Slow Energy Exchange Research Articles

Identifying internal resonance regimes in free-vibrating systems with multiple autoparametric couplings

This paper introduces a method for identifying internal resonance regimes in free-vibrating systems with multiple nonlinear couplings, illustrated using a chain of two Rott’s pendula. The method based on decomposing Hamiltonian and subsequent treatment of the coupling energies allows for a comprehensive understanding of autoparametric system behaviour. A novel signal-processing-based coupling energy evaluation significantly enhances the effectiveness of the method, revealing internal resonances with slow energy exchange. In the theoretical part, a general step-by-step procedure applicable to any conservative coupled oscillators is introduced. We demonstrate that concerning such free-vibrating systems, the proposed method effectively uncovers internal resonances influenced by system parameters, initial conditions, and the initial energy of the system. Through an illustrative example of the system with multiple quadratic couplings, we also show that slow energy exchange occurs for sum and difference internal resonances.

Microfluidic Oxidation of Graphite in Two Minutes with Capability of Real-Time Monitoring.

Graphite oxide and its exfoliated counterpart, graphene oxide, are important precursors for the large-scale production of graphene-based materials and many relevant applications. The current batch-style preparation of graphite oxide suffers from safety concern, long reaction time, and nonuniform product quality, due to the large volume of reactors and slow energy exchange. Reaction in microchannels can largely enhance the oxidization efficiency of graphite due to the enhanced mass transfer and extremely quick energy exchange, by which the controllable oxidization of graphite is achieved in ≈2 min. Comprehensive characterizations show that the graphene oxide obtained through the microfluidic strategy has features like those prepared in laboratory beakers and industrial reactors, yet with the higher oxidization degree and more epoxy groups. More importantly, the microfluidic preparation allows for on-line monitoring of the oxidization by Raman spectroscopy, ready for the dynamical control of reaction condition and product quality. The capability of continuous preparation is also demonstrated by showing the assembly of fibers and reduction of graphene oxide in microfluidic channels, and the applicability of graphene oxide prepared from the microfluidic strategy for thermally and electrically conductive films.

The fidelity of exponential and IMEX integrators for wave turbulence: Introduction of a new near-minimax integrating factor scheme

Exponential and IMEX integrators are considered for wave turbulence problems, which are stiff systems that often require long-time simulations. Applying a multi-scale asymptotic analysis to the difference equations for a simple wave turbulence problem reveals that Exponential Runge-Kutta methods incorrectly model interactions between resonant waves unless they are specifically designed otherwise, and IMplicit-EXplicit (IMEX) integrators introduce dispersive and dissipative errors that severely degrade the accuracy of slow energy exchange between resonant waves. The same analysis shows Integrating Factor (IF) methods to be the most suitable. Numerical experiments performed on the resonant triad model and the Majda-McLaughlin-Tabak (MMT) model verify the discrete-asymptotic analysis. In the MMT problem, Exponential Time Differencing methods require a time step more than two orders of magnitude smaller than IF methods to achieve comparable accuracy, and IMEX methods require more than six orders of magnitude smaller time steps. A novel near-minimax rational approximation of the matrix exponential is proposed to be used for integrating factor methods, and is demonstrated to be as accurate as the IF method but without the need to compute a matrix exponential.

Implicit-Explicit Variational Integration of Highly Oscillatory Problems

In this paper, we derive a variational integrator for certain highly oscillatory problems in mechanics. To do this, we take a new approach to the splitting of fast and slow potential forces: rather than splitting these forces at the level of the differential equations or the Hamiltonian, we split the two potentials with respect to the Lagrangian action integral. By using a different quadrature rule to approximate the contribution of each potential to the action, we arrive at a geometric integrator that is implicit in the fast force and explicit in the slow force. This can allow for significantly longer time steps to be taken (compared to standard explicit methods, such as Störmer/Verlet) at the cost of only a linear solve rather than a full nonlinear solve. We also analyze the stability of this method, in particular proving that it eliminates the linear resonance instabilities that can arise with explicit multiple-time-stepping methods. Next, we perform some numerical experiments, studying the behavior of this integrator for two test problems: a system of coupled linear oscillators, for which we compare the stability of this method against the resonance behavior of the r-RESPA method, and slow energy exchange in the Fermi–Pasta–Ulam (FPU) problem, which couples fast linear oscillators with slow nonlinear oscillators. Finally, we prove that this integrator accurately preserves the slow energy exchange between the fast oscillatory components, which explains the numerical behavior observed for the FPU problem.

Anomalously slow phase transitions in self-gravitating systems.

The kinetics of collapse and explosion transitions in microcanonical self-gravitating ensembles is analyzed. A system of point particles interacting via an attractive soft Coulomb potential and confined to a spherical container is considered. We observed that for 100-200 particles collapse takes 10(3) - 10(4) particle crossing times to complete; i.e., it is by two to three orders of magnitude slower than the velocity relaxation. In addition, it is found that the collapse time decreases rapidly with an increase of the soft-core radius. We found that such an anomalously long collapse time is caused by the slow energy exchange between a higher-temperature compact core and relatively cold diluted halo. The rate of energy exchange between the faster modes of the core particles and slower-moving particles of the halo is exponentially small in the ratio of the frequencies of these modes. As the soft-core radius increases and the typical core modes become slower, the ratio of core and halo frequencies decreases and the collapse accelerates. Implications for astrophysical systems and phase transition kinetics are discussed.

Representative Electrons and Energy Exchange in Strong Laser Fields

The problem of electron-ion collisions in an intense linearly polarized laser radiation field is numerically analyzed. The main contribution to the effective cross section of inelastic scattering is shown to be made by a small part of electrons---representative electrons. It is found out that at velocities lower than the oscillatory one these cross sections may multiply exceed the traditional cross sections. The effect is treated in terms of adiabatic attraction of an electron repeatedly oscillating close to an ion and instantaneous or slow energy exchange with the field.

SPECIAL FEATURE: TUTORIAL Slow Heating Methods in Tandem Mass Spectrometry

Several approaches to ion activation in tandem mass spectrometry have been developed in recent years for use in ion trapping instruments that allow for conditions to be reached wherein rates of ion activation and deactivation are comparable. These approaches are defined as slow heating methods and include continuous-wave laser infrared multiphoton dissociation, dissociation driven by blackbody radiation, quadrupole ion trap collisional activation and sustained off-resonance irradiation in ion cyclotron resonance mass spectrometry. In the limiting case in which ion activation and deactivation rates are equal, a steady-state parent ion internal energy distribution is achieved and the kinetics of dissociation can be interpreted in analogy with thermal dissociation. This discussion describes the thermal analogy and the limiting conditions of rapid energy exchange and slow energy exchange along with the possible ramifications for dissociation rates and product ion spectra. The figures of merit that the various slow heating methods share as a class of activation methods are also discussed. The purpose of this perspective is to provide a frame-of-reference from which slow heating methods can be considered. Such methods are seeing increasing use as the number of ion trapping instruments grows and have shown remarkable success with dissociation of high-mass ions. © 1997 by John Wiley & Sons, Ltd.

SPECIAL FEATURE:TUTORIAL Slow Heating Methods in Tandem Mass Spectrometry

Several approaches to ion activation in tandem mass spectrometry have been developed in recent years for use in ion trapping instruments that allow for conditions to be reached wherein rates of ion activation and deactivation are comparable. These approaches are defined as slow heating methods and include continuous-wave laser infrared multiphoton dissociation, dissociation driven by blackbody radiation, quadrupole ion trap collisional activation and sustained off-resonance irradiation in ion cyclotron resonance mass spectrometry. In the limiting case in which ion activation and deactivation rates are equal, a steady-state parent ion internal energy distribution is achieved and the kinetics of dissociation can be interpreted in analogy with thermal dissociation. This discussion describes the thermal analogy and the limiting conditions of rapid energy exchange and slow energy exchange along with the possible ramifications for dissociation rates and product ion spectra. The figures of merit that the various slow heating methods share as a class of activation methods are also discussed. The purpose of this perspective is to provide a frame-of-reference from which slow heating methods can be considered. Such methods are seeing increasing use as the number of ion trapping instruments grows and have shown remarkable success with dissociation of high-mass ions. © 1997 by John Wiley & Sons, Ltd.

On Non‐Equilibrium Effects in Solid State Rate Processes

AbstractAt temperatures high enough to make thermally activated hopping sufficiently fast and at correspondingly high frequencies, inelastic or dielectric relaxation (damping) due to point defects may show the following deviations from the predictions of equilibrium theory: a) the main damping maximum appears at a frequency lower than predicted by equilibrium theory, b) the intensity of the main maximum is reduced, c) secondary damping maxima appear at eigenfrequencies of the energy exchange between defect and surroundings. The analysis shows that experimental detection of these effects is most easily realizable on defect systems with slow energy exchange with the surrounding crystal, and low activation energy.

On the trapping of wave energy round islands

It is shown that islands can trap long-wave energy in a way similar to the capture of a particle by an atomic nucleus. The frequencies of the captured waves form a discrete set, being determined by the shape of the island and the contours of the surrounding sea bed. If the depth at great distances tends to a constant value, the trapped modes must leak some energy to infinity, though the consequent rate of decay may be exceedingly small. The initial energy of the trapped modes may be absorbed from incident radiation of the same frequency or from a sharp pulse. The particular example of a rectilinear pulse incident on a circular island is discussed in some detail.The effect of the rotation of the Earth is to split the frequencies of a pair of waves progressing in opposite directions round the island. The splitting of the frequencies produces slow beats in the waves as seen at any fixed point. Slight asymmetry in the island induces a slow exchange of energy between each pair of progressive modes.The present investigation was suggested by the occurrence of regular oscillations having a period of 6 min and a beat period of about 3 h in long-wave records taken at Macquarie Island, in the Southern Ocean.

On the Origin of the Ultrasonic Absorption in Liquids. III

In “Kneser” liquids the mechanism of excess ultrasonic absorption is the slow exchange of energy between internal and external degrees of freedom, just as in gases. Eight such liquids are selected; in two the relaxation time in the liquid is known from measurements, in the others it is calculated from low-frequency absorption. In comparing the relaxation time for the liquids with that in the vapors of the same substances, the latter is found to be 100 to 600 times that in the liquid. This is related to the ratio of the collision times.

The Origin of the Ultrasonic Absorption in Liquids. II

Excess absorption in nonassociated liquids has been explained by the same mechanism as is responsible for molecular absorption in gases, namely, slow exchange of energy between internal and external degrees of freedom. In this paper the rate of energy exchange is calculated for benzene and found to be three times too rapid. The Lennard-Jones interaction potential and model of the liquid are used; the fourth-order terms in the interaction provide the coupling between internal vibrations and Debye waves.

The Absorption and Dispersion of Sound in Oxygen as a Function of the Frequency-Pressure Ratio

The velocity and absorption of ultrasonic waves in oxygen were measured by means of an improved ultrasonic interferometer in the range from one to one hundred Mc per atmosphere. Dispersion of the velocity ranged from 333.14 m/sec, to 357.22 m/sec at 30°C. The ratio (αexper/αclass) dropped from 3.68 to 2.05, and the corresponding value of Cv/R dropped from 2.50 to 1.61. The increase in velocity and the decrease in (αexper/αclass) is interpreted as caused by the slow exchange of energy between the translational and rotational degrees of freedom. The relaxation time for the rotation was 5.24×10−9 second.

Dispersion of Velocity and Anomalous Absorption of Ultrasonics in Nitrogen

By means of the interferometer, the velocity and absorption of ultrasonic waves in nitrogen were measured at the frequency of 2.992 mc/sec, in the pressure range of 2.09 to 76 cm of Hg at a temperature of 29°C. A dispersion of velocity was found ranging from 354.3 meters/sec to 364.4 meters/sec. The ratio αexp/αclass dropped from 1.40 to 1.32, and the corresponding value of Cv/R dropped from 2.50 to 2.08. Theoretical values for the change in velocity and in the absorption ratio, calculated by applying the equations for the exchange of energy between translational and vibrational degrees of freedom, show good agreement with the observed values. The increase in velocity and the decrease in αexp/αclass is interpreted as due to the slow exchange of energy between the translational and rotational degrees of freedom. The relaxation time for rotation was found to be 1.2×10−9 second.

The Absorption of Ultrasonic Waves in Liquids and its Relation to Molecular Constitution

Measurements of the absorption of ultrasonic waves in water and ethyl alcohol were made by the pulse method, and are more accurate than those made by other methods. The absorption coefficient α in water was found to vary as the square of the frequency ν over a wide range of temperatures. The measurements in water were made at frequencies between 7.5 and 67.5 Mc/s. and temperatures between 0 and 95° c. The absorption decreased with increasing temperature by a factor of 8 between freezing and boiling points. The observed absorption was about three times that calculated from Stokes formula and this ratio was nearly independent of temperature. In ethyl alcohol measurements were made at 52.4 Mc/s., and at temperatures between - 50° and + 60° c. The absorption decreased by a factor of 3 in this range and observed values were about twice those given by Stokes' formula. Values of (α/ν2) at 25° c. are 22.0 x 10-17 sec2/cm. for water and 50.5 x 10-17 sec2/cm. for ethyl alcohol. Published values of the absorption coefficients of a wide range of pure liquids and of some mixtures are reviewed, and a classification proposed. The experimental results suggest that the absorption in associated liquids is due to a different mechanism from that in non-associated liquids. In the latter the absorption is probably due to the same phenomenon as in gases: a slow exchange of energy between different degrees of freedom. The relaxation frequencies must be above 200 Mc/s. for most liquids. A simple theory is given to explain the observed variation of the absorption with concentration in mixtures of non-associated liquids. The semi-quantitative agreement of theory with experiment lends support to the relaxation hypothesis.