A general orientation distribution function for clay-rich media

The role of the preferential orientation of clay platelets on the properties of a wide range of natural and engineered clay-rich media is well established. However, a reference function for describing the orientation of clay platelets in these different materials is still lacking. Here, we conducted a systematic study on a large panel of laboratory-made samples, including different clay types or preparation methods. By analyzing the orientation distribution functions obtained by X-ray scattering, we identified a unique signature for the preferred orientation of clay platelets and determined an associated reference orientation function using the maximum-entropy method. This new orientation distribution function is validated for a large set of engineered clay materials and for representative natural clay-rich rocks. This reference function has many potential applications where consideration of preferred orientation is required, including better long-term prediction of water and solute transfer or improved designs for new generations of innovative materials.


Supplementary Note 1. Relation between the angle  on the detector and the orientation angle  of the normal to a clay platelet and implications on the extracted values of order parameters.
The geometrical description of the X-ray experiments we performed is summarized in Supplementary  Fig. 1 within the framework of the Ewald sphere construction. Notations are detailed in Supplementary  Fig. 1. In brief, the Ewald sphere construction is a geometric construction which allows one to visualize the occurrence of Bragg peaks. One draws a sphere of radius 2 / (the incident wave-vector modulus) around the sample. The origin of the reciprocal space is at the intersection of the transmitted X-ray beam and of this sphere. The conservation of wave-vectors implies that Bragg peaks are observable whenever a reciprocal lattice point lies exactly on the Ewald sphere.
The orientation of the normal ⃗ to a platelet with respect to the mean orientation axis z is given by an angle , as shown in Fig. 1a. The 001 Bragg reflections from all the platelets, whose orientations are given by the orientation distribution function f(), result in a ring on the planar detector. The position of a point P on this ring is given by the angle  between Cz and ⃗⃗⃗⃗⃗ , and the intensity at point P is written ( ). There is a one-to-one correspondence between point P and a wave-vector ⃗ 001 ≡ ⃗⃗⃗⃗⃗⃗ on the front of the Ewald sphere. This wave-vector is parallel to the normal ⃗ of diffracting platelets; thus, one finds the following simple relation: In the (Oxyz) frame and using Supplementary Fig. 1, the relationship between the angles and is obtained as follows: where is the Bragg angle verifying the well-known relation: Projection of ⃗ 001 on the Oz axis thus also leads to: 001, = 4 sin ( )cos ( ) Following Supplementary Eq. (2) and Supplementary Eq. (4), 2 sin(2 ) cos( ) = 4 sin ( )cos ( ), wherefrom the general relation between the angles and is as follows: For sufficiently small values of , its cosine can be approximated as 1 so that Supplementary Eq. (5) gives: Taking into account the normalization of the ODF in Eq.
(3), one thus obtains Eq. (4): To validate the = approximation in Supplementary Eq. (6), let us consider the 001 values at 0.4, 0.42, 0.44, 0.63, and 0.88 Å -1 for smectite, vermiculite, chlorite, mica, and kaolinite, respectively. The worst scenario corresponds to the highest 001 value (i.e., 001 =0.88 Å -1 for kaolinite). When =1.5418 Å, ≈ 6.2° and Supplementary Eq. (5) becomes: Supplementary Fig. 2 reports = f( ) and shows that can only be considered equal to in the range from ~10 to ~170°. Moreover, the diffraction condition, which is the location of the wave-vector ⃗ 001 on the Ewald sphere in Supplementary Fig. 1, cannot be satisfied when < , which is also evidenced in Supplementary Fig. 2. Consequently, even if one takes into account the exact relation between and in Supplementary Eq. (5), our X-ray scattering experiments will only give access to the ODF in a limited range between and 180°-, i.e., between 6.3 and 173.7° in the present case.
To evaluate the effect of the approximation we used in the article by taking = between 0 and 180°, let us consider the following general form for the intensity distribution: ( ) ∝ exp ( 2 2 (cos( )) + 0.005(λ 2 ) 5 4 (cos ( ))) which is based on the general ODF we determined for clay platelets in Eq. (21) using the maximumentropy method. Let us now consider the exact relation between and for ∈ [ , − ], so that according to Supplementary Eq. (1) and Supplementary Eq. (5), the ODF is written as: ( ) ∝ exp ( 2 2 (cos( ) /cos ( )) + 0.005 λ 2 5 4 (cos( ) /cos ( ))) Let us also assume that ( ) = ( ) for < or > − , which will lead to an underestimation of the calculated order parameters. The calculated order parameters 〈 2 〉, 〈 4 〉, and 〈 6 〉 are compared for the worst scenario (i.e., for kaolinite) in Supplementary Fig. 3 to those calculated using the ODF in Eq. (21) with = between 0 and . The very good agreement observed fully justified our approximation. A reason for such a good agreement is the presence of the sinus of the angle in the calculation of the order parameters (Eq. (10)), which minimizes the contributions close to 0 and . Supplementary Fig. 1. Geometric description of the X-ray scattering experiment. The laboratory frame is (xyz). An incident X-ray beam with wavelength is antiparallel to the x-axis. The sample is at centre E of the Ewald sphere, and the radius is 2 ⁄ . The origin of the reciprocal space is O, and wavevector ⃗ 001 , which gives the position of the Bragg peak 001 associated with a clay particle with orientation with respect to the z-axis, intersects the Ewald sphere at point M ( ⃗ 001 = ⃗⃗⃗⃗⃗⃗ ). By varying the platelet orientation, point M describes a circle of centre B on the Ewald sphere. The (EM) line intersects the detector plane (Cyz) at point P; ⃗⃗⃗⃗⃗ makes an angle with the Cz axis.

Supplementary Note 2. Correlation between Lagrange multipliers  and  .
Using Eqs. (19) and (20), we determined the values of the Lagrange multipliers 2 and 4 for pairs of experimental <P2> and <P4> values. The results are shown in Supplementary Fig. 4. The relation between 2 and 4 is well fitted with a power law of the form  4 =A( 2 ) with A=0.005±0.002 and B=5±0.4, and the general ( ) ODF for clay minerals (Eq. (21)) can be expressed as a function of a single parameter, 2.

Supplementary Note 3. Experimental assessment of sample slicing along the main stratigraphic direction.
A random selection of 8 samples with various degrees of anisotropy was used to validate the assumption that the X-ray scattering (XRS) measurements were performed with the preferred orientation axis perpendicular to the incident X-ray beam, as drawn in Fig. 1b and Supplementary Fig. 1. To do so, two slices were prepared in the longitudinal and transverse direction to the poly(tetrafluoroethylene) cylinders, assuming that the preferential orientation axis is the axis of the cylinder. The sample slices were mounted on a goniometer head. All samples were kept perpendicular to the incident X-ray beam ( Supplementary Fig. 5a). The first slice provides the description of preferential orientation, whereas the second slice should, in principle, show isotropic features according to the transverse isotropy of clay media. Supplementary Fig. 5b and 5c show the corresponding scattering pictures on the detector as well as the angular dependence of the XRS intensities at = 0.88Å −1 (Fig. 1) on the 001 diffraction ring of a centrifuged kaolinite. The intensity of the 001 diffraction ring is strongly modulated for the slice prepared in the longitudinal direction of the cylinder; the sample presents a relatively high anisotropy degree (〈 2 〉~0.6). The intensity along the 001 diffraction ring is constant for the slice prepared in the transverse direction. This result shows that the sample preparation and cutting process allow retrieval of