Seeds of imperfection rule the mesocrystalline disorder in natural anhydrite single crystals

Significance

Many people perceive crystals as the embodiment of perfect order, although in reality, it is well understood that monocrystals possess imperfections. By considering large anhydrite crystals from the famous Naica Mine (“Cueva de los cristales”), an extended picture begins to emerge, revealing a suite of correlated self-similar void defects spanning multiple length scales. These flaws in the macroscopic crystal likely stem from “seeds of imperfection” originating from a particle-mediated nucleation pathway. Hence, building a crystal could be viewed as nature stacking blocks in a game of Tetris, while slowly forgetting the game’s core concept and failing to fill rows completely.

Abstract

In recent years, we have come to appreciate the astounding intricacies associated with the formation of minerals from ions in aqueous solutions. In this context, a number of studies have revealed that the nucleation of calcium sulfate systems occurs nonclassically, involving the aggregation and reorganization of nanosized prenucleation species. In recent work, we have shown that this particle-mediated nucleation pathway is actually imprinted in the resultant micrometer-sized CaSO₄ crystals. This property of CaSO₄ minerals provides us with the unique opportunity to search for evidence of nonclassical nucleation pathways in geological environments. In particular, we focused on large anhydrite crystals extracted from the Naica Mine in Mexico. We were able to shed light on this mineral's growth history by mapping defects at different length scales. Based on this, we argue that the nanoscale misalignment of the structural subunits, observed in the initial calcium sulfate crystal seeds, propagates through different length scales both in morphological, as well as in strictly crystallographic aspects, eventually causing the formation of large mesostructured single crystals of anhydrite. Hence, the nonclassical nucleation mechanism introduces a “seed of imperfection,” which leads to a macroscopic “single” crystal whose fragments do not fit together at different length scales in a self-similar manner. Consequently, anisotropic voids of various sizes are formed with very well-defined walls/edges. However, at the same time, the material retains in part its single crystal nature.

The formation, transformation, and dissolution of minerals in aqueous solutions plays a key role in the natural and engineering evolution of the Earth’s surface. These factors control geological processes as diverse as the mass transfers within the lithosphere, elemental cycling, natural water composition, soil formation and biomineralization in living organisms, sequestration of CO₂, (sea)water desalination, geological nuclear waste storage, the setting of cement, and the synthesis of advanced functional materials, to name just a few. Although there are long standing theories to explain these mineral processes, in recent times a vast amount of new evidence has surfaced challenging these traditional views. In the particular case of nucleation, a number of precursor and intermediate solute/solid species, both stable and metastable, have been identified. The observation of these prenucleation phases extends the classical view of one-step nucleation toward multistep “non-classical” models (1–3). Although these recent insights have significantly expanded our view of mineral formation, only a reduced matrix of physicochemical parameters have been explored in the laboratory, which may not be fully representative of conditions prevalent in natural or even engineered environments. Consequently, it remains uncertain whether the multistep pathways observed in a laboratory setting are truly universal or only incidental.

To evaluate the general applicability of these so-called nonclassical mechanisms to out-of-the laboratory environments, mineral formation should ideally be observed in situ. Nevertheless, this is a daunting task taking into account the inherent stochastic nature of the nucleation process, combined with the close to equilibrium conditions usually prevailing in natural environments. One way to circumvent these difficulties is to identify evidence/remnants of the nonclassical nucleation pathways within the subsequent crystals, similar to what has been reported for several biominerals (4). Recently, we have shown that the particle-mediated nucleation pathway of CaSO₄ (5–7) is imprinted in the resultant single crystals (8), which are almost universally mesocrystalline (9–11) in nature. This property of calcium sulfate minerals provides us with the unique tool to search for evidence of nonclassical nucleation pathways in different geological processes spanning timescales of thousands of years and beyond (12). In particular, we focused on single crystals of natural anhydrite (i.e., anhydrite AII phase). Together with gypsum, anhydrite is commonly encountered in evaporitic, and also hydrothermal, environments on the Earth’s surface (7). Although there are different polymorphs (13–15) (AI, AII, and AIII; reference SI Appendix, Supplementary Note 1 and Fig. S1), only anhydrite AII crystallizes from aqueous solutions (16–18), either directly or via a dissolution and reprecipitation process from gypsum. In the Naica Mine (Cueva de los cristales or Cave of the Crystals or Giant Crystal Cave) in Mexico, anhydrite samples exceeding 10 cm in length (19–21) can be found (reference SI Appendix, Figs. S2 and S3), with these large natural crystals of anhydrite AII containing unique fingerprints of the growth processes that have taken place over a period of millennia.

We used a multitechnique approach to extract the growth history of an anhydrite single crystal from the Naica Mine to better understand its internal structure at different length scales and correlate it with the particle-mediated crystallization model of calcium sulfate (22–24). In particular, we argue that the nanoscale misalignment of the structural subunits, observed in the initial calcium sulfate crystal seed, propagates through different length scales both in morphological as well as strictly structural aspects, eventually leading to the formation of large, mesostructured, single crystals of anhydrite.

Results

Local Heterogeneities Observed in “Single” Anhydrite Crystals at the Nanoscale.

Fig. 1 shows high-resolution transmission electron microscopy (HRTEM) lattice fringe images of two focused ion beam (FIB) foils obtained from arbitrary locations, several micrometers apart from each other, on a single anhydrite crystal (Fig. 1 A and B). Both images display the same crystal orientation and their near-identical fast Fourier transforms (FFTs), showing only individual maxima (FFT-calculated diffraction spots), confirming that the material is indeed single crystalline in nature (Insets I in Fig. 1A and II in Fig. 1B). The relatively long distance between both fields of view further indicates that we are dealing with a continuous single crystal, which is in agreement with X-ray diffraction measurements (Materials and Methods). However, the individual spots in the FFTs are not circular in shape but elliptical. This directly implies mosaicity at the length scales corresponding to the field of view of the HRTEM images (i.e., nanoscale <100 nm). To emphasize any structural variations, we used Fourier filtering (8) to calculate the inverse FFT real-space images corresponding to the HRTEM micrographs (Fig. 1 C and D). In addition, a false-color map was applied to highlight the structural features. The obtained images indicate that the HRTEM regions shown in Fig. 1 A and B are composed of areas with lattice fringes aligned in the same manner (areas marked with pink rectangles in complementary representations in Fig. 1). However, the extent of order within the crystal lattice continuously fluctuates, with better ordered domains separated by less ordered ones. If we trace selected fringes across the field of view (straight lines in Fig. 1C), it becomes apparent that some domains are slightly misaligned by <1° (see arrows in Inset III in Fig. 1C). Small disordered areas act as transitional parts between the ordered areas (see arrows in Inset IV in Fig. 1D). Based on HRTEM, these order–disorder structural modulations occur typically at a length scale of ∼10 to 20 nm.

Fig. 1.

HRTEM analysis of an FIB foil from the anhydrite single crystal. (A and B) Two similar HRTEM images of the small sections of the foil, which are >10 µm apart from each other. Inset I in A and II in B show FFTs calculated to the respective images; selected reflections in FFTs are indexed for anhydrite; flux: ∼8 × 10⁵ e⁻Å⁻²s⁻¹, estimated received fluence ∼1 × 10²⁷ e⁻m⁻²; pink rectangles in A and B correspond to the same regions-of-interest (ROIs) in C and D, respectively. (C) Inverse FFT and filtered image, which highlights the order–disorder in A with a fake-color palette applied; light-blue lines trace the lattice fringes in two directions; Inset III shows 2× magnified overlap between the ROIs marked by the pink rectangles so that the light-blue lines remain continuous; the arrows point to an apparent lattice fringe shift (pseudodislocation) along the selected light-blue line. (D) Inverse FFT and filtered image, which highlights the order–disorder in B with a fake-color palette applied; Inset IV shows the ROI contained in a pink rectangle and 3× magnified; the arrows in IV point to highly disordered regions in the crystal.

We also collected selected-area electron diffraction (SAED) patterns from the FIB foils using an effective aperture of ∼1 μm (SI Appendix, Fig. S4A). The obtained data confirms the single-crystalline nature of the anhydrite specimen, while the elliptical diffraction spots indicate strong mosaicity in the [0 1 0] direction. This indicates that the observed order–disorder regions are anisotropic, with shorter length scales perpendicular to the [0 1 0] plane and longer ones parallel. This is also directly visible in real space (Fig. 1), but the HRTEM images represent significantly smaller areas of the crystal than that probed for SAED measurements. Therefore, the latter indicates that the alignment of the anisotropic defects extends over a length scale of at least 1 μm. It is also important to note, the SAED also contains a significant (i.e., above the noise background) diffuse scattering contribution, for example, a raster of streaks parallel to the [1 0 0] and [0 1 0] directions passing through the diffraction spots. SI Appendix, Fig. S4A shows that scattering in the [0 1 0] direction has higher intensity than that in the [100] direction. Although it might be difficult to unequivocally interpret the origin of the diffuse scattering without in-depth structural modeling (25–30), it appears reasonable to conclude that the observed streaks are related to the presence of the aforementioned order–disorder modulation regions (Fig. 1). In SI Appendix, Fig. S4B, we show a dark-field image collected with one of the diffracted beams from SI Appendix, Fig. S4A. Such a low-magnification dark-field image from a near-perfect single crystal should exhibit a uniform contrast (8). This is not the case here, with the image highlighting an order–disorder modulation over a distance of hundreds of nanometers. The visible domains are also oriented in the same way as that seen in Fig. 1, though SI Appendix, Fig. S4B shows that this extends over a greater length scale.

Finally, it should be noted that FIB-prepared TEM lamellae, in general, may have up to 30 nm of amorphous material on both sides (31–34). With a sample thickness of ∼150 nm, the scattering volume of the undisturbed crystal is substantially larger than the amorphous or FIB affected layer on both sides of the foil. Furthermore, this effect should reveal itself as a decrease in the diffraction contrast because the amorphous layer is perpendicular to the beam during TEM imaging (34). Conversely, in our thin sections the order–disorder modulation is parallel to the beam, as observed in transmission, which indicates that it is highly unlikely to be a result of the FIB thinning. Furthermore, the observed structural effects correlate well to those obtained from other techniques as we show.

Structural Heterogeneity at the Nanoscale—A Global Perspective.

The internal homogeneity of an anhydrite single crystal was also probed by means of transmission X-ray scattering measurements at scattering vector q-ranges corresponding to small-angle X-ray scattering (SAXS; ∼1 to 70 nm) and wide-angle X-ray scattering (WAXS; <1 nm). In contrast to HRTEM, the signal measured by scattering originates from a large sample volume of an order of 1 mm³, and as such it provides statistically global information about the nanostructure. The general considerations on how an idealised single crystal ought to scatter in SAXS and WAXS are summarized in SI Appendix, Supplementary Note 2. Fig. 2 presents two-dimensional (2D) SAXS patterns for an anhydrite crystal in position S at 0° and 21° tilt, respectively (see also SI Appendix, Fig. S3). For both of the crystal orientations, distinctly different anisotropic scattering patterns were observed. The patterns show that the anhydrite crystal structure is heterogeneous at a length scale <∼70 nm (q > 0.1 nm⁻¹) and that these structural heterogeneities are orientated. The fact that we can observe such structural features in the first place means that their average electron density is different from the one of the surrounding matrix. It can be either higher, which is unlikely since anhydrite is the densest phase of CaSO₄, or lower if there is disorder and/or empty voids/pores. In Fig. 2A, the direction of the high-intensity scattering pattern is approximately parallel to the apparent vertical long axis of the crystal (Inset in Fig. 2A and SI Appendix, Fig. S3A Z-axis). This suggests that the nanosized structural features, from which this pattern originates, have their longer dimensions aligned approximately perpendicularly to the crystal vertical long axis (and thus their shorter dimensions parallel to the long axis, SI Appendix, Fig. S3B X-axis). When the crystal is tilted, the resulting cross-shaped scattering patterns represent cross-sectional components of the anisotropic objects. The obtained cross-sections present a more complex scattering pattern (Fig. 2B). The intensity in Fig. 2B does not extend as far toward high-q as that seen in Fig. 2A (compare q_x and q_y scales). Hence, the scattering in Fig. 2B originates from larger scattering features, than the one in Fig. 2A. Such a cross-shaped pattern for the tilted orientation, together with the one from Fig. 2A, implies that the scattering objects are rod- or platelet-like in nature, with the long axis of these objects aligned perpendicularly to the long axis of the crystal. Furthermore, the scattering profile in the direction perpendicular to the long axis of the crystal exhibits small-angle diffraction peaks (indicated with arrows in Fig. 2B). Such peaks most likely originate from a regular arrangement of mesoscale features in the crystal along this direction. A more complete interpretation is presented further in the text in the context of the microtomography (μCT) data (see the section Structural Heterogeneity at the Micrometer Scale and Beyond).

Fig. 2.

SAXS patterns from the anhydrite crystal. (A) 2D SAXS pattern of the anhydrite crystal in Position S at 0° (see also SI Appendix, Fig. S3A). (B) 2D SAXS pattern of the anhydrite crystal in Position S at 21° tilt (see also SI Appendix, Fig. S3C). In A and B, the angular directions marked with dotted lines indicate integration directions based on the polar coordinate representations shown in SI Appendix, Fig. S5. The Insets show the two orientations of the single crystal in accordance with SI Appendix, Fig. S5; approximate long Z-axis direction of the crystal is indicated with green arrows. (C and D) Direction-dependent scattering curves integrated from A and B, respectively (see also SI Appendix, Fig. S5). Fitted scattering dependencies in a form of I(q)∝q^-a, where −a is a scattering exponent, are indicated with dotted red lines; the Porod-scattering (smooth interface) I(q)∝q⁻⁴ is shown with dashed black lines. (C) Position S at 0°, averaged high-intensity direction (black), and low-intensity direction (purple); curves are obtained by integrating intensity profiles with centroids of azimuthal angles as written in the legends, and based on A and II in SI Appendix, Fig. S5A; each of two symmetric profiles are averaged together. (D) Position S at 21° tilt; three characteristic scattering directions are shown I (orange), II (cyan and purple), III (black); curves are obtained by integrating intensity profiles with centroids of azimuthal angles as written in the legends and based on B and II in SI Appendix, Fig. S5B.

In order to compare intensities, 2D patterns were converted to polar coordinates (SI Appendix, Fig. S5 A and B), based on which we calculated one-dimensional (1D) scattering profiles (SI Appendix, Supplementary Note 3) from the selected directions (dotted lines in Fig. 2 A and B). The resulting 1D scattering profiles (Fig. 2C) have the form of straight lines in a log–log representation, with both converging to background scattering at ∼1,000 cts. Both profiles scale proportionally to I(q)∝q ^<−3, but the exponents are higher than the typical smooth interface dependence of I(q)∝q⁻⁴ [Porod interface (35, 36)]. These relatively feature-poor forms indicate that scattering arises from objects extending beyond the available q-range and/or exhibit high polydispersity, meaning that the objects are >∼70 nm. In addition, the scattering exponents between −3 and −4 point to rough interfacial, surface-fractal–like scattering (37), but due to the limited q-range this cannot be unequivocally confirmed. The high-intensity profile (black curve, Fig. 2C) converges to background scattering at q of ∼1 nm⁻¹, whereas the low-intensity scattering (purple curve, Fig. 2C) converges at ∼0.3 nm⁻¹. This implies that the length-scale aspect ratio between the perpendicular scattering features is >3:1. For the tilted orientation, shown in Fig. 2D, the perpendicular scattering patterns (I and II) are similar in terms of intensity and their characteristic length scales. In both cases, the dominant interfacial scattering exponent is <−3, and the patterns converge to the background level at ∼0.6 nm⁻¹. However, scattering profiles in II, in addition to an interfacial-type scattering profile (i.e., a straight line), also exhibit relatively well-pronounced small-angle diffraction peaks corresponding to d-spacings of 22, 19, 13, and 12 nm. Profile III in Fig. 2D is calculated from a “streak” in Fig. 2B in the direction of 0° (see also SI Appendix, Fig. S5B and Supplementary Note 3) and contains a straight-line interfacial component, but its scaling follows a less steep dependence of I(q)∝q^−2.5 and small-angle diffraction peaks at 13, 9, and 7 nm. The presence of the peaks in II and III indicates the contribution of a scattering structure factor S(q), which describes the regular, paracrystalline arrangement of the scattering objects (38, 39), where the normalized structure factor is defined as S(q) → 1 for q → ∞. The occurrence of such a paracrystalline structure factor has two major implications: 1) it potentially explains why the observed interfacial scattering exponents are less steep than the expected −4 ; 2) S(q) is direction dependent and the contributing scattering features are closely spaced/correlated only along the directions where the peaks are observed.

In summary, the SAXS data tell us that 1) the single crystal is structurally heterogeneous at the mesoscale; 2) the heterogeneities are highly anisotropic and preferentially orientated, with their longer dimensions perpendicular to the apparent long axis of the crystal; and 3) the tilt of the crystal reveals the presence of a paracrystalline structure factor only in some cross-sectional directions. Although the exact crystallographic alignment of S(q), with respect to the crystal structure, is not known, its presence could be explained by the fact that the anisotropic scattering features are, to a certain degree, regularly arranged in the plane perpendicular to the long axis, where the actual scattering profiles with peaks (II and III in Fig. 2D) originate from cross-sections (between 0 and 90°) of such anisotropic superstructure.

Fig. 3A shows a four-panel composite WAXS diffraction pattern in polar coordinates from an anhydrite crystal at 0° relative tilt. The patterns were measured for four orientations (SI Appendix, Fig. S3B) resulting from a rotation of the crystal around the goniometer’s vertical axis, Z. Consecutive panels correspond to the following positions: S (starting), N (180° clockwise), W (90° clockwise), E (270° clockwise). The pattern consists of individual diffraction spots, which again confirms that we are dealing with a single-crystalline material. The diffraction spots are, in general, very similar in shape and broadening, and at first glance they do not reveal any obvious structural defects, such as strong mosaicity. However, in our WAXS measurements, we probed only a very limited set of crystal orientations. Furthermore, there are some very apparent exceptions from this trend, most noticeable for the peaks at q ∼28 nm⁻¹ (marked with arrows in Fig. 3A). The strongest of these reflections is at 166° in panel S and has the form of a cross (dashed white rectangle in Fig. 3A). The reflection appears to be broadened both in q and the azimuthal-angle direction, while also being composed of several subreflections (Fig. 3B). This cross also has an asymmetric counterpart for the same q at 76° in the same panel (i.e., 90° apart in the azimuthal angle) and weaker analogs at 5° and −85° in panel N, due to the rotation of a crystal by 180° around the Z-axis (SI Appendix, Fig. S3B). The reflections in panel S are also accompanied by long streaked lines extending at 90° in directions parallel to the cross in Cartesian coordinates (Fig. 3C), which makes them appear as curves in polar coordinates (Fig. 3B). For a given peak, the two considered broadening values in polar coordinates are essentially independent from each other because they are related to different structural effects. The increased broadening in the q-direction is correlated with structural effects, which typically affect the lattice d-spacing, for example, caused by strain. The broadening in the azimuthal direction expresses the structural coherence of a crystal and is a measure of mosaicity, as is the case for the reflections at q ∼28 nm⁻¹. The observed occurrence of such mosaic effects only for a single group of diffraction spots implies that the potential defects in the crystal structure are strongly anisotropic. Furthermore, the observed streaks are indicative of diffuse scattering, which signifies the presence of anisotropic disordered features at the mesoscale, associated with the structural defects. Both effects (strong mosaicity and diffuse scattering) are in agreement with the interpretation of the SAXS and TEM data presented earlier.

Fig. 3.

WAXS patterns from the anhydrite crystal. (A) Composite WAXS 2D diffraction pattern, which consists of four panels. The data are plotted in polar coordinates (the “cake plot”). The diffraction patterns were measured for four orientations resulting from a rotation of a crystal around the goniometer’s vertical axis, Z, where consecutive panels correspond to positions in Fig. 1B: I (starting), II (180° clockwise), III (90° clockwise), and IV (270° clockwise). Furthermore, each of the panels comprises five subpatterns obtained by moving the detector in a plane perpendicular to the beam, that is, emulating a larger area detector. For the overlapping pixels among such subpatterns, the intensities were averaged out. The discussed peaks at q ∼28 nm⁻¹ are indicated with arrows; the cross-shaped reflection is marked with a dashed rectangle. (B) Close-up 2D WAXS in polar coordinates, and profile plots of the reflection marked in A; an arrow points to a diffuse scattering streak. (C) 2D WAXS in Cartesian coordinates of the two peaks at q ∼28 nm⁻¹ from the first panel in A; the diffuse scattering streaks are well pronounced; the Cartesian-coordinates representation is recalculated from the polar coordinates.

Structural Heterogeneity at the Micrometer Scale and Beyond.

We further evaluated the structure of a single crystal at the micrometer-length scale using X-ray μCT. This technique allows for the three-dimensional (3D) visualization of a structure, based on the absorption contrast that is, differences in the electron density at a resolution of 550 × 550 × 550 nm³ voxel size. In Fig. 4A, two selected 3D projections of a reconstructed crystal derived from ring-uncorrected slices are shown (Materials and Methods). These images reveal the presence of voids within the crystal volume. In Videos S1 and S2, complete 360° rotations of the crystal are shown: around the Z-axis in the X–Y plane, and around the Y-axis in the X–Z plane, respectively. These renderings show that the crystal volume contains objects of different sizes, spanning from several tens of microns, down to objects at the voxel-size resolution limit. Fig. 4B shows two of the, ring-corrected and cropped, X–Y slices of the crystal (the Z direction is out of plane) that contain a number of these objects (in blue), surrounded by a relatively homogenous matrix (in brown). Considering the origin of contrast in μCT, the objects have significantly lower absorption than that of the surrounding crystal matrix and are thus attributed to voids/pores within the crystal. These voids exhibit very straight and well-defined edges in the X–Y plane (as indicated by the intensity profile function in the Inset in Fig. 4B). Fig. 4C shows a selected projection of a segmentation highlighting the voids; a full rotation of these objects around the Z-axis in the X–Y plane can be found in Video S3. Overall, voids are clustered into channel-like features parallel to the Z-axis (long axis) of the crystal. Along the Z-axis, the largest voids (∼25 μm in X–Y) form continuous regular structures, whereas the smaller ones, although not necessarily connected with each other, are grouped in pillar-like arrangements. To further evaluate the properties of the void arrangements, we performed statistical analysis on all slices (Fig. 4D and Materials and Methods for details). This analysis highlights what is directly observable in Fig. 4C, and in Video S3, that is, on average the features are anisotropic, with a width <∼5 μm and a length >∼10 μm in X–Y and <∼5 μm for width and >∼5 μm for length in Y–Z. In addition, they exhibit preferred orientation in the Y direction of the X–Y plane (at 90° to X, hence parallel to Y) and in the Z direction in the Z–Y plane (at 0° to Z, hence normal to Y), as can be seen in Fig. 4E. The void walls thus appear to approximate morphologically preferred faces of anhydrite, that is, {0 1 0} and {1 0 0}.

Fig. 4.

Microtomographic reconstructions and analysis. (A) Selected projections from a 3D reconstruction of an uncorrected (Materials and Methods) μCT dataset collected for the anhydrite single crystal; the projections show the two sides of the crystal (Left and Right) and highlight internal defect structure; see also Video S1, overview 360° rotation in the X–Y plane around Z, and Video S2, overview 360° rotation in the X–Z plane around Y. (B) Selected processed (Materials and Methods) cross-sections in the X–Y plane for two arbitrary Z-values; the voids are shown in blue; an Inset graph on the Left shows the overall abruptness of the contrast transition between the void and the surrounding crystal matrix; the cocentric rings are a typical artifact of the reconstruction processes, and in the image, they are already partially suppressed (Materials and Methods). (C) A projection of a segmentation, which shows the void structure within the crystal, derived from the processed data such as those in B; see also Video S3, 360° rotation in the X–Y plane around Z, which highlights the void structure. (D) Distribution of voids’ dimensions in X–Y and Z–Y planes calculated from the corrected data, which demonstrates the anisotropic character of the defects. (E) Distribution of voids’ orientations in X–Y and Z–Y planes calculated from the corrected data, which demonstrates the preferred orientation of the defects along Z and Y.

Bridging Length Scales: Nanometers to Microns.

We further characterized the voids exposed at the crystal facets using atomic force microscopy (AFM; Fig. 5). Detailed observation of different crystallographic faces revealed that on the top face (Fig. 5A), nano- to macroscopic sized porosity is observed (Fig. 5 B and C), while on the side faces (Fig. 5A) no such porosity is found (Fig. 5D). This multiscale surface imaging reveals that surface voids are rectangular/regular in shape and in the size range of nano- to micrometers (∼50 nm to 25 μm). Although, with AFM the depth of these pores cannot be probed, the obtained surface images directly complement the observations made at different length scales by SAXS and μCT. A quantitative analysis of the topographical AFM images reveals a porosity of 7.4 ± 1.6%, obtained as the area of the pores divided by the total area of the AFM images. Furthermore, we measured the dimensions of the pores directly from the height profiles of a wide range of AFM images (from 1 to 120 µm field of view), which were acquired at different locations of the same sample. Fig. 5E shows the as-obtained pore/void size distribution, which covers a wide range of pore sizes up to ∼60 µm. However, pores larger than 10 µm only constitute a minor contribution to the total porosity. In fact, the highest contribution corresponds to the pores smaller than 0.5 µm. The pore size distribution at the nanoscale highlights that a dominant pore size exists at ∼85 nm as shown in Fig. 5F. The AFM study is intrinsically limited to the external crystal surfaces; however, considering the observations from SAXS and µCT, it is inferred that the void size distribution, within the crystal, is analogous to that at the surface.

Fig. 5.

AFM characterization of the crystal facets. (A) View of a typical hydrothermal anhydrite sample from Naica; a side of the black square, on which the crystal is laid, is 1 cm. AFM images of the selected sides of the anhydrite crystal, which show topographical details of (B) the top face at low magnification; (C) the top face at high magnification; and (D) the side face at low magnification. The smallest observed voids exhibit well-defined edges and are <100 nm in size; (E) the pore/void size distribution on the anhydrite surface obtained from the quantitative analysis of the AFM topographical images. (F) The pore/void size distribution at the nanometer-length scale, where a maximum can be observed at ∼85 nm; an average void size is calculated as (length + width)/2.

We also probed the surface of the studied single crystal at the intermediate length scales using electron backscatter diffraction (EBSD), which bridges findings between X-ray scattering and μCT by providing crystallographic and lattice distortion information. In Fig. 6A, we show an EBSD orientation map of one of the crystal facets of the studied single crystal. This map represents an area of 51 × 51 µm², has a quasi-uniform cyan color, and corroborates the idea that, at the macroscale, the studied sample is a single crystal. Crystallographic orientation plots indicate that the EBSD mapped facet is parallel to the anhydrite (9 5 3) crystal plane. However, when we calculate the kernel average misorientation (KAM) assuming a maximum misorientation of 2.5°, it becomes apparent that the probed area is composed of heterogeneous regions (Fig. 6B). This is further evidenced by low-angle grain boundaries, with many of them completely closed and looking like small “cells” that are neighbored by other cells that have slightly mismatched angles between each other down to the 100 nm resolution used in the map (Fig. 6C). The degree of their misorientation is very small, as seen in the histogram of Fig. 6D, with a mismatch between blue-to-green regions in the range of 0.3° to <1°, which is in agreement with TEM data. The mean misorientation is 0.35° with an SD of 0.22°. Furthermore, the observed misorientation does not have a random character. In Fig. 6 B and D, one can observe that the upper half of the image (mean misorientation 0.40°, SD 0.26°) contains more pronounced disorder than the lower half (mean 0.30°, SD 0.15°).

Fig. 6.

EBSD maps of the selected crystal facet. (A) Orientation map from the (9 5 6) facet parallel to Z in SI Appendix, Fig. S3. The corresponding pole figure is shown in SI Appendix, Fig. S7. The total viewed area is 51 × 51 µm² at a resolution of 100 nm. A quasi-uniform cyan color indicates that the studied sample is a single crystal. (B) KAM map for the first neighbors with a 2.5° threshold, which highlights a heterogeneous character of A. (C) A cropped image from B, which further illustrates the intrinsic disorder down to ∼100 nm. (D) Distribution of orientations in B for the complete area (black), the upper half of the image (pink), and the lower half of the image (blue); in B and C, the intensity is expressed using a Green-Fire-Blue palette (black–blue–green–yellow) from ImageJ2 (77), where green codes misorientation of ∼0.9° and black of 0°.

Discussion

Anhydrite: Single, Poly-, or Mesocrystalline?

Although an ideal single crystal is assumed to have a continuous, perfectly ordered structure, real-life crystalline materials often contain point, line, and/or planar defects. If the amount and/or extension of defects are significant enough to physically separate crystalline domains by forming grain boundaries, the material is considered to be polycrystalline. When no long-range order is observed anymore, the solid phase is referred to as amorphous. Surprisingly, our anhydrite samples cannot be strictly classified into either of these groups. On the one hand, the diffraction measurements and the external appearance of the investigated mineral sample seem to indicate that we are dealing with a regular single crystal. Accordingly, its structure can be solved using standard methods of single crystal X-ray diffraction (see Materials and Methods and Supplementary Information Dataset S1 for the solved/refined structure). On the other hand, our detailed structural characterization at different length scales revealed that the anhydrite crystal contains numerous types of intercorrelated structural defects, which separate coaligned crystalline areas identifiable at the nanoscale. This corresponds with the general definition provided for so-called mesocrystals (9, 11, 40), which is a relatively novel concept, originally derived from experimental data and observations focusing primarily on biomineralization. Importantly, most mesocrystals identified so far formed in the presence of templating (macro)molecules and are linked to bioinspired mineral systems. Consequently, these mesocrystals are composites of crystalline material and templates. However, we observe for anhydrite crystals a segmented structure that appears to have formed in the absence of a template, as no template remnants are found in the crystal nor are organic templates expected considering the formation environment (41). Moreover, we previously observed the same phenomenon for micron-sized “single” crystals of gypsum and bassanite (8) formed under controlled laboratory conditions (i.e., in the absence of templating molecules). Hence, in the case of calcium sulfate, it seems to further blur the already murky distinction between single- and mesocrystals (42).

Seeds of Imperfection: An Origin of the Mesocrystallinity?

At the nanoscale, that is, <∼100 nm, HRTEM imaging reveals order–disorder modulation with crystallographic domains 10 to 20 nm in size. This modulation exhibits smooth transitions among the neighboring regions and can be interpreted as a reminiscence of the particle-mediated nucleation/crystallization. We proposed that tiny misalignments (<1°) between the crystalline regions, in combination with the disordered areas, act as “seeds of imperfection” for the further crystal growth, resulting in the formation of a macroscopic mesocrystal. The misorientations between the crystalline areas, of the order of the 1° observed here, were also reported for other nonclassical systems (e.g., bismuth or magnetite) growing via particle-mediated processes, such as orientated attachment (23, 43). Μicrometer-sized anhydrite mesocrystals were also found to be a byproduct of the bacterial dehydration of gypsum under very dry conditions (44), where particle-mediated nucleation is suggested as the reason behind its mesocrystallinity. Our earlier studies (5, 6, 45) on the nucleation of gypsum and bassanite, as well as those conducted by other groups (46–53), show that the crystalline phases of CaSO₄ nucleate and form within a micrometer-sized amorphous matrix of aggregated primary species (45, 52), several nanometers in size (for gypsum sub-3 nm). This constitutes the first stage of crystallization, which produces the initial micrometer-sized mesocrystalline seeds. Importantly, the restructuring processes within the amorphous aggregates do not continue until a near-perfect, homogeneous single crystal is obtained but instead comes to a halt during the observation window, because any mass transport processes inside such aggregates must be subject to slow diffusion when compared with those associated with ion transportation through the bulk aqueous solution. This early-stage crystallization leads to the formation of several-micrometer-sized single-crystalline seeds. What happens afterward is unclear, but the particle-mediated stage may be followed by ion-by-ion growth, which is the prevailing mechanism under thermodynamic conditions close to equilibrium. During this secondary growth stage, the initial mesostructured single crystals would act as the aforementioned seeds of imperfection. In fact, in situ AFM experiments conducted in a controlled laboratory environment using chemically pure solutions have shown so far that ion-by-ion growth can take place on the {1 0 0} and {0 1 0} facets at room temperature (54) and on the {1 0 0} facet between 70 to 120 °C (55). It is generally assumed that this growth mode should lead to rather continuous crystal structures. But, this does not correspond with what we observe in our anhydrite samples. This leaves open two scenarios: 1) the anhydrite samples examined by us did not experience significant ion-by-ion growth or 2) this growth mode can replicate the preexisting, or underlying, mesocrystalline matrix.

Voids Are a Key Feature to Keep the Crystal Structure Together.

Irrespective of the prevailing growth mechanism occurring during the postnucleation stage, the inherent misalignment and disorder in the crystalline matrix has to be compensated for if a crystal is to grow to macroscopic dimensions. We hypothesize that voids (SI Appendix, Fig. S8) are formed to compensate for the lattice strain/internal stress during growth. This would explain why voids are observed at all length scales. The fact that these voids, over time, develop straight facets to minimize their surface free energy, and thus lower the overall excess energy of the crystals, seems to corroborate our hypothesis.

The scattering behavior observed in SAXS can be explained by the presence of voids at the length scale of ∼100 nm. Tilt-dependent SAXS patterns actually correspond to structural features, which could very well look exactly like those in µCT, if it was not for the fact that computed tomography (CT) probes length scales >500 nm, whereas SAXS in our configuration is limited to <<500 nm, where the largest structures (>70 nm) could only be partially observed, if at all. This implies that the void topology extends to significantly smaller length scales than those measured with µCT, in a self-similar manner. This is confirmed by our AFM images (Fig. 5), which show that the voids can be <100 nm, with an average size of 85 nm. HRTEM revealed orientated disordered regions that can be correlated with the diffuse scattering/streaks observed in SAED and WAXS. In this regard, SAXS is likely to produce structural features from both the voids and the disordered nanosized regions, since they would all exhibit lower electron density than the crystalline anhydrite matrix, and therefore contribute to the scattering contrast. The oriented and anisotropic character of the voids indicates, in this context, that they form as semiregular errors in the replication/growth processes in a similar manner, as the order–disorder modulation is both regular and anisotropic. Such a behavior also explains the presence of the orientation-dependent small-angle diffraction peaks from semiperiodic structures. Indeed, we see large semiperiodic structures in µCT, and thus we infer that they have analogs at shorter length scales. Finally, the EBSD results bridge our interpretation of the scattering data with what we observe in μCT and other methods. This technique accesses the intermediate length scales of several tens to several hundreds of nanometers and highlights a heterogenous crystallographic character of the single crystal. Hence, the original seeds of imperfection are expressed through the length scales both in morphological (e.g., voids) as well as strictly structural (e.g., crystallographic) aspects of the actual single crystal.

In the context of the discussion above, it is also worth considering whether the observed voids might simply constitute dissolution etch pits (56). The voids observed here extend into the volume of the crystal whereas etch pits are typically limited to the surfaces of crystals. The deepening of etch pits leads to the formation of inverse pyramid dissolution patterns (56, 57) and not vertical wells/channels as those observed in our anhydrite samples. Moreover, etch pits should be filled up again when the crystal surface continues to grow, so it is highly unlikely that the remnants of a large number of etch pits should be preserved within the bulk of the crystal.

Conclusions and Outlook

Ongoing investigations into the nucleation of various mineral systems seem to hint that nonclassical pathways are much more prevalent than was considered a decade or two ago. In fact, it is even possible that some materials can nucleate both classically and nonclassically, a phenomenon that was recently demonstrated for calcium oxalate (58). Here, we show that the intrinsic mesocrystallinity seen in anhydrite (8, 44) is imprinted evidence of nonclassical, particle-mediated nucleation. This nucleation mechanism introduces seeds of imperfection, which still leads to the formation of a macroscopic single crystal, though its fragments do not ideally fit together at different length scales in a self-similar manner. This results in the formation of anisotropic voids of various sizes with very well-defined walls/edges, which approximately follow the anhydrite faces with the lowest surface free energy. This resembles nature playing a game of Tetris, which in some ways it is losing.

Materials and Methods

Anhydrite Single Crystals.

Macroscopic well-formed translucent anhydrite samples were obtained from the Naica Mine, Chihuahua, Mexico (municipality of Saucillo, the mine is owned by Industrias Peñoles). This mining area is located on the northern side of the Sierra de Naica and constitutes one of the main lead and silver deposits in the world. Hydrothermal fluid circulation associated with Tertiary dikes formed these Ag-Pb-Zn deposits (59). During the late hydrothermal stage, sulfuric acid formed by oxidation of the underlying sulfides and reacted with the available limestone to form calcium-sulfate–rich waters that eventually precipitated anhydrite masses (21). These specimens (SI Appendix, Figs. S2 and S3) are famous for their high purity, light blue color, and large size (single crystals can easily reach >10 cm). Almost all the experiments were performed on the same selected single crystal shard shown in SI Appendix, Fig. S3, which was chipped off from a bigger body of crystals similar to the one shown in SI Appendix, Fig. S2. The only exception was in the case of the AFM characterization (see AFM below), for which we used another pristine shard from the same group.

Single Crystal X-Ray Diffraction.

Single crystal X-ray diffraction experiments were performed on a Bruker D8 Venture system with graphite-monochromatic Mo-Kα radiation (λ = 0.71073 Å). For the diffraction experiments, the crystal (SI Appendix, Fig. S3) was not modified in any way, to ensure sample preservation for further analyses. Data reduction was performed with Bruker AXS SAINT (60) and SADABS (61) packages. The structure was solved in the space group Cmcm using direct methods and completed using differential Fourier maps calculated with SHELXL 2018 (62). Full matrix least-squares refinements were performed on F² using SHELXL 2018 (62) with anisotropic displacement parameters for all atoms. All programs were run under the WinGX (version 1.80) system (63). Visualization for Electronic and STructural Analysis (VESTA) program (version 3.5.7) was used for structure visualization (64). The resulting crystal information file (CIF) is included as a part of Supplementary Information, Dataset S1. Diffraction unequivocally confirmed the single-crystalline character of the investigated anhydrite sample.

Scattering Methods.

SAXS/WAXS measurements were conducted using the MOUSE instrument (65). X-rays were generated from a microfocus X-ray tube, followed by multilayer optics to parallelize and to monochromatise the X-ray beams to wavelength of Mo Kα (λ = 0.71073 Å). Scattered radiation was detected on an in-vacuum Eiger 1M detector (Dectris), which was placed at multiple distances between 52 to 2,354 mm from the sample. Beam parameters were kept consistent for all sample-to-detector distances used with a spot size of 643 µm (full width at half maximum).

The anhydrite single crystal was placed on a goniometer for data collection to allow multiple orientations to be probed in SAXS and WAXS (SI Appendix, Fig. S3). The initial crystal orientation was arbitrary, and the goniometer was set to null positions. This starting orientation with an X–Z tilt of 0° constituted position S (SI Appendix, Fig. S3A). WAXS was measured starting from S and at further positions N–W–E, 90° apart from each other, corresponding to a rotation of the crystal around its axis in the X–Y plane (SI Appendix, Fig. S3B). For SAXS, the crystal was returned to S (SI Appendix, Fig. S3A) and measured in a second step in position S with an additional X–Z tilt of 21° (SI Appendix, Fig. S3C).

The resulting data were processed using the Data Analysis WorkbeNch (DAWN) software package (version 2.20) in a standardized complete 2D correction pipeline with uncertainty propagation (66, 67). These included, among other steps, essential corrections for sample transmission and the instrument background subtraction. For SAXS, in order to compare the intensities in different directions the 2D patterns were also converted to polar coordinates (“cake” plots). Such a representation allows for an easy integration of the direction dependent–scattering intensities to 1D scattering curves. The azimuthal positions of the intensity directions of interest, as well as their angular widths, are directly obtained from the mean intensity profiles.

In the case of WAXS, such a polar representation was the only one used, due to the fact that version 2.20 of DAWN applies a small-angle approximation to scale the q_x- and q_y-axis in 2D patterns. This issue does not affect SAXS, but at higher scattering angles the resulting scales are incorrect for 2D images in Cartesian coordinates. However, the small-angle approximation is not utilized for calculating the “cake” plots, hence they are rendered correctly for all angular ranges. In those cases when 2D WAXS in Cartesian coordinates were required, we back calculated them from the “cake” plots rather than use outputs from DAWN so that q_x and q_y were expressed correctly. Further processing and analysis of reduced 2D scattering datasets was performed in Python using NumPy, SciPy, and Pandas (68–71). The dataset in a Nexus format is deposited at Zenodo (72).

TEM.

In order to analyze single crystals under a TEM, we prepared ∼15 × 4 µm thin foils (∼100- to 150-nm thickness) using the FIB technique (FEI FIB200) following a standard procedure (31, 32). Neither did the crystal show any signs of alteration under the vacuum of the instrument during cutting/milling nor did the foils when imaged in TEM. For TEM imaging and SAED, a Tecnai F20 XTWIN TEM was used at 200 kV, equipped with a field-emission gun electron source. SAED patterns were collected using an aperture with an effective diameter of ∼1 μm, and the diffraction plates were developed in a high-dynamic range Ditabis Μm scanner. To correctly interpret any preferred orientation or texture-related effects in the TEM images, the objective stigmatism of the electron beam was corrected by ensuring the FFT was circular over the amorphous carbon film.

X-Ray μCT.

μCT was performed with an EASYTOM (RX Solutions) equipped with a LaB₆ filament. The final resolution was set to 0.55 μm obtained by applying a voltage of 100 kV and a current of 100 μA and by collecting 2,816 sinograms over a 360° rotation. With these settings, the collection time was set to ∼38 h. A 3D reconstruction was performed by the software provided by the manufacturer (RX Solutions). The collected 2,816 sinograms were reconstructed using the Back Projection Algorithm into a 3D tomogram, where the Y–Z plane was 1,447 × 1,718 pixels² and the Y–X plane was 1,447 × 1,716 pixels². The reconstructed tomogram constituted raw data, which were processed by reslicing the dataset into the X–Y and X–Z planes. For quantitative analysis, the 3D data were further processed by applying artifact correction and data restoration algorithms and scripts described in refs. 73 through 75. The images in the X–Y plane revealed typical ring artifacts from the reconstruction processes, which were partially suppressed following the referenced method (73). In the next step, the images were corrected for an illumination drift using histogram matching through the Z-axis in the stack. A median filter with a three-pixel 2D kernel together with a nonlocal means filter to suppress noise were applied, which was necessary to perform feature analysis, because the rings and noise were hindering the deduction of the solid-void threshold parameter. Such as-calculated images were trimmed in order to remove the edge/background parts of the slices leaving only the measured crystal. We calculated gradients distribution of features that measured along which axes the defects were dominant, to see if the defects exhibited any anisotropy. This was performed separately for the X–Y and Y–Z plane because of the memory limitations and demonstrated that the features were strongly anisotropic. Finally, an ellipse fit on the features was performed to further characterize their dimensions, which showed features were longer than wider. This was also readily visible from a visual inspection of the 3D projections. An example of uncorrected and corrected images is shown in SI Appendix, Fig. S6. The dataset and the uncompressed Videos S1–S3 are deposited at Zenodo (72).

EBSD.

We used EBSD in a scanning electron microscope (SEM) to characterize the general orientation and the intracrystalline distortion of the studied anhydrite single crystal. This was done on an FEI Quanta200 F SEM with EDAX EBSD/EDS (energy-dispersive X-ray spectroscopy) detectors and Team/OIM Analysis software. The orientation map was collected from one of the single crystal facets, which exhibited very high apparent smoothness and hence did not require any special sample preparation. The crystal was orientated in the vacuum chamber in a way that the elongated direction of the single crystal (∼Z in SI Appendix, Fig. S3) was horizontal in the EBSD orientation maps and in the pole figures. The SEM operating conditions included an accelerating voltage of 20 kV, beam current of 8 nA, working distance of 15 mm, and a step size of 100 nm, in an uncoated sample, with the SEM working under low vacuum (30 Pa H₂O). Postacquisition processing included confidence index (CI) standardization with a grain tolerance angle of 5°, followed by one iteration of CI neighbor correlation considering only grains with CI > 0.1. Afterward, we removed all the pixels with CI < 0.2 and image quality below 25% to ensure that the orientations presented here were correct. From the orientation map, we then calculated the KAM map calculated in relation to a fixed distance between neighbors, which showed the average misorientation of a given pixel in the map in comparison with all its neighbors. The as-obtained map had a threshold misorientation angle of 2.5° and was calculated in relation to the first neighbor pixel. The crystal orientation data were plotted in the upper hemisphere of an equal-angle stereographic projection (SI Appendix, Fig. S7). Here, we plotted the three main crystal directions of anhydrite ([1 0 0], [0 1 0], and [0 0 1]) plus the direction <5 8 6>, which had one of the four symmetrically equivalent directions ([5 8 6], [-5 -8 6], [-5 8 -6] and [5 -8 -6]) plotted right in the middle of the pole figure and thus indicate which crystal direction is approximately normal to the reader when looking at the orientation map. That indicated that this direction was normal to the facet, which was equivalent to a crystal plane within the {9 5 3} group ((9 5 3), (-9 -5 3), (-9 5 -3) or (9 -5 -3)).

AFM.

Topographical features of the studied anhydrite sample (see also Anhydrite Single Crystals) were evaluated using AFM operating in contact mode using an MFP-3D microscope from Asylum Research. The maximum range of the piezo scanner was 120 µm in the planar direction (X–Y) and 15 µm in the vertical direction (Z). All the AFM images were acquired by using triangular silicon nitride cantilevers (Pyrex-Nitride Probe – Triangular Cantilevers from NanoWorld) with a nominal spring constant of 0.08 N ⋅ m⁻¹. Before each experiment, the used cantilever was routinely calibrated using the thermal method. All the obtained images were processed using the AR and WSxM software (76). Images collected at a field-of-view length scale of ∼120 µm were postprocessed by applying a thresholding algorithm that filtered out all topographical features above 20 nm (set as a threshold value). As a result, filtered AFM images were obtained showing almost exclusively the pores/voids, which allowed us to perform quantitative calculations of the volume, surface, and perimeter of the pores.

Data Availability

All study data are included in the article and/or supporting information. The original unprocessed and raw data sets from Scattering Methods and X-Ray μCT are public and deposited in Zenodo at https://zenodo.org (DOI: 10.5281/zenodo.4943234).

Change History

December 02, 2021: The article text and SI Appendix have been updated.