Supercooled liquids analogous fractional Stokes–Einstein relation in NaCl solution above room temperature

The Stokes–Einstein relation D～T/η and its two variants D～τ~(-1) and D～T/τ follow a fractional form in supercooled liquids, where D is the diffusion constant, T the temperature, η the shear viscosity, and τ the structural relaxation time.The fractional Stokes–Einstein relation is proposed to result from the dynamic heterogeneity of supercooled liquids.In this work, by performing molecular dynamics simulations, we show that the analogous fractional form also exists in sodium chloride(NaCl) solutions above room temperature.D～τ~(-1) takes a fractional form within 300–800 K; a crossover is observed in both D～T/τ and D～T/η.Both D～T/τ and D～T/η are valid below the crossover temperature T_x,but take a fractional form for T T_x.Our results indicate that the fractional Stokes–Einstein relation not only exists in supercooled liquids but also exists in NaCl solutions at high enough temperatures far away from the glass transition point.We propose that D～T/η and its two variants should be critically evaluated to test the validity of the Stokes–Einstein relation.     

Fractional Stokes–Einstein relation in TIP5P water at high temperatures

Fractional Stokes–Einstein relation described by D ～(τ/T)~ξ is observed in supercooled water, where D is the diffusion constant, τ the structural relaxation time, T the temperature, and the exponent ξ =τ~(-1). In this work, the Stokes–Einstein relation in TIP5 P water is examined at high temperatures within 400 K–800 K. Our results indicate that the fractional Stokes–Einstein relation is explicitly existent in TIP5P water at high temperatures, demonstrated by the two usually adopted variants of the Stokes–Einstein relation, D ～τ~(-1)τand D ～ T/τ, as well as by D ～ T/η, where η is the shear viscosity. Both D ～τ~(-1)τand D ～ T/τ are crossed at temperature T_x= 510 K. The D ～τ~(-1)τis in a fractional form as D ～τξwith ξ =-2.09 for T ≤ T_xand otherwise ξ =τ~(-1).25. The D ～ T/τ is valid with ξ =τ~(-1).01 for T ≤ T_xbut in a fractional form for T T_x. The Stokes–Einstein relation D ～ T/η is satisfied below T_x = 620 K but in a fractional form above T_x. We propose that the breakdown of D ～ T/η may result from the system entering into the super critical region, the fractional forms of D ～τ~(-1)τand D ～ T/τ are due to the disruption of the hydration shell and the local tetrahedral structure as well as the increase of the shear viscosity.     

A Kinetic Transition from Low to High Fragility in Cu-Zr Liquids

Researchers have reported that Cu-Zr liquids are kinetically strong at the best glass-forming compositions. Here we systematically study the temperature dependence of viscosity and diffusion of Cu-Zr liquids using molecular dynamics simulations, and the results illustrate that the better glass formers are actually more fragile close to the glass transition. There is a kinetic transition from low to high fragility when the optimal glass-forming liquids are quenched into glass states. This transition is associated with the more rapid decrease of the excess entropy of the liquids above and close to the glass transition temperature, Tg, compared to other compositions. Accompanied by the transition to high fragility, peaks in the thermal expansivity and specific heat are observed at the optimal compositions. Furthermore, the Stokes Einstein relation is examined over a wide composition range for Cu-Zr alloys, and the results indicate that glass-forming ability closely correlates with dynamical heterogeneity.     

Measurement scheme to detect α relaxation time of glass-forming liquid

A measurement scheme for detecting the α relaxation time(τ) of glass-forming liquid is proposed, which is based on the measured ionic conductivity of the liquid doped with probing ions by low-and middle-frequency dielectric spectroscopy and according to the Nernst–Einstein, Stokes–Einstein, and Maxwell equations. The obtained τ values of glycerol and propylene carbonate by the scheme are consistent with those obtained by traditional dielectric spectroscopy, which confirms its reliability and accuracy. Moreover, the τ of 1,2-propanediol in a larger temperature range is compared with existing data.     

New analytical exact solutions of time fractional KdV KZK equation by Kudryashov methods

In this paper, new exact solutions of the time fractional KdV–Khokhlov–Zabolotskaya–Kuznetsov(KdV–KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann–Liouville derivative is used to convert the nonlinear time fractional KdV–KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV–KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics.The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV–KZK equation.     

Refractive index of ionic liquids under electric field:Methyl propyl imidazole iodide and several derivatives

Ionic liquids have received wide attention due to their novel optoelectronic structures and devices as an optical means of regulating electricity.However,the quantitative testing and analysis of refractive index of ionic liquids under electric field are rarely carried out.In the present study,an experimental apparatus including a hollow prism is designed to measure the refractive indices of ionic liquids under different electric fields.Five groups of imidazole ionic liquids are experimentally investigated and an inversion is performed to determine the refractive indices under electric fields.The error propagation analysis of the apex angle and the minimum deflection angle are conducted,and the machining accuracy requirements of the hollow prism are determined.The results show that the refractive indices of imidazole ionic liquids change with the light wavelength,following a downward convex parabola.Furthermore,the refractive index decreases with the carbon chain length of ionic liquid at a given wavelength,presenting an order of C3MImI>C4MImI>C5MImI>C3MImBr>C3MImBF4.Notably,the refractive index of imidazole ionic liquid exhibits a nonlinear change with the applied voltage at 546 nm and a monotonical decrease at 1529 nm.Besides,the variation of refractive index at 1529 nm with the applied voltage is larger than that at 546 nm and 1013 nm.Importantly,the variation of refractive index is contrary to that of absorption coefficient under electric field.This study illustrates that the theory of electrode and carrier transport can be used to explain the law of variation of n–k value of ionic liquid under the electric field,and provides the support for the evaluation of physical properties of ionic liquids,the measurement of optical functional parameters and the regulation of electric–optic performances of optical devices.     

Revisiting the breakdown of Stokes-Einstein relation in glass-forming liquids with machine learning

The Stokes-Einstein(SE) relation has been considered as one of the hallmarks of dynamics in liquids. It describes that the diffusion constant D is proportional to(τ/T)~(–1), where τ is the structural relaxation time and T is the temperature. In many glassforming liquids, the breakdown of SE relation often occurred when the dynamics of the liquids becomes glassy, and its origin is still debated among many scientists. Using molecular dynamics simulations and support-vector machine method, it is found that the scaling between diffusion and relaxation fails when the total population of solid-like clusters shrinks at the maximal rate with decreasing temperature, which implies a dramatic unification of clusters into an extensive dominant one occurs at the time of breakdown of the SE relation. Our data leads to an interpretation that the SE violation in metallic glass-forming liquids can be attributed to a specific change in the atomic structures.     

Analytical Approach to Space- and Time-Fractional Burgers Equations

A scheme is developed to study numerical solution of the space- and time-fractional Burgers equations under initial conditions by the homotopy analysis method. The fractional derivatives are considered in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.     

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.     

Bright and dark soliton solutions for some nonlinear fractional differential equations

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney(m BBM) equation, the time fractional m Kd V equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense.     

Exp-Function Method and Fractional Complex Transform for Space-Time Fractional KP-BBM Equation

In the present article, He's fractional derivative, the ansatz method, the(G′/G)-expansion method, and the exp-function method are used to construct the exact solutions of nonlinear space-time fractional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony(KP-BBM). As a result, different types of exact solutions are obtained. Also we have examined the relation between the solutions obtained from the different methods. These methods are an efficient mathematical tool for solving fractional differential equations(FDEs) and it can be applied to other nonlinear FDEs.     

Landau damping and frequency-shift of a quadrupole mode in a disc-shaped rubidium Bose–Einstein condensate

The damping and frequency-shift in Landau mechanism of a quadrupole mode in a disc-shaped rubidium Bose–Einstein condensate are investigated by using the Hartree–Fock–Bogoliubov approximation. The practical relaxations of the elementary excitations and the orthometric relation among them are taken into account to obtain advisable calculation formula for damping as well as frequency-shift. The first approximation of Gaussian distribution function is employed for the ground-state wavefunction to suitably eliminate the divergence of the analytic three-mode coupling matrix elements.According to these methods, both Landau damping rate and frequency-shift of the quadrupole mode are analytically calculated. In addition, all the theoretical results agree with the experimental ones.     

Quantum Entanglement of Many Distant Bose–Einstein Condensates in an Optical Lattice by Interference

We propose a scheme to generate maximally entangled states of two distant Bose–Einstein condensates,which are trapped in different potential wells of a one-dimensional optical lattice. We show how such maximally entangled state can be used to test the Bell inequality and realize quantum teleportation of a Bose–Einstein condensate state. The scheme proposed here is based on the interference of Bose-Einstein condensates leaking out from different potential wells of optical lattice. It is briefly pointed out that this scheme can be extended to generate maximally entangled Greenberger–Horne–Zeilinger(GHZ) states of 2m(m 1) distant Bose–Einstein condensates.     

Dark Soliton Solutions of Space-Time Fractional Sharma–Tasso–Olver and Potential Kadomtsev–Petviashvili Equations

Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivative and fractional complex transform. After reducing both equations to nonlinear ODEs with constant coefficients, the tanh ansatz is substituted into the resultant nonlinear ODEs. The coefficients of the solutions in the ansatz are calculated by algebraic computer computations. Two different solutions are obtained for the Sharma–Tasso–Olver equation as only one solution for the potential Kadomtsev–Petviashvili equation. The solution profiles are demonstrated in 3D plots in finite domains of time and space.     

Pressure Effects on the Transport and Structural Properties of Metallic Glass-Forming Liquid

Transport and structural properties of metallic glass-forming liquid Cu_(50)Zr_(50) are investigated by molecular dynamics simulation,under high pressures from 1 bar to 70 GPa.The following results have been obtained:(i)reversals of component diffusion coefficients(D_(Cu) and D_(Zr)) are observed at the reversion pressure.At low pressures below the reversion pressure,D_(Cu) and D_(Zr) decreases from about 1.4 to 1.0.At high pressures above the reversion pressure,D_(Cu) and D_(Zr) decreases more rapidly from 1.0 to about 0.7.(ii) Component diffusion coefficients decay exponentially with pressure up to reversion pressure,then the strength of the exponential dependence changes,while the pressure-dependent behavior of viscosity can be well described by a single exponential relation over the full range of pressure.(iii) The Stokes–Einstein relation(SER) works well at low pressures and starts to be violated at the breakdown pressure.For glass-forming liquid Cu50 Zr50 along the 2000 K isotherm,the breakdown pressure equals the reversion pressure of component diffusion coefficients and is about 35 GPa.(iv) The pressure dependences of the ratio between component diffusion coefficients can be used to predict the breakdown pressure of SER along isotherm.The validity of SER and the reversals of component diffusion coefficients are found to be related to the pressure dependence of the relative total fractions of predominant Voronoi polyhedrons around individual components.     

Solutions to Forced and Unforced Lin–Reissner–Tsien Equations for Transonic Gas Flows on Various Length Scales

The Lin–Reissner–Tsien equation is useful for studying transonic gas flows, and has appeared in both forced and unforced forms in the literature. Defining arbitrary spatial scalings, we are able to obtain a family of exact similarity solutions depending on one free parameter in addition to the model parameter holding the scalings. Numerical solutions compare favorably with the exact solutions in regions where the exact solutions are valid. Mixed wave-similarity solutions, which describe wave propagation in one variable and self-similar scaling of the entire solution, are also given,and we show that such solutions can only exist when the wave propagation is sufficiently slow. We also extend the Lin–Reissner–Tsien equation to have a forcing term, as such equations have entered the physics literature recently. We obtain both wave and self-similar solutions for the forced equations, and we are able to give conditions under which the force function allows for exact solutions. We then demonstrate how to obtain these exact solutions in both the traveling wave and self-similar cases. There results constitute new and potentially physically interesting exact solutions of the Lin–Reissner–Tsien equation and in particular suggest that the forced Lin–Reissner–Tsien equation warrants further study.     

Dynamic crossover in [VIO~(2+)][Tf_2N~-]_2 ionic liquid

Ionic liquids usually behave as fragile liquids,and the temperature dependence of their dynamic properties obeys supper-Arrhenius law.In this work,a dynamic crossover is observed in([VIO²⁺][Tf₂N-]₂) ionic liquid at the temperature of 240-800 K.The diffusion coefficient does not obey a single Arrhenius law or a Vogel-Fulcher-Tammann(VFT) relation,but can be well fitted by three Arrhenius laws or a combination of a VFT relation and an Arrhenius law.The origin of the dynamic crossover is analyzed from correlation,structure,and thermodynamics.Ion gets a stronger backward correlation at a lower temperature,as shown by the fractal dimension of the random walk.The temperature dependence function of fractal dimension,heterogeneity order parameter,and thermodynamic data can be separated into three regions similar to that observed in the diffusion coefficient.The two crossover temperatures observed in the three types of data are almost the same as that in diffusion coefficient fitted by three Arrhenius laws.The results indicate that the dynamic crossover of[VIO2+][Tf2 N-]2 is attributed to the heterogeneous structure when it undergoes cooling.     

Equation of state for aluminum in warm dense matter regime

A semi-empirical equation of state model for aluminum in a warm dense matter regime is constructed. The equation of state, which is subdivided into a cold term, thermal contributions of ions and electrons, covers a broad range of phase diagram from solid state to plasma state. The cold term and thermal contribution of ions are from the Bushman–Lomonosov model, in which several undetermined parameters are fitted based on equation of state theories and specific experimental data. The Thomas–Fermi–Kirzhnits model is employed to estimate the thermal contribution of electrons. Some practical modifications are introduced to the Thomas–Fermi–Kirzhnits model to improve the prediction of the equation of state model. Theoretical calculation of thermodynamic parameters, including phase diagram, curves of isothermal compression at ambient temperature, melting, and Hugoniot, are analyzed and compared with relevant experimental data and other theoretical evaluations.     

Approximate analytical solutions and mean energies of stationary Schr¨odinger equation for general molecular potential

The Schrodinger equation is solved with general molecular potential via the improved quantization rule.Expression for bound state energy eigenvalues, radial eigenfunctions, mean kinetic energy, and potential energy are obtained in compact form. In modeling the centrifugal term of the effective potential, a Pekeris-like approximation scheme is applied. Also, we use the Hellmann–Feynman theorem to derive the relation for expectation values. Bound state energy eigenvalues, wave functions and meanenergies of Woods–Saxon potential, Morse potential, Mobius squared and Tietz–Hua oscillators are deduced from the general molecular potential. In addition, we use our equations to compute the bound state energy eigenvalues and expectation values for four diatomic molecules viz. H_2, CO, HF, and O_2. Results obtained are in perfect agreement with the data available from the literature for the potentials and molecules. Studies also show that as the vibrational quantum number increases, the mean kinetic energy for the system in a Tietz–Hua potential increases slowly to a threshold value and then decreases. But in a Morse potential, the mean kinetic energy increases linearly with vibrational quantum number increasing.