2D general nonlinear parabolic equation for unstable wave package envelope in structural macrokinetics

We reduce the mathematical model for a chemical reaction in a moving medium to the general nonlinear parabolic equation (GNPE) for the complex amplitude of envelope wave to analyse weakly nonlinear interactions in supercritical regions. We use the method of many scales, wave packages, modification of Mandelshtam method and take into account the group velocity of envelope wave that is typical for nonlinear dispersive medium. GNPE describes the long-term system behaviour after stability loss and its coefficients are explicitly expressed through parameters of the initial nonlinear partial differential equations. It is taking into account the location of wave packet centre out of harmonics with maximum increment.     

Fourier transform rheology as a universal non-linear mechanical characterization of droplet size and interfacial tension of dilute monodisperse emulsions

A new protocol to gain interfacial tension and droplet size of dilute monodisperse emulsions from Fourier Transform Rheology (FTR), is proposed. Specifically, a universal dimensionless quantity E was found at small strain amplitudes to correlate with the droplet size of the emulsion where E is inversely related to the square of the capillary number Ca and directly proportional to the relative intensities of the fifth and third harmonics, I(5)/I(3). The limiting value E(0) at small strain deformations can be used as a universal parameter to calculate different emulsion properties. Different morphological constitutive models for emulsions were used to establish the universality of the parameter E(0). Preliminary analysis on experimental data confirms the validity of this approach for the characterization of emulsion properties, including the estimation of interfacial tension and droplet radius.     

Scaled equations for the coexistence curve,the capillary constant and the surface tension of n-alkanes

A review of experimental data of several fluids shows that their coexistence curve follows a power law in reduced temperature at the approach of the critical point, with an universal exponent equal to 0.325, their capillary constant a power law with an universal exponent equal to 0.925 and their surface tension a power law with an universal exponent equal to 1.26. In the critical region, the concept of two-scale-factor universality was used to predict the density difference amplitude, the capillary constant amplitude, and the surface tension amplitude between near critical vapor and liquid phases. A comparison with amplitudes determined from experimental data is given. In order to extend this universality all along the liquid–gas coexistence curve from the triple point to the critical point for n-alkanes, a mean field approximation was used far away from T_C. We show that the density difference, the capillary constant and the surface tension can be calculated with a reasonable accuracy by generalized scaled equations adding only two empirical constants. A comparison between calculated and experimental data is presented.     

Hofmeister effect on the interfacial dynamics of single polymer molecules

The Hofmeister effect on interfacial dynamics has been discovered for single charged polymer molecules (sodium polystyrene sulfonate) adsorbed on a hydrophobic surface from an aqueous solution. The presence of ions in the aqueous solution affects the surface diffusivity, and its amplitudes and the surface friction follow the Hofmeister series-the kosmotropic ions slowed down the surface diffusivity and the chaotropic ions speeded it up. The amplitude of the surface friction exhibits a good correlation with the surface tension increment, indicating the interfacial feature of the Hofmeister effect.     

Nonlinear alternating current responses of electrorheological solids

When a composite containing nonlinear dielectric particles suspended in a host medium is subjected to a sinusoidal alternating current (ac) electric field, the dielectric response of the composite will generally consist of ac fields at frequencies of higher order harmonics. For an electrorheological (ER) solid under structure transformations due to external fields, we apply the Ewald-Kornfeld formulation to derive the local electric fields and induced dipole moments explicitly, and then we perform the perturbation expansion method to extract their fundamental and third-order harmonics analytically. It is shown that the degree of anisotropy of the ER solid can affect these harmonics significantly. Our results are well understood in the spectral representation theory. Thus, it seems possible to perform a real-time monitoring of the structure transformation by measuring the nonlinear ac responses of ER solids.     

Turing pattern amplitude equation for a model glycolytic reaction-diffusion system

For a reaction-diffusion system of glycolytic oscillations containing analytical steady state solution in complicated algebraic form, Turing instability condition and the critical wavenumber at the Turing bifurcation point, have been derived by a linear stability analysis. In the framework of a weakly nonlinear theory, these relations have been subsequently used to derive an amplitude equation, which interprets the structural transitions and stability of various forms of Turing structures. Amplitude equation also conforms to the expectation that time-invariant amplitudes are independent of complexing reaction with the activator species.     

Theory of nonlinear optical properties of small metallic spheres

We calculate the nonlinear optical properties of small metallic spheres using electromagnetic theory and assuming that the local response of the conduction electrons is the same as for a plane surface. Electromagnetic Mie-resonances cause a strong increase of the second and higher harmonics in the reflected light. Detailed results are given for the second and third harmonic generation, its dependence on the frequency and polarization of the incident light, and on the cluster size. An enhancement of the second harmonic generation by a factor of about 5000 is obtained for small spherical metallic clusters. This is in good agreement with experiments on artificially roughened metal surfaces.     

Nonlinear EHD Stability of the Interfacial Waves of Two Superposed Dielectric Fluids

The slow modulation of the interfacial capillary–gravity waves of two superposed dielectric fluids with uniform depths and solid horizontal boundaries, under the influence of a normal electric field and in the absence of surface charges at their interface, is investigated by using the multiple-time scales method. It is found that the complex amplitude of quasi-monochromatic traveling waves can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. The stability characteristics of a uniform wave train are examined analytically and numerically on the basis of the nonlinear Schrödinger equation, and some limiting cases are recovered. Three cases appear, depending on whether the depth of the lower fluid is equal to, greater than, or less than the depth of the upper fluid. The effect of the normal electric field is determined for the three stability regions of the pure hydrodynamic case. It is found that the normal electric field has a destabilizing influence in the first stability region and a stabilizing effect in the second and third stability regions. Moreover, one new unstable region or two new stable and unstable regions appear, all of which increase when the electric field increases. On the other hand, the complex amplitude of quasi-monochromatic standing waves near the cutoff wavenumber is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation makes it possible to estimate the nonlinear effect on the cutoff wavenumber.     

Experimental demonstration and simulation of electrochemical non-linear responses to glucose and its interferents with an amperometric sensor

A novel sensing system based on the multi-dimensional information contained in a dynamic non-linear response is proposed. A sinusoidal potential was applied to an amperometric-type glucose sensor and the resulting current of the sensor was analyzed by fast Fourier transformation (FFT). The amplitudes of the higher harmonics of FFT characterize the non-linear properties of the response. The amplitudes of the higher harmonics of FFT exhibit characteristic changes which depend on the concentration and the kinetics of the reactions of glucose and its interferents at the sensor surface. The essential features of the current-potential curve were reproduced by a computer simulation based on the kinetics of electrochemical reactions.     

Alternative reference system in the thermodynamics of solid elastic electrodes

Basic quantities in the thermodynamics of the solid elastic electrode are the surface tension tensor g _mn and the work needed for the formation of the surface (interface) γ. It is scarcely mentioned explicitly anywhere that these intensive (specific) quantities are related to the surface of the elastically deformed electrode. On the other hand, in the thermodynamics of the volume elasticity, the free energy density of the deformed solid is related to the volume of the undeformed solid. In this paper, we introduce equivalently the undeformed surface of the solid elastic electrode as reference for both the surface tension tensor and the work of formation of the surface. Generalizing the analysis of two model systems, we deduce the corresponding alternative form of the Shuttleworth equation, where the two quantities appear as generalized force and generalized potential, and discuss consequences for the formulation of the differential of the surface excess of the internal energy.Dedicated to the memory of W. Schwabe     

Fatigue behavior of polystyrene (PS) analyzed from the Fourier transform (FT) of stress response: First evidence of I2/1(N) and I3/1(N) as new fingerprints

The fatigue behavior of polystyrene (PS) in strain controlled torsion rectangular oscillatory tests was analyzed via Fourier transform (FT) to better understand the time evolution of linear and nonlinear mechanical parameters, and to establish fingerprints related to failure onset. The tests were performed under large amplitude oscillatory shear (LAOS), so the stress response was no longer perfectly sinusoidal and higher harmonics could be detected and quantified in the FT spectra as a function of time or number of cycles N. A linear parameter, the storage modulus (G′(N)), was analyzed, as well as nonlinear parameters of the ratios of the second (I₂(N)) and the third (I₃(N)) harmonics over the fundamental one (I₁(N)). The nonlinear parameter I_2/1(N) is very low for undamaged samples, but its intensity was found to increase when defects were created in the structure to a point where cracks became visible in the sample. On the other hand, the I_3/1(N) parameter increased steadily during a test up to a local maximum where a macroscopic crack occurs in the sample. Both parameters I_2/1(N) and I_3/1(N) are proposed as new criteria to detect the onset of part failure under the conditions tested and can be used as safety limits for partial damage.     

Experimental observation and prediction of interfacial tension and viscoelastic emulsion model behavior in novel phosphate glass-polymer hybrids

The interfacial tension of hybrids composed of a tin-based phosphate glass (Pglass) and thermoplastic polymers, low-density polyethylene (LDPE), polystyrene (PS), and polypropylene (PP) was investigated using pendant drop and droplet deformation methods. High surface tension values were determined for the pure Pglass and subsequently used to obtain interfacial tension values that were found to be greater than that of most polymer blends reported in the literature. Small amplitude oscillatory shear data were fitted to the Choi-Schowalter and Palierne emulsion models in order to estimate the interfacial tension and to validate the accuracy (or lack thereof) of using a polymer emulsion model on the special Pglass-polymer systems. Although some of the hybrids showed satisfactory agreement with the emulsion models, wide ranges of interfacial tensions were obtained, suggesting that a more complicated theory that explicitly takes the Pglass-polymer interactions, shape factor, and size distributions of the dispersed Pglass phase into account may be necessary for more accurate modeling of these special hybrid systems with enhanced benefits.     

A universal criterion of melting

Melting is analyzed dynamically as a problem of localization at a liquid-solid interface. A Lindemann-like criterion of melting is derived in terms of particular vibrational amplitudes, which turn out to equal a universal quotient (about one-tenth) of the molecular spacing, at the interface. The near universality of the Lindemann ratio apparently arises owing to strongly overdamped dynamics near melting, and despite the anharmonic interactions being system-specific. A similar criterion is derived for structural displacements in the bulk of the solid, in particular the premelted layer; the criterion is no longer strictly universal, but still depends only on the harmonic properties of the solid. We further compute the dependence of the magnitude of the elemental molecular translations, in deeply supercooled fluids, on the temperature and the high frequency elastic constants. We show explicitly that the surface tension between distinct liquid states, near the glass transition of a supercooled liquid, is nearly evenly split between entropic and energetic contributions.     

Oscillating drop/bubble tensiometry: effect of viscous forces on the measurement of interfacial tension

The oscillating drop/bubble technique is increasingly popular for measuring the interfacial dilatational properties of surfactant/polymer-laden fluid/fluid interfaces. A caveat of this technique, however, is that viscous forces are important at higher oscillation frequencies or fluid viscosities; these can affect determination of the interfacial tension. Here, we experimentally quantify the effect of viscous forces on the interfacial-tension measurement by oscillating 100 and 200 cSt poly(dimethylsiloxane) (PDMS) droplets in water at small amplitudes and frequencies ranging between 0.01 and 1 Hz. Due to viscous forces, the measured interfacial tension oscillates sinusoidally with the same frequency as the oscillation of the drop volume. The tension oscillation precedes that of the drop volume, and the amplitude varies linearly with Capillary number, Ca=DeltamuomegaDeltaV/gammaa(2), where Deltamu=mu(D)-mu is the difference between the bulk Newtonian viscosities of the drop and surrounding continuous fluid, omega is the oscillation frequency of the drop, DeltaV is the amplitude of volume oscillation, gamma is the equilibrium interfacial tension between the PDMS drop and water, and a is the radius of the capillary. A simplified model of a freely suspended spherical oscillating-drop well explains these observations. Viscous forces distort the drop shape at Ca>0.002, although this criterion is apparatus dependent.     

The molecular dynamics problem in multiphoton excitation

While the molecular dynamics for weakly coupled, harmonic oscillators undergoing infinitesimal amplitude displacements are well described by normal modes, and the other extreme — strongly coupled anharmonic, large amplitude oscillating units can be treated as though they are ergodic systems, so that only state densities are needed, intermediate cases are very difficult to analyze. Here we propose that some intermediate cases, wherein we seek relative rates for reaction induced by specific frequency excitation (three-center displacement reactions or dissociation), may be approximately estimated from a normal mode analysis by calculating increment in root mean square amplitudes for a suitable internal coordinate, upon specific multiple photon absorption. A test case is presented as an illustration.     

Thermal fluctuations of clusters with the long-range interaction

Analysis of surface fluctuation spectra is performed for a large cluster of particles interacting via a sum of the short-range Lennard-Jones potential and long-range ±1/r potential, where the positive sign corresponds to the gravity, and negative corresponds to the electrostatic interaction. The spectral amplitudes of thermally driven capillary modes in a self-consistent field induced by cluster particles including the modes with no axial symmetry are derived in the approximation of small amplitudes. It is demonstrated that within used approximation, the surface tension is independent of the field strength. The low wave vector amplitudes are damped by attracting field that compresses the cluster and magnified by repulsing field leading to cluster fission. The fission threshold is found to be different from that found by Bohr and Wheeler and Frenkel due to the replacement of the ordinary surface tension by the bare one. Molecular dynamics study of a cluster with the long-range interaction in the vapor environment is performed using a novel integrator for a multiscale system. Simulation scheme implies rotation of the long-range components of forces acting on cluster particles thus vanishing an artificial torque. Simulation results justify theoretical conclusion of modes damping and independence of the surface tension of the field strength. Fission threshold evaluated from simulation data is in a good agreement with theory.     

Molecular dynamics simulation of hydrated DPPC monolayers using charge equilibration force fields

We present results of molecular dynamics simulations of a model DPPC-water monolayer using charge equilibration (CHEQ) force fields, which explicitly account for electronic polarization in a classical treatment of intermolecular interactions. The surface pressure, determined as the difference between the monolayer and pure water surface tensions at 323 K, is predicted to be 22.92 ±1.29 dyne/cm, just slightly below the broad range of experimental values reported for this system. The surface tension for the DPPC-water monolayer is predicted to be 42.35 ±1.16 dyne/cm, in close agreement with the experimentally determined value of 40.9 dyne/cm. This surface tension is also consistent with the value obtained from DPPC monolayer simulations using state-of-the-art nonpolarizable force fields. The current results of simulations predict a monolayer-water potential difference relative to the pure water-air interface of 0.64 ±0.02 Volts, an improved prediction compared to the fixed-charge CHARMM27 force field, yet still overestimating the experimental range of 0.3 to 0.45 Volts. As the charge equilibration model is a purely charge-based model for polarization, the current results suggest that explicitly modeled polarization effects can offer improvements in describing interfacial electrostatics in such systems.     

The non-linear ion trap. Part 5. Nature of non-linear resonances and resonant ion ejection

The superposition of higher order multipole fields on the basic quadrupole field in ion traps generates a non-harmonic oscillator system for the ions. Fourier analyses of simulated secular oscillations in non-linear ion traps, therefore, not only reveal the sideband frequencies, well-known from the Mathieu theory, but additionally a commonwealth of multipole-specific overtones (or higher harmonics), and corresponding sidebands of overtones. Non-linear resonances occur when the overtone frequencies match sideband frequencies. It can be shown that in each of the resonance conditions, not just one overtone matches one sideband, instead, groups of overtones match groups of sidebands.The generation of overtones is studied by Fourier analysis of computed ion oscillations in the direction of thez axis. Even multipoles (octopole, dodecapole, etc.) generate only odd orders of higher harmonics (3, 5, etc.) of the secular frequency, explainable by the symmetry with regard to the planez = 0. In contrast, odd multipoles (hexapole, decapole, etc.) generate all orders of higher harmonics. For all multipoles, the lowest higher harmonics are found to be strongest. With multipoles of higher orders, the strength of the overtones decreases weaker with the order of the harmonics.Forz direction resonances in stationary trapping fields, the function governing the amplitude growth is investigated by computer simulations. The ejection in thez direction, as a function of timet, follows, at least in good approximation, the equation

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wheren is the order of multipole, andC is a constant.This equation is strictly valid for the electrically applied dipole field (n = 1), matching the secular frequency or one of its sidebands, resulting in a linear increase of the amplitude. It is valid also for the basic quadrupole field (n = 2) outside the stability area, giving an exponential increase. It is at least approximately valid for the non-linear resonances by weak superpositions of all higher odd multipoles (n = 3,5,…), showing hyperbolically increasing amplitudes, whereas the even multipoles strongly suppress their ownz direction non-linear resonances. The hyperbolic increase of the amplitude, having a mathematical pole, explains the fast ejection processes possible with non-linear resonances.     

Improved 1.5-sided model for the weakly nonlinear study of Bénard-Marangoni instabilities in an evaporating liquid layer

Bénard-Marangoni instability, with coupled gravity and surface tension effects, in an evaporating liquid layer surmounted by its vapor and an inert gas is investigated theoretically. We show that this system can be described by a model that consists in the liquid layer equations plus the diffusion equation for the vapor in the gas (the so-called 1.5-sided model) and that this model is equivalent to a one-sided model when the vapor mass fraction field can be considered as quasi-stationary, provided that the equivalent Biot number is a nonlocal function of the interface temperature. A comparison of weakly nonlinear results for the 1.5-sided model with a previous one-sided model [M. Dondlinger, J. Margerit, P.C. Dauby, J. Colloid Interface Sci. 283 (2) (2005) 522-532] that considered a Biot number depending on the wavenumber evaluated at the threshold is performed. Very good agreement is found between both models. For this reason, the present analysis can also be considered as a detailed theoretical justification for the use of a one-sided model in the study of evaporative thermoconvection.     

On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions

The correlation energy in the direct random phase approximation (dRPA) can be written, among other possibilities, either in terms of the interaction strength averaged correlation density matrix, or in terms of the coupled cluster doubles amplitudes obtained in the direct ring approximation (drCCD). Although the corresponding dRPA correlation density matrix on the one hand, and the drCCD amplitude matrix on the other hand, differ significantly, they yield identical energies. Similarly, the analogous RPA and rCCD correlation energies calculated from antisymmetrized two-electron integrals are identical to each other despite very different underlying working equations. In the present communication, a direct correspondence between amplitudes and densities is established and investigated with perturbation theory arguments. Our analysis also sheds some light on the properties of recently proposed RPA/rCCD variants which use antisymmetrized integrals in part of the equations and nonantisymmetrized integrals in others.