Cell Dynamical System Research Articles

Time-dependent analysis of agglomerating Pt thin films on YSZ single crystals

Abstract The controlled assembly of nanostructures via shape instability mechanisms is a potential alternative to traditional top-down processes like e-beam lithography for nanostructuring surfaces. In this contribution, the dynamics of the nanostructuresʼ assembly via thin film agglomeration have been analyzed. Pt thin films with a thickness of 50 nm were deposited via magnetron sputtering on yttria-stabilized zirconia (YSZ) single crystals and subjected to heat treatments at 1023 K for times ranging from 10 to 130 min. The morphological evolution of Pt thin films has been investigated by means of scanning electron microscopy (SEM) and atomic force microscopy (AFM), obtaining the hole growth dynamics and morphological parameters like the lateral correlation length and the nanostructuresʼ Minkowski functionals. The experimentally obtained morphology evolution is compared to the simulated evolution of thin film structures resulting from a cell dynamical system (CDS) model. Three main observations have been made. (i) The hole radius is found to scale as function of time t with a rate proportional to t − 3 4 [ log 3 t ] . This is in agreement with Srolovitzʼs instability theory describing hole growth predominated by surface diffusion. (ii) The morphological evolution of the Pt thin films has been analyzed as function of time t by means of Minkowski measures and the lateral correlation length. A discontinuity in the lateral correlation length and a significant deviation of the Minkowski functionals from the expected Gaussian behavior was found to be coupled with the coalescence of holes. (iii) By using the Ginzburg–Landau equation for the description of the fundamental diffusion process, the CDS model allows a computational reproduction of the experimentally obtained film morphologies in the early stages of agglomeration.

Numerical study of the phase separation in binary lipid membrane containing protein inclusions under stationary shear flow

The phase separation of lipids is believed to be responsible for the formation of lipid rafts in biological cell membrane. In the present work, a continuum model and a particle model are constructed to study the phase separation in binary lipid membrane containing inclusions under stationary shear flow. In each model, employing the cell dynamical system (CDS) approach, the kinetic equations of the confusion-advection process are numerically solved. Snapshot figures of the phase morphology are performed to intuitively display such phase evolving process. Considering the effects from both the inclusions and the shear flow, the time growth law of the characteristic domain size is discussed.

Angle Dependence of Nematic Liquid Crystal Dynamics with Conical Boundary Condition

In the manufacture of the thin film for liquid crystal devices, we can often find the situation in which the directors are equilibrated under the constraint that fixes only their polar angles on the film surface. We call this surface constraint the “conical boundary condition”. In this study, we investigate the dynamics of nematic liquid crystal under the conical boundary condition by a cell dynamical system (CDS) method. In particular, we focus on the anchoring and conical polar angle dependences of the annihilation time of topological defects and the time needed for equilibration. The angle dependence on the time characteristic of phase ordering is recognized and explained using the simplified scheme of director configuration under the hybrid condition of the conical and anchoring boundaries.

Modeling the phase separation in binary lipid membrane under externally imposed oscillatory shear flow

By adding external velocity terms, the two-dimensional time-dependent Ginzburg–Landau (TDGL) equations are modified. Based on this, the phase separation in binary lipid membrane under externally imposed oscillatory shear flow is numerically modeled employing the Cell Dynamical System (CDS) approach. Considering shear flows with different frequencies and amplitudes, several aspects of such a phase evolving process are studied. Firstly, visualized results are shown via snapshot figures of the membrane shape. And then, the simulated scattering patterns at typical moments are presented. Furthermore, in order to more quantitatively discuss this phase-separation process, the time growth laws of the characteristic domain sizes in both directions parallel and perpendicular to the flow are investigated for each case. Finally, the peculiar rheological properties of such binary lipid membrane system have been discussed, mainly the normal stress difference and the viscoelastic complex shear moduli.

Numerical simulation of the phase separation in binary lipid membrane under the effect of stationary shear flow

A numerical simulation of the phase separation in binary lipid membrane under the effect of stationary shear flow is performed. We numerically solved the modified two-dimensional time-dependent Ginzburg–Landau (TDGL) equations with an external velocity term, employing the CDS (i.e., Cell Dynamical System) technique. In the present simulation, stationary shear flows with different shear rates are taken into account. The evolution process of the phase separation is illustrated macroscopically via the snapshot figures and simulated scattering patterns at several typical moments. For each case, the growth exponents of the characteristic domain sizes in both directions parallel and perpendicular to the flow are studied, and the domain area as well. Also, the behavior of the excess viscosity has been investigated, which is a peculiar rheological indicator of such a membrane system with domain structures.

Simulation on the phase behavior of binary blends contacted with external surfaces under two-step quench conditions

The morphology evolution of binary blends subjected to a two-step quench in the presence of external surfaces, such as a static substrate or immobile fillers, was studied using the cell dynamical system (CDS) simulation method. When the binary blends were first annealed at a shallow quench for a long time followed by a second further deep quench into a two-phase region, transient lamellar structures were formed in the blends with both critical and off-critical compositions. The occurrence of the lamellar structure was found to be correlative with the compositional difference between the separated phases formed during the first shallow quench as well as the quench depth of the second quench. For blends contacted with a selective substrate or fillers, well-defined layer structures or target-like compositional waves which were paralleled to the external surfaces were formed during the two-step quench. In these systems, the bulk lamellar structure was affected by the concentration of components and fillers in the blend. The simulation by CDS demonstrated that the combination of various quench conditions and external surfaces play an important role in the pattern formation of binary blend, and provide a new approach in tailoring the morphologies of binary blends in addition to the other available methods.

Nanoscale Block Copolymer Templates Decorated by Nanoparticle Arrays

We present a novel one-step assembly process that encompasses in-situ formation of silver nanoparticles (NP), directed assembly of block copolymers into a perpendicular lamellar morphology, and NP organization into patterned arrays. This process utilizes organometallic precursors that thermally decompose and form spatially organized NP arrays on the free surface of the block copolymer. As NP concentration increases, the conversion of perpendicular to parallel lamellar morphology at the free surface slows down. A cell dynamical system (CDS) simulation captures the dynamics of this reorganization. The final structure is attractive for producing nanodevices because the surface is decorated by NP that are arranged in parallel arrays with tunable NP separation.

Effect of anisotropic current of the order parameter onInxGa1−xNcompositional fluctuation caused by phase separation

We studied details of the anisotropic spatial fluctuation in In compositions numerically as described by an early stage in the time evolution of InGaN phase separation using the cell dynamical system (CDS) approach. We extended the previous isotropic CDS approach to the anisotropic case of current of the order parameter corresponding to the two-dimensional anisotropic diffusion of the constituents. The results of this numerical-based work indicate that the anisotropic current of the order parameter, which consists of a reduction of the current in one direction and a nonreduction in the opposite direction, suppresses the In compositional fluctuation. This suppression arises from the unidirectional current of the order parameter, which relates to the enhancement of anisotropic diffusion of the constituents. Thus we substantiate the homogeneous growth of the InGaN film by using the surface of an off-axis substrate consisting of uniformly spaced steps with high density.

Diffusion coefficient of the order parameter in the early stages of the time evolution of InGaN phase separation

We have carried out a study involving a model for the diffusion mechanism of group III atoms in an InGaN alloy semiconductor that uses a cell dynamical system (CDS) approach. In this CDS analysis, we focused on gathering information concerning the In-rich region with a dot-like structure that is formed at an early stage in the time evolution of phase separation of this alloy. A similar dot-like structure is known to appear in the active layer of InGaN-based multiquantum well laser diodes. Thus, we can determine the diffusion coefficient of group III atoms in InGaN by comparing the dot-like structures observed by cross-sectional transmission electron microscopy with those obtained by the CDS numerical approach. In order to perform this comparison quantitatively such that we could use it to calculate the diffusion coefficient, we derived a relationship between the rule of mapping used in CDS, and the discretization of the Cahn–Hilliard–Cook equation, which describes the dynamics of phase separation with a conserved order parameter. The value obtained for the diffusion coefficient is 4×10−16 cm2/s at a growth temperature of 800 °C. From this result, we can conclude that the diffusion of group III atoms arises from atomistic migration on the surface of an InGaN active layer during growth, instead of diffusion within the solid phase.

Numerical analysis of InGaN dot-like structure and compositional fluctuation caused by phase separation:: comparison with experiment

We studied numerically the details of In compositional fluctuations and a spontaneous formation of a dot-like structure of InGaN alloy semiconductor due to phase separation by using the cell dynamical system (CDS) approach focusing on the details of the stage of time-evolution of phase separation. The numerical results in this work resemble the experimental results on inhomogeneouos alloying of InGaN reported so far and in addition predict the diffusion which previous experiments could not determine quantitatively. The In-rich regions with a dot-like structure appearing in InGaN-based multi-quantum-well (MQW) laser diodes (LDs) with their emission wavelength below 450 nm are composed of InGaN showing small spatial fluctuations of In composition formed at an early stage of phase separation. The large blueshift of the static electroluminescence (EL) peak observed in blue and green InGaN-based single-quantum-well (SQW) light-emitting diodes (LEDs) above 450 nm also relates to the large spatial fluctuations of In composition grown at an early stage of phase separation. For the higher In composition corresponding to the emission wavelength longer than 450 nm, the time-evolution of phase separation proceeds rapidly, indicating the large In compositional fluctuation and thus the large blueshift. In order to obtain a spatially uniform In composition, we should realize a uniform distribution of In composition during the growth of InGaN layer before phase separation proceeds.

Early Stage Process of Pattern Formation Phenomena

Pattern formation phenomena have been observed universally in various systems, such as metal alloys and polymer blends systems. 2) Entropy dominates at high temperatures, leading to a homogeneous, disordered phase, while energy becomes crucial with decreasing temperature, and this leads to the formation of the ordered phases. One of the most successful models is known as the Cahn-Hilliard (CH) equations. The familiar spinodal decomposition (SD) processes under a deep quench to an unstable state have been investigated by numerical simulations based on the above time-dependent Ginzburg-Landau ansatze. 4) Under a shallow quench, on the other hand, the fluctuations change the nature of phase transitions from the second to the first order. By using the Hartree approximation, Brazovskii has firstly shown about this effect under the long-range interaction caused by the nonzero wave number. 6) One of the important essences for this Brazovskii effect is implied here that the Landau free-energy fL(ψ) = −τLψ/2 + uLψ/4 is transformed as the effects of the fluctuations into fH(ψ) = −τHψ/2+uHψ/4+wHψ/36, where τL, uL, τH, uH and wH are phenomenological parameters. It leads to that the fluctuation-induced first order transition changes the early stages of ordering to the nucleation and growth (NCG) manner. In spite of that there exist many analytical predictions of the dynamics about fluctuationinduced pattern formation far from equilibrium [see the reference], this is not the case for numerical studies on the morphology transitions under a shallow quench. The purpose of this short note is to propose the simple numerical model, in which the Landau free-energy is extended to the Hartree free-energy so that the freeenergy can have another minimum between the local minima of the double well free-energy. In terms of the cell dynamical system (CDS) approach, Oono and Puri have proposed the hyperbolic tangent form of one-toone map, ∂fL/∂ψ = −α tanh(ψ) + ψ, by taking the Landau free-energy functional fL into account. 3) On the basis of the Hartree free-energy functional fH, in the present study, the CDS map is modified to ∂fH/∂ψ = −α tanh(βψ) + ψ. Notations α and β are adjustable parameters which are determined such that the modified map has the five intersection points with ∂f/∂ψ = 0; i.e.,

Numerical Study of InGaN Pattern Formation Caused by Phase Separation

We performed a numerical study of the two-dimensional pattern formation of InGaN driven by a phase separation using the cell dynamical system (CDS) approach to reveal the detailed nature of a self-assembled InGaN dot-like structure showing characteristic optical properties. We calculated two-dimensional pattern formations under various conditions. First, over a wide In compositional range (0<In<0.6), a dot-like structure appears arising from a phase separation as well as due to the effects of strains and defects. These results are consistent with those in the case of experimentally observed structures reported earlier. Thus, we can conclude that in the present study, the pattern formation by a phase separation affects the emission mechanism of InGaN-based semiconductor light-emitting device because of a compositional fluctuation of In. Second, the calculated results suggest that the pattern formations depend on the experimental parameters, such as growth temperature, the average and local fluctuation of In composition, the lattice mismatch of InGaN and GaN, and defects.

Cell Dynamical System Approach for a Microemulsion System

We study the phase separation of water and oil in a water-oil-surfactant microemulsion, from a totally disordered phase in two dimensions by means of the cell dynamical system (CDS) simulations on the basis of the time-dependent Ginzburg-Landau (TDGL) model with two order parameters. From our results of the bicontinuous and droplet patterns, the scaling law and the Porod’s law of the structure factor are observed.

Some recent applications of cell dynamical modelling to phase ordering dynamics

We study experimentally relevant effects in phase ordering dynamics using Cell Dynamical System (CDS) models. In particular, we present representative numerical results for phase ordering in random magnets and phase separation in binary fluids.

Scattering function of the disordered phase of block copolymers under shear flow

The scattering function of the disordered phase of a block copolymer (BCP) melt under a steady shear flow is investigated by the cell dynamical system (CDS) approach. Agreement with previous theoretical calculations indicates the applicability of the CDS approach to the disordered phase as well as the ordered phase.

Scaling of hysteresis in the Ising model and cell-dynamical systems in a linearly varying external field.

We present results from simulations of the hysteresis loops in the two-dimensional (2D) Ising model and a cell-dynamical system (CDS) in a linearly, rather than sinusoidally, varying external field. We find in the CDS a disorder-induced transition, which has behavior similar to the critical point in the 2D Ising model. Below the critical point, the area of the hysteresis loops, representing the dissipation per cycle, scales with the rate of the driving field H\ifmmode \dot{}\else \.{}\fi{} as A=${\mathit{A}}_{0}$+aH${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}^{\mathrm{\ensuremath{\alpha}}}$, with a nearly constant \ensuremath{\alpha}\ensuremath{\sim}0.36\ifmmode\pm\else\textpm\fi{}0.08 for the Ising model and 0.66\ifmmode\pm\else\textpm\fi{}0.02 for the CDS. Thus, the CDS belongs to the class of mean-field models, which is different from that of the Ising model. Above the critical point, both the Ising model and the CDS give ${\mathit{A}}_{0}$=0 and an \ensuremath{\alpha} that increases with temperature and disorder.

Spinodal decomposition in 3-space.

Using a cell-dynamical system (CDS) to model the separation of phases, we present the results of large-scale simulations of spinodal decomposition in 3-space of a symmetric binary alloy and incompressible binary fluid at critical quench. Our main results are (1) the reliable determination of the asymptotic or the almost asymptotic form factors for these systems, and (2) an understanding of the preasymptotic behavior of growth laws in terms of dispersion relations of interface fluctuations. To achieve (1) it is indispensable to have (i) methods to analyze the data, (ii) a scheme to simulate a system with fluid dynamic interactions, and (iii) a demonstration of the self-averaging nature of spherically averaged quantities. Item (iii) allows us to study sufficiently large systems with currently available computational resources. A careful stability analysis of our CDS model (such as artificial pinning, anisotropy, etc.) is also given. We must point out that the conventional data analysis coupled with artificial pinning and finite-size effects may spuriously give the theoretically desired results, especially in binary fluids. The asymptotic form factors are estimated in collage form from our large-scale late-time simulation results and various theoretical asymptotic results such as Tomita's sum rule. We may conclude that for binary fluids, the agreement between our form factor and ones obtained experimentally is excellent. This also clearly demonstrates that apart from time- and space-scale changes binary polymer systems so far studied and low-molecular-weight systems are not distinct; that is, no polymer effect has been observed in the form factor. In the case of binary alloys, the agreement with our form factor and various experimental results is not as good as the binary fluid case. This may be ascribed to various complications in solids, anisotropy, elastic effects, etc.

Simulation of Domain Growth in a System with Random Impurities

Domain growth in a system with random impurities for both conserved and nonconserved order parameter is investigated by means of the cell-dynamical system (CDS) method. We find the following. (i) The characteristic length in domain growth obeys l ( t )∼(log t ) α where α is smaller than the value predicted by Huse and Henley [Phys. Rev. Lett. 55 (1985) 2708]. (ii) The exponent α seems to depend on the impurity concentration and temperature. (iii) Dynamical scaling of the structure factor is satisfied and the form of the scaling function depends on the impurity concentration.

Nonequilibrium autocorrelations in phase-ordering dynamics.

We investigate the ordering process of an unstable system governed by a nonconserved scalar order parameter using a theoretical approach developed previously. The two-time order-parameter correlation function is shown to obey asymptotic dynamical scaling. The temporal evolution of the autocorrelation function exhibits power-law decay with a nonequilibrium exponent. We find a relation between this exponent and a nonlinear eigenvalue controlling the equal-time structure factor. This relation, as well as the predicted values of the exponent, is compared with direct numerical simulations of a cell-dynamical-system (CDS) model of the ordering process.

Non-algebraic domain growth in random magnets: a cell dynamical approach

The authors develop a novel numerical approach, based on a computationally efficient cell dynamical system (CDS) model, for studying the kinetics of ordering in systems (described by a non-conserved order parameter) with quenched disorder, evolving from unstable initial states. They use this model to study the kinetics of domain growth in a coarse-grained version of the random exchange Ising model. Their numerical data strongly indicate quantitative agreement with the theoretically predicted asymptotic growth law over a limited range of disorder amplitudes. They also compare their observations with those in laboratory experiments and make important predictions regarding dynamical scaling in these systems.