Nanoscale structural order from the atomic pair distribution function (PDF): There's plenty of room in the middle

Emerging materials of scientific and technological interest are generally complex and often nanostructured: they have atomic orderings that extend on nanometer length-scales. These can be discrete nanoparticles; bulk crystals with nanoscale chemical or displacive order within them; mesoporous materials that are bulk materials containing nanoscale holes; and nanocomposites that are intimate heterogeneous mixtures of nano-sized constituents. As always, a quantitative knowledge of the atomic structure within these materials is a prerequisite to understanding and engineering their properties. Traditional crystallographic methods for obtaining this information break down at the nanoscale, sometimes referred to as “the nanostructure problem”. We describe here some emerging methods for studying nanoscale structure. We present some examples of recent successes. Finally, we discuss future directions and opportunities and draw attention to limitations and potential problems.     

Synthesis and structural characterisation using Rietveld and pair distribution function analysis of layered mixed titanium-zirconium phosphates

Crystalline metal (IV) phosphates with variable zirconium-to-titanium molar ratios of general formula (Ti_1−xZr_x)(HPO₄)₂·H₂O have been prepared by precipitation of soluble salts of the metals with phosphoric acid and heating the amorphous solids in 12 M H₃PO₄ in an autoclave. The new materials are structurally characterised by Rietveld analysis of synchrotron X-ray powder diffraction data and pair distribution function (PDF) analysis of high energy synchrotron X-ray total scattering data. A broad range of zirconium-titanium phosphate solid solutions were formed showing isomorphous substitution of titanium by zirconium in the α-titanium phosphate lattice and vice versa for titanium substitution into the α-zirconium phosphate lattice. In both cases the solubility is partial with the coexistence of two substituted phases observed in samples with nominal compositions between the solubility limits.     

Quantum correction to the pair distribution function

We report a numerical technique that allows the quantum effects of zero-point motion to be incorporated into Pair Distribution Functions calculated classically for molecules using Monte Carlo or Molecular Dynamics simulations. We establish the basis for this approximation using a diatomic molecule described by a Morse potential. The correction should significantly improve the agreement between modeled and experimental data, and facilitate conclusions about inter- and intra-molecular motion and flexibility. We describe a similar approach to obtain the energy and the specific heat.     

Structural and mechanistic revelations on an iron conversion reaction from pair distribution function analysis

Not simply small particles: pair distribution function analysis yields comprehensive insights into the electrochemical reaction of α-Fe(2)O(3) with lithium. The metallic Fe formed in this reaction was found to be defect-rich nanoparticles that restructure continuously without growing-an unusual characteristic likely linked to its highly reversible capacity.     

Improving the picture of atomic structure in nonoriented polymer domains using the pair distribution function: A study of polyamide 6

Investigations of the atomic structures within polyamides started over 80 years ago and continue today. These weakly ordered materials diffract X-rays poorly and typically require postprocessing to obtain idealized samples for structural studies. An important goal remains to develop techniques to study the local structure in its natural state, with atomic resolution, and with sensitivity to subtle changes due to synthesis conditions or other technologically relevant processing procedures. Here, we compare the structures of as-produced, nonoriented polyamide 6 ([C₆H₁₁NO] _n) from both hydrolytic and anionic processes. A total scattering pair distribution function approach is used to elucidate information about the atomic bonding, molecular conformation, chain packing, crystallite size, and ratio of ordered to disordered domain content. The results are compared with those from standard analytical methods.     

The analysis of electron pair distribution functions in molecules

Electron pair distribution functions are analyzed for a variety of SCF+CI wavefunctions, for a range of simple molecules. The statistical correlation between electrons of like spin introduced by the antisymmetry requirement on the many-electron wavefunction is contrasted with the manner in which unlike-spin electron correlation is introduced through the inclusion of configuration interaction.     

Schulz distribution function and the polydispersity of the binary suspension of charged macroions

Bidispersity of binary suspensions of charged macroions due to different sizes and charges are reduced into one-component model (OCM) using Schulz distribution function. Ordering in charged macroions has been studied using rescaled mean spherical approximation (RMSA) method with modified Derjaguin, Landau, Verwey and Overbeek (DLVO) potential. The results obtained are compared with the experiment, weight-average and Roger–Young (RY) schemes. It is inferred that Schulz distribution function is a plausible model to average out size and charges of macroions to study the structural behavior of the binary suspension of macroions. An ordering with co-ordination number 12 has been reported in the binary suspension of charged macroions.     

The mechanism of perylene photo-oxidation in a water-soluble polymeric photocatalyst

The photo-oxidation of perylene in aqueous solutions of a polymeric photocatalyst was investigated to probe the mechanism of polycyclic aromatic hydrocarbon degradation. Perylene and other hydrophobic molecules are efficiently solubilized in aqueous polymer solutions with distribution coefficients as high as 4 x 10⁶. The rate of perylene photo-oxidation was much more rapid in aqueous polymer solutions than in organic solvents. In organic solvents, ¹0₂ sensitizers (rose bengal) had little effect on the reaction, but electron acceptors, such as dicyanobenzene, caused an acceleration in rate. Naphthoquinone was suggested as a potential electron acceptor in the naphthalene-containing polymer, and it was shown to be formed in small concentrations by polymer oxidation. It was concluded that the polymer plays several key functions in perylene photo-oxidation: (1) solubilization of the hydrophobic molecule; (2) energy migration through the polymer coil and energy transfer, providing additional photochemical energy to the reactants; (3) the enhancement of oxidation by photoinduced electron transfer via provision of an electron acceptor and facilitation of charge separation.     

Evaluation of high-frequency elastic moduli and shear relaxation time of the Lennard-Jones fluid using three known analytical expressions for radial distribution function

High-frequency elastic moduli (G_∞ and K_∞) for a Lennard-Jones (LJ) fluid have been calculated using three known analytical expressions for radial distribution function (RDF) at different temperatures and densities and compared with the corresponding values of these properties obtained from molecular dynamics (MD).     

Elasticity of syndiotactic polypropylene: Insights from temperature and time dependence

The origin of the unusual and puzzling elasticity of drawn sPP samples was investigated. The mechanism responsible of the elasticity was studied for drawn samples characterized by a very simple structural organization, where there are no involvements of crystallographic modifications with different chain conformation. The elastic behavior of the drawn samples, valued through the hysteresis cycles, was determined at different temperatures. At room temperature the samples show remarkable elastic properties, whereas decreasing the temperature the elastic behavior becomes worse and worse, disappearing at temperatures lower than 0 °C. The elasticity also disappears in drawn samples after a long aging under tension. Thermomechanical and structural investigations, as well as shrinkage as a function of temperature and aging at room temperature of the fixed drawn samples, support the idea that the elastic behavior of sPP can be explained by the model of the plastic deformation of semicrystalline polymers. The interpretation is based on the presence of “tie” molecules axially connecting the crystals in the oriented samples, whose extension, chain conformation and/or crystallization determine the retractive stress of the oriented sample, as well as other mechanical properties. We show that many experiments on the drawn samples, either fixed or relaxed, are strictly connected to the morphology of the drawn sample, derived by the transition between the lamellar initial and the fibrillar final structure. By applying the model we can answer the questions derived from the experimental facts not yet well clarified, giving a new insight into the interesting elasticity of sPP.     

Approach on Tsallis statistical interpretation of hydrogen-atom by adopting the generalized radial distribution function

This paper revisits the statistical interpretation of the hydrogen atom within the framework of Tsallis Statistical Mechanics in the Canonical Ensemble. The convergence of the partition function does not exhibit for all the temperatures, while the well-known T → T′ transformation method of Tsallis Statistics fails, since non-monotonicity is observed between the ordinary temperature, T, and the auxiliary one, T′. Here we re-examine the inconsistency of T → T′ transformation method, in the case where the partition function converges for all the temperatures, by considering the generalized radial distribution function. We find that both the transformation method inconsistency and the partition function divergence can be recovered for all the temperatures, if the hydrogen atom is restricted within a critical radius R _c  ≤ 4.832 bohr, while Tsallis entropic index values are given by . An erratum to this article can be found at     

Preparation of polystyrene particles with narrow particle size distribution by γ-ray initiated miniemulsion polymerization stabilized by polymeric surfactant

Co-60 γ-ray initiated miniemulsion polymerization of styrene stabilized by an alkali soluble polymeric surfactant (ASPS) was studied in this work. The affecting factors, including absorbed dose, dose rate, surfactant concentration, and hexadecane (HD) concentration, were systematically studied. The particle size and particle size distribution (PSD) of final latexes was determined by transmission electron microscope (TEM). The results indicated that polystyrene (PS) particles with narrow PSD could be easily prepared by this method, and the particle size could be controlled from 50 nm to 250 nm by adjusting the concentration of HD and ASPS. HD concentration, surfactant concentration, dose rate and total absorbed dose strongly affected the particle size and PSD of PS latexes.     

Monte Carlo investigations of dense copolymer systems, 1. Pivot algorithm and pair distribution function

ABA triblock copolymers (the solvent being a θ-solvent for B and an athermal one for A) as well as their constituent homopolymers were investigated by means of Monte Carlo simulations in a 5-choice cubic lattice. The polymer concentration (represented by the fraction ϕ of sites occupied by the polymer) varied from 0 (isolated chain) to 0.80. An investigation of the usefulness of the pivot algorithm in dense systems resulted in reasonable acceptance fractions up to a volume fraction of 0.12 (athermal chain and ABA copolymer) and 0.05 (θ-chain). For larger volume fractions only the chain-ends remain mobile. The overlap concentration of the polymers defined by several quantities was approximately 0.07 (athermal chain), 0.12 (θ-chain) and 0.09 (ABA copolymer). At a polymer fraction of 0.32, the chains had the same number of inter- and intramolecular contacts on average. At higher concentrations, the behaviour of the chains was primarily determined by intermolecular interactions. Contrary to isolated pairs, the pair distribution function g(r) of two athermal chains in a (cubic) box exceeded unity at intermediate distances because of the influence of the finite size of the box. The larger the size of the box, the smaller the (positive) deviation was. In the limit of r → 0 the pair distribution function g(r) – being smaller than unity – increased with increasing concentration while the maximum at intermediate distances simultaneously decreased. Ultimately, at the highest concentration, the pair distribution function resembled that of isolated θ-chains. The concentration dependence of the θ-pair distribution function itself, however, is negligible. At low concentrations the pair distribution function of the ABA triblock copolymer behaved like that of an athermal chain (up to ϕ ≈ 0.40) where characteristic oscillations around 1 developed. This might be taken as indicative of the formation of a lamellar phase.     

Parametrization of the radial distribution function of a Yukawa fluid

We give semi-empirical expressions for five terms in an inverse temperature expansion of the radial distribution function of a Yukawa fluid. The parameters in these terms are chosen to fit what we believe should be an accurate equation of state when either the energy or pressure routes is used. Thus, a measure of self consistency is achieved. The equation of state which is the basis of our fit is the inverse temperature expansion of the Yukawa fluid free energy, obtained from the mean spherical approximation but modified to give reasonable results at low densities.     

用指数函数拟合长链分子超极化率时目标函数的选择

长链体系的轴向单元(超)极化率的饱和值Δp(∞),需要对有限的计算数据用函数拟合,再根据拟合函数外推求得。以a-be-cn作为拟合函数,有两拟合公式:p(n)-p(n-1)=a-be-cn和p(n)/n=a-be-cn。从理论上探讨了两公式目标函数不同存在的本质区别,结论是应该使用p(n)-p(n-1)作为目标函数,即合理的公式是前者。并用氢分子链,氟化氢分子链和聚乙烯为例,表明后者在链增长时没有正确的渐近行为,原因是当后者还原为对p(n)的拟合函数时含有一个ne-cn的项,而这一项会影响拟合后外推的饱和值的正确性。     

Self-consistent field equations in the distribution function theory of polymeric liquids

In the distribution function approach to the conformational and thermodynamic properties of polymeric liquids site-site (pair) distribution functions are essential components of the theory. These site-site pair distribution functions are basically mean fields obeying integral equations. In our recent works, a set of self-consistent field equations has been proposed for site-site pair correlation functions which allow us to study conformational and thermodynamic properties of polymeric liquids. In this article, we present a short review of the theory and its applications to a number of aspects of polymeric liquids we have made until now. We also present a self-consistent version of the polymer reference interaction site model where the integral equations for the intramolecular site-site correlation functions are obtained from the Kirkwood hierarchy on the basis of the present theory. The present theory is shown to predict correctly the scaling properties associated with swollen and collapsed polymers in good and poor solvents, respectively. At finite densities, self-consistent solutions of the intra- and intermolecular equations yield the structures and thermodynamics of polymer melts which are favorably compared with Monte Carlo simulation results. Self-consistent theory results are found to be more accurate than the non-self-consistent approaches that use an ideal Gaussian chain conformation distribution function. © 1995 John Wiley & Sons, Inc.     

Dynamics simulation on the associative properties of amphiphilic functional monomer modified polyacrylamide copolymers

A kind of amphiphilic functional monomer was selected to modify polyacrylamide(PAM) or partially hydrolyzed polyacrylamide(HPAM). The relative properties of the modified polyacrylamide(HM-PAM) and modified partially hydrolyzed polyacrylamide(HM-HPAM) such as radius of gyration(Rg), hydrodynamic radius(RH), and radial distribution functions(RDFs) have been studied to find the intrinsic relation between the microstructure of the polymer chain and the intrinsic viscosities with changing the amount of modified monomers from 1% to 4%. The simulation results show that, compared to HPAM, HM-HPAM has a better performance in increasing viscosity when the percentage of modified monomers is 2% and has a stronger salt tolerance when the modified monomers is 4%. Furthermore, a complex hydrogen bonding network was revealed with the analysis of radial distribution functions(RDFs) and the pair correlation function was used to investigate the diffusivity of Na+ and carbon atoms in the COO― group.     

Relaxation experiments of polymeric systems confined in very narrow gaps

A new way of investigating the relaxation processes of polymeric systems confined between solid surfaces is presented here. The test is based on draining experiments by means of a surface force apparatus and consists in monitoring the hydrodynamic force release following a drainage motion. Experiments carried out with solutions of high polymers exhibit two distinct relaxation processes. The longest one is connected to the relaxation of the chains adsorbed onto the solid surfaces and which are pinned in the area of closest distance between the solid surfaces. Its variations as a function of spacing are consistent with the bridging of some macromolecules. The fastest process is connected to the flow of the solvent molecules through the pseudonetwork formed by the adsorbed layers carried by the solid surfaces. These results have been compared favorably with those obtained by oscillatory measurements as far as relaxation time and viscosity are concerned. The accuracy of the experimental relaxation function is not sufficient for describing reasonably the viscoelastic behavior of the confined fluid. © 1996 John Wiley & Sons, Inc.     

Application of interface distribution functions to the evaluation of the small angle x-ray scattering of multicomponent polymer systems

A method for correcting the small angle x-ray intensity of multi-component polymer systems to compute the interface distribution function is presented. The resolution of this function can be improved to a great extent by evaluating the interference function from the measured intensity corrected for fluctuation scattering by use of a function which is proportional to the gas scattering of the investigated system. The method is applied successfully to polymer blends composed of components having similar morphology.Presented at the Spring Meeting of the Deutsche Physikalische Gesellschaft, Hamburg, March 14–16, 1988.     

Recent advances in the formation of ultrathin polymeric membranes for gas separations

During the past 20 years membrane systems have been applied to a limited number of commercial gas separations. To further advance membrane-based gas separations, current research efforts focus on optimization of (i) membrane materials, (ii) membrane structures and (iii) membrane system design. In this overview, recent developments in the formation of high-performance gas separation membranes are discussed. The gas separation properties of state-of-the-art integrally skinned asymmetric membranes and thin-film composite membranes are summarized. Future directions for the preparation of advanced gas separation membranes are highlighted.