Mean-field Estimates Research Articles

Kinematic Diffusion in Quasi-Static Granular Deformation

A statistical-mechanical model is proposed for the quasi-static deformation of granular assemblies, in which particle motion is decomposed into a mean-field contribution, given by the macroscopically imposed deformation, together with fluctuations representing stochastic multiparticle mechanics. This leads to the notion of kinematic diffusion and the postulate of a convection-diffusion (Fokker-Planck) equation for various configurational probability distributions. Based on statistics obtained from numerical simulation of a frictional-sphere assembly, self diffusivities and pair diffusivities are derived for various homogeneous deformations, including 'cubical-triaxial' strains as well as simple shear. Among the important findings are (i) diffusive motions are found generally to be small relative to convection, suggesting that the mean-field approximation should be quite accurate, and (ii) pair correlations are weak, implying that two-particle and higher-order cluster diffusivities follow from single-particle diffusivities. Based on the idea of negligible diffusion, a semi-theoretical model of granular plasticity with fabric evolution is proposed, as an extension of the exact mean-field model of Jenkins and Strack. It is concluded, however, that even weak diffusion effects might have important consequences for certain continuum properties, because of the influence on unstable equilibrium configurations. This is supported by comparison of various mean-field kinematic estimates of Reynolds dilatancy to a more accurate estimate obtained from the mechanics simulation for a dense random packing of spheres.

Critical Temperature of Ferromagnetic Transition in Three-Dimensional Double-Exchange Models

Ferromagnetic transition in three-dimensional double-exchange models is studied by the Monte Carlo method. Critical temperature $T_{\rm c}$ is precisely determined by finite-size scaling analysis. Strong spin fluctuations in this itinerant system significantly reduce $T_{\rm c}$ from mean-field estimates. By choosing appropriate parameters, obtained values of $T_{\rm c}$ quantitatively agree with experiments for the ferromagnetic metal regime of (La,Sr)MnO$_{3}$, which is a typical perovskite manganite showing colossal magnetoresistance. This indicates that the double-exchange mechanism alone is sufficient to explain $T_{\rm c}$ in this material. Critical exponents are also discussed.

Mean field estimates of the response of fiber composites with nonlinear interface

This paper presents a theory of the effective response of composites containing general, nonlinearly separating inclusion–matrix interfaces. The direct method of composite materials theory is utilized to pass from local nonlinear behavior of a solitary inclusion problem to nonlinear aggregate response. Interaction effects at finite volume concentration are captured in the representative solitary problem by employing the Mori–Tanaka mean field estimate. The resulting model falls within the conceptual framework of continuum damage mechanics in that nonlinear effective response depends on internal variables that are governed by local evolution equations. The damage variables turn out to be the expansion coefficients arising in an eigenfunction representation of the displacement jump at a representative inclusion–matrix interface. Interfaces are generally modeled according to a nonlinear force-separation law that allows for both normal and shear decohesion (X-.P. Xu, A. Needleman, Modell. Simul. Mater. Sci. Eng. 1 (1993) 111). Detailed calculations of effective stress–strain response are carried out for the case of transverse shear and plane dilatation of unidirectional fiber composites at various values of concentration and interface constitutive constants. The effects of the various parameters on bifurcation of equilibrium separation in the solitary inclusion problem and on overall composite stability are demonstrated as well.

Critical Temperature of Ferromagnetic Transition in Three-Dimensional Double-Exchange Models

Ferromagnetic transition in three-dimensional double-exchange models is studied by the Monte Carlo method. Critical temperature T c is precisely determined by finite-size scaling analysis. Strong spin fluctuations in this itinerant system significantly reduce T c from mean-field estimates. By choosing appropriate parameters, obtained values of T c quantitatively agree with experiments for the ferromagnetic metal regime of (La,Sr)MnO 3 , which is a typical perovskite manganite showing colossal magnetoresistance. This indicates that the double-exchange mechanism alone is sufficient to explain T c in this material. Critical exponents are also discussed.

Magnetization and susceptibility of coupled ferromagnetic trilayers calculated with a Green’s function type theory

The effect of the interlayer exchange interaction Jinter in coupled Co/Cu/Ni trilayers is studied theoretically, as motivated by recent experiments. In these systems, Jinter induces a considerable Ni magnetization for temperatures above the Curie temperature TC,Ni of the Ni film. We will show that this strong effect can only be understood if magnetic fluctuations in these 2D-like systems are taken into account. A mean field estimate, neglecting these fluctuations, yields a ten times smaller increase of the Ni magnetization. The induced magnetization increases for both a ferromagnetic (Jinter>0) and an antiferromagnetic (Jinter<0) interlayer coupling. The Ni susceptibility, exhibiting a resonance-like peak at T≳TC,Ni, looks different for these two cases. The calculations are performed with the help of a Heisenberg Hamiltonian and a Green’s function approach.

Enhanced induced magnetization in coupled magnetic trilayers in the presence of spin fluctuations

Motivated by recent experiments, the effect of the interlayer exchange interaction $J_{inter}$ on the magnetic properties of coupled Co/Cu/Ni trilayers is studied theoretically. Here the Ni film has a lower Curie temperature $T_{C,\rm Ni}$ than the Co film in case of decoupled layers. We show that by taking into account magnetic fluctuations the interlayer coupling induces a strong magnetization for $T\gtsim T_{C,\rm Ni}$ in the Ni film. For an increasing $J_{inter}$ the resonance-like peak of the longitudinal Ni susceptibility is shifted to larger temperatures, whereas its maximum value decreases strongly. A decreasing Ni film thickness enhances the induced Ni magnetization for $T\gtsim T_{C,\rm Ni}$. The measurements cannot be explained properly by a mean field estimate, which yields a ten times smaller effect. Thus, the observed magnetic properties indicate the strong effect of 2D magnetic fluctuations in these layered magnetic systems. The calculations are performed with the help of a Heisenberg Hamiltonian and a Green's function approach.

Miscibility behavior and single chain properties in polymer blends: a bond fluctuation model study

Computer simulation studies on the miscibility behavior and single chain properties in binary polymer blends are reviewed. We consider blends of various architectures in order to identify important architectural parameters on a coarse grained level and study their qualitative consequences for the miscibility behavior. The phase diagram, the relation between the exchange chemical potential and the composition, and the intermolecular pair correlation functions for symmetric blends of linear chains, blends of cyclic polymers, blends with an asymmetry in cohesive energies, blends with different chain lengths, blends with distinct monomer shapes, and blends with a stiffness disparity between the components are discussed. For strictly symmetric blends the Flory-Huggins theory becomes quantitatively correct in the long chain length limit, when the χ parameter is identified via the intermolecular pair correlation function. For small chain lengths composition fluctuations are important. They manifest themselves in 3D Ising behavior at the critical point and an upward parabolic curvature of the χ parameter from small-angle neutron scattering close to the critical point. The ratio between the mean field estimate and the true critical temperature decreases like √χ/(ρb3) for long chain lengths. The chain conformations in the minority phase of a symmetric blend shrink as to reduce the number of energeticaly unfavorable interactions. Scaling arguments, detailed self-consistent field calculations and Monte Carlo simulations of chains with up to 512 effective segments agree that the conformational changes decrease around the critical point like 1/√N. Other mechanisms for a composition dependence of the single chain conformations in asymmetric blends are discussed. If the constituents of the blends have non-additive monomer shapes, one has a large positive chain-length-independent entropic contribution to the χ parameter. In this case the blend phase separates upon heating at a lower critical solution temperature. Upon increasing the chain length the critical temperature approaches a finite value from above. For blends with a stiffness disparity an entropic contribution of the χ parameter of the order 10–3 is measured with high accuracy. Also the enthalpic contribution increases, because a back folding of the stiffer component is suppressed and the stiffer chains possess more intermolecular contacts. Two aspects of the single chain dynamics in blends are discussed: (a) The dynamics of short non-entangled chains in a binary blend are studied via dynamic Monte Carlo simulations. There is hardly any coupling between the chain dynamics and the thermodynamic state of the mixture. Above the critical temperatures both the translational diffusion and the relaxation of the chain conformations are independent of the temperature. (b) Irreversible reactions of a small fraction of reactive polymers at a strongly segregated interface in a symmetric binary polymer blend are investigated. End-functionalized homopolymers of different species react at the interface instantaneously and irreversibly to form diblock copolymers. The initial reaction rate for small reactant concentrations is time dependent and larger than expected from theory. At later times there is a depletion of the reactive chains at the interface and the reaction is determined by the flux of the chains to the interface. Pertinent off-lattice simulations and analytical theories are briefly discussed.

Avalanches in breakdown and fracture processes.

We investigate the breakdown of disordered networks under the action of an increasing external-mechanical or electrical-force. We perform a mean-field analysis and estimate scaling exponents for the approach to the instability. By simulating two-dimensional models of electric breakdown and fracture we observe that the breakdown is preceded by avalanche events. The avalanches can be described by scaling laws, and the estimated values of the exponents are consistent with those found in mean-field theory. The breakdown point is characterized by a discontinuity in the macroscopic properties of the material, such as conductivity or elasticity, indicative of a first-order transition. The scaling laws suggest an analogy with the behavior expected in spinodal nucleation.

Irreversibility lines in the H- T phase diagram of re-entrant amorphous ferromagnets

The longitudinal components of the `zero-field-cooled' and `field-cooled' magnetization of amorphous ( TM= Co, Ni) and re-entrant ferromagnetic alloys have been measured in the static mode and at different thermal cycling rates varying from to in constant magnetic fields (H) ranging between 1.5 Oe and 15 kOe. The difference, , is taken to be the direct measure of irreversibility in magnetization. The onset of weak irreversibility and a crossover from weak to strong irreversibility are observed at the temperatures and , respectively, for fields ; depends on x and y. and follow the relations and , where ( stands for or ), which have the same form as those predicted by the modified versions of the Gabay-Toulouse (GT) and de Almeida-Thouless (AT) mean-field (MF) theories that include non-vanishing spontaneous magnetization, but the observed values of the coefficients and are several orders of magnitude larger than the MF estimates. is relatively insensitive while is extremely sensitive to thermal cycling rates (TCR) if they exceed a threshold value. The physical implications of extremely large magnitudes of and , and of the observed TCR-induced shifts in the GT and AT irreversibility lines in the H-T plane, have been brought out clearly while discussing these results in terms of the existing theoretical models.

Microphase Separation and Rheological Properties of Polyurethane Melts. 1. Effect of Block Length

A series of polyesterurethanes with differing block length and constant composition have been synthesized for rheological studies. Hard segments based on isophorone diisocyanate and 1,4-butanediol and soft segments based on polycaprolactone to ensure high thermal stability and to prevent high melting point crystallinity enabled long-duration rheological characterization at high temperatures. DSC and SAXS revealed that, at any fixed temperature above the polyester melting point, the degree of microphase separation increased with block length, with the shortest block lengths being almost single-phase. Temperature-resolved SAXS experiments demonstrated gradual microphase mixing of the microphase-separated materials as the temperature increased. In addition, the SAXS data for one material were shown to obey the predictions of the mean field theory, allowing a mean field estimate of the spinodal temperature to be calculated. Frequency sweep dynamic mechanical experiments show viscoelastic behavior characterist...

The simulation of switching in polycrystalline ferroelectric ceramics

A polarization switching model for polycrystalline ferroelectric ceramics has been developed. It is assumed that a single ferroelectric crystallite in a ceramic, which is subjected to an electric field and/or a stress, undergoes a complete polarization change and a corresponding strain change if the resulting reduction in potential energy exceeds a critical value per unit volume of switching material. The crystallite’s switch causes a change in the interaction of its field and stress with the surrounding crystallites, which is modeled by the Eshelby inclusion method to provide a mean field estimate of the effect. Thus the model accounts for the effects of the mean electric and stress fields arising from the constraints presented by surrounding crystallites as well as the externally applied mechanical and electrical loads. The switching response of the ceramic polycrystal is obtained by averaging over the behavior of a large number of randomly oriented crystallites. The model, along with the linear dielectric, elastic, and piezoelectric behavior of the material, is implemented in a computer simulation. A fit to experimental electric displacement versus electric field, strain versus electric field, and strain versus stress curves of a ceramic lead lanthanum zirconate titanate PLZT at room temperature is used to obtain material parameters. The model then successfully predicts the electric displacement and strain hysteresis loops for the PLZT under varying electric fields with a constant applied stress.

Effect of tree-level and mean-field improvement on the light-hadron spectrum in quenched QCD

We compute the light-hadron mass spectrum at $\ensuremath{\beta}=5.7$ using the $O(a)$-improved Sheikholeslami-Wohlert fermion action with two choices of the clover coefficient: the classical value $c=1$ and a mean-field or tadpole-improved estimate $c=1.57$. We compare our results with those of the GF11 Collaboration who use the Wilson fermion action $(c=0)$. We find that changing $c$ from zero to 1 and 1.57 leads to significant differences in the masses of the chirally extrapolated and strange pseudoscalar and vector mesons, the nucleon, the $\ensuremath{\Delta}$, and also in the Edinburgh plot. A number of other quantities, for example, ${m}_{V}^{2}\ensuremath{-}{m}_{P}^{2}, J, {\mathrm{am}}_{K}{/am}_{\ensuremath{\rho}},$ and ${\mathrm{am}}_{{K}^{*}}{/am}_{\ensuremath{\rho}}$ do not appear to change significantly. We also investigate the effect of changing the lattice volume from approximately (2 ${\mathrm{f}\mathrm{m})}^{3}$ to (2.6 ${\mathrm{f}\mathrm{m})}^{3}$. We find that the meson masses are consistent to within one standard deviation and baryon masses are consistent to within two standard deviations.

Origin of pseudospin symmetry.

A many-particle operator that affects a transformation to the pseudospin basis in heavy nuclei is identified. Both mean-field and many-particle estimates demonstrate that in the helicity-transformed representation the nucleons move in a finite-depth nonlocal potential with a reduced spin-orbit strength. Because of the close relation between the helicity and chirality operations, the results suggest that the pseudospin symmetry in heavy nuclei yields to the chiral symmetry of massless hadrons in the high energy region.

Asymptotic critical behavior of Ni.

The values \ensuremath{\beta}=0.395(10), \ensuremath{\gamma}=1.345(10), \ensuremath{\delta}=4.35(6) for the asymptotic critical exponents, \ensuremath{\mu}(0)${\mathit{h}}_{0}$/${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{C}}$=1.35(10), ${\mathit{DJ}}_{0}^{\mathrm{\ensuremath{\delta}}}$/${\mathit{h}}_{0}$=1.20(55), ${\mathit{a}}_{\mathit{M}}^{\mathrm{\ensuremath{-}}}$/${\mathit{a}}_{\mathrm{\ensuremath{\chi}}}^{+}$=-0.19(6) for the universal ratios and the ratio ${\mathit{J}}_{0}$/${\mathit{J}}_{\mathit{S}}$(0)=1.70(16), involving asymptotic and correction-to-scaling amplitudes, have been deduced from the bulk magnetic polarization data in the critical region near the ferromagnetic (FM)-paramagnetic (PM) phase transition of polycrystalline Ni samples of different shapes through an elaborate data analysis. These values, though close to those predicted by the renormalization-group calculations for a three-dimensional isotropic short-range Heisenberg ferromagnet, are shifted towards the mean-field estimates. Such a shift is taken to be evidence for a crossover to the fixed point corresponding to isotropic long-range exchange interactions. In accordance with the theoretical expectations, nonanalytic corrections (originating from the nonlinear irrevelant scaling fields) to the singular behavior at ${\mathit{T}}_{\mathit{C}}$ (Curie point) dominate over the analytic ones (arising on account of the nonlinear relevant scaling fields) in the critical region but the reverse is true for T\ensuremath{\gg}${\mathit{T}}_{\mathit{C}}$. Initial susceptibility follows the generalized Curie-Weiss law [Eq. (14) of the text with a${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{\ensuremath{\chi}}1}$=0] from ${\mathit{T}}_{\mathit{C}}$ to 1.4${\mathit{T}}_{\mathit{C}}$ and the Curie constant permits an accurate determination of the atomic moment in the PM state. Not all but only about 80% of the moments (spins) in Ni actually participate in the FM-PM phase transition.

Demagnetizing field in periodic models of granular film structures

Previous explicit results for the demagnetizing field and energy in in-plane periodic models of granular thin film structure are extended to comprise structure non-uniform along the film normal. Numerical results for oblique-column structures show good agreement with early mean-field estimates by Hara in the case of relatively large film thickness and low porosity.

Fluctuation effects on microphase separation in random copolymers

We study random copolymers consisting of two kinds of monomers with attraction between similar kinds. The mean-field analysis of this system indicates a continuous phase transition into a phase with periodic microdomain structure. It is shown that the inverse of the renormalized propagator has a minimum at non-zero wave numbers. Consequently, there is an anomalously large contribution of fluctuations that make the disordered phase locally stable. However, below a certain temperature, the ordered phase is shown to be locally stable and a weak first-order transition is possible, with qualitative differences from the weak crystallization theory developed by Brazovskii. These differences are due to the peculiar form of the fourth-order vertex of the effective Hamiltonian for the system. We calculated the temperature region for the validity of the mean-field estimates of the amplitude and the scale of microphase separation.

Monte Carlo study of interfacial properties in an amphiphilic system.

A three-component model representing a balanced, amphiphilic system at a three-phase coexistence of oil-, water-rich, and disordered phases is simulated, and interfacial tensions between all phases are obtained from histogram methods. We find that the disordered phase wets the oil-water interface, and that its tension is reduced due to the presence of the amphiphile by a factor of 93. The determination of the structure function shows that the system is between disorder and Lifshitz lines. These three factors are all consistent with the simulated amphiphile being weak. Fluctuations do not reduce the tension significantly from mean-field estimates, but do alter the location of the wetting transition. The latter is in accord with the expected behavior of a system in which the interfacial tension is small.

Structure formation and transport in dissipative drift-wave turbulence

Numerical flow simulations are used to study the effect of coherent structures on transport in the context of a two-field, two-dimensional model of dissipative drift-wave turbulence. The presence and nature of structures are found to depend on the adiabaticity parameter α=k∥2 VT2/2νeiωs which controls the degree to which the electrons respond to parallel electric fields. Transport estimates based on quasilinear and mixing-length models are compared with the simulations. In the regime with long-lived coherent structures, the turbulent particle transport predicted by a standard quasilinear or mean-field estimate is found to exceed that actually observed in the presence of coherent structures.

Effect of temperature and small-scale defects on the strength of solids

Using a statistical-thermodynamic formulation, we investigate the failure of ideal and almost-ideal solids at finite temperature. We propose that the onset of failure in a defect-free crystal is associated with the loss of a metastable minimum in the free energy at a critical value of the applied tensile force. Using a mean-field approximation, we estimate the free energy of the two-dimensional Lennard-Jones crystal under stress and derive the temperature dependence of its ideal strength and other properties. These results are compared to Monte Carlo simulations of this system, and the mean-field estimate of the ideal strength is shown to be an upper bound to the values observed via simulation. We also show that atomic-scale defects such as vacancies and substitutional impurities significantly reduce the crystal’s strength as a result of stress enhancement effects. While the overall strength of a defective crystal depends strongly on both temperature and the nature of the defects, the maximum local stress that the crystal can sustain without failure is essentially independent of these factors.

Spin waves and Heisenberg exchange constants for α-iron

Using a density functional theory for non-collinear spin arrangements, we model magnetization changes with well-defined wave vectors and calculate self-consistently the corresponding total-energy changes. From this we obtain the anisotropic spin wave dispersion relation for long and medium wavelengths, finding the spin wave stiffness constant to be in good agreement with experimental values. We also derive Heisenberg-exchange constants that give a reasonable mean-field estimate of the Curie temperature.