Chemical diffusion in molybdenum disulphide

Ceramic molybdenum disulphide (MoS₂) was equilibrated at an ambient sulphur vapour partial pressure p(S₂), 10 Pa<p(S₂)<1000 Pa. After the step change of p(S₂) to a new value, the equilibration kinetics was monitored by measuring electrical conductivity. The application of the solution of Fick's second law (with the initial condition: no concentration gradient in specimen and the boundary condition: surface concentration constant) to the kinetic data gave the chemical diffusion coefficient. The chemical diffusion coefficient, D_chem, determined at 1273 K, was D_chem=(3.20±0.32)*10⁻⁷ cm² s⁻¹ and was found to be independent of sulphur vapour partial pressure. The usefulness of transient electrical conductivity method for determining real values of diffusion data was discussed in terms of defect structure of the studied material.     

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

Exact solutions for a diffusion equation with a nonlinear external force

This work is devoted to investigate the solutions of the one-dimensional diffusion equation by taking the nonlinear external force F(x,t;ρ)=−k(t)x+K/x+κx|x|^α−1η[ρ(x,t)] into account. Our investigation is first performed by considering the case α=0 and η=1, which results in a Burgers like equation with a spatial and time dependent external force. After, we consider the case α≠0 and η=α+1 and show that the solution found may be expressed in terms of the q-exponential functions present in the Tsallis formalism. In addition, we also discuss the stationary solution for α and η arbitraries.     

Critical behavior of the two-dimensional first passage time

We study the two-dimensional first passage problem in which bonds have zero and unit passage times with probabilityp and 1–p, respectively. We prove that as the zero-time bonds approach the percolation thresholdp _c, the first passage time exhibits the same critical behavior as the correlation function of the underlying percolation problem. In particular, if the correlation length obeys(p) ¦p–p _c¦^–v, then the first passage time constant satisfies(p)¦p–p _c¦^v. At p_c, where it has been asserted that the first passage time from 0 tox scales as ¦x¦ to a power with 0<<1, we show that the passage times grow like log ¦x¦, i.e., the fluid spreads exponentially rapidly.     

The effect of duration of diffusion on Ag diffusion coefficients in YBa2Cu3O7

The Ag diffusion in superconducting YBa₂Cu₃O₇ (YBaCuO) ceramic has been studied over the duration of the diffusion range 5-24 h in the temperature range 700-850 °C by the energy-dispersive X-ray fluorescence (EDXRF) technique. For the excitation of silver atoms, an annular Am-241 radioisotope source (50 mCi) emitting 59.543 keV photons was used. The temperature dependences of silver diffusion coefficients in grains (D₁) and over the grain boundaries in the range 700-850 °C (D₂) are described by the relations D₁=1.4×10⁻² exp[−(1.18±0.10)/kT] and D₂=3.1×10⁻⁴ exp[−(0.87±0.10)/kT].     

Fractional nonlinear diffusion equation and first passage time

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passage time distribution, the mean first passage time, and the mean squared displacement corresponding to different time-dependent diffusion coefficient. In addition, we compare our results for the first passage time distribution and the mean first passage time with the one obtained by usual linear diffusion equation with time-dependent diffusion coefficient.     

Causal particle detectors and topology

We investigate particle detector responses in some topologically non-trivial spacetimes. We extend a recently proposed regularization of the massless scalar field Wightman function in four-dimensional Minkowski space to arbitrary dimension, to the massive scalar field, to quotients of Minkowski space under discrete isometry groups and to the massless Dirac field. We investigate in detail the transition rate of inertial and uniformly accelerated detectors on the quotient spaces under groups generated by (t, x, y, z) ? (t, x, y, z + 2a), (t, x, y, z) ? (t, −x, y, z), (t, x, y, z) ? (t, −x, −y, z), (t, x, y, z) ? (t, −x, −y, z + a) and some higher dimensional generalizations. For motions in at constant y and z on the latter three spaces the response is time dependent. We also discuss the response of static detectors on the RP³ geon and inertial detectors on RP³ de Sitter space via their associated global embedding Minkowski spaces (GEMS). The response on RP³ de Sitter space, found both directly and in its GEMS, provides support for the validity of applying the GEMS procedure to detector responses and to quotient spaces such as RP³ de Sitter space and the RP³ geon where the embedding spaces are Minkowski spaces with suitable identifications.     

Time decay of excitations in the one-dimensional trapping problem

An exact solution is obtained for the survival fraction in the one-dimensional diffusion problem with randomly distributed deep traps. The time decay is studied both with and without a bias field. The small concentration (x) long time (t) decay behaves as exp[-(x ² t/t ₀)^1/3]. The exact results are compared with the coherent potential approximation (CPA) and the first passage time approach (FPT). We find that in most cases of practical interest the FPT is superior to the CPA.     

First-Passage and Extreme-Value Statistics of a Particle Subject to a Constant Force Plus a Random Force

We consider a particle which moves on the x axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point x ₁ of a particle which starts at x ₀ with velocity v ₀. The probability that the particle has not yet arrived at x ₁ after a time t, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement m=max  _t [x(t)].     

Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation

We study the Cauchy problem of generalized Boussinesq equation u_tt−u_xx+(u_xx+f(u))_xx=0, where f(u)=±|u|^p or ±|u|^p−1u, p>1. By introducing a family of potential wells we obtain invariant sets, vacuum isolating and threshold result of global existence and nonexistence of solution.     

Size-dependent microstructure and europium site preference influence fluorescent properties of Eu-doped Ca10(PO4)6(OH)2 nanocrystal

In this study, Eu³⁺-doped nanocrystalline Ca₁₀(PO₄)₆(OH)₂ (Ca_10−xEu_x(PO₄)₆(OH)₂) with different particle sizes have been prepared by the thermal decomposition of precursors. Size-dependent microstructure could be observed in nanocrystalline Ca_10−xEu_x(PO₄)₆(OH)₂. The lattices of Ca_10−xEu_x(PO₄)₆(OH)₂ nanocrystals were more distorted in comparison with the bulk, and the smaller the particle size, the more distorted the lattices. Room temperature photoluminescence showed europium site preference was also size-dependent, with the majority of Eu³⁺ ions occupying Ca(II) sites in the bulk, but more and more Eu³⁺ ions occupying Ca(I) sites in Ca_10−xEu_x(PO₄)₆(OH)₂ with decreasing particle size. Fluorescent properties of Ca_10−xEu_x(PO₄)₆(OH)₂ were considered to be influenced by both microstructure and site preference of Eu³⁺ ions. An abnormal strong intensity of ⁵D₀-⁷F₀ transition was observed in bulk and larger Ca_10−xEu_x(PO₄)₆(OH)₂ nanocrystals, but the relative intensities of ⁵D₀-⁷F₀ transition to ⁵D₀-⁷F_1,2,3,4 transition of Eu³⁺ became weaker as the particle sizes decreased. As the particle sizes became smaller, the ratios of the red emission transition (⁵D₀-⁷F₂) to the orange emission transition (⁵D₀-⁷F₁) (R/O values) first increased by comparing the bulk sample with 96 nm sample, and then decreased by comparing 96 nm sample to 57 nm sample. The quenching concentrations of Ca_10−xEu_x(PO₄)₆(OH)₂ samples increased with decreasing particle size. Possible mechanisms responsible for these phenomena were proposed. Since nanosized Ca_10−xEu_x(PO₄)₆(OH)₂ showed higher fluorescent intensities, higher R/O values and higher quenching concentrations, this material is considered to be a promising phosphor.     

Random time averaged diffusivities for Lévy walks

We investigate a Lévy walk alternating between velocities ±v ₀ with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is ?x ²? ∝ t ², the latter to enhanced diffusion with ?x ²? ∝ t ^ν, 1 < ν < 2. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.     

On the first passage time and leapover properties of Lévy motions

We investigate two coupled properties of Lévy stable random motions: the first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly attracted any attention. Considering a particle that starts at the origin and performs random jumps with independent increments chosen from a Lévy stable probability law λ_α,β(x), the FPT measures how long it takes the particle to arrive at or cross a target. The FPL addresses a different question: given that the first passage jump crosses the target, then how far does it get beyond the target? These two properties are investigated for three subclasses of Lévy stable motions: (i) symmetric Lévy motions characterized by Lévy index α(0<α<2) and skewness parameter β=0, (ii) one-sided Lévy motions with 0<α<1, β=1, and (iii) two-sided skewed Lévy motions, the extreme case, 1<α<2, β=−1.     

A parallel algorithm for solving the 3d Schrödinger equation

We describe a parallel algorithm for solving the time-independent 3d Schrödinger equation using the finite difference time domain (FDTD) method. We introduce an optimized parallelization scheme that reduces communication overhead between computational nodes. We demonstrate that the compute time, t, scales inversely with the number of computational nodes as t ∝ (N_nodes)^{−0.95 ± 0.04}. This makes it possible to solve the 3d Schrödinger equation on extremely large spatial lattices using a small computing cluster. In addition, we present a new method for precisely determining the energy eigenvalues and wavefunctions of quantum states based on a symmetry constraint on the FDTD initial condition. Finally, we discuss the usage of multi-resolution techniques in order to speed up convergence on extremely large lattices.     

Calculation of the diffusion coefficient in acoustic turbulence

The first (Born) approximation commonly used to calculate the diffusion coefficient D_T of a passive scalar in acoustic turbulence is shown to be insufficient. Even for a small main parameter—the Mach number, M?1—the next approximation gives a larger contribution to D_T than does the first approximation, but negative in sign. We present a procedure for correctly calculating D_T based on the solution of a nonlinear DIA (direct interaction approximation) equation for the mean Green’s function of the problem. We include an additional term in the general formula for D_T that directly describes the compressibility of acoustic turbulence. This term has not been known previously and has been disregarded even in the Born approximation. A positive value was obtained for D_T=CM³u₀/p₀. The spectrum E(x) was assumed to be smooth at distances Δ x～M²?1.     

Usefulness of diffusion derived vessel density computed from a simplified IVIM imaging protocol: An experimental study with rat biliary duct blockage induced liver fibrosis

ObjectivesLiver vessel density can be evaluated by DDVD (diffusion derived vessel density): DDVD_(b0b1) = Sb0/ROIarea0 – Sb1/ROIarea1, where Sb0 and Sb1 refer to the liver signal when b is 0 or 1 s/mm². Sb1 and ROIarea1 may be replaced by other b-values. With a rat biliary duct ligation (BDL) model, this study assesses the usefulness of liver DDVD computed from a simplified IVIM imaging protocol using b = 25 and b = 50 to replace b = 1 s/mm², alone and in combination with other IVIM parameters.MethodsMale Sprague-Dawley rats were used. The rat number was 5, 5, 5, and 3 respectively, for the timepoints of 7, 14, 21, 28 days post-BDL surgery. 12 rats had partial biliary duct recanalization performed after the rats had BDL for 7 days and then again followed-up for a mean of 14 days. Liver diffusion MRIs were acquired at 3.0 T with a b-value distribution of 0, 25, 50, 75, 100, 150, 300, 700, 1000 s/mm². DDVD_mean (control rats n = 6) was the mean of DDVD_(b0b25) and DDVD_(b0b50). IVIM fitting started from b = 0 s/mm² with segmented fitting and a threshold b of 50 s/mm² (n = 5 for control rats). Three 3-D spaces were constructed using a combination of the four diffusion parameters.ResultsThe control rats and BDL rats (n = 18) had a liver DDVD_mean of 84.0 ± 26.2 and 44.7 ± 14.4 au/pixel (p < 0.001). All 3-D spaces totally separated healthy livers and all fibrotic livers (n = 30, BDL rats and recanalization rats). The mean relative distance between healthy liver cluster and fibrotic liver cluster was 0.331 for PF, D_slow, and D_fast; 0.381 for PF, D_fast, and DDVD_mean; and 0.384 for PF, D_slow, and DDVD_mean.ConclusionA combination of PF, D_slow, and D_fast allows total separation of healthy livers and fibrotic livers and the integration of DDVD improved the separation.     

Reconnection dynamics for quantized vortices

By analyzing trajectories of solid hydrogen tracers in superfluid ⁴He, we identify tens of thousands of individual reconnection events between quantized vortices. We characterize the dynamics by the minimum separation distance δ(t) between the two reconnecting vortices both before and after the events. Applying dimensional arguments, this separation has been predicted to behave asymptotically as δ(t)≈A(κ|t−t₀|)^1/2, where κ=h/m is the quantum of circulation. The major finding of the experiments and their analysis is strong support for this asymptotic form with κ as the dominant controlling feature, although there are significant event to event fluctuations. At the three-parameter level the dynamics may be about equally well-fit by two modified expressions: (a) an arbitrary power-law expression of the form δ(t)=B|t−t₀|^α and (b) a correction-factor expression δ(t)=A(κ|t−t₀|)^1/2(1+c|t−t₀|). The measured frequency distribution of α is peaked at the predicted value α=0.5, although the half-height values are α=0.35 and 0.80 and there is marked variation in all fitted quantities. Accepting (b) the amplitude A has mean values of 1.24±0.01 and half height values of 0.8 and 1.6 while the c distribution is peaked close to c=0 with a half-height range of −0.9 s⁻¹ to 1.5 s⁻¹. In light of possible physical interpretations we regard the correction-factor expression (b), which attributes the observed deviations from the predicted asymptotic form to fluctuations in the local environment and in boundary conditions, as best describing our experimental data. The observed dynamics appear statistically time-reversible, which suggests that an effective equilibrium has been established in quantum turbulence on the time scales (≤0.25 s) investigated. We discuss the impact of reconnection on velocity statistics in quantum turbulence and, as regards classical turbulence, we argue that forms analogous to (b) could well provide an alternative interpretation of the observed deviations from Kolmogorov scaling exponents of the longitudinal structure functions.     

A diffusion model of metal surface modification during plasma nitriding

A diffusion model of metal surface modification by plasma nitriding has been developed. This model takes into account the erosion effects at the plasma/solid interface occurring due to the ion bombardment of the surface. For constant sputtering rate, which is the usual situation during plasma nitriding, the growth of the sub-layers is well described by the analytical expressiong(t) =g ₀,f ^–1 (t/t ₀), whereg(t) is the sub-layer thickness at timet,g ₀ andt ₀ are parameters which depend on the treated material and plasma characteristics,f ^–1 is the inverse of the function — In(1 - x) + x), 0 x 1. Under negligible erosion effects, the expression forg(t) reduces to the parabolic law. The diffusion zone (substratum) growth does not follow the parabolic law as well. However, the deviation occurs after long plasma nitriding time. The model can be used for experimentally determining the effective diffusion coefficients and the erosion rate during plasma nitriding of metal surfaces.