Nonconserved Order Parameter Research Articles

Magnetic Field Dependence of Critical Attenuation in Ferromagnet

Many aspects of sound attenuation above and below critical point have been discussed in the literature [1, 2]. However, little attention has been paid so far to critical attenuation in the presence of the magnetic eld. A notable exception are the meaneld theory results [3 5]. The aim of this paper is to provide a dynamical matching formalism of the renormalization group which allows us to calculate the ultrasonic attenuation coe cient α(t, ω,H) by means of the e expansion not only in the asymptotic region H = 0 but also in the whole magneticeld range. We implement the method used in the calculation of the dynamic susceptibility in the ordered phase [6] for the evaluation of the sound attenuation coe cient in an external magnetic eld but use a modi ed matching condition suitable for H = 0. One of the best method for computation of the scaling functions is the method introduced by Nelson [7] for computation of the static correlation function above the critical temperature. It was a generalization of the renormalization group technique developed by Nelson and Rudnick [8] used to obtain the equation of state to rst order in e = 4 − d as well as the scaling functions for the susceptibility, speci c heat and etc. both in the ordered and disordered phase. Later this method was generalized by Achiam and Kosterlitz [9] to calculate the static momentum-dependent correlation function for arbitrary temperature and magnetic eld. Dengler et al. [10] were rst who generalized this method into dynamic correlation function in the disordered phase. In a recent paper Pawlak and Erdem [6] extended these results by obtaining (to rst order in e) the expressions for the dynamic susceptibility and correlation function for nonconserved Ising order parameter both above and below the critical temperature in zero magnetic eld. Generalization of this method to nonzero elds will be used for calculation of the ultrasonic attenuation.

Geometrical properties of the Potts model during the coarsening regime

We study the dynamic evolution of geometric structures in a polydegenerate system represented by a q-state Potts model with nonconserved order parameter that is quenched from its disordered into its ordered phase. The numerical results obtained with Monte Carlo simulations show a strong relation between the statistical properties of hull perimeters in the initial state and during coarsening: The statistics and morphology of the structures that are larger than the averaged ones are those of the initial state, while the ones of small structures are determined by the curvature-driven dynamic process. We link the hull properties to the ones of the areas they enclose. We analyze the linear von Neumann-Mullins law, both for individual domains and on the average, concluding that its validity, for the later case, is limited to domains with number of sides around 6, while presenting stronger violations in the former case.

Effects of turbulent transfer on critical behavior

Using the field theory renormalization group, we study the critical behavior of two systems subjected to turbulent mixing. The first system, described by the equilibrium model A, corresponds to the relaxational dynamics of a nonconserved order parameter. The second system is the strongly nonequilibrium reaction-diffusion system, known as the Gribov process or directed percolation process. The turbulent mixing is modeled by the stochastic Navier-Stokes equation with a random stirring force with the correlator ∞ δ(t − t′)p 4−d−y, where p is the wave number, d is the space dimension, and y is an arbitrary exponent. We show that the systems exhibit various types of critical behavior depending on the relation between y and d. In addition to known regimes (original systems without mixing and a passively advected scalar field), we establish the existence of new strongly nonequilibrium universality classes and calculate the corresponding critical dimensions to the first order of the double expansion in y and ɛ = 4 − d (one-loop approximation).

Modeling ternary mixtures by mean-field theory of polyelectrolytes: Coupled Ginzburg–Landau and Swift–Hohenberg equations

The purpose of this work is to model ternary mixtures using the theory of pattern formation and of polyelectrolytes, with mean-field approximations. The model has two local, non-conserved order parameters. In the free energy short-range and long-range nonlocal interactions between elements of the mixture are considered. The spatiotemporal dynamics of the system is described by coupling the time-dependent Ginzburg–Landau equation and the Swift–Hohenberg equation. These non-linear partial differential equations are solved with numerical methods to study the emergent spatially stable configurations. The model shows a large diversity of patterns, which permit an interpretation of the behavior of some biological systems and presents different growth lengths within its spatial structures.

Theoretical investigation of antiferroelectric (SmCA⁎) subphases by hydrodynamical approach

We provide a hydrodynamical approach utilizing time dependent Landau–Ginzburg model (L–G) and the Cahn–Hilliard model (C–H) to investigate antiferroelectric liquid crystals (AFLCs) exhibiting different chiral phases between paraelectric smectic A (SmA⁎) phase and antiferroelectric smectic CA⁎ phase (SmCA⁎). Introducing conserved and non-conserved order parameters in C–H and L–G models, we have predicted the appearance of a chiral smectic C (SmC⁎) phase and a ferrielectric SmCFI1⁎ phase (three layers SmCA⁎) in an antiferroelectric phase sequence. The three layers periodicity for SmCFI1⁎ phase is studied in detail with a non-uniform layer interactions among smectic layers with strong experimental support. Finally, we provide some theoretical basis for the non-uniformity of our proposed layer interactions.

A phase-field model for polycrystalline thin film growth

In this work we introduce and apply a diffuse interface model to the problem of polycrystalline evolution in zeolite thin film growth. The phase-field model is based on a Ginzburg–Landau free energy functional, where a set of non-conserved order parameters is used to capture geometry and temporal evolution of the solid grains and the liquid phase. Special attention is given to the modelling of anisotropic interface properties resulting in a growth shape which corresponds to experiments for hydrothermally grown MFI zeolites. The model parameters used in the simulations are determined from available physical data, taking into account the experimental length and energy scales. To produce sound results with a minimum of seed crystals, a method to cover the unit sphere in three dimensions with angularly equidistant grain orientations is developed. Simulations in 2D and 3D are carried out to study competitive growth of equisized seeds with randomly distributed orientations on a flat substrate. In each case, the evolution of the orientation distribution leads to a fibre texture, selecting the direction of fastest growth perpendicular to the substrate. The dynamics of front growth velocity and of the average grain diameter is evaluated and compares well to experimental and theoretical findings. No significant differences in the competitive growth process using either solid–liquid interface energy anisotropy or anisotropy in the kinetic coefficient were found. This allows the conclusion that under the imposed conditions no major effect of interface energy balances at triple junctions on structure evolution is present, as it is the case for grain growth. The simplifying assumptions imposed in the modelling of the process are discussed, and possible refinements of the phase-field model are proposed.

Renormalization-Group for Amplitude Equations in Cellular Pattern Formation with and without Conservation Law

A proper version of the proto renormalization-group scheme is presented to derive amplitude equations in striped pattern formation with conserved and nonconserved order parameter. In the conserved case, the result preserves the conservation law as well as the rotational invariance of the physical system. This makes a great contrast with existing singular perturbation theories such as the multiple-scales method, in which the perturbation expansion truncated at any finite order destroys those symmetries.

Effects of turbulent mixing on the critical behavior

Effects of strongly anisotropic turbulent mixing on the critical behavior are studied by means of the renormalization group. Two models are considered: the equilibrium model A, which describes purely relaxational dynamics of a nonconserved scalar order parameter, and the Gribov model, which describes the nonequilibrium phase transition between the absorbing and fluctuating states in a reaction-diffusion system. The velocity is modelled by the d-dimensional generalization of the random shear flow introduced by Avellaneda and Majda within the context of passive scalar advection. Existence of new nonequilibrium types of critical regimes (universality classes) is established.

Effects of turbulent mixing on critical behaviour in the presence of compressibility: renormalization group analysis of two models

Critical behaviour of two systems, subjected to the turbulent mixing, is studied by means of the field theoretic renormalization group. The first system, described by the equilibrium model A, corresponds to the relaxational dynamics of a non-conserved order parameter. The second one is the strongly non-equilibrium reaction–diffusion system known as Gribov process and equivalent to the Reggeon field theory. The turbulent mixing is modelled by the Kazantsev–Kraichnan ‘rapid-change’ ensemble: time-decorrelated Gaussian velocity field with the power-like spectrum ∝k−d − ξ. Effects of compressibility of the fluid are studied. It is shown that, depending on the relation between the exponent ξ and the spatial dimension d, both the systems exhibit four different types of critical behaviour, associated with four possible fixed points of the renormalization group equations. Three fixed points correspond to known regimes: Gaussian fixed point, original model without mixing and passively advected scalar field. The most interesting fourth point corresponds to a new type of critical behaviour, in which both nonlinearity and turbulent mixing are relevant, and the critical exponents depend on d, ξ and the degree of compressibility. The critical exponents and regions of stability for all the regimes are calculated in the leading order of the double expansion in two parameters ξ and ε = 4 − d. For both models, compressibility enhances the role of the nonlinear terms in the dynamical equations: the region in the ε–ξ plane, where the new nontrivial regime is stable, is getting much wider as the degree of compressibility increases. For the incompressible fluid, the most realistic values d = 3 and ξ = 4/3 (Kolmogorov turbulence) lie in the region of stability of the passive scalar regime. If the compressibility becomes strong enough, the crossover in the critical behaviour occurs, and these values of d and ξ fall into the region of stability of the new regime, where both advection and the nonlinearity are important. In its turn, turbulent transfer becomes more efficient due to combined effects of the mixing and the nonlinear terms.

Coarsening in the Potts model: out-of-equilibrium geometric properties

Geometric properties of polymixtures after a sudden quench in temperature are studied through the q-states Potts model on a square lattice, and their evolution with Monte Carlo simulations with non-conserved order parameter. We analyse the distribution of hull enclosed areas for different initial conditions and compare with exact and numerical findings for the q = 2 (Ising) case.

Short-time dynamics of finite-size mean-field systems

We study the short-time dynamics of a mean-field model with non-conserved orderparameter (Curie–Weiss with Glauber dynamics) by solving the associated Fokker–Planckequation. We obtain closed-form expressions for the first moments of the order parameter,near to both the critical and spinodal points, starting from different initial conditions. Thisallows us to confirm the validity of the short-time dynamical scaling hypothesis in bothcases. Although the procedure is illustrated for a particular mean-field model, ourresults can be straightforwardly extended to generic models with a single orderparameter.

Curvature-driven coarsening in the two-dimensional Potts model

We study the geometric properties of polymixtures after a sudden quench in temperature. We mimic these systems with the q -states Potts model on a square lattice with and without weak quenched disorder, and their evolution with Monte Carlo simulations with nonconserved order parameter. We analyze the distribution of hull-enclosed areas for different initial conditions and compare our results with recent exact and numerical findings for q=2 (Ising) case. Our results demonstrate the memory of the presence or absence of long-range correlations in the initial state during the coarsening regime and exhibit superuniversality properties.

The corrections to scaling within Mazenko’s theory in the limit of low and high dimensions

We consider corrections to scaling within an approximate theory developed by Mazenko for nonconserved order parameter in the limit of low (d → 1) and high (d → ∞) dimensions. The corrections to scaling considered here follows from the departures of the initial condition from the scaling morphology. Including corrections to scaling, the equal time correlation function has the form: C(r, t )= f0(r/L)+L −ω f1(r/L)+ ··· ,w hereL is a characteristic length scale (i.e. domain size). The correction-to-scaling exponent ω and the correction-to-scaling functions f1(x) are calculated for both low and high dimensions. In both dimensions the value of ω is found to be ω = 4 similar to 1D Glauber model and OJK theory (the theory developed by Ohta, Jasnow and Kawasaki).

Phase-field simulations of partial melts in geological materials

A diffuse interface description based on a multi-phase-field model for geological grain microstructures is introduced, especially useful in the treatment of partially molten structures. Each grain as well as different phases are represented by individual non-conserved order parameters, the phase fields φ α , which are defined on the complete simulation domain. The derivation of the model from a Ginzburg–Landau type free energy density functional is briefly shown and all occurring energy contributions are discussed. Also, the nondimensionalisation necessary to relate the simulation to real experiments and a brief overview of the numerical methods used in the simulations is given. To illustrate the applicability of the method to large grain systems a simulation of normal grain growth was carried out. The results on the dynamics of the process are in close agreement with theory. The extension of the phase-field model to incorporate phases with conserved volume is described next. This capacity is exploited for the liquid phase in a partially molten structure, where melt and a solid mineral phase are in equilibrium. Simulations with isolated melt inclusions within a grain structure in 2D and 3D are presented, resulting in the formation of the correct dihedral angles corresponding to the solid/solid and solid/liquid interface energies γ SS and γ SL . The case of complete wetting of the grain boundaries, if γ SL / γ SS < 0.5 is shown. Taking the structure of an analogue experiment as initial data, a simulation of a grain structure with a melt fraction of 3.6% and a dihedral angle of 10 ∘ were performed using the phase-field model. The comparison with a sharp interface front-tracking model for this case results in a highly comparable microstructure evolution.

Experimental Test of Curvature-Driven Dynamics in the Phase Ordering of a Two Dimensional Liquid Crystal

We study electric field driven deracemization in an achiral liquid crystal through the formation and coarsening of chiral domains. It is proposed that deracemization in this system is a curvature-driven process. We test this prediction using the recently obtained exact result for the distribution of hull-enclosed areas in two-dimensional coarsening with nonconserved scalar order parameter dynamics [J. J. Arenzon et al., Phys. Rev. Lett. 98, 145701 (2007)]. The experimental data are in very good agreement with the theory. We thus demonstrate that deracemization in such bent-core liquid crystals belongs to the Allen-Cahn universality class, and that the exact formula, which gives us the statistics of domain sizes during coarsening, can also be used as a strict test for this dynamic universality class.

Morphology of spinodal decompositions in liquid crystal–colloid mixtures

We study the morphology of spinodal decompositions (SDs) in mixtures of a liquid crystal and a colloidal particle by solving time-dependent Landau-Ginzburg equations for a conserved order parameter (concentration) and two nonconserved order parameters (orientation and crystallization). We numerically examine the coupling between concentration, nematic ordering, and crystalline ordering in two dimensional fluid mixtures, coexisting a nematic and a crystalline phase. On increasing the concentration of colloidal particles, we have three different SDs: a nematic order-induced SD, a phase-separation-induced SD (PSD), and a crystalline-order-induced SD (CSD). In NSD, the phase ordering can lead to fibrillar and cellular networks of the minority colloidal-particle-rich phase in early stages. In the PSD, we find a bicontinuous network structure consisting of a nematic phase rich in liquid crystal and a crystalline phase rich in colloidal particles. In the CSD, nematic droplets can be formed in a crystalline matrix. Asymmetric mixtures of a liquid crystal and a colloidal particle lead to rich varieties of morphologies.

Kinetics of Polydomain Ordering at Second-Order Phase Transitions (by the Example of the AuCu3 Alloy)

Kinetics of polydomain spinodal ordering is studied in alloys of AuCu3 type. We introduce four non-conserved long-range order parameters whose sum, however, is conserved and, using the statistical approach, follow the temporal evolution of their random spatial distribution after a rapid temperature quench. A system of nonlinear differential equations for correlators of second and third order is derived. Asymptotical analysis of this system allows to investigate the scaling regime, which develops on the late stages of evolution and to extract additional information concerning the rate of decrease of the specific volume of disordered regions and the rate of decrease of the average thickness of antiphase boundaries. Comparison of these results to experimental data is given. The quench below the spinodal and the onset of long-range order may be separated by the incubation time, whose origin is different from that in first-order phase transitions. Numerical integration of equations for correlators shows also, that it is possible to prepare a sample in such a way that its further evolution will go with formation of transient kinetically slowed polydomain structures different from the final L12 structure.

Existence of Traveling Waves of Invasion for Ginzburg–Landau-type Problems in Infinite Cylinders

We study a class of systems of reaction–diffusion equations in infinite cylinders which arise within the context of Ginzburg–Landau theories and describe the kinetics of phase transformation in second-order or weakly first-order phase transitions with non-conserved order parameters. We use a variational characterization to study the existence of a special class of traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation. Our main result is a simple verifiable criterion for existence of these traveling waves under the very general assumptions of non-linearities. We also prove boundedness, regularity, and some other properties of the obtained solutions, as well as several sufficient conditions for existence or non-existence of such traveling waves, and give rigorous upper and lower bounds for their speed. In addition, we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized. We give a sample application of our results using a computer-assisted approach.

Numerical approximation of the Ginzburg–Landau equation with memory effects in the dynamics of phase transitions

We consider the out-of-equilibrium time evolution of a nonconserved order parameter using the Ginzburg–Landau equation including memory effects. Memory effects are expected to play important role on the nonequilibrium dynamics for fast phase transitions, in which the time scales of the rapid phase conversion are comparable to the microscopic time scales. We consider two numerical approximation schemes based on Fourier collocation and finite difference methods and perform a numerical analysis with respect to the existence, stability and convergence of the schemes. We present results of direct numerical simulations for one and three spatial dimensions, and examine numerically the stability and convergence of both approximation schemes.

Monte Carlo simulation of film growth in a phase separating binary alloy model

Using a kinetic Monte Carlo method, we simulate binary film ( A 0.5 B 0.5 / A ) growth on an L × L square lattice with the focus on the domain growth behaviour. We compute the average domain area, A ( t ) , as a measure of domain size. For a sufficiently large system, we find that A ( t ) grows with a power law in time with A ( t ) ∼ t 2 / 3 after the initial transient time. This implies that the dynamic exponent for domain growth with non-conserved order parameter is z = 3 , a value which was theoretically predicted for the conserved order parameter case. Further analysis reveals that such a power-law behaviour emerges because the order parameter is approximately conserved after the early stage of growth.