Time Scale Set Research Articles

Formation and decay of vorticity in coupled helium-II flow

The kinematic viscosity of helium II below the transition temperature can be defined with the total density of the liquid or the normal-fluid density alone. These two definitions have very different temperature dependences. In this work we considered the decay of Gaussian distributions of vorticity in helium II using the two fluid Hall-Vinen-Bekharevich-Khalatnikov equations. Our calculation shows that in these two-dimensional axisymmetric flows both fluids decay together with a viscous time scale set by the total fluid density and not by the normal-fluid density alone. We also considered the spin-up of each fluid and observed that for all initial conditions considered both fluids tend towards a matched vorticity state.

Specific heat of around the glass transition

Dynamic calorimetric measurements are performed for the quaternary metallic glass Zr65Al7.5Cu17.5Ni10 in order to analyse the dependence on different heating rates for the glass transition temperature Tg. We compare two different temperature programs used for sample relaxation, to estimate the influence of the thermal history on Tg. A lower limit for the glass transition temperature Tg was calculated according to two different models based on the fact, that width and temperature of the glass transition depend on the experimental time scale set by the heating rate: One model assumes a Vogel-Fulcher-Tammann type behaviour, as used to describe more or less “fragile” glass formers and the other assumes an Arrhenius-like behaviour, which is related to “strong” glass formers. The values obtained from both models differ by about 80K. From additional absolute specific heat capacity measurements we calculate the Kauzmann temperature TK, as a lower limit for the temperature of the glass transition from thermodynamic aspects. Comparing TK with the temperature values obtained from the two evaluation models we can classify the quaternary metallic glass Zr65Al7.5Cu17.5Ni10, to behave more like a “strong” glass former.

Long-Lived Dynamic Heterogeneity in a Relaxor Ferroelectric

The polydispersive dielectric response of the relaxor ferroelectric lead magnesium niobate was investigated using nonresonant spectral hole-burning experiments. Using large alternating electric fields, specific parts of the relaxation-time distribution could be addressed selectively. This provides evidence for the dynamic heterogeneity of the dipolar reorientation process. The refilling of single as well as of double holes was much slower than the time scale set by the pump frequencies. Upon refilling, the holes exhibited no signs of spectral broadening. The present findings suggest a speedup of polarization response associated with a domain-wall depinning that is induced by the hole-burning event.

Rouse and Reptation Dynamics of Linear Polybutadiene Chains Studied by 2H NMR Transverse Relaxation

Deuterium NMR has been used to investigate two different types of dynamics of linear polybutadiene chains in the melt. The transverse relaxations of short Rouse chains of molecular weight 640- 3000 were biexponential, which has been attributed to separate decays of the methylene and methine deuterons. Interpretation of the transverse relaxation rates using a model for Rouse dynamics, combined with molecular simulations, gave the shortest Rouse unit as approximately 4.4 monomers and the shortest Rouse time as 8.3 × 10 -7 s. The reptation dynamics of higher molecular weight entangled chains were investigated using ABA isotopic triblock copolymers, of total molecular weight 14000-135000, where A is protonated polybutadiene of molecular weight greater than the entanglement molecular weight and B is a deuterated block. These polymers were specifically synthesized so that the fast motion of the Rouselike chain ends should not complicate the signal. The fundamental parameters found for the Rouse chain were used in the reptation model, assuming fast dynamics, and gave an entanglement molecular weight, M e , of 5380 or approximately 21 Rouse units. This M e is more than twice the conventional value, obtained from rheology, and is more suggestive of the critical molecular weight M c , consistent with previous NMR work. The theoretical analysis used in this work is based on the assumption that the chain dynamics are fast on the time scale set by the NMR deuterium quadrupolar interaction. The highest molecular weight samples were found to not satisfy this criterion and indicate the molecular weight at which a new theoretical approach is needed.

Shear-induced ordering kinetics of a triblock copolymer melt

Disorder-to-order transitions, in which an isotropic system acquires a spatially periodic structure, are common to a number of phenomena in materials science. Here, we employ small-angle neutron scattering to probe the effects of reciprocating shear on the isotropic-to-lamellar transition of a triblock copolymer composed of poly(ethylene-co-propylene) (PEP) and poly(ethylethylene) (PEE). Prior work has shown that the transition temperature decreases with increasing shear rate, implying that the isotropic state can be produced at low temperatures through application of a flow field [T. Tepe et al., J. Rheol. 41, 1147 (1987)]. Removing this field results in a “jump” to conditions under which the lamellar phase is stable with a time scale set by the relaxation of concentration fluctuations. We describe the ordering process which results, concentrating on the possible existence of a stability limit for the initial (isotropic) state.

Nonresonant dielectric hole burning spectroscopy of supercooled liquids

The nonexponential response of propylene carbonate and glycerol near their glass transitions could be selectively altered using nonresonant spectral hole burning (NSHB) experiments. This observation provides evidence of the existence of a distribution of relaxation times in these supercooled liquids. NSHB is based on a pump, wait, and probe scheme and uses low-frequency large amplitude electrical fields to modify the dielectric relaxation. The temporal evolution of the polarization of the sample is then measured subsequent to a small voltage step. By variation of a recovery time inserted between pump and probe, the refilling of the spectral features could be monitored and was found to take place on the time scale set by the peak in the distribution. The recovery time and pump frequency dependences of the spectral modifications were successfully simulated using a set of coupled rate equations.

Protoplanet Migration by Nebula Tides

The tidal interaction of a protoplanet with a circumstellar gaseous disk results in mutual angular momentum exchange that modifies both the disk and the orbit of the secondary. There are two, conceptually distinct circumstances wherein nebula torques can cause a secular variation in a protoplanet's semimajor axis. (I) The net torque exerted on a secondary by an undisturbed disk is not, in general, zero. Although gradients in the disk density and temperature can contribute to this torque, it is mostly due to asymmetries in the disk/planet interaction that are inherent to a Keplerian disk. For a relatively thick disk, the differential torque can be a significant fraction of the torque from either its exterior or its interior portion. In this case, the protoplanet drifts relative to disk material on a time scale inversely proportional to its mass. (II) The protoplanet opens a gap in the disk that establishes a flow barrier to disk material. A density discontinuity that locks the protoplanet into the disk's viscous evolution develops across the orbit. The protoplanet migrates on a time scale set by the disk's viscosity. A unified model that clarifies the relationship between these migration types and reveals under what circumstances a given type is selected is presented. In both cases, orbital decay is the prevailing outcome, although type I migration rates can be between one and two orders of magnitudes faster than type II. Estimates of orbital lifetimes are given and are generally shorter than the expected lifetime of the disk. Some implications to the issue of planetary formation are discussed.

Continuum contribution to excitonic four-wave mixing due to interaction-induced nonlinearities: A numerical study

We present a theoretical investigation of ultrafast transient four-wave mixing (FWM) of GaAs quantum wells for coherent excitation of excitons and a large number of continuum states. It is shown that in this case the line shape of the FWM signal is drastically altered due to an interaction-induced coupling of the exciton to all the excited continuum states. The signal is dominantly emitted at the spectral position of the exciton and decays, as a function of delay, on a time scale set by the duration of the laser pulse rather than by the intrinsic dephasing time. Nevertheless, the spectral width of the exciton line in the FWM spectrum and in the decay of the time-resolved FWM signal in real time are governed by the intrinsic excitonic dephasing rate. It is shown that for pulse durations of \ensuremath{\sim} 100 fs (for GaAs quantum wells) this behavior can be explained as the influence of the Coulomb exchange interaction, while for even shorter pulses this behavior is dominantly caused by nonlinear polarization decay.

Validity of the rapid buffering approximation near a point source of calcium ions

In the presence of rapid buffers the full reaction-diffusion equations describing Ca2+ transport can be reduced using the rapid buffering approximation to a single transport equation for [Ca2+]. Here we simulate the full and reduced equations, exploring the conditions necessary for the validity of the rapid buffering approximation for an isolated Ca2+ channel or a cluster of channels. Using a point source and performing numerical simulations of different durations, we quantify the error of the rapid buffering approximation as a function of buffer and source parameters as well as the time and spatial scale set by the resolution of confocal microscopic measurements. We carry out simulations of Ca2+ "sparks" and "puffs," both with and without the indicator dye Ca2+ Green-1, and find that the rapid buffering approximation is excellent. These calculations also show that the traditional calculation of [Ca2+] from a fluorescence signal may grossly underestimate the true value of [Ca2+] near a source. Finally, we use the full model to simulate the transient Ca2+ domain near the pore of an open Ca2+ channel in a cell dialyzed with millimolar concentrations of 1,2-bis(2-aminophenoxy)ethane-N,N,N,N-tetraacetic acid or EGTA. In this regime, where the rapid buffering approximation is poor. Neher's equation for the steady-state Ca2+ profile is shown to be a reliable approximation adjacent to the pore.

Time and distance scales of membrane domain organization.

Time and distance scales in membranes are discussed in relation to the inference of domain structure from spectroscopic measurements. Each type of spectroscopy has a natural time scale set by the magnitude of the interactions that determine the spectral width. For fluid membranes, the lateral diffusion of the lipid molecules then implies an associated distance scale over which the measurements are averaged. These factors have an influence on the interpretation of spectroscopic measurements and on whether or not the spectroscopic technique is capable of distinguishing neighbouring domains from each other. NMR occupies a special place among spectroscopies because its time scale is so long. Some examples are given of the conceptual role of spectroscopic time and distance scales with regard to the inference of domains in membranes from NMR or other spectroscopic studies, or the apparent failure to detect domains believed to be present. These examples include mixtures of phospholipids with cholesterol and/or protein molecules. In addition to time scales being set by line width and line shape considerations, the study of relaxation times within a given spectroscopy carries its own characteristic insights into motional correlation times.

Modes of Tropical Variability under Convective Adjustment and the Madden–Julian Oscillation. Part II: Numerical Results

Abstract Convective interaction with dynamics (CID) dictates the structure and behavior of the eigenmodes of the tropical atmosphere under moist convective adjustment (MCA) when the convective adjustment time scale, τc, is much smaller than dynamical time scales, as examined analytically in Part I. Here, the modes are reexamined numerically to include the effects of finite τc, again for a primitive equation model with the Betts-Miller MCA parameterization. The numerical results at planetary scales are consistent with the analytical approach, with two well-separated classes of vertical modes: one subset evolves at the cumulus time scale, while the other subset evolves at a time scale set by the large-scale dynamics. All modes are stable for homogeneous basic states in the presence of simple mechanical damping effects. Thus, there is no CISK at any scale under MCA. However, the finite τc effect has the property of selectively damping the smallest scales while certain vertical modes at planetary scales decay...

Broadband noise and perturbation response in o-TaS3: Fluctuation-dissipation and driven-noise considerations.

The slow response (as distinguished from the fast current overshoot) of the charge-density wave in o-${\mathrm{TaS}}_{3}$ after a reversal in bias current was measured as a function of perturbation parameters. The response roughly decayed as a power of time, up to a time scale set by the length of the perturbation. This result fits a quasiequilibrium picture of the broadband noise (BBN) and other slow relaxational phenomena. The form of the relaxation does not fit pictures in which BBN comes from a large response to external perturbations.

Chiral Organometallic Reagents, VIII. On the Configurational Stability of α‐Hetero‐Substituted Benzyllithium Compounds

AbstractA test based on kinetic resolution was applied to probe the configurational stability of various α‐substituted benzyllithium compounds 1. It was found that the trimethylsilyl‐ (1b), phenylsulfenyl‐ (1c), and the phenylselenyl‐substituted (1d) benzyllithium compounds enantiomerized more rapidly than they added to the chiral aldehyde 6. In contrast, the carbamoyloxy‐ (1e), (16), and the phenylsulfonyl‐substituted (1g) benzyllithium compounds were found to be configurationally stable on the time scale set by their addition to the aldehyde 6.

Decoupling of time scales of motion in polybutadiene close to the glass transition.

Studying the dynamic structure factor of polybutadiene in the first valley of S(Q) we observed a decoupling of microscopic and macroscopic time scales about 40\ifmmode^\circ\else\textdegree\fi{} above ${\mathit{T}}_{\mathit{g}}$. At higher temperatures the time scale set by the viscosity scales with the microscopic scale, while at lower T strong decoupling effects occur. We present evidence that the decoupled low-temperature relaxation is distinctly different from the high-frequency ``\ensuremath{\beta} process.'' The decoupling temperature coincides with the critical temperature of the mode coupling theory.

On the embedding‐dimension analysis of AE and AL time series

Several authors have employed the embedding‐dimension method to analyze time series of geomagnetic indices, with differing results for the value of the correlation dimension ν. It is argued that these differences may arise from corresponding differences in the length and construction of the various data sets used. Practical application of the method to sets of discretized data requires use of a delay time scale set by the autocorrelation time of the data set. It is found that a particular data set containing 35 days of AE exhibits an autocorrelation time τc longer by an order of magnitude than that of a short‐duration (<5 days) set, raising the possibility that extant analyses of long‐duration sets may have employed delay times shorter than τc. In addition, the power spectrum of AE reveals modulation at a period of 24 hr. A numerical experiment on the logistic map shows that such modulation introduces an extra degree of freedom in the data, resulting in an augmented correlation dimension.

An exact expression for the transverse nuclear magnetic resonance relaxation of a dynamic scale invariant polymer chain governed by a single relaxation time

An exact analytic calculation of the transverse nuclear magnetic resonance (NMR) relaxation function, due to dipolar interactions, is presented for a polymer chain considered at the scale invariant level of description. The calculation is possible for the particular case where the dynamics of the bond vectors are governed by a single relaxation time. This exact result is used to check the accuracy of two approximation methods when they are applied to the single relaxation time case. The well known second moment approximation is shown to be seriously wrong when the time scale of the bond dynamics is comparable or greater than the NMR time scale set by the dipolar interactions. A more recent method, based on a representation of the chain dynamics in terms of hybrid Fourier components defined over the experimental time interval, is shown to give excellent agreement with the exact result over the entire range of relaxation times.

A Scale Analysis of Deep Moist Convection and Some Related Numerical Calculations

Abstract A scale analysis valid for deep moist convection is carried out. The approximate equations of motion are anelastic with the time scale set by the Brunt- Vaisala frequency. A new assumption is that the base state potential temperature is a slowly varying function of the vertical coordinate. It is this assumption that eliminates the energetic inconsistency discussed by Wilhelmson and Ogura (1972) for a non-isentropic base state. Another key result is that the dynamic pressure is an order of magnitude smaller than the first-order temperature and potential temperature. In agreement with observations, the kinetic energy is found to be an order of magnitude smaller than the first-order thermodynamic energy. A set of six numerical simulations representing moderately deep moist convection is carried out. The base state is an idealized maritime tropical sounding with no vertical wind shear. The first calculation (Run A) shows the growth and dissipation of a typical shower cloud. The remaining calculations...

Multivariable Expansions Using Time Scales Generated from Differential Equations

We deal with the equation $(1 + \varepsilon t)y'' + 2\varepsilon y' + y = 0$ and show how a set of nonlinear time scales may be derived from the equation itself. Using these time scales multitime expansions of the solution of the equation satisfying the initial conditions $y(0) = 0,y'(0) = 1$ are found. Proof of the uniform asymptotic convergence of the expansions is given. The technique used is an extension of the method used by Reiss [1] and has been successfully applied to a broad class of second order linear differential equations with slowly varying coefficients.

Motion and Decay of a Vortex in a Nonuniform Stream

The motion of a vortex in a two-dimensional incompressible nonuniform stream is studied by including the viscous effects in the inner core of the vortex. A systematic procedure is presented by the introduction of two different sets of length and time scales with the larger scales identified with the typical scales of the outer nonuniform flow. The ratios of the scales are powers of the small parameter ε which is inversely proportional to the square root of the Reynolds number of the vortex. It is shown that the leading term of the solution to the Navier-Stokes equations are composed of the classical inviscid solution matched with the solution of a decaying axially symmetric vortex. Thus the singularity in the center of the vortex associated with the classical inviscid theory is removed. The next-order solution shows that the time average of the velocity of the center of the vortex over a period of the order of the larger time scale should agree at least to the order of ε2 with the local space mean of the velocity of the outer inviscid flow which has been assumed to be the velocity of the center of the vortex in the classical inviscid theory of vortex motion. The nonuniformity of the outer flow will induce a nonaxially symmetric solution for the inner viscous region of the order ε2 or higher, which in turn will introduce an additional term to the outer flow of the order of ε4 or higher.