Mean-field Regime Research Articles

Dynamic structure factor of homogeneous diblock copolymers solutions

The dynamics of homogeneous diblock copolymer solutions in the mean field regime sensitively depends on the composition polydispersity at constant chain length polydispersity. It was observed by photon correlation spectroscopy that the decay of the order parameter fluctuations in high molecular mass diblock copolymers solutions in a common solvent can change from a pure relaxational to a pure diffusive character. A rationalization of this behavior is presented.

Dynamic Structure Factor of Diblock Copolymer Solutions in the Disordered State. 2. Effect of Composition Polydispersity

The dynamic structure factor of entangled solutions of one symmetric and two asymmetric high molar mass styrene-b-ethylene-alt-propylene (SEP) and the parent styrene-b-isoprene (SI) diblock copolymers in the common solvent toluene was thoroughly studied by photon correlation spectroscopy. The incomplete hydrogenation transforms the I-block of SI into a random heteropolymer block. In turns out that this acts as enhanced composition polydispersity on both static and dynamic structure factor. In the mean-field regime of the disordered phase, the decay of the composition fluctuations in SI and SEP changes from a pure relaxational (q-independent rate) for SI to a pure diffusive (q2-dependent rate) character for SEP in a controlled way.

Response of non-equilibrium systems at criticality: ferromagnetic models in dimension two and above

We study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We show that these systems are aging in the long-time regime, i.e., their two-time autocorrelation and response functions and associated fluctuation-dissipation ratio are non-trivial scaling functions of both time variables. This is exemplified by the exact analysis of the spherical model in any dimension D>2, and by numerical simulations on the two-dimensional Ising model. We show in particular that, for $1\ll s$ (waiting time) $\ll t$ (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value $X_\infty$, which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain $X_\infty=1-2/D$ for 2<D<4, and $X_\infty=1/2$ for D>4 (mean-field regime). For the two-dimensional Ising model we measure $X_\infty\approx0.26\pm0.01$.

Janssen's law and stress fluctuations in confined dry granular materials

In order to clarify the mechanisms which determine the Janssen's law for the pressure distribution at the bottom of a silo we reconsider the so-called q-model describing an assembly of granular particles on a lattice, confined between vertical walls. We find that the expected macroscopic behavior with the correct scaling is obtained whenever a mechanism able to transfer the weight from the interior of the silo to the walls in an efficient way is present, i.e., in mean field regime. Deviations from the Janssen law's found in lattice models are due to the absence of this efficient mechanism. We investigate the scaling properties of a stick-slip model recently introduced, and find that relative fluctuations do not disappear for large systems and are of the order of average values. Finally we observe that an exponential local weight distribution at the bottom of the silo is independent of the model considered.

Critical dynamics at incommensurate phase transitions and NMR relaxation experiments

We study the critical dynamics of crystals which undergo a second-order phase transition from a high-temperature normal phase to a structurally incommensurate (IC) modulated phase. We give a comprehensive description of the critical dynamics of such systems, e.g. valid for crystals of the A_2BX_4 family. Using an extended renormalization scheme, we present a framework in which we analyze the phases above and below the critical temperature T_I. Above T_I, the crossover from the critical behavior to the mean-field regime is studied. Specifically, the resulting width of the critical region is investigated. In the IC modulated phase, we consider explicitly the coupling of the order parameter modes to one-loop order. Here the Goldstone anomalies and their effect on measurable quantities are investigated. We show their relation with the postulated phason gap. While the theory can be applied to a variety of experiments, we concentrate on quadrupole-perturbed nuclear magnetic resonance (NMR) experiments. We find excellent agreement with these dynamical measurements and provide answers for some questions that arose from recent results.

First-order wetting of rough substrates and quantum unbinding

Replica and functional renormalization group methods show that, with short range substrate forces or in strong fluctuation regimes, wetting of a self-affine rough wall in 2D turns first-order as soon as the wall roughness exponent exceeds the anisotropy index of bulk interface fluctuations. Different thresholds apply with long range forces in mean field regimes. For bond-disordered bulk, fixed point stability suggests similar results, which ultimately rely on basic properties of quantum bound states with asymptotically power-law repulsive potentials.

Multi-field effective Hamiltonian theory

Ideally one would like to analyse the properties of inhomogeneous fluids/Ising-like magnets (e.g., wetting of a fluid phase at a wall or confinement in thin film geometries) using a microscopic Hamiltonian H[ m], with m( r ) the local order-parameter (number density/magnetization). For many problems, however, this is too difficult and traditionally one has to introduce effective interfacial models based on a collective coordinate l( y ) measuring the position of the fluid interface. We review progress made in unifying these approaches using multi-field effective Hamiltonian theory which is a powerful new investigative tool. We emphasize: (i) a systematic method for recovering order-parameter correlations G( r 1, r 2) from collective coordinate theory, (ii) the role of coupled fluctuations at three dimensional wetting transitions leading to (a) an observable increment to the value of the wetting parameter at complete wetting and (b) an inflation of the mean field regime for local surface response functions at critical wetting, (iii) the derivation of new identities relating moments of G at different positions in the fluid, and (iv) the development of a linear response theory of fluid adsorption at a non-planar wall which predicts roughness-induced first-order wetting transitions. The relevance of these predictions for long-standing controversies surrounding Ising model simulation studies is discussed.

From individual to collective pinning: Effect of long-range elastic interactions

We study the effect of long-range elastic interactions in the dynamical behavior of an elastic chain driven quasi-statically in a quenched random pinning potential and in the strong pinning limit. This is a generic situation occuring in solid friction, crack propagation, wetting front motion, ... Tuning the exponent of the algebraic decay of the elastic interaction with the distance is shown to give rise to three regimes: a Mean-Field (MF) regime, a Laplacian (L) regime and an intermediate regime where the critical exponents interpolate continuously between the MF and L limit cases. The effect of the driving mode on the avalanche statistics is also analyzed.

Constrained free-energy functional of deformed polymer systems

The constrained free-energy functional for deformed polymer systems is derived in a regime of small fluctuations. The free energy contains an elastic contribution due to chain deformation as well as the usual Ginzburg–Landau form. Up to terms quadratic in fluctuations, the linear elastic modulus tensor is obtained. We find that the shear modulus scales with the concentration in the same way as that suggested by scaling arguments in the mean-field regime. The elastic Poisson ratio is 1/2 for the dilute system and decreases as the volume fraction increases. In the melt limit, where the volume fraction is close to unity, the Poisson ratio is down to 0.2.

Gaussian, three-dimensional-XY,and lowest-Landau-level scalings in the low-field fluctuation magnetoconductivity ofBi2Sr2CaCu2O8

Systematic measurements of low-field fluctuation magnetoconductivity in a single crystal of ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}$ are reported. Gaussian, critical, and lowest-Landau-level scalings are observed. In the Gaussian regimes, large intervals corresponding to low-dimensional fluctuations are evidenced. Far above ${T}_{c},$ effects of disorder produces a fluctuation spectrum characterized by a fractal topology. Decreasing the temperature, at first a homogeneous two-dimensional behavior is observed. Then, near ${T}_{c}$ a crossover occurs to a narrow three-dimensional (3D) mean-field regime. Still closer to ${T}_{c},$ a scaling consistent with the predictions of the full dynamic 3D-$\mathrm{XY}$ universality class is clearly evidenced. This genuine critical regime is destroyed upon the application of magnetic fields above a few mT. For fields above a certain limit and in large temperature intervals, fluctuation magnetoconductivity scales as predicted by the lowest-Landau-level approximation of the Ginzburg-Landau theory.

Crossover from mean field to three-dimensional Ising critical behavior in a three-component microemulsion system.

Critical density fluctuations of water droplets in an oil-rich three-component microemulsion system have been studied by small-angle neutron scattering as a function of temperature near and far from the boundary of phase decomposition. The observed data in the one-phase region are well described in terms of the asymptotic crossover expression calculated by Belyakov et al. The data are found exclusively in the crossover region between the universality class of three-dimensional Ising and mean field regimes. The Ginzburg number is found to be between one and two orders of magnitude less than that for low molecular weight liquids. \textcopyright{} 1996 The American Physical Society.

Dynamics of dislocations in the Twist-Grain-Boundary-A phase of liquid crystals

Abstract The superconductor-smectic A analogy by de Gennes has been successfully applied by Renn and Lubensky to predict the existence and mean field properties of the Twist-Grain-Boundary-A phase, which was independently discovered experimentally by Goodby and coworkers. We explore the possibility of extending this powerful analogy to aspects of the Twist-Grain-Boundary-A phase which require treatments going beyond the static mean field regime. Careful analysis reveals (1)the motion of a grain boundary as a liquid crystal analogue of the current-induced flux flow of the Abrikosov vortices, which can be realized either by applying the electric field in the helical axis direction, or by imposing a temperature gradient, and (2) the dynamical renormalization of effective Frank elastic constants resulting from the motion of the grain boundary. Implications of our findings for the vulnerability of the TGBA phase to thermal fluctuation is briefly discussed.

Finite-size corrections to the correlation function of the spherical model at d >= 4.

Finite-size effects in the correlation function G(R,T;L)\ensuremath{\equiv}〈\ensuremath{\Phi}(0)\ensuremath{\cdot}\ensuremath{\Phi}(R)〉 of a spherical-model ferromagnet, confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\ensuremath{'}}$ (d\ensuremath{\ge}4, ${\mathit{d}}^{\ensuremath{'}}$\ensuremath{\le}2) and subjected to twisted boundary conditions, are analyzed. Focusing attention on the region of first-order phase transition (T${\mathit{T}}_{\mathit{c}}$), we examine the influence of the twist parameter \ensuremath{\tau} on the function G(R,T;L) in different regimes of the distance parameter \ensuremath{\varepsilon}(=R/L). Two complementary methodologies are employed: (i) dimensional regularization involving the limit d\ensuremath{\rightarrow}4 as approached from the mean-field regime at short separations \ensuremath{\varepsilon}\ensuremath{\ll}1, and (ii) \ensuremath{\zeta}-function regularization at comparably larger separations \ensuremath{\varepsilon}=O(1) using generators from more simplified classes of sums so as to properly handle the singular features of the correlation function at d=4. Results following from the two methodologies are found to be completely consistent with one another.

Measurement and interpretation of nonuniversal critical exponents in disordered conductor-insulator composites.

The dc resistivity of carbon-black--polymer composites as a function of carbon-black concentration has been measured and fit to a power law, yielding a conductivity critical exponent t=2.9\ifmmode\pm\else\textpm\fi{}0.1. This value is in excellent agreement with the mean-field value t=3. The sample-to-sample variations in resistivity at each concentration also show power-law behavior. If interpreted as due to correlation length fluctuations, these data yield a correlation-length critical exponent \ensuremath{\nu}=0.51\ifmmode\pm\else\textpm\fi{}0.02. This value is in excellent agreement with the mean-field value \ensuremath{\nu}=1/2. These results suggest that nonuniversal (t\ensuremath{\ne}2) transport behavior in this and similar composites is caused by the composite being outside the universal scaling regime and in the mean-field regime.

Superconducting fluctuation study of the 110 K phase in polycrystalline Bi 1.6Pb 0.4Sr 2Ca 2Cu 3O γ high- Tc superconductor

Characterization of the Bi 1.6Pb 0.4Sr 2Ca 2Cu 3O γ composition of high- T c superconductor, using XRD, SEM/EDAX, EPR, χ AC and ϱ( T) techniques, has confirmed that the sample is of single phase. The real and imaginary parts along with their higher-order harmonics of magnetic susceptibility ( χ= χ′+i χ″) have been investigated. The temperature dependence of the imaginary part exhibits a peak whereas the real part undergoes a sharp diamagnetic transition and the second harmonic a cusp-like peak, all at the same T c=108 K. High-resolution electrical resistivity ϱ( T) data on the 110 K phase sample have been taken for investigating critically the superconducting fluctuations. Using the Aslamazov and Larkin (AL) and Lawrence and Doniach (LD) models of excess conductivity, several physical parameters of the models have been estimated. The salient features, displayed by the excess conductivity analysis, are the two-dimensional superconducting fluctuations in the mean field regime (−4.0<In ϵ<−2.0) with critical exponent ( λ)=−(0.96±0.02) and a short coherence length ( ζ 0) of the order of a few Å. λ varies from −0.74 to −1.22 with the variation in the choice of T c between (108±1) K. Two other critical regions, beyond the mean-field one, have been observed, giving exponents −0.70 for −5.2<In ϵ<−4.0 and −0.33 for −0.6<In ϵ<−5.2. The plot of ϱ( T) has reflected exclusively one single and sharp transition for the 110 K phase. The T c and the width ( ΔT c), estimated from d ϱT( T) plots, are =108 and 3 K, respectively.

Nucleation phenomena in polymeric systems

Materials formed from long flexible macromolecules differ from their small-molecule analogs, because corresponding collective length scales are distinctly larger and many dynamical phenomena are very much slower; in addition, the variation of chain length N yields a control parameter that leaves intermolecular forces invariant, but allows a stringent test of theories. These concepts are exemplified in a discussion of nucleation barriers for symmetrical polymer ( A, B)-mixtures (chain lengths N A = N B = N) near the critical temperature T c , and for symmetrical block copolymers near the (fluctuation-induced) first order transition between the disordered melt and the lamellar mesophase. While in the latter case for N → ∞ the transition becomes second-order and the order of magnitude of the nucleation barrier vanishes as N − 1 3 , for the polymer mixtures it increases as N 1 2 in the mean-field critical regime. Experiments and simulations, however, both show that very long chains are needed to fully reach this mean-field critical regime. For asymmetrical block copolymers {f= N A (N A+ N) ≠ 1 2 } the nucleation barrier scales as N 1 2 |f− 1 2 | 5 .

SPINODAL NUCLEATION

The aim of this paper is to present the theoretical foundations of spinodal nucleation, by reviewing key theoretical and numerical work. The basic ideas of classical nucleation theory are first presented: the classical droplet model, and the Becker-Döring theory, as these concepts are important to the development of the field theoretical formulation of nucleation. The field theoretical framework for classical nucleation is exposed in some detail, followed by the presentation of a similar framework, extended to nucleation in the proximity of a spinodal (non-classical nucleation), in the presence of long-range Ising interactions. The non-classical nucleating droplet is found to be diffuse, hence to strongly depart from the classical prediction of a compact object with a well-defined surface. The fact that the non-classical nucleating droplet is identified with a ramified object prompts the development of an appropriate cluster description. The basic principles of percolation theory are outlined, and some lattice percolation models introduced. The Kastaleyn-Fortuin mapping, which establishes a connection between a particular percolation model and a limit of the Potts model, is briefly described. This mapping is crucial to the development of a second mapping (Coniglio-Klein) of the Ising spinodal point into a percolation model, where the long-range Ising interactions are translated into a long-range connectivity in the appropriate percolation model. The final result consists of the most powerful tool available to identify precisely the non-classical nucleating droplet in numerical simulations of nucleation in proximity of a spinodal. Numerical simulation results are presented, which support the field theoretical formulation of non-classical nucleation. As the numerical results seem to support the fact that the non-classical nucleating droplet is also a percolation cluster, its fractal structure is investigated by considering the mean-field regime of the percolation model, i.e. a percolation model with long-range connectivity. This leads to an apparent contradiction between the field theory and the mean-field percolation model predictions concerning the mass (or density) scaling of the nucleating droplet. This inconsistency is resolved by postulating that the mean-field percolation clusters cannot be non-classical nucleating droplets, and proposing that the non-classical nucleating droplet is in fact the result of a coalescence of many such clusters. Finally, the calculation of the static prefactor in the nucleation rate by assuming a Becker-Döring dynamics for the coalescence mechanism is outlined. The result is found to be consistent with the predictions of the field theory for the static prefactor. Numerical results are also presented in support of the hypothesized coalescence mechanism.

Irreversible multilayer adsorption

AbstractRandom sequential adsorption (RSA) models have been studied [1] due to their relevance to deposition processes on surfaces. The depositing particles are represented by hard‐core extended objects; they are not allowed to overlap. Numerical Monte Carlo studies and analytical considerations are reported for 1D and 2D models of multilayer adsorption processes. Deposition without screening is investigated; in certain models the density may actually increase away from the substrate. Analytical studies of the late stage coverage behavior show the crossover from exponential time dependence for the lattice case of the power law behavior in the continuum deposition. 2D lattice and continuum simulations rule out some “exact” conjectures for the jamming coverage. For the deposition of dimers on a 1D lattice with diffusional relaxation we find that the limiting coverage (100%) is approached according to the ∼1/√t power‐law preceded, for fast diffusion, by the mean‐field crossover regime with the intermediate ∼1/t behavior. In case of k‐mer deposition (k&gt;3) with diffusion the void fraction decreases according to the power‐law t−1/2. In the case of RSA of lattice hard squares in 2D with diffusional relaxation the approach to the full coverage is ∼t−1/2. In case of RSA‐deposition with diffusion of two by two square objects on a 2D square lattice the coverage also approaches 1 according to the power law t−1/2, while on a finite periodic lattice the final state is a frozen random regular grid of domain walls connecting single site defects.

Line tension at wetting: finite-size effects and scaling functions

We study the effect of finite system size on the tension 7 of a three-phase contact line. This is especially important in systems with forces that are sufficiently long-ranged for 7 to diverge in the thermodynamic limit. Near wetting phase transitions we propose an anisotropic finite-size scaling hypothesis, which takes into account the effect of thermal capillary-wave fluc- tuations. The singular behaviour of T near a first-order wetting transition is analyzed. Scaling functions are calculated explicitly in the mean-field regime, which includes the physically most common systems with van der Wads forces in dimension d = 3.

Superconducting properties of YBa 2(Cu 1− xM x) 3O 7− δ(M=Ti, V)

High-quality polycrystalline YBa 2(Cu 1− x M x ) 3O 7− δ (M=Ti, V) compounds have been synthesized by solid state reaction. The structure (orthorhombic), the T c, and the Δ T c remain almost unaffected up to x=0.1. This is in contrast to several doping studies where generally a decrease, increase or almost no change in T c associated with a varying degree of broadening of Δ T c is reported. DTA, SEM, EDXA, resistivity, and thermopower analyses suggest (indirectly) that the solubility of Ti and V in Y-Ba-Cu-O is very limited and, in any case, is not more than 2.5 at.%. Any further addition results in Ti- or V-rich second phase formation. The fact that the T c is almost unaffected may be of technological interest in view of the need for a non-poisonous, high melting point substrate desirable for high-temperature processing of Y-Ba-Cu-O. The Anderson-Zou fit to the data is found to work as well in polycrystals as in single crystals. Paraconductivity studies show a 2D to 3D crossover while passing from high-temperature to mean-field regime. Well above the T c, the thermopower was found to follow a relation of the type S= A/ T+ BT. The band-filling and the phonon drag mechanisms with the involvement of two-carrier species seem operative in these superconductors.