Phase behavior of a suspension of hard spherocylinders plus ideal polymer chains

We study isotropic-isotropic and isotropic-nematic phase transitions of fluid mixtures containing hard spherocylinders (HSC) and added non-adsorbing ideal polymer chains using scaled particle theory (SPT). First, we investigate isotropic-nematic (I -N phase coexistence using SPT in the absence of polymer. We compare the results obtained using a Gaussian form of the orientational distribution function (ODF) to minimize the free energy versus minimizing numerically. We find that formal numerical minimization gives results that are much closer to computer simulation results. In order to describe mixtures of HSC plus ideal chains we studied the depletion of ideal chains around a HSC. We analyze the density profiles of ideal chains near a hard cylinder and find the depletion thickness δ is a function of the ratio of the polymer's radius of gyration R_g and the cylinder radius R_c. Our results are compared with a common approximation in which the depletion thickness is taken equal to the radius of gyration of the polymer chain. We incorporate the correct depletion thickness into SPT and find that for R _g/R _c < 1.56 using ideal chains gives phase transitions at smaller polymer concentrations, whereas for R _g/R _c > 1.56 , which is a common experimental situation, the phase transitions are found at larger polymer concentrations with respect to δ = R _g . The differences are significant, especially for R _g ≫ R _c , so we can conclude it is essential to take into account the properties of ideal polymer chains and the resulting depletion near a cylinder. Finally, we present phase diagrams for rod-polymer mixtures which could be realized under experimental conditions.     

Exact Analytical Solution of the Schrödinger Equation for an N-Identical Body-Force System

A mathematical method is presented for solving the Schr?dinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated within the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyperradius x = (ξ₁² + ξ₂² + ⋯ + ξ_N−1²)^1/2 only, where ξ₁,ξ₂,…,ξ_N−1 are Jacobi relative coordinates which are functions of N-particle relative positions r₁₂,r₂₃,…,r_N1. The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax² + bx − (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schr?dinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles.     

On the lower critical field and phase diagram of a thin cylindrical type-II superconductor

The lower critical field H _c1 ^cyl (T) of a superconducting cylinder with radius r ₀∼ξ(T)≪λ(T) is found on the basis of the Ginzburg-Landau theory with various boundary conditions. These results together with the well-known results for the upper critical field are used to construct phase diagrams in terms of the field versus the reduced radius r ₀∼ξ(T) variables. The jump in the average magnetization at H _c1 ^cyl (T) is calculated as a function of the reduced radius. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 8, 537–542 (25 April 1999)     

Anomalous light-induced drift of lithium atoms in a mixture of noble gases

This paper is a theoretical study of the spectral features of the velocity of light-induced drift (LID) of lithium atoms (⁷Li and ⁶Li) in a binary mixture of noble gases: Ne + Ar, Ne + Kr, and Ne + Xe. The spectral shape of the LID signal is predicted to depend strongly on the fraction ξ of neon in the buffer mixture in the range ξ≈0.8–0.9 (ξ=N _Ne/N _b, where N _Ne is the neon concentration, and N _b is the total concentration of the buffer particles). When the velocity of anomalous LID is treated as a function of the radiation frequency, it is found to have one, three, five, or seven zeros and to differ substantially from the dispersion-curve-like behavior with one zero predicted by the standard LID theory with velocity-independent transport collision rates. The reason for these additional zeros of the drift velocity is the alternating-sign dependence on the lithium-atom velocity of the relative difference of transport rates of collisions between buffer particles and excited and unexcited atoms. What is also established is that the anomalous LID of lithium atoms can be observed at almost all temperatures, depending on the value of ξ. At a fixed temperature, anomalous LID can be observed only in a narrow range of values of the fraction of neon in the buffer mixture, Δξ≈0.02. The results make possible highly precise testing in the LID experiments of the interatomic potentials used in calculations of the velocity spectrum of anomalous LID. Zh. éksp. Teor. Fiz. 116, 1587–1600 (November 1999)     

Annihilation of nematic point defects: Pre-collision and post-collision evolution

The annihilation of the nematic hedgehog and anti-hedgehog within an infinite cylinder of radius R is studied. The semi-microscopic lattice-type model and Brownian molecular dynamics are used. We distinguish among the i) early pre-collision, ii) late pre-collision, iii) early post-collision, and iv) late post-collision stages. In the pre-collision stage our results agree qualitatively with the existing experimental observations and also continuum-type simulations. The core of each defect exhibits a ring-like structure, where the ring axis is set perpendicular to the cylinder symmetry axis. For ξ⁽⁰⁾_d/(2R) > 1 the interaction between defects is negligible, where ξ⁽⁰⁾_d describes the initial separation of defects. Consequently, the defects annihilate within the simulation time window for ξ⁽⁰⁾_d/(2R) < 1. For close enough defects their separation scales as ξ_d (t_c - t)^0.4±0.1, where t_c stands for the collision time. In elastically anisotropic medium the hedgehog is faster than the anti-hedgehog. In the early pre-collision stage the defects can be treated as point-like particles, possessing inherent core structure, that interact via the nematic director field. In the late pre-collision stage the cores reflect the interaction between defects. After the collision a charge-less ring structure is first formed. In the early post-collision stage the ring adopts an essentially untwisted circular structure of the radius ξ_r. In the late post-collision stage we observe two qualitatively different scenarios. For μ = ξ_r/R < μ_c ∼ 0.25 the ring collapses leading to the escaped radial equilibrium structure. For μ > μ_c the chargeless ring triggers the nucleation growth into the planar polar structure with line defects.     

Measuring the Michel parameter ξ'

Unlike the majority of Michel parameters which are consistent with the Standard Model V-A interaction, the experimental value of ξ″(=0.65±0.36) [1] is poorly known. Our experiment will measure the longitudinal polarization, P _L , of positrons emitted from the decay of polarized muons. The value of P _L , equal to unity in the Standard Model, will decrease for high energy positrons emitted antiparallel to the muon spin if the combination of Michel parameters ξ″/ξξ′ − 1 deviates from the Standard Model value of zero. This revised version was published online in August 2006 with corrections to the Cover Date.     

Large Deviations for the Fermion Point Process Associated with the Exponential Kernel

For the fermion point process on the whole complex plane associated with the exponential kernel , we show the central limit theorem for the random variable ξ(D _r , the number of points inside the ball D _r of radius r, as r → ∞ and we establish the large deviation principle for the random variables {r ⁻²ξ (D _r ), r > 0}.     

Viscosity at small scales in polymer melts

Flows around small colloidal particles of diameter b, or in thin films, capillaries, etc., cannot always be described in terms of the macroscopic polymer viscosity. We discuss these features for entangled polymer melts, where two distinct regimes can be found: (a) the thin regime where b is smaller than the coil radius R₀, but larger than the diameter of the Edwards tube; (b) the ultrathin regime, where . We consider (i) non adsorbing particles, where slippage may occur between the melt and the solid surface; (ii) “hairy” particles, which carry some bound polymer chains. We obtain scaling predictions for mobilities of spheres, of needles, and of clusters of particles. We also discuss translational and rotational diffusion of needles. Received 19 April 1999     

Study on B → φK * and B d → φK S decays in a flavor changing Z′ model

We study the polarization anomaly of the B _d → φK ^*0 decay and the time-dependent CP asymmetry S _φKS of the B _d → φK _S decay in a Z′ model associated with flavor changing neutral currents at the tree level. We obtain the results of 0.0113 < ξ < 0.0127 and 0.0226 < ξ < 0.0255 for the two decay processes when setting ξ _LL = ξ _RL = ξ _LR = ξ _RR = ξ, ξ _LL = ξ _RL = ξ, and ξ _LR = ξ _RR = 0, respectively. These results are consistent with the constraints and assumptions on the model parameters in some references. Supported by the National Natural Science Foundation of China (Grant No. 10575029), the Natural Science Foundation of Henan Province of China (Grant No. 0611050300), the Natural Science Foundation of Henan Provincial Educational Committee (Grant No. 2007140013), and the High-qualified Talents Scientific Research Startup Foundation of Nanyang Normal University (Grant No. nytc2006k92)     

The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

Study of CdSe/CdS quantum dots in solutions and gels by small-angle X-ray scattering

The sizes of semiconductor nanocrystals of CdSe/CdS quantum dots (QDs) synthesized by the colloidal method were estimated using small-angle X-ray scattering. The distribution of QD nanocrystals in organic solvents of different polarities and in polymer gels and matrices is studied. Structural invariants of scattering QD particles (heterogeneities of the electron density)—namely, inertia radii and sizes, forms, and dispersive composition of particles—are determined. The contribution of scattering by QDs in solutions and gels is calculated. The effective sizes of particles and their aggregates are determined, and the parameters of the distribution over the QD sizes in organic solvents and polymer matrices are estimated. The typical distance between particles in samples is determined. The position of the maximum at the beginning of the small-angle scattering curves corresponds to the distance d _m = 2π/h ₀ between the planes (here, h ₀ is the position of the maximum on the scale h). It is 74–76.9 ? for solutions, 60 ? for gels, and 99 ? for polymer matrices with concentrations of up to 0.15% and 77 ? for those with the concentrations exceeding 0.15%, which is close to the estimation of the sizes of separate CdSe QDs that was obtained from the distribution histograms (60–80 ?). This result shows that CdSe/CdS QDs introduced in the polymer matrices disperse to form either separate particles or small aggregates and located at a distance on the order of 80 ? from each other.     

Phase behaviour of a dispersion of charge-stabilised colloidal spheres with added non-adsorbing interacting polymer chains

We present a theory for the phase behaviour of mixtures of charge-stabilised colloidal spheres plus interacting polymer chains in good and θ -solvents within the framework of free-volume theory. We use simple but accurate combination rules for the depletion thickness around a colloidal particle and for the osmotic pressure up to the semi-dilute concentration regime. Hence, we obtain expressions for the free energy for mixtures of charged colloidal particles and non-adsorbing interacting polymers. From that, we calculate the phase behaviour, and discuss its topology in dependence on the competition between the charge-induced repulsion and the polymer-induced attraction. The homogeneous mixture of colloids and polymers becomes more stabilised against demixing when increasing the electrostatic repulsion. This charge-induced stabilisation is strongest for small polymer-to-colloid size ratios and is more pronounced for charged colloids mixed with polymers in a good solvent than for polymers in a θ -solvent. For the weakly charged regime we find that the phase diagram becomes salt-concentration-independent in the protein limit for charged colloids plus polymers in a θ -solvent. The liquid window, i.e., the concentration regimes where a colloidal liquid exists, is narrowed down upon increasing the charge-induced repulsion. Also this effect is more pronounced when charged colloids are mixed with polymer chains in a good solvent. In summary, we demonstrate that the solvent quality significantly influences the phase behaviour of mixtures of charged colloids plus non-adsorbing polymers if the range of the screened electrostatic repulsion becomes of the order of the range of the depletion-induced attraction.     

Three-dimensional correlated-fermion phase separation from analysis of the geometric mean of the individual susceptibilities

A quasi-Gaussian approximation scheme is formulated to study the strongly correlated imbalanced Fermions thermodynamics, where the mean-field theory is not applicable. The non-Gaussian correlation effects are understood to be captured by the statistical geometric mean of the individual susceptibilities. In the three-dimensional unitary fermions ground state, a universal nonlinear scaling transformation relates the physical chemical potentials with the individual Fermi kinetic energies. For the partial polarization phase separation to full polarization, the calculated critical polarization ratio is δ _C = [1−(1−ξ)^6/5]/[1+(1−ξ)^6/5] ≐ 0.34. ξ = 4/9 gives the ratio of the symmetric ground state energy density to that of the ideal fermion gas. Supported by the National Natural Science Foundation of China (Grant Nos. 10875050 and 10675052), the Fund of Central China Normal University, and the Foundation of Ministry of Education of China (Grant No. IRT0624)     

Universal relationships for the phonon spectra in BCC,FCC, and HCP crystals with a short-range interatomic interaction

The frequencies of the phonon branches that correspond to the vibrations of the close-packed atomic planes in bcc, fcc, and hcp crystals with short-range interatomic interaction are shown to be described by a universal relationship, which only contains two parameters for each branch, for any polarization λ. These phonon branches correspond to the (ξ, ξ, 0) direction in bcc crystals, the (ξ, ξ, ξ) direction in fcc crystals, and the (0, 0, ξ) direction in hcp crystals. This universal relationship can only be violated by long-range interactions, namely, the interactions outside the sixth coordination shell in a bcc crystal, the fifth coordination shell in an fcc crystal, and the eleventh or tenth coordination shell in an hcp crystal. The effect of these long-range interactions for each phonon branch can be quantitatively characterized by certain parameters Δ _nλ, which are simply expressed in terms of the frequencies of three phonons of the branch. The values of these parameters are presented for all bcc, fcc, and hcp metals whose phonon spectra are measured. In most cases, the proposed relationships for the frequencies are found to be fulfilled accurate to several percent. In the cases where the Δ _nλ parameters are not small, they can give substantial information on the type and scale of long-range interaction effects in various metals.     

Critical behaviour in solutions of branched polymers and swollen gels as studied by static and dynamic light scattering

Summary We present results of static and dynamic light scattering study of the critical phenomena in semi-dilute solutions and gels of poly(methyl methacrylate) (PMMA) prepared by radical copolymerization of methyl methacrylate and ethylene dimethacrylate in the θ-solvent (4-heptanone). The correlation length ξ and the extrapolated zero-angle scattering intensityI _c,0 of the solutions with degree of cross-linkingf _c≤1% diverge at the pseudospinodal temperature with exponents ofv=0.5 and γ=1, respectively, characteristic of mean-field behaviour. The values ofv and y for samples withf _c≥1.5% are about 0.6 and 1.2, respectively, indicating that the phase transition in solutions of branched PMMA molecules and PMMA gels can be classified to the 3D Ising model. The pseudospinodal temperaturesT _s, for semi-dilute solutions of branched molecules and gels were found as a function of the degree of cross-linking,f _c. The time autocorrelation functions consist of two contributions of which the faster one is attributed to the cooperative motions of the polymer matrix, while the broad slow mode is due to the dynamics of the partly penetrating polydisperse clusters and can be interpreted in terms of a simple cluster diffusion model. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Many-body Hamiltonian with screening parameter and ionization energy

We prove the existence of a Hamiltonian with ionization energy as part of the eigenvalue, which can be used to study strongly correlated matter. This eigenvalue consists of total energy at zero temperature (E ₀) and the ionization energy (ξ). We show that the existence of this total energy eigenvalue, E ₀±ξ, does not violate the Coulombian atomic system. Since there is no equivalent known Hamilton operator that corresponds quantitatively to ξ, we employ the screened Coulomb potential operator (Yukawa-type), which is a function of this ionization energy to analytically calculate the screening parameter (σ) of a neutral helium atom in the ground state. In addition, we also show that the energy level splitting due to spin-orbit coupling is inversely proportional to ξ eigenvalue, which is also important in the field of spintronics.     

Flory radius of polymers in a periodic field: An exact analytic theory

We found an exact expression for the Flory radius R _F of Gaussian polymers placed in an external periodic field. This solution is expressed in terms of the two parameters η and a that describe the reduced strength of an external field and the period of the field to the polymer gyration radius ratio, respectively. R _F is found to be a decaying function of η for any values of a . Provided that the gyration radius is of the order of the period of an external field or less, the ground-state (GS) approximation of the exact result for R _F is shown to give qualitatively incorrect results. In addition to the “ground-state” contribution, the exact solution for R _F contains an additional term that is overlooked by the GS approximation. This term gives rise to the fact that R _F as a function of η exhibits power law behavior (rather than exponential decay obtained from the GS result) once η exceeds the threshold value η_con .     

Immobilization of luminescent nanosilicon in a microfine polytetrafluoroethylene matrix by means of supercritical carbon dioxide

With the use of supercritical carbon dioxide (SC-CO₂), the matrix immobilization of photoluminescent silicon nanocrystals (nc-Si) in polytetrafluoroethylene microparticles (mp-PTFE) is performed, which leads to the formation of mp-PTFE/nc-Si photoluminescent nanocomposite containing ∼10³–10⁴ nc-Si particles per mp-PTFE particle (1–2 μm in size). This approach is based on the effect of polymer swelling in SC-CO₂, efficient SC-CO₂-assisted transport of nanoparticles into the internal free volume of the polymer, and contraction of the nanocomposite after the release of CO₂, an effect that prevents the subsequent agglutination of nanoparticles. Particles of nc-Si photoluminescent in the visible spectrum were synthesized from silicon suboxide powder (SiO _x , x ≈ 1) heated at various temperatures within 25–950°C and then etched in concentrated hydrofluoric acid. The hydrosilylation procedure was used to graft 1-octadecene molecules to the surface of nc-Si particles. As a result, the photoluminescence intensity of nc-Si increased substantially. According to TEM images and small angle X-ray scattering data, the maximum size of nc-Si particles did not exceed 5 nm and 7 nm, respectively, and the core of these nanoparticles consisted of crystalline silicon. The structure and spectral properties of the initial nc-Si particles and synthesized mp-PTFE/nc-Si photoluminescent nanocomposite microparticles were studied.     

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G _D(r,r′, N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r′ . The expression for G(r,r′, s) is found to be the sum of the commonly accepted Dirichlet result G _D(r,r′, N) = G ₀(r,r′, N) - G ₀(r,r”, N) , where r” is the mirror image of r′ , and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.     

Convergence and gauge dependence properties of the resummed one-loop quark-quark scattering amplitude in perturbative QCD

The one-loop QCD effective charge α _s ^eff for quark-quark scattering is derived by diagrammatic resummation of the one-loop amplitude using an arbitrary covariant gauge. Except for the particular choice of gauge parameter ξ = −3, α _s ^eff is found to increase with increasing physical scale, Q, as lnQ or ln² Q. For ξ = −3, α _s ^eff decreases with increasing Q and satisfies a renormalization group equation. Also, except for the case ξ = 19/9, convergence radii of geometric series are found to impose upper limits on Q. The text was submitted by the author in English.