shape factor’ is used Φ, which is Φ = FWHM/β, where Φ ranges from Φ_C = 0.63662 for a Lorentzian profile, to Φ_G = 0.93949 for a Gaussian profile. This shape factor gives a measure of the relative weight of the Gaussian and Cauchy components in the profile. The ratio of Gaussian and Cauchy contributions to the Voigt profile can be expressed using the integral breadth

$$\frac{{\beta }_G}{\beta } \left(Gauss\right), \frac{{\beta }_C}{\beta } \left(Cauchy\right)$$

where β_G and β_C are the integral breadths of the Gaussian and Lorentzian profile, respectively.

To determine the value of each contribution in the diffraction profile, the Voigt analysis relationships of the diffraction peaks (empirically obtained in [71]) can be used, obtaining:

$${\beta }_C=\beta [2.0207-0.4803·\Phi -1.7756·{\Phi }^2]$$

(22)

$${\beta }_G=\beta [0.6420+1.4187{(\Phi -2/\pi )}^{1/2}-2.2043·\Phi +1.8706·{\Phi }^2]$$

(23)

Conversely, it is also possible to obtain β and Φ from β_C and β_G by analogous expressions.

These relationships apply to the experimental profile h(x) and to the instrumental profile g(x). Assuming that h(x), g(x) and f(x) are Voigt functions, and knowing for each profile the shape factor Φ and the integral breadth β, then β_C and β_G (β_C^h, β_C^g, β_G^h and β_G^g) can be calculated. Once β_C and β_G are known, the Gaussian and Cauchy components of the diffraction profile of f(x) are obtained by deconvolution using the correction formulae based on Equations (1) and (2):

For a Cauchy profile:

$${\beta }_C^f={\beta }_C^h+{\beta }_C^g$$

(24)

And for a Gauss profile:

$${\left({\beta }_G^f\right)}^2={\left({\beta }_G^h\right)}^2+{\left({\beta }_G^g\right)}^2$$

(25)

Separation of size and strain effects.

It is shown [71] that the effect of the size of the crystallites on the broadening is fully represented by the Cauchy component of the real profile, while the contribution of the micro-strains is included in the Gaussian component. The apparent size of the crystallites is then obtained from Equation (3)

$$〈D〉=\frac{\lambda }{{\beta }_C^f cos\theta }$$

(26)

and the parameter of micro-strains from (4) is:

$$\mathit{e}=\frac{{\beta }_G^f}{4 tg\theta }$$

(27)

where λ is the wavelength used (Å) and θ is the Bragg angle (radians). In both cases, β is expressed on the θ scale. The value of e can be considered proportional to the integral breadth of the distribution of the lattice strains, considering the hypothesis of a Gaussian distribution curve. This hypothesis is part of a specific microstructural model, which may differ from that used in other methods such as that of Warren and Averbach, which will be detailed below. The parameter e according to the Voigt function method may therefore behave differently from the corresponding lattice strains in another method.

The procedures for correction of instrumental effects and microstructural analysis in this method can be summarized in the following scheme (Figure 2).

3.

Implementation in Rietveld refinement of the whole diffraction powder pattern.

Bish [72] showed line broadening analysis as one of the applications of the Rietveld method [73] to clay minerals. Thus, information can be extracted from broadened reflections using the Scherrer equation relating peak broadening and crystallite size. Different analytical peak-shape functions are included in Rietveld refinements, to fit the observed peak shapes by different possible functions. Although earlier Rietveld refinement programs used simple profile functions (Gauss and Cauchy), more recent programs introduced other functions as pseudo-Voigt (combination of Gaussian and Cauchy), Pearson VII or others combined with an angle-dependent peak-width function, e.g., the above-mentioned function of Caglioti [50], using U, V and W as refinable parameters. The GSAS program [74] uses a pseudo-Voigt function that contains separate expressions for Gaussian and Cauchy broadening and his formulation allows separation of crystallite size and strain contributions to reflections with anisotropic broadening.

Explained examples of microstructural analyses via the methods of Scherrer, Williamson and the Hall and Voigt function were provided [75] in a review of crystallite size determination via powder X-ray diffraction and complementary techniques in nanomaterials.

Fourier Transform Methods

The method of Stokes [62] allows one to obtain the coefficients of the Fourier development of f(x) in order to synthesize the real diffraction profile, using relationships derived from the properties of Fourier transforms, with the starting data being the following: Hr, Hi = Fourier coefficients of cosine and sine corresponding to the experimental profile h(x); Gr, Gi = Fourier coefficients of cosine and sine corresponding to the instrumental profile g(x). An extended explanation can be found, e.g., in [67,76].

The iterative method of Ergun [63] proceeds by successive convolutions. Considering that the convolution product is h = g ∗ f, an increment u_n is defined such that u_n = h (g ∗ f_n), where h is taken as the initial value of f. The iteration ends with the relation f _n+1 = f _n + u_n, which approaches the real profile after reaching a convergence criterion.

The LWL method [64] is based on treating the convolution product, h(x) = g(x) ∗ f(x), as a system of linear equations whose solution is reached with the stabilization of the convolution operator.

Method of Warren and Averbach

A said before, the broadening of the diffraction peaks is contributed by the length of the domain in the direction perpendicular to the diffracting planes. This length or thickness can be described as the average length of the cell columns perpendicular to the diffraction plane, which for mathematical convenience is in all cases taken to be the (00 l) plane. In this way, each reflection of a crystal of any symmetry can be considered of type 00l. This condition, which can be achieved with an appropriate transformation of the crystallographic axes, does not affect the general applicability of the method.

The microstructural study via analysis of Fourier series of the peak profiles via this method is described, e.g., in item 5.5.3. of the book of Ginebretière [76]. The powder diffraction peak profiles can be expressed as Fourier series. The cosine Fourier coefficients may be used to obtain information relating to the domain size (or crystallite size) and strain from the deconvoluted pure profiles

Considering the column of lattice cells in the direction of a₃

$$f\left(h_3\right)=\sum _{n=-\infty }^{+\infty }(A_{n}cos2\pi n{h}_3+B_{n}sen2\pi n{h}_3)$$

(28)

where

$$A_n=\frac{N_n}{N_3}〈cos2\pi l{Z}_n〉$$

(29)

$$B_n=\frac{N_n}{N_3}〈sen2\pi l{Z}_n〉$$

(30)

where f(h₃) is the intensity distribution corresponding to the reflection (00l), h₃ = 2 a₃ sin θ/l and a₃ = cell length in the crystallographic direction (00l).

The Fourier coefficients A_n contain information related to the size of the crystallites and the lattice strains. Indeed, it is found that these coefficients are the product of two terms that can be grouped into a size A_n^S coefficient, and a strain coefficient A_n^D:

$$A=\left(A_n^S|A_n^D\right)$$

(31)

The size coefficient A_n^S is independent of the order of reflection, contrary to what occurs with the distortion coefficient A_n^D:

$$A_n^S=\frac{N_n}{N_3}$$

(32)

$$A_n^D=〈cos2\pi l{Z}_n〉$$

(33)

In these expressions, N_n = number of cells in the whole sample that have an nth neighbour in the same column, N₃ = average number of cells per column in the a₃ direction. If N is the total number of cells in the sample, and N_col is the number of columns, N₃ = N/N_col (representing the mean dimension of the domain in unit cells).

Regarding Z_n, it is a magnitude associated with displacements, in the direction perpendicular to the (00l) plane, which can affect the positions of the cells in a column. Thus, the product Z_n a₃ is the difference between the displacements of two cells separated by n cells, or in other words it is the change in the length of a column of original length na₃. Z_n is usually replaced in calculations by the strain, ε_L (Z_n = ε_L n), which is defined as

$${\epsilon }_L=\frac{∆L}{L}$$

(34)

with

$$∆L=a_3{Z}_n$$

(35)

And taking A_n^D as a function of ε_L,

$$A_n^D=<cos 2 \pi ln\epsilon L>$$

(36)

Note that A_n^S only contains information about the size of the domains, while A_n^D contains only information about the lattice distortion.

Regarding the Fourier series coefficients B_n, if the positive and negative values of Z_n are equally probable for a given value of n (symmetrical peak), they disappear in the total computation, so they are not taken into account in the calculation that follows.

Separation of size and strain effects.

By taking logarithms in An:

$$ln{A}_n=ln{A}_n^S+ln{A}_n^D$$

(37)

If in A_n^D we develop the cosine function in series, discarding the terms of order higher than two (for small l and n values), we obtain

$$ln{A}_n=ln{A}_n^S+\mathrm{l}\mathrm{n}(1-2{\pi }^2{l}^2{n}^2〈{\epsilon }_L^2〉)$$

(38)

Taking into account that for small x it is true that ln(1 − x) ≈ x,

$$ln{A}_n=ln{A}_n^S+\mathrm{l}\mathrm{n}{A}_n^S-2{\pi }^2{l}^2{n}^2〈{\epsilon }_L^2〉$$

(39)

This equation is the basis for calculating the size of the domains and the distortions. Since lnA_n is a function of the order of reflection l, if at least two orders of a reflection are available (for example, 002 and 004, 111 and 222 ) the values of A_n^S and <ε_L²> can be determined from the graphical representation of lnA_n vs. l². This representation gives us a series of lines, one for each known value of n. From the ordinate at the origin of these lines, lnA_n^S is obtained, while <ε_L²> can be calculated from the slope.

The magnitude <ε_L²> is the root mean square of the strain calculated over all the columns of cells in the sample crystallites (values of ε_L can be either positive or negative; thus, ε_L² is taken). In practice, the square root of this expression is used as the parameter that indicates the mean lattice distortion <ε_L²>^1/2, abbreviated RMS (Root Mean Square Strain). Since from the representation of lnA_n vs. l² a distribution of the RMS values can be obtained, for different values of n (or of L), the RMS value for L is usually taken as the characteristic parameter of strain in the sample for L = 50 D.

The average dimension of the domains is obtained from the coefficients A^S previously calculated. According to Warren and Averbach, the first derivative of A^S with respect to n, for n = 0, is:

$${\left[\frac{\mathsf{\partial }{A}_n^S}{\mathsf{\partial }n}\right]}_{n\to 0}=-\frac{1}{N_3}$$

(40)

Therefore, N₃ can be calculated from the slope at the origin of the curve A^S vs. n. In practice, n is replaced by the length L; that is, A_L^S vs. L is plotted; thus, the above equation can be written as:

$${\left[\frac{\mathsf{\partial }{A}_L^S}{\mathsf{\partial }L}\right]}_{L\to 0}=-\frac{1}{\overline{D_{eff}}}$$

(41)

D_eff = N₃ a₃ is the obtained size parameter. This parameter represents the length of the cell columns in the direction perpendicular to the diffracting planes, averaged over the whole sample (within each domain and between all of them). This is a surface mean value (<Ds>). The subscript ‘eff’—effective or apparent—indicates the possibility that this parameter includes the effects of lattice macro-strains.

Finally, the second derivative of the A_L^S coefficients is proportional to the distribution function of the length of the columns of cells in the sample, P(L):

$$\left[\frac{\mathsf{\partial }{A}_L^S}{\mathsf{\partial }{L}^2}\right]=\propto P(L)$$

(42)

The function P(L) is a measure of the relative frequency with which columns of cells appear in the sample:

$$P_L=\frac{N_L}{\sum _{L=0}^{\infty }{N}_L}$$

(43)

where N_L is the number of columns of length L in the sample.

The graphic representation that is usually used to visualize the size distribution is the relative frequency curve P(L) vs. L. When the profile f(x) is affected only by size effects, the distribution of the lengths L can be obtained using the method of LeBail and Louër [77].

The MudMaster program [78,79] was developed to study crystallite thickness distributions and strain distributions in minerals, particularly in clay minerals, by the Warren–Averbach method (named Bertaut–Warren–Averbach by the authors of the program).

Whole Powder-Pattern Fitting

The fitting of particular line profiles may be difficult in the case of overlapping peaks, in monophase or multiphase samples. By adopting an adequate structural model for the studied crystalline phase, the problem can be solved via Rietveld method-based programs. Examples of those programs are GSAS [74], BGMN [80], MarqX [81], Fullprof.2k [82], adapted to lattice-parameter refinement and line profile analysis using peak breadths and shapes; the profile shape parameters can be used as tuneable parameters for each reflection or constrained by a suitable function describing their angular (or hkl) dependency. Several contributions to modeling X-ray diffraction patterns, the Rietveld method and particular cases of clay minerals are included in the book edited by Reynolds and Ferrell [83]. Sybilla program Version 2.2.2 software (Chevron ETC proprietary) [84] was used to fit experimental basal XRD reflections of clay minerals.

3. X-ray Diffraction Microstructural Analysis in Clay Minerals

3.1. General Features of Structure and Classification of Clay Minerals

The clay minerals are hydrous layer lattice silicates appearing in clays; the fundamental structural scheme common to layer silicates as well as the description of their chemical variability and identification was provided by Warshaw and Roy [85] after reviewing detailed features described in many previous papers and books. The main points are briefly described below.

The fundamental structural units in layer silicates are tetrahedral silica layers and octahedral (of brucite or gibbsite) layers. The former (T layer) is made of Si-O tetrahedra connected at three corners in the same plane producing a two-dimensional network of near hexagonal rings. The fourth corner, unjoined oxygen corners, all point in the same direction. The brucite or gibbsite layers consist of hydroxyl ions in two planes, above and below a plane of magnesium or aluminum ions which are octahedrally co-ordinated by the hydroxyls. The difference between both octahedral layers (brucite and gibbsite) is that in the former (trioctahedral) all the octahedral cation positions are filled, whereas in the latter (dioctahedral) only two thirds of these positions are filled.

Three mains groups were distinguished for structures with 2, 3 or 4 layers (respectively, 1:1 or T-O layers, 2:1 or T:O:T layers and 2.1:1 or T-O-T + O layers) characterized by basal spacings of near 7, 10 and 14 Å, with subgroups by the type of octahedral layer. Some of the mixed-layer clay minerals consist of combinations of fundamental units of more than one main group. The fourth group of clay minerals, including sepiolite and palygorskite (or attapulgite), was considered in the identification scheme of the paper, but without a particular discussion about their structural units.

Compositional variations in the clay minerals are controlled by simple crystal-chemical principles. The simplest substitutions involve the replacement of an ion by another of similar ionic radius but with identical charge, e.g., Fe²⁺ replacing Mg2⁺. However, many more interesting combinations are possible by ‘balanced substitution’ in the various positions in the layer lattice. The charge lost in one layer is compensated by an identical gain in another layer.

A third criterion in the classification is the manner and perfection of stacking, allowing subtle distinctions. Both ‘mixed layering’ (heteropolytypism) and ‘polytypism’ (homopolytypism) of layer silicates are examples of a general feature of polymorphism in layer structures which are considered in the paper.

Structures of layer silicates are described in chapters of reference books, e.g., [8,23,24,25,26,27,28,29,83,85,86,87]. The updated classification of the AIPEA Nomenclature Report for 2006 [88] includes additional characterizations at the group level of the classification of planar hydrous phyllosilicates by type of interlayer material and net layer charge per formula unit, as well as a classification of non-planar hydrous phyllosilicates (considering sepiolite and palygorskite, among others) and criteria for interstratification nomenclature.

A mixed-layer mineral is considered [79] as an assemblage of statistically weighted crystallites (or coherent scattering domains, CSDs) and the thickness of crystallites is determined by the number of interstratified layers parallel to each other in the ab plane and the crystallites of the mixed-layer minerals have quite different structure and composition. All XRD files were transferred into the format of the NEWMOD program. A theory of crystal size measurement from XRD line broadening is shown in this paper. As an alternative to the instrumental broadening correction from a standard (difficult to find), they used theoretical XRD patterns generated via the NEWMOD© program (below referenced in the software section) for samples with the same particular experimental setting conditions and not accounting for combined instrumental broadening produced by the geometry of the goniometer; the sample and the spectral characteristics not modelled by NEWMOD are considered negligible for very broad reflections.

Structures of clay minerals can exhibit different types of order/disorder described in detail by Brindley [89], Moore and Reynolds [28], and in the book of Drits and Tchoubar [90] concerning the structure analysis of lamellar crystals, and a main tool is to compare experimental diffractograms with simulated models considering all the factors (instrumental factors included). The paper of Drits [91] deals with methodological approaches to study defective layer structures.

Chapters about interstratified clay minerals were included in books on XRD clay mineralogy [26,27] and in special publications [92,93] including different chapters with updated reviews on structures, defective structures, intercalation, delamination, surface properties and advanced techniques in research of sheet silicates and other layered structures. More recently a thorough contribution by Sakharov and Lanson [94] provides a detailed description of concepts involved in diffraction by mixed-layer structures from calculated intensities, statistical description of layer stacking by matrix formalism. Some relevant points are summarized next.

Distribution of crystal thickness and intensity calculations include a parameter δ for the shape of size distribution and N, the total number of layers in a crystal and the probability of layers with a thickness. The outer surface layers (OSLs) are considered because of the very small crystal sizes as well as the influence of intra-crystalline defects [95,96]. There are new insights about hydration and expansion heterogeneity and intercalation, and about the actual structure of mixed layers as intra-crystalline defects and outer surfaces of crystals. The use of the whole-pattern profile-modelling approach, such as the multi-specimen approach [95,96,97], rather than the position of the main diffraction lines to determine the present clay minerals, allowed the uncovering of their frequent polyphasic character of the mineral assemblage.

It can be said that the identification of the minerals present in the studied sample and particularly of mixed layers, in the low angle range of the X-ray diffraction pattern, must be performed to avoid possible contribution of lines of other minerals to the line profiles of the mineral selected for microstructural analysis.

Two more recent books [29,98] include recent advances in research on fibrous clay minerals and nanosized tubular clay minerals of the kaolinite group, respectively

3.2. Crystallinity of Clay Minerals

Different forms of order and disorder occur in phyllosilicates affecting stacking sequences, isomorphous substitutions of ions or vacancies, in particular coordination sites, or in the interlayer, or in stacking of two or more kinds of layers in mixed-layer minerals as well as interstitial impurities, dislocations and other defects. Those structural features can contribute to line profiles and to the whole powder diffraction pattern of minerals. In practice, ‘crystallinity’ is defined as the degree of perfection of translational periodicity as determined by some experimental method. The meaning of the term crystallinity was discussed by the AIPEA Nomenclature Report for 2001 [99].

The powder diffraction patterns of kaolinite can be significantly different as a result of different structural disorders. Thus, comparisons of features of the X-ray diffraction patterns of kaolinites were used as qualitative descriptions of order. Sharp and narrow peaks are found in ordered kaolinite, whereas disordered kaolinite shows less well-defined, broad and asymmetrical peaks, mainly for hkl reflections with k ≠ 3n (where n is an integer) [100,101].

Increasing disorder in XRD patterns of minerals of kaolinite group from kaolinite to halloysite was found by Brindley and Robinson [95]; in this paper the classification of four representative experimental X-ray diffraction patterns based on the variations of intensity and shape of the peaks corresponding to hk bands was considered useful for a quick routine assessment of ‘crystallinity’. XRD patterns of kaolinites were classified according to ‘crystallinity’ by number, definition and sharpness of peaks in XRD powder patterns [96]. Those criteria were used to distinguish different triclinic and monoclinic kaolinites (T, T partially disordered, pM-T, pM partially ordered and pM disordered kaolinite), as intermediate between ordered triclinic kaolinite and pseudomonoclinic (pM) disordered kaolinite [102].

The Hinckley index was proposed as an empirical measurement of the degree of disorder of a kaolinite, based on the ratio (B + C)/A, calculated by considering the intensities of peaks 1-10 (B) and 11-1 (C) measured on the line joining the background noise between peaks 020 and 1-10 and the background noise just after the 11-1 peak and the intensity of the 1-10 peak (A) measured over the overall background noise [103,104].

The Range and Weiss index [105] has been widely used. It compares the area of the diffraction band between the 11-1 and 02-1 peaks to the total area of a rectangle formed with the height of the 11-1 peak and the distance separating the 11-l and 02-1 peaks as the base.

The intercalation of various kinds of molecules among the layers of minerals belonging to the group of the kaolinite was applied to the study of ‘crystallinity’ as well as to the identification of species. Hydrazine intercalation is especially interesting for studying crystallinity, because of the influence of structural order; thus, kaolinites with an X-ray diffraction powder pattern showing high crystallinity intercalate hydrazine quantitatively but those with structural disorder do not [106].

The relationship between morphology, particle size and crystallinity of kaolinites was compared in several works [101,106,107,108,109,110,111]. It was found that crystallinity is not a function of the size fraction. A good crystallinity of the raw whole sample usually corresponds also to good crystallinity in the fine <0.25 micron fraction.

In a paper dealing with a set of eight kaolinite samples [106], different ‘crystallinity’ indices (those of Hinckley, of Range and Weiss and others) were described. The correlations and utility of their values were considered as well as the results obtained with the ‘expert system’ of Plancon and Zacharie [112] and the FWHM values of 001 reflection of kaolinite [113]. This last paper included an extended bibliography about different methods of evaluation of ‘crystallinity’ of kaolinites, and was focused on X-ray microstructural analyses from 001 and 002 kaolinite reflections.

In experiments of wet grinding of kaolinite of API standard Nº9 [114], it was noted that diffraction patterns of the wet-ground materials evidenced increasing disorder that could be modelled best as a physical mixture of low- and high-defect material. Grinding of kaolinite does not produce a progressive increase in disorder for all of the crystallites present in the sample but apparently increased amounts of a disordered kaolinite coexisting with relatively unaffected material.

The mentioned AIPEA Nomenclature Report for 2001 [99] noted that 001 and 002 diffraction peaks have been widely used to produce ‘crystallinity indices’ mainly for illite and chlorite, and provided historical background and comments about that use, as shown below.

Weaver [115] was the first to realize regular changes in shape of the first 001 basal reflection of illite related to burial (temperature and pressure change) among uses of clay minerals in search for oil. The ‘sharpness ratio’ of Weaver (‘Weaver index’) was defined by the ratio of intensities measured at the peak maximum (near 10 Å) and at 10.5 Å it was related to increasing depth in sedimentary basins.

Kübler [116] used FWHM of 001 illite peak to differentiate the ‘anchimetamorphic zone’ [117,118] hoping to identify the transition between the dry-gas zone and the unproductive zone by using the FWHM of the 001 illite peak (named ‘illite crystallinity’, frequently abbreviated as IC in papers). Many other authors used this index subsequently.

It was shown in different investigations that the Kübler index is mainly influenced by the mean crystallite size and eventually at lower (diagenetic) grades by the amount of swelling of interstratified components [94]. Two papers using TEM [119,120] stated that it is largely controlled by crystallite thickness of illite-muscovite.

In a similar way, FWHM values of chlorite basal reflections have been used as a ‘chlorite crystallinity index’ in low metamorphic grade research as shown in two reviews [121,122] of 1987 and 1995, respectively.

An empirical method was used for description of ‘crystallinity’ of smectites in series of smectitic clays by a parameter v/p of X-ray diffraction pattern of glycolated oriented aggregates of clay, where p was the height of the 17 Å peak above the background and v the depth of the ‘valley’ in the low angle side of the peak [123]. Subsequently, five ‘crystallinity’ classes were described [124] based on (1) the shape, symmetry and intensity of that reflection and (2) the occurrence/absence and shape of the 002 reflection about 8 Å and of the 003 reflection about 5.4 Å.

3.3. X-ray Powder Diffraction Microstructural Analysis of Some Particular Groups of Clay Minerals

Next, we show different contributions about X-ray powder diffraction microstructural analysis of phyllosillicates arranged by main groups of phyllosilicates as industrial minerals.

3.3.1. Kaolinite

The above-mentioned paper of Brindley [30] showed a survey of particle sizes (lengths and widths) and habits of clay mineral of different groups from electron microscopy. The author noticed the measurable broadening distinguishable from instrumental effects in X-ray powder diffraction by crystals of colloidal and near colloidal size, and reviewed the existing knowledge about line broadening. He noted also the possibility of obtaining average volume weighted particle size from β = λ/Lcosθ and the surface weighted distribution of particles sizes via Fourier analysis of the broadened profile via the method of Warren and Averbach [36].

The introduction of a paper by Cizel and Krantz [125] summarized the results about kaolinite grinding from the bibliography at that time in two points. (1) The grinding of kaolinite causes delamination of the clay crystals and thinning in the direction of the c-axis, with secondary importance of rupture perpendicular to the 001 plane. (2) Dry grinding causes a decrease in the degree of crystallinity of kaolinite crystals with more rapid degradation of kaolinite than during wet grinding and also includes the rupture of bonds between tetrahedral and octahedral sheets. It was noted in the introduction of the paper that previous studies about montmorillonite grinding concerned only dry grinding and monitoring of structure decomposition via different techniques. The involved processes were described as delamination and progressive amorphization of the mineral but without precise details of the involved reactions.

In a study [126] about mechanical activation via dry ball milling of two kaolins from Burela and Alcañiz (Spain), it was found that 001 crystallite size via the Scherrer method was much larger in the unground Burela kaolin (238 Å) than in that of Alcañiz (117 Å) and decreased with increasing grinding time, more markedly in the former (173 Å after 65 h grinding) than in the latter (94 Å over the same time).

Changes caused by pressure grinding in the range 0–20 Kbar on two samples were studied [127,128] and it was found that increased pressure caused loss of crystallinity, shown by a decrease in the Hinckley, Lietard and reference intensity ratio indices, as well as in the crystallite size measured from the 001 reflection.

XRD line broadening of kaolinite Te from from Teruel (Spain) was compared with that of kaolinites GP and GW (KGa-1 and KGa-2 of the Reference Clay Source), using the Voigt function method, and that of Warren and Averbach [129]. The order of crystallite size was GW > GP > Te, in both methods, and in agreement with that of morphogical thickness via SEM, and opposite that of micro-strains e (measured via the Voigt function method).

In an experiment of grinding by steel balls planetary mill of a well crystallized kaolinite [130], the decrease in crystallite size with grinding time (0 to 10 h grinding time) was monitored by increasing FWHM values of 001 diffraction peaks.

The 001 crystallite sizes of kaolinite in two kaolins from Indonesia and Western Australia were compared [131]. The values obtained via the Warren–Averbach method were in the ranges 5–6 nm and 9.7–13.4 nm, respectively. Sizes from the Scherrer equation were approximately twice those values but the results showed the same pattern of variation.

A method allowing easy measurement of the thickness of kaolinite particles (flakes), reported as crystallites, from FESEM images was described and a good correlation was found between these values and those of XRD average apparent crystallite sizes from 001 reflections estimated via the Voigt function method [132].

Results of high energy vibrating milling were compared for two kaolinites: a well crystallized kaolinite (the reference kaolinite KG-A1 from the Source Clay Repository) and a poor crystallized kaolinite (from Queensland, Australia) [133]. Good linear correlation was found between the surface-weighted average apparent crystallite size obtained via the Warren–Averbach method and the volume-weighted average apparent crystallite sizes provided via the Voigt function method, in the wide range of crystallite sizes studied. Continuous decrease in <Dv> (mean apparent Voigt 001 crystallite size) with milling time was observed in both sample series, and greater strain values were found in KG-A1 milled samples.

Crystallite size evolution of kaolinite in vibrating cup milling of the reference kaolinite KG-A1b of the Source Clay Repository was studied via the Warren–Averbach method (001 and 002 reflections) and the Voigt function method (00l reflections) [134]. A good correlation was found for results of both methods, as well as with crystallite size measurements from images from field emission scanning electron microscopy at high magnification using the method of [132]. Extensive microstructural degradation was achieved within seconds, showing the high efficiency of the used strain comminution method.

The influence of hydrothermal and meteoric alteration processes on the formation of kaolins of the Iberian Massif (Spain) was studied [135] via the Warren–Averbach method using the X Powder program. The lognormal parameters of crystallite size distributions allowed recognition of the different growth mechanism of hydrothermal and meteoric kaolinites. The same program was used in an analysis of kaolinites obtained via hydrothermal precipitation from solutions of amorphous aluminosilicates, at 200 and 250 °C and at different aging times [136]. Strongly correlated values were found between Warren–Averbach crystallite sizes from 001 and 002 reflections and those of the Voigt function method using 001 reflection.

Experiments of milling in a planetary high-energy ball mill of a kaolinite at defined fixed conditions and variable time (1 to 9 h) were performed and XRD microstructural parameters were obtained via the Voigt function method [137]. Mean apparent crystallite size decreased from 30 ± 2 nm for unmilled powder to 7 ± 1.5 nm for ground material and the internal strain increased from 0.08 ± 0.03% for unmilled powder to 0.35 ± 0.05%. The paper included an extended bibliography about mechanochemical treatment of kaolinites and other materials.

3.3.2. Muscovite, Illite, Vermiculite

Microcrystalline muscovites were studied [138] via a Fourier analysis method, and it was concluded that line broadening of 00l reflections was due not only to a small particle size effect, but also to structural disorders involving the variation of the interlayer spacings, considering that the observed distortions were mainly attributed to non-uniform interlayer spaces between silicate layers arising from an irregular distribution of interlayer cations.

A set of 43 samples of illites of drilled samples from a borerhole of the Paris Basin and 10 additional samples of different sources were studied [139] performing XRD line profile analysis, and a 0.990 linear regression coefficient was found between full width at half maximum of crystallite size distribution with mean crsytallite size via the Warren–Averbach method.

Crystallinity indices, apparent mean crystallite sizes, lattice strain values and crystallite thickness distributions of illite-muscovite and chlorite from metapelitic rocks were determined via powder XRD and/or HRTEM [140] and a reasonably good correlation was recognized between TEM- and XRD-crystallite sizes determined via the Scherrer method

By using the methodology of [141] a set of samples of a prograde sequence of pelitic rocks in the Gaspé Peninsula (Quebec Appalachians) was studied [142] considering the following: (1) crystallite size and strain via XRD profile analysis in whole sample powder and in clay fraction separates, (2) correlation of these data with those from TEM measurements and (3) correlation of microstructural states of illite in clay separates and whole rocks, and illite crystallinity as a function of metamorphic grade. It was shown that similar mean values were determined via the XRD profile analysis and TEM methods.

In the above-mentioned paper [79], a set of well characterized (including thickness distributions from TEM measurements) monomineralic samples of illite and illite-smectite mixed layers was studied. The paper showed the detailed procedure to obtain both mean sizes by avoiding the effect mixed layering. The ‘Kubler index’ (FWHM of the 001 illite reflection on air-dried samples), was affected by mixed layering in a variable degree, depending on expandability, relative humidity and the interlayer cation.

The dehydrated I-S mixed layer was measured via the BWA method [78], and results were very correlated with those obtained via the integral peak width method. However, the refined Warren–Averbach method of MudMaster adopted for the analysis of PVP-10 (polyvinylpyrrolidone, molecular weight 10,000) saturated clays produced profiles not matching the TEM-determined distributions [142].

In a study dealing with six standard samples of crystallinity, using as instrumental standard a muscovite flake [143], an agreement was found between XRD-determined area-weighted mean crystallite sizes calculated from integral breadth via the integral peak-width method and mean thicknesses of crystallites measured via TEM, whereas disagreement was found using the Warren–Averbach method according to the procedure of [78], disagreement probably reflecting inadequate treatment of both background and instrumental broadening effects when dealing with narrow reflections from phyllosilicates.

The use of X-ray diffraction crystallite size distribution from the Bertaut–Warren–Averbach method in deduction of mechanisms of formation of clay minerals was critized for different reasons [144] but subsequently [145] the mentioned method was shown to be powerful, reliable and useful in the analysis of clay minerals’ growth and particularly alteration of rocks.

Thickness distributions of illite crystallites via the Warren–Averbach method (MudMaster program) and high-resolution transmission electron microscopy in shales were compared [146] noting that HRTEM measurements underestimate the proportion of coarse crystallites, which was attributed to the effect of using number-weighted (rather than area-weighted) distributions and to low counting statistics in HRTM.

The illite area-weighted crystallite size distribution of assemblages with mixed smectite layers in shales and bentonites was obtained by using the MudMaster program on X-ray diffraction data of mounts of PVP-10 intercalated Na-saturated clays [147]. The results were compared with those from number-weighted distributions via HRTEM measurements.

Mineralogy of clays related with processing of Atabasca oil sands was studied and kaolinite and illite, both with significant amounts of smectite interstratification, were identified [148]. Mean crystallite sizes were determined via the Williamson–Hall method and the calculated surface areas from the crystallite sizes were comparable to the total surface area determined from methylene blue adsorption tests.

Fifteen samples from the Lower Palaeozoic from the Gaspé Peninsula (Quebec Appalachians) were studied [149]. Crystallite sizes and lattice strains were determined using the Winfit program with different approaches, and it was found that the resultant size distributions showed log normal distributions.

The expandability of anchizonal illite and chlorite was studied [150] and mixed-layer illite/smectite containing >80% illite were found using powder XRD and HRTEM lattice-fringe images of samples treated with n-alkylammonium cations. It was noted that the improvement of illite crystallinity with grade metamorphism was the result of the decrease in expandable layers and also of the decrease in the number of lattice defects and of the increase in crystallite size. Stacking faults would be the main source of lattice distortion only when phyllosilicates consist of thick non-expandable layer silicates.

Mineralogical and physico-chemical properties of illite du Puy material (Puy-en-Velay, France), envisaged as a model clay system in the European Joint Programme on Radioactive Waste, were studied [151]. Gradual decrease in the mean crystallite size in ethylene-glycolated mount was found from the coarsest (0.2–2 μm fractio) to the <0.05 μm size fraction, with slight increase in the smectite layer content, and with no significant changes in particle aspect ratios.

Different methods used for delamination of macroscopic vermiculite crystals were reviewed and the effects of ultrasound treatment on the mean particle size, crystal structure and crystallite dimensions in different directions were studied in vermiculite of Santa Olalla (Spain) [152].

The 001 crystallite sizes of two vermiculites (from Ojén and Santa Olalla, Spain) were compared after sonication times from 10 to 100 h; the ranges of values were 30–15 and 100–40 nm, respectively [153]. The sample from Santa Olalla maintained the typical laminar shape after sonication, while that of Ojén showed a morphological change. The effect of sonication was related to the different nature, i.e., layer charge, of the vermiculite samples. Samples with lower layer charge delaminate to a larger extent than those with higher layer charge and samples yielding thin layers could produce nanotubes by scrolling the layers.

Comminution by grinding or by ultrasound treatment of vermiculite were compared [154] and it was found that grinding treatment produces a decrease in particle size, amorphization and agglomeration of the particles, whereas the ultrasound treatment only produces a decrease in particle size.

Ultrasonic comminution of a vermiculite from Yuli (China) was used to obtain micron/submicron particles under low-frequency ultrasound (20 kHz) irradiation in water or in hydrogen peroxide solution [155]. The obtained crystallite sizes were in the range 11–3.2 nm, and these results were achieved in shorter times (few hours) than those of previous papers, e.g., [154].

Purification of three commercial vermiculites via alcohol treatment and subsequent irradiation with microwave was reported [156]. Results in colour, delamination and expansibility with the different treatments were compared. Crystallite sizes and micro-strains were evaluated with X’Pert Plus software. The changes caused in the vermiculites treated with alcohol would be related to structural order–disorder and hydration–dehydration due to the entry of alcohol into the vermiculites and the loss of water content.

Thin sheets of a natural vermiculite were irradiated at room temperature with 80 MeV O6+ ions at varying fluence with a 15 UD Pelltron accelerator in the Material Science Beam line at the Inter University Accelerator Centre, New Delhi, India; the influence of irradiation on optical, dielectric, structural, chemical and thermoluminescence (TL) properties was studied [157] and a decrease was found in mean crystallite size and an increase in micro-strain with increasing ion fluence.

Particle-sized fractions (10–20, 1–2 and 0.1–0.2 μm) separated after 10 h of sonication of a purified sample of swelling vermiculite derived from a well crystallized phlogopite were analyzed via different techniques [158]. Results of distribution of geometrical parameters obtained via AFM (atomic force microscopy) showed no significant differences.

3.3.3. Chlorite

The above-mentioned papers [120,121,122,140] showed crystallinity indices, apparent mean chlorite crystallite sizes, lattice strain values and crystallite size distributions from TEM observations and referred to the use of FWHM of basal reflection of chlorite as crystallinity indices. Some additional references are shown below in Section 4.2.1 about use in evaluation of low-grade metamorphism of pelites.

3.3.4. Smectites

Crystallite size distributions of a set of smectites (in the Na-, glycolated form) were obtained via the BWA method using the MudMaster program [159]; mean crystallite thickness were in the range 5.7–12.3 nm. The distributions were lognormal in the whole investigated range of thicknesses and the parameters of the lognormal distributions (α and β2) related to the mean crystallite size. At higher humidity, the mean thicknesses decrease, because of the splitting of some layers or sets of layers, but the differences in mean thickness were maintained between samples, with beidellites having the greater values. Smectite crystallites were found to be dispersed into individual layers during infinite osmotic swelling and rebuilt through coagulation, recovering their original mean thickness and thickness distribution.

Water sorption and crystallite thickness calculated from surface area measurements were studied in reference smectites of the Source Clays Repository [160] obtaining results, which for the case of calcium smectites were similar to those obtained via X-ray diffraction in [159]. Decrease (delamination) of the particle thickness was found on (i) prolonged storage at a high water content, (ii) grinding and (iii) other mechanical treatments. The paper just cited reported increases of crystallite size with time (particle collapse, contraction) (i) if stored in dry state, (ii) with freezing–thawing cycles, (iii) with consolidation stress.

Differences in the X-ray diffraction microstructural evolution of talc and smectite with different time (1–90 min) of dry ball milling were studied [161]. The 001 crystallite size of talc decreased continuously (from 22.5 to 12.9 nm), whereas there was no change for 001 crystallite size of smectite (around 10 nm). Crystallite size distribution (using MudMaster program) was lognormal both in the original state and after grinding of smectites, and multimodal in the case of talc, with migration of the maximum frequency of the distribution to smaller sizes.

Two fractions of two reference clays (kaolinite or montmorillonite from the Source Clay Repository) were studied using the Voigt function method [162]. A rough agreement was found between crystallite sizes determined from X-ray diffraction patterns and from images via field-emission scanning electron microscopy. Two factors were found to affect the crystallite size variation in ethylene-glycol-treated clay minerals: (i) the increase in the unit cell in the [001] direction due to the interlayer absorption of ethylene glycol molecules in the case of swelling minerals and (ii) the physisorption at the surfaces of the crystallites. Both effects operate in the case of montmorillonite, whereas just the last one operates in kaolinite.

Experiments of fine grinding of Na montmorillonite in mortar grinder, at variable grinding time (up to 90 min) and fixed pressure were performed [163], showing increases of FWHM of 001 montmorillonite reflection with increasing time and with enhanced oxidant capacity and conductivity in the ground material.

Experiments of fine grinding of Na montmorillonite in mortar grinder, at variable grinding time (up to 90 min) and fixed pressure, were performed [163], showing increases of FWHM of 001 montmorillonite reflection with increasing time and with enhanced oxidant capacity and conductivity in the ground material.

Investigation of the organization of interlayer water and cations in smectite was reviewed by Ferrage [164] starting with the original models about organization of water and ions in smectites as developed in the 1930s via XRD analysis of the smectite structure. Some relevant contents of this article are outlined below.

Effect of hydration heterogeneity and crystallite size is considered for a theoretical basis for calculating the 00l reflections in smectite XRD patterns, showing complications in the extraction of true d001 values and potential ways to overcome it. The different smectite hydration states with structures of dehydrated mono-hydrated, bi-hydrated or tri-hydrated smectite showing characteristic d001 spacings in the ranges (9.6–10.7 Å), (11.8–12.9 Å), (14.5–15.8 Å) and (18.0–19.5 Å) is reviewed and extensively discussed.

The apparent shift of the 001 reflection from its theoretical position is described as also related to the small crystallite size reviewed. This effect is very remarkable for the 001 reflection position and for structures with high hydration states by considering calculations of scattering factor and Lorentz polarization factor at 2θ < 10⁰.

Few smectites consist of 100% of one type of layer and the first reflection could gradually migrate along hydration experiments. Basics for calculating an XRD pattern 00l reflection for a homogeneous smectite structure, and also for mixed layering, were considered.

The review about refinement of water and ion organization in smectites at different hydration states includes the results from X-ray and neutron diffraction.

According to the multi-specimen method [97], a consistent structural model is obtained when the stacking sequences and proportions of the different layer types are nearly identical in the different performed treatments.

Recent examples of the application of such a procedure to collate experimental diffraction data and molecular simulations are presented for the specific case of deciphering the molecular organization of interlayer water and cations in the different smectite hydrates (mono-, bi-, and tri-hydrated layers). The extension of this approach to the interlayer refinement of organo-clays is also detailed, and perspectives regarding the characterization of other lamellar compounds are discussed

3.3.5. Pyrophyllite, Talc

Increases in mean lattice micro-strains and crystallite size decreases along the 00l diffraction direction were observed with increases in grinding time of pyrophyllite. The maximum surface area of ground samples was reached after 30 min of grinding, showing marked line broadening, mean crystallite size of ∼15 nm and lattice micro-strains 0.68 [165].

Crystallite size evolution in dry grinding experiments of pyrophyllite in a planetary agata balls mill was studied via XRD (Warren–Averbach method) and HRTEM [166]. It was found that mean crystallite size decreases with grinding time and with energy applied per unit mass, and thus in experiments with the smaller weight of sample, the mean crystallite size dropped from 19.9 nm at 1 h grinding to 3.8 nm at 6 h grinding. The Warren–Averbach results were checked via HRTEM measurements and good fits were obtained for samples having small mean sizes. Particles initially having lognormal size distribution were first delaminated randomly, and then some particles were delaminated preferentially, producing polymodal size distributions and finally all particles undergo delamination showing a lognormal size distribution.

The mentioned paper [154] showed the crystallite size evolution in grinding of talc compared to that of smectite as described above. The initial delamination was followed by destruction of talc crystallites. Both crystallite and grain size decreased when increasing grinding time.

Comminution of a talc from Puebla de Lillo (Spain) by dry grinding in a vibratory cup mill during 5 min was studied [167]. A reduction of crystallite size (calculated via the Scherrer equation) from 40 to 10 nm was found. It was shown that the performed grinding caused a decrease in the temperatures of the release of structure bound OH groups as well as a decrease in the crystallization of the non-crystalline phase into orthorhombic enstatite.

Products of ball milling (0 to 20 h) of a commercial talc were characterized [168]. Progressive increases in surface area were found above 2 h grinding. Crystallite sizes obtained with the MudMaster program showed progressive decreasing from 25 to 5 nm. Lineal decreases of CIELAB whiteness (L*) values with grinding time (R² = 0.99), and with crystallite size (R² = 0.77) were found.

Sonication treatment of a commercial talc from China of 18.73 μm mean particle size at constant temperature for 2 h, 10 h and 40 h in acid medium was studied [169]. Obtained 001 crystallite sizes exhibited progressive reduction from 34.02 to 12.31 nm after 40 h of sonication, whereas for other diffraction directions there was no reduction after 2 h sonication.

Sonication of a talc in both water and acid media up to 100 h and 40 h, respectively, was studied [170]. Severe delamination and reduction of the plate diameter was found in sonicated samples; 001 crystallite sizes were in the range 12–4 nm (decreasing with sonication time), with smaller crystallite size and with small significant differences in lattice strain in acid media.

Crystallite size and lattice strain of talc samples produced in a test work of operational conditions of fine grinding in a jet mill were found in the ranges 1477 nm to 3538 nm and 0.08 to 0.2 nm, respectively [171].

Investigation at structural level of the transformation from an amorphous talc precursor to crystalline synthetic talc synthesized at 100–300 °C for 1 or 6 h was conducted [172]. No particular line profile analysis was performed, but increases in peak definition and decreases in widths of 00l reflections of talc peaks with increasing temperature synthesis can be recognized in the shown X-ray powder diffraction patterns.

3.3.6. Sepiolite-Palygorskite

The chemical and structural stability of a Spanish sepiolite from Vallecas (Spain) in water and in alkaline solution was studied [173]. The specific surface was determined via the BET method and the crystallite size via X-ray powder diffraction. High chemical stability in neutral and alkaline media was found but in alkaline media greater dissolution of silica was observed. The observed values of 110 crystallite size were in the range 12–16 nm, and their evolution with time was consistent with evolution of measured values of BET surface area.

The above-mentioned method of Kojdecki [41,42] was applied [43] to ground sepiolite obtained by high energy grinding (vibrating cup mill with agate set elements, at different times, from 3–60 s) of the sepiolite NEV-1 of the Source Clay Repository. The modelled microstructural parameters were as follows: prevalent crystallite shape (parameters A, B and C in directions [100], [010] and [001]), volume-weighted crystallite size, distribution and second-order crystalline lattice strain distribution. These parameters were determined for each sample from a set of selected reflections (110, 060, 131 and 260) of the diffraction pattern and fitting the simulated pattern to each measured set. Strict correlations of microstructure parameters with grinding time and with specific surface area were observed. A parallelepiped with edge-length ratios almost independent of grinding time (for longer times) was found as the predominant crystallite shape. Considering the volume-weighted mean standardized crystallite size, larger needle-like crystallites were found at the first stage of grinding (0–3 s), and a subsequent crack into small fragments was evident but slow, without significant variation of crystallite shape (determined from values of the ratios B/A and C/A). The crystallite size distributions were found to be close to logarithmic normal ones, with the mean values decreasing with increasing grinding time and the standard-deviation-to-mean-value ratios approximately constant.

A subsequent study [174] was performed with the set of samples of the previous paper [43]. The crystallite size evolution was studied through X-ray diffraction via the methods of Warren–Averbach and of the Voigt function in different diffraction directions and through FESEM images and surface area measurements. A previous morphological modelling allowed the interpretation of FESEM images. The apparent crystallite sizes determined via the Voigt function method were in agreement with measurements of widths of the prevalent pinacoidal and prismatic prevalent faces (of {110} and {010} forms) on FESEM images. The found ranges for FESEM width values in rectangular shaped faces included the XRD crystallite sizes found for [110] and [010] directions or diffraction. So the observed decrease in XRD crystallite size through comminution is in agreement with FESEM micromorphological and surface area measurements. The apparent crystallite sizes determined via XRD of the same set of samples (in the range 10–85 nm, via the Voigt function method) were in agreement with values from extended observations on FESEM images [167]. The sepiolite aggregates were shown in [175] as lath-shaped agglutinations of prisms and pinacoids elongated along [001], each lath including several crystallites in that direction. The surface area values were in the range of previous experimental measurements of other sepiolites. The results obtained showed the effectiveness of the used vibromilling as a procedure for comminution of sepiolite.

Leguey et al. [176] studied Sepiolite microbialites from the Miocene Madrid Basin (Spain) and found that textural organization of sepiolite fibers is similar to the fabric of cellulose fibers produced by microorganisms. Two types of fiber association were recognized: short and straight fibers (<2 μm length) intertwined in compact bundles and long curved fibers (>10 μm length) forming thin tissues that mimic cellulose and other exopolymers produced by bacteria. The similarity is extended to the nanometric scale. Sepiolite was composed of 6 to 10 nm thick fibers forming mesocrystals (oriented nanocrystal aggregates) from 50 to 200 nm in length. Based on an XRD line analysis (obtained mean crystallite size in the range 6 nm, MudMaster program) of the crystal size distribution, this average fibril thickness is the same as that of cellulose.

Products of grinding by high-speed vibration milling (at times 10 to 220 s) of a purified sepiolite from Dafang county (China) were studied in [177]. Average crystallite sizes obtained via the Williamson–Hall method (from 101, 002, 301, 211, 212 and 611 reflections) were in the range 31.8–22.9 nm. The lengths of crystals obtained from Dinamic Light Scattering (DLS) were in the range 1599–359 nm, and the ratio of these values to that of crystallite size (used as an aspect ratio) were in the range 50.30–15.70.

3.4. Useful Software in X-ray Powder Diffraction Microstructural Analysis of Clay Minerals

An extended list of crystallographic software including programs on X-ray powder diffraction analysis can be found on web pages of the International Union of Crystallography https://www.iucr.org/resources/other-directories/software/ (accessed on 28 January 2024). It can be also found at http://ccp14.cryst.bbk.ac.uk/solution/peakprofiling/index.html/ (accessed on 28 January 2024). This list shows details and the accessibility of the programs, and includes programs frequently referenced in papers about clay minerals (e.g., BGMN, Winfit, XPowder, High Score Plus, Diffrac EVA, Topas, Breadth, etc.)

The mentioned MudMaster program [78] is available at https://pubs.usgs.gov/of/1996/of96-171/ (accessed on 28 January 2024). Its advantages over many commercial programs are indicated in [142]. In addition to a description of NEWMOD^® [178], other contributions can be found in the book of Reynolds and Ferrell [83]. PATISSIER, a program allowing estimation of smectite content and number of consecutive illite layers in mixed-layer illite-smectite using illite crystallinity data, is described in [179]. Newmod Plus [180] was a version of Newmod to interpret XRD patterns of clay minerals.

Descriptions of Rietveld-based programs frequently used in clay mineral analysis are provided in [181]. Table 2 of the review of [2] includes a summary of programs frequently used for quantitative analysis of clay minerals, which can be used in production of XRD microstructural parameters.

4. Some Examples of Use of XRD Microstructural Parameters in Particular Applications

Section 3.3.1, Section 3.3.2, Section 3.3.3, Section 3.3.4, Section 3.3.5 and Section 3.3.6 include references to quite advanced X-ray powder diffraction microstructural analysis of the main groups of clay minerals and phyllosilicates used as industrial minerals, concerning mainly experimental works about grinding. Next, we show examples of the use of microstructural parameters for the more referenced groups of minerals and for particular fields of research more or less developed.

4.1. For Kaolinite

References of different papers showing features of X-ray diffraction powder patterns related with structural order–disorder in phyllosilicates and described by ratios of intensities between particular peaks in kaolinites, used as ‘crystallinity’ indices (mentioned above in Section 3.1), were eventually used to compare properties or behaviour of kaolinites [104,105,122,123]. In a set of industrial kaolins from France and the United Kingdom [104,105], the decrease in crystallinity of kaolinite was accompanied by a reduction in crystallite size.

Hydrothermal syntheses of kaolinite with distilled water or with acidic solutions using a mixture of silica-gel derived from alkoxide and gibbsite with a Si/AI ratio of 1:1 as the starting material was studied in [182], and smaller 001 crystallite size and lower Hinckley index were found in the first case.

Structural content of Fe in kaolinites was investigated [183] and crystallite size decreasing and strain increasing were observed when increasing Fe₂O₃ contents.

As an example of environmental application, the relationship between kaolinite 001 crystallite size and Cs-137 uptake mechanism onto the clay was investigated in environmental dispersion of radionuclides [184] and the best adsorption capacity was related to the kaolinite content and to the lowest crystallite size.

The increased pozzolanic activity of kaolin as a function of milling time compared to that obtained via thermal treatment (700 °C) was shown [185], and the FWHM values of kaolinite reflections (001 and 002) were used among other methods to estimate the ‘crystallinity’ of kaolinite.

The relationship between 001 Scherrer crystallite size and ‘crystallinity’ indices of Hinckley and of Lietard was found in kaolins of the Carboniferous and Cretaceus Ages from the West-Central Sinai Peninsula [186].

The processing of two Egyptian Kaolin Powders, one with ordered kaolinite (Hinckley Index HI > 0.7, crystallite size D001 > 30 nm) and the other with disordered kaolinite (HI < 0.7, crystallite size < 30 nm), was studied [187] and it was found that powder flow characteristics of both kaolinite-rich samples were improved via thermal treatment.

4.2. For Muscovite, Illite, Chlorite

As mentioned in Section 3.1, the FWHM values of 001 reflections of mica-like minerals and chlorite were widely used in earth sciences as a crystallinity index. Results for different standards of illite or chlorite crystallinity can be correlated with reasonably narrow ranges of mean crystallite size as a well-behaved function of metamorphic grade [139]. In the report of AIPEA about the use of the term crystallinity [94], it was stated that the Kübler index and the FWHM values of chlorite, as well as the microstructural parameters (e.g., mean crystallite size values) and chemical characterization of these metastable phases, cannot serve as the geothermometer, and that these parameters are only qualitative indicators of the stages reached for the studied minerals through a series of metastable mineral reactions, but the utility of those parameters as indicators of diagenesis and low-temperature metamorphism in different geotectonic regimes was stated [188,189].

4.2.1. Diagenesis, Basin Evolution, Low Grade Metamorphism

The chapter of Frey [121] included an item concerning ‘crystallinity’ indices, frequently referring to illite, but also determining other sheet silicates, for example, chlorite, or pyrophillite, and it referred to different indices of ‘crystallinity’ used for illite: [110,111,190,191] determined from X-ray diffraction patterns, noting the successful application of the Kübler index by more than one hundred authors, in that moment, and referring to the compilations and results of Kisch [192,193,194].

Relationships between different indices of illite crystallinity (Weaver, Kübler, Weber) used to define low-grade metamorphic zones were studied [195], noting a transformation from one index to the others, involving strictly only the datasets from which they were taken.

In the Cinco Villas massif (Spanish Pyrenees), a metamorphic zonation of Devonian–Carboniferous rocks towards the granitic Aya massif and a concentric pattern around the intrusive body were found using illite crystallinity and other proxies [196].

Fairly strong (r = 0.75–0.85) positive linear correlations were found between crystallinity indices (peak widths) measured on 001 and 002 reflections of chlorite and those of illite-muscovite clay fractions of a representative shale-slate-phyllite series from Palaeozoic and Mesozoic formations of northeast Hungary [197]. Chlorite crystallinity values measured on air-dried and ethylene-glycol-solvated samples suggested that the effect of expandable interlayers was negligible, especially in the higher-grade part of the series. No correlation was found between crystallinity and changes in chlorite composition and thus an increase in crystallite size and a decrease in lattice strain with increasing grade (approximately temperature) would be main factors affecting chlorite crystallinity

Simultaneous measurements of chlorite and illite crystallinity were used in monitoring low- to very low-grade metamorphisms [117,194,198,199], and it was also observed [190] that chlorite crystallinity could be used to determine the metamorphic grade of meta-igneous rocks lacking indicative mineral assemblages. Many examples of the use of the crystallinity index of illite and chlorite are shown in contributions to thematic symposia on diagenesis and low-grade metamorphism in different meetings and in subsequent published papers.

Equations relating the indices of Kübler, Weaver and Weber were provided [191] to be applied in the following: (i) anchizone boundary conversion; (ii) approximate illite domain size calculation for the Weaver and Weber indices: (iii) peak profile analysis and qualitative lattice strain/domain size evaluation; (iv) identification of the presence of illite/smectite from XRD profile in air-dried state.

The thermal maturity of the Lower Cretaceous Sindong and Hayang groups in Korea was investigated using the Kübler illite ‘crystallinity’ index (KI) [200]. Depth of burial was not a major factor controlling Kl variation in the basin because Hayang mudrocks have higher thermal maturity than the underlying Sindong mudrocks. Short-lived heating by the emplacement of the Upper Cretaceous plutonic rocks was considered responsible for the higher thermal maturity in the Hayang mudrocks.

Crystallite size distributions of illite were determined for a set of pelitic samples from Palaeozoic rocks [201]. Two main types of distributions were found, asymptotic and lognormal, with evolution related to increasing metamorphic grade by two growth stages: (1) the first stage of simultaneous nucleation and growth, during which the asymptotic profiles were established, and (2) a subsequent stage of surface-controlled growth without further nucleation, producing the lognormal shapes. The transition from the diagenesis zone to the anchizone, as determined from the Kubler index, is determined by the change in shape of crystallite size distribution from asymptotic to lognormal.

Illite crystallinity and conditions of deformation of slates and phyllites were studied [202] and it was found that metamorphism during D1 deformation occurred under anchizone to epizone-grade conditions at temperatures probably near 300 °C.

The synthetic graph of metapelitic zones of illite crystallinity showing related litholgies, microfabrics and different mineral transitions according to [188] was detailed for mineral transitions of the kaolinite-pyrophillite group [203].

The application of the ratio Kübler index/Boron illite content to perform calibration of depth before constraining the roots and paths of mud volcanism in the Northern Apennines was recommended using representative collections of clay samples, formation waters and temperatures from stratigraphic sections [204]. The method must be also advisable for all environments where high-B fluids were trapped within sedimentary successions associated with mud volcanism.

The Beni Mezala antiform (in Internal Zone of the Rif Cordillera, N Morocco) was characterized by a complete clastic sequence ranging from diagenetic to epizone conditions favored by the tectonic stress of the Alpine orogeny. The reaction progress was evidenced [205] through the evolution of clay minerals (mineral phases, illite crystallinity values, polytypes and textural and chemical features).

The main traditional criteria and complementary tools (organic matter reflectometry, geothermometry, etc.) used together with crystallinity indices for research of reaction progress in diagenesis to low grade metamorphism paths were referenced to in an editorial article [206].

4.2.2. Structural Geology and Tectonics

In works of structural geology, illite crystallinity can be used to show correlation or no correlation with a finite tectonic strain, e.g., [207], in which no correlation was established between finite strain, lattice strain and/or crystallite size of illite. Illite crystallinity has been also used to detect gradients and breaks in metamorphic grade along the tectonostratigraphic profiles, e.g., [208], or to constrain limits of tectonics units, e.g., [209].

Fifty-four pelitic samples of defined mineral facies from the well-studied rocks of the South Island of New Zealand were analyzed [210]. The good correspondence between Kübler and Árkai index values and metamorphic mineral zones confirmed that clay ‘crystallinity’ indices provide a useful method for mapping regional grades of low temperature metamorphism and the general state of cleavage development.

Retarded illite crystallinity induced by stress was described by Giorgetti et al. [211].

4.3. Research of Mineral Resources

The onset of illite crystallinity research [115,116] was related to oil research as said above. Subsequently, crystallinity indices of clay minerals were used as a tool in research of other mineral resources, and some examples are shown below.

In exploration of Cu deposits at SW Gaspe (Quebec) [212], it was found that several deposits were surrounded by halos with increased illite crystallinity in the deposits.

Enhanced neoformation and/or recrystallization of illite at the time of mineralization in Kupferschiefer of the Polish Zechstein [213] suggested the formation of diagenetic illite induced by the mineralizing event.

Within the mineralized alteration zone in Dexing porphyry Cu deposit (East China), the lower values of the illite Kübler index were found related to the higher copper grade [214].

Systematic variation of crystallite size distributions of illite with changes in hydrothermal alteration type, fracture density and attendant mineralization in a large acid-sulfate/Mo-porphyry hydrothermal system was shown in a zoned hydrothermal deposit of Lake City, Colorado [215].

Illite and chlorite ‘crystallinities’ were considered related to shear deformation and alterations in the Jinshan gold deposit, East China [216].

Black oolithic talcs from Jiangxi Province (China) showed lower 001 crystallite size (in the range 21–28 nm) compared to other studies [217].

The same change pattern was found for illite crystallinity (Kübler index) and spectral reflectance and thus very low-grade metamorphic belts could be subdivided using spectral indices of clay minerals measured by using field portable spectroradiometers [218].

The highest values of apparent crystallite sizes, measured by volume weighted Dv values from 001 kaolinite reflection (via the Voigt function method), was in one of the four types of the classifications of bauxites of NE Spain [219].

The aim of a recent paper [220] was to contribute to the knowledge of methane accumulation and related coalbed methane generation by using vitrinite reflectance and illite crystallinity as main tools.

Illite morphology, pore-size and crystallite-size distributions were studied in Aeolian Rotliegend Sandstones [221]. It was found that crystal growth occurred by a continuous nucleation and growth mechanism probably controlled by the multiple influx of potassium-rich fluids during late Triassic and Jurassic times. Detailed analysis of textural varieties of illite, their crystal growth and calculated permeabilities provided important constraints for understanding fluid flow in tight reservoir sandstones.

4.4. Weathering, Soils, Provenance

The illite crystallinity values and other proxies eventually with chemical weathering index data have been used to evaluate weathering and/or to constrain sediment provenances or to evaluate climate evolution, e.g., [222,223,224,225,226,227,228,229,230,231,232,233].

Complex mineral assemblages with mixed-layer clay minerals involved in pedogenetic evolution were deciphered from different fractions of soils in [234] via the multi-specimen method [96] using the above-mentioned Sybilla program [84] for XRD analysis of air-dried and ethylene-glycol-treated mounts.

Soils developed on different geological formations were studied and it was noted that the crystallographic parameters of the kaolinite (FWHM, layer-d spacing, mean crystallite size and mean number of layers) varied, revealing a distinct pedogenetic evolution, related to different degrees of weathering as well as to differences in the source materials [235].

AFM image analysis of kaolinite in hard sub-superficial horizons of Brazilian soils [236] showed average particle size between 80 and 250 nm and average AFM height between 60 and 80 nm, including between 4 and 13 crystallites measured from powder XRD data presenting between 88 and 112 layers.

Clays of soils play a very important role for enzyme adsorption. It was found that average crystallite size increased after acid phosphatase enzyme adsorption for studied clays of different types of soils from India [237].

Crystallite size of mica, kaolinite and smectite of soils were determined and related to hydraulic conductivity of the studied soils [238]. Crystallite size of kaolinite was greater than that of mica or smectite. The influence of smectites of smaller crystallite size (3.5–5.5 nm) on reducing the saturated hydraulic conductivity of the studied soils due to high dispersion and swelling was noted.

The experimental reduction of illite (IMt-1) of the Source Clay Repository of the Clay Minerals Society was investigated to compare biotic and abiotic variation of illite crystallinity and extension of Fe reduction in bioreduced sediments [239]. Variations in the illite crystallinity value of bioreduced samples were greater than those of abiotic reduced samples considered as indicative of no significant microstructural modification in abiotic reduction.

4.5. Growth Mechanisms

Deduction of crystal growth mechanisms from crystal size distributions was described [240,241] and subsequently applied for an analysis of crystallite growth mechanisms in clay minerals [242,243,244,245] and in other papers mentioned in previous sections [135,136,201].

As an example, in [245] the crystal growth of NH4-illite (NH4-I) was deduced in the hydrothermal system of Harghita Bai (Eastern Carpathians) from the shapes of crystallite size distributions by using MudMaster and Galoper programs showing simultaneous nucleation and growth, followed by surface-controlled growth without simultaneous nucleation.

4.6. Synthesis, Thermal Processing, Ceramics and Related Fields

The thermal transformation of kaolinites in the temperature range of primary mullite formation was studied in order to monitor nucleation and growth of mullite. The lower crystallinity of the starting kaolinite was found to enhance mullite growth [246].

A swelling Na-mica was synthesized from a mixture of kaolinite, magnesium nitrate (or ultrafine MgO) and NaF at 850 °C. Decreasing the mass of NaF flux during the crystallization of Na-4-mica reduced the crystallite size from 2 to 0.2 µm. As the crystallite size decreased, the average particle size of the mica decreased from 5 to 2 µm [247].

Significant relationships between kaolinite crystallite size and properties (as water absorption and apparent density) of kaolinitic casting clays after firing were found [248].

The influence of thermal treatments below the dehydroxilation temperature of clay minerals of ball clays (kaolinite and illite) on the Atterberg limits of these treated materials was studied [249]. The observed variations (slight decreases in the plastic limit and more important decreases in the liquid limit) were attributed to the increase in crystallite size of the present main clay minerals (kaolinite and illite), as was found through a simplified X-ray diffraction microstructural analysis and via field emission scanning electron microscopy.

As mentioned above [167], a reduction of crystallite size of talc raw material produced a decrease in their temperature of thermal transformation to enstatite.

Variation in thermal behavior and properties of kaolinite, talc and Ca-rich montmorillonite, with grinding by planetary ball milling working in vacuum (P = 0.13 Pa) at room temperature (25 °C), was investigated in [250]. The temperature at which there was maximum dehydroxylation and weight losses of the intralayer OH was found to be linearly related to an increase in the FWHM of the 001 peaks.

The influence of Cu(II) on the hydrothermal and thermal transformations of a synthetic hectorite was investigated in [251]. The presence of Cu(II) during the hydrothermal treatment increased the crystallite size of the produced hectorite.

Conditions of hydrothermal synthesis of kaolinite were monitored via XRD microstructural analysis trough Warren–Averbach and Voigt function methods, as mentioned above [136].

Microstructures of mullite in porcelain bodies obtained from the same compositions but using two different kaolinite raw materials (B of greater crystallite size and M of lower crystallite size) were compared [252]. The higher maximum frequency distribution of 110 crystallite size of mullite observed at the same firing temperature was found using blends with M kaolinite, suggesting a clearer crystallite growth of mullite in this blend.

4.7. Crystallinity in Polymers and Polymer Composites

Usually polymers are semi-crystalline (composed of crystalline and amorphous regions) and degradation takes place preferentially in amorphous regions owing to easy intrusion of aqueous solutions in irregularly arranged molecular chains. Their performances and the long-term stability of the final articles and applications depend upon three major groups of factors: (1) polymer structure and monomer composition, (2) additivation and blending with secondary constituents and (3) processing parameters [253].

The degree of crystallinity of polymers is frequently related with thermal behaviour, and thus the evolution of crystallinity is tested via thermal analysis techniques. The X-ray diffraction methods developed to determine the crystallinity (degree of crystallinity) of polymers via different techniques have recently been reviewed [254].

Sheet silicates with high rigid behaviour are frequently used as a reinforcing phase to improve mechanical properties of polymers. The layered mineral may promote nucleation sites for the crystallization of molecular chains of the polymer during the sintering process or not, e.g., [255,256,257,258] among other papers.

Good dispersibility and high orientation of smectite platelets result in higher performances in composite compared with non-composite polymers, e.g., the case of saponite-cellulose films [255].

From what has been said, the XRD research on polymer-composites focuses on the structure and microstructure of the polymer. XRD microstructural parameters of the sheet silicates are considered in the paper [259], where it was noted that the studied properties were the result of dispersion effects with marginal effects of XRD crystallite sizes of polymer and sheet silicates (in the considered case of 110 reflection of polyhetylene and 001 reflection of montmorillonite).

4.8. Carriers of Active Agents

Sheet silicates and clay minerals, particularly smectites, are frequently used as carriers to accommodate and stabilize active agents for catalysts or other uses. Delamination and the construction of pillars in the interlayer space are solutions to increase mesoporosity and surface area of smectites and natural and synthetic smectites have been investigated as catalyst supports for different reactions. Pillared clay minerals are widely used in catalysis, mainly for petroleum cracking as well as for applications in environmental protection, molecular sieves, selective adsorbent, thermal insulators and electrochemical and optical devices, as shown in a general review [260].

Several examples were provided in a paper [261] showing the use of crystallite sizes to compare delamination, and the best catalytic activity was found to be related to the most optimal copper oxide dispersion on the high surface area of delaminated saponite and to the lower support acidity.

5. Concluding Remarks

Powder X-ray diffraction microstructural analysis has been particularly useful in the study of clay minerals due to its typically small particle size, and also in other non-clay phyllosilicates in the study of comminution.

Many of the cited works pay attention to the comparison between XRD crystallite size and sizes obtained via different techniques, among which the one with the best correspondence must be that of measurements via HRTEM. This technique allows a direct measurement of the number of layers that determine the size of a crystallite of the studied mineral, and with increased number of measured crystallites it has better reliability with respect to the obtained statistic value.

The use of FWHM values of 00l lines, particularly in the case of illite, has been enormous because of its abundance in the lithosphere, and this started, as mentioned above, in its initial use in oil research. Soil science and weathering studies are possible fields for growth in terms of the use of these parameters, and of other XRD microstructural characteristics.

Some examples have been cited concerning the use of crystallite size distributions related to growth mechanisms, which can be applied to research of synthesis experiments as well as to mechanical and thermal evolution of the microstructure of raw materials and products in industrial processing.

Improvements of properties linked to greater adsorption, to increased ability to transport active agents or to more dispersion thereof can be associated with variations in average crystallite size or with crystallite size distribution of platelets, or related shapes, or tubular-shaped submicronic particles typical of different sheet silicates and clay minerals used as fillers or in formulated composites. Thus, increases in XRD microstructural research of sheet silicates and clay minerals can be expected in these fields of applications.