Favored local structures in amorphous colloidal packings measured by microbeam X-ray diffraction

Significance

Local structure and symmetry in amorphous materials and glasses may play a critical role in their formation and properties yet are notoriously hard to measure. Here, we demonstrate a direct method for measuring the proportions of polyhedra with different local point symmetries in amorphous colloidal packings, using small-volume transmission diffraction patterns. We show that local order is tuned by the interaction potential between microspheres and the method of preparation. This methodology can be readily applied to a broad range of disordered materials and packings to probe for universal features in their structure. It also has the potential to quantify local order in liquid, undercooled, and liquid–crystal systems approaching a phase transition.

Abstract

Local structure and symmetry are keys to understanding how a material is formed and the properties it subsequently exhibits. This applies to both crystals and amorphous and glassy materials. In the case of amorphous materials, strong links between processing and history, structure and properties have yet to be made because measuring amorphous structure remains a significant challenge. Here, we demonstrate a method to quantify proportions of the bond-orientational order of nearest neighbor clusters [Steinhardt, et al. (1983) Phys Rev B 28:784–805] in colloidal packings by statistically analyzing the angular correlations in an ensemble of scanning transmission microbeam small-angle X-ray scattering (μSAXS) patterns. We show that local order can be modulated by tuning the potential between monodisperse, spherical colloidal silica particles using salt and surfactant additives and that more pronounced order is obtained by centrifugation than sedimentation. The order in the centrifuged glasses reflects the ground state order in the dispersion at lower packing fractions. This diffraction-based method can be applied to amorphous systems across decades in length scale to connect structure to behavior in disordered systems with a range of particle interactions.

The role of local structures in amorphous materials in formation and properties is still contested. For example, the importance of local structure, or structures, in promoting glass formation and circumventing crystallization remains unclear (1, 2). The structural signatures that dictate material rheology and shear response (3–6) and demarcate the glassy state from a random close-packed state have yet to be unambiguously identified (7). Progress in addressing these fundamental questions has been hindered by the limitations of simulation and the experimental difficulty of measuring glassy structure beyond pair correlations (1).

It has long been acknowledged that unlike crystal structures, the structures of glassy and disordered assemblies are not adequately specified by the pair-correlation function (8) obtained by inverting broad-beam diffraction patterns that sample many local configurations. Higher order correlation functions can be accessed by using diffraction geometries in which the probe beam is comparable in size to the short-range structural correlations (9, 10). These measurements have been made using visible light on granular systems (11), X-rays scattered into small angles by colloidal systems (12), and electrons in atomic glasses (13). The intensity variations that arise can be used to calculate an intensity variance (14) or an angular correlation function (13), sensitive to different complex functions of the 2-, -3-, and 4-body correlations. The data can be compared with simulations from models (13) or used to refine models (15). Inverting the data to obtain directly the higher order correlation functions is an outstanding and apparently intractable problem (14), although there has been recent progress in extracting a function that shows the angular correlations in different coordination shells (16).

The bond-orientational order (BOO) parameters introduced by Steinhardt, Nelson, and Ronchetti (17) have been invaluable tools for assessing the local order (first coordination shell) in disordered, glassy, or liquid assemblies. In brief, these are the set of rotationally invariant parameters based on spherical harmonics that can be used as fingerprints of different archetypal short-range clusters (face-centered cubic, FCC; body-centered cubic, BCC; hexagonal close-packed, HCP; icosahedral, ICO; simple cubic, SC). Recently we have recalculated these parameters in the projection geometry, appropriate for transmission diffraction using high-energy radiation (18). We found that the projected parameters are still strong identifiers of the order and symmetry of the cluster. We hypothesized that these new projected BOO parameters could be used to directly quantify local structures in isotropic, glassy assemblies from the average angular symmetries in the first diffraction ring of an ensemble of limited-volume diffraction patterns (18). While confocal optical microscopy provides particle positions for a limited volume of colloidal packings of microspheres, a measurement using transmission diffraction offers the possibility of better statistics and more convenient dynamic sampling. Further, by tuning the energy and/or radiation, decades in particle size can be accessed, allowing structure to be examined using the same method across atomic glasses, colloidal glasses, and granular systems.

In this contribution, we show that the average angular symmetries of $\mu$SAXS diffraction patterns from monodisperse amorphous colloidal packings do indeed demonstrate considerable variation with potential-tuning additives. These average angular symmetries can successfully be decomposed into linear contributions from the projected symmetries of the archetypal clusters, providing a powerful and direct measurement of local structure in such materials. Using this technique, we found that centrifuged glasses possess local structures that reflect the equilibrium order of the dispersion at lower packing fractions. Through comparison with simulation from published data, we find that centrifugation results in glasses with strong order signatures, in contrast to sedimentation, which, at the same packing fraction, does not exhibit distinct local environments (7). This measurement may be useful in defining a “random close-packed” state (7, 19).

Experimental Method and Symmetry Analysis

An ensemble of $\mu$SAXS patterns were obtained by scanning thin glass specimens (Fig. S1) comprising 300-nm diameter silica particles in water with or without added surfactant or salt present, with an X-ray probe defined by a near-field aperture (see Fig. 1). The $\mu$SAXS patterns show well-resolved diffracted intensities (Fig. S2); the form factor is modulated by a local structure factor arising from structural correlations from within the limited volume (10). The interactions between the 300-nm diameter ${\text{SiO}}_2$ particles were tuned using additives. Salt screens the repulsive electrical double-layer force between particles that arises from surface charging in water, reducing the range of interaction. Added surfactant (nonionic; Tween20) results in a short-range attractive component to the potential due to an osmotic depletion force from the presence of micelles (20, 21).

Fig. S1.

Transmission optical micrographs of $20\mathrm{\mu }$m-thick colloidal specimens mounted between Kapton layers. (A) A glassy specimen. (B) A polycrystalline specimen showing opalescence. (Scale bar, 100 $\mathrm{\mu }$m.)

Fig. 1.

Schematic of experiment and analysis. An ensemble of $\mu$SAXS patterns are collected from a thin, colloidal glass. (i) The angular autocorrcorrelation function is calculated from each pattern, and then (ii) the magnitude of each symmetry is calculated by Fourier decomposition. (iii) The symmetry magnitudes in the $q$-range corresponding to the first peak in the structure factor are averaged and mapped as a function of transverse specimen distance $(x\text{,}y)$. (iv) The symmetry maps are averaged to give a fingerprint of the projected local symmetries in the glass. Errors are standard errors.

Fig. S2.

Raw SAXS data. (A) SAXS from a colloidal glass. (B) SAXS from the circular aperture. (C) $\mu$SAXS from a glass. (D) Average $\mu$SAXS from a glass. The angular symmetry magnitudes of the aperture were calculated from this and subtracted directly from the average symmetry magnitudes from the glass. (E) Simulated $\mu$SAXS pattern from the confocal dataset from Kurita and Weeks (7). (F) Simulated $\mu$SAXS pattern from a completely randomly placed assembly of particles at the same volume fraction as the experimental glasses studied. (Scale bar, 0.005 ${\text{Å}}^{-1}$.)

Fig. 1 demonstrates the analysis method (18). The angular autocorrelation function (Fig. 1i) $C(q,\mathrm{\Delta })$) of each pattern is calculated according to:

$$C(q,\mathrm{\Delta })=\frac{{⟨I(q,\varphi )I(q,\varphi +\mathrm{\Delta })⟩}_{\varphi }-{⟨I(q,\varphi )⟩}_{\varphi }^2}{{⟨I(q,\varphi )⟩}_{\varphi }^2}.$$ [1]

Here $|\overrightarrow{q}|=q=\sqrt{q_x^2+q_y^2}=4\pi \text{sin}\theta /\lambda$ is the scattering vector magnitude where $\theta$ is the scattering angle and $\lambda$ is the X-ray wavelength. $\varphi$ is the azimuthal angle in the diffraction plane. Normalized symmetry magnitudes (Fig. 1ii) $\frac{c_q^n}{c_q^0}$) are calculated by Fourier cosine decomposition of the angular autocorrelation at each value of $q$ (18). The average of these normalized symmetry magnitudes from a $q$-range corresponding to the first peak in the structure factor is used to construct a map (Fig. 1iii) of each local n-fold symmetry. These maps can be averaged to obtain a spectrum of the average symmetry magnitudes (Fig. 1iv). These average symmetry magnitudes can be corrected for dynamical diffraction (10, 22) to obtain a set of symmetries that reflects the orientationally averaged projected symmetries of the favored nearest neighbor clusters in the isotropic glass.

Local Structures in Centrifuged Glasses and a Sedimented Packing

Fig. 2 displays the measured structure factor ($S(q)$) and pair-distribution function ($g(r)$) for each glass. The volume fraction ($\eta$) of the glasses was obtained by fitting this structure factor and measured to be within the range of $\eta =0.6\pm 0.05$ for each glass. The structure factor and pair distribution function for the glass with no additives are similar to other colloidal and metallic glasses (12, 23). The high-$q$ shoulder on the second peak has been shown to be consistent with ICO local order (23). The structure factors of the glasses with added salt and surfactant almost have a double-peak structure, with greater magnitude on the high-$q$ side. This is consistent with the screening of the repulsive double-layer potential by salt and the fact that surfactant adds a short-range attraction to the potential (20, 21). The pair-distances encompassed by the broad first-nearest neighbor peak generally arise from a set of polyhedra with different populations. In these one-dimensional functions, there is only indirect structural information that is insufficient to deduce the prominent local order (12).

Fig. 2.

Volume-averaged pair-correlations. (A) Structure factors of the centrifuged glasses and (B) corresponding pair-distribution functions. Structure factors and pair-distribution functions are offset in the vertical axis by the addition of 2 and 4, respectively, for clarity.

Fig. 3 reproduces the projected symmetry fingerprints (18) of some archetypal short-range polyhedra with the distinctive BOO that have been used to characterize local structure in many model, colloidal, and hard sphere glasses (17). These projected symmetries are rotationally invariant; they are exactly what would be obtained by averaging the angular symmetries of the projected structure of a given cluster over all rotations (18). In Fig. 4 we show average angular symmetry magnitudes from experiment (Fig. 4 A–C) and simulation (Fig. 4 D and E). The experimental symmetry magnitudes are in agreement in magnitude with the ideal calculations (see Figs. S3 and S4). In electron diffraction, instrumental diminishment of diffraction contrast (e.g., finite convergence angle, partial coherence, camera noise) and/or specimen instability (24) result in a difference between measurement and theory. We do not observe such a discrepancy here.

Fig. 3.

Projected BOO parameters. Calculated projected symmetries of ideal unit clusters used to decompose the average symmetry magnitudes from the centrifuged glasses and the corresponding unit polyhedra used for the calculations.

Fig. 4.

Decomposition of the average projected symmetries into polyhedral and random components. Centrifuged colloidal glasses with (A) no additives, (B) added salt, and (C) added surfactant. Simulated average symmetry magnitudes and decompositions for (D) an amorphous colloidal packing formed by sedimentation (7) and (E) a simulated, completely random ensemble of particles. Error bars represent SE.

Fig. S3.

Experimental symmetry magnitudes calculated using fractions of the available 3,600 diffraction patterns. The magnitudes vary slightly when calculating from different fractions of the whole dataset, demonstrating that variation in specimen thickness makes a small contribution to systematic error. The magnitudes should be multiplied by 2x2xNx$\pi$/N, which brings them into agreement with the ideal calculated values shown in the main text. These factors correspond to Parseval’s theorem that requires the discrete Fourier transform to be scaled by the resolution of the original function (2x$\pi$/N) and the definition of the Fourier transform implemented in Interactive Data Language (2xN). Compared with the ideal angular symmetry magnitudes from isolated clusters shown in the main text, the magnitudes from close-packed assemblies have a significant and seemingly constant offset above 12-fold. This is due to structural noise and may be accounted for as a constant offset in the fit. Error bars indicate standard deviations.

Fig. S4.

Simulated symmetry magnitudes using a range of near-field aperture sizes. The magnitude of this constant offset decreases with aperture size due to coherent superposition of waves from different nearest neighbor clusters with large transverse separations and hence large fixed phase difference. Error bars indicate standard deviations.

The magnitude of the angular symmetries from isolated ideal clusters (see Fig. 3) rapidly declines after the 12-fold symmetry, but the symmetry magnitudes from large ensembles of close packed particles have a large constant offset that continues beyond this (Fig. S3). The Fourier coefficients that measure the magnitude of the angular symmetries are always positive, and so signals with no true angular correlation would give rise to such a constant offset, such as camera noise, shot noise, beam or specimen thickness fluctuations, or “structural” noise. We do not expect the camera noise to be a large contribution for a direct detection camera. The systematic error due to Poisson noise is ${\sigma }_p=1/{⟨I(q,\varphi )⟩}_{\varphi }$ (25), which for this experiment was 0.015 photons. The variation due to beam instability or specimen fluctuations is low (Fig. S5). Thus, we attribute this constant offset to structural noise. This additive structural noise arises from two sources: spurious angular correlations from particles that are in the same diffracting volume (defined by the aperture) but far enough apart in depth to have no real structural correlation, or local configurations that are highly distorted and possess less inherent symmetry. The magnitude of this constant offset depends on the size of the near-field aperture (Fig. S4). Angular symmetries from different first-nearest neighbor clusters from the same projected area seem to “add” up in the final diffraction pattern (9). In larger apertures the final diffraction pattern is produced by coherent superposition of the plane waves diffracted from each cluster in the aperture. Large transverse separations add fixed phase differences between the X-rays diffracted by different clusters, resulting in interference that degrades the symmetry of the diffraction pattern. Thus, for larger apertures the symmetry magnitudes decrease, and the magnitude of the constant offset can only be compared for datasets with the same instrument resolution. These observations suggest that we can model the data by assuming that the structure consists of symmetric polyhedra that give rise to a structured symmetry fingerprint and uncorrelated particles that contribute to a constant background. These components are similar to the local clusters whose BOO parameters calculated from 3D position data allow them to be readily assigned to an ideal polyhedron type and those whose parameter values vary too greatly from the ideal for designation (7, 17).

Fig. S5.

Sources of uncertainty in the symmetry magnitudes. (A) Experimental map of sixfold symmetry magnitudes for a centrifuged glass and (B) corresponding histogram of values. There are systematic and random variations in symmetry magnitude. The error in the average incorporates both sources in the following way. The weighted mean of two 600 pixel areas was used as the average value. The SE of this weighted mean was used as the estimate of the systematic error. This was added in quadrature to the SE calculated directly from the values in the two areas. The simulated data were treated in the same way.

We therefore use the projected symmetry fingerprints of the ideal clusters and a constant offset (RAN) as a nonorthogonal linear basis to decompose the average symmetry magnitudes from the $\mu$SAXS patterns and show the results of these fits in Fig. 4 A–C for the glasses with no additives, added salt, and added surfactant, respectively (see Materials and Methods). Initial trial fits ruled out the need for the SC and HCP structures, reducing the degrees of freedom of the fit. The fits can account for the data quite well and demonstrate significant differences in the local order of the different glasses. All of the fits have a reduced chi-squared value greater than 1 (Table S1). Systematic and random contributions to the uncertainty were estimated (Materials and Methods and Fig. S5). Thus, the reduced chi-squared values indicate that our structural model can be improved. The data for the glass with no additives and added salt have two-fold symmetries with magnitudes slightly too great to be accounted for by the linear combination of symmetries from this set of polyhedra, perhaps indicating that other, less symmetric, polyhedra need to be included in the basis set. This discrepancy highlights the advantages of this direct measurement technique. Atomic models generated by refinement or empirical potentials may reveal an atomic-level structure that was not anticipated, but devising tests to check the validity of the input assumptions is difficult. Using conventional BOO parameters on 3D coordinate datasets can test for the presence of a particular kind of local order, but indicative ranges for the BOO parameters need to be chosen. Our technique uses a fit with fingerprints of local point symmetries and is thus a powerful and fast way to test different structural models.

Table S1.

Values of the reduced ${\chi }^2$ for fits in Fig. 4

Material	Reduced ${\chi }^2$
Centrifuged—no additives	26
Centrifuged—added salt	157
Centrifuged—added surfactant	24
Simulated—random	41
Simulated—sedimented	269

The glass with no additive (Fig. 4A) has pronounced BCC and FCC order. This mixture shows a competition between BCC and FCC clusters, reflecting the equilibrium ordered phases of the dispersion at lower packing fractions (26, 27). The BCC phase minimizes the ratio of the surface area of the Voronoi cell to its volume and is the stable phase for long-range repulsive forces (27). For these slightly charged particles (see Materials and Methods) and for a short Debye screening length, the BCC phase is the stable crystal phase over the approximate range $0.1<\eta <0.5$ (26, 27). For larger values of the Debye screening length, the ground state of the system will be FCC crystal (26, 27). Potentially, there is a broad region of coexistence between the BCC and FCC phases in the equilibrium phase diagram, explaining why we see a distinct competition between these two types of local order in the glass with no additives. The addition of salt reduces the long-range repulsive potential and increases the Debye screening length. This trend is clearly seen in the local order decomposition of Fig. 4B whereupon the addition of salt, the amount of BCC order decreases and FCC order increases. The reduction in the long-range repulsive potential may also stabilize local ICO order in the liquid phase (28, 29) and explain why ICO order is present in this glass. Thus, the local order quenched into the glass reflects the stable phase in the liquid and in the dispersion at lower packing fractions. The addition of surfactant at sufficient concentration creates a short-range attractive component in the potential (20, 21), and the more efficiently packed ICO clusters are the most populous in the glass, as we see in Fig. 4C. The local structural differences that are measured may be signatures of “packing” glasses versus “bonding” glasses (30). These results confirm the observations from the structure factors that the glasses with salt and surfactant contain different populations of polyhedra compared with the glass with no additives. The ICO order in the glasses with additives is consistent with the increased magnitude in the high-$q$ side of the first peak of the structure factor. Using the symmetry fingerprints to decompose the populations allows access to information about local symmetry and structure not available from the pair-correlations.

We simulated angular correlation magnitudes from a structure with completely randomly placed particles at the same volume fraction as the centrifuged glasses (Fig. 4E). As predicted, this structure produced a predominantly flat profile. We also simulated a sedimentary amorphous assembly of sterically stabilized colloids with comparable packing fraction to our glasses, from the published confocal dataset of Kurita and Weeks (7) (Fig. 4D). Our analysis of order in the sedimentary amorphous material as being low, with no ICO order, and a very low amount of other polyhedral order is similar to a traditional BOO analysis on the 3D confocal data (7). We find the sedimented packing has a nonnegligible proportion of FCC order, in contrast to the traditional BOO analysis. This may reflect the range of BOO parameters chosen to demarcate FCC clusters in the original work. In contrast, we fit the data with the projected symmetries of ideal clusters. Interestingly, our analysis has no problem distinguishing low order and flat profiles from ICO order. The ICO fingerprint, while flat, has depressed two-, four-, and eightfold symmetries (see Fig. 3). Comparing Fig. 4C to parts Fig. 4 D and E, we see the ICO fingerprint is distinct enough to distinguish ICO symmetry from random local environments.

We summarize our results in Fig. 5 (all contributions normalized to sum to 1—see Fig. S6 for unnormalized results). Centrifugation quenches in local structures from lower density dispersions. In the case of a long-range repulsive potential, there is competition between the polyhedron corresponding to the equilibrium phase of the dispersion at lower packing fraction. Diminishing the range of this repulsion introduces ICO order into the polyhedral population. The introduction of a short-range attractive potential results in local ICO order only. Amorphous packings created by sedimentation of sterically stabilized particles have little BOO. Our data and analysis strikingly demonstrate the difference between a glass quenched from the liquid and a more random close-packed state created closer to equilibrium. We show conclusively that local order at a given packing fraction is a sensitive function of interparticle potential and method of preparation. Measuring the magnitude of the RAN component is itself an interesting parameter that shows whether there are any local environments in the material with a distinct point symmetry. Such a measurement may be able to explain glass-forming ability in different systems with no need to know the details of any distinct local environments that do exist (31). It may also be a useful way to distinguish and demarcate random close-packed states (7, 19).

Fig. 5.

Populations of different local structures measured by decomposing the average projected symmetry magnitudes from disordered colloidal assemblies into contributions from archetypal polyhedra. Shown are magnitudes of polyhedral and random components for the centrifuged glasses, a sedimentary amorphous colloidal material, and a simulated random dispersion at the same packing fraction. The parameter confidence ranges are determined using $\mathrm{\Delta }{\chi }^2=1$.

Fig. S6.

Unnormalized decomposition of symmetry magnitudes into contributions from different clusters for the centrifuged colloidal glasses and simulated sedimentary and random colloidal models. Parameter confidence ranges were obtained by varying parameters individually and calculating the values corresponding to a variation in ${\chi }^2$ of 1.

Angular correlations in reduced-probe diffraction data are much more sensitive to local symmetries and structures than the corresponding pair-correlations. These local symmetries will provide key insights into the glass-formability and rheology of different systems. By tuning the radiation, the same order parameters can be used to measure local symmetries in disordered materials across decades in length scale—from atoms to grains—providing opportunities to compare the role of thermal fluctuations, hydrodynamic forces, gravity, and friction in glassy behavior. This direct measurement method opens avenues to understanding the role of structure in the behavior and creation of glassy and amorphous materials.

Materials and Methods

$\mu$SAXS patterns were obtained by scanning a specimen with an ultra-low-divergence 5-keV X-ray beam shaped by cleaved Ge slits (32) and limited by a near-field aperture ($18\text{-}\mathrm{\mu }$m diameter tungsten; Lennox Laser). The 300 nm-diameter colloidal ${\text{SiO}}_2$ glass specimens (polydispersity 5%; Bangs Laboratories) were sandwiched between two Kapton polymer film layers (see Fig. S1) with a spacer layer to control the thickness to $20\hspace{0.17em}\mathrm{\mu }$m to limit dynamical scattering and the number of nearest neighbor clusters that would contribute to the pattern (10). Scans were conducted with 5-$\mathrm{\mu }$m steps covering an area of 150 $\mathrm{\mu }$m × 150 $\mathrm{\mu }$m and collecting $\approx$4,000 diffraction patterns using a Dectris–Pilatus 1M direct detection camera.

The particles were dispersed in water. Interparticle potentials were tuned using salt (0.1 M NaCl) and a surfactant (Tween20 at a concentration of 0.0081 M). Zeta potentials were measured using a NanoBrook Omni instrument (Brookhaven Instrument Corporation) operating in phase-analysis light scattering mode. The Smoluchowski equation was used to determine the zeta potential from the electrophoretic mobility, and the zeta potential was converted to surface charge density ($\sigma$) using the Grahame equation. For the system with no additives, we obtained $\zeta =-31\pm 2.4$ mV and $\sigma =0.0022\pm 0.0001$ ${\text{C/m}}^2$. For the system with added salt, we obtained $\zeta =-20.2\pm 0.8$ mV and $\sigma =\mathrm{\hspace{0.17em}0.0147}\pm 0.0006$ C/m². These values correspond to contact values of the pair-potential of $125\pm 10$ and $59\pm 2$ (per $k_{B}T$, where $k_B$ is the Boltzmann constant and $T$ is temperature) for the dispersion with no additives and added salt, respectively.

Glasses were centrifuged for 10 min at 10,000 rpm (Eppendorf 5804R F-34-6-38 rotor). Raw SAXS and $\mu$SAXS intensities are shown in Fig. S2. Imperfections in the aperture gave rise to scatter. The magnitudes of the angular correlations from this scatter were calculated from the averaged $\mu$SAXS patterns from each specimen and directly subtracted from the average symmetry magnitudes of the glasses. Both random and systematic sources of error contributed to the uncertainty in the symmetry magnitudes. The systematic error due mainly to long-range thickness fluctuations (Fig. S5) was estimated by calculating the average symmetry magnitudes from two different areas of the scanned array. The quoted average is the weighted mean of these values. The systematic error is given by the SE of this weighted mean. The total SE is the SE of this weighted mean added in quadrature to the SE of the values in the measured areas, representing the systematic and random components.

Simulated amorphous structures were created in the following way. The completely random structure was generated by placing particles randomly in a simulation cell, with no consideration for particle overlap, at the same packing fraction as the measured glasses. The sedimentary amorphous colloidal packing particle coordinates were made available from a published work (7). These were scaled according to particle diameter to match our experiments. Simulated $\mu$SAXS patterns were obtained using the phase object approximation and Fourier propagation (Fig. S2).

Fitting was performed using a nonnegative least squares algorithm (33) implemented in IDL (34) according to the following equation:

$$C^n={\left(\begin{array}{c}\hfill C_{FCC}^n\hfill \\ \hfill C_{BCC}^n\hfill \\ \hfill C_{ICO}^n\hfill \\ \hfill C_{RAN}^n\hfill \end{array}\right)}^T\left(\begin{array}{c}\hfill f_{FCC}\hfill \\ \hfill f_{BCC}\hfill \\ \hfill f_{ICO}\hfill \\ \hfill f_{RAN}\hfill \end{array}\right);f_{FCC},f_{BCC},f_{ICO},f_{RAN}\ge 0$$ [2]

Here $C^n$ is a vector containing the average projected symmetries (2 to 12) from the amorphous packings. This is decomposed as a weighted sum of the averaged projected symmetries from ideal clusters denoted $C_{cluster}^n$, where “cluster” indicates $FCC$, $BCC$, $ICO$, and $RAN$ and the weights are denoted $f_{cluster}$. Confidence ranges for the values of the weights $f_{cluster}$ were obtained by varying these parameters individually and calculating the values corresponding to a variation in ${\chi }^2$ of 1. The total SE as calculated above was used to calculate the ${\chi }^2$.

Data are available at https://doi.org/10.4225/03/5966c964c1215.