Systems Of Finite Thickness Research Articles

Minimal model for the onset of slip pulses in frictional rupture.

We present a minimal one-dimensional continuum model for the transition from cracklike to pulselike propagation of frictional rupture. In its nondimensional form, the model depends on only two free parameters: the nondimensional prestress and an elasticity ratio that accounts for the finite height of the system. The model predicts stable slip pulse solutions for slip boundary conditions, and unstable slip pulse solutions for stress boundary conditions. The results demonstrate that a mechanism based solely on elastic relaxation and redistribution of initial prestress can cause pulselike rupture, without any particular rate or slip dependences of dynamic friction. This means that pulselike propagation along frictional interfaces is likely a generic feature that can occur in systems of finite thickness over a wide range of friction constitutive laws.

Hopping transport in systems of finite thickness or length

Variable-range hopping transport along short one-dimensional wires and across the shortest dimension of thin three-dimensional films and narrow two-dimensional ribbons is studied theoretically. Geometric and transport characteristics of the hopping resistor network are shown to depend on temperature T and the dimensionality of the system. In two and three dimensions the usual Mott law applies at high T where the correlation length of the network is smaller than the sample thickness. As T decreases, the network breaks into sparse filamentary paths, while the Mott law changes to a different T-dependence, which is derived using the percolation theory methods. In one dimension, deviations from the Mott law are known to exist at all temperatures because of rare fluctuations. The evolution of such fluctuations from highly-resistive "breaks" at high T to highly-conducting "shorts" at low T is elucidated.

The fracture analysis of a graded coating/substrate system of finite thickness with arbitrary spatial variations of coating properties: Plane deformation

A new multi-layered model for functionally graded materials (FGMs) with continuously varying elastic properties is developed. The model divides the FGM into multiple layers. In each layer the material properties vary linearly and are continuous on the sub-interfaces. With this new multi-layered model, we solve the crack problems of an FGM coated strip under the in-plane deformation. The method employs the Fourier integral transform technique and singular integral equation theory. The stress intensity factors are calculated. Comparisons between the present model and other existing models show some advantages of the new model: (i) it involves no discontinuities of the material properties at the sub-interfaces; and (ii) it can be used to analyze the crack problems of FGMs with properties of arbitrary variations.

Simulation of quasi-two-dimensional dipolar systems

A brief survey is given of recent simulation work of quasi-two-dimensional (Q2D)systems, monolayers and systems of finite thickness, involving dipolarinteractions. Representative examples include colloidal particles at an interface,electrorheological fluids, magnetic thin films, nanocrystals deposited on asubstrate etc. Methodological aspects related to the long range of the dipolarinteraction and its directional nature (energy calculation, cluster moves) aresummarized. Taking as an example a bilayer of dipolar hard spheres, wedemonstrate that the energy and structural properties of a thin slab of dipolarparticles periodic in two dimensions and finite in the third can be accuratelycalculated by means of the usual three-dimensional Ewald sum. Some newsimulation results for the structural properties of Q2D dipolar hard spheres withadditional attraction, in an external field or in bilayers, are presented.

A linear stability analysis for miscible displacements

A linear stability analysis has been performed for a miscible displacement in a semi-infinite system of finite thickness and unbounded width. A more general description of dispersion has been adopted than those used by previous workers. It is shown that, when there is a step change in concentration and the mobility ratio is unfavorable, the displacement can be unstable at the injection boundary. But, if the concentration is changed sufficiently slowly with time at the entrance to the system, the displacement is stable to infinitesimal perturbations, no matter how unfavorable the mobility ratio. When the mobility ratio is favorable, the displacement is unconditionally stable.

First-order transition in systems of finite thickness.

Film systems, limited by two parallel disrupting surfaces and undergoing a first-order bulk transition, are shown to possess a number of specific features essentially governed by the ratio of the film thickness $L$ to the "characteristic" length ${L}_{0}$ of the transition. For disordering boundaries, it is found that (1) the transition temperature scales as ${L}^{\ensuremath{-}1}$ for $\frac{L}{{L}_{0}}\ensuremath{\gg}1$, (2) the continuous transition of the surface order parameter, predicted to occur in semi-infinite systems, transforms into a weak first-order transition, and (3) below a critical thickness, no transition occurs. Comparisons with experimental systems are discussed.

Destruction of a first-order transition by dimensional crossover

The q=3-state Potts model undergoes a first-order transition in three dimensions, and a continuous one in two dimensions. For a system of finite thickness L, the transition will be either first order (for L>Lx) or continuous (for L<Lx). For some particular thickness Lx, the tricritical three-state Potts behaviour is obtained. An experimental system is proposed for which an external magnetic field plays the role of temperature, and physical temperature in turn corresponds to variation of the effective thickness. The system may provide the first experimental values for three-state Potts tricritical exponents in two dimensions.

Critical indices for spin correlation functions in ferromagnetic films

Using the Ginzburg–Landau–Wilson (GLW) Hamiltonian, we obtain, with the mean-field approximation, explicit expressions for the spin–spin correlation function χ(z,z′) of a film of thickness L above the phase transition temperature Tc and the spontaneous magnetization [Formula: see text] below Tc. The boundaries are treated using a temperature-independent extrapolation length Λ. From our results, we verify explicitly that for finite L, the critical indices associated with the spin–spin correlation functions and the surface magnetization are identical with those for the analogous two-dimensional system, for both the ordinary (Λ &gt; 0) and surface (Λ &lt; 0) transitions. Our model results nicely exhibit the fact that as long as L is finite, when the temperature T approaches sufficiently close to Tc, there is a crossover from behaviour characteristic of a single surface to two-dimensional behaviour. Within the one-loop, Hartree self-consistent field approximation, we study the effects of mode–mode coupling on the surface layer susceptibility in films of varying thicknesses. The singular behaviour obtained in the mean-field approximation is found to be completely removed in systems of finite thickness, the susceptibilities only exhibiting a finite cusp at the transition.

Ground subsidence due to nonlinear flow through deformable porous media

An analytical solution is presented for the problem of time rate of ground subsidence and piezometric pressure distribution caused by a sudden increase in the effective stress due to an instantaneous lowering of the groundwater table in an aquifer-aquitard-aquiclude system of finite thickness, obeying a nonlinear flow law of the type v = Mi n . Unlike for Darcian flow the degree of subsidence and the nondimensional pore pressure distribution are found to be dependent on both the amount of groundwater fall and the thickness of the deforming medium.

Critical Phenomena in Systems of Finite Thickness. III. Specific Heat of an Ideal Boson Film

The specific heat at constant density of an ideal-Bose-fluid film of thickness $L=l{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, but infinite lateral extent, is calculated analytically to order ${l}^{\ensuremath{-}2}$. Both hard-wall and periodic boundary conditions are considered. Good agreement with the numerical calculations of Goble and Trainor for the total specific heat under hard-wall conditions is obtained down to $l\ensuremath{\simeq}10$. In the critical region, the large-$l$ behavior accords with the scaling theory of finite-size effects. The appropriate scaling functions and the surface specific heat are explicitly calculated.

Critical phenomena in systems of finite thickness I. The spherical model

The critical behavior of d-dimensional spherical models is investigated for hypercubical lattices which are infinite in d − 1 dimensions ( d ⩾ 3) but finite with n “layers” in the d-th dimension. Boundary conditions, ( a) periodic, ( b) antiperiodic, and ( c) free-surface, are applied in the finite dimension. The shift in the critical temperature, T c, d ( n), varies for n ⪢ 1 as ( a) n −( d−2) (all d); ( b) n −1 ( d = 3), n −2 ln n ( d = 4), n −2 (for d ⩾ 5); and ( c) n −1 ln n ( d = 3), n −1 (for d ⩾ 4). The critical temperature is depressed below the bulk value T c (∞), under conditions ( a) and ( b), but the presence of surfaces ( c) enhances T c ( n) above T c (∞) for n ⩾ 1. The bulk critical region is asymptotically “rounded” in all cases on a scale x = n 2 φ = O(1) where φ is proportional to the deviation in spherical field; this effect, unlike the critical point shift, corresponds to the bulk correlation length matching the thickness n. Finally, the behavior of the susceptibility per spin is studied for boundary conditions ( a) and ( c) and found to be in accord, on the whole, with the finitesize scaling theory developed by Fisher. The appropriate scaling functions are explicitly calculated.

Critical Phenomena in Systems of Finite Thickness. II. Ideal Bose Gas

The grand potential pressure of an ideal boson film which is infinite in d-1 dimensions (d &gt; 2) but of finite thickness L in the dth dimension is analysed in the limit L -+ CX), to order L -d. Across the film (i) periodic, (ii) antiperiodic, and (iii) hard wall boundary conditions are applied. The corresponding shifts in the critical temperature Te.iL) are found to vary, for large L, as (i) L -d+2 for all d ;;:. 3, (ii) L-1, L -2lnL, and L -2 for d = 3,4, and ;;:. 5 respectively, and (iii) L -1lnL and L -1 for d = 3 and ;;:. 4. With hard wall boundary conditions, the critical temperature of a sufficiently thick film exceeds that of the bulk, while the other boundary conditions depress Te. The bulk critical region is asymptotically "rounded" on a scale set by the bulk correlation length matching the thickness L in accord with the scaling theory of finite-size effects.

Decay of Order in Isotropic Systems of Restricted Dimensionality. II. Spin Systems

The ordering of one- and two-dimensional spin systems of finite thickness and cross section is considered in the presence and absence of a symmetry-breaking magnetic field. The exchange interactions are allowed to vary randomly or regularly throughout the lattice. It is shown rigorously by applying Bogoliubov's inequality to a subdomain of the system that, provided the (suitably averaged) exchange interactions do not fall off too slowly, no spontaneous ordering can occur. Explicit bounds on the spin-spin correlation function, summed over the sites in a subdomain, are obtained which indicate how the short-range order decays with distance. Detailed numerical plots for the order as a function of the subdomain size are presented for various realistic values of the temperature. Conditions under which these curves yield bounds on the spatial decay of the spin-spin correlation function are also discussed.