Dynamic Scaling Law Research Articles

Cell Dynamical Approach to the Ordering Process of the Three-Dimensional Heisenberg System

The ordering process of the quenched time-dependent Ginzburg-Landau model for the three-component nonconserved order-parameter in the three-dimensional system is numerically investigated. The form factor is found to obey a dynamical scaling law with a characteristic length growing as t 1/2 . The density of topological defects and the energy density exhibit a power decay that is consistent with the scaling of the form factor. The interaction of the defects is briefly discussed.

Dynamic light scattering of swollen rubber vulcanizates and the swelling mechanism

The cooperative diffusion constant D coop of end-linked poly(dimethylsiloxane) (PDMS) model networks and dicumyl peroxyde (DCP)-cured natural rubber (NR) was determined. Variations of D coop with cross-link density for both PDMS networks and DCP-cured NR approximately corresponded to the prediction expected from both the dynamic scaling law and the so called C* theorem. It is shown that the correlation length ξ calculated from D coop can be regarded as the mesh size of a swollen network

Dynamic Scaling Laws for Bifurcations of Chaos in Dissipative Differential Systems

It is shown that the dynamic scaling law for the spectrum ψ(Λ) of the coarse-grained expansion rates Λ holds for type I intermittency and attractor-merging in the driven damped pendulum.

Dynamic Scaling Laws for Bifurcations of Chaos in Dissipative Differential Systems

It is shown that the dynamic scaling law for the spectrum ψ(Λ) of the coarse-grained expansion rates Λ holds for type I intermittency and attractor-merging in the driven damped pendulum.

Ergodic convergence properties of supercooled liquids and glasses.

The dynamical properties of two measures to probe ergodic behavior in Hamiltonian systems with a large number of degrees of freedom are investigated. The measures, namely, the energy metric d(t) and the fluctuation metric \ensuremath{\Omega}(t) are both based on the time-averaged energies of the individual particles comprising the system. The energy metric d(t) is obtained by following the dynamics of the system using two independent configurations, whereas \ensuremath{\Omega}(t) is expressed in terms of the properties of a single trajectory. Both measures obey a dynamical scaling law for long but finite times. The scaling law for d(t), which was previously established numerically, and for \ensuremath{\Omega}(t) is derived for systems that are effectively ergodic. The scaling function suggests that the configuration space is explored by a ``diffusive'' process in the space of the variables used to construct d(t) and \ensuremath{\Omega}(t). Furthermore, a single parameter, namely ${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\ensuremath{\Omega}}}$, characterizes the time scales needed for effective ergodicity to be obtained. These ideas are used to study the behavior of supercooled and glassy states of soft-sphere binary alloys using microcanonical molecular dynamics. The scaling forms for d(t) and \ensuremath{\Omega}(t) are explicitly demonstrated. The temperature dependence of the numerically computed diffusion constants (${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\ensuremath{\Omega}}}$) reveals that the nature of phase-space exploration changes both near the fluid-solid transition and near the liquid-to-glass transition. The changes in ${\mathit{D}}_{\mathit{E}}$ and ${\mathit{D}}_{\mathrm{\ensuremath{\Omega}}}$ with temperature are well described by a Vogel-Fulcher equation close to the glass transition temperature. In addition, the distribution of the time-averaged individual particle energies P(e;t), moments of which are related to \ensuremath{\Omega}(t), is shown to be broad in the glassy state. We argue that the time dependence of P(e;t) can be used to probe local structural relaxations in glasses.

Dynamic Scaling Laws for the Intermittent Switching between Two Bands

The bifurcations of chaos exhibit various fascinating features. I ),2) At the bifurcation point the chaotic attractor suffers a drastic change, and anomalous fluctuations emerge for various quantities. Many investigations have been devoted to the study of the bifurcations, where dynamic scaling laws in certain sense are expected to hold, as shown in the previous papers.)-6) A cascade of band merging after the onset of chaos in the logistic map leads to a chaotic attractor which consists of two bands. Furthermore, the two bands collide with an unstable fixed point and continuously merge. into one band. I ),2) Then a remarkable linear slope is formed in the spectrum ¢(~) for fluctuations of a coarsegrained coordinate ~.7),2) As the bifurcation parameter is further increased, ¢(~) deviates from the linearity but this deviation obeys a dynamic scaling, as. will be shown in the present paper. Recently the characterization of chaos has successfully been made by the use of dynamic structure functions, which reflect the local structures of a strange attractor.) When bands merge, the q-weighted avergage ~(q) of ~ exhibits a q-phase transition at q=l in the limit r---'>oo, where r is the mean lifetime of the intraband motions in each band. For a finite but large r, ~(q) obeys a dynamic scaling for q around q=l with rlq-11:S1, as will be shown later. We take the logistic map,

In situ observation of spinodal decomposition in In-35 at.% Tl-13.5 at.% Pb alloys

In situ observation of a spinodal decomposition in an In-35 at.% Tl-13.5% Pb alloys was made by transmission electron microscopy. When the alloy is aged at room temperature, the spinodal decomposition from fcc to f.c.t.( c/ a > 1) + f.c.t. ( c/ a < 1) takes place, followed by the f.c.t.(c/a > 1) + f.c.t.(c/a < 1) → f.c.c. ∗ + f.c.t. ∗(c/a > 1) separation. The in situ observation clearly confirmed that in the spinodal decomposition the formation of interfaces due to the growth of a compositional fluctuation takes place in an early stage and their disappearance in a later stage. Two types of elementary processes for the disappearance of the interfaces occur in the later stage. Further, both the power law for the increase in the wavelength, λ ∝ t 1 3 , and the dynamic scaling law for the space correlation function are found in the later stage on the analysis of the in situ observation. On the basis of the experimental results, the one-dimensional collision model is proposed for the later stage and further the frequency of the collision between a kink and an antikink, ω′, is shown to obey a power law of ω′ ∝ t −1.

Dynamics of supercooled water and simple liquids

Abstract The measurement of the p, T-dependence of the self diffusion coefficients in simple liquids, alcohols and supercooled water permits a critical test of the models offered for the description of the translational dynamics. For simple liquids rough hard sphere models permit a quantitative description. For cold liquid alcohols and high-pressure supercooled water the isobaric temperature dependence of D is given by the VTF-equation. In supercooled water for p < 150 MPa the isobars of D are best described by a dynamic scaling law. The studies on supercooled H2O and D2O are complemented by spin-lattice-relaxation time measurements of the deuterons, protons and oxygen-17-nuclei. These permit the evaluation of the rotational dynamics of the single water molecules.

Q-Phase Transitions and Dynamic Scaling Laws at Attractor-Merging Crises in the Driven Damped Pendulum

q-Phase Transitions and Dynamic Scaling Laws at Attractor-Merging Crises in the Driven Damped Pendulum

Collective Motion of an Assembly of Disclinations in Non-Orthogonally Twisted Nematics Quenched below the Clearing Point. II

The disclination pattern, emerging in the ordering process after the non-orthogonally twisted nematics is quenched, showed quite different features from that of orthogonally twisted nematics. Dynamical scaling law found in the orthogonally twisted nematics does not hold. By introducing, however, the characteristic time and characteristic length from the equation of motion of single disclination line, we have found a scaling property in these systems.

Scaling tests of late stages of spinodal decomposition in PC/PMMA blends

AbstractTime‐resolved light scattering studies were undertaken to elucidate the kinetics of phase separation in polycarbonate (PC)/polymethyl methacrylate (PMMA) blends. The 40:60 PC/PMMA blend undergoes thermally induced phase separation through spinodal decomposition. Temperature jump experiments were carried out from a single‐phase to a two‐phase temperature region. The general trend of spinodal decomposition in this blend system is nonlinear in character and obeys the power laws. The time evolution of scattering curves was analyzed in accordance with dynamical scaling laws for self‐similarity and the shape of scaled structure functions.

Thermal conductivity, diffusivity, and heat-capacity studies at the smectic-A-nematic transition in alkylcyanobiphenyl liquid crystals.

Thermal conductivity, diffusivity, and specific-heat studies were carried out simultaneously over the smectic-A--nematic transition in alkylcyanobiphenyl liquid crystals with different nematic ranges. The critical exponents of all the thermal parameters have been calculated and discussed in terms of the dynamic scaling law of the asymmetric planar magnet model, which applies to three-dimensional XY-like transitions.

Q-Phase Transitions and Critical Phenomena just after Band Crisis

Just after a band crisis, a new type of q-phase transition occurs, and the fluctuation spectrum of coarse-grained local expansion rates has a remarkable linear part. As the crisis point is approached from above, the deviation from the linear part obeys a dynamic scaling law.

Pattern evolution of ferroelectric domain structure in TGS quenched below phase transition point

Abstract The temporal evolution of the domain structure in TGS (Triglycine sulphate) quenched from the paraelectric phase to the ferroelectric phase has been investigated by means of the liquid crystal method. The sequential photographs obtained were analyzed by the image processing technique. From the spatial correlation functions of the patterns, the correlation is found to depend on the direction, but the dynamical scaling law holds irrespective of the direction. It is clarified by detailed observation of the pattern that the system behaves like the one with a conserved order parameter, though the order parameter in this system is the polarization.

Dynamic Scaling Law of Order-q Time Correlation Function

The symmetry dynamics near the band·splitting point in chaotic dynamics is studied with the aid of the fluctuation spectrum and the order-q time correlation function introduced by the present authors for clarifying the static and statistical characteristics of temporal fluctuations. The clear-cut evidence of the long-lived time correlation is shown to be described as the dynamic scaling law of the order-q time correlation function.

Late stage spinodal decomposition in an oligomeric blend of polystyrene/polybutadiene: A test of the scaling law for the structure function

The late stages of phase separation by spinodal decomposition have been studied by light scattering in an oligomeric blend of polystyrene (PS) with polybutadiene (PB) as a function of both quench depth and composition. The scattering function I(q,t) follows a dynamical scaling law of the form S(q,t)=qm(t)−3S̃[q/qm(t)] where the wave vector qm corresponds to the maximum in I(q,t). The profiles of the normalized scaled structure functions are compared with that predicted by Furukawa for critical mixtures (percolation regime) which is given by S̃(x)=4x2/(3+x8)[S̃(1)=1]. Apart from one exception, Furukawa’s function adequately describes the form of the normalized structure functions for a 60% PS mixture. However there are problems with some of the comparisons with the 50% and 70% PS blends. These disparities may in part be attributable to the fact that some of the data may lie outside the ‘‘true’’ late stages.

A small-angle neutron scattering study of the kinetics of phase separation in a supersaturated Ni-12.5 at. pct Si alloy

The kinetic behavior of precipitation in a supersaturated Ni-12.5 at. pct Si alloy single crystal has been studied by the small-angle neutron scattering (SANS) technique to supplement earlier transmission electron microscopy (TEM) and wide-angle X-ray diffraction (XRD) work. The SANS measurements performed at room temperature on quenched specimens subjected to isothermal anneals at 400, 450, 505, and 550 °C for various amounts of time have revealed the presence of an interference peak in the scattering function. The particle size, determined according to the Guinier approximation, is found to grow in accordance with the diffusion controlled model put forth by Lifshitz and Slyozov, and independently by Wagner. The activation energy for solute diffusion is determined using the rate constants governing the growth of particle size and the variation of the mean interparticle distance. Results are in agreement with the values given in the literature. Transition from an earlier growth stage has been observed, and enhanced diffusion is noted at temperatures below 505 °C; both observations are consistent with the previous X-ray results. The dynamical scaling law appears to be followed by the data obtained in the coarsening stage. A disruption of scaling occurs at the point when the particle growth changes from a parabolic rate behavior to a cubic coarsening rate. Dynamical scaling offers the potential for projecting the service lifetimes for components from experimental measurements carried out over a much shorter time interval. Discrepancies in the size parameters determined by different techniques are discussed.

Ferroelectric Domain Pattern Evolution in Quenched Triglycine Sulphate

Time evolution of domain patterns in a quenched ferroelectric triglycine sulphate (TGS) crystal is investigated in real time and in real space. The effect of applying an electric field after quenching is studied. The characteristic length, defined as a mean domain width, is found to obey the power law with respect to time. From the experimentally obtained time dependence of the area fraction of the domains with positive and negative polarization, it is found that this system behaves like one with a conserved order parameter in the studied time scale, though the order parameter in TGS is the polarization. For the system, to which the electric field is applied, it is found that the dynamical scaling law holds for the domain size distribution function.

Ergodic behavior in supercooled liquids and in glasses.

Ergodic behavior in liquids, supercooled liquids, and glasses is examined with a focus on the time scale needed to obtain ergodicity. A measure, d(t), which is based on the time-averaged energies of the individual particles and which is referred to as the ``energy metric,'' is introduced to probe the approach to ergodic behavior. We suggest that d(t) obeys a dynamical scaling law for long but finite times and that it can be used to characterize the degree of stochasticity in measure preserving systems with large numbers of degrees of freedom. Examination of d(t) indicates that the configuration space is explored by a ``diffusive'' process in the space of the dynamical energy variables used in constructing the energy metric. The characteristic diffusion constant associated with this process is argued to be analogous to the well-known maximal Lyapunov exponent which is often used to characterize stochasticity in systems with few degrees of freedom. Based on the long-time behavior of d(t) it is shown that ergodicity is effectively broken in the glassy state. In addition to broken ergodicity, the possibility that a subtle symmetry is broken as the liquid-to-glass transition takes place is examined. It is suggested that a ``discrete'' symmetry, to be referred to as the statistical symmetry, is broken in the glassy phase. This is illustrated by analyzing the distribution of the energy of the particles. Based on this, we expect long-time dynamics and structural relaxation in glasses to be dominated by fluctuations in domains of finite length within which the particles are highly correlated. This is in accord with the ideas of Adams and Gibbs. All of the above arguments are illustrated with the aid of molecular-dynamics simulations of soft-sphere mixtures.

Static and dynamic scaling laws near the symmetry-breaking chaos transition in the double-well potential system

The fluctuation of dynamics near the symmetry-breaking chaos transition in the double-well potential system under an external periodic excitation is studied with the fluctuation spectrum theory and the order- q time correlation function. The anomalous elongation of temporal correlation near the transition point, associated with the development of the intermittency characteristic, is shown to be described especially by the dynamic scaling law of the oder- q time correlation function.