Introduction

Clay minerals are one of the fundamental constituents of numerous terrestrial and marine environments on Earth and are systematically present in soils and sedimentary rocks1. The sub-micrometric particle sizes of clay particles most often account for a major proportion of the specific surface area of the whole material, leading to strong interactions with organic macromolecules or dissolved cations and anions. Owing to their lamellar shape, the preferred orientation of clay platelets gives rise to anisotropy in the morphology of the pore network2. Both effects have considerable influence on retention and anisotropic transfer of water and solutes in clay-rich media, causing clay minerals to play a pivotal role in the exploration and management of critical resources for human beings.

In sedimentary rocks such as mudrocks or shales, clay minerals are common markers for paleoenvironmental reconstructions3,4 and the preferred orientation of clay platelets associated with nanometer-sized pores provides low permeability values to the whole clay-rich material. This latter process has a major effect on the development of oil, gas, water reservoirs, and is at the origin of the consideration of these clay-rich media as a natural barrier for CO2 capture and nuclear waste storage5,6,7. The omnipresence of clay minerals in soil environments is also involved in the development of thin crusts at the top surface. The significant preferred alignment of clay platelets in this environment8 and the low permeability of the material exert a key effect on soil erosion and directional dependence of water and pollutant migration9. Clay minerals are also used in engineered materials given their low cost, abundance and the availability of natural deposits. Tuning the desirable fluid and gas barrier efficiency of these engineered materials through orientation control of clay platelets offers a variety of innovative sealing solutions ranging from rigid gaskets for high-temperature industrial applications10 to flexible and transparent films for packaging applications in food and electronic industries11,12,13,14.

Although anisotropic features in clay platelet alignment have demonstrated their importance in numerous natural or engineered materials, a general function predicting preferred orientation for clay platelets is still lacking. Preferred orientation is generally studied through the analysis of the orientation distribution (OD) of crystallites by diffraction methods8,15,16,17,18,19,20. The extent of anisotropy in clay platelet orientation is then most often illustrated by global descriptors of the experimental orientation distribution function (ODF; e.g., the width of the distribution or its maximum value expressed as a multiple of random distribution)8,15,16,17,18,21,22 or by an adjustable spreading parameter of conventional distribution functions (Gaussian, Bingham, etc.18,21,23,24) fitted to the experimental data. Though providing a more complete description of the shape of the OD, these latter analyses remain targeted towards a limited number of clay minerals and preparation methods or are restricted to a narrow range of anisotropy degrees. To date, no conventional distribution function has been shown to be applicable for a large set of data. This lack of a reference function describing OD for clay minerals thus leads to significant uncertainties in the prediction of long-term transfers of critical resources in compacted clay-rich systems5,25 or for driving the design of high-performance engineered materials26.

Herein, our objective is to fill this crucial gap by providing a general reference function of clay platelet alignment in a large number of clay-rich engineered and natural materials. We first develop a large set of laboratory clay-based engineered materials covering the most characteristic clay types (kaolinite, mica, chlorite, vermiculite, and smectite) and a broad panel of preparation methods (compaction, sedimentation, and centrifugation). Experimental orientation distributions of clay mineral platelets are then determined by X-ray scattering (XRS) measurements. Through detailed analysis of the different momenta of the experimental data, we unravel, for the first time, the existence of a specific orientation signature independent of the type of clay platelet or preparation method. We also demonstrate that none of the conventional distribution models used so far allows for the reproduction of the overall set of measured ODFs. Based on the maximum-entropy method, we propose a new function that allows a complete description of the experimental OD for clay platelets throughout the whole range of anisotropy degrees. Finally, the efficiency of this new distribution function in modeling the preferred orientation of clay platelets is illustrated for a representative set of engineered samples, as well as for three natural systems, i.e., a shale, a soil crust, and a schist, representative of the main different clay-rich environments on Earth’s surface or subsurface.

Results

Basics of the orientation distribution function and its determination from X-ray scattering experiments

The orientation of a clay platelet in an orthogonal framework Oxyz is determined by the orientation of its normal Oz′, which is defined by its spherical coordinates θ and φ (Fig. 1a). Clay systems are transverse isotropic media with the z-axis as the unique symmetry axis15. The orientation distribution function f(θ, φ) thus only depends on the angle θ where:

$$f(\theta ) \ge 0$$
(1)
$$f\left( \theta \right) = f\left( {\pi - \theta } \right)$$
(2)
$${\int_0^\pi} f \left( \theta \right)\sin \left( \theta \right){\mathrm{d}}\theta = 1$$
(3)
Fig. 1
figure 1

Experimental measurement of the orientation distribution function. a Spherical coordinates (θ, φ) defining the orientation of an individual clay platelet (in yellow) with respect to the laboratory frame. b Schematic representation of an X-ray scattering experiment. The sample is studied in transmission, and scattered intensities are recorded on a planar detector where the color scale ranges from dark blue for the smallest intensity values to red for the largest values. The image here corresponds to a kaolinite sample. The rings corresponding to 00l Bragg reflections and hk reflections (i.e., 02 and 11 reflections) have maximum intensities along the vertical z direction and the horizontal x direction, respectively. c A typical experimental ODF deduced from the angular scan of the 001 diffraction ring using Eq. (4), with the calculated 〈P2〉, 〈P4〉, and 〈P6〉 order parameter values from Eq. (10).

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Any distribution function is positive and can be normalized to unity27, which yields Eqs. (1) and (3), respectively. The platelet orientation is equivalently characterized by a unit vector parallel to Oz′ and by its opposite, defined by the spherical coordinates (θ, φ) and (π − θ, φ + π) from Eq. (2). In the XRS experiments performed here, scattered intensities are recorded on a planar detector (Fig. 1b). Due to the preferred orientations of the platelets, the intensity of the 001 diffraction ring is modulated angularly28. Assuming that the angle τ on the detector is equal to the spherical coordinate θ (see Supplementary Note 1, Supplementary Figs. 1–3), the ODF f(θ) (Fig. 1c) can be fully determined from the angular modulation of intensity I along the 001 diffraction ring:

$$f(\theta) = \frac{I(\theta)}{\smallint_{0}^{\pi} I(\theta)\sin (\theta)\mathrm{d}\theta}$$
(4)

Momenta analysis of the experimentally determined orientation distribution functions for clay platelets

The ODF f(θ) can be developed as an infinite series of Legendre’s polynomial functions Pl(cosθ) as:

$$f\left( \theta \right) = \mathop {\sum }\limits_{l = 0}^\infty \frac{{2l + 1}}{2}\left\langle {P_l({\mathrm{cos}}\theta}) \right\rangle P_l({\mathrm{cos}}\theta )$$
(5)

The sum can be restricted to even values of l as Eq. (2) leads to null values of odd terms of the series. The first Legendre polynomials in the expansion are expressed as follows:

$$P_0\left( {{\mathrm{cos}}\theta } \right) = 1$$
(6)
$$P_2\left( {{\mathrm{cos}}\theta } \right) = \frac{1}{2}\left( {3{\mathrm{cos}}^2\theta - 1} \right)$$
(7)
$$P_4\left( {{\mathrm{cos}}\theta } \right) = \frac{1}{8}\left( {35{\mathrm{cos}}^4\theta - 30{\mathrm{cos}}^2\theta + 3} \right)$$
(8)
$$P_6\left( {{\mathrm{cos}}\theta } \right) = \frac{1}{{16}}\left( {231{\mathrm{cos}}^6\theta - 315{\mathrm{cos}}^4\theta + 105{\mathrm{cos}}^2\theta - 5} \right)$$
(9)

The different momenta of the distribution function, also defined as order parameters, are:

$$\left\langle {P_l} \right\rangle = \left\langle {P_l\left( {{\mathrm{cos}}\theta } \right)} \right\rangle = {\int_0^\pi} P_l\left( {{\mathrm{cos}}\theta } \right).f\left( \theta \right){\sin} \left( \theta \right){\mathrm{d}}\theta$$
(10)

Analysis of orientation anisotropy most often relies on the value of the second momentum 〈P2〉. This parameter is known as H, the Hermans parameter in polymer and composites sciences29, or S, the nematic order parameter in colloid science30,31. This parameter has a value of 0 for isotropic organization and 1 for perfectly aligned particles. However, 〈P2〉 also has a null value when all particles are oriented at the magic angle θ = acos(1/√3) ≃ 54.7°. This specificity illustrates the limit of using only the 〈P2〉 order parameter to characterize the anisotropy. The actual fingerprint of the shape of the ODF and the whole set of momenta 〈Pl〉 are accessible from XRS analysis according to Eqs. (4) and (10). This contrasts with other vibrational techniques such as infrared or Raman spectroscopies, which provide only 〈P2〉 or 〈P2〉 and 〈P4〉 momenta32,33, respectively, thus limiting the number of Legendre polynomials to the fourth rank order in Eq. (5).

In an effort to obtain information that is as general as possible on the ODF for clay-rich media, a large set of different clay types (kaolinite, mica, chlorite, vermiculite, and smectite) and preparation methods (compaction, sedimentation, and centrifugation) are considered here. These different clay types represent the most ubiquitous clay minerals on the Earth’s subsurface1,3. The three preparation protocols are considered here to be representative of the main different formation mechanisms of clay-rich natural or engineered materials. These protocols include the compaction process in dried conditions (i.e., compaction method), the settling of clay particles in suspension (i.e., sedimentation method), and a coupled sedimentation-compaction method in water saturated conditions mimicking the early diagenesis process of minerals (i.e., centrifugation method). A total of 100 XRS analyses, complemented by recent anisotropy measurements on four sedimented artificial clay samples34, provide a large experimental dataset for the analysis of ODF and a wide range of anisotropy degrees (i.e., order parameter values). Among the whole set of 〈Pl〉 order parameters accessible by XRS analysis, Fig. 2 shows the first momenta 〈P4〉 and 〈P6〉 as a function of 〈P2〉 for all samples investigated (with ODF determined from Eq. (4) and order parameters from Eq. (10)) to assess a potential common signature of ODFs. Regardless of the nature of the clay mineral (i.e., kaolinite, mica, chlorite, vermiculite, and smectite) or the preparation method (i.e., compaction, sedimentation, or centrifugation), univocal relationships are evident for the three first momenta of the ODF, revealing a specific signature for the preferred orientation of clay platelets.

Fig. 2
figure 2

Relationships between the experimental 〈P2〉, 〈P4〉, and 〈P6〉 order parameters. Momenta of the experimental ODF for kaolinite, mica, chlorite, vermiculite, smectite, and artificial clay samples made of disc packings34 are shown in pink, orange, grey, blue, green, and dark green, respectively. Sample preparation based on compaction, centrifugation, and sedimentation methods are depicted by squares, circles, and diamonds, respectively. Experimental order parameters are compared with calculated parameters, with conventional Gaussian (G), Lorentzian (L), Fisher (F), Bingham/Maier-Saupe (B/MS), and mechanical compaction (MC) distribution models shown as orange, dark blue, light blue, green, and gray dotted lines, respectively.

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For simplicity, experimental OD profiles are commonly fitted in the literature with conventional distribution models. For clay minerals, the most commonly used distribution functions are the Gaussian18,21,35,36 (fG(θ)), Lorentzian37 (fL(θ)), Fisher24,38 (fF(θ)), Bingham or Maier-Saupe23,24,38 (fB/MS(θ)), and the mechanical compaction models24,39 (fMC(θ)), taking the following form:

$$f_G\left( \theta \right) = k_G(m)\,{\mathrm{exp}}\left( { - \frac{{\theta ^2}}{m}} \right)$$
(11)
$$f_L\left( \theta \right) = k_L(m)\frac{m}{{\theta ^2 + m}}$$
(12)
$$f_F\left( \theta \right) = k_F(m)\,{\mathrm{exp}}(m\,{\mathrm{cos}}\theta )$$
(13)
$$f_{{\mathrm{B/MS}}}\left( \theta \right) = k_{{\mathrm{B/MS}}}(m)\,{\mathrm{exp}}(m\,{\mathrm{cos}}^2\theta )$$
(14)
$$f_{{\mathrm{MC}}}\left( \theta \right) = k_{{\mathrm{MC}}}(m)\frac{m}{{({{\mathrm{cos}}}^2\theta + m\,{{\mathrm{sin}}}^2\theta )^{3/2}}}$$
(15)

Note that to satisfy the symmetry of the ODF stated in Eq. (2), the Gaussian, Lorentzian and Fisher functions fi(θ) (with i = G, L, F) are summed up with fi(π − θ)24. The prefactor ki(m), with i = G, L, F, B/MS, or MC, is a normalization constant to satisfy Eq. (3), and m is the parameter governing the ODF spread.

For each function and by varying the parameter m, we calculated order parameters for l = 2, 4, and 6. The calculated 〈P4〉−〈P2〉 and 〈P6〉−〈P2〉 curves are reported in Fig. 2. It is clear that none of the standard models are able to correctly reproduce our experimental dataset. The Lorentzian and mechanical compaction distributions indeed clearly overestimate the 〈P4〉 and 〈P6〉 order parameters through the entire range of 〈P2〉 values. The Gaussian and Fisher models provide good agreement with experimental data for small and large 〈P2〉 values, i.e., below 0.2 and above 0.9, but they significantly underestimate the 〈P4〉 and P6 order parameter values in the intermediate range of 〈P2〉 values. Finally, the Bingham or Maier-Saupe distribution model provides the closest agreement with our data but still fails to account for 〈P4〉 and 〈P6〉 contributions in the 0.4 < 〈P2 〉 < 0.9 region. The present results clearly demonstrate the limitations of conventional distribution models in reproducing the full signature of the clay mineral ODFs across the whole range of anisotropy degrees.

Determination of a general orientation distribution function based on the maximum-entropy method

To describe the entire ODF for clay-rich media, an alternative to the limitation of existing distribution models is to consider the maximum-entropy method (MEM) for reconstructing the full orientation function. The MEM relies on information theory (in statistics) and provides the most probable ODF that is consistent with a set of few known 〈Pl〉 values27,32,40,41. One estimates the equilibrium ODF and thus the entire range of 〈Pl〉 parameters by maximizing the information entropy defined as:

$$S\left( {f\left( \theta \right)} \right) = - {\int_0^\pi} f \left( \theta \right)\ln \left( {f\left( \theta \right)} \right)\sin \left( \theta \right)d\theta$$
(16)

The ODF can then be written as:

$$f_{{\mathrm{MEM}}}\left( \theta \right) = k_{{\mathrm{MEM}}}\,{\mathrm{exp}}\left( {\mathop {\sum}\limits_{i = 2}^n {{\uplambda}_i} P_i({\mathrm{cos}}\theta )} \right)$$
(17)

where n is the index of the higher order parameter used as a constraint (the sum over i in Eq. (17) runs on even numbers between 2 and n), λi represents the Lagrange multipliers, and kMEM(λ2, …, λn) is the normalization constant that satisfies the relation in Eq. (3). The index ‘MEM’ refers to the maximum-entropy method. The MEM approach has been proven to be very efficient when only a few order parameters are known40,41. Indeed such a reduced knowledge yields Eq. (5) inoperable for high degree of anisotropy. The situation differs here as XRS analysis provides the full f(θ) ODF and thus the entire set of order parameters. The objective in the present work is to beneficiate from MEM theory to derive a reference distribution function based on a minimum number of parameters (i.e., λi coefficients) so that the whole set of 〈Pl〉 momenta would be automatically accounted for. By limiting the reconstruction of the fMEM(θ) function to the two first λi parameters (i.e., n = 4), the ODF takes the following form:

$$f_{{\mathrm{MEM}}}\left( \theta \right) = k_{{\mathrm{MEM}}}({\uplambda}_2,{\uplambda}_4)\,{\mathrm{exp}}({\uplambda}_2P_2\left( {{\mathrm{cos}}\theta } \right) + {\uplambda}_4P_4({\mathrm{cos}}\theta ))$$
(18)

with Lagrange multipliers λ2 and λ4 calculated to satisfy the experimental constraints:

$${\int_0^\pi} {P_2} \left( {{\mathrm{cos}}\theta } \right).f_{{\mathrm{MEM}}}\left( \theta \right)\sin \left( \theta \right){\mathrm{d}}\theta = \left\langle {P_2} \right\rangle$$
(19)
$${\int_0^\pi} {P_4} \left( {{\mathrm{cos}}\theta } \right).f_{{\mathrm{MEM}}}\left( \theta \right)\sin \left( \theta \right){\mathrm{d}}\theta = \left\langle {P_4} \right\rangle$$
(20)

According to MEM theory, each pair (λ2, λ4) defines a full set of 〈Pl〉 momenta. Figure 3 reports the interrelation between Lagrange multipliers and the first 〈P2〉 or 〈P4〉 momenta and illustrates the associated shapes of ODFs41.

Fig. 3
figure 3

The 〈P4〉−〈P2〉 order parameters interrelations based on maximum-entropy theory. According to MEM theory, each pair (λ2, λ4) defines a full set of 〈Pl〉 momenta and thus a given shape of the ODF. Here, selected representative ODFs (calculated from Eq. (18)) are shown in inserts as red lines in the case of λ2 = 0 or λ4 = 0 for 〈P2〉 values of 0.00, 0.33, 0.66, and ~1.

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The isotropic configuration of particles is represented by a horizontal line associated with λ2 = λ4 = 0 and leads to 〈P2〉 = 〈P4〉 = 0. The increase in λ2 and λ4 implies a sharpening of the fMEM(θ) function. For λ4 = 0, the ODF in Eq. (18) is identical to the Bingham/Maier-Saupe distribution in Eq. (14), which was successfully applied in liquid crystals to describe the condensation of the nematic phase within a mean-field theory42. Herein, the comparison between the fB/MS(θ) (or Eq. (18) with λ4 = 0) and the experimental dataset of order parameters in Fig. 2 calls for a positive contribution of λ4, especially in the 0.4 < 〈P2〉 < 0.9 region. In contrast, in the case of rod-like organic 4,4′-dimethoxyazoxybenzene compound, Humpries et al.43 proposed an alternative model leading to an ODF similar to that in Eq. (18), with a negative contribution of λ4. Moreover, these authors considered a fixed ratio between λ2 and λ4 in their model. For the clay minerals studied here, the correlation between λ2 and λ4 is achieved by first extracting the exact values of (λ2−λ4) pairs associated with each experimental 〈P2〉 − 〈P4〉 pair and then by deriving the λ4−λ2 relationship (see Supplementary Note 2, Supplementary Fig. 4). A good fit is obtained for λ4 = 0.005(λ2)5, and the general ODF for clay platelets becomes:

$$f_{{\mathrm{CM}}}\left( \theta \right) = k_{{\mathrm{CM}}}\left( {{\uplambda}_2} \right){\mathrm{exp}}[{\uplambda}_2P_2\left( {{\mathrm{cos}}\theta } \right) + 0.005\left( {{\uplambda}_2} \right)^5P_4\left( {{\mathrm{cos}}\theta } \right)]$$
(21)

where λ2 is a single parameter controlling the spreading parameter of the ODF and kCM(λ2) is the normalization constant that satisfies Eq. (3). The index ‘CM’ refers to clay minerals, whereas P2(cosθ) and P4(cosθ) are the first two orders of the Legendre polynomial series reported in Eqs. (7) and (8), respectively. We show in Fig. 4 that the fCM(θ) ODF based on the MEM well reproduces the correlation between the 〈P2〉 and 〈P4〉 order parameters for our experimental OD dataset. Such an excellent agreement does not guarantee, however, that the entire set of other 〈Pl〉 momenta are correctly accounted for by this new function. The adequacy of our model for the prediction of the 〈P6〉 order parameter (Fig. 4) provides, however, good indications that a cut-off of n = 4 in Eq. (17) is relevant and that λi parameters for i ≥ 6 are null. A crucial test for the validation of the fCM(θ) function through its capacity to reproduce the full set of 〈Pl〉 order parameters is performed below on the basis of a direct comparison between the experimental and predicted ODF curves.

Fig. 4
figure 4

Determination of the general orientation distribution function for clay minerals. Experimental momenta 〈P4〉−〈P2〉 or 〈P6〉−〈P2〉 relationships (same notations as Fig. 2) are compared to calculated momenta using the general ODF for clay minerals (fCM(θ) using Eq. (21)) (red line) based on the maximum-entropy method.

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Comparison between experimental and predicted orientation distribution functions

The full validation of the general fCM(θ) function for reproducing the entire signature of the experimental OD profiles is shown in Fig. 5 for representative engineered clay samples. To do so, the parameter λ2 in the fCM(θ) function is chosen so that the calculated 〈P2〉 value is equal to the experimental value. The experimental and calculated ODFs are then compared without further adjustments (Fig. 5). The improvement provided by the fCM(θ) function compared to standard models is also highlighted. This comparison is limited to the Fisher (fF(θ)), Bingham/Maier-Saupe (fB/MS(θ)), and mechanical compaction (fMC(θ)) distribution models, which provided the closest 〈P4〉−〈P2〉 relations with respect to our experimental order parameters (Fig. 2). Here again, calculations are performed by adjusting the variable m in Eqs. (12–14) to fit the experimental 〈P2〉 value. As expected from Fig. 2, all these standard models reproduce rather satisfactorily the experimental OD profile for 〈P2〉 = 0.2 (Fig. 5a). For the Bingham/Maier-Saupe model at 〈P2〉 = 0.4 (Fig. 5b), the calculated 〈P4〉 value is consistent with the measured value in Fig. 2. However, a systematic discrepancy is observed near θ = 0 (or θ = π) for higher degrees of anisotropy (Fig. 5c, d). Benefitting from the MEM theory, we attribute the excellent agreement between our experimental and predicted OD dataset to the consideration of the λ4 contribution in the general fCM(θ) ODF.

Fig. 5
figure 5

Validation of the general orientation distribution function for clay minerals. Comparison between experimental ODFs (black circles) and predicted ODFs (red lines) using the general orientation distribution function for clay minerals (fCM(θ) using Eq. (21)). The selected samples display a gradual increase in anisotropy value with a compacted kaolinite with 〈P2〉 = 0.20, b compacted kaolinite with 〈P2〉 = 0.40, c centrifuged kaolinite with 〈P2〉 = 0.63, and d sedimented smectite (fluorohectorite) with 〈P2〉 = 0.86. Isotropic organization is shown as a horizontal dotted line for reference. Conventional distribution models are shown in blue, green, and gray for the Fisher (fF(θ)), Bingham/Maier-Saupe (fB/MS(θ)), and mechanical compaction (fMC(θ)) models, respectively.

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Extension to natural clay-rich systems

To go further, we investigate whether the proposed general ODF could be extended to the analysis of the anisotropy of clay platelet orientation in three complex natural systems, i.e., a shale, a soil crust, and a schist. These selected samples are representative of different clay-rich environments on Earth’s surface or in the subsurface1 where orientation of clay minerals is of particular interest. Preferred orientation of clay minerals in soils and shales plays indeed a key role on water and solutes migration5,6,7,15,25 whereas anisotropy in clay particle orientation in schists is a marker of strains during metamorphism44,45,46. Compared to previous samples used to determine the shape of the general fCM(θ) function, the organization of clay platelets in these three samples originates from extremely contrasted conditions in terms of depth, temperature, or pressure. Moreover, these samples display a complex assemblage of both clay and non-clay particles, as well as a wide range of particle sizes. Because of this complexity, the collected experimental XRS diagrams (Fig. 6) appear noisier than the monomineralic clay samples (Fig. 5). However, fitting the experimental ODFs (extracted from the 001 reflection of mica or illite clay minerals) with the general fCM(θ) function leads to a very good agreement for the three natural samples investigated (Fig. 6). Such an agreement further confirms the ability of the new fCM(θ) reference function to describe anisotropy in clay platelet orientation for complex natural media. The robustness of this function is evidenced for clay minerals formed under extremely different natural conditions and leading to a wide range of anisotropy degrees (i.e., for 〈P2〉 values ranging from 0.22 to 0.89; Fig. 6).

Fig. 6
figure 6

Application of the general orientation distribution function for natural clay-rich media. Experimental and best-fitted ODFs using the general fCM(θ) ODF for clay minerals (fCM(θ) using Eq. (21)) are shown as circles and red lines, respectively. The anisotropy in particle orientation described by the 〈P2〉 order parameter is extracted from the calculated fCM(θ) functions. The samples are displayed with a gradual increase in anisotropy value with a a shale from Paris Basin with 〈P2〉 = 0.22, b a soil crust from the Versailles experimental site with 〈P2〉 = 0.36, and c a schist from the Armorican Hercynian belt with 〈P2〉 = 0.89. Isotropic organization is shown as a horizontal dotted line for reference.

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Discussion

This work provides evidence for the existence of a reference function describing the ODF of clay platelets and is applicable for a large set of clay types, anisotropy ranges, and formation processes in both engineered and natural materials. Our methodology based on the MEM to derive this new numerical function contains, within experimental precision, all relevant information on the ODF (i.e., the full set of 〈Pl〉 order parameters) from the coupling between λ2 and λ4 parameters. By relating both entities, the unique general function fCM(θ) in Eq. (21), with only one variable parameter, can be applied to describe the whole range of anisotropy degrees. Accordingly, this new function provides clear improvements compared to conventional distribution models (Gaussian, Bingham/Maier-Saupe, etc.), which are limited to certain anisotropy domains. Furthermore, this general fCM(θ) function paves the way for unifying the various global descriptors of the experimental OD profiles (e.g., maximum intensity or width of the distribution) used in the literature to interpret the preferred orientation of clay platelets.

One of the important aspects of this approach is that the fCM(θ) function relies on the consideration of homogeneous samples. In the case of highly heterogeneous samples, the difference between experimental and predicted ODF should help in elucidating the coexistence of spatial regions (or particle populations) with contrasted anisotropic features15,18. Another important property of the fCM(θ) function relies on its construction through the coupling between the λ2 and λ4 parameters. Here, a positive contribution of the λ4 parameter is required to satisfy the 〈P4〉−〈P2〉 relation for clay platelets, whereas a negative λ4 contribution was considered in the case of rod-like organic compounds43. One may thus tentatively hypothesize that (λ2, λ4) coupling may be somehow linked to the shape of the particle. Future work could thus aim at assessing the extent of the general function for other lamellar materials (e.g., graphene, MXenes, MoS2, and layered double hydroxides)47, as well as for other types of grain morphologies (tubes, rods, and wires) in an effort to generalize the methodology developed here to improve the design of engineered materials for the electronics, energy, or catalysis fields (e.g., ZnO48, Co3O449, carbon nanotubes50,51, halloysite51, or imogolite52,53). Beyond this fundamental knowledge, a better description of the preferential orientation of clay platelets, as proposed here on the basis of the fCM(θ) function, represents a key parameter for driving the design of new generations of innovative engineered materials in the field of packaging or high-temperature industrial applications with improved mechanical resistance and drastic fluid and gas permeability reduction10,11,12,13.

We demonstrated here that the fCM(θ) function is not only valid for pure clay-based samples but also mimics well the OD of clay platelets in three natural media (i.e., soil, shale, and schist). This result provides evidence for the applicability of this reference function to clay minerals in the case where these minerals are not the major mineralogical components. Accordingly, one may thus consider that this generic function represents a good proxy for the quantitative description of preferred orientation in the main different superficial environments of the Earth where clay minerals are found. In that regard, the fCM(θ) function in Eq. (21) could be incorporated into current predictive models of caprock transport properties for oil, gas, or water resources. As far as clay barriers for waste repositories are concerned, the obtained diffusion coefficient for water can vary by a factor of 3–4 when considering either purely isotropic or perfectly anisotropic orientation of clay platelets25. Correctly accounting for the actual preferred orientation using the numerical function determined can be expected to considerably improve the predictive power of these transport models.

Methods

Samples

Two types of sample series were considered in this study, i.e., monomineralic reference clay samples and clay-rich natural polymineralic samples. The selected reference clay minerals (i.e., kaolinite, mica, chlorite, vermiculite, and smectite) represent the most common different types of clay minerals in natural environments. Kaolinite KGa-2 with [(Al7.60Ti0.26Fe3+0.14)(Si7.68Al0.32)O10(OH8)] as the structural formula54 was obtained from the Source Clays Repository of the Clay Minerals Society. Large monocrystals of mica clay minerals (i.e., muscovite) from the University of Poitiers collection were first sonicated to reduce particle size, and the <1 and 0.1–0.2 µm fractions were then collected by conventional centrifugation55. Chlorite sample from the T300 sample series of the Trimouns deposit (sample T340C, French Pyrenées) with (Al2.27Mg8.37Ca0.20Na0.14K0.01Fe2+0.57Fe3+0.02)(Si7.68Al0.32)O20(OH15.90 F0.10)] as structural formula56 were first micro-milled and the <2 µm clay fraction was then collected by conventional centrifugation55. Vermiculite samples with [(Mg4.92Fe2+0.43Al0.59Ti0.04)(Si5.66Al2.34)O20(OH4)]Na1.64 as the structural formula57 and in the form of 0.1–0.2, 1–2, and 10–20 µm size fractions were obtained from a previous study55. Three types of smectite clay minerals considered here were obtained from previous studies. The first is a natural Wyoming MX80 montmorillonite, with the structural formula58 [(Al3.10Mg0.56Fe3+0.18Fe2+0.16)(Si7.96Al0.04)O20(OH)4]Na0.76. The two other smectites correspond to hydrothermally synthetized hydroxylated and fluorinated hectorite59 endmembers with a generic structural formula [(Mg5.2Li0.8)(Si8.0)O20(OH,F)4]Na0.8. The three natural clay-rich polymineralogic samples selected (i.e., soil crust, shale and schist) correspond to environments with contrasted formation conditions where clay minerals are commonly found. The first sample is a centimeter-sized soil crust from the long-term experimental site in Versailles, France60, with ~15% of <2 µm clay fraction. This crust is formed at the surface of the soil by soft sedimentation of clay particles (mainly mica-type and kaolinite) after rainwater. The second sample is a shale from the eastern Paris basin, France, extracted from the EST104 borehole at 453 m depth and belonging to the Callovo-Oxfordian formation61. This claystone contains approximately 25–50% clay minerals (mainly mica-type and smectite) resulting from an original sedimentation event followed by a compaction process under soft lithostatic temperature and pressure conditions. The third sample is a schist from Les Sables-d’Olonne, France, located in the south Armorican Hercynian belt46. This sample was formed under Barrowian metamorphic conditions corresponding to a folding process at a temperature up to 300 °C and leading to crystallization of mica-type clay minerals.

Preparation of pure clay samples with different organizations

Different preparation methods (compaction, centrifugation, and sedimentation) were considered to maximize the range of anisotropy degree in particle orientation for monomineralic engineered clay samples investigated here. All powders were first dried at 60 °C overnight and then sieved through <50 µm to limit the size of particle aggregates. The compaction method involved uniaxial compression of a given amount of powder in poly(tetrafluoroethylene) (PTFE) cylinders with a diameter of 0.64 cm. The centrifuged samples were obtained using the following procedure: dispersion of clay particles in water by sonication, successive centrifugation of the suspension directly in the PTFE cylinders under gravitational fields ranging from 4500 to 18,000 × g, and drying of the settled sample at 60 °C overnight. For the sedimentation procedure applied to the smectites that allowed thin, self-standing clay films to be obtained, the clay particles were first dispersed in water by sonication and then allowed to settle and dry at room temperature on a flat polymer substrate.

Resin induration and sample slice cutting

For resin impregnation, all self-standing samples were first dried overnight at 60 °C and placed in a cell under a primary vacuum to remove the water. Porosity was then filled with Methyl MethAcrylate (MMA) over a time span ranging from 3–14 days for monomineralic samples and up to 1 month for natural samples. The MMA resin was then polymerized in a 55 °C bath for at least 24 h using benzoyl peroxide (BPO) as a thermal initiator (BPO/MMA ratio of 0.5 wt.%62). The impregnated clay samples were then sawed perpendicularly to the main stratigraphic direction (i.e., longitudinally to the PTFE cylinders), and the thickness of the obtained lamellas was then reduced to ~500 µm by a successive polishing procedure before XRS analysis.

X-ray scattering measurements

The XRS experiments were carried out at the Laboratoire de Physique des Solides at Orsay, France. The X-ray beam is delivered by a copper rotating anode generator (RU H3R, Rigaku Corporation, Japan) and monochromatized (λCuKα = 1.5418 Å) by a multilayer W/Si mirror (Osmic), providing a monochromatic beam of 600 × 600 µm² at the sample position. Two-dimensional XRS patterns were collected on a MAR345 detector (marXperts GmbH, Germany), with a 150 µm pixel size, placed at a sample-to-detector distance D of 250 mm or 350 mm. The beam stop was placed 30 mm behind the sample. This setup allows us to reach a scattering vector modulus down to Qmin = 0.2 Å−1 (Q = 4π/λ sin(θB), where λ is the incident wavelength and 2θB is the scattering angle), i.e., d-spacing up to 31.5 Å (d = 2π/Q). The sample slices were mounted on a goniometer head so that the axis of the main stratigraphic direction was perpendicular to the incident X-ray beam. The assumption regarding the cutting of sample slices longitudinally to the main stratigraphic direction was verified on selected samples (see Supplementary Note 3, Supplementary Fig. 5). The ODFs were deduced from the angular modulation of intensity I along the 001 diffraction ring (Eq. (4)) after background subtraction.