Spinodal Region Research Articles

Kinetics of spinodal decomposition in mixtures of low molecular weight liquid crystals and flexible polymers

AbstractIn this paper, the time‐dependent Ginzburg‐Landau model for mixtures containing nematogens has been applied to mixtures of low molecular weight liquid crystals and flexible polymers. Dynamic equations for the time evolution of concentration and orientation fluctuations and the structure factors for these fluctuations are given. It is shown that the coupling between concentration and orientation fluctuations is absent in the isotropic spinodal region, thus the evolution of the structure factors for the concentration fluctuations falls into the Cahn‐Hilliard classic category and it exhibits no maximum in the structure factors of orientation fluctuations. We should emphasize that, in the anisotropic spinodal region, both concentration and orientation structure factors possess a maximum but not coincide with each other and both are shifting to smaller wave numbers according to the scaling relation, qmax ˜ τ−α, as time increases. The value of a closely correlates to the interfacial free‐energy parameters.

Thermodynamic Analysis of the Cubic–Tetragonal Phase Equilibria in the System ZrO2–YO1.5

The cubic‐tetragonal (c‐t) phase equilibria in the system ZrO2‐YO1.5 are thermodynamically analyzed from Landau's phenomenological theory. The calculated c‐t two‐phase field is depicted as a miscibility gap with a sharp maximum and the spinodal region as originally predicted by Hillert and Sakuma. However, the observed c‐t two‐phase field and the spinodal region are better described by the present model. In addition, this model can be used to discuss the nature of the c‐t diffusionless transformation from the order parameter in contrast with the original model. The predicted change in the tetragonality of t‐ZrO2 with YO1.5 content is slightly different from that in the c/a axial ratio estimated from X‐ray diffraction analysis. The displacement of cations and anions may not take place simultaneously during the c‐t transformation.

Chemical freezing of phase separation in immiscible binary mixtures

We discuss the thermodynamic and kinetic conditions under which chemical reactions may prevent the coarsening terminating spinodal decomposition and freeze the unmixing of binary mixtures at some early, pattern forming, stage of evolution. Under very general conditions, we establish that (i) this pattern freezing phenomenon can only occur in nonequilibrium systems the level of dissipation of which exceeds a finite, nonzero threshold value; (ii) at least two independent chemical processes must take place; (iii) chemistry must be destabilizing, which requires that at least one of these processes must be autocatalytic; (iv) pattern formation is possible even outside of the spinodal region, i.e., without involving a phase separation phenomenon, in unsymmetrical mixtures where the potential energies between pairs of identical particles are sufficiently different. This latter condition replaces for nonideal chemically reacting binary mixtures the unequal diffusion coefficients condition which governs the appearance of Turing patterns in the classical reaction-diffusion theory.

Low-temperature phase decomposition in metal from iron, stony-iron, and stony meteorites

The microstructure of the outer taenite rim and cloudy zone of meteoritic metal was investigated using high resolution scanning electron microscopy and analytical electron microscopy. The micro-structure is formed by a series of complex phase transformations at <-400°C. These transformations are interpreted using the Fe-Ni phase diagram. The outermost subzone (zone 1) of the outer taenite rim is composed of two single-phase layers, Ni 3Fe and FeNi (tetrataenite). The Ni 3Fe phase appears at the border between kamacite and the outer taenite rim. The inner two subzones (zones 2 and 3) in the outer taenite rim are two-phase with a matrix of tetrataenite and low Ni bcc precipitates. The three zone structure is a general feature in the outer taenite rim of metal particles in all major types of meteorites. The cloudy zone is composed of an island region and a bcc single phase honeycomb region containing ∼9.0 wt% Ni. The island region is a two-phase mixture of a L1 0 ordered fee tetrataenite matrix and low-Ni bee precipitates. The high-Ni taenite region, γ 2, resulting from the monotectoid reaction, γ 1 → α + γ 2 at 400°C remains as a supersaturated solid solution to lower temperatures and corresponds metallographically to the outer taenite rim. The width of the outer taenite rim varies inversely with the cooling rate of the meteorite. The γ′, Ni 3Fe, phase is formed when the high-Ni γ 2 phase containing ∼53 wt% Ni undergoes an eutectoid reaction, γ 2 → α + γ′ at about 345°C. The cloudy zone develops by spinodal decomposition below 350°C. Metastable γ 2 island region of high Ni content and metastable γ 1 honeycomb region of low Ni content are formed. The size of the constituents of the cloudy zone increases with increasing Ni content because the higher Ni regions enter the spinodal region at higher temperatures. The precipitates in the island region of the cloudy zone and in zones 2 and 3 of the outer taenite rim have essentially the same Ni content (14.2–14.7 wt%) and form at lower temperatures when the Ni content of γ 2 is located in the γ 1 + γ 2 miscibility gap but outside the spinodal. If cooling rates or metamorphic reheating is to be studied at or below 400°C, the low temperature phase transformations must be considered. Improvement in simulation models will be difficult to make, however, because the diffusion rates in ordered FeNi, tetrataenite, and ordered Ni 3Fe are not known.

Nonequilibrium dynamics of symmetry breaking inλΦ4theory

The time evolution of O$(N)$ symmetric $\ensuremath{\lambda}({\ensuremath{\Phi}}^{2}{)}^{2}$ scalar field theory is studied in the large $N$ limit. In this limit the $〈{\mathbf{\ensuremath{\Phi}}}_{i}〉$ mean field and two-point correlation function $〈{\mathbf{\ensuremath{\Phi}}}_{i}{\mathbf{\ensuremath{\Phi}}}_{j}〉$ evolve together as a self-consistent closed Hamiltonian system, characterized by a Gaussian density matrix. The static part of the effective Hamiltonian defines the true effective potential ${U}_{\mathrm{eff}}$ for configurations far from thermal equilibrium. Numerically solving the time evolution equations for energy densities corresponding to a quench in the unstable spinodal region, we find results quite different from what might be inferred from the equilibrium free energy potential $F$. Typical time evolutions show effectively irreversible energy flow from the coherent mean fields to the quantum fluctuating modes, due to the creation of massless Goldstone bosons near threshold. The plasma frequency and collisionless damping rate of the mean fields are calculated in terms of the particle number density by a linear response analysis and compared with the numerical results. Dephasing of the fluctuations leads also to the growth of an effective entropy and the transition from quantum to classical behavior of the ensemble. In addition to casting some light on fundamental issues of nonequilibrium quantum statistical mechanics, the general framework presented in this work may be applied to a study of the dynamics of second order phase transitions in a wide variety of Landau-Ginsburg systems described by a scalar order parameter.

Spinodal decomposition in elastically anisotropic inhomogeneous systems in the presence of an applied traction

The nonlinear diffusion equation governing mass flow in an elastically anisotropic, inhomogeneous system with applied tractions is developed for arbitrary crystal symmetry. The linearized version of the equation is solved using standard Fourier methods and the spinodal region of the material is identified and examined for systems with either cubic or isotropic symmetry in the presence of hydrostatic, uniaxial, and pure shear stresses. The resulting change in the onset of the spinodal instability and the morphology of the decomposed system is discussed. Using this approach, we derive a nonlinear Cahn - Hilliard equation for an elastically inhomogeneous system.

Self-organization in two- and three-dimensional adsorbates with attractive lateral interactions

Self-consistent lattice gas models for the dynamics of monolayer and multilayer adsorbates with attractive lateral interactions are investigated. The existence of spinodal regions with uphill diffusion is shown. For monolayer growth, periodic stationary distributions of adsorbate density are found analytically. Generalized kinetic BET-model for multilayer growth is studied numerically. A possibility for surface transformation into self-organized arrays of quantum dot-like structures is shown.

Pattern formation of the stationary Cahn-Hilliard model

We investigate critical points of the free energy Eε(u) of the Cahn–Hilliard model over the unit square under the constraint of a mean value ü. We show that for any fixed value ü in the so-called spinodal region and to any mode of an infinite class, there are critical points of Eε(u) having the characteristic symmetries of that mode provided ε &gt; 0 is small enough. As ε tends to zero, these critical points have singular limits forming characteristic patterns for each mode. Furthermore, any singular limit is a stable critical point of E0(u)). Our method consists of a global bifurcation analysis of critical points of the energy Eε(u) where the bifurcation parameter is the mean value ü.

A revision of the Fe-Ni phase diagram at low temperatures (&lt;400 °C)

The low-temperature Fe-Ni phase diagram was assessed experimentally by investigating Fe-Ni regions of meteorites using high resolution analytical electron microscopy techniques. The present phase diagram differs from the available experimental phase diagram based on observations of meteorite structure, but it is consistent with the available theoretical diagram in that α/Ni3Fe equilibrium was found at low temperatures. The a phase containing 3.6 wt.% Ni is in local equilibrium with the γ′ (Ni3Fe) phase containing 65.5 wt.% Ni, while the γ′' (FeNi) phase is present as a metastable phase. The new phase diagram incorporates a monotectoid reaction (γ1 → α + γ2, where (γ1 is a paramagnetic fcc austenite, a is a bcc ferrite, and γ2 is a ferromagnetic fcc austenite) at about 400 °C, a eutectoid reaction (γ2 → α + γ′) at about 345 °C, and a miscibility gap associated with a spinodal region at low temperatures. The miscibility gap is located between 9.0 and 51.5 wt. % Ni at ∼200 °C. The new low-temperature Fe-Ni phase diagram is consistent with all the phases observed in the metallic regions of meteorites.

Replication of interphase boundaries under spinodal decomposition

We discuss a peculiar kinetic phenomenon that can occur after a two-step and sufficiently deep quench of a phase-separating system into the spinodal instability region: interphase boundaries “replicate” generating approximately periodic patterns. We also discuss conditions for experimental observations of this self-organization phenomenon.

Nonlinearities of the Vlasov equation in the spinodal region

We estimate the importance of the nonlinear terms in the Vlasov equation for the development of the unstable modes. The results allow to identify the region of wavelength where the linear evolution is justified.

Contribution of density fluctuations to the damping of giant resonances

A generalized kinetic equation of Lennard-Balescu type is derived with retardation effects. By explicit consideration of the density-density fluctuations we give the contribution of fluctuations to the spreading width of giant resonances besides the collisional part. For small frequencies and long wavelengths, approximate formulas for the damping of giant resonances are derived. The density fluctuations couple the equation of state to the damping rate and give rise to an overall enhancement factor. This density and temperature dependent factor can reproduce the experimental values of the damping. Inside the spinodal region the factor turns out to be negative, indicating unstable modes. Explicit comparisons are made with the experimental values of giant monopole and dipole resonances. It is shown that the experimental widths can be described assuming finite nuclei since surface effects are important for the influence on density fluctuations. The recently reported saturation of the dipole width with increasing energy is explained as a transition between zero and first sound. Within this treatment we find an overcritical range of fluctuations, above which the nucleus is destroyed and no collective behavior is possible at all. {copyright} {ital 1996 The American Physical Society.}

Gauge-invariant effective potential: Equilibrium and nonequilibrium aspects.

We propose a gauge invariant formulation of the effective potential in terms of a gauge invariant order parameter, for the Abelian Higgs model. The one-loop contribution at zero and finite temperature is computed explicitly, and the leading terms in the high temperature expansion are obtained. The result is contrasted to the effective potential obtained in several covariant gauge-fixing schemes, and the gauge invariant quantities that can be reliably extracted from these are identified. It is pointed out that the gauge invariant effective potential in the one-loop approximation is complex for {\em all values} of the order parameter between the maximum and the minimum of the tree level potential, both at zero and non-zero temperature. The imaginary part is related to long-wavelength instabilities towards phase separation. We study the real-time dynamics of initial states in the spinodal region, and relate the imaginary part of the effective potential to the growth rate of equal-time gauge invariant correlation functions in these states. We conjecture that the spinodal instabilities may play a role in non-equilibrium processes {\em inside} the nucleating bubbles if the transition is first order.

Chaos vs linear instability in the Vlasov equation: A fractal analysis characterization.

In this paper we discuss the most recent results concerning the Vlasov dynamics inside the spinodal region. The chaotic behavior which follows an initial regular evolution is characterized through the calculation of the fractal dimension of the distribution of the final modes excited. The ambiguous role of the largest Lyapunov exponent for unstable systems is also critically reviewed. This investigation seems to confirm the crucial role played by deterministic chaos in nuclear multifragmentation. \textcopyright{} 1996 The American Physical Society.

Instabilities in Nuclear Multifragmentation

The evolution of dynamical perturbations is examined in nuclear multifragmentation in the frame of Vlasov equation. Both plane wave and bubble type of perturbations are investigated in the presence of surface (Yukawa) forces. An energy condition is given for the allowed type of instabilities and the time scale of the exponential growth of the instabilities is calculated. The results are compared to the mechanical spinodal region predictions.

Phase transitions, correlations and fluctuations of nuclear multifragmentation

Five different phases of the nuclear equation of state are described: the gas, the liquid, the super heated liquid, the super cooled gas and the mechanically unstable spinodal regions, both for nuclear matter and for the finite nucleus197Au, with a density, temperature, and momentum dependent mean field derived from the extended Skyrme force. From realistic simulations in r-space and in p-space for these different phases at a finite temperature T=6 MeV, one can sort all kinds of nuclear clusters by a coalescence model. Correlation analyses of the nuclear clusters demonstrate that, both multifragmentation and intermittent patterns come only from the spinodal region, critical behaviour comes mainly from the spinodal region while there is also a small contribution from the super heated liquid and the liquid regions.

Phase instability of inhomogeneous gels crosslinked within the spinodal region

We consider theoretically the gels crosslinked in the early stage of spinodal decomposition. A weakly crosslinked and homogeneous gel is brought into the thermodynamically unstable state, and additional crosslinks are introduced while the spinodal decomposition takes place. The inhomogenization of gel is ‘‘recorded’’ as position dependent distribution of chain conformations. The model is based on the theory of permanent set and that of nonequilibrium thermodynamics of gels. We investigate the effect of the rate of crosslinking on the process of spinodal decomposition. We also analyze the stability of the inhomogeneous gel thus processed. Our study shows that the spinodal instability may still occur in an inhomogeneous ‘‘opaque’’ gel, and a possible experimental test is proposed.

Structure Pinning During Phase-Separation of Polyamide/Ionomer Blends

ABSTRACTBlends of lightly sulfonated polystyrene and poly(N,N'-dimethylethylene sebacamide) (Li-SPS/mPA) are miscible as a result of strong ion-amide complexation. The blends exhibit LCST phase behavior and an increase of the sulfonation level from 4 to 9.5 mol% raises the critical temperature by 150°C. Phase separation may be thermally induced and isthermodynamically reversible. The phase separation kinetics that occur following a temperature-jump deep into the spinodal region of the phase diagram deviate from conventional Cahn-Hilliard theory and the phase separation process stalls after a couple of hours, essentially pinning the structure at that point. The extent of phase-separation that occurs before pinning is temperature-dependent.

On the density of volume states in the isobaric ensemble

The conventional implementation of the isobaric (fluctuating volume) ensemble neglects the density of volume states. Here it is shown that the latter is proportional to the inverse of the volume. It is argued that this expression should always be used, and that it has measurable consequences in simulations of small systems, in simulations that seek an accurate equation of state, and in simulations of the critical, spinodal, and coexistence regions.

Comment on the presence of chaos in mean field dynamics inside the spinodal region

In this letter we re-analyze the role of chaos in the mean field simulations of spinodal decomposition. We demonstrate that, conversely to recent publications, the RPA unstable modes are only weakly non-linear and weakly coupled during the early clusterization of the system. The Lyapunov exponents are shown to be nothing but the largest imaginary RPA frequencies and the early mean field evolution, even of a randomly initialized trajectory, is shown to be mostly understandable within the framework of linear instabilities. Finally we recall how the time scales are related to the use of a reasonable range for the nuclear force.