Maxwell Relaxation Time Research Articles

Vertical Crustal Deformation Due To Viscoelastic Earthquake Cycles at Subduction Zones: Implications for Nankai and Cascadia

AbstractDespite significant progress in studying subduction earthquake cycles, the vertical deformation is still not well understood. Here, we use a generic viscoelastic earthquake‐cycle model that has recently been validated by horizontal observations to explore the dynamics of vertical earthquake‐cycle deformation. Conditioned on two dimensionless parameters (i.e., the ratio of earthquake recurrence interval T to mantle Maxwell relaxation time tM (T/tM) and the ratio of downdip seismogenic depth D to elastic upper‐plate thickness Hc (D/Hc)), the modeled viscoelastic deformation exhibits significant spatiotemporal deviations from the simple time‐independent elastic solution. Caution thus should be exercised in interpreting fault kinematics with vertical observations if ignoring Earth's viscoelasticity. By systematically exploring these two parameters, we further investigate three metrics that characterize the predicted vertical deformation: the coastal pivot line (CPL), the uplift zone (UZ) landward above the downdip seismogenic extent, and the secondary subsidence zone (SSZ) in the back‐arc region. We find that these metrics can all be time‐dependent, subject to D/Hc and T/tM. The CPL location and the UZ width are mainly controlled by D/Hc and T/tM, respectively. The presence of the SSZ is prevalent during the interseismic phase due to viscous mantle flow driven by ongoing plate convergence. Contemporary vertical deformation in Nankai and Cascadia is largely consistent with the model predictions and features differences mainly related to contrasting D/Hc values in the two margins. These findings suggest that vertical crustal deformation bears fruitful information about subduction‐zone dynamics and is potentially useful for inversions of key subduction‐zone parameters, deserving properly designed monitoring.

Viscoelastic relaxation and topological fluctuations in glass-forming liquids.

A method for characterizing the topological fluctuations in liquids is proposed. This approach exploits the concept of the weighted gyration tensor of a collection of particles and permits the definition of a local configurational unit (LCU). The first principal axis of the gyration tensor serves as the director of the LCU, which can be tracked and analyzed by molecular dynamics simulations. Analysis of moderately supercooled Kob-Andersen mixtures suggests that orientational relaxation of the LCU closely follows viscoelastic relaxation and exhibits a two-stage behavior. The slow relaxing component of the LCU corresponds to the structural, Maxwellian mechanical relaxation. Additionally, it is found that the mean curvature of the LCUs is approximately zero at the Maxwell relaxation time with the Gaussian curvature being negative. This observation implies that structural relaxation occurs when the configurationally stable and destabilized regions interpenetrate each other in a bicontinuous manner. Finally, the mean and Gaussian curvatures of the LCUs can serve as reduced variables for the shear stress correlation, providing a compelling proof of the close connection between viscoelastic relaxation and topological fluctuations in glass-forming liquids.

Rheological properties and microstructure of thermodynamically stable microemulsions as factors influencing the release rate of liposoluble vitamins

Microemulsions (ME) are stable nano-sized emulsions that constitute one of the most effective ways to incorporate lipophilic active compounds into water-based food matrices. Great stability, colloidal domain droplets, and optical isotropy are some of the advantages of ME. The present work analyzed the influence of rheological behavior and microstructural characteristics of food microemulsions on the release rate of lipophilic vitamins E and D. A fluid ME was compared to a ME with a higher dispersed phase content and a bicontinuous structure (gel-ME). Additionally, carboxymethyl cellulose (CMC) was added to fluid ME (3.5 and 10 g/100 g), obtaining systems with different apparent viscosities and microstructure. The percentage of released vitamins was determined by HPLC, and mathematically modeled. All ME were characterized using TEM microscopy FTIR, rheological and viscoelastic analysis. Fluid and gel-ME reached higher percentages of vitamin release in a very fast manner, while systems containing CMC showed a matrix-driven nature of the release. Those systems with similar zero-shear viscosity and different microstructure exhibited significantly different viscoelastic behavior. Microemulsions with CMC exhibited a viscoelastic solid type behavior with G’ > G’’. Mechanical spectra were satisfactorily fitted with the Generalized Maxwell model and relaxation time spectra were determined. In the quiescent state, gel-ME exhibited a higher plateau modulus than those containing large amount of thickener in the continuous phase. The relaxation time behavior of the structure could explain the kinetic release of liposoluble vitamins from the inner lipid phase of the microemulsions.

A Computational Framework for Time‐Dependent Deformation in Viscoelastic Magmatic Systems

AbstractTime‐dependent ground deformation is a key observable in active magmatic systems, but is challenging to characterize. Here we present a numerical framework for modeling transient deformation and stress around a subsurface, spheroidal pressurized magma reservoir within a viscoelastic half‐space with variable material coefficients, utilizing a high‐order finite‐element method and explicit time‐stepping. We derive numerically stable time steps and verify convergence, then explore the frequency dependence of surface displacement associated with cyclic pressure applied to a spherical reservoir beneath a stress‐free surface. We consider a Maxwell rheology and a steady geothermal gradient, which gives rise to spatially variable viscoelastic material properties. The temporal response of the system is quantified with a transfer function that connects peak surface deformation to reservoir pressurization in the frequency domain. The amplitude and phase of this transfer function characterize the viscoelastic response of the system, and imply a framework for characterizing general deformation time series through superposition. Transfer function components vary with the frequency of pressure forcing and are modulated strongly by the background temperature field. The dominantly viscous region around the reservoir is also frequency dependent, through a local Deborah number that measures pressurization period against a spatially varying Maxwell relaxation time. This near‐reservoir region defines a spatially complex viscous/elastic transition whose volume depends on the frequency of forcing. Our computational and transfer function analysis framework represents a general approach for studying transient viscoelastic crustal responses to magmatic forcing through spectral decomposition of deformation time series, such as long‐duration geodetic observations.

A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals

ABSTRACT Parameter estimation for generalized Maxwell models for viscoelastic materials can become ill-posed when insufficient experimental data is available. In this article, we introduce a rheological model containing Maxwell elements, define the associated forward operator and the inverse problem in order to determine the number of Maxwell elements and the material parameters of the underlying viscoelastic material. We simulate a relaxation experiment by applying a strain to the material and measure the generated stress. Since the mechanical response varies with the number of Maxwell elements, the forward operator of the underlying inverse problem depends on parts of the solution. Thereby, the forward problem consists in computing stress responses for a given number of Maxwell elements, stiffness parameters and relaxation times. The inverse problem means to compute these parameters from given stress measurements, where an additional difficulty lies in the fact that the forward mapping changes with the number of Maxwell elements and, thus, with a quantity to be computed as part of the solution. Under the assumption that every relaxation time is located in one temporal decade we propose a clustering algorithm to resolve this problem. We provide the calculations that are necessary for the minimization process and conclude by investigating unperturbed as well as noisy data. Different reconstruction approaches for the stiffnesses and relaxation times based on minimizing a least squares functional are presented. We look at individual stress components to analyze different strain rates and displacement rates, respectively, and study how experimental duration affects the identified material parameters.

Generalized hydrodynamics of the Lennard-Jones liquid in view of hidden scale invariance.

In recent years lines along which structure and dynamics are invariant to a good approximation, so-called isomorphs, have been identified in the thermodynamic phase diagrams of several model liquids and solids. This paper reports computer simulation data of the transverse and longitudinal collective dynamics at different length scales along an isomorph of the Lennard-Jones system. Our findings are compared to corresponding results along an isotherm and an isochore. Confirming the theoretical prediction, the reduced-unit dynamics of the transverse momentum density is invariant to a good approximation along the isomorph on all time and length scales. Likewise, the wave-vector dependent shear-stress autocorrelation function is found to be isomorph invariant (with minor deviations at very short times). A similar invariance is not seen along the isotherm or the isochore. Using a spatially nonlocal hydrodynamic model for the transverse momentum-density time-autocorrelation function, the macroscopic shear viscosity and its wave dependence are determined, demonstrating that the shear viscosity is isomorphic invariant on all length scales studied. This analysis implies the existence of a length scale that is isomorph invariant in reduced units, i.e., which characterizes each isomorph. The transverse sound-wave velocity, the Maxwell relaxation time, and the rigidity shear modulus are also isomorph invariant. In contrast to the isomorph invariance of all aspects of the transverse dynamics, the reduced-unit dynamics of the mass density is not invariant on length scales longer than the interparticle distance. By fitting to a generalized hydrodynamic model, we extract values for the wave-vector-dependent thermal diffusion coefficient, sound attenuation coefficient, and adiabatic sound velocity. The isomorph variation of these quantities in reduced units on long length scales can be eliminated by scaling with the density-scaling exponent, a fundamental quantity in the isomorph theory framework; this is an empirical observation that remains to be explained theoretically.

Remarks on acoustic propagation in inhomogeneous fluids: Single-phase shock regularization under the Maxwell model

The atmospheric shock problem considered in a recent article by Jordan (2020) is revisited. We show that the amplitude blow-up suffered by upward propagating acoustic shocks in an atmosphere whose ambient temperature is constant can also be eliminated by regarding air as a Maxwell fluid (i.e., a type of viscoelastic fluid), thus obviating the need for the dual-phase model assumed by Jordan (2020). Using an approach based on the Laplace transform, we then establish the existence of a range of values of the Maxwell relaxation time parameter for which the shock amplitude tends to zero and, moreover, that the upper bound of this range can be determined exactly by solving a certain quartic polynomial. And while linear, the equation of motion studied herein is shown to exhibit behaviors remarkably similar to those of certain PDEs that arise in nonlinear acoustics theory.

Fluctuation theorem convergence in a viscoelastic medium demonstrated experimentally using a dusty plasma.

The convergence of the steady-state fluctuation theorem (SSFT) is investigated in a shear-flow experiment performed in a dusty plasma. This medium has a viscoelastic property characterized by the Maxwell relaxation time τ_{M}. Using measurements of the time series of the entropy production rate, for subsystems of various sizes, it is discovered that the SSFT convergence time decreases with the increasing system size until it eventually reaches a minimum value of τ_{M}, no matter the size of the subsystem. This result indicates that the convergence of the SSFT is limited by the energy-storage property of the viscoelastic medium.

The temporal viscoelastic model of flexural isostasy for estimating the elastic thickness of the lithosphere

SUMMARY The (effective) elastic thickness of the lithosphere defines the strength of the lithosphere with respect to a load on it. Since the lithosphere is buoyant on a viscous mantle, its behaviour with respect to a load is not fully elastic, but rather viscoelastic. Fennoscandia is a well-known area in the world where the lithosphere has not yet reached its isostatic equilibrium due to the ongoing uplift after the last glacial period at the end of the Pleistocene. To accommodate for this changing property of the lithosphere in time, we present the flexural model of isostasy that accommodates temporal variations of the lithospheric flexure. We then define a theoretical model for computing the elastic thickness of the lithosphere based on combining the flexural and gravimetric models of isostasy. We demonstrate that differences between the elastic and viscoelastic models are not that significant in Fennoscandia. This finding is explained by a relatively young age of the glacial load when compared to the Maxwell relaxation time. The approximation of an elastic shell is then permissible in order to determine the lithospheric structure and its properties. In this way, the elastic thickness can be estimated based on combining gravimetric and flexural models of isostasy. This approach takes into consideration the topographic and ocean-floor (bathymetric) relief as well as the lithospheric structural composition and the post-glacial rebound. In addition, rheological properties of the lithosphere are taken into consideration by means of involving the Young modulus and the Poisson ratio in the model, both parameters determined from seismic velocities. The results reveal that despite changes in the Moho geometry attributed to the glacial isostatic adjustment in Fennoscandia are typically less than 1 km, the corresponding changes in the lithospheric elastic thickness could reach or even exceed ±50 km. The sensitivity analysis confirms that even small changes in input parameters could significantly modify the result (i.e. the elastic thickness estimates). The reason is that the elastic thickness estimation is an inverse problem. Consequently, small changes in input parameters can lead to large changes in the elastic thickness estimates. These findings indicate that a robust estimation of the elastic thickness by our method is possible if comprehensive information about structural and rheological properties of the lithosphere as input parameters are known with a relatively high accuracy. Otherwise, even small uncertainties in these parameters could result in large errors in the elastic thickness estimates.

The race between relaxation and nucleation in supercooled liquid and glassy BaS – A molecular dynamics study

Details of the microscopic mechanisms and dynamics of most processes in supercooled liquids (SCL) and glasses remain blurred due to their complexity and the extraordinarily long or inaccessibly short timescales, depending on the supercooling degree. In this paper, we determined the structural relaxation times in an intermediate supercooling window (0.67–0.78 Tm, Tm = melting point) of barium sulfide liquid, used as a model material, through molecular dynamics (MD) simulations based on a reliable two-body interatomic potential. We also inferred these dynamics in the glassy state using extrapolated data and well-accepted models. The average structural α-relaxation times, <tα>, were determined via the intermediate scattering function. Relaxation kinetics were also estimated by the Maxwell relation via the equilibrium shear viscosity and shear modulus, obtained by MD. We found that the viscosity derived (Maxwell) relaxation times are significantly shorter than the intrinsic α -relaxation times, corroborating two recent experimental studies of other substances. For the simulated system, with 36,000 particles and average volume of 1.4x103 nm3, the structural relaxation and nucleus birth time curves cross at the kinetic spinodal temperature, TKS (=0.33 Tm), which is significantly below the glass transition temperature, Tg (=0.44 Tm). However, for larger sample sizes, the TKS occurs at higher temperatures. As for temperatures above TKS, crystal nucleation starts after structural relaxation, and vice-versa, to understand and describe crystal nucleation one must necessarily take the relaxation process into account, however, this is not included in nucleation theories. These discoveries shed light on some obscure aspects of supercooled liquids, i.e., they challenge a critical assumption of nucleation theories (that crystal nucleation always occurs in a fully relaxed SCL), and show that crystallization is BaS ultimate fate, corroborating studies with other substances.

Near-infrared holographic photorefractive recording under applied electric field in undoped Bi12TiO20 sillenite crystal

Abstract Direct holographic recording in undoped Bi12TiO20 crystal at 1064 nm is investigated aiming the characterization of diffraction efficiency under action of applied dc electric field (E0). An enhancement of 12-fold in the diffraction efficiency was revealed when E0 increased from 0.0 to 4.2 kV/cm. The theoretical dependence of the diffraction efficiency and the holographic erasure time upon E0 was investigated using the standard model of Kukhtarev for photorefractivity and the results showed a good experimental data fitting in both cases, allowing the computation of the effective trap concentration (ND)eff ≈ 5.62 × 1015 cm−3 which is responsible by the recording mechanism into the crystal sample. In particular, the theoretical-experimental investigation using the holographic erasure time also allowed to determine others relevant parameters for Bi12TiO20 such as the diffusion transport length LD, the Maxwell relaxation time τ M , and the quantum efficiency Φ that can explain the relatively slow dynamics of photorefractive recording and erasure in near-infrared region. The type of charge carrier involved in the recording mechanism in the near-infrared region is also discussed.

Experimental determination of the temperature-dependent Van Hove function in a Zr80Pt20 liquid

Even though the viscosity is one of the most fundamental properties of liquids, the connection with the atomic structure of the liquid has proven elusive. By combining inelastic neutron scattering with the electrostatic levitation technique, the time-dependent pair-distribution function (i.e., the Van Hove function) has been determined for liquid Zr80Pt20. We show that the decay time of the first peak of the Van Hove function is directly related to the Maxwell relaxation time of the liquid, which is proportional to the shear viscosity. This result demonstrates that the local dynamics for increasing or decreasing the coordination number of local clusters by one determines the viscosity at high temperature, supporting earlier predictions from molecular dynamics simulations.

Maxwell relaxation time for nonexponential α‐relaxation phenomena in glassy systems

AbstractWe examine the mean relaxation time predicted by the Maxwell relation for stress and structural α‐relaxation phenomena. We express this relation using the Markov network framework and present an expression for the average relaxation time under equilibrium and nonequilibrium conditions that is rooted in the energy landscape of a material. We show thatstructuralrelaxation times calculated using the Maxwell relation must systematically underpredict the relaxation time. Finally, we report experimental evidence suggesting that the relaxation time obtained from shear viscosity measurements must correspond to astressrelaxation time.

Non-Stationary Photovoltage in Materials Science of Wide Band-Gap Semiconductors of Modern Adaptive Optics and Optoelectronics

Non-stationary photovoltage excitation in silicon carbide and gallium oxide crystals is studied. The signal transient responses are measured, and the photoelectric parameters necessary for developing adaptive photodetectors, i.e., the specific photoconductivity, Maxwell relaxation time, and diffusion length of carrier transport, are determined. The sensitivity of the photodetector based on gallium oxide is determined.

Linear viscoelastic dynamics in holography

We study the mechanical response under time-dependent sources of a simple class of holographic models that exhibit viscoelastic features. The ratio of viscosity over elastic modulus defines an intrinsic relaxation time scale -- the so-called Maxwell relaxation time $\tau_M$, which has been identified traditionally with the relaxation time scale. We compute explicitly the relaxation time in our examples and that it differs from $\tau_M$. At high temperatures $\tau_M$ over-estimates the actual relaxation time, although not by much and moreover it still captures reasonably well the temperature behaviour. At sufficiently low temperatures the situation is reversed: $\tau_M$ underestimates the actual relaxation time, in some cases quite drastically. Moreover, when $\tau_M$ under-estimates the real-time response exhibits an overshoot phenomenon before relaxation. We comment on the $T = 0$ limit, where the relaxation is power-law because our models exhibit criticality.

Rheology-determined critical conditions for shear-induced crystallization of biosynthesized polyhydroxyalkanoates

Shear-induced crystallization plays a crucial role in the manufacturing process of polymers. In this work, crystallization kinetics of biosynthesized polyhydroxyalkanoates (PHA) under different shear conditions were systematically investigated by rheometers. First, rheological properties of PHA melts were performed at different temperature to obtain mastercurves via the time-temperature superposition principle at 170 °C as a reference temperature. Then the stretch relaxation time and corresponding critical shear rate at different temperatures for the flow regime transition were calculated via the discrete Maxwell relaxation time spectrum and Arrhenius equation. Finally, the influence of shear temperature (Ts), shear time (ts) and shear rate (γ̇) on the crystallization process of PHA were discussed. The results showed the crystallization rate of PHA was improved significantly under high shear rate and long shear time. Interestingly, the t1/2 reached the minimum value when the γ̇ or the ts was large enough, which reached around 450 s at isothermal crystallization condition of 100 °C. Moreover, the nucleation density for PHA increased by appx.100 times than that under quiescent conditions. Therefore, this work may provide a useful theoretical guidance on the shear-induced crystallization of PHA.

Directed Drift Fluxes and Electric Domains in Plasma

It is shown that the electric domains in plasma are generated due to the inequality of the fluxes of the directed drift of charged particles. The time of their nucleation is approximately equal to the Maxwellian relaxation time of the space charge. It has been theoretically obtained that the frequency of space-charge waves and the time constant of the rise (decay) of oscillations during the domain instability are determined by the differential conductivity of the plasma. There are modes of electric domains generation in a gas-discharge plasma that are characteristic of a semiconductor plasma.

Maxwell relaxation time for argon and water

A new method for the determination of the Maxwell relaxation time (MRT) τM for argon, argon-like liquids and water is presented. It is based on the study of the spectral density for the velocity auto-correlation function (VACF) of a molecule. Two inequalities for the MRT having the general character are established. It is noted that the majority of literary data do not satisfy to them. The recommended values of the MRT, obtained with the help of our new approach, for water and argon are presented. It is shown that the notion of the MRT takes meaning only within the temperature intervals: 85K < T < 110K for argon and 273K < T < 400K for water. Improved estimates for the collective contributions to the self-diffusion coefficients of water and argon are presented. A new method for determination of the kinematic shear viscosity with the help of the VACF or its spectral density is proposed.

Shear modulus of two-dimensional Yukawa or dusty-plasma solids obtained from the viscoelasticity in the liquid state.

Langevin dynamical simulations of two-dimensional (2D) Yukawa liquids are performed to investigate the shear modulus of 2D solid dusty plasmas. Using the known transverse sound speeds, we obtain a theoretical expression of the shear modulus of 2D Yukawa crystals as a function of the screening parameter κ, which can be used as the candidate of their shear modulus. The shear relaxation modulus G(t) of 2D Yukawa liquids is calculated from the shear stress autocorrelation function, consisting of the kinetic, potential, and cross portions. Due to their viscoelasticity, 2D Yukawa liquids exhibit the typical elastic property when the time duration is much less than the Maxwell relaxation time. As a result, the infinite frequency shear modulus G_{∞}, i.e., the shear relaxation modulus G(t) when t=0, of a 2D Yukawa liquid should be related to the shear modulus of the corresponding quenched 2D Yukawa solid (with the same κ value), with all particles suddenly frozen at their locations of the liquid state. It is found that the potential portion of the infinite frequency shear modulus for 2D Yukawa liquids at any temperature well agrees with the shear modulus of 2D Yukawa crystals with the same κ obtained from the transverse sound speeds. Thus, we find that the shear modulus of 2D Yukawa solids can be obtained from the motion of individual particles of the corresponding Yukawa liquids using their viscoelastic property.

Kinetic criteria of vitrification and pressure-induced glass transition: dependence on the rate of change of pressure

Kinetic criteria are formulated describing vitrification and devitrification of liquids in response to variations of the rate of change of external pressure. General analytical relations are derived allowing one to determine the pressure at the glass transition, pg, and the width of the glass transition interval, δpg, in dependence on the rate of change of pressure. In addition, a direct correlation between the pressure at the glass transition and the width of the glass transition interval is established. It is shown that the ratio δpg/pg is determined generally by an appropriately defined fragility index being a measure of the steepness of change of relaxation time at the glass transition, pg, relevant for the selected rate of change of pressure. Similar relations are derived also for dynamic glass transitions in response to oscillations of pressure. All these dependencies are widely determined by the Maxwellian relaxation time of the liquid under consideration. However, they are shown to hold independently of any particular assumptions concerning the particular type of dependence of viscosity and/or relaxation time on pressure and the set of structural order-parameters describing the effect of deviations from (metastable) thermodynamic equilibrium. The results are illustrated here utilizing a recently developed new relation for the pressure dependence of the Newtonian viscosity, respectively, the relation for the pressure dependence of the relaxation time corresponding to it. Similarly to vitrification caused by variations of temperature, the pressure at the glass transition and the width of the glass transition range are shown to be logarithmic functions of the rate of change of pressure. The analytical results are confirmed by numerical computations employing a simple statistical model of glass-forming liquids.