Entropic Comparison of Atomic-Resolution Electron Tomography of Crystals and Amorphous Materials

Electron tomography bears promise for widespread determination of the three-dimensional arrangement of atoms in solids. However, it remains unclear whether methods successful for crystals are optimal for amorphous solids. Here, we explore the relative difficulty encountered in atomic-resolution tomography of crystalline and amorphous nanoparticles. We define an informational entropy to reveal the inherent importance of low-entropy zone-axis projections in the reconstruction of crystals. In turn, we propose considerations for optimal sampling for tomography of ordered and disordered materials.

Fluctuation microscopy analysis of amorphous silicon models

Using computer-generated models we discuss the use of fluctuation electron microscopy (FEM) to identify the structure of amorphous silicon. We show that a combination of variable resolution FEM to measure the correlation length, with correlograph analysis to obtain the structural motif, can pin down structural correlations. We introduce the method of correlograph variance as a promising means of independently measuring the volume fraction of a paracrystalline composite. From comparisons with published data, we affirm that only a composite material of paracrystalline and continuous random network that is substantially paracrystalline could explain the existing experimental data, and point the way to more precise measurements on amorphous semiconductors. The results are of general interest for other classes of disordered materials.

Flexibility mechanisms in ideal zeolite frameworks

Zeolites are microporous crystalline aluminosilicate materials whose atomic structures can be usefully modelled in purely mechanical terms as stress-free periodic trusses constructed from rigid corner-connected SiO4 and AlO4 tetrahedra. When modelled this way, all of the known synthesized zeolite frameworks exhibit a range of densities, known as the flexibility window, over which they satisfy the framework mechanical constraints. Within the flexibility window internal stresses are accommodated by force-free coordinated rotations of the tetrahedra about their apices (oxygen atoms). We use rigidity theory to explore the folding mechanisms within the flexibility window, and derive an expression for the configurational entropic density throughout the flexibility window. By comparison with the structures of pure silica zeolite materials, we conclude that configurational entropy associated with the flexibility modes is not a dominant thermodynamic term in most bulk zeolite crystals. Nevertheless, the presence of a flexibility window in an idealized hypothetical tetrahedral framework may be thermodynamically important at the nucleation stage of zeolite formation, suggesting that flexibility is a strong indicator that the topology is realizable as a zeolite. Only a small fraction of the vast number of hypothetical zeolites that are known exhibit flexibility. The absence of a flexibility window may explain why so few hypothetical frameworks are realized in nature.

Examination of a polycrystalline thin-film model to explore the relation between probe size and structural correlation length in fluctuation electron microscopy

We examine simulated electron microdiffraction patterns from models of thin polycrystalline silicon. The models are made by a Voronoi tessellation of random points in a box. The Voronoi domains are randomly selected to contain either a randomly-oriented cubic crystalline grain or a region of continuous random network material. The microdiffraction simulations from coherent probes of different widths are computed at the ideal kinematical limit, ignoring inelastic and multiple scattering. By examining the normalized intensity variance that is obtained in fluctuation electron microscopy experiments, we confirm that intensity fluctuations increase monotonically with the percentage of crystalline grains in the material. However, anomalously high variance is observed for models that have 100% crystalline grains with no imperfections. We confirm that the reduced normalized variance, V(k,R) - 1, that is associated with four-body correlations at scattering vector k, varies inversely with specimen thickness. Further, for probe sizes R larger than the mean grain size, we confirm that the reduced normalized variance obeys the predicted form given by Gibson et al. [Ultramicroscopy, 83, 169-178 (2000)] for the kinematical coherent scattering limit.

Density of mechanisms within the flexibility window of zeolites

By treating idealized zeolite frameworks as periodic mechanical trusses, we show that the number of flexible folding mechanisms in zeolite frameworks is strongly peaked at the minimum density end of their flexibility window. 25 of the 197 known zeolite frameworks exhibit an extensive flexibility, where the number of unique mechanisms increases linearly with the volume when long wavelength mechanisms are included. Extensively flexible frameworks therefore have a maximum in configurational entropy, as large crystals, at their lowest density. Most real zeolites do not exhibit extensive flexibility, suggesting that surface and edge mechanisms are important, likely during the nucleation and growth stage. The prevalence of flexibility in real zeolites suggests that, in addition to low framework energy, it is an important criterion when searching large databases of hypothetical zeolites for potentially useful realizable structures.

Designability of Graphitic Cones

We show that, with topologically flexible seeds which are allowed to explore different growth modes, graphitic cones are inherently more “designable” than flat graphitic disks. The designability of a structure is the number of seed topologies encoding that structure.

We illustrate designability with a simple model, where graphite grows onto Cn (5≤n≤30) ring seeds. For a wide range of ring sizes, cones are the most likely topological outcome. Results from the model agree well with data from special cone-rich carbon black samples.

The concept of designability allows entropy to be incorporated into the “pentagon road” model of the formation of curved graphitic structures.

On the Propagation of Twin-Fault-Induced Stress in Platelet Fau-Framework Zeolites

A model describing the propagation of a rhombohedral distortion in platelet CSZ–1 zeolites is presented. It is proposed that internal stress gradients grown into CSZ–1 platelets at synthesis are responsible for this distortion of the cubic FAU framework, where the spacings of 111 planes parallel to the platelet surfaces are elongated relative to the {111} planes. The presence of inhomogeneities is suggested by the presence of thin bands of twin faults which are invariably observed near the central layers of each platelet. Elastic modelling confirms that the effects of any stress associated with such twin faults will be most pronounced in the thinnest platelets, where the effects of elastic relaxation are minimal, and where the width of the fault zone relative to the platelet thickness is maximal. Platelet CSZ–3 and Y-type zeolites, which are considerably thicker, are therefore not expected to show significant rhombohedral distortion despite the presence of similar twin fault bands.

Substantial crystalline topology in amorphous silicon

Using electron correlograph analysis we show that coherent nanodiffraction patterns from sputtered amorphous silicon indicate that there is more local crystallinity in unannealed amorphous silicon than was previously suspected. By comparing with simulations for various models we show that within a typical unannealed amorphous silicon film a substantial volume fraction (>50%) is topologically crystalline with correlation lengths up to 2 nm. Electron correlograph analysis is a variant of the fluctuation electron microscopy technique and its sensitivity to local crystalline ordering is derived from its sensitivity to four-body correlations.

On the collapse of locally isostatic networks

We examine the flexibility of periodic planar networks built from rigid corner-connected equilateral triangles. Such systems are locally isostatic, since for each triangle the total number of degrees of freedom equals the total number of constraints. These nets are two-dimensional analogues of zeolite frameworks, which are periodic assemblies of corner-sharing tetrahedra. If the corner connections are permitted to rotate, as if pin-jointed, there is always at least one collapse mechanism in two dimensions (and at least three mechanisms in three dimensions). We present a number of examples of such collapse modes for different topologies of triangular net. We show that the number of collapse mechanisms grows with the size of unit cell. The collapsible mechanisms that preserve higher symmetry of the network tend to exhibit the widest range of densities without sterical overlap.

When structural noise is the signal: Speckle statistics in fluctuation electron microscopy

In a recent paper, Jungk et al. [Ultramicroscopy 104 (2005) 206-219] conclude that standard fluctuation electron microscopy speckle analysis, which examines the normalized variance of scattered intensity as a function of scattering angle, does not provide more information about disordered materials than does classical diffraction. In this letter, I point out flaws in both their experiments and their modeling that led them to this erroneous conclusion. In their experiments, they use resolutions that are excessively high relative to the length scale of the medium-range ordering in their samples, and consequently the diffraction fluctuations they seek are very weak. In their modeling, they attempt to correct this by applying low-pass image processing filters in order to broaden the effective point-spread function (i.e. to lower the image resolution after the experiment). Such filters, when applied to image intensities, do not emulate properly the evolution of coherent speckle as the point-spread function becomes wider. I affirm that fluctuation microscopy does indeed provide more information about disordered materials than classical diffraction when the experiments and modeling are conducted appropriately.

Fluctuation X-ray microscopy: a novel approach for the structural study of disordered materials

Measuring medium-range order is a challenging and important problem in the structural study of disordered materials. We have developed a new technique, fluctuation x-ray microscopy, that offers quantitative insight into medium-range correlations in disordered materials at nanometre and larger length scales. In this technique, which requires a spatially coherent x-ray beam, a series of speckle patterns are measured at a large number of locations in a sample using various illumination sizes. Examination of the speckle variance as a function of the illumination spot size allows the structural correlation length to be measured. To demonstrate this technique we have studied polystyrene latex spheres, which serve as a model for a dense random-packed glass, and for the first time have measured the correlation length in a disordered system by fluctuation X-ray microscopy. We discuss data analysis and procedures to correct for shot noise and detector noise. This approach could be used to explore medium-range order and subtle spatial structural changes in a wide range of disordered materials, from soft matter to nanowire arrays, semiconductor quantum dot arrays and magnetic materials.

The flexibility window in zeolites

Today synthetic zeolites are the most important catalysts in petrochemical refineries because of their high internal surface areas and molecular-sieving properties. There have been considerable efforts to synthesize new zeolites with specific pore geometries, to add to the 167 available at present. Millions of hypothetical structures have been generated on the basis of energy minimization, and there is an ongoing search for criteria capable of predicting new zeolite structures. Here we show, by geometric simulation, that all realizable zeolite framework structures show a flexibility window over a range of densities. We conjecture that this flexibility window is a necessary structural feature that enables zeolite synthesis, and therefore provides a valuable selection criterion when evaluating hypothetical zeolite framework structures as potential synthetic targets. We show that it is a general feature that experimental densities of silica zeolites lie at the low-density edge of this window--as the pores are driven to their maximum volume by Coulomb inflation. This is in contrast to most solids, which have the highest density consistent with the local chemical and geometrical constraints.

A systematic topological search for the framework of ZSM-10

The application of a new technique for zeolite framework structure solution is described that exhaustively enumerates every possible topology consistent with known unit-cell dimensions and space-group symmetry. It is shown that computer-generated on-line databases of hypothetical crystal structures can radically augment structure building in the pre-refinement stage.