The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. I. Droplet Density

This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion, also known as the Ohta–Kawasaki model, in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. In this paper we show that an appropriate setting for Γ-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta–Kawasaki energy functional Γ-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain an asymptotic characterization of the energy minimizers in the diffuse interface model.     

Vortices for a Rotating Toroidal Bose–Einstein Condensate

We construct local minimizers of the Gross–Pitaevskii energy, introduced to model Bose–Einstein condensates (BEC) in the Thomas–Fermi regime which are subject to a uniform rotation. Our sample domain is taken to be a solid torus of revolution in with starshaped cross-section. We show that for angular speeds ω_ε = O(|ln ε|) there exist local minimizers of the energy which exhibit vortices, for small enough values of the parameter ε. These vortices concentrate at one or several planar arcs (represented by integer multiplicity rectifiable currents) which minimize a line energy, obtained as a Γ-limit of the Gross–Pitaevskii functional. The location of these limiting vortex lines can be described under certain geometrical hypotheses on the cross-sections of the torus.     

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

We study global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W ^1,2, to a global minimizer predicted by the Oseen–Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen–Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau–De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau–De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.     

Giant Vortex and the Breakdown of Strong Pinning in a Rotating Bose-Einstein Condensate

We consider a two-dimensional model for a rotating Bose-Einstein condensate (BEC) in an anharmonic trap. The special shape of the trapping potential, negative in a central hole and positive in an annulus, favors an annular shape for the support of the wave function u. We study the minimizers of the energy in the Thomas-Fermi limit, where a small parameter ɛ tends to 0, for two different regimes of the rotational speed Ω. When Ω is independent of ɛ, we observe that the energy minimizers acquire vorticity beyond a critical Ω, but the vortices are strongly pinned in the central hole where the potential is negative. In this regime, minimizers exhibit no vortices in the annular bulk of the condensate. There is a critical rotational speed Ω=O(|lnɛ|) for which this strong pinning effect breaks down and vortices begin to appear in the annular bulk. We derive an asymptotic formula for the critical Ω, and determine precisely the location of nucleation of the vortices at the critical value. These results are related to very recent experimental and numerical observations on BEC.     

Invertibility and Non-Invertibility in Thin Elastic Structures

The nonlinear elastic energy of a thin film of thickness h is given by a functional E ^h . Friesecke, James and Müller derived the Γ-limits, as h → 0, of the functionals h ^−α E ^h for α ≧ 3. In this article we study the invertibility properties of almost minimizers of these functionals, and more generally of sequences with equiintegrable energy density. We show that they are invertible almost everywhere away from a thin boundary layer near the film surface. Moreover, we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages. We construct examples showing that these bounds are sharp. In particular, for all α ≧ 3 there exist Lipschitz continuous low energy deformations which are not locally invertible.     

纳米压痕法测磁控溅射铝薄膜屈服应力

为了在考虑残余应力下测量出磁控溅射铝薄膜的屈服应力,提出了一种实验测量方法,通过曲率测试法和球形压头纳米压痕法测出磁控溅射铝薄膜的屈服应力．建立球形压痕力学模型,并用ANSYS对球形压痕进行力学有限元仿真,利用直流磁控溅射技术在硅基上淀积一层1 μm厚的铝薄膜,首先通过曲率测试法测量膜内等双轴残余应力,再利用最小二乘曲线拟合法从薄膜/基底系统的球形压头纳米压痕实验数据中提取出铝薄膜的屈服应力,测得磁控溅射铝薄膜的屈服应力为371 MPa．该方法也可以用来研究其他材料的薄膜和小体积材料的力学特性．     

Equilibrium Configurations of Epitaxially Strained Crystalline Films: Existence and Regularity Results

Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and substrate, flat configurations are in general energetically unfavorable and a corrugated or islanded morphology is the preferred growth mode of the strained film. After specifying the functional setup in which the existence problem can be properly framed, a study of the qualitative properties of the solutions is undertaken. New regularity results for volume-constrained local minimizers of the total free energy are established, leading, as a byproduct, to a rigorous proof of the zero-contact-angle condition between islands and wetting layers.     

Film thickness measurements in liquid–liquid slug flow regimes

At present there is significant interest in the development of small scale medical diagnostic equipment. These devices offer faster processing times and require smaller sample volumes than equivalent macro scale systems. Although significant attention has been focused upon their outputs, little attention has been devoted to the detailed fluid mechanics that govern the flow mechanisms within these devices. Conventionally, the samples in these small scale devices are segmented into distinct discrete droplets or slugs which are suspended in an organic carrier phase. Separating these slugs from the channel wall is a very thin film of the organic carrier phase.The magnitude of this film is the focus of the present study and the effects of sample slug length and carrier phase fluidic properties on the film are examined over a range of Capillary numbers. A non-intrusive optical technique was used to capture images of the flow from which the magnitude of the film was determined.The experimental results show that the film is not constant along the length of the slug; however above a threshold value for slug length, a region of constant film thickness exists. When compared with existing correlations in the literature, the experimental data showed reasonable agreement with the Bretherton model when the Capillary number was calculated based on the mean two phase flow velocity. However, significant differences were observed when the Capillary number was redefined to account for the mean velocity at the liquid interface, i.e., the mean slug velocity.Analysis of the experimental data revealed that it fell into two distinct flow regimes; a visco-capillary regime and a visco-inertial regime. A modified Taylor expression is presented to estimate the magnitude of the film for flows in the visco-capillary regime while a new model is put forward, based on Capillary and Weber numbers, for flows in the visco-inertial regime. Overall, this study provides some novel insights into parameters, such as aqueous slug length and carrier phase fluidic properties, that affect the thickness of the film in liquid–liquid slug flow regimes.     

Aerodynamic control of bridge cables through shape modification: A preliminary study

This paper examines the viability of modifying bridge cable shape and surface for the purpose of controlling wind-induced vibrations. To this end, an extensive wind-tunnel test campaign was carried out on various cable shapes about the critical Reynolds number region. Cable shapes were chosen to passively modify the flow in a particular manner. Tested shapes included those which have some form of waviness, faceting and shrouding. Section models were tested using a static inclined rig, allowing them to be installed at yawed cable-wind angles for both smooth and turbulent flow conditions. The aerodynamic damping of the tested cylinders is evaluated by applying both 1- and 2-dof quasi-steady aerodynamic instability models. This allows for the prediction of regions of aerodynamic instability, as a function of flow angle and Reynolds number. Whilst the plain, wavy and faceted cylinders are predicted to suffer from either dry inclined galloping, “drag crisis” or Den Hartog galloping, the shrouded cylinder is found to be stable for all angles of attack, albeit with an increase in drag at typical design wind velocities. Finally, turbulent flow is found to introduce an increased amount of aerodynamic damping mainly by providing a more constant lift force over tested Reynolds numbers.     

Equilibrium Configurations of Epitaxially Strained Elastic Films: Second Order Minimality Conditions and Qualitative Properties of Solutions

We consider a variational model introduced in the physical literature to describe the epitaxial growth of an elastic film over a thick flat substrate when a lattice mismatch between the two materials is present. We study quantitative and qualitative properties of equilibrium configurations, that is, of local and global minimizers of the free-energy functional. More precisely, we determine analytically the critical threshold for the local minimality of the flat configuration and we also prove several results concerning its global minimality. The non-occurrence of singularities in non-flat global minimizers is also addressed. One of the main results of the paper is a new sufficient condition for local minimality, which provides the first extension of the classical criteria based on the positivity of the second variation to the context of functionals with bulk and surface energies.     

A piezoelectric field-effect strain gage

A new and highly sensitive strain transducer has been developed using a thin-film semiconductor deposited on a polished piezoelectric ceramic substrate. Field-effect coupling has been found to exist between the substrate and film in which the number of mobile carriers in the semiconductor is dependent on the electric-displacement vector of the substrate. Therefore, the conductivity of the semiconducting film can be altered by piezoelectric charge due to a strain applied to the substrate material. An effective gage constant has been calculated in terms of the piezoelectric and elastic constants of the substrate and electronic properties of the film. Experimental devices were constructed by depositingp type tellurium on polished lead-zirconate-titanate ceramic resulting in experimentally observed gage factors as high as 5800 compared to 100–200 for conventional semiconductor gages. The semiconductor film exhibits an electronic instability that limits its use, at present, to transient measurements with frequencies above 1 Hz. Data will also be presented to show that the gage constant is continuously variable between a positive and negative maximum value by altering the magnitude and direction of the substrate-polarization vector. It is believed that these gages will be useful in those cases where extremely small strains (～10^?7) are to be measured or when moderate strains (～10^?4) are to be determined in an electrically noisy background.     

电化学聚合沉积膜的摩擦磨损性能研究

利用电化学聚合沉积法分别制备了聚吡咯和聚队甲基吡咯薄膜，并对其摩擦磨损性能进行了试验研究，结果表明，两种聚合物薄膜的摩擦学性能与其单体类型、电解质成分、化学状态和膜厚等均有密切关系：聚Ｎ－甲基吡咯薄膜的摩擦磨损性能比聚吡咯薄膜的好，而且波动性也比后者的小；膜厚较小时，氧化态的聚吡咯薄膜之摩擦性能比还原态的好，但其耐磨性却远比后者的差，而两者在膜厚较大时的摩擦磨损性能之优劣与膜厚较小时的恰好相反，研究发现，高聚物薄膜在膜厚较小时发生的是粘着磨损，而在膜厚较大时既有粘着磨损，也有疲劳磨损．     

Action-Minimizing Orbits in the Parallelogram Four-Body Problem with Equal Masses

In this paper we prove the existence of a new periodic solution for the planar Newtonian four-body problem with equal masses. On this orbit two mass points travel on one star-shaped closed curve while the other two travel on another and have the opposite orientation. The configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time. Our proof is based on a variational approach inspired by a recent work of Chenciner &; Montgomery [5]. By choosing an appropriate subspace of the Sobolev space H ¹([0,T],V), where V is the configuration space, we show that the action functional restricted to this subspace attains its infimum and any minimizer solves the Newtonian four-body problem. The orbits we found are indeed extensions of these minimizers. By studying the behavior of minimizers in reduced configuration space and comparing their action with rhomboid motions, we show that these minimizers do not experience any collision.     

Cartesian currents,weak diffeomorphisms and existence theorems in nonlinear elasticity

Summary In this paper we introduce some new classes of functions, among these a class of weak diffeomorphisms. In these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals. In particular, we prove the existence of one-to-one orientation-preserving maps that minimize suitable energies associated with hyperelastic materials. The minimizers are also proved to satisfy equilibrium equations. Finally radial deformations are discussed in connection with cavitation.     

Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three-body problem in three-dimensional space. First we give a finite complete list of symmetry groups fitting to the minimization of the action, with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives. A second step is to prove that the resulting (local and global) symmetric action-minimizers are always collisionless (when they are not already bound to collisions). Furthermore, we prove some results which address the question of whether minimizers are planar or non-planar; as a consequence of our theory we will give general criteria for a symmetry group to yield planar or homographic minimizers (either homographic or not, as in the Chenciner-Montgomery eight solution). On the other hand we will provide a rigorous proof of the existence of some interesting one-parameter families of periodic and quasi-periodic non-planar orbits. These include the choreographic Marchal's P₁₂ family with equal masses – together with a less-symmetric choreographic family (which anyway probably coincides with the P₁₂ family).     

Variational Problems¶on Multiply Connected Thin Strips II:¶Convergence of the Ginzburg-Landau

Let M be a planar embedded graph whose arcs meet transversally at the vertices; Let ?(M) be a strip-shaped domain around M, of width M except in a neighborhood of the singular points. Assume that the boundary of ?(M) is smooth. We consider the Ginzburg-Landau energy functional for superconductivity on ?(M). We prove that its minimizers converge in a suitable sense to the minimizers of a simpler functional on M. The supercurrents in ?(M) are shown to converge to one-dimensional currents in M.     

A novel method of supporting gold nanoparticles on MWCNTs： Synchrotron X-ray reduction

Gold nanoparticles decorating the surface of multiwalled carbon nanotubes （MWCNTs） are prepared by photochemical reduction. The gold clusters form different interesting geometrical faceted shapes in accordance to time duration of synchrotron X-ray irradiation. The shape of nanogold could be spherical, rod-like, or triangular. Carbon nanotubes serve as optimal templates for the heterogeneous nucleation of gold nanocrystals. These nanocrystal structures are characterized by transmission electron microscope （TEM） and element analysis by energy dispersive spectroscopy （EDS）. 2007 Chinese Society of Particuology and Institute of Process Engineering,Chinese Academy of Sciences. Published by Elsevier B.V.     

ZDDP作用下钨/钼膜的摩擦化学润滑机理研究

利用SRV高温磨损试验机考察了钨膜、钼膜以及钨-钼合金膜与ZDDP的交互作用规律,并使用XPS和SEM等表面分析技术分析摩擦反应膜的成分与化学态.结果表明:不同材料的薄膜在加入ZDDP后,其表面的摩擦系数的变化趋势与基础油润滑作用下相同,均从0.11降到0.08左右.而磨损量表现出不同的变化规律:钨膜表面的磨损量在加入ZDDP前后均保持在3.7×10-13m3左右;钼膜和钨-钼膜表面的磨损量在加入ZDDP后明显降低,而且钨-钼膜表面的磨损量降低幅度最大,达74.5%.这与不同材料表面的硬度值和生成的表面摩擦反应膜有关.     

A Variational Rod Model with a Singular Nonlocal Potential

The classical theory of elastic rods does not account for the possibility that large deformations may involve distinct points along the rod occupying the same physical space. We develop an elastic rod model with a pairwise repulsive potential such that, if two non-adjacent points along the rod are close in physical space, there is an energy barrier which prevents contact while for points nearby along the rod the potential is describable classically. This framework is developed to prove the existence of minimizers within each homotopy class, where the idea of topological homotopy of a curve is generalized to elastic rods as framed curves. Finally, the relevant first-order optimality conditions are derived and used to investigate the regularity of minimizers.