Analytic extensions of the Debye–Hückel approximation to the Poisson–Boltzmann equation

The Poisson–Boltzmann equation (P-B) is used as an analytic model in a wide variety of fields in chemistry and physics, because it describes the charge distribution in a solute. Being highly nonlinear, there are only a few known solutions for simple boundary geometries and, beyond, iterative numerical schemes are often employed. This study, on the other hand, presents a systematic perturbation solution of the P-B using a non-dimensional electrokinetic–thermal energy ratio λ which, when it approaches zero, reduces the P-B to the Debye–Hückel approximation. Perturbation-series solutions are obtained for five basic examples, and lead to the surprising result that, even when λ is as large as 3 or larger, the perturbation solution is very accurate with only a few terms included in the series. This is because the perturbation analysis generates very rapidly vanishing coefficients at higher-order approximations. This result has the important implication that the perturbation method presented in this study could be applied quite generally for investigating more complicated problems.     

On symmetries of the Fokker–Planck equation

This paper is devoted to Lie point symmetries of the Fokker–Planck (FP) equation. It describes the relation between symmetries and first integrals of stochastic differential equations (SDEs) and symmetries of the associated FP equations. This relation is illustrated on symmetries of (1 + 1)-dimensional FP equations specified by Lie group classification of the scalar SDE. Further, it is used to find symmetries of (1 + 2)-dimensional FP equations specified by Lie group classification of the system of two SDEs.     

A generalisation of the Wilshire–Scharning methodology to creep life prediction with application to 1Cr–1Mo–0.25 V rotor steel

Wilshire–Scharning have recently developed a new methodology that has been demonstrated to deliver accurate longer term creep life predictions, and so offers the prospect of cost-effective acquisition of long-term creep design data. This methodology differs from existing approaches to creep life prediction by normalising the applied stress through the appropriate tensile strength. This article develops a generalisation of this Wilshire–Scharning model that has the potential to increase the predictive accuracy of this methodology—which will be so essential if it is to be adopted as a way of economising on the acquisition of creep design data. When applied to 1Cr–1Mo–0.25 V steel, it was found that this generalisation reduced the average error in prediction from 2.5 under the Wilshire–Scharning specification to 1.8 years when extrapolating from 5,000 out to over 100,000 h. Further, over this time scale the generalised model produces a mean absolute percentage error of 28%. This compares to 47% obtained using the traditional 4Θ projection technique and 26% using a modification of this methodology as recently proposed by Evans.     

A semianalytic meshless approach to the transient Fokker–Planck equation

In this paper, a semianalytic partition of unity finite element method (PUFEM) is presented to solve the transient Fokker–Planck equation (FPE) for high-dimensional nonlinear dynamical systems. Meshless spatial discretization of PUFEM with local p$p$-refinement (discussed in a previous paper) is employed to develop linear ordinary differential equations for the time varying coefficients of local shape functions. A similarity transformation to modal coordinates is shown to reveal numerous spurious modes in the eigenspace of the discretized FPE operator. Identification and elimination of these modes leads to an analytical solution of ODEs obtained from spatial discretization in terms of the remaining admissible modes, and a significant reduction in the size of the discretized transient problem. Initial equation error resulting from the set of admissible modes is shown to be an upper bound for all time, meaning that the reduced admissible set is sufficient for the FPE approximation for all time.     

Effect of the Electron–Phonon Coupling on the Effective Thermal Conductivity of Metallic Bilayers

Systems consisting of metallic layers are commonly used in many applications for microelectronics, data storage, protection coatings, and microelectro-mechanical systems. The physical properties of such systems are strongly determined by the flow of the electron and phonon gases and their interactions. In this study, the effective thermal conductivity of a metal–metal bilayer system is studied using the two-temperature model of heat conduction. By defining the total interfacial thermal resistance, it is shown that the thermal conductivity of the bilayer system depends on the ratio between the thicknesses of the metallic layers and their intrinsic coupling length and it has a simple interpretation as the sum of thermal resistances in series. It is demonstrated that the total interfacial thermal resistance can be minimized by choosing appropriately the thermal and geometrical properties of the component layers. The proposed approach could be useful for thermally characterizing and guiding the design of novel metal–metal-layered systems involved in diverse technological applications.     

Identification of an extended Bouc–Wen model with application to seismic protection through hysteretic devices

In this paper is proposed an extended Bouc–Wen model for improving its capability to approximate experimental symmetric hysteretic loops. On the basis of the generalized equation there are defined integral and differential conditions that describe the essential geometric properties of a hysteretic curve. Next, a new method based on Genetic Algorithms is developed to identify the Bouc–Wen model parameters from experimental hysteretic loops obtained from periodic loading tests. The performance of presented approach is illustrated for two types of seismic protection devices with hysteretic characteristics: elastomeric base isolators and buckling restrained dissipative braces. The applicability of proposed method is highlighted by using the derived models to analyse by numerical simulation the efficiency of these devices for reducing seismic response of a three stories civil structure.     

Effect of heat treatment with the mould on the super-elastic property of Ti–Ni alloy castings for dental application

Tensile property of Ti–50.85Ni (mol %) alloy castings was investigated quantitatively in relation to the thermal behavior accompanied with phase transformation to evaluate the effect of heat treatment after casting with the mould in air. The heat treatment temperature was 713 or 773 K, and the period was 0.9, 1.8, or 3.6 ks. Apparent proof stress of the castings decreased with increasing period of heat treatment, and the decrease was larger with the treatment at 773 K. Residual strain also decreased by the heat treatment, however, it was low with the treatment for relatively short period, i.e. 713 K-0.9 and 1.8 ks, and 773 K-0.9 ks treatments. From the thermal behavior measured by differential scanning calorimetry (DSC), the ascent in the transformation temperatures and the increase in the thermal peak height appeared to influence the changes in the tensile property. These changes by heat treatment were believed to be effective to utilize more flexibility, less stress and less permanent deformation in dental castings.     

Universal Features of the Electron Transport in Tungsten–Carbon Nanocomposites

The wide-temperature-range (4.2–300 K) electron transport had being studied in tungsten–carbon nanocomposites in tungsten concentration interval 0.1–0.45. It is shown that electron transport in the nanocomposites possesses the features of the universality, manifested in the form of power-law dependences of the conductivity on temperature in the two characteristic temperature intervals. The critical temperature separating the intervals is about 25–30 K and has no appreciable dependence on the value of tungsten concentration in nanocomposites. The power exponents of the temperature dependences of the conductivity in both temperature intervals are the non-monotonic functions of the tungsten concentration and vary in the range 0–2 with a wide minimum at 0.2 and 0.25 of tungsten content in the high- and low-temperature intervals, respectively. The observed power-law temperature corrections to the conductivity are simulated and discussed within the effective medium approximation in the framework of the model of the inelastic tunneling of the electrons between the conducting clusters in the tungsten–carbon nanocomposites.     

New application of plate–fin heat exchanger with regenerative cryocoolers

A design idea is newly proposed and investigated for the application of plate–fin heat exchanger (PFHX) with regenerative cryocoolers. The role of this heat exchanger is to effectively absorb heat from the stream of coolant and deliver it to the cold-head of a cryocooler. While various types of tubular HX’s have been developed so far, a small PFHX could be more useful for this purpose by taking advantage of compactness and design flexibility. In order to confirm the feasibility and effectiveness, a prototype of aluminum-brazed PFHX is designed, fabricated, and tested with a single-stage GM cryocooler in experiments for subcooling liquid nitrogen from 78 K to 65–70 K. The results show that the PFHX is 30–50% more effective in cooling rate than the tubular HX’s. Several potential applications of PFHX are presented and discussed with specific design concepts.     

Electrical Resistivity of the Mg(B1−x C x )2 Superconductors: Role of Electron–Electron, and Electron–Phonon Interactions

In this paper, we undertake a quantitative analysis of temperature-dependent resistivity Mg(B_1?x C _x )₂ superconductors. Due to inherent two energy gaps, the elastic scatterings of electrons from impurities have first been estimated and within a two-band picture, the impurity-limited resistivity due to π band carriers $\rho_{0}^{\pi} $ is larger as compared to the contribution from σ band carriers. An effective inter-ionic interaction potential (EIoIP) with the long-range Coulomb, van der Waals interaction and the short-range repulsive interaction within the Hafemeister and Flygare approach have allowed us to determine the Debye and Einstein temperature. An investigation exhibiting the mechanism of Mg(B_1?x C _x )₂ (0.0≤x≤0.125) was accomplished by comparing to the resistivity estimated by considering both phonons, with that of reported metallic resistivity, accordingly ρ _diff.=[ρ _exp.?{ρ ₀+ρ _e-ph (=ρ _ac+ρ _op)}] have been analysed through electron–electron scattering. The quadratic temperature dependence of $\rho_{\mathrm {diff}} = [\rho_{\mathrm{exp}.} - \{\rho_{0} + \rho_{\text{e-ph}}\ (= \rho ^{\sigma}_{\text{e-ph}} + \rho^{\pi}_{\text{e-ph}})\}]$ is understood in terms of inelastic electron–electron scattering. The comparison of transport parameter with single crystal data appears consistent within the two-band scheme for resistivity that we have presented.     

Stability analysis of the Fourier–Bessel method for the Cauchy problem of the Helmholtz equation

This paper is concerned with the Cauchy problem connected with the Helmholtz equation in a smooth-bounded domain. The Fourier–Bessel method with Tikhonov regularization is applied to achieve a regularized solution to the problem with noisy data. The convergence and stability are obtained with a suitable choice of the regularization parameter. Numerical experiments are also presented to show the effectiveness of the proposed method.     

Schrödinger wave equation for confinement of photon in a waveguide

In the present work, electromagnetic field confinement in a subwavelength waveguide structure is obtained using concepts of quantum mechanics and uncertainty principle. Semi-macroscopic considerations of field interaction at the dielectric interfaces are used in this work. The modal field profile in the subwavelength waveguide is obtained by considering the photon as a particle in the waveguide having finite probability of tunneling. Thus, uncertainty of position is assigned to it. The momentum uncertainty is calculated from position uncertainty. Schrödinger wave equation for the photon is written by incorporating position-momentum uncertainty. The equation is solved and the field distribution in the waveguide is obtained. The field distribution and power confinement is compared with conventional waveguide theory. They were found in good agreement with each other.     

Improved accuracy of frequency-domain photon migration close to light source using delta–P1 approximation of the adjustable parameter f

The δ–P1 approximation can improve the accuracy of the diffusion equation in the near field. Usually, the method to determine the fractional forward-scattering peak f in the δ–P1 approximation is to compare the first two moments of the Henyey–Greenstein phase function and those of the δ–Eddington phase function. The error relative to Monte Carlo estimation still exists within a mean free path using the value f above. To overcome this defect, we put forward a new method to determine the parameter f for improving the accuracy of frequency-domain photon migration in the δ–P1 approximation if the detector is close to a light source. Results demonstrate that the adjustable f δ–P1 approximation can accurately predict photon fluence rate at distances less than half the transport mean free paths from the source.     

Comment on ‘Wigner equation of motion for time-dependent potentials’

Abstract

A correction is pointed out to a statement made by Galleani and Cohen (2002, J. mod. Optics, 49, 561) regarding the impossibility of deriving an equation of evolution for the Wigner function corresponding to the Schrödinger equation with a time-dependent potential.     

The Neumann series as a fundamental solution of the two-dimensional convection–diffusion equation with variable velocity

The fundamental solution of the two-dimensional convection–diffusion equation with variable coefficients and its adjoint equation are obtained in complex form in terms of the unknown density of two equivalent uniquely solvable Volterra integral equations of the second kind whose analytical solutions are given explicitly as convergent Neumann series. The Volterra integral equations are obtained by integrating the complex form of the original differential equations, without additional change of variables as proposed by previously authors. In the numerical examples, cases corresponding to non-self-adjoint operators are considered. As a validation, the proposed approach is used to derive the fundamental solution of the adjoint to the convection–diffusion equation with constant velocity. In this case, the series solution can be evaluated analytically. For more general velocity fields, the recursive terms of the series can be evaluated by symbolic computation or numerical integration.     

Effect of hydrolysis on the phase evolution of water-based sol–gel hydroxyapatite and its application to bioactive coatings

In a previous report, we demonstrated a successful synthesis of crystalline hydroxyapatite (HA) through the use of a water-based sol–gel process. It was shown that the apatite can be obtained at temperatures generally below 400 °C, providing a great advantage for practical bioactive coating purposes. The influence of hydrolysis of phosphorus sol solution on the phase evolution of the resulting HA is the focus of this investigation. Experimental results show that, in the absence of acid catalyst, a long-term hydrolysis, i.e. >4 h, is required for better evolution of apatitic phase. Such a phase evolution is mainly attributed to an increased concentration of apatitic phase, rather than improved crystallinity in the calcined gels. With the aid of acid catalyst, we found that a well-crystalline HA can be synthesized over a time period shorter by 2–3 orders of magnitude than those without catalyst, i.e. a few minutes. In almost all cases, a small amount of tricalcium phosphate (TCP) was detected, which may be explainable by the formation of oligomeric derivatives of the phosphorus sol during synthesis, where calcium phosphate derivatives with lower Ca/P ratio than stoichiometry can be developed. By selecting an optimal sol as a dipping source, highly-porous dental root specimens were coated and a thin, dense, adhesive (upon finger-nail scratching test) coating was achieved after calcinations at 375 °C. An in vitro test also shows a bioactive character of the coating.     

Extension of the Reiner–Riwlin equation to determine modified Bingham parameters measured in coaxial cylinders rheometers

The determination of the exact rheological properties, in fundamental units, of cementitious materials has become gradually a necessary step in the domain of concrete science. Several types of rheometers and their corresponding transformation equations are described in the literature. In this paper, the Reiner–Riwlin transformation equation, valid for coaxial cylinders rheometers, is developed for the modified Bingham model, which is an extension of the Bingham model with a second order term in the shear rate. The established transformation is shown to be compatible with the Reiner–Riwlin equation for the Bingham and Herschel–Bulkley models. Its validation is further proven by means of numerical simulations applied on experimental data. The yield stress values for the three rheological models (applied on the same experimental data) are compared with the yield stress calculated by means of slump flow values. Results showed that the modified Bingham model results in the most stable yield stress values, which are independent of the non-linear behaviour.