Theory Of Spinodal Decomposition Research Articles

Structure factor of a phase separating binary mixture with natural and forceful interconversion of species

Using a modified Cahn-Hilliard-Cook theory for spinodal decomposition in a binary mixture that exhibits both diffusion and interconversion dynamics, we derive the time-dependent structure factor for concentration fluctuations. We compare the theory and obtain a qualitative agreement with simulations of the temporal evolution of the order parameter and structure factor in a nonequilibrium Ising/lattice-gas hybrid model in the presence of an external source of forceful interconversion. In particular, the characteristic size of the steady-state phase domain is predicted from the lower cut-off wavenumber of the amplification factor in the generalized spinodal-decomposition theory. • A source of forceful interconversion may produce microphase separation. • The time evolution of the phase formation is characterized through two wavenumbers. • Steady-state domain size scales with square root of forceful interconversion rate • Results match with Monte Carlo simulations of a nonequilibrium hybrid model. • Source of forceful interconversion may be connected to an external energy source.

Phase transitions affected by natural and forceful molecular interconversion.

If a binary liquid mixture, composed of two alternative species with equal amounts, is quenched from a high temperature to a low temperature, below the critical point of demixing, then the mixture will phase separate through a process known as spinodal decomposition. However, if the two alternative species are allowed to interconvert, either naturally (e.g., the equilibrium interconversion of enantiomers) or forcefully (e.g., via an external source of energy or matter), then the process of phase separation may drastically change. In this case, depending on the nature of interconversion, two phenomena could be observed: either phase amplification, the growth of one phase at the expense of another stable phase, or microphase separation, the formation of nongrowing (steady-state) microphase domains. In this work, we phenomenologically generalize the Cahn-Hilliard theory of spinodal decomposition to include the molecular interconversion of species and describe the physical properties of systems undergoing either phase amplification or microphase separation. We apply the developed phenomenology to accurately describe the simulation results of three atomistic models that demonstrate phase amplification and/or microphase separation. We also discuss the application of our approach to phase transitions in polyamorphic liquids. Finally, we describe the effects of fluctuations of the order parameter in the critical region on phase amplification and microphase separation.

Chirality Enhancement of Porphyrin Supramolecular Assembly Driven by a Template Preorganization Effect

AbstractCationic polylysine promotes, under neutral conditions, the spontaneous aggregation of opposite charged ZnTPPS in water. Spectroscopic investigations evidence a different preorganization of ZnTPPS onto the polypeptide matrix depending on the chain length. Spinodal decomposition theory in confined geometry is used to model this mechanism by considering the time evolution of a homogeneous distribution of randomly adsorbed particles (porphyrins) onto a rodlike polyelectrolyte (polymer) of variable length L.

Chirality Enhancement of Porphyrin Supramolecular Assembly Driven by a Template Preorganization Effect.

Cationic polylysine promotes, under neutral conditions, the spontaneous aggregation of opposite charged ZnTPPS in water. Spectroscopic investigations evidence a different preorganization of ZnTPPS onto the polypeptide matrix depending on the chain length. Spinodal decomposition theory in confined geometry is used to model this mechanism by considering the time evolution of a homogeneous distribution of randomly adsorbed particles (porphyrins) onto a rodlike polyelectrolyte (polymer) of variable length L.

About the damage to building materials and structures

We consider the damage to building materials and structures as the initial stage of dynamic process of its destruction. For the mathematical description of the initial stage of material destruction we apply recently proposed mathematical model of a non-equilibrium phase transitions formulated in terms of the Cahn-Hillard theory of spinodal decomposition. The representation of the damage process as a non-equilibrium phase transitions is carried out on the principles of thermodynamics and taking into account the specific geometric forms of the investigated construction. This allows to describe the damage process in details. So, the changes in building material under load due to the mechanisms of creep, plasticity, ductile or brittle fracture are considered as a non-equilibrium phase transitions. A mathematical model is a one-dimensional spatial variables initial-boundary problem for a system of four differential equations of second and fourth orders. Because of the nonlinearity of the system its analytical solution is difficult. So for it solution, we apply numerical methods. For the numerical solution of the problem a finite-difference scheme of second order accuracy is used. The numerical results confirm the conclusion that the mathematical model describes the main features of the process. Numerical study of one-dimensional models allows to go to the two-dimensional case and compared obtained results with experimental data

Spinodal Decomposition in Multilayered Fe-Cr System: Kinetic Stasis and Wave Instability

Used as fuel cladding in the Gen IV fission reactors, ODS steels would be held at temperatures in the range of 350°C to 600°C for several months. Under these conditions, spinodal decomposition is likely to occur in the matrix, resulting in an increase of material brittleness. In this study, thin films consisting of a modulated composition in Fe and in Cr in a given direction have been elaborated. The time evolution of the composition profiles during aging at 500°C has been characterized by atom probe tomography, indicating an apparent kinetic stasis of the initial microstructure. A computer model has been developed on the basis of the Cahn–Hilliard theory of spinodal decomposition, associated with the mobility form proposed by Martin (1990). We make the assumption that the initial profile is very close to the amplitude-dependent critical wavelength. Our calculations show that the thin film is unstable relative to wavelength modulations, resulting in the observed kinetic stasis.

Periodic spinodal decomposition as a self-limited instability

Discussed are details of the charged liquid surface reconstruction in an external field, enabling one to draw a deep analogy between the development of the charged liquid surface instability and the spinodal decomposition in the theory of phase transitions. Various consequences of this analogy for the theory of spinodal decomposition are considered.

Effect of epitaxial strain on phase separation in thin films

In epitaxially grown alloy thin films, spinodal decomposition may be promoted or suppressed depending on the sign of the epitaxial strain. We study this asymmetry by extending Cahn’s linear theory of spinodal decomposition to systems with a composition dependent lattice parameter and modulus (represented by Vegard’s law coefficients, and y, respectively), and an imposed (epitaxial) strain (e). We show analytically (and confirm using simulations) that the asymmetric effect of epitaxial strains arises only in elastically inhomogeneous systems. Specifically, we find good agreement between analytical and simulation results for the wave number of the fastest growing composition fluctuation. The asymmetric effect due to epitaxial strain also extends to microstructure formation: our simulations show islands of elastically softer (harder) phase with (without) a favourable imposed strain. We discuss the implications of these results to GeSi thin films on Si and Ge substrates, as well as InGaAs films on GaAs substrates.

A theory of spinodal decomposition stabilized by coherent strain in binary alloys

A phenomenological theory is developed which predicts the possible existence of meta-stable chemical composition modulations in the spinodal (instable) region of the phase diagram of binary alloys. This is accomplished through a modified elastic energy term in the theory of diffuse interfaces by Cahn and Hilliard: Analogously to Cahn’s expansion of the molar chemical free energy, also the (un-relaxed) internal stress tensor due to chemical composition modulations is expanded in a Taylor series of the spatial derivatives of the local composition up to second order terms. This mathematical approach is confirmed by atomistic modeling.

Theory of spinodal decomposition assisted crystallization in binary mixtures

We propose a general theory of crystallization of one component from an unstable mixture of two components by addressing the coupling between the spinodal decomposition associated with concentration fluctuations in the mixture and the nucleation kinetics for the crystallization. We propose that the domains created spontaneously by spinodal decomposition then present interfaces on which heterogeneous nucleation of the crystallizable component takes place with a much reduced nucleation barrier. Combining the theories of heterogeneous nucleation and spinodal decomposition kinetics, we present an analytic calculation of the nucleation rate as a function of the allowed duration of spinodal decomposition as well as the spinodal quench depth. In the present theory, shorter time evolution of spinodal decomposition or, equivalently, larger propensity of heterogeneous nucleation, results in faster crystallization. This is in contrast to the expectation of faster nucleation in more pure phases at later stages of spinodal decomposition. The analytic formula is found to correspond well with the recent experimental results on polymer mixtures for the late stage of spinodal decomposition kinetics. More detailed experiments are required to verify our prediction for the nucleation rate for early times and for different quench depths.

Simultaneous use of temperature and concentration gradients to control polymer solution morphology development during thermal-induced phase separation

Temperature and initial concentration gradients have been used independently during thermal-induced phase separation to create anisotropic functional porous polymeric materials.In this paper, we developed and implemented a mathematical model that combines both linear temperature and initial linear concentration gradients during the thermal-induced phase separation process. The numerical results show that the ensuing anisotropic morphology depends largely on the directions of both the temperature and concentration gradients. This allows for a better understanding and control of the fabrication process of porous functional polymeric materials. The numerical results are explained using existing spinodal decomposition theory.

Spinodal decomposition and phase separation kinetics in nanoclay–biopolymer solutions

AbstractIntensity of light, I(q,t), scattered from homogeneous aqueous solutions, of nanoclay (Laponite) and protein (gelatin‐A), was studied to monitor the temporal and spatial evolution of the solution into a phase‐separated nanoclay–protein‐rich dense phase, when the sample temperature was quenched below spinodal temperature, Ts (=311 ± 3 K). The zeta potential data revealed that the dense phase comprised charge‐neutralized intermolecular complexes of nanoclay and protein chains of low surface charge. The early stage, t < 500 s, of phase separation could be described adequately through Cahn‐Hilliard theory of spinodal decomposition where the intensity grows exponentially, I(q, t) = I0 exp.(2R(q)t). The wave vector, q dependence of the growth parameter, R(q) exhibited a maxima independent of time. Corresponding correlation length, 1/qc = ξc was found to be ≈75 ± 5 nm independent of quench depth. In the intermediate regime, anomalous growth described by I(q, t) ∼ tα with α = 0.1 ± 0.02 independent of q was observed. Rheological studies established that there was a propensity of network structures inside the dense phase. Isochronal temperature sweep studies of the dense phase determined the melting temperature, Tm = 312 ± 4 K, which was comparable with the spinodal temperature. The stress‐diffusion coupling prevailing in the dense phase when analyzed in the Doi‐Onuki model yielded a viscoelastic correlation length, ξv determined from low‐frequency storage modulus, G ′0 ≈ kB T/ξ, which was ξv ≈ 35 ± 3 nm indicating 2ξv ≈ ξc. It is concluded that the early stage of phase separation in this system was sufficiently described by linear Cahn‐Hilliard theory, but the same was not true in the intermediate stage. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 555–565, 2010

Gelation mechanism of ultra-high-molecular-weight polyethylene (UHMWPE) chains in dispersion solutions containing multiwall carbon nanotubes (MWNTs) analyzed in terms of liquid-liquid phase separation

Abstract Gelation mechanisms of ultra-high-molecular-weight polyethylene (UHMWPE) in dispersion solutions containing multiwall carbon nanotubes (MWNTs) were investigated in terms of liquid-liquid phase separation mechanisms. When an incident beam of He-Ne gas laser was directed to the dispersion solution quenched to a desired temperature, the logarithm of scattered intensity increased linearly with elapsing time and tended to deviate from this linear relationship. The two different increasing behaviors of the scattered intensity were termed as initial and latter stages, respectively, of the phase separation. The linear increase in logarithmic light-scattered intensity was analyzed within the framework of the linear theory of spinodal decomposition in terms of the viewpoint that the quenched solution becomes thermodynamically unstable and tends to incur phase separation to resolve the unstable state. The growth rate of the concentration fluctuation of neat UHMWPE solution (without MWNTs) calculated from the slope of linear increase was faster than that of the dispersion solution. In spite of the rapid growth rate, however, the gelation time of the dispersion solution was much shorter than that of the neat UHMWPE solution and the gelation and quasi-spinodal temperatures were slightly higher. These phenomena were in contradiction with the result of the concentration fluctuation. This contradiction was thought to be due to the fact that diffused UHMWPE chains in the dispersion solution were gelled easily on the MWNT surface rather than the gelation by self-coagulation.

Freezing of phase separation in spatiotemporal stripe patterns formed by Ag and Sb co-electrodeposition

Various spatiotemporal patterns of dark and light stripes are formed in the Ag and Sb co-electrodeposition system. In this research, we investigate the time-evolution of the stripe width of a spatiotemporal stripe pattern with a size of ca. 10 μm. The results show that the stripe width increases with time and is eventually saturated. Such a phenomenon is predicted by some theories of spinodal decomposition of materials with chemical reactions in a nonequilibrium system. This suggests that the pattern is formed by the spinodal decomposition of metal alloys, Ag and Sb, with chemical reactions in a nonequilibrium system, which has not yet been reported in a metallic system.

A conserved phase field system with memory and relaxed chemical potential

In this paper we propose a conserved phase-field model with memory and relaxed chemical potential. Our model could be considered also in the frame of Cahn–Hilliard's theory of spinodal decomposition of a binary mixture, where there is a coupling to a slowly relaxing variable, e.g., in phase separation of polymer mixtures near a glass transition. The global well-posedness in an L p -setting is obtained under suitable relaxation functions and under homogeneous conditions.

Morphology control in symmetric polymer blends using two-step phase separation

This paper studied through modeling and computer simulation the two-step thermal-induced phase separation phenomenon in a symmetric polymer blend via spinodal decomposition. The two-step phase separation phenomenon involves the two-step process where the initial quench is allowed to phase separate for a certain period of time before the second quench takes place into the unstable region. The second quench occurs at the transition between the early and intermediate stages of spinodal decomposition, which is the time frame when functional polymeric materials with predefined material properties are fabricated. The one-dimensional model consisted of the Cahn–Hilliard theory for spinodal decomposition, and incorporated the Flory–Huggins–deGennes free energy equation, the slow mode mobility theory and reptation model for polymer diffusion. The numerical results replicated frequently reported experimental observations published in the literature. This includes the observation that secondary phase separation occurs only if the second quench is sufficiently deep. Furthermore, the numerical results indicate that a dimensionless diffusion coefficient may be used as a parameter to control the formation and evolution of the phase-separated regions during spinodal decomposition as a means to tailor-make functional polymeric materials with predefined material properties.

Morphology control in symmetric polymer blends using spinodal decomposition

This paper studied, through modeling and computer simulation, the thermal-induced phase separation phenomenon in a symmetric polymer blend via spinodal decomposition. The one-dimensional model consisted of the Cahn–Hilliard theory for spinodal decomposition, and incorporated the Flory–Huggins–deGennes free energy equation, the slow mode mobility theory and reptation model for polymer diffusion. The numerical results replicated frequently reported experimental observations published in the literature for the early and intermediate stages of spinodal decomposition for symmetric polymer blends. Furthermore, the numerical results indicate that a dimensionless diffusion coefficient may be used as a parameter to control the formation and evolution of the phase-separated regions during spinodal decomposition as a means to customize functional polymeric materials with predefined material properties.

Phase Separation of Several Kinds of Polyethylene Solution under the Gelation/Crystallization Process

The gelation mechanism of polyethylene solutions was investigated by using light scattering and X-ray diffraction techniques in terms of the liquid-liquid phase separation. Three kinds of polyethylene, ultrahigh molecular weight polyethylene (UHMWPE), low molecular weight linear polyethylene (L-LMWPE), and low molecular weight branched polyethylene (B-LMWPE), were used as test specimens. When an incident beam of He-Ne gas laser was directed to the UHMWPE and B-LMWPE solutions quenched to a desired temperature, the logarithm of scattered intensity increased linearly in the initial stage and tended to deviate from this linear relationship in the latter stage. If the linear increase in the initial stage can be analyzed within the framework of the linear theory of spinodal decomposition proposed by Cahn, the gelation is obviously attributed to the phase separation leading to the concentration fluctuation of the solution. Furthermore, in the later stage showing the deviation from the linear relationship, light scattering from the UHMWPE solution with 0.5% concentration under Hv polarization condition showed an X-type pattern, indicating the appearance of optically anisotropic rods. On the other hand, the scattering from the B-LMWPE solution with 3% concentration showed a four-leaf clover pattern, indicating the appearance of optically anisotropic spherulites. No crystallite was confirmed by the X-ray diffraction measurements, when the rods and spherulites appeared. With further lapse of time, the Hv patterns became clearer and the corresponding X-ray diffraction intensity curves showed a very small diffraction peak from the (110) plane. In contrast, for the L-LMWPE solutions, the logarithm of scattered intensity against time showed a rapid increase and tended to level off because of rapid change from slightly transparent to whiten gels. In such a process, any quantitative analysis of the scattered light intensity was impossible. The corresponding X-ray diffraction revealed the strong reflections from the (110) and (200) planes, indicating the rapid formation of stable crystallites. Through a series of experiments for the three kinds of polyethylene solution, it turned out that gelation mechanism of polyethylene solutions is strongly affected by molecular weight and the degree of branching.

EVOLUTION OF THE STRUCTURE FACTORS IN PURE SU(N) LATTICE GAUGE THEORY AND EFFECTIVE SPIN MODELS

We consider model A dynamics for a quench from the disordered into the ordered phase of SU (3) lattice gauge theory and the analogue 3d 3-state Potts model. For the gauge model this corresponds to a rapid heating from the confined to the deconfined phase. The exponential growth factors of low-lying structure function modes are numerically calculated. The linear theory of spinodal decomposition is used to determine the critical modes. This allows for the Debye screening mass estimation in an effective phenomeno-logical model. The quench leads to competing vacuum domains, which make the equilibration of the QCD vacuum after the heating non-trivial. The influence of such domains on the gluonic energy density is studied.

Observation of One-dimensional Spinodal Decomposition in a Nematic Liquid Crystal

The zigzag instability coarsening of splay-bend walls formed in a nematic liquid crystal under external fields is investigated experimentally as a study of a one-dimensional analog of the spinodal decomposition in a conserved order parameter system. A sinusoidal modulation first appears, prior to the establishment of the zig-zag instability. Then higher harmonics of this modulation progressively grow up with time and the shape of wall changes to zigzag. We have analyzed the time evolution of the sinusoidal modulation in the early stage of coarsening and found that the sinusoidal modulation develops exponentially up to an equilibrium value as expected in the theory of spinodal decomposition. Basing our study on the linear stability analysis of the wall equation of motion, we discuss here the exponential growth and the growth rate of the sinusoidal modulation.