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Xray cross correlation analysis uncovers hidden local symmetries in disordered matter

Communicated by Philip H. Bucksbaum, Stanford University, Menlo Park, CA, May 21, 2009 (received for review February 25, 2009)
Abstract
We explore the different local symmetries in colloidal glasses beyond the standard pair correlation analysis. Using our newly developed Xray cross correlation analysis (XCCA) concept together with brilliant coherent Xray sources, we have been able to access and classify the otherwise hidden local order within disorder. The emerging local symmetries are coupled to distinct momentum transfer (Q) values, which do not coincide with the maxima of the amorphous structure factor. Four, 6, 10 and, most prevalently, 5fold symmetries are observed. The observation of dynamical evolution of these symmetries forms a connection to dynamical heterogeneities in glasses, which is far beyond conventional diffraction analysis. The XCCA concept opens up a fascinating view into the world of disorder and will definitely allow, with the advent of free electron Xray lasers, an accurate and systematic experimental characterization of the structure of the liquid and glass states.
Disordered matter, such as glasses and liquids, does not exhibit translational symmetry and in turn is able to accommodate different local symmetries in the same system, among them the icosahedral local order, which belongs to the forbidden motifs in periodic structures. This mysterious and so far experimentally inaccessible localized order within disorder has been fascinating scientists for many decades (1–5), because it is held responsible for the undercooling of liquids and the existence of the glass state. Similarly, nonperiodic materials have always attracted the attention of materials scientists, because they do carry—through these structural degrees of freedom—a unique potential to display novel smart functions (6–8).
The microscopic understanding of the structure and properties of crystals has advanced rapidly during the last decades. The translational invariance of the crystalline state allowed the introduction of the Brillouin Zone concept, thus enabling an elegant and powerful theoretical description of the thermal, electronic and magnetic properties. At the same time, crystal diffraction has continuously been developed to such a fine art that even complex biological structures can be solved today with atomic resolution (when forced to form a crystal). In severe contrast to this, the local microscopic structure of disordered matter has remained a challenge and a mystery (1–3). Our lack of knowledge on the local order within disorder constrains the development of a better understanding of the properties of liquids and glasses (9). In turn, the open question of how the structure of the liquid and amorphous states can be accessed experimentally has become one of the holy grails in condensed matter science (10).
The fundamental limits of conventional (Xray, neutron, electron) diffraction from disordered materials are accountable for this situation, because such techniques only allow to extract the pair distribution function g(r) = n_{0}^{−2}〈ρ(0)ρ(r)〉 of the single particle density ρ(r) = Σ_{m}δ(r − R_{m}) averaged over the illuminated sample area. Here, and in the rest of the article, R_{m} denotes the positions of the particles, n_{0} the average particle density and r the interparticle distance. The associated structure factor 〈S(Q)〉 = 1 + n_{0}∫(g(r) − 1) e^{iQr}dr, which gives the scattering intensity I(Q), depends on the momentum transfer Q. 〈S(Q)〉 shows rather similar features for all disordered structures (see Fig. 1B), thus carrying a quite limited information, i.e., the probability to find another atom in a certain distance from a given “average atom.” In particular, 〈S(Q)〉 provides no direct answer on the local symmetries in the system (11), which are intimately related to the local bonding and to bonding angles.
Our experimental and theoretical approach to solve this ageold problem has followed the guiding principle that the intrinsic spatial (and temporal) averaging mechanism performed in conventional (i.e., partially coherent) diffraction has to be eliminated experimentally. Then, a properly defined higherorder angular correlation function has to be devised and applied to data in order to disclose the hidden local symmetries of disordered matter. The first results obtained from different colloidal glasses deliver a new realm of structural details, which already shed a fascinating new light on the origin of glasses and on the glass forming mechanism.
Configurational averaging can be avoided if the coherence volume of a partially coherent Xray beam is equal to the illuminated sample volume (12, 13). In case of extreme forward scattering only the transverse coherence is relevant (14). Additionally, temporal averaging can be neglected for exposure times shorter than the onset time for speckle dynamics (15). Modern highbrilliance synchrotron radiation facilities and—in the future, even more so—free electron Xray laser facilities deliver sufficient coherent Xray flux for carrying out (nonaveraging) coherent diffraction experiments from disordered matter that produce the wellknown laser speckle patterns on a 2Ddetector (14–16) (see Fig. 1A). By applying a temporal correlator to one selected speckle spot associated with a fixed Qvector, the Qdependent temporal relaxation behavior of colloidal glasses has been extracted (14) [Xray photon correlation spectroscopy (XPCS)] (see for example Fig. 2A).
Results and Discussion
Colloidal glasses can be studied with commercially available 2Ddetectors, whereas suitable 2Ddetectors, which cover the Qrange required to study molecular or metallic glasses yet have to be developed. Here, we use a similar experimental setup as for XPCS (Fig. 3A): The coherent Xray beam prepared at station ID10A of the European Synchrotron Radiation Facility (ESRF) hits a colloidal glass sample and produces a full 2π speckle pattern at the 2Ddetector covering a Qrange up to 0.2 nm^{−1}. A typical dataset displays the expected isotropic granular “intensity rings” that are mediated by the random local order within the system (Fig. 1A). The deduced angular averaged structure factor shows the standard radial intensity distribution (Fig. 1B).
To transcend this information, we must unravel correlations in the angular distribution of the Xray speckles. At first sight this is not evident at all, because the angular variation of the intensity exhibits an isotropic distribution as anticipated for amorphous systems (Fig. 1A). However, as we shall show, this is possible by considering the generic intensityintensity cross correlation and the associated 4point correlation function. where 〈…〉 means a statistical average.
Angular Cross Correlation Function.
Here, we introduce a first simple subset of these new types of higherorder correlations, i.e., the instantaneous local angular correlations with respect to a given azimuth Δ.* They are found by performing the average in Eq. 1 over ϕ with the vectors Q and Q′ separated by Δ on the intensity ring with modulus Q (see Fig. 3B) and t = t′. We define the normalized angular 4point crosscorrelation function C_{Q}(Δ), with Fig. 1C shows the result after applying C_{Q}(Δ) (Eq. 3) to data in Fig. 1A for selected Q values. Most fascinating is that C_{Q}(Δ) clearly reveals a very pronounced anisotropy with 5fold symmetry specifically for the intensity ring associated with Q = 0.04 nm^{−1} (marked in black in Fig. 1A), which points to a so far hidden local symmetry in the colloidal system. The other Q values shown in Fig. 1C present a C_{Q}(Δ), which most closely resembles 4fold and 10fold symmetry; and 6fold symmetry can also be found (Fig. 2B). The fluctuations of the data points are (mostly) due to its speckle origin and not due to counting statistics (see numerical simulation results below). The solid lines are only guides to the eye to point out the dominant symmetry in C_{Q}(Δ).
Encouraged by this discovery, we have systematically evaluated the speckle patterns of many different colloidal systems, which all produced pronounced features in C_{Q}(Δ). We focus in this report on a hard sphere polymethylmethacrylate (PMMA) system with radius of 117 nm. Details on the sample preparation can be found in Materials and Method.
We have made several systematic observations in all colloidal glass systems. The emerging local symmetries are coupled to distinct Q values, which do not coincide with the maxima of the amorphous structure factor. The most prevailing symmetry is 5fold.^{†} For a basic understanding of these observations we assume our sample to consist of locallyfavored structures with icosahedral symmetry embedded randomly in disordered regions (19). Then, C_{Q}(Δ) (Eq. 3) has the simple form with ρ_{Q}^{i}(ϕ) as the structure amplitude of the ith LFS. Eq. 6 shows that C_{Q}(Δ) is a superposition of the angular intensityautocorrelation function of individual local structures, each contributing an intensity pattern ρ_{Q}^{i}(ϕ)^{2} as the one in Fig. 4B. In the coherent speckle pattern produced by the random ensemble of such clusters, this pattern becomes completely obscured. Only the application of XCCA recovers its symmetries. In the SI we present a straightforward discussion of the Qdependent behavior of C_{Q}(Δ) based on an analytical derivation of the structure factor of icosahedral clusters (20–22). We note that angular correlations between local clusters, so called mediumrange correlations, are also accessible by XCCA. This requires a different 4point correlator in Eq. (3), which takes Qvectors with Q′ ≠ Q into account.
We have performed calculations of the full expression (Eq. 3) for 8000 randomly oriented icosahedra (Fig. 4C) on a cubic lattice. The resulting correlation patterns confirm very nicely the local angular symmetries of our experiment (Fig. 4A) and show that the sprinkle of the data points is due to the coherent scattering process (speckles).
Dynamical Heterogeneity.
A further most fascinating observation is associated with the temporal relaxation behavior of C_{Q}(Δ) (Fig. 2B). Our new 4point correlator unveils a continuous change of the locallyfavored structures within the first 600 s from initially 6fold symmetry to 5fold symmetry, with no obvious symmetry in between. Apparently, icosahedral clusters reorganize in different orientations, form either out of local nanocrystals (hexagonal/fcc) or disorder, all of which involves a breaking and forming of bonds. Such behavior is known from molecular dynamics simulations as “dynamical heterogeneity” (19). Within the observed time frame, the temporal autocorrelation drops below 30% (see Fig. 2A). This new Xray crosscorrelation analysis concept reveals now that the processes, which are responsible for these relaxations, are accompanied by distinct changes of the local structure. These cooperative processes are the βrelaxation at shorter and αrelaxation at longer time scales (23), where the βrelaxation describes the rattling of individual particles trapped in transient cages formed by their neighbors and the αrelaxation the structural rearrangement of these cages (24). In future, this technique can be used to interrogate the time dependence of the local structures in glassy and supercooled states and new local structural motifs in strong and fragile glass formers by employing even more sophisticated crosscorrelators between intensities at different times.
Conclusions
We have introduced and applied a particularly simple 4point cross correlator that enables us to unveil otherwise hidden symmetries in a colloidal glass. Our approach is general enough to accommodate many more complex cross correlators derived from Eq. 1. In particular, the orientational pair–pair correlations will allow access to midrange order. Preliminary yet promising calculations employing a modecoupling ansatz and an appropriate atomic potential (25) show a dominant instability for 6 and 5fold symmetry (associated with closepacked structures) and a multitude of incommensurate wave vector instabilities (associated with random arrangements). This should enable to extract in the future the relevant interaction potentials from experimental datasets.
The availability of shortpulse XFEL radiation in the 0.1nm regime and with 100fs pulse length will open up the fascinating option to analyze the local structure of liquids (in particular also water) by applying the new concept of Xray cross correlation analysis (XCCA) to single laser shot speckle diffraction pattern. The combination of shortpulse XFEL radiation and large 2Ddetector arrays will also open up the window for the study of nanopowders and transient complex molecular structures in solution.
Materials and Methods
Samples.
The colloidal particles were synthesized via a polymerization method for preparing lattices of polymethylmethacrylate (PMMA) with a covalently surface linked stabilizer of poly(12hydroxystearicacid) (PHSA) following the method of Antl et al. (26). These fluorescent PMMA particles are obtained by copolymerization of methyl methacrylate, methacrylic acid and the fluorescent monomer, 7nitrobenzo2oxa1,3diazolemethyl methacrylate (NBDdyed) (27). To obtain highly concentrated, glassy systems, particle suspensions of ≈20–30% in volume were filled into quartzglass capillaries of 1 mm and centrifuged at 1,360 × g for ≈2 days. The supernatant decalin has been removed and the capillaries sealed.
Data Treatment.
The dark images were averaged and subtracted from the data. The colloidal form factor f_{S}(Q) has been obtained from a diluted PMMA sample. The form factor allowed to deduce the static structure factor S(Q) of the concentrated sample presented in this work. A multispeckle temporal intensity autocorrelation analysis yielded the dynamic correlation function of the sample. The dynamical correlation function, which relates to the diffusive motion of the colloidal particles, shows no decay within the first 100 s. Therefore, for the XCCA analysis we averaged 50–100 CCD images. Rings of intensity with constant Q and 1 to 2pixel widths have been extracted and an unbiased cross correlation function according to Eq. 3 has been calculated.
Note also that a diffraction pattern obtained via a 2D detector (i.e., mapping an Ewald sphere) will not satisfy Friedel's law at higher momentum transfers (see SI).
Acknowledgments
We thank M. Rauscher for discussions. Samples were synthesized by A. Schofield. This work was supported by the A. v. Humboldt Foundation (A.D.O.).
Footnotes
 ^{1}To whom correspondence should be addressed. Email: desydirector{at}desy.de

Author contributions: P.W., V.B., and H.D. designed research; P.W., C.G., T.A., A.D., F.Z., and G.G. performed research; C.G., T.D., and A.D.O. analyzed data; and P.W., A.D.O., and H.D. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/cgi/content/full/0905337106/DCSupplemental.

↵* A formalism, which is applicable only to highly diluted solutions studied with incoherent radiation, was developed in refs. 17 and 18 (see also ref. 10).

↵† The observation of odd symmetries appears to be in conflict with Friedel's law (I(−Q) = I(Q)), which holds for any plane in reciprocal space that intersects the origin. The observation of odd symmetries in our scheme is due to the deviation from the farfield (Fraunhofer) limit (13), because the sample is located as close as 20 cm to the 10μm entrance slit, thereby adding an imaginary part to the phase factor of each particle.

Freely available online through the PNAS open access option.
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