Universality in diffusion front growth dynamics

We have studied the scaling properties of diffusion fronts by numerical calculations based on the mean field approach in the context of a lattice gas model, performed in a triangular lattice. We find that the height-height correlation function scales with time t and length l as C(l, t) ≈l ^α f (t/l ^α/β) with α = 0.62±0.01 and β = 0.39±0.02. These exponent values are identical to those characterising the roughness of the diffusion fronts evolving through a square lattice [1,2], thus confirming their universality. Received 14 November 2001 / Received in final form 20 April 2002 Published online 31 July 2002     

Director field configurations around a spherical particle in a nematic liquid crystal

We study the director field around a spherical particle immersed in a uniformly aligned nematic liquid crystal and assume that the molecules prefer a homeotropic orientation at the surface of the particle. Three structures are possible: a dipole, a Saturn-ring, and a surface-ring configuration, which we investigate by numerically minimizing the Frank free energy supplemented by a magnetic-field and a surface term. In the dipole configuration, which is the absolutely stable structure for micron-size particles and sufficiently strong surface anchoring, a twist transition is found and analyzed. We show that a transition from the dipole to the Saturn ring configuration is induced by either decreasing the particle size or by applying a magnetic field. The effect of metastability and the occurrence of hysteresis in connection with a magnetic field are discussed. The surface-ring configuration appears when the surface-anchoring strength W is reduced. It is also favored by a large saddle-splay constant K₂₄. A comparison with recent experiments [#!itapdb:Poulin1997!#,#!itapdb:Poulin1998!#] gives a lower bound for W, i.e., for the interface of water and pentylcyanobiphenyl (5CB) in the presence of the surfactant sodium dodecyl sulfate. Received 2 November 1998     

Swelling behavior and viscoelasticity of ultrathin grafted hyaluronic acid films

In this paper we study the effect of monovalent and divalent ions on the swelling behavior and viscoelastic parameters of ultrathin layers of the natural polyelectrolyte hyaluronic acid covalently coupled to glass substrates. A colloidal probe technique is applied for this purpose based on latex beads, hovering over the polymer cushion. By analyzing the vertical Brownian motion of these beads with reflection interference contrast microscopy (RICM) we determined the equilibrium layer thickness (with 3 nm vertical resolution), the interfacial interaction potential, and the characteristic mesh size limiting the hydrodynamic flow within the polyelectrolyte film as a function of the ionic strength. The experimental results are interpreted in terms of three different theoretical models: the polyelectrolyte brush approximation of Pincus [#!ref1!#], a modified polyelectrolyte brush approximation in the high salt concentration limit of Ross and Pincus [#!ref2!#] and the simple scaling approximation for neutral adsorbed polymers of de Gennes [#!ref3!#]. Within experimental error all of these different models fit our experimental data and yield comparable results for the equilibrium layer thickness. Moreover we determine a thickness dependent, effective surface coverage from both brush models. The hydrodynamic properties of the films are interpreted in terms of the Brinkmann model of elastic porous media by assuming an effective mesh size, which depends linearly on the Debye screening length. The salt induced condensation of the polyelectrolyte films can be described microscopically in terms of a progressive contraction of the mesh size with increasing salt concentration. Received 10 September 1998 and Received in final form 30 November 1998     

Magnetization dynamics below of EuO and EuS

Between 4.2 K and the Curie temperatures of the cubic Heisenberg ferromagnets EuS and EuO, their homogeneous dynamic susceptibilities have been investigated by means of a broad-band reflectometer operating from 0.1 GHz to 40 GHz. For internal magnetic fields larger than the anisotropy fields H A ( T ) of both materials, their static susceptibilities exhibit a -divergence, which reveals quantitatively the dominance of dipolar-anisotropic spin-wave fluctuations. displays a Lorentzian shape the damping frequency of which obeys scaling in terms of .The scaling function agrees quantitatively with work by Frey and Schwabl [#!FS88!#] for dipolar Heisenberg ferromagnets at temperatures above T_c. Building upon their approach, the resonance frequency of the Lorentzian can be related to a memory effect in the damping determined by the large value of the relaxation rate of the longitudinal magnetization fluctuations . For EuS, this relation is substantiated directly by inelastic neutron scattering. All these features reveal the hitherto uncovered importance of the dipolar anisotropic fluctuations below T_c of ferromagnets. Received: 4 March 1998 / Accepted: 12 May 1998     

Van der Waals interaction between flux lines in high- superconductors

In anisotropic or layered superconductors thermal fluctuations as well as impurities induce a van der Waals (vdW) attraction between flux lines, as has recently been shown by Blatter and Geshkenbein in the thermal case [#!BlatterGeshkenbein!#] and by Mukherji and Nattermann in the disorder dominated case [#!NattermannMukherji!#]. This attraction together with the entropic or disorder induced repulsion has interesting consequences for the low field phase diagram. We present two derivations of the vdW attraction, one of which is based on an intuitive picture, the other one following from a systematic expansion of the free energy of two interacting flux lines. Both the thermal and the disorder dominated case are considered. In the thermal case in the absence of disorder, we use scaling arguments as well as a functional renormalization of the vortex-vortex interaction energy to calculate the effective Gibbs free energy on the scale of the mean flux line distance. We discuss the resulting low field phase diagram and make quantitative predictions for pure BiSCCO (Bi₂Sr₂CaCu₂O₈). In the case with impurities, the Gibbs free energy is calculated on the basis of scaling arguments, allowing for a semi-quantitative discussion of the low-field, low-temperature phase diagram in the presence of impurities. Received: 9 February 1998 / Accepted: 17 April 1998     

Brownian rotation of classical spins: dynamical equations for non-bilinear spin-environment couplings

Given two strings X and Y of N and M characters respectively, the Longest Common Subsequence (LCS) Problem asks for the longest sequence of (non-contiguous) matches between X and Y. Using extensive Monte-Carlo simulations for this problem, we find a finite size scaling law of the form for the average LCS length of two random strings of size N over S letters. We provide precise estimates of for .We consider also a related Bernoulli Matching model where the different entries of an array are occupied with a match independently with probability 1/S. On the basis of a cavity-like analysis we find that the length of a longest sequence of matches in that case behaves as where r=M/N and . This formula agrees very well with our numerical computations. It provides a very good approximation for the Random String model, the approximation getting more accurate as S increases. The question of the “universality class” of the LCS problem is also considered. Our results for the Bernoulli Matching model show very good agreement with the scaling predictions of [#!HwaLassig96_PRL!#] for Needleman-Wunsch sequence alignment. We find however that the variance of the LCS length has a scaling different from Var in the Random String model, suggesting that long-ranged correlations among the matches are relevant in this model. We finally study the “ground state” properties of this problem. We find that the number of solutions typically grows exponentially with N. In other words, this system does not satisfy “Nernst's principle”. This is also reflected at the level of the overlap between two LCSs chosen at random, which is found to be self averaging and to approach a definite value q _S <1 as . Received: 23 April 1998 / Revised: 30 July 1998 / Accepted: 14 August 1998     

Reaction kinetics in polymer melts

We study the reaction kinetics of end-functionalized polymer chains dispersed in an unreactive polymer melt. Starting from an infinite hierarchy of coupled equations for many-chain correlation functions, a closed equation is derived for the 2nd order rate constant k after postulating simple physical bounds. Our results generalize previous 2-chain treatments (valid in dilute reactants limit) by Doi [#!doi:inter2!#], de Gennes [#!gennes:polreactionsiandii!#], and Friedman and O'Shaughnessy [#!ben:interdil_all_aip!#], to arbitrary initial reactive group density n₀ and local chemical reactivity Q. Simple mean field (MF) kinetics apply at short times, .For high Q, a transition occurs to diffusion-controlled (DC) kinetics with (where x_t is rms monomer displacement in time t) leading to a density decay . If n₀ exceeds the chain overlap threshold, this behavior is followed by a regime where during which k has the same power law dependence in time, , but possibly different numerical coefficient. For unentangled melts this gives while for entangled cases one or more of the successive regimes ,t ^-3/8 and t ^-3/4 may be realized depending on the magnitudes of Q and n₀. Kinetics at times longer than the longest polymer relaxation time are always MF. If a DC regime has developed before then the long time rate constant is where R is the coil radius. We propose measuring the above kinetics in a model experiment where radical end groups are generated by photolysis. Received: 2 June 1998 / Revised: 9 July 1998 / Accepted: 10 July 1998     

Phases hexagonales colonnaires thermotrope et lyotrope : défauts observés par cryofracture et caractère anomal du thermotrope

Electron microscopy observations of replicas of freeze-fractured samples of two columnar hexagonal phases of different nature (a lyotropic one, the inverse AOT in water; a thermotropic one, ) yield very different results: most defects at microscopic scales are screw dislocations in the lyotropic phase, longitudinal edge dislocations in the thermotropic phase. A possible way to interpret these differences is as follows: in the lyotropic the Lamé coefficients and μ and the bend modulus K₃ would not display any anomaly compared to expected values; in the thermotropic the shear modulus μ would be ten times smaller than the compressibility modulus , while K₃ would still be comparable to (but larger than) the bend modulus of a small molecules liquid crystal. We present an elementary theoretical model of the latter case which could explain the anomalous measurements of K₃ and of the longitudinal compressibility (Ref. [#!ref10!#]) without contradicting more recent measurements of (Refs. [#!ref17!#,#!ref22!#]). Essentially, the hexagonal phase would be a phase with defects (longitudinal dislocations) akin to an hexatic phase but with some differences. Re?u : 26 mai 1997 / Révisé : 20 Janvier 1998 / Accepté : 27 avril 1998     

Anomalous viscosity exponent in a discretized model for a chain diffusion in one dimension

We present a one-dimensional Monte Carlo simulation for the diffusion motion of a chain of N beads. We found that the scaling exponent for the viscosity can be smaller or greater than 3. This anomalous behavior cannot be attributed to the diffusivity scaling or the length fluctuations but is due to the chain dynamics details during diffusion in which the end beads play the key role. The viscosity exponent 3 and its expected relation with the diffusivity exponent are recovered in the asymptotic regime (N ↦∞). Received 24 September 2001 and Received in final form 28 January 2002     

Ultrasound and light scattering from a suspension of reversible fractal clusters in shear flow

Shear break-up of reversible fractal clusters is investigated by ultrasound and multiple light scattering in the low shear regime. We consider a dense suspension of Rayleigh scatterers (particles or clusters) with acoustic properties close to those of the surrounding liquid so that the attenuation of the ultrasonic coherent field is weak and multiple scattering is negligible. The concept of variance in local particle volume fraction is used to derive an original expression of the ultrasound scattering cross-section per unit volume for Rayleigh fractal clusters. On the basis of a scaling law for the shear break-up of aggregates, then we derive the shear stress dependence of the ultrasound scattered intensity from a suspension of reversible fractal clusters. In a second part, we present rheo-acoustical experiments to study the shear break-up of hardened red cell aggregates in plane-plane flow geometry and we examine both the self consistent field approximation and the scaling laws used in microrheological models. We further compare the ability of acoustical backscattering and optical reflectometry techniques to estimate the critical disaggregation shear stress and the particle surface adhesive energy. Finally, the microrheological model from Snabre and Mills [#!ref5!#] based on a fractal approach is shown to describe the non Newtonian behavior of a dense distribution of hardened red cell aggregates. Received 12 November 1998 and Received in final form 17 May 1999     

Study of transients in the propagation of nonlinear waves in some reaction diffusion systems

We study the transient dynamics of single species reaction diffusion systems whose reaction terms f(u) vary nonlinearly near u ≈ 0, specifically as f(u) ≈ u² and f(u) ≈ u³. We consider three cases, calculate their traveling wave fronts and speeds analytically and solve the equations numerically with different initial conditions to study the approach to the asymptotic front shape and speed. Observed time evolution is found to be quite sensitive to initial conditions and to display in some cases nonmonotonic behavior, ascribable to the disparity in time scales between the evolution of the front interior and the front tail.     

Steps and facets at the surface of soft crystals

We consider the shape of crystals which are soft in the sense that their elastic modulus μ is small compared to their surface tension γ, more precisely μa < γ where a is the lattice spacing. We show that their surface steps penetrate inside the crystal as edge dislocations. As a consequence, these steps are broad with a small energy which we calculate. We also calculate the elastic interaction between steps a distance d apart, which is a 1/d ² repulsion. We finally calculate the roughening temperatures of successive facets in order to compare with the remarkable shapes of lyotropic crystals recently observed by Pieranski et al. [#!Pieranski!#,#!EPJ!#]. Good agreement is found. Received 25 June 2001     

Metastability of a circular o-ring due to intrinsic curvature

An o-ring takes spontaneously the shape of a chair when strong enough torsion is applied in its tangent plane. This state is metastable, since work has to be done on the o-ring to return to the circular shape. We show that this metastable state exists in a Hamiltonian where curvature and torsion are coupled via an intrinsic curvature term. If the o-ring is constrained to be planar (2d case), this metastable state displays a kink-anti-kink pair. This state is metastable if the ratio is less than , where C and A are the torsion and the bending elastic constants [#!landau!#]. In three dimensions, our variational approach shows that . This model can be generalized to the case where the bend is induced by a concentration field which follows the variations of the curvature. Received: 27 August 1997 / Revised: 23 October 1997 / Accepted: 12 November 1997     

Viscosity at small scales in polymer melts

Flows around small colloidal particles of diameter b, or in thin films, capillaries, etc., cannot always be described in terms of the macroscopic polymer viscosity. We discuss these features for entangled polymer melts, where two distinct regimes can be found: (a) the thin regime where b is smaller than the coil radius R₀, but larger than the diameter of the Edwards tube; (b) the ultrathin regime, where . We consider (i) non adsorbing particles, where slippage may occur between the melt and the solid surface; (ii) “hairy” particles, which carry some bound polymer chains. We obtain scaling predictions for mobilities of spheres, of needles, and of clusters of particles. We also discuss translational and rotational diffusion of needles. Received 19 April 1999     

Noisy covariance matrices and portfolio optimization

According to recent findings [#!bouchaud!#,#!stanley!#], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [#!bouchaud!#], e.g., it is reported that about 94% of the spectrum of these matrices can be fitted by that of a random matrix drawn from an appropriately chosen ensemble. In view of the fundamental role of covariance matrices in the theory of portfolio optimization as well as in industry-wide risk management practices, we analyze the possible implications of this effect. Simulation experiments with matrices having a structure such as described in [#!bouchaud!#,#!stanley!#] lead us to the conclusion that in the context of the classical portfolio problem (minimizing the portfolio variance under linear constraints) noise has relatively little effect. To leading order the solutions are determined by the stable, large eigenvalues, and the displacement of the solution (measured in variance) due to noise is rather small: depending on the size of the portfolio and on the length of the time series, it is of the order of 5 to 15%. The picture is completely different, however, if we attempt to minimize the variance under non-linear constraints, like those that arise e.g. in the problem of margin accounts or in international capital adequacy regulation. In these problems the presence of noise leads to a serious instability and a high degree of degeneracy of the solutions. Received 31 December 2001     

Non-linear evolution of step meander during growth of a vicinal surface with no desorption

Step meandering due to a deterministic morphological instability on vicinal surfaces during growth is studied. We investigate nonlinear dynamics of a step model with asymmetric step kinetics, terrace and line diffusion, by means of a multiscale analysis. We give the detailed derivation of the highly nonlinear evolution equation on which a brief account has been given [6]. Decomposing the model into driving and relaxational contributions, we give a profound explanation to the origin of the unusual divergent scaling of step meander (where F is the incoming atom flux). A careful numerical analysis indicates that a cellular structure arises where plateaus form, as opposed to spike-like structures reported erroneously in reference [6]. As a robust feature, the amplitude of these cells scales as t ^1/2, regardless of the strength of the Ehrlich-Schwoebel effect, or the presence of line diffusion. A simple ansatz allows to describe analytically the asymptotic regime quantitatively. We show also how sub-dominant terms from multiscale analysis account for the loss of up-down symmetry of the cellular structure. Received 4 May 2000 and Received in final form 8 September 2000     

Structure and rheology of concentrated wormlike micelles [4]at the shear-induced isotropic-to-nematic transition

We have investigated the simple shear flow behavior of wormlike micelles using small-angle neutron scattering and mechanical measurements. Ternary surfactant solutions made of cetylpyridinium chloride, hexanol and brine (0.2 M NaCl) and hereafter abbreviated as CPCl-Hex were studied in the concentrated regime, . In a preliminary report (Berret et al. [#!ref16!#]), the discontinuity of slope observed in the shear stress versus shear rate curve was interpreted in terms of first-order phase transition between an isotropic state and a shear-induced nematic state ( transition). At the transition rate, , the solution exhibits a macroscopic phase separation into viscous and fluid layers (inhomogeneous shear flow). Above a second characteristic shear rate, the flow becomes homogeneous again, the sheared solution being nematic only. The neutron patterns obtained in the two-state inhomogeneous region have been re-examined. Based on a consistent analysis of both orientational and translational degrees of freedom related to the wormlike micelles, we emphasize new features for the transition. In the present paper, the shear rate variations of the relative proportions of each phase in the two-state region, as well as the viscosity ratio between isotropic and nematic phases are derived. We demonstrate in addition that slightly above the transition rate, the shear induced nematic phase is already strongly oriented, with an order parameter P ₂ = 0.65. The orientational state is that of a nematic flow-oriented monodomain. Finally, from the locations of the neutron scattering maxima for each isotropic and nematic contributions, we evaluate the concentrations for each phase and and derived a dynamical phase diagram of CPCl-Hex, in terms of the stress versus and . According to the classification by Schmitt et al. [#!ref22!#], the transition observed in CPCl-Hex micellar solutions could result from a positive flow-concentration coupling, in agreement with the observed monotonically increasing shear stress in the two-phase region. Received: 16 February 1998 / Revised: 18 February 1998 / Accepted: 24 May 1998     

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided. Received 22 May 2000