Buckling of a compressed elastic membrane: a simple model

The buckling of a folded membrane submitted to a bi-axial compression is studied in the framework of the continuum non-linear elasticity theory. We show that the formation of the fold patterning can be quantitatively well described with a simple non-linear model. As a matter of fact, with this model, we recover the experimental phase diagram of a secondary buckling instability with a very good precision. In addition, depending on the anisotropy of the applied compressive stress, we find that the buckling coarsening dynamics can be described as a 1D spinodal decomposition (for a uni-axial stress) or as a 2D XY model (for an isotropic bi-axial stress) with an irrotational non-scalar order parameter. For an isotropic bi-axial stress, we indeed recover the famous coarsening exponent: n=1/4. This exponent has to be confirmed experimentally.     

界面润湿性及固相体积分数对颗粒粗化动力学影响的相场法研究

利用多相场模型模拟了液-固两相体系中固相颗粒的粗化过程, 分析了界面润湿性及固相体积分数对粗化指数、粗化速率及颗粒尺寸分布的影响.结果表明, 不同固相体积分数下粗化指数基本不变, 但粗化速率常数及尺寸分布与固相体积分数及界面润湿性密切相关.在完全润湿条件下, 随着固相体积分数的增加, 粗化速率常数逐渐增大; 而非完全润湿条件下, 随着固相体积分数的增加, 粗化速率常数增大速度变缓, 且当润湿性较低、 固相分数较大时, 粗化速率常数还将随体积分数的增加而下降. 此外, 模拟结果表明各种润湿条件下颗粒的尺寸分布均随着固相分数增加而变宽, 分布峰值降低, 但非完全润湿条件下峰值下降变缓.模拟结果为理解不同实验观测结果之间的分歧提供了依据. 关键词： 粗化 相转变 相场法 润湿性     

Persistence in systems with algebraic interaction

Persistence in coarsening one-dimensional spin systems with a power-law interaction r(-1-sigma) is considered. Numerical studies indicate that for sufficiently large values of the interaction exponent sigma (sigma > or =1/2 in our simulations), persistence decays as an algebraic function of the length scale L, P(L) approximately L(-theta). The persistence exponent theta is found to be independent on the force exponent sigma and close to its value for the extremal (sigma-->infinity) model, theta =0.175 075 88. For smaller values of the force exponent (sigma < 1/2), finite size effects prevent the system from reaching the asymptotic regime. Scaling arguments suggest that in order to avoid significant boundary effects for small sigma, the system size should grow as [O(1/sigma)](1/sigma).     

A correlated majority model

We extend and solve analytically a bichromatic majority model including color correlation between nearest neighbor plaquettes. The behaviour of the one color, say black, dominant cluster size is studied as a function of the black concentration and the correlation degree. This mean value shows a divergence with a critical exponent v. We also define an order parameter m for the model whose critical exponent is β. We find v = β = 1 independent of the correlation degree. We present a Monte Carlo simulation of the process which gives a good agreement with the theoretical results.     

Noise-assisted mound coarsening in epitaxial growth

Two types of mechanisms are proposed for mound coarsening during unstable epitaxial growth: stochastic, due to deposition noise, and deterministic, due to mass currents driven by surface energy differences. Both yield the relation H=(RWL)² between the typical mound height W, mound size L, and the film thickness H. An analysis of simulations and experimental data shows that the parameter R saturates to a value which discriminates sharply between stochastic () and deterministic () coarsening. We derive a scaling relation between the coarsening exponent 1/z and the mound-height exponent which, for a saturated mound slope, yields . Received: 11 November 1997 / Revised in final form: 28 November 1997 / Accepted: 28 November 1997     

Stages of homogeneous nucleation in solid isotopic helium mixtures

We have made pressure and NMR measurements during the evolution of phase separation in solid helium isotopic mixtures. Our observations indicate clearly all three stages of the homogeneous nucleation-growth process: (1) creation of nucleation sites; (2) growth of the new-phase component at these nucleation sites; and (3) coarsening: the dissolution of subcritical droplets with the consequent further late-stage growth of the supercritical droplets. The time exponent for the coarsening, a=1/3, is consistent with the conserved order parameter Lifshitz-Slezov evaporation-condensation mechanism.     

Nearest neighbor ising model with added local 4-spin interaction

The effects of adding a local 4-spin term to the nearest neighbor Ising model are studied. The relevant parameter isx, the ratio of the four- to the two-spin interaction strength. We have evaluated the first six terms of the high temperature susceptibility expansion and found them to be consistent with the value 1.75 of the critical exponent γ. The critical exponent α is shown to remain zero to first order inx, provided that the free energy is analytic inx. We also demonstrate the equivalence of the model examined to an elastic spin system with pure two-spin interactions.     

Dynamics of aeolian sand ripples

We analyze theoretically the dynamics of aeolian sand ripples. In order to put the study in the context, we first review existing models. This paper is a continuation of two previous papers (Z. Csahók et al., Physica D 128, 87 (1999); A. Valance et al., Eur. Phys. J. B 10, 543 (1999)), the first one is based on symmetries and the second on a hydrodynamical model. We show how the hydrodynamical model may be modified to recover the missing terms that are dictated by symmetries. The symmetry and conservation arguments are powerful in that the form of the equation is model-independent. We then present an extensive numerical and analytical analysis of the generic sand ripple equation. We find that at the initial stage the wavelength of the ripple is that corresponding to the linearly most dangerous mode. At later stages the profile undergoes a coarsening process leading to a significant increase of the wavelength. We find that including the next higher-order nonlinear term in the equation leads naturally to a saturation of the local slope. We analyze both analytically and numerically the coarsening stage, in terms of a dynamical exponent for the mean wavelength increase. We discuss some future lines of investigations. Received 20 January 2000     

Scaling behavior of grain-rotation-induced grain growth

Recent investigations of grain growth in nanocrystalline materials have revealed a new growth mechanism: grain-rotation-induced grain coalescence. Based on a simple model employing a stochastic theory and using computer simulations, here we investigate the coarsening of a polycrystalline microstructure due solely to the grain-rotation coalescence mechanism. Our study demonstrates that this mechanism exhibits power-law growth with a universal scaling exponent. The value of this universal growth exponent is shown to depend on the assumed mechanism by which the grain rotations are accommodated.     

The kinetics of hill-and-valley faceting of oxygen-covered tungsten

The coarsening kinetics of steplike faceted O/W(332) surface is investigated using field ion microscopy. The spatial density of facets is measured as a function of time (10–320 s) and temperature (1100–1400 K). The results are discussed in terms of the temporal coarsening exponent and the activation energy for the coarsening process.     

Energy avalanches in a rice-pile model

We investigate a one-dimensional rice-pile model. We show that the distribution of dissipated potential energy decays as a power law with an exponent α = 1.53. The system thus provides a one-dimensional example of self-organized criticality. Different driving conditions are examined in order to allow for comparisons with experiments.     

Etch front roughening in wrinkly metal

Scaling behavior of wrinkle evolution due to chemical etching of Au on stretchable substrate is studied by atomic force microscopy. The surfaces were etched with a small drop of KI solution. Scaling exponent α decreased with the etch time, from 0.93 to 0.62. For dynamic exponent β up to 30 min we find it to be 0.16 ± 0.05. Within the same time frame, the wavelength increases as ∼t^0.23+−0.05 but drops at 45 min as ∼t^{−1.22+−0.42} and saturates. At this stage, we observe percolation island type features along with some wrinkles. The measured value of α at this stage is 0.62 ± 0.02; consistent with the percolation model. The wavelength coarsening at this stage suggests the existence of chemically driven spinodal decomposition. Our obtained value of α and β is consistent with the value of z obtained from correlation length which is given by ξ = t^1/z. We find z = 4.87 ± 0.50. Our experiment suggests wrinkle formation in thin films on viscoelastic surface occurs in three stages.     

Critical behaviour of random-bond Potts models: a transfer matrix study

We study the two-dimensional Potts model on the square lattice in the presence of quenched random-bond impurities. For q > 4 the first-order transitions of the pure model are softened due to the impurities, and we determine the resulting universality classes by combining transfer matrix data with conformal invariance. The magnetic exponent β/v varies continuously with q, assuming non-Ising values for q > 4, whereas the correlation length exponent ν is numerically consistent with unity. We present evidence for the correctness of a formerly proposed phase diagram, unifying pure, percolative and non-trivial random behaviour.     

Block persistence

We define a block persistence probability p _l (t) as the probability that the order parameter integrated on a block of linear size l has never changed sign since the initial time in a phase-ordering process at finite temperature T<T _c . We argue that in the scaling limit of large blocks, where z is the growth exponent (), is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent for large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large-n model, and generically it can be used to determine easily from simulations of coarsening models. We also argue that and the scaling function do not depend on temperature, leading to a definition of at finite temperature, whereas the local persistence probability decays exponentially due to thermal fluctuations. These ideas are applied to the study of persistence for conserved models. We illustrate our discussions by extensive numerical results. We also comment on the relation between this method and an alternative definition of at finite temperature recently introduced by Derrida [Phys. Rev. E 55, 3705 (1997)]. Received: 25 February 1998 / Revised: 24 July 1998 / Accepted: 27 July 1998     

Exact Scaling in the Expansion-Modification System

This work is devoted to the study of the scaling, and the consequent power-law behavior, of the correlation function in a mutation-replication model known as the expansion-modification system. The latter is a biology inspired random substitution model for the genome evolution, which is defined on a binary alphabet and depends on a parameter interpreted as a mutation probability. We prove that the time-evolution of this system is such that any initial measure converges towards a unique stationary one exhibiting decay of correlations not slower than a power-law. We then prove, for a significant range of mutation probabilities, that the decay of correlations indeed follows a power-law with scaling exponent smoothly depending on the mutation probability. Finally we put forward an argument which allows us to give a closed expression for the corresponding scaling exponent for all the values of the mutation probability. Such a scaling exponent turns out to be a piecewise smooth function of the parameter.     

Damage spreading and critical exponents for “model A” Ising dynamics

Using damage spreading and heat bath dynamics, we study the Ising model in 2 and 3 dimensions with non-conservative dynamics. Our algorithm differs in some important points from previous ones, which makes it rather efficient. We give estimates for the exponent z which seem to be the most precise published so far (2.172 ± 0.006 for d = 2, 2.032 ± 0.004 for d = 3). We also give precise estimates of the exponent θ′ introduced by Janssen et al. (Z. Phys. B 73 (1989) 539) and of analogous but in principle independent exponents. We find surprisingly that some of the latter agree with θ′, and give an explanation for this.     

Hurst exponent determination for digital speckle patterns in roughness control of metallic surfaces

In this paper, we present a study of metallic surface roughness using the Hurst exponent calculated from speckle pattern. A set of samples was prepared using polishing techniques and the roughness was directly measured by means of an optical profilometer. To study the H exponent, an experiment was performed by illuminating the samples using an expanded laser beam and the surface image was captured by a CCD camera. We applied techniques of the Hurst exponent calculation, traditionally calculated from surface profile, in the digitalized speckle patterns generated by the rough surfaces. We showed a clear dependence of the H exponent on roughness of the samples. We demonstrated that this tool is very sensitive to defects in the surfaces and can be used for roughness control.     

Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D

We introduce a geometric generalization of the O(N)-field theory that describes N-colored membranes with arbitrary dimension D. As the O(N)-model reduces in the limit N → 0 to self-avoiding polymers, the N-colored manifold model leads to self-avoiding tethered membranes. In the other limit, for inner dimension D → 1, the manifold model reduces to the O(N)-field theory. We analyze the scaling properties of the model at criticality by a one-loop perturbative renormalization group analysis around an upper critical line. The freedom to optimize with respect to the expansion point on this line allows us to obtain the exponent ν of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large N limit and Hartree approximation, can also be applied to this model. By comparison of low- and high-temperature expansions, we arrive at a conjecture for the nature of droplets dominating the 3d Ising model at criticality, which is satisfied by our numerical results. We can also construct an appropriate generalization that describes cubic anisotropy, by adding an interaction between manifolds of the same color. The two parameter space includes a variety of new phases and fixed points, some with Ising criticality, enabling us to extract a remarkably precise value of 0.6315 for the exponent ν in d = 3. A particular limit of the model with cubic anisotropy corresponds to the random bond Ising problem; unlike the field theory formulation, we find a fixed point describing this system at 1-loop order.     

Do the extended states in random dimer chains survive under perturbations?

We study the delocalization properties of the resonant extended states in the random dimer model (RDM) with various perturbations added. The divergent localization length ξ is found to display a power-law decay ξ ∝ W^α as a function of the disorder W in the dimer or the background energies, with an exponent α around 2 depending on the location of the resonant state. For randomness in the dimer structure the exponent becomes α_p = 1 and for the case of an applied electric field it reaches higher values α_e > 2. These effects on the resonant states are discussed in connection with the electronic transport properties of realistic disordered systems.     

Roughness evolution of Si surfaces upon Ar ion erosion

We studied the roughness evolution of Si surfaces upon Ar ion erosion in real time. Following the theory of surface kinetic roughening, a model proposed by Majaniemi was used to obtain the value of the dynamic scaling exponent β from our data. The model was found to explain both the observed roughening and the smoothening of the surfaces. The values of the scaling exponents α and β, important for establishing a universal model for ion erosion of (Si) surfaces, have been determined. The value of β proved to increase with decreasing ion energy, while the static scaling exponent α was found to be ion energy independent.