Model Of Spinodal Decomposition Research Articles

Improved capping layer growth towards increased stability of InGaN quantum dots

AbstractImprovements in GaN capping layer growth towards increased stability of InGaN quantum dots (QDs) will be presented. Surface quality is enhanced with changing the carrier gas during GaN capping layer growth from nitrogen to hydrogen. Without a dissolution of QDs it is possible to increase the temperature to 1050 °C applying a temperature ramp during growth of GaN.Dot stacks with an uniform and up to room temperature bright photoluminescence emission spectra is reported qualifying the structure to be incorporated into LED and laser structures. Finally, the evolution of the nucleation layer into flat and broad island‐like structures will be explained by the spinodal decomposition model (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Deterministic and Stochastic Dynamics in Spinodal Decomposition of a Binary System

A model for diffusion and phase separation, which takes into account hyperbolic relaxation of the solute diffusion flux, is developed. Such a ‘hyperbolic model’ provides analysis of ‘hyperbolic evolution’ of patterns in spinodal decomposition in systems supercooled below critical temperature. Analytical results for the hyperbolic model of spinodal decomposition are summarized in comparison with outcomes of classic Cahn−Hilliard theory. Numeric modelling shows that the hyperbolic evolution leads to sharper boundary between two structures of a decomposed system in comparison with prediction of parabolic equation given by the theory of Cahn and Hilliard. Considering phase separation processes in stochastic systems with a field-dependent mobility and an internal multiplicative noise, we study dynamics of spinodal decomposition for parabolic and hyperbolic models separately. It is that the domain growth law is generalized when internal fluctuations are introduced into the model. A mean field approach is carried out in order to obtain the stationary probability, bifurcation and phase diagrams displaying re-entrant phase transitions. We relate our approach to entropy-driven phase-transitions theory.

Verified Homology Computations for Nodal Domains

Homology has long been accepted as an important computational tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurately the resulting homology can be computed. In this paper we present an algorithm for correctly computing the homology of one- and two-dimensional nodal domains. The approach relies on constructing an appropriate cubical approximation for the nodal domain based on the behavior of the defining function at the vertices of a fixed grid. Betti numbers for these cubical sets are readily computable using [T. Kaczynski, K. Mischaikow, and M. Mrozek, Computational Homology, Springer-Verlag, New York, 2004; W. Kalies, M. Mrozek, and P. Pilarczyk, Computational Homology Project, http://chomp.rutgers.edu/ (2006)]. Here, we present a technique to verify that the cubical representation is homeomorphic to the nodal domain and therefore preserves homology. To illustrate this approach we consider examples from three classes of nodal domains, including the time-dependent patterns generated by the Cahn–Hilliard model for spinodal decomposition. We use these results to examine the probability of correct homology computations given specific grid sizes as related to the analytic estimates presented in [S. Day et al., Electron. Res. Announc. Amer. Math. Soc., 13 (2007), pp. 60–73; K. Mischaikow and T. Wanner, Ann. Appl. Probab., 17 (2007), pp. 980–1018; K. Mischaikow and T. Wanner, submitted; P. Niyogi, S. Smale, and S. Weinberger, Discrete Comput. Geom., 39 (2008), pp. 419–441].

Mechanical Model for Filament Buckling and Growth by Phase Ordering

A mechanical model of open filament shape and growth driven by phase ordering is formulated. For a given phase-ordering driving force, the model output is the filament shape evolution and the filament end-point kinematics. The linearized model for the slope of the filament is the Cahn-Hilliard model of spinodal decomposition, where the buckling corresponds to concentration fluctuations. Two modes are predicted: (i) sequential growth and buckling and (ii) simultaneous buckling and growth. The relation among the maximum buckling rate, filament tension, and matrix viscosity is given. These results contribute to ongoing work in smectic A filament buckling.

Experimental test for the hyperbolic model of spinodal decomposition in the binary system

A model has been presented for the physical decay with the relaxation of the diffusion flux described by the hyperbolic diffusion equation. The analysis of such a hyperbolic model provides the predictions for the critical parameters of the decay, which are compared with the conclusions of the Cahn-Hilliard theory. It has been shown that the hyperbolic model predicts the nonlinearity of the dispersion curve for the spinodal decay, which is controlled by the ratio of the diffusion and correlation lengths. The predicted behavior of the dispersion curve is compared with the experimental data on phase separation in binary glasses.

Oscillatory solutions to the system of Allen-Cahn and Cahn-Hilliard equations: A spinodal decomposition model

The boundary-value problem for the system of Cahn-Novick-Cohen equations is analyzed. This problem is a quasi-continuum model of the corresponding lattice model for an Fe-Al alloy and simultaneously describes the processes of phase separation (spinodal decomposition) and atomic ordering in sublattices. It is demonstrated that the evolution of the system can occur according to the following three scenarios. (i) Against the background of a disordered state v = 0, spatially nonuniform distributions of the concentration u with respect to a stationary distribution u = um that is dependent on the mean mass m are developed at long times t → ∞. (ii) Against the background of a stationary distribution of the concentration u = um, spatially nonuniform distributions of the order parameter are developed at t → ∞. (iii) The first and second scenarios can proceed simultaneously for a specific set of parameters (for example, with the dimensionless temperature ϑ = T/Tc, where Tc is the critical temperature). The results of the calculations performed with so-called asymmetric boundary conditions of wetting in a constant magnetic field for a thin quasi-one-dimensional film consisting of a binary mixture are compared with data obtained from numerical and real experiments.

Investigation of nanoscale composition fluctuations in InGaN using optical transmission spectroscopy and near-field scanning optical microscopy

Nanoscale composition fluctuations in InGaN alloy epitaxial layers have been investigated by optical transmission spectroscopy and near-field scanning optical microscopy. A spatial variation of the optical transmission at 488nm is observed, which is attributed to an inhomogeneous indium distribution. Both the characteristic wavelength and the normalized root mean square magnitude of the compositional fluctuations increase with increasing In concentration. The fluctuations are well described in terms of a spinodal decomposition model.

An approach to the formation mechanism of the composition fluctuation in GaInNAs quantum wells

This work is focused on the study by transmission electron microscopy of the composition inhomogeneity of GaInNAs quantum wells. A variation of the contrast along the quantum well is reported which is related to phase separation in the alloy. This variation becomes more intense on increasing the growth temperature. From the intensity profiles taken from the images, we have calculated the energy of activation for surface diffusion in the alloy. The results suggest that the diffusion of indium controls the process of formation of phase separation. With regard to the thermodynamics of the process, we have extended the spinodal decomposition model of Ipatova to the quaternary alloy GaInNAs. Our calculations have shown that composition fluctuations of indium are stimulated by the introduction of nitrogen into the ternary alloy GaInAs. It is proposed that the observed compositional inhomogeneity of GaInNAs is composed mainly of a fluctuation of the indium content but secondly that N bonds preferentially to Ga-rich regions in the alloy.

Depletion−Flocculation in Oil-in-Water Emulsions Using Fibrillar Protein Assemblies

This paper shows that low concentrations of beta-lactoglobulin fibrils can induce depletion-flocculation in beta-lactoglobulin-stabilized oil-in-water emulsions. The minimum required fibril concentration for flocculation was determined experimentally for fibril lengths of about 3 and 0.1 microm. The minimum fibril concentration for flocculation is two orders of magnitude higher for the short fibrils than for the long ones. These experimental results correspond well with a theoretical prediction based on a model of spinodal decomposition. In addition, rheological measurements were performed, verifying that flocculation was induced by a depletion mechanism. The results of this study show that the minimum concentration required for depletion-flocculation can be tuned by varying the length of the fibrils.

Dynamics of nanoscale self-assembly of ternary epilayers

This paper models the self-assembly dynamics of a ternary epilayer on an elastic substrate. The phases coarsen to reduce the phase boundary energy and refine to reduce the elastic energy. They may order into nanoscale patterns to minimize the total free energy. We combine the continuum phase field model of spinodal decomposition and the surface stress. A set of nonlinear diffusion equations couple the two concentration fields in the epilayer and the stress field in the substrate. Numerical simulations reveal remarkably rich dynamics and suggest the exciting possibility of controlling a nanoscale self-assembly process.

Nonlinear growth of periodic patterns.

We study the growth of a periodic pattern in one dimension for a model of spinodal decomposition, the Cahn-Hilliard equation. We particularly focus on the intermediate region, where the nonlinearity cannot be neglected anymore, and before the coalescence dominates. The dynamics is captured through the standard technique of a solubility condition performed over a particular family of quasistatic solutions. The main result is that the dynamics along this particular class of solutions can be expressed in terms of a simple ordinary differential equation. The density profile of the stationary regime found at the end of the nonlinear growth is also well characterized. Numerical simulations correspond satisfactorily to the analytical results through three different methods and asymptotic dynamics are well recovered, even far from the region where the approximations hold.

Universal growth exponent at the early nonlinear stage of spinodal decomposition

The nonlinear Cahn's model of spinodal decomposition is considered within the framework of the Langer's ‘mean-field approximation’. It is argued that a universal growth exponent of 1/4 characterizes the early nonlinear stage of spinodal decomposition, regardless of the space dimensionality and the spectral characteristics of the gradient energy matrix. This is a contrary to our earlier predictions made for isotropic and anisotropic decomposition (Mitlin, 1989, 1990). The temporal dependence of the magnitude of the structure factor follows a power law with the exponent of d/2 (d is the space dimensionality). The temporal dependence of the semi-width of the structure factor follows a power law with the exponent of −1/4 (with logarithmic corrections).

Fragment size correlations in finite systems: Application to nuclear multifragmentation

We present a new method for the calculation of fragment size correlations in a discrete finite system in which correlations explicitly due to the finite extent of the system are suppressed. To this end, we introduce a combinatorial model, which describes the fragmentation of a finite system as a sequence of independent random emissions of fragments. The sequence is accepted when the sum of the sizes is equal to the total size. The parameters of the model, which may be used to calculate all partition probabilities, are the intrinsic probabilities associated with the fragments. Any fragment size correlation function can be built by calculating the ratio between the partition probabilities in the data sample (resulting from an experiment or from a Monte Carlo simulation) and the 'independent emission' model partition probabilities. This technique is applied to charge correlations introduced by Moretto and collaborators. It is shown that the percolation and the nuclear statistical multifragmentaion model ({\sc smm}) are almost independent emission models whereas the nuclear spinodal decomposition model ({\sc bob}) shows strong correlations corresponding to the break-up of the hot dilute nucleus into nearly equal size fragments.

Symmetry breaking in self-assembled monolayers on solid surfaces: Anisotropic surface stress

This paper models the self-assembly dynamics of a two-phase monolayer on an elastic substrate. The two phases coarsen to reduce the phase boundary energy and refine to reduce the elastic energy. To minimize the total free energy, the two phases can order into nanoscale patterns. We combine the continuum phase field model of spinodal decomposition and the anisotropic surface stress. The numerical simulation shows various patterns, such as interwoven stripes, parallel stripes, triangular lattice of dots, and herringbone structures. The surface stress anisotropy causes a transition from the parallel stripes to the herringbone structures. We show that this symmetry breaking mesophase transition obeys the classical theory of Landau.

Modelling spinodal decomposition at the atomic scale: beyond the Cahn - Hilliard model

In this paper three models are used to describe spinodal decomposition in Fe - Cr alloys. The challenge presented by this particular alloy system is that the wavelength of the decomposition is at the nanometre scale. The scale of the microstructure is too small to expect the continuum approximations of the Cahn - Hilliard approach to apply, and, indeed, we show that the approach fails to describe the kinetics of the experimentally observed decomposition. On the other hand it has already been established in our earlier work that the dynamic Ising model describes the kinetics of this decomposition remarkably well. In this paper we show that the Penrose model of spinodal decomposition can also account for the kinetics quantitatively. The Penrose model is deterministic and is essentially a mean-field approximation to the dynamic Ising model. In addition we show that the form of the initial distribution of amplitudes of compositional variations within the high-temperature random alloy has a significant effect on the subsequent kinetics of phase separation.

Dispersion relation around a kink solution in binary fluids undergoing spinodal decomposition.

A numerical study is presented of the dispersion relation for the linearized operator about the kink or interfacial wall solution to a model of spinodal decomposition of an incompressible binary fluid. The essential spectrum of the linearized operator does not stay well separated from the Nambu-Goldstone- (NG-) like mode representing purely interfacial motion. For large wave vector k along the interface, the NG-like mode decays as \ensuremath{\omega}\ensuremath{\sim}${\mathit{k}}^{3}$, while for small wave vector it decays as \ensuremath{\omega}\ensuremath{\sim}k. However, the bottom of the essential spectrum decays like \ensuremath{\omega}\ensuremath{\sim}${\mathit{k}}^{2}$ and at small enough k, it intersects the point spectrum, the NG-like mode. The dispersion relation of the NG-like modes as one varies the viscosity indicates a nonuniversal crossover behavior from \ensuremath{\omega}\ensuremath{\sim}${\mathit{k}}^{3}$ to \ensuremath{\omega}\ensuremath{\sim}k, due to the interaction between NG-like mode and the essential spectrum.

Interdiffusion in binary polymer mixtures

The interface between two partially miscible polymer species (A+B) is characterized theoretically under different thermodynamic conditions using the standard model for spinodal decomposition and interdiffusion. Two quantities are introduced to characterize the mass transport process across the interface between the A-rich and B-rich layers. One is the interfacial width W defined as related to the reciprocal of the maximal gradient in composition and is most sensitive to the local structure cline to the A/B interface. The other is mass transport M due to diffusion and is mint insensitive to detailed profiling of the composition

Micellar formation and correlation in the cavity of porous silica glass.

Small-angle neutron scattering (SANS) is used to study the scattering length density correlations in a dry porous silica glass (Vycor) and in the glass filled with anionic and cationic surfactant molecules dispersed in water. SANS intensity distributions from empty pores of the glass can be satisfactorily analyzed using a standard late stage spinodal decomposition model in which the porous structure is described by a superposition of random density waves. This model, however, agrees with experiment only if one allows for a suitable distribution of wave numbers for the density waves. Micellar formation is observed in Vycor filled with surfactant solutions. We discuss the interpretation of the intermicellar correlations, measured by SANS, within the pores of the Vycor glass.

Anomalous small-angle X-ray scattering study of the early stages of decomposition in Cu–15wt%Ni–8wt%Sn

The decomposition process of the supersaturated solid solution Cu–15wt%Ni–8wt%Sn prepared by powder metallurgy has been studied by anomalous small-angle scattering of X-rays. The ageing has been followed at 623 K. The analysis of the small-angle scattering intensities shows that the spinodal decomposition model accounts neither for the measured contrast of the electronic densities nor for the anomalous effects observed near the absorption edges of the elements contained in the sample. The measured intensity variation is large and of opposite sign when the Cu and Ni edges are approached from the lower energy side: a decrease of 30% for an energy variation from 292 to 8 eV below the Ni K-absorption edge and a sharp increase of 140% from 300 to 7 eV below the Cu K edge. A comparison with previous studies on the same system indicates that the decomposition process investigated in our samples corresponds more to the nucleation and growth regime than to the spinodal one. We found that the assumption of (Cu0.52Ni0.48)3Sn precipitates having D022 structure is in agreement with our experimental results.

Multiscaling and multifractality

A general scaling argument is presented which predicts for the two points correlation function for growth phenomena and critical phenomena a scaling form (multiscaling) of the type g(rR)=r−d+DrRArR, where r is the distance from the origin, R a characteristic length and A(x) is an amplitude. A connection between multifractality and multiscaling is shown. The numerical data of Plischke and Racz on diffusion-limited aggregation and a soluble model for spinodal decomposition are found in agreement with such multiscaling form.