Microstructural Degeneracy associated with a Two-Point Correlation Function and its Information Content

Two-point correlation functions provide crucial yet incomplete characterization of microstructures because different microstructures may have the same correlation function. In an earlier Letter [Phys. Rev. Lett. 108, 080601 (2012)], we addressed the degeneracy question: What is the number of microstructures compatible with a specified correlation function? We computed this degeneracy, i.e., configurational entropy, in the framework of reconstruction methods, which enabled us to map the problem to the determination of ground-state degeneracies. Here, we provide a more comprehensive presentation and additional results. Since the configuration space of a reconstruction problem is a hypercube on which a Hamming distance is defined, we can calculate analytically an energy profile corresponding to the average energy of all microstructures at a given Hamming distance from a ground state. The steepness of this profile is a measure of the roughness of the energy landscape, which can be used as a proxy for ground-state degeneracy. The relationship between roughness metric and ground-state degeneracy is calibrated using a Monte Carlo algorithm for determining the degeneracy of a variety of microstructures, including hard disks and Poisson point processes as well as those with known degeneracies (single disks of various sizes and a particular crystalline microstructure). We show that our results can be expressed in terms of the information content of the two-point correlation functions. From this perspective, the a priori condition for a reconstruction to be accurate is that the information content, expressed in bits, should be comparable to the number of pixels in the unknown microstructure. We provide a formula to calculate the information content of any two-point correlation function, which makes our results broadly applicable to any field in which correlation functions are employed.Comment: Accepted for publication in Physical Review

Inelastic Neutron Scattering Analysis with Time-Dependent Gaussian-Field Models

Converting neutron scattering data to real-space time-dependent structures can only be achieved through suitable models, which is particularly challenging for geometrically disordered structures. We address this problem by introducing time-dependent clipped Gaussian field models. General expressions are derived for all space- and time-correlation functions relevant to coherent inelastic neutron scattering, for multiphase systems and arbitrary scattering contrasts. Various dynamic models are introduced that enable one to add time-dependence to any given spatial statistics, as captured e.g. by small-angle scattering. In a first approach, the Gaussian field is decomposed into localised waves that are allowed to fluctuate in time or to move, either ballistically or diffusively. In a second approach, a dispersion relation is used to make the spectral components of the field time-dependent. The various models lead to qualitatively different dynamics, which can be discriminated by neutron scattering. The methods of the paper are illustrated with oil/water microemulsion studied by small-angle scattering and neutron spin-echo. All available data - in both film and bulk contrasts, over the entire range of $q$ and $\tau$- are analyzed jointly with a single model. The analysis points to static large-scale structure of the oil and water domains, while the interfaces are subject to thermal fluctuations. The fluctuations have an amplitude around 6 nm and contribute to 30 % of the total interface area.Comment: The following article has been accepted by Journal of Chemical Physics. After it is published, it will be found at https://aip.scitation.org/journal/jcp

CONEX, a program for angular calibration and averaging of two-dimensional powder scattering patterns

CONEX is a Windows application for converting series of two-dimensional X-ray powder patterns measured on flat two-dimensional detectors into one-dimensional scattering patterns. It is based on the rigorous use of scattering patterns of calibration samples to determine the three-dimensional position of the detector, with respect to the sample and to the beam. This enables correction of the data for geometric distortions, even when the detector is highly tilted and not centred on the beam.status: publishe

Stochastic analysis of capillary condensation in disordered mesopores

Most mesoporous materials of practical interest are inherently disordered, which has a significant impact on the condensation and evaporation of vapours in their pores. Traditionally, the effect of disorder is theoretically analyzed in a perturbative approach whereby slight elements of disorder (constriction, corrugation) are added to geometrically ideal pores. We propose an alternative approach, which consists of using a stochastic geometrical model to describe both the porous material and the condensate within the pores. This is done through a multiphase generalisation of the standard Gaussian random field model of disordered materials. The model parameters characterising the condensate provide a low-dimensional approximation of its configuration space, and we use a Derjaguin–Broekhoff–de Boer approximation to calculate the free-energy landscape. Our analysis notably questions the existence of vapour-like metastable states in realistically disordered mesoporous materials. Beyond capillary condensation, our general methodology is applicable to a broad array of confined phenomena

2-Point correlation function of nanostructured materials via the grey-tone correlation function of electron tomograms: A three-dimensional structural analysis of ordered mesoporous silica

Electron tomography is a unique technique for imaging the microstructure of materials with a nanometer resolution. The signal-to-noise ratio of electron tomograms is, however, often too low for a reliable segmentation-based image analysis. We derive a general relation between the grey-tone correlation function of the tomograms and the 2-point correlation function of the morphology, which enables us to analyse quantitatively the grey-tone correlation function with a morphological model of the material. The methodology is applied to SBA-15 ordered mesoporous silica. The three-dimensional grey-tone correlation function obtained from electron tomography is analysed in terms of a hexagonal array of Gaussian independent pores. The model enables us to relate the morphology obtained from the 2-point correlation function to macroscopic characterization data of the material, notably small-angle X-ray scattering and nitrogen adsorption

A more thorough analysis of water rockets: Moist adiabats, transient flows, and inertial forces in a soda bottle

Although water rockets are widely used to illustrate first year physics principles, accurate measurements show that they outperform the usual textbook analysis at the beginning of the thrust phase. This paper gives a more thorough analysis of this problem. It is shown that the air expansion in the rocket is accompanied by water vapor condensation, which provides an extra thrust; the downward acceleration of water within the rocket also contributes to the thrust, an effect that is negligible in other types of rockets; the apparent gravity resulting from the acceleration of the rocket contributes as much to water ejection as does the pressure difference between the inside and outside of the rocket; and the water flow is transient, which precludes the use of Bernoulli’s equation. Although none of these effects is negligible, they mostly cancel each other, and the overall accuracy of the analysis is only marginally improved. There remains a difference between theory and experiment with water rockets

Modeling the Morphology and Mechanical Behavior of Shape memory Polyurethanes based on Solid State NMR and Synchrotron SAXS/WAXD

status: publishe

Condensation-Induced Decrease of Small-Angle X-ray Scattering Intensity in Gelling Silica Solutions

peer reviewedWe propose a mathematical modeling of the total SAXS intensity in silica sol-gel processes in terms of hydrolysis and condensation reactions, as well as of microsyneresis. The results are used to rationalize previously published SAXS data of TEOS solutions reacting with organically modified trialkoxysilanes. We notably show that the decrease in SAXS intensity reported for these samples at the end of gelation is a consequence of condensation reactions. The water released by the latter reactions contributes to reduce the electron density difference between the silica and the solvent phases of the gel