Percus-Yevick Model Research Articles

Electron scattering and transport in simple liquid mixtures

The theory for electron transport in simple liquids developed by Cohen and Lekner (1967 Phys. Rev. 158 305), is extended to simple liquid mixtures. The focus is on developing benchmark models for binary mixtures of hard-spheres, using the Percus-Yevick model (Lebowitz 1964 Phys. Rev. A 133 895, Hiroike 1969 J. Phys. Soc. Japan 27 1415) to represent the density structure effects. A multi-term solution of Boltzmann’s equation is employed to investigate the effect of the binary mixture structure on hard-sphere electron scattering cross-sections and transport properties, including the drift velocity, mean energy, longitudinal and transverse diffusion coefficients. Benchmark calculations are established for electrons driven out of equilibrium by a range of reduced electric field strengths ( 0.1−100 Td).

Electron transport and negative streamers in liquid xenon

In this work we investigate electron transport, transition from an electron avalanche into a negative streamer, and propagation of negative streamers in liquid xenon. Our standard Monte Carlo code, initially developed for dilute neutral gases, is generalized and extended to consider the transport processes of electrons in liquids by accounting for the coherent and other liquid scattering effects. The code is validated through a series of benchmark calculations for the Percus–Yevick model, and the results of the simulations agree very well with those predicted by a multi term solution of Boltzmann’s equation and other Monte Carlo simulations. Electron transport coefficients, including mean energy, drift velocity, diffusion tensor, and the first Townsend coefficient, are calculated for liquid xenon and compared to the available measurements. It is found that our Monte Carlo method reproduces both the experimental and theoretical drift velocities and characteristic energies very well. In particular, we discuss the occurrence of negative differential conductivity in the E/n0 profile of the drift velocity by considering the spatially-resolved swarm data and energy distribution functions. Calculated transport coefficients are then used as an input in fluid simulations of negative streamers, which are realized in a 1.5 dimensional setup. Various scenarios of representing the inelastic energy losses in liquid xenon, ranging from the case where the energy losses to electronic excitations are neglected, to the case where some particular excitations are taken into account, and to the case where all electronic excitations are included, are discussed in light of the available spectroscopy and photoconductivity experiments. We focus on the way in which electron transport coefficients and streamer properties are influenced by representation of the inelastic energy losses, highlighting the need for the correct representation of the elementary scattering processes in the modeling of liquid discharges.

USAXS analysis of concentration-dependent self-assembling of polymer-brush-modified nanoparticles in ionic liquid: [I] concentrated-brush regime.

Using ultra-small angle X-ray scattering (USAXS), we analyzed the higher-order structures of nanoparticles with a concentrated brush of an ionic liquid (IL)-type polymer (concentrated-polymer-brush-modified silica particle; PSiP) in an IL and the structure of the swollen shell layer of PSiP. Homogeneous mixtures of PSiP and IL were successfully prepared by the solvent-casting method involving the slow evaporation of a volatile solvent, which enabled a systematic study over an exceptionally wide range of compositions. Different diffraction patterns as a function of PSiP concentration were observed in the USAXS images of the mixtures. At suitably low PSiP concentrations, the USAXS intensity profile was analyzed using the Percus-Yevick model by matching the contrast between the shell layer and IL, and the swollen structure of the shell and "effective diameter" of the PSiP were evaluated. This result confirms that under sufficiently low pressures below and near the liquid/crystal-threshold concentration, the studied PSiP can be well described using the "hard sphere" model in colloidal science. Above the threshold concentration, the PSiP forms higher-order structures. The analysis of diffraction patterns revealed structural changes from disorder to random hexagonal-closed-packing and then face-centered-cubic as the PSiP concentration increased. These results are discussed in terms of thermodynamically stable "hard" and/or "semi-soft" colloidal crystals, wherein the swollen layer of the concentrated polymer brush and its structure play an important role.

Approximating the nonlinear density dependence of electron transport coefficients and scattering rates across the gas–liquid interface

This study reviews the neutral density dependence of electron transport in gases and liquids and develops a method to determine the nonlinear medium density dependence of electron transport coefficients and scattering rates required for modeling transport in the vicinity of gas–liquid interfaces. The method has its foundations in Blanc’s law for gas-mixtures and adapts the theory of Garland et al (2017 Plasma Sources Sci. Technol. 26) to extract electron transport data across the gas–liquid transition region using known data from the gas and liquid phases only. The method is systematically benchmarked against multi-term Boltzmann equation solutions for Percus–Yevick model liquids. Application to atomic liquids highlights the utility and accuracy of the derived method.

Monte Carlo study of coherent scattering effects of low-energy charged particle transport in Percus-Yevick liquids.

We generalize a simple Monte Carlo (MC) model for dilute gases to consider the transport behavior of positrons and electrons in Percus-Yevick model liquids under highly nonequilibrium conditions, accounting rigorously for coherent scattering processes. The procedure extends an existing technique [Wojcik and Tachiya, Chem. Phys. Lett. 363, 381 (2002)], using the static structure factor to account for the altered anisotropy of coherent scattering in structured material. We identify the effects of the approximation used in the original method, and we develop a modified method that does not require that approximation. We also present an enhanced MC technique that has been designed to improve the accuracy and flexibility of simulations in spatially varying electric fields. All of the results are found to be in excellent agreement with an independent multiterm Boltzmann equation solution, providing benchmarks for future transport models in liquids and structured systems.

On the approximation of transport properties in structured materials using momentum-transfer theory

In this paper, we present a fluid model for electrons and positrons in structured and soft-condensed matter utilizing dilute gas phase cross-sections together with a structure factor for the medium. Generalizations of the Wannier energy and Einstein (Nernst–Townsend) relations to account for coherent scattering effects present in soft-condensed matter are presented along with new expressions directly relating transport properties in the dilute gas and the structured matter phases. The theory is applied to electrons in a benchmark Percus–Yevick model and positrons in liquid argon, and the accuracy is tested against a multi-term solution of Boltzmann's equation (White and Robson 2011 Phys. Rev. E 84 031125).

Multiterm solution of a generalized Boltzmann kinetic equation for electron and positron transport in structured and soft condensed matter

In this paper, we generalize the semiclassical Boltzmann kinetic equation for dilute gases to consider highly nonequilibrium electrons and positrons in soft condensed matter, accounting rigorously for all types of interactions, including positronium formation, and allowing for both coherent and incoherent scattering processes. The limitations inherent in the seminal paper of Cohen and Lekner [M. H. Cohen and J. Lekner, Phys. Rev. 158, 305 (1967); Y. Sakai, J. Phys. D 40, R441 (2007)] are avoided by solving the kinetic equation using a "multiterm" spherical harmonic representation of the velocity distribution function, as well as formulating a necessarily nonperturbative treatment of nonconservative collisional processes such as positronium formation. Numerical calculations of transport properties are carried out for a Percus-Yevick model of a hard-sphere system, and for positrons in liquid argon. New phenomena are predicted, including structure-induced negative conductivity and anisotropic diffusion.

Small-angle neutron scattering study of nanophase separated epoxy hydrogels

Abstract A series of hydrophilic epoxy networks was prepared by end-linking reaction of α,ω-diamino terminated poly(oxypropylene)-block-poly(oxyethylene)-block-poly(oxypropylene) of average molar mass ca. 600 g mol−1 and diglycidyl ether of Bisphenol A propoxylate at various values of stoichiometric ratio, r, of amino to epoxy groups. Epoxy hydrogels were obtained by swelling of the epoxy networks in deuterated water. Nanophase separated structure of all hydrogels has been revealed by small-angle neutron scattering. The structure is much finer than that found in the similar system investigated previously which was prepared using much longer diamine (molar mass ca. 2000 g mol−1). Unlike the previous system, the scattering data from the present system cannot be fitted properly by Percus–Yevick model but Teubner–Strey model based on the locally lamellar order of nanophases dictated by conservation of polymer network topology proves well in the fitting. The values of parameters of the nanophase separated structure – two characteristic lengths: periodicity and persistence length of the locally lamellar order and their dependence on the network composition are determined and analyzed in detail.

Ultra-small-angle neutron scattering: A tool to study packing of relatively monodisperse polymer spheres and their binary mixtures

We measured ultra-small-angle neutron scattering (USANS) from polymethylmethacrylate spheres tamped down in air. Two slightly polydisperse pure sphere sizes (1.5 and 7.5 microm diameters) and five mixtures of these were used. All were loose packed (packing fractions of 0.3-0.6) with nongravitational forces (e.g., friction) important, preventing close packing. The USANS data are rich in information on powder packing. A modified Percus-Yevick fluid model was used to parametrize the data-adequately but not well. The modifications required the introduction of small voids, less than the sphere size, and a parameter reflecting substantial deviation from the Percus-Yevick prediction of the sphere-sphere correlation function. The mixed samples fitted less well, and two further modifying factors were necessary. These were local inhomogeneities, where the concentration of same-size spheres, both large and small, deviated from the mean packing, and a factor accounting for the presence within these "clusters" of self-avoidance of the large spheres (that is, large spheres coated with more small spheres than Percus-Yevick would predict). The overall deviations from the hard-sphere Percus-Yevick model that we find here suggest that fluid models of loose packed powders are unlikely to be successful but lay the ground work for future theoretical and computational works.

Structure and swelling behaviour of hydrophilic epoxy networks investigated by SANS

Structure of hydrophilic epoxy networks swollen in deuterated water has been investigated by small-angle neutron scattering (SANS). Two series of the networks were prepared by end-linking reaction of α,ω-diamino terminated poly(oxypropylene)- block-poly(oxyethylene)- block-poly(oxypropylene) (POP–POE–POP), α,ω-diamino terminated poly(oxypropylene) (POP) and diglycidyl ether of Bisphenol A propoxylate (PDGEBA). In the first series, prepared using functionalized POP–POE–POP and PDGEBA, hydrophilicity of the networks was controlled via cross-linking density of the networks, changing the initial ratio of reactives. In the second series, POP–POE–POP was gradually substituted by POP of the same molecular weight, enabling a control of hydrophilicity of the networks at essentially constant crosslinking density in this way. Microphase separated structure of the networks swollen in deuterated water with composition variations on nanometer scale was found by SANS. The scattering data were fitted to the Percus–Yevick model, approximating the swollen network by an idealized system of spheres dispersed in the matrix. Satisfactory fits in the region of the interference scattering peak were obtained giving values 30–70 Å for the average radius of the spherical regions. Radius of the spheres increases with increasing hydrophobicity of the networks (growing POP content and/or network crosslinking density).

Particle sizing in dense suspensions with multiwavelength photon migration measurements

AbstractCharacterization of dense polydisperse suspensions, with mean diameters 143 and 226 nm and volume fractions ranging from 0.03 to 0.27, was conducted using time‐dependent measurements of multiply scattered light. Frequency‐domain photon migration measurements of isotropic scattering coefficients at wavelengths between 488 and 828 nm were found to agree with those predicted from the Mie theory and the polydisperse hard‐sphere Percus–Yevick model. The wavelength‐dependent isotropic scattering data were then used to successfully recover the particle size distribution and volume fraction of the suspensions, by minimizing a nonlinear least‐squares problem. The mean particle size and volume fractions were recovered within an error range of 0–15.53 and 0–24%, respectively, compared with dynamic light scattering results and experimentally measured volume fractions. © 2005 American Institute of Chemical Engineers AIChE J, 2005

A review of small-angle scattering models for random segmented poly(ether-urethane) copolymers

Small-angle X-ray scattering data from several experimental and commercial poly(ether urethane) formulations was used to test various scattering models based on different morphologies. The best fits were generally found with ‘globular’ scattering models based on a distorted one-dimensional lattice or the Percus–Yevick model of liquid structure. Whilst this is not conclusive proof of the morphologies exhibited by the materials studied, these scattering models are roughly consistent with many AFM and TEM studies, which indicated discrete globular or elongated cylindrical microdomains. Moreover, reasons why meandering elongated or finite lamellar microdomains may behave more like globular scattering bodies are discussed. Hence, these models are proposed as a basis for interpreting scattering data from polyurethanes. Other models were unable to fit the observed scattering data adequately. A model based on the weak segregation of copolymers was rejected on the basis of deviations from the observed scattering behaviour in the Porod region. A model based on stacks of ideal (infinite, parallel, flat) lamellae was ruled out, since it was unable to reproduce the observed peak width. The Teubner–Strey model, based on microemulsion structure, was found to be incapable of fitting the data at low q, particularly for strained samples. However, this model appeared to be more successful at reproducing the scattering observed at elevated temperatures. One possible inference is that the morphologies became more akin to microemulsions during heating.

Structure and dynamics of solutions of a polystyrene–polybutadiene pentablock copolymer in a styrene‐selective solvent

AbstractWe have examined solutions of a polystyrene–polybutadiene pentablock copolymer in 1,4‐dioxane, a slightly selective solvent for polystyrene and a θ solvent for polybutadiene, with static light scattering (SLS), dynamic light scattering (DLS), and small‐angle neutron scattering (SANS). The SANS data have been analyzed with the Percus–Yevick model to represent the scattering from interacting cores, approximated as hard spheres, and with a Lorentzian function to represent the scattering from unassociated and associated polymer chains. The SANS data at 25 °C clearly reveal interacting domains, approximately 6 nm in radius, formed by the association of the insoluble polybutadiene block in the 20% sample. The 4% sample does not show such domains, whereas the 7% sample represents an intermediate situation, with both unassociated polymer and associated polymer. At higher temperatures, the domains dissolve. The DLS data for samples with concentrations of 2–22% show two diffusive modes: a fast mode corresponding to the cooperative dynamics of concentration fluctuations and a slow mode corresponding to the diffusion of clusters. The large length‐scale heterogeneities, indicated by the strong angular dependence of SLS, implies that the small microdomains of about 10–15 polybutadiene blocks are bridged by the polystyrene chains, forming large aggregates with randomly distributed crosslinks on length scales much larger than the domain size. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2807–2816, 2002

Assessment of Small-Angle and Angle-Averaged Structure Factor for Monitoring Electrostatic Colloidal Interactions Using Multiply Scattered Light

The isotropic scattering coefficients of 143-nm diameter polystyrene latex suspensions were measured using frequency-domain photon migration (FDPM) at 687 and 828 nm as a function of volume fraction (0.05–0.3) and ionic strength (1.0 to 120 mM NaCl equivalents) in order to derive the angle-integrated structure factor, 〈S(q)〉, and structure factor at zero wave vector, S(0). The effective surface charges of the dispersions were estimated by fitting the measured isotropic scattering coefficients at each wavelength as a function of volume fraction to the solution of the Orstein–Zernike integral equation using the hard sphere Yukawa potential model and mean spherical approximation as a closure relation. The estimates of surface charges were comparable at both wavelengths, but decreased with ionic strength. At 120 mM NaCl equivalents, the values of S(0) obtained from FDPM matched those predicted by the Percus–Yevick model, and decreased with volume fraction, consistent with prediction by the Carnahan–Starling equation.

Investigation of Particle Interactions in Concentrated Colloidal Suspensions Using Frequency Domain Photon Migration: Monodisperse Systems

Frequency domain photon migration (FDPM) measurements were conducted to assess particle interactions of concentrated, monodisperse, polystyrene samples obtained directly from industry by using multiple scattering light. The angle-integrated static structure factor, 〈S(q)〉, and static structure factor at small wave vector q, S(0), were obtained from FDPM measurements at high volume fractions ranging from 0.05 to 0.3, and were compared with those obtained from the monodisperse hard sphere Percus-Yevick (HSPY) model. Effects of different colloid sizes on structure factor evaluated at two different wavelengths were also investigated. Results show that the monodisperse HSPY model is suitable for accounting for particle interactions and local microstructures in these colloidal suspensions. Upon using the HSPY model, particle sizes of polystyrene suspensions were recovered from FDPM measurements at high volume fractions (up to 0.3), which agree well with the DLS measurement of diluted sample (∼0.001). The study of polydispersity effect shows that the FDPM method can be successfully used for recovering the mean particle size of polydisperse colloidal suspension with low polydispersity (<16%) under the assumption of monodisperse hard sphere systems.

Micellar structure and mechanical properties of block copolymer‐modified epoxies

AbstractAmphiphilic block copolymers provide a unique means for toughening epoxy resins because they can self‐assemble into different inclusion shapes before epoxy curing. The two examples reported here are spherical micelles and vesicles, which form in blends containing epoxy and symmetric or asymmetric poly(ethylene oxide)–poly(ethylene‐alt‐propylene) (PEO–PEP) block copolymer with PEO volume fractions of 0.5 and 0.26, respectively. The vesicles and spherical micelles were characterized by transmission electron microscopy and small‐angle X‐ray scattering (SAXS), respectively. SAXS data from the spherical micelles were fit to the Percus–Yevick model for a liquid‐like packing of spheres with hard‐core interactions. Mechanical properties of spherical‐micelle‐modified and vesicle‐modified epoxies in the dilute limit are compared. The glass‐transition temperature and Young's (storage) modulus were tested with dynamic mechanical spectroscopy, and compact‐tension experiments were performed to determine the critical plane‐strain energy release rate for fracture. Vesicles were most effective in improving the epoxy fracture resistance. © 2001 John Wiley &amp; Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2996–3010, 2001

Characterization of concentrated colloidal suspensions using time-dependent photon migration measurements

A method is presented to characterize concentrated suspensions of particles using frequency-domain photon migration in order to demonstrate its potential for on-line monitoring in the pharmaceutical and chemical based industries. In this study, the isotropic scattering coefficient was determined from FDPM measurements conducted at wavelengths between 448 and 835 nm using surfactant stabilized polystyrene latices of 195–335 nm mean diameter, 7–19% polydispersity and concentrations between 1 and 40% by volume. Upon using the Percus–Yevick model for static structure of monodisperse spheres to account for hindered scattering in the inverse solution, determination of the volume fraction and mean particle diameter were considerably improved size at solids volume fractions of 10% or greater. Owing to (i) the relative refractive index of polystyrene in water, (ii) the inaccuracy of the first order particle interaction model for suspensions with polydispersity >8%, and (iii) the minimal number of wavelengths used, it was difficult to recover the variance of the unimodal particle size distributions. Nonetheless, this work demonstrates the use of time-dependent measurements of multiply scattered light to characterize suspensions with solids volume fractions greater than 10%.

A small-angle scattering study of the ceramization process from organic polymer precursors

Amorphous polyborosil azane bulk ceramics are prepared by heat treatment above 1273 K from organometallic bulk precursors which are formed by sintering precursor powders using cold isostatic pressing. A structural change during the conversion process of the polyborosilazane bulk precursors causes a change of small-angle X-ray scattering (SAXS) profiles in the high q region. The analysis of the SAXS intensities indicates that the size of the precipitates increases monotically with increasing heat-treated temperature and that they develop to small particles with sharp interfaces distributed in the amorphous phase after heat treatments above 1173 K. All the scattering profiles of the polyborosil azane bulk specimens throughout the conversion process can be fitted using the PercusYevick model of particle scattering including the concept of adepleted zone. The SAXS measurements show that the structural inhomogeneities originating from organometallic polymers arechanged by the heat-treatment above about 973 K to structures where the nano-sized precipitates embedded in the amorphous matrix occupy smaller volume fractions compared with those heat treated below 873 K.

Investigation of static structure factor in dense suspensions by use of multiply scattered light

Near-infrared, frequency-domain photon migration measurements of phase shift are used to derive accurate values of isotropic scattering coefficients in concentrated, interacting suspensions of aqueous polystyrene microspheres with volume concentrations ranging from 1% to 45% by solids and mean diameters ranging from 135 to 500 nm. Under conditions of high ionic strength, the isotropic scattering coefficient can be quantitatively predicted by the Percus-Yevick model for hard-sphere interactions and Mie theory. In addition, the attractive interactions between scatterers arising from the addition of soluble poly(ethylene glycol) with molecular weights of 100 and 600 K cause hindered scattering. The increases in static structure and decreases in isotropic scattering coefficient agree with that predicted by Mie theory and the depletion interaction model developed by Asakura and Oosawa [J. Chem. Phys. 22, 1255 (1954)]. These results demonstrate the success of monitoring interaction between particles by use of multiple-scattered light and the necessity of incorporating models for these interactions when predicting scattering of dense, concentrated suspensions.

A SAXS study of the ceramization process of Si–C organic precursor

Amorphous Si–Ti–O–C ceramic fibers are prepared from organic precursor fibers by heat treatment above 1000°C. Small-angle X-ray scattering (SAXS) measurements have detected a structural change accompanying an increase in mechanical strength of the fibers arising from an increase in heat-treatment temperature. The scattering profiles of Si–Ti–O–C fibers heat treated at temperatures below 800°C differ from those heat-treated at higher temperatures above 1000°C. The former profiles are fit by Gaussian functions, while the latter profiles can be fitted using the Percus–Yevick model of particle scattering including the concept of a depleted zone. The SAXS measurements showed that structural changes occur over a length scale greater than 1.5 nm for fibers heat-treated at temperatures between 500°C and 1000°C.