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

Edited by Steve Granick, IBS Center for Soft and Living Matter, Ulju-gun, Ulsan, Republic of Korea, and approved August 22, 2017 (received for review May 5, 2017)

## 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 (*μ*SAXS) 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

## Experimental Method and Symmetry Analysis

An ensemble of

Fig. 1 demonstrates the analysis method (18). The angular autocorrelation function (Fig. 1*i*) *ii*) *iii*) of each local n-fold symmetry. These maps can be averaged to obtain a spectrum of the average symmetry magnitudes (Fig. 1*iv*). 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 (

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.

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

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 *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.

The glass with no additive (Fig. 4*A*) 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 *B* 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. 4*C*. 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-

We simulated angular correlation magnitudes from a structure with completely randomly placed particles at the same volume fraction as the centrifuged glasses (Fig. 4*E*). 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. 4*D*). 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).

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

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 (^{2}. These values correspond to contact values of the pair-potential of

Glasses were centrifuged for 10 min at 10,000 rpm (Eppendorf 5804R F-34-6-38 rotor). Raw SAXS and

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

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

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

## Acknowledgments

We thank Adrian Hawley and Nigel Kirby for experimental assistance. A.C.Y.L. acknowledges discussions with Joanne Etheridge, Laure Bourgeois, Andrew Martin, Alessio Zaccone, and Peter Berntsen during manuscript preparation. A.C.Y.L. acknowledges the Margaret Clayton Women in Research Fellowship and the Science Faculty, Monash University. T.C.P. and A.C.Y.L. acknowledge support from the Monash Center for Electron Microscopy. The SAXS experiments were conducted at the SAXS/WAXS beamline at the Australian Synchrotron, Clayton, Victoria, Australia.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: amelia.liu{at}monash.edu.

Author contributions: A.C.Y.L., R.F.T., M.D.d.J., S.T.M., and T.C.P. designed research; A.C.Y.L., R.F.T., M.D.d.J., S.T.M., and T.C.P. performed research; A.C.Y.L., R.F.T., and T.C.P. analyzed data; and A.C.Y.L., R.F.T., M.D.d.J., S.T.M., and T.C.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Data deposition: Data have been deposited in the Monash fig-.share repository, https://doi.org/10.4225/03/5966c964c1215.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1707198114/-/DCSupplemental.

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