Initial Stages Of Spinodal Decomposition Research Articles

Rayleigh–Benard Instability: a Study by the Methods of Cahn–Hillard Theory of Nonequilibrium Phase Transitions

This article is an attempt to study the process of Rayleigh–Benard convective instability by the methods used for mathematical modeling of critical phenomena as nonequilibrium phase transitions in their initial stages of spinodal decomposition. We show that it is possible to extend the formalism adopted in the Cahn–Hillard theory of nonequilibrium phase transitions and perfected on problems of highgradient crystallization to other types of problems, in particular, those pertaining to the Rayleigh–Benard convective instability. For the initial stage of instability, a model is constructed that represents it as a nonequilibrium phase transition due to diffusive stratification. It is shown that the Gibbs free energy of deviation from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginsburg–Landau potential. Numerical experiments, by means of boundary temperature control, have been conducted with regard to self-excitation of the homogeneous state. Numerical analysis shows that convective flows may appear and proceed from regular forms (the so-called regular structures) to nonregular flows through a chaotization of the process. External factors, such as temperature growth, may lead to chaos via period doubling bifurcations.

The Effect of Nonisothermality on the Early Stages of Spinodal Decomposition

A nonisothertmal model of spinodal decomposition is proposed for a binary system described by the Ginzburg–Landau energy. The initial stages of spinodal decomposition are investigated for a system of simultaneous equations describing the impurity and temperature redistribution. The dispersion equation and the instability growth rate of spinodal structures are obtained. The temperature dependence of the growth rate and the wave number corresponding to the maximum instability is found. The observed difference in the dispersion relations of isothermal and nonisothermal models shows that a criterion for the effect of temperature fluctuations on the spinodal decomposition is given by a dimensionless parameter Γ; for large values of Γ (Γ ≥ 1000), temperature fluctuations in the form of noise are certainly insufficient for spinodal decomposition due to a change in the dispersion structure of the equations, which suggests the need to take into account the nonisothermal behavior of the system.

Pattern formation mechanisms in sphere-forming diblock copolymer thin films

The order-disorder transition of a sphere-forming block copolymer thin film was numerically studied through a Cahn-Hilliard model. Simulations show that the fundamental mechanisms of pattern formation are spinodal decomposition and nucleation and growth. The range of validity of each relaxation process is controlled by the spinodal and order-disorder temperatures. The initial stages of spinodal decomposition are well approximated by a linear analysis of the evolution equation of the system. In the metastable region, the critical size for nucleation diverges upon approaching the order--disorder transition, and reduces to the size of a single domain as the spinodal is approached. Grain boundaries and topological defects inhibit the formation of superheated phases above the order--disorder temperature. The numerical results are in good qualitative agreement with experimental data on sphere-forming diblock copolymer thin films.
 Received: 22 August 2017, Accepted: 12 December 2017; Edited by: R. Dickman; Reviewed by: A. Peters, Dept. Chemical Engineering, Louisiana Tech Univ., Ruston, USA; DOI: http://dx.doi.org/10.4279/PIP.100001
 Cite as: L R Gómez, N A García, R A Register, D A Vega, Papers in Physics 10, 100001 (2018)
 This paper, by L R Gómez, N A García, R A Register, D A Vega, is licensed under the Creative Commons Attribution License 4.0.

Effects of elastic energy on the spinodal decomposition in InAlGaN materials

AbstractThe purpose of our study is to determine the influence of elastic energy on the spinodal decomposition ranges of InAlGaN alloys lattice‐matched to GaN. The distinction in lattice constants of the binary compounds constituting these alloys gives rise to their deformation energies. Such energies provide the tendency to disintegration that can lead to an appearance of the thermodynamically unstable states with respect to the decomposition. The initial stage of spinodal decomposition of the InAlGaN quaternary alloys is accompanied by transfers of atoms over distances of a lattice parameter. Accordingly, these transfers produce a thin two‐layer region with negligibly small differences in their compositions. Formation of these layers causes the elastic energy. The alloys are represented as quaternary regular solutions. The internal deformation energy is calculated with the interaction parameters estimated by the valence force field model. The spinodal decomposition ranges of these materials are demonstrated up to 580 °C (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase separation of an asymmetric binary-fluid mixture confined in a nanoscopic slit pore: Molecular-dynamics simulations

As a generic model system of an asymmetric binary-fluid mixture, hexadecane dissolved in carbon dioxide is considered, using a coarse-grained bead-spring model for the short polymer, and a simple spherical particle with Lennard-Jones interactions for the carbon dioxide molecules. In previous work, it has been shown that this model reproduces the real phase diagram reasonably well, and also the initial stages of spinodal decomposition in the bulk following a sudden expansion of the system could be studied. Using the parallelized simulation package ESPResSo on a multiprocessor supercomputer, phase separation of thin fluid films confined between parallel walls that are repulsive for both types of molecules are simulated in a rather large system ( 1356 x 1356 x 67.8 A{3} , corresponding to about 3.2-million atoms). Following the sudden system expansion, a complicated interplay between phase separation in the directions perpendicular and parallel to the walls is found: In the early stages the hexadecane molecules accumulate mostly in the center of the slit pore, but as the coarsening of the structure in the parallel direction proceeds, the inhomogeneity in the perpendicular direction gets much reduced. Studying then the structure factors and correlation functions at fixed distances from the wall, the densities are essentially not conserved at these distances, and hence the behavior differs strongly from spinodal decomposition in the bulk. Some of the characteristic lengths show a nonmonotonic variation with time, and simple coarsening described by power-law growth is only observed if the domain sizes are much larger than the film thickness.

Domain formation and growth in spinodal decomposition of a binary fluid by molecular dynamics simulations

The two initial stages of spinodal decomposition of a symmetric binary Lennard-Jones fluid have been simulated by molecular dynamics simulations, using a hydrodynamics-conserving thermostat. By analyzing the growth of the average domain size R(t) with time, a satisfactory agreement is found with the R(t) proportional t1/3 Lifshitz-Slyozov growth law for the early diffusion-driven stage of domain formation in a quenched homogeneous mixture. In the subsequent stage of viscous-dominated growth, the mean domain size appears to follow the linear growth law predicted by Siggia.

Initial stage of spinodal decomposition in a rigid-rod system

The initial stage of spinodal decomposition is investigated for a rigid-rod system. Spinodal decomposition proceeds through either of two mechanisms: (1) The randomly aligned rods rotate toward a common director with no inherent length scale. (2) The rods diffuse axially and segregate into regions of common alignment with a selected length scale [script-l]. Previous studies on spinodal decomposition yielded radically different conclusions about which mechanism is dominant. A computational method is employed to analyze the growth rate and properties of the dominant fluctuation mode through an eigenvalue analysis of the linearized Doi diffusion equation in Fourier space. The linearized operator is discretized in Fourier mode and orientation space (k,theta,phi) space, and the maximum eigenvalue and corresponding eigenvector of the operator are calculated. Our analysis generalizes the results of previous studies and shows that the conflicts seen in those studies are due to differences in the diffusivities for rotational motion, motion perpendicular to the rod axis, and motion along the rod axis. For a given system density, a plot of the dominant perturbation wave number k(max) as a function of the diffusivity ratios shows two separate regions corresponding to mechanisms (1) and (2). High rotational diffusivity corresponds to mechanism (1), whereas high axial diffusivity corresponds to mechanism (2). The transition between the two mechanisms depends on the ratio of diffusivities and on system density. Also, the dominant wave number increases with increasing density indicating that a deeper quench into the spinodal regime leads to a smaller characteristic length scale.

Does coarsening begin during the initial stages of spinodal decomposition?

The initial stages of spinodal decomposition were studied by subjecting a critical blend of model polyolefins to a pressure quench and monitoring the evolution by time-resolved small angle neutron scattering. Contrary to the predictions of the widely accepted Cahn-Hilliard-Cook theory, we demonstrate that coarsening of the phase-separated structure begins immediately after the quench and occurs throughout the initial stages of spinodal decomposition.

On the initial stage of spinodal decomposition

A consistent theory describing the initial stage of spinodal decomposition of a two-component system is proposed. It is shown that the structure factor S(q, t) has two maxima as a function of wavenumber at this stage. The main maximum point q+ varies with time, first moving from qsaddle to qm (given by expressions (22) and (12), respectively) and then back after the turning point in time given by (26) is passed. The other maximum point is localized at q ≈ 0, and the corresponding peak amplitude is virtually independent of time. The characteristics of the main maximum are sensitive to the existence of the “zero” peak. Available experimental observations support the predictions of the theory.

Spinodal decomposition and coarsening of stressed thin films on compliant substrates

The microstructural evolution and long-time coarsening behavior of a thin film attached to a compliant substrate is investigated for a spinodally decomposing, binary, two-phase alloy using a Cahn–Hilliard type equation. Elastic fields arise because of composition dependence of the lattice parameter (compositional self-strain) as well as an average misfit between the film and substrate. This leads to a local mass flux within the film that depends on the global composition field as well as the thickness and elastic properties of the substrate. Three distinct coarsening regimes are observed: a rapid initial stage of spinodal decomposition into alternating phases, a transition region where internal interfaces grow and coalesce, and a final coarsening regime where the outermost layers grow with little change in the internal structure. An asymptotic expression for the long-time coarsening rate is derived as a function of the compositional self-strain, the average misfit strain, the film thickness and the total (film plus substrate) thickness. The analytic predictions are in good agreement with numerical simulations for a variety of film and substrate thicknesses and compositional and average film misfit strains.

Gelation Process and Phase Separation of PVA Solutions As Studied by a Light Scattering Technique

Time-resolved light scattering measurements were performed on gelation processes of poly(vinyl alcohol) (PVA) solutions at 25 °C in mixtures of dimethyl sulfoxide (DMSO) and water (H2O) with volume ratios of 60/40 and 80/20. It was found that the scattering profile I(Q) from a 60/40 (DMSO/H2O) solution has a maximum whose position Qm is independent of time and that the intensity I(Q) at Q = Qm increases exponentially with time. These observations agree with the theoretical predictions for the initial stage of spinodal decomposition (SD) type phase separation, while the scattering profile I(Q) from a 80/20 solution has no peak. It is therefore concluded that the SD type microphase separation occurs before gelation in the 60/40 solution, but not in the 80/20 solution. On the basis of present, as well as past results, a possible mechanism of the gelation process for PVA solutions has been proposed.

A constitutive relation describing the shear-banding transition.

An additional contribution to the standard expression for the shear stress must be considered in order to describe shear banding. A possible extension of the standard constitutive relation is proposed. Its physical, purely hydrodynamic origin is discussed. The corresponding Navier-Stokes equation is analyzed for the two-plate geometry, where flow gradients are assumed to exist only in the direction perpendicular to the two plates. The linearized Navier-Stokes equation is shown to be very similar to the Cahn-Hilliard equation for spinodal decomposition, with a similar term that stabilizes rapid spatial variations. Only slowly varying flow gradients are unstable. Just as in the initial stage of spinodal decomposition there is a most rapidly growing wavelength in the initial stage of the shear-banding transition, leading to a predictable number of shear bands. A modified Maxwell equal area construction is derived, which dictates the stress and the shear rates in the bands under controlled shear conditions, and which shows that under controlled stress conditions no true shear bands can coexist. The kinetics of the shear-banding transition is studied numerically. For the two-plate geometry it is found that there exist multiple stationary states under controlled shear conditions, depending on the initial state of the flow profile. Shear banding occurs not only when the system is initially unstable, but can also be induced outside the unstable region when the amplitude of the initial perturbation is large enough. The shear-banding transition can thus proceed via "spinodal demixing" (from an unstable initial state) or via "condensation." Under controlled stress conditions no stationary state is found. Here, coupling with flow gradients extending in other directions, not perpendicular to the two plates, should probably be taken into account.

Shear induced fluctuations and fluctuation-length transition in the quasiflexible polymer in a solution of a hydrogen-bond enhancing solvent

Dynamics of cellulose diacetate (a typical polysaccharide) in N,N-dimethylacetamide (a hydrogen-bond enhancing solvent) at 27 °C was investigated in a shear field of velocity gradient γ=0–5.3 s−1 through dynamic light scattering under circular Couette flow. The polymer mass concentration was set around the crossover of dilute and semidilute regimes. Growth of extra-large length-scale fluctuations and their transitions with shear were detected, which were originated by the long-ranged interaction, or the hydrogen bond, operating on the OH and CO groups in the glucose residues of cellulose acetates via high electronegative solvent. Two modes of fluctuations with fast and slow decay rates Γf and Γs were induced and each Γ value changed stepwise and linearly with γ, respectively. The decay rate distribution, which was estimated from the electric-field time correlation function expressed in the KWW-type of stretched exponential decay form g(1)(t)=A exp[−(t/τ)β] with β the index of decay rate distributions, was changed from 1/2 (one-sided stable distribution) to 3/2 (Holtzmark distribution) around γ=0.8–1 s−1 for the fast mode, while β=2 (Gaussian, or the sharpest distribution) over the full range of 0.40<γ<5.3 s−1 for the slow mode. The usual Cauchy distribution (β=1) at γ=0 (the quiescent state) was destroyed above a threshold γ*=0.033 s−1. At γ*<γ<0.2 s−1, fluctuations of various length-scale modes (0<β<1/2) were created but finally converged to two modes of fixed β(=1/2, 3/2, 2) at γ>0.4 s−1. In addition, oscillatory feature was conspicuous in g(1)(t) at every γ of 0.40<γ<5.3 s−1 and periodic fluctuations became remarkable in the scattered intensity IVv at γ>3.7 s−1, which may come from the competition between shear flow distortion and the Brownian diffusion of polymer chain in solution and can be grasped qualitatively by the initial stage of spinodal decomposition under the control of long-ranged attractive interactions.

Surface effects on spinodal decomposition in binary mixtures and the interplay with wetting phenomena.

The phase separation of binary mixtures in a semi-infinite geometry is investigated both by a phenomenological theory and by numerical calculations using a discrete equivalent of the descriptive equations. In the framework of ``model B'' (which describes solid binary mixtures), attention is paid to a proper treatment of the boundary conditions at the free surfaces. We confine ourselves to short-range surface forces and consider parameter values that correspond to both nonwet and wet surfaces in thermal equilibrium. During the initial stages of spinodal decomposition, after a quench from considering an initial condition that corresponds to a completely random concentration distribution, one finds a rather rapid growth of a thin surface enrichment layer of the component that is energetically preferred by the surface. This layer stabilizes a growing concentration wave in the direction normal to the surface, whose oscillations are damped to zero as one goes into the bulk. Studying concentration correlations in the direction parallel to the surface at a distance z from it, one can define a length scale ${\mathit{l}}_{\mathrm{\ensuremath{\parallel}}}$(z,t) describing the coarsening of the growing domains. We find that ${\mathit{l}}_{\mathrm{\ensuremath{\parallel}}}$(z,t)\ensuremath{\approxeq}A(z)+B(z)${\mathit{t}}^{\mathit{a}}$ for large t, with an amplitude B(z) that increases as z\ensuremath{\rightarrow}0 and an exponent a which is close to that for Lifshitz-Slyozov growth, viz., a\ensuremath{\approxeq}1/3. This increase is due to an orientation of the growing elongated domains parallel to the surface near z=0. There is surprisingly little influence of the wetting transition on these phenomena--even for a wet surface the growth of the wetting layer is only logarithmic in time for short-range surface forces, and hence does not significantly affect phenomena on the faster time scales of spinodal decomposition. Experimental findings on the interplay of spinodal decomposition and wetting are critically discussed.

Monte Carlo simulation of polymeric materials: Recent progress

AbstractMonte Carlo simulations are presented, dealing with phase diagrams of block copolymer melts and polymer blends, including the unmixing kinetics of the latter systems. The theoretical background is briefly reviewed: Ginzburg‐type criteria reveal that in mixtures of long flexible polymers a “crossover” occurs from mean‐field behavior (as described by Flory‐Huggins theory) to nonclassical Ising‐type behavior, and spinodal curves can be unusually sharp. This crossover is demonstrated by large scale simulations of the bond fluctuation model, and it is also shown that for symmetric mixtures the critical temperature scales with chain length as Tc α N. The prefactor in this relation is distinctly smaller than predicted by Flory‐Huggins, but the Curro‐Schweizer integral equation theory prediction Tc α √N is clearly ruled out. Tests of the Cahn theory on the initial stages of spinodal decomposition of polymer blends will also be reported.To conclude, the mesophase formation in block copolymers is discussed, and it is shown that the simulations agree very well with experiment. The pronounced chain stretching that already occurs in the disordered phase is compelling evidence against the validity of simple random phase approximation concepts for these systems. This shows how Monte Carlo simulations can assist in better understanding large classes of polymeric materials.

Scattering function and the dynamics of phase separation in polymer mixtures under shear flow

The phenomenological mean-field theory describing concentration fluctuations and spinodal decomposition of binary mixtures of long flexible macromolecules is generalized to mixtures under steady shear flow. This shear flow leads to a partial orientation and stretching of the coils, as well as to an anisotropic deformation of concentration fluctuations. Generalizing the approach of Onuki and Kawasaki, we obtain the collective scattering function describing these concentration fluctuations in the mixture under shear flow. Both the steady-state situation in the one-phase region and the initial stages of spinodal decomposition for concentrations inside of the spinodal curve are considered.

Influence of a continuous quenching procedure on the initial stages of spinodal decomposition

Instead of the standard assumption in the theory of phase separation where an instantaneous quench from an initial equilibrium state to the final state in the two-phase region is assumed, we consider the more realistic situation that the change of the external control parameter (e.g. temperature) can only be performed with finite rates. During the initial stages of spinodal decomposition the system then has some “memory” of the states intermediate between the initial and the final one. This influence of the finite quench rate in continuous quenching procedures is studied within the linearized theory of spinodal decomposition, with the Langer-Baron-Miller decoupling, and with Monte Carlo simulations. Both the case of thermally activated mobilities (applicable to solid metallic alloys) and the case of nearly temperature-independent mobilities (applicable to fluid polymer mixtures) are treated, and possible experimental applications are discussed. We find drastic deviations from the standard instantaneous quench situations in all cases of experimental interest.

Dynamic effects in spinodal decomposition

We present a preliminary study of wave propagation in elastic media which can undergo spinodal decomposition. Spinodal decomposition is a phase separation process which takes place by non-linear diffusion. The induced deformation gradients, however, can remain small and the theory of linear elasticity usually suffices for their determination. In this work, attention is confined to initial stages of spinodal decomposition where the dynamic counterpart of the linear coupled diffusion-deformation theory earlier developed by Aifantis seems to be appropriate in analyzing the relevant phenomena.

Diffuse and Small Angle Neutron Scattering Studies of Clustering and Decomposition in Electron-Irradiated Cu-Ni-Fe Alloys

ABSTRACTEarlier investigations of short range clustering (SRC) and long range decomposition (LRD) in Cu-Ni alloys upon electron irradiation [1,2] have been extended to alloys containing 0.5 and 2 at.% Fe. Moreover, quenching instead of furnace cooling was applied to achieve a well defined homophase equilibrium state with a small degree of SRC. Under subsequent irradiation at temperatures between 373 K and 623 K the decomposition kinetics of the alloys have been followed by diffuse and small angle neutron scattering experiments within a momentum transfer range of 0.001 < κ(Å−1) < 8. The addition of Fe is found to extend the miscibility gap to higher temperatures. The observed scattering curves are discussed in terms of the existing models for the initial stages of spinodal decomposition, irradiation enhanced diffusion and phase stability under irradiation.

Phase Separation in Fluid Systems by Spinodal Decomposition: A Molecular-Dynamics Simulation

Using the molecular-dynamics technique, we describe the dynamical evolution of phase separation by spinodal decomposition for a Lennard-Jones fluid quenched to the unstable region. This is the first direct observation of the initial stages of spinodal decomposition.