Normalized Structure Factors Research Articles

A novel iterative solution to the phase problem

A new Fourier cycling phasing method is proposed based on the mathematical principle of the global minimization. In reciprocal space, the Fourier coefficient is of a mixed form of the normalized structure factors (2E(o)(2) - E(c)(2))E(c), while in direct space the Fourier map is modified with a peak-picking procedure. This method does not use any preliminary information and does not rely on any critical parameter; it can start with either randomly assigned phases or fixed phases (all zeros). This method performs significantly better than the commonly used forms of Fourier cycling.

Directed Assembly of Block Copolymers by Sparsely Patterned Substrates

Self-consistent field theory and dissipative particle dynamics simulations are used to demonstrate the controlled positioning and alignment of cylindrical domains by chemical surface patterns in an asymmetric slit. Self-assembly of the copolymer melt is simulated near dense and sparse surface patterns, both with one and two characteristic length scales. In particular, we compare the effect of a hexagonal pattern, sparse rectangular pattern, and triangle pattern. The parallel alignment of microdomains between preferentially attractive homogeneous surfaces is shown to transform into the stable perpendicular hexagonal phase in the case of the substrate patterns, which commensurate with the hexagonal morphology in the bulk. For the sparse rectangular pattern, when two different length scales of interactions are involved, and for the sparse hexagonal pattern, we observe that the self-assembly of diblocks multiplies the density of the surface patterns by a factor of 2 and 4, respectively. A 2-fold resolution enhancement was found for sparse triangle pattern with two characteristic length scales. The analysis of the circularly averaged structure factor allowed us to distinguish the structure orientation in a film. Additional peak in the region of small lateral wave numbers is attributed to undulational deformations of cylinder domains under preferable film boundaries. This theoretical work serves to rationalize modern nanolithographic fabrication of high-spatial-frequency arrays using lower-spatial-frequency templates.

Direct phasing from Patterson syntheses by δ recycling

The direct methods origin-free modulus sum function [Rius (1993). Acta Cryst. A49, 406-409] includes in its definition the structure factor G(Φ) of the squared crystal structure expressed in terms of Φ, the set of φ phases of the normalized structure factors E's of the crystal structure of unit-cell volume V. Here the simpler sum function variant S'(P) = ∑(H)E(-H)∫(V)δ(P,Δ)(Φ)exp(i2πHr)dV extended over all H reflections is introduced which involves no G's and in which the δ(P,Δ) function corresponds to δ(P) = FT(-1){(E(2)(H) - <E(2)>)exp[iφ(H)(Φ)]} (where FT = Fourier transform) with all values smaller than Δ = 2.5σ(P) equated to zero (σ(2)(P) is the variance of δ(P) calculable from the experimental intensities). The new phase estimates are obtained by Fourier transforming δ(P,Δ). This iterative phasing method (δ recycling) only requires calculation of Fourier transforms at two stages. Since δ(M) ≃ δ(P)/2, similar arguments are valid for δ(M) = FT(-1)[(E(H) - <E>)exp(iφ(H))] from which the corresponding S'(M) phasing function follows.

Theory of Absorbance Transients of an Optically Transparent Rough Electrode

Theory for the potentiostatic absorbance transient for a species generated at the rough electrode/electrolyte interface is developed. Absorbance transients are strongly affected by electrode geometries, particularly the effect of electrode roughness which is the least understood area, is dealt here. A general operator structure between concentration, surface absorbance density, absorbance transients and arbitrary roughness profile of electrode is emphasized. The statistically averaged absorbance transients is obtained by ensemble averaging over all possible surface roughness configurations. An elegant mathematical formula between the average spectroelectrochemical absorbance transient and surface structure factor or power spectrum of roughness is obtained. This formula is used to obtain an explicit equation for the absorbance on an approximately self-affine (or realistic) fractal and nonfractal electrode. Our results are also applicable to chronocoulometric response as its directly related to chronoabsorptometric response. Limiting behavior in short time, is attributed to the electrode roughness factor and surface curvatures. Limiting behavior in long time is like a smooth electrode response but the transition region is attributed to the mean square width of roughness. In the intermediate time, the absorbance has anomalous power law behavior which is attributed to the diffusion length weighted spatial frequency features of roughness and becomes almost time independent for large roughness electrodes. Finally, this theory provides a quantitative description of the role of roughness on potential step transmission chronoabsorptometry and chronocoulometric measurements, therefore opening up the way to explore and understand various anomalies in their response.

Theory of potentiostatic current transients for coupled catalytic reaction at random corrugated fractal electrode

We developed a mathematical model for the first order homogeneous catalytic chemical reaction coupled with an electron transfer (EC′) on a rough working electrode. Results are obtained for the various roughness models of electrode corrugations, viz., (i) roughness as an exact periodic function, (ii) roughness as a random function with known statistical properties, and (iii) roughness as a random function with statistical self-affine fractality over a finite range of length scales. Method of Green's function is used in the formulation to obtain second-order perturbation (in roughness profile) expressions for the concentration, the local current density and the current transients. A general operator structure between these quantities and arbitrary roughness profile is emphasized. The statistically averaged (randomly rough) electrode response is obtained by an ensemble averaging over all possible surface configurations. An elegant mathematical formula between the average electrochemical current transient and surface structure factor or power-spectrum of roughness is obtained. This formula is used to obtain an explicit equation for the current on an approximately self-affine (or realistic) fractal electrode with a limited range of length scales of irregularities. This description of realistic fractal is obtained by cutoff power law power-spectrum of roughness. The realistic fractal power-spectrum consists of four physical characteristics, viz., the fractal dimension ( D H ), lower ( ℓ ) and upper ( L) cutoff length scales of fractality and a proportionality factor ( μ), which is related to the topothesy or strength of fractality. Numerical calculations are performed on final results to understand the effect of catalytic reaction and fractal morphological characteristics on potentiostatic current transients.

Theory of Generalized Cottrellian Current at Rough Electrode with Solution Resistance Effects

A theory for effect of uncompensated solution resistance on Nernstian (reversible) charge transfer at an arbitrary rough electrode is developed. The significant deviation from the Cottrellian behavior is explained, as arising from the resistivity of the solution and geometric irregularity of the interface. Results are obtained for various electrode roughness models, viz., (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, and (iii) random functions with limited self-affine fractal properties. Expressions for concentration, current density, and total current transients have a systematic operator structure in Fourier transformed (deterministic) surface profile function. For a randomly rough electrode, the statistically averaged response is obtained by ensemble averaging over all possible surface configurations. An elegant mathematical formula between the average electrochemical current transient and surface structure factor of roughness is obtained. Realistic fractal roughness is characterized as self-affine scaling property over limited length scales. Limiting behavior in short time, is attributed to resistive effects of the electrolyte, roughness factor and surface curvatures. In the intermediate time, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness. Our results help with the quantitative understanding of generalized Cottrellian response of moderately supported electrolytic solution at rough electrode/electrolyte interface.

The difference electron density: a probabilistic reformulation

The joint probability distribution function P(E, E(p)), where E and E(p) are the normalized structure factors of the target and of a model structure, respectively, is a fundamental tool in crystallographic methods devoted to crystal structure solution. It plays a central role in any attempt for improving phase estimates from a given structure model. More recently the difference electron density rho(q) = rho - rho(p) has been revisited and methods based on its modifications have started to play an important role in combination with electron density modification approaches. In this paper new coefficients for the difference electron density have been obtained by using the joint probability distribution function P(E, E(p), E(q)) and by taking into account both errors in the model and in measurements. The first applications show the correctness of our theoretical approach and the superiority of the new difference Fourier synthesis, particularly when the model is a rough approximation of the target structure. The new and the classic difference syntheses coincide when the model represents the target structure well.

Scaling behavior of a wormlike polyelectrolyte chain by mesoscopic Brownian dynamics simulations

The scaling prediction of semiflexible wormlike chain was examined by applying Brownian dynamics simulation, which goes beyond other simulations as they do not consider both the hydrodynamic interaction between pairs of beads and long-range screening effect. The rheological behavior of the intrinsic viscosity was properly im- plemented by combining with optimized model parameters for the polyelectrolyte xanthan, and the validity of the sim- ulation was previously confirmed. Scaling plots present that the structure and diffusion of polyelectrolyte chains depend sensitively on the Debye screening effect. In the scaling of end-to-end distance RE, radius of gyration RG, and trans- lational diffusivity DT with respect to the number of beads Nb, the Flory-Edwards exponent ν was estimated as up to 1.0, which should be a really higher level compared to the case of the flexible neutral chains with self-avoiding walk in good solvent. Unlike the case of spherically averaged structure factor, scaling plots in that parallel to the first principal axis of gyration could not clearly be identified with a well defined exponent. With increasing screening effect, pro- nounced oscillations observed in the case of lower screening tend to smear out.

Atomistic simulations for the non-equilibrium surface premelting and melting of Nb(1 1 0) plane

In the present paper molecular dynamics (MD) simulations have been preformed to investigate the surface melting process and microscopic mechanism of Nb(1 1 0) plane in the atomic scale with a modified analytic embedded atom method (MAEAM). On the basis of the MD relaxation dependence of averaged internal energy and layer structure factor at given temperatures, the melting point of the sample has been estimated to be 2510 K. Then by the above results the Nb(1 1 0) plane melting process has been approximately divided into two stages: first the layer-by-layer premelting phase in the surface region and then a simultaneous abrupt melting transition for the inner layers. According to the variation of the averaged internal energy of the inner atomic layer, the melting latent heat has been calculated and the result is in good agreement with the experimental value. The simulated snapshots of atomic configuration for Nb(1 1 0) plane have indicated that the dynamically microscopic mechanism of melting nucleation during the melting transition.

Microstructure and Stability of a Lamellar Liquid Crystalline and Gel Phase Formed by a Polyglycerol Ester Mixture in Dilute Aqueous Solution

The self-assembly behavior of a commercial mixture of polyglycerol fatty acid esters (PGE) and water is investigated as a function of temperature and surfactant content. The phase diagram of this pseudo-binary mixture was characterized using a combination of cross-polarized light and freeze-fracture electron microscopy (cryo-SEM), X-ray diffraction (XRD), small-angle neutron scattering (SANS), and differential scanning calorimetry (DSC). Our experiments show that the morphology of the supramolecular aggregates is lamellar and present in the form of a continuous or dispersed phase (multilamellar vesicles) depending on the water content of the system. Under the effect of temperature, the short- and long-range order of the bimolecular layers successively changes from a biphasic surfactant dispersion to a lamellar liquid-crystalline (Lalpha) and a stable lamellar gel phase (Lbeta) upon cooling; this transition is found to be irreversible. Formation of the lamellar aggregates can be related to the average molecular structure and shape factor of PGE. The stability of the resulting gel phase (Lbeta) appears to be due to the presence of small amounts of unreacted ionic co-surfactant, namely, fatty acid soaps, in this per se nonionic commercial mixture.

Effect of Temperature and Surfactant Type on the Stability and Aggregation Behavior of Styrene−Acrylate Copolymer Colloids

The colloidal stability, aggregation kinetics, and cluster structure of two styrene-acrylate copolymer latexes, stabilized with an aliphatic sulfonate and an aliphatic carboxylate surfactant, respectively, have been investigated experimentally in the temperature range between 283 and 323 K. The main objective of this study is to investigate the role of temperature and surfactant type on the aggregation kinetics and cluster structure. For this, the values of the Fuchs stability ratio and the time evolutions of the average radius of gyration, hydrodynamic radius, and structure factor of the clusters have been determined using static and dynamic light scattering techniques at different temperatures. It is found that although the two latexes exhibit a somewhat different dependence of the colloidal stability on temperature, all of the values of the average radius of gyration (or hydrodynamic radius) measured at different temperatures and surfactant types, which are plotted as a function of a properly defined dimensionless time, collapse to form a single master curve. Similarly, all of the measured average structure factors also collapse to form a single master curve when they are plotted as a function of the wavevector normalized using the average radius of gyration. These results indicate that, at least for the conditions investigated in this work, the aggregation mechanism and cluster structure are independent of temperature and surfactant type.

Quantum Monte Carlo modeling of the spherically averaged structure factor of a many-electron system

The interaction and exchange-correlation contributions to the ground-state energy of an arbitrary many-electron system can be obtained from a spherical average of the wave-vector-dependent diagonal structure factor (SF). We model the continuous-k spherically averaged SF using quantum Monte Carlo calculations in finite simulation cells. We thus derive a method that allows us to substantially reduce the troublesome Coulomb finite-size errors that are usually present in ground-state energy calculations. To demonstrate this, we perform variational Monte Carlo calculations of the interaction energy of the homogeneous electron gas. The method is, however, equally applicable to arbitrary inhomogeneous systems.

Discontinuous molecular dynamics simulation study of polymer collapse

Discontinuous molecular dynamics simulations were used to study the coil-globule transition of a polymer in an explicit solvent. Two different versions of the model were employed, which are differentiated by the nature of monomer-solvent, solvent-solvent, and nonbonded monomer-monomer interactions. For each case, a model parameter lambda determines the degree of hydrophobicity of the monomers by controlling the degree of energy mismatch between the monomers and solvent particles. We consider a lambda-driven coil-globule transition at constant temperature. The simulations are used to calculate average static structure factors, which are then used to determine the scaling exponents of the system in order to determine the theta-point values lambda(theta) separating the coil from the globule state. For each model we construct coil-globule phase diagrams in terms of lambda and the particle density rho. Additionally, we explore for each model the effects of varying the range of the attractive interactions on the phase boundary separating the coil and globule phases. The results are analyzed in terms of a simple Flory-type theory of the collapse transition.

Structural properties of molten dilute aluminium–transition metal alloys

The short-range order in liquid binary Al-rich alloys (Al–Fe, Al–Ti) was studied by x-raydiffraction. The measurements were performed using a novel containerless technique whichcombines aerodynamic levitation with inductive heating. The average structure factors,S(Q),have been determined for various temperatures and compositions in the stable liquid state. FromS(Q), the pair correlationfunctions, g(r), have been calculated. The first interatomic distance is nearly temperature-independent,whereas the first-shell coordination number decreases with increasing temperature for allthe alloys investigated. For the Al–Fe alloys, room-temperature scanning electronmicroscropy (SEM) studies show the formation of a microstructure, namely the existence ofAl13Fe4 inclusions in the Al matrix.

Simulated annealing study of gyroid formation in diblock copolymer solutions

Conditions for the formation of gyroid structures in diblock copolymer solutions are examined using a simulated annealing technique. The simulations were performed on diblock copolymer systems of A(NA)-b-B(NB) (with NA<NB) in solvents that are selective to the A blocks. It is shown that gyroid structures form in a narrow range of block copolymer concentrations between the hexagonally packed cylindrical and the lamellar phases and at an almost constant B-monomer concentration. It is also shown that the gyroid structure is especially sensitive to the B-solvent interaction (epsilonBS) and the length of the B block (NB). Phase diagrams for the diblock copolymer solutions are constructed. These predicted results are consistent with previous experimental observations. The three-dimensional isosurface contour plots of the simulated gyroid structure shows two interpenetrating strut networks. The projection along the [111] direction of the simulated gyroid structure and the spherically averaged structure factor are in good agreement with previous experimental results.

Investigation of aggregation, breakage and restructuring kinetics of colloidal dispersions in turbulent flows by population balance modeling and static light scattering

Quantitative modeling of aggregating colloidal systems is the underlying problem in many industrial processes, such as micro and nanoparticle processing, crystallization or flocculation. Population balance models with various aggregation and breakage kernels have been proposed in order to describe aggregating systems, but they have been rarely validated against appropriate experimental data. Typically, model parameters are fitted against a single measured moment of the cluster distribution which can usually be equivalently described using several variations of the set of parameters underlying the relevant aggregation, breakage and restructuring kernels. In order to discriminate among alternative models we propose an approach based on measurement and quantitative modeling of multiple moments of the cluster mass distribution, such as those obtained from static light scattering measurements. This approach is applied to aggregation processes in turbulent conditions in order to test alternative kernels for aggregation, breakage, and restructuring kinetics. We present a detailed study on the sensitivity of measurables from static light scattering with respect to commonly used aggregation and breakage kinetic models. In particular, we analyze the dynamic and steady state behavior of two measurables: the average radius of gyration and the average zero angle intensity which represent two independent moments of the cluster mass distribution. In addition, we discuss the effect of cluster structure and mass distribution on the average structure factor and the apparent fractal dimension measured by static light scattering, in order to assess what structural information can be reliably extracted from such measurements.

Pressure-induced amorphization, crystal-crystal transformations, and the memory glass effect in interacting particles in two dimensions

We study a model of interacting particles in two dimensions to address the relation between crystal-crystal transformations and pressure-induced amorphization. On increasing pressure at very low temperature, our model undergoes a martensitic crystal-crystal transformation. The characteristics of the resulting polycrystalline structure depend on defect density, compression rate, and nucleation and growth barriers. We find two different limiting cases. In one of them the martensite crystals, once nucleated, grow easily perpendicularly to the invariant interface, and the final structure contains large crystals of the different martensite variants. Upon decompression almost every atom returns to its original position, and the original crystal is fully recovered. In the second limiting case, after nucleation the growth of martensite crystals is inhibited by energetic barriers. The final morphology in this case is that of a polycrystal with a very small crystal size. This may be taken to be amorphous if we have only access (as experimentally may be the case) to the angularly averaged structure factor. However, this `X-ray amorphous' material is anisotropic, and this shows up upon decompression, when it recovers the original crystalline structure with an orientation correlated with the one it had prior to compression. The memory effect of this X-ray amorphous material is a natural consequence of the memory effect associated to the underlying martensitic transformation. We suggest that this kind of mechanism is present in many of the experimental observations of the memory glass effect, in which a crystal with the original orientation is recovered from an apparently amorphous sample when pressure is released.

Scaling of the Kinetics of Slow Aggregation and Gel Formation for a Fluorinated Polymer Colloid

The aggregation and gelation kinetics in moderately concentrated (0.004 </= phi(0) </= 0.1) colloidal dispersions of fluorinated polymer particles has been studied. The aggregation was adjusted to proceed slowly enough to allow a convenient characterization of the kinetics through static and dynamic light scattering on quenched and diluted samples. A population balance model based on second-order aggregation rates is developed to compute the time evolution of the cluster mass distribution, from which we calculate the values of the average radii and structure factor measured by light scattering, so as to allow a direct comparison between measured and calculated quantities. The model suggests the introduction of a dimensionless time which allows the scaling of all the aggregation data on unique master curves defined by only two parameters: the exponent of the power-law aggregation kernel, lambda, and the aggregate fractal dimension, d(f). The predicted master curves were observed experimentally, which confirms the validity of the aggregation model and allows the unique determination of the kinetic and structural parameters of the aggregation process. The cluster growth behavior, although significantly slower than DLCA, shows power-law kinetics rather than the exponential one typical of RLCA and the cluster structure is characterized by an unexpectedly small fractal dimension, d(f) = 1.7. The occurrence of gelation has been characterized using small amplitude oscillatory shearing to monitor the time evolution of the elastic modulus. It is found that also these curves, together with the gel time value, scale with the stability ratio of primary particles for a given solid volume fraction. We further use the model to calculate the cumulative occupied volume fraction of the growing aggregates and quantify in this way the increasing space filling, which is solid volume fraction dependent. The experimentally determined dimensionless gel times, which are also solid volume fraction dependent, scale then directly with the dimensionless time to reach a certain degree of the space filling in the model. This finding suggests that, like the aggregation kinetics, the gelation kinetics is governed by a second-order rate process.

On the Theory of Diffraction of Light in Photonic Crystals with Allowance for Interlayer Disordering

Electrodynamic Green’s functions are used to construct an analytical theory of the Bragg diffraction of polarized light in photonic crystals having a close-packed structure. For opal-based photonic crystals, the Bragg diffraction intensity is calculated with allowance for permittivity periodic modulation and for the presence of an optical crystal boundary and interlayer disordering, which usually appears during sample growth. A comprehensive study is made of the effect of the structure disorder caused by the random packing of growth layers on diffraction. For a random constructed twinned fcc structure, the average structure factor and the scattering (diffraction) cross sections (which are dependent on the linear polarization of the incident and scattered waves) are calculated. Numerical examples are used to show that the theory developed can be applied to analyze and process experimental diffraction patterns of real photonic crystals having a close-packed structure disordered in one direction.

Simulation study of the coil-globule transition of a polymer in solvent

Molecular dynamics simulations are used to study the coil-globule transition for a system composed of a bead-spring polymer immersed in an explicitly modeled solvent. Two different versions of the model are used, which are differentiated by the nature of monomer-solvent, solvent-solvent, and nonbonded monomer-monomer interactions. For each case, a model parameter lambda determines the degree of hydrophobicity of the monomers by controlling the degree of energy mismatch between the monomers and solvent particles. We consider a lambda-driven coil-globule transition at constant temperature. The simulations are used to calculate average static structure factors, which are then used to determine the scaling exponents of the system in order to determine the theta-point values lambdatheta separating the coil from the globule states. For each model we construct coil-globule phase diagrams in terms of lambda and the particle density rho. The results are analyzed in terms of a simple Flory-type theory of the collapse transition. The ratio of lambdatheta for the two models converges in the high density limit exactly to the value predicted by the theory in the random mixing approximation. Generally, the predicted values of lambdatheta are in reasonable agreement with the measured values at high rho, though the accuracy improves if the average chain size is calculated using the full probability distribution associated with the polymer-solvent free energy, rather than merely using the value obtained from the minimum of the free energy.