Percus-Yevick Hard Sphere Research Articles

Theoretical investigation on the structural and vibrational properties of some alkali liquid metals: A pseudopotential approach

Along with a few elastic constants, a recently created pseudopotential is employed to examine the vibrational properties of some simple alkali metals, i.e. Li, Na, K, Rb and Cs. The phonon dispersion curves (PDC) are computed in the text. (ωL and ωT), longitudinal sound velocity (vl), transverse sound velocity (vt), isothermal bulk modulus (B), modulus of rigidity (G), Young’s modulus (Y ), Debye temperature (θD) of some liquid alkali metals. The second order technique used in the current work, using Hubbard and Beeby (HB) equations, is based on pseudopotential theory. The various Percus-Yevick hard sphere (PYHS) and one-component plasma (OCP) structure factors used in the current investigation are used to construct the pair correlation function g(r). The current paper makes use of three distinct forms of local field correction functions developed by Hartree (H), Taylor (T), and Nagy (N). The current findings are found to qualitatively agree with the experimental and theoretical data that is currently available.

Tagged-particle motion of Percus-Yevick hard spheres from first principles.

We develop a first-principles-based generalized mode-coupling theory (GMCT) for the tagged-particle motion of glassy systems. This theory establishes a hierarchy of coupled integro-differential equations for self-multi-point density correlation functions, which can formally be extended up to infinite order. We use our GMCT framework to calculate the self-nonergodicity parameters and the self-intermediate scattering function for the Percus-Yevick hard-sphere system based on the first few levels of the GMCT hierarchy. We also test the scaling laws in the α- and β-relaxation regimes near the glass-transition singularity. Furthermore, we study the mean-square displacement and the Stokes-Einstein relation in the supercooled regime. We find that qualitatively our GMCT results share many similarities with the well-established predictions from standard mode-coupling theory, but the quantitative results change, and typically improve, by increasing the GMCT closure level. However, we also demonstrate on general theoretical grounds that the current GMCT framework is unable to account for violation of the Stokes-Einstein relation, underlining the need for further improvements in the first-principles description of glassy dynamics.

Shape and Structure Formation of Mixed Nonionic-Anionic Surfactant Micelles.

Aqueous solutions of a nonionic surfactant (either Tween20 or BrijL23) and an anionic surfactant (sodium dodecyl sulfate, SDS) are investigated, using small-angle neutron scattering (SANS). SANS spectra are analysed by using a core-shell model to describe the form factor of self-assembled surfactant micelles; the intermicellar interactions are modelled by using a hard-sphere Percus–Yevick (HS-PY) or a rescaled mean spherical approximation (RMSA) structure factor. Choosing these specific nonionic surfactants allows for comparison of the effect of branched (Tween20) and linear (BrijL23) surfactant headgroups, both constituted of poly-ethylene oxide (PEO) groups. The nonionic–anionic surfactant mixtures are studied at various concentrations up to highly concentrated samples ( ≲ 0.45) and various mixing ratios, from pure nonionic to pure anionic surfactant solutions. The scattering data reveal the formation of mixed micelles already at concentrations below the critical micelle concentration of SDS. At higher volume fractions, excluded volume effects dominate the intermicellar structuring, even for charged micelles. In consequence, at high volume fractions, the intermicellar structuring is the same for charged and uncharged micelles. At all mixing ratios, almost spherical mixed micelles form. This offers the opportunity to create a system of colloidal particles with a variable surface charge. This excludes only roughly equimolar mixing ratios (X≈ 0.4–0.6) at which the micelles significantly increase in size and ellipticity due to specific sulfate–EO interactions.

Generalized mode-coupling theory of the glass transition. I. Numerical results for Percus-Yevick hard spheres.

Mode-coupling theory (MCT) constitutes one of the few first-principles-based approaches to describe the physics of the glass transition, but the theory's inherent approximations compromise its accuracy in the activated glassy regime. Here, we show that microscopic generalized mode-coupling theory (GMCT), a recently proposed hierarchical framework to systematically improve upon MCT, provides a promising pathway toward a more accurate first-principles description of glassy dynamics. We present a comprehensive numerical analysis for Percus-Yevick hard spheres by performing explicitly wavenumber- and time-dependent GMCT calculations up to sixth order. Specifically, we calculate the location of the critical point, the associated non-ergodicity parameters, and the time-dependent dynamics of the density correlators at both absolute and reduced packing fractions, and we test several universal scaling relations in the α- and β-relaxation regimes. It is found that higher-order GMCT can successfully remedy some of MCT's pathologies, including an underestimation of the critical glass transition density and an overestimation of the hard-sphere fragility. Furthermore, we numerically demonstrate that the celebrated scaling laws of MCT are preserved in GMCT and that the predicted critical exponents manifestly improve as more levels are incorporated in the GMCT hierarchy. Although formally the GMCT equations should be solved up to infinite order to reach full convergence, our finite-order GMCT calculations unambiguously reveal a uniform convergence pattern for the dynamics. We thus argue that GMCT can provide a feasible and controlled means to bypass MCT's main uncontrolled approximation, offering hope for the future development of a quantitative first-principles theory of the glass transition.

Stochastic dynamics and thermodynamics around a metastable state based on the linear Dean–Kawasaki equation

The Dean–Kawasaki equation forms the basis of the stochastic density functional theory (DFT). Here it is demonstrated that the Dean–Kawasaki equation can be directly linearized in the first approximation of the driving force due to the free energy functional of an instantaneous density distribution , when we consider small density fluctuations around a metastable state whose density distribution is determined by the stationary equation with denoting the chemical potential. Our main results regarding the linear Dean–Kawasaki equation are threefold. First, (i) the corresponding stochastic thermodynamics has been formulated, showing that the heat dissipated into the reservoir is negligible on average. Next, (ii) we have developed a field theoretic treatment combined with the equilibrium DFT, giving an approximate form of that is related to the equilibrium free energy functional. Accordingly, (iii) the linear Dean–Kawasaki equation, which has been reduced to a tractable form expressed by the direct correlation function, allows us to compare the stochastic dynamics around metastable and equilibrium states, particularly in the Percus–Yevick hard sphere fluids; we have found that the metastable density is larger and the effective diffusion constant in the metastable state is smaller than the equilibrium ones in repulsive fluids.

Multiple and dependent scattering by densely packed discrete spheres: Comparison of radiative transfer and Maxwell theory

The radiative transfer equation (RTE) has been widely used to deal with multiple scattering of light by sparsely and randomly distributed discrete particles. However, for densely packed particles, the RTE becomes questionable due to strong dependent scattering effects. This paper examines the accuracy of RTE by comparing with the exact electromagnetic theory. For an imaginary spherical volume filled with randomly distributed, densely packed spheres, the RTE is solved by the Monte Carlo method combined with the Percus–Yevick hard model to consider the dependent scattering effect, while the electromagnetic calculation is based on the multi-sphere superposition T-matrix method. The Mueller matrix elements of the system with different size parameters and volume fractions of spheres are obtained using both methods. The results verify that the RTE fails to deal with the systems with a high-volume fraction due to the dependent scattering effects. Apart from the effects of forward interference scattering and coherent backscattering, the Percus–Yevick hard sphere model shows good accuracy in accounting for the far-field interference effects for medium or smaller size parameters (up to 6.964 in this study). For densely packed discrete spheres with large size parameters (equals 13.928 in this study), the improvement of dependent scattering correction tends to deteriorate. The observations indicate that caution must be taken when using RTE in dealing with the radiative transfer in dense discrete random media even though the dependent scattering correction is applied.

Time domain finite element estimates of dynamic stiffness of viscoelastic composites with stiff spherical inclusions

A weighted residual time domain unstructured mesh finite element approach was used to obtain numerical estimates for the dynamic stiffness of medium and high stiffness contrast composites consisting of viscoelastic epoxy matrix filled with stiff spherical inclusions. Both random and regular microstructure composites were studied. It was shown that over the broad temperature and inclusion fraction ranges studied, the generalized self-consistent model ( Christensen and Lo, 1979 ) provided accurate predictions for the effective dynamic stiffness of random composites obeying classical Percus–Yevick hard sphere statistics. Assuming sphere flocculation, we have studied the role of microstructural effects and found that upon increasing stiffness contrast, they become progressively more important for both the effective storage and loss moduli. As a consequence, for the reliable predictions of effective stiffness of high contrast composites with rubber like matrices, one should necessarily use homogenization models accounting not only for the inclusion concentration but also for the finer microstructural details. However, it was also found that for such high stiffness contrast viscoelastic composites, the ratio of their effective storage and loss shear moduli (the damping factor) remains essentially unchanged as compared to that of pure unfilled matrix so it is the invariability of the damping factor that can be viewed as a special signature of underlying micromechanical mechanism of reinforcement.

Controlled accuracy finite element estimates for the effective stiffness of composites with spherical inclusions

Mixed form finite element method was used to study the microstructural effects in the effective stiffness of composites reinforced or weakened by regular and random dispersions of non-overlapping identical spheres. Controlled accuracy stiffness estimates were achieved by using the unstructured mesh polynomial extension and asymptotic nested grid extension of the finite element method. Periodic regular and random microstructure computer models with up to a hundred spheres were considered. The significance of the elastostatic bridges separating near-to-touch spheres was studied and their impact on the effective stiffness was assessed. It was found that for a variety of practically relevant composites with moderate stiffness contrast between the phases, the effective stiffness estimates were remarkably insensitive to the underlying composite microstructure. The obtained finite element estimates were compared with predictions of the generalized self-consistent (GSC) model ( Christensen and Lo, 1979 ) and it was shown that the model was accurate over the entire range of practically relevant sphere fractions. We studied a high stiffness contrast composite consisting of a near incompressible rubber matrix filled with rigid spherical inclusions. In this case strong microstructural effects were observed but, nevertheless, for the random composite obeying the hard sphere Percus–Yevick statistics the GSC model remained strikingly accurate over the whole range of practically relevant inclusion fractions. The asymptotic Einstein limit GSC solution was derived and it was demonstrated that it gave accurate micromechanics-based solution of the classical Einstein homogenization problem of a random dispersion of perfectly rigid spherical inclusions in a fully incompressible matrix.

Mixture of cholesterol end-capped polyethylene glycol with DSPC liposomal

The dynamic of network of self-assembled liposome by end-capped polymer was investigated using dynamic light scattering. The liposome network, physically cross-linked by mixed liposome solutions with three different length scale of cholesterol end-capped polyethylene glycol. The network of liposome is dependent on both the polymer concentration and length scale. In the pure liposome, one motion at low time scale is observed by DLS. In the higher concentration of polymer in liposome, several motion is observed that the fast motion is alpha relaxation and other two slow motion are beta and gamma relaxations. The distance between diffusion coefficient of fast and slow relaxation is increased with increase of length scale of endcapped polymers. The SAXS data is fitted with a Percus-Yevick hard sphere model and it shows that the size of liposome increasing with increase of polymer length scale in the mixture system.

Mechanistic origins of multi-scale reinforcements in segmented polyurethane-clay nanocomposites

The objective of the present work is to get insights into the mechanistic origin of the reinforcement effects of nanoclay on a segmented polybutadiene polyurethane-urea system. To this end, a convergent analysis of the hard domain morphology and conformational state of soft segment in the nanocomposites was carried out by using a combination of complementary characterization techniques, namely, Fourier transform infrared spectroscopy, small angle neutron scattering, transmission electron microscopy, modulated differential scanning calorimetry and dynamic mechanical analysis. Analysis of small angle neutron scattering data by a combination of Percus–Yevick hard sphere and Zernike-Ornstein model coupled with direct visualization of the dispersed hard domain morphology from transmission electron microscopy provided insight on clay induced changes in the hard domain morphology. A monotonic decrease in the domain size as well as the average interdomain distance was observed with increasing nanoclay content in the polymer matrix. Analysis of the carbonyl stretching region from FTIR showed increased degree of hydrogen bonding for the urethane carbonyl groups of the nanocomposites compared to the neat matrix. A combination of calorimetric and dynamic mechanical analysis revealed the existence of a constrained amorphous region; quantified to be ≈ 16% at the highest clay content experimented. The manifestation of these morphological and conformational changes on the nano-, micro- and macro scale reinforcements in the nanocomposites was investigated by mechanical properties at these length scales using nanoindentation, DMA and tensile testing, respectively.

Atomic Transport Properties of 3D Liquid Transition Metals Using Different Reference Systems

Atomic transport properties like self diffusion coefficient (D), viscosity coefficient (η) of 3d liquid transition metals are studied. Here we have applied our own model potential to describe electron ion interaction with different reference system like Percus - Yevick Hard Sphere (PYHS), One Component Plasma (OCP) and Charge Hard Sphere (CHS) systems. We have investigated the effect of different correction function like Hartree (H), Vashishta-Singwi (VS), Hubbard-Sham (HS), Sarkar et al (S), Ichimaru-Utsumi(IU), Taylor (T) and Farid et al (F) on atomic transport properties. The proper choice of the model potential alongwith the local field correction function and reference system plays a vital role in the study of the atomic transport properties of 3d liquid transition metals.

Theoretical Investigation of Atomic Transport Properties of 4d Transition Metals in Liquid Phase

Present article deals with atomic transport properties like self-diffusion coefficient (D) and viscosity coefficient (η) of 4d transition metals in liquid state. To describe structural information we have used different reference systems like Percus - Yevick Hard Sphere (PYHS), One Component Plasma (OCP) and Charge Hard Sphere (CHS) systems alongwith our newly constructed parameter free model potential. To see the effect of different correction functions on atomic transport properties, we have used different local field correction functions like Hartree (H), Vashishta-Singwi (VS), Hubbard-Sham (HS), Sarkar et al (S), Ichimaru-Utsumi (IU), Taylor (T) and Farid et al (F). From the present results we conclude that our newly constructed model potential successfully calculated atomic transport properties of 4d transition metals in liquid phase.

Theoretical Investigation of Thermodynamical and Structural Properties of 3d Liquid Transition Metals Using Different Reference Systems

The present paper deals with the theoretical investigation of thermodynamical and structural properties like internal energy (E), entropy (S), Helmholtz free energy (F), isothermal compressibility (χT), specific Heat (CV), structure factor S(q), and long wave length limit S(0) of structure factor of 3d liquid transition metals. To describe electron-ion interaction we have used our newly constructed parameter free model potential. To perform this task, we have used different reference systems like Percus Yevick Hard Sphere (PYHS), One Component Plasma (OCP), and Charged Hard Sphere (CHS) reference systems. We have also seen the influence of different local field correction functions like Hartree (HR), Taylor (TR), and Sarkar et al. (SR) on thermodynamical properties of 3d liquid transition metals. Finally we conclude that the proper choice of the model potential along with reference system plays a vital role in the study of thermodynamical and structural properties of 3d liquid transition metals.

Electronic transport properties of liquid lanthanides

The electronic transport properties like electrical resistivity (ρ), thermoelectric power (Q) and thermal conductivity (σ) of liquid lanthanides are calculated through Ziman’s formulism. To describe electron-ion interaction we have used our newly constructed parameter free model potential alongwith Percus Yevick hard sphere (PYHS) reference system. To see the influence of exchange and correlation effect on the electronic transport properties, we have used five different local field correction functions like Hartree (H), Sarkar et al. (S), Taylor (T), Ichimaru Utsumi (IU) and Farid et al. (F). The obtained results have found good in agreement with available experimental data as well as other theoretical data. Lastly, we conclude that our newly constructed model potential is capable to explain the electronic transport properties of liquid lanthanides.

Electronic transport properties of some transition liquid metals

The electronic transport properties of electrical resistivity (ρ), thermoelectric power (Q) and thermal conductivity (σ) of some transition liquid metals (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Zr, Pd, Ag, Cd, Pt, Au and Hg) are evaluated using Ziman's formula, along with our newly constructed parameter-free model potential. To describe the structural information, the structure factor S(q) from the Percus–Yevick Hard Sphere (PYHS) reference system is used. The various local-field correction functions, namely Hartree (H), Vashishta–Singwi (VS), Hubbard–Shamm (HS), Sarkar et al. (SS), Ichimaru–Utsumi (IU), Taylor (T) and Farid et al. (F), have been incorporated to see the influence of exchange and correlation effects on electronic transport properties. The proper choice of the model potential, along with the local field correction function and the oxidation state (valency Z), plays a vital role in the study of the electronic transport properties of some transition liquid metals.

Thermodynamic properties of liquid metals using Percus–Yevick (PY) hard sphere reference system

The Gibbs-Bogoliubov inequality is used to evaluate the Helmholtz free energy of some liquid metals. The structural contribution to this energy is investigated by adopting the Percus-Yevick (PY) hard sphere fluid. The well-known single parametric empty core local pseudopotential of Ashcroft is applied to describe the electron-ion interaction along with five different local field correction functions, viz, Hartree, Taylor, Ichimaru-Utsumi, Farid et al., and Sarkar et al. The difference of 2.18 to 28.00% in the influence on the Helmholtz free energy is due to these local field correction functions with respect to the static Hartree dielectric function. A qualitative agreement with experimental or theoretical results is achieved.

Analytic methods for the Percus-Yevick hard sphere correlation functions

Analytic methods for the Percus-Yevick hard sphere correlation functions

Thermodynamics of liquid d- and f-shell metals: a variational approach

The Gibbs–Bogoliubov variational method is applied to investigate the thermodynamic properties of some d- and f-shell metals in liquid phase namely; Rh, Ir, Cu, Ag, Ni, Pd, Pt, Au, La, Yb, Ce, and Th using pseudopotential theory. For description of the structure, Percus-Yevick hard sphere model is used as a reference system. Ashcroft's well-known empty core model potential is used to describe the electron-ion interaction. The local field correction function proposed by Hartree, Taylor, Ichimaru–Utsumi, Farid et al., and Sarkar et al. are applied to introduce the exchange and correlation effects in the study of thermodynamics of these metals. The comparison with available experimental or theoretical findings is found in qualitative agreement and establishes the use of the local field correction functions in such study.

Study on the cluster of floating dross before nucleating during hot-dipping process

During hot-dipping in molten Zn-55AI, the forming of “floating” dross is closely related with the interaction between the Al and the Fe atoms. Utilizing the thita-thita X-ray diffraction and the Percus-Yevick hard sphere model, this paper investigates the structure of molten AI3Fe that is an approximant of partial melt in practical Zn-55AI alloy melt. It is found that in the melt there exist clusters that resemble the structure of intermetallic compound Af3Fe. The strong interaction between Al and Fe atoms is also indicated by the deficiency of the coordination number from 12 and the shrinkage of averaged atomic diameter. Based on such information, it is speculated that in the molten Zn-55AI, with the accumulation of Fe, Al3Fe-like clusters form with priority, then grow into dross. On the other hand, after deflating of the effective hard sphere diameter of pure Al atom and enlarging that of the pure Fe atom, the modified model can achieve reasonable fit to the experimental structure factor. Hard sphere modeling also gave a good estimation of the mass density 3.65 g/cm3 of liquid Al3Fe alloy at 1550°C. Compared with the density of 3.7 g/cm3 of Zn-55AI, the reason for dross floating is satisfactorily understood.

Investigation of structure factors in dense colloidal suspensions using frequency domain photon migration: polydisperse systems

Frequency domain photon migration (FDPM) technique was employed to investigate the structure factors of dense, polydisperse colloidal suspensions. The angle-integrated structure factors, 〈 S( q)〉, extracted from FDPM measurements of scattering properties at volume fractions ranging from 0.05 to 0.4, were compared with the values predicted from the polydisperse hard sphere Percus–Yevick (HSPY) model, as well as decoupling approximation (DA) and local monodisperse approximation (LMA) models that incorporated independently measured particle size information. Results show that the polydisperse HSPY model is the most suitable for accounting for particle interactions which predominately arise from volume exclusion effects. Furthermore, the influence of size polydispersity upon 〈 S( q)〉 is most significant at high volume fractions. The static structure factors at small wave vector q, S(0), were also assessed from dual wavelength FDPM measurements by using the small wave number approximation as well as the local monodisperse approximation. The measured S(0) agrees well with the values predicted by the polydisperse HSPY model.