Landau theory for biaxial nematic liquid crystals with two order parameter tensors

We present a phenomenological theory for the homogeneous phases of nematic liquid crystals constituted by biaxial molecules. We propose a general polynomial potential in two macroscopic order parameter tensors that reproduces the mean-field phase diagram confirmed by Monte Carlo simulations [De Matteis et al. in Phys Rev E 72:041706 (2005)] and recently recognized to be universal [Bisi et al. in Phys Rev E 73:051709 (2006)] for dispersion force molecular pair-potentials enjoying the D _2h symmetry. The requirement that the phenomenological theory comply uniquely with this phase diagram reduces considerably the admissible phenomenological coefficients, both in their number and in the ranges where they can vary.      

Dynamics of shear aligning of nematic liquid crystal monodomains

The equations of linear and angular momentum for nematic liquid crystals have been described with Ericksen's transversely isotropic fluid [TIF] model and solved for start-up of shear flow at constant rate and varying initial alignment conditions. An analytical solution for the rotation provides predictions of the nematic director which closely agree with experimental results of Boudreau et al. (1999), supporting the validity of Ericksen's TIF model. The solution is limited to flows where the effects of director gradients are negligible. Received: 13 September 1999/Accepted: 24 January 2000     

Wrinkle patterns of anisotropic crystal films on viscoelastic substrates

This paper presents a nonlinear mathematical model for evolution of wrinkle patterns of an anisotropic crystal film on a viscoelastic substrate layer. The underlying mechanism of wrinkling has been generally understood as a stress-driven instability. Previously, theoretical studies on wrinkling have assumed isotropic elastic properties for the film. Motivated by recent experimental observations of ordered wrinkle patterns in single-crystal thin films, this paper develops a theoretical model coupling anisotropic elastic deformation of a crystal film with viscoelastic deformation of a thin substrate layer. A linear perturbation analysis is performed to predict the onset of wrinkling instability and the initial evolution kinetics. An energy minimization method is adopted to analyze wrinkle patterns in the equilibrium states. For a cubic crystal film under an equi-biaxial compression, orthogonally ordered wrinkle patterns are predicted in both the initial stage and the equilibrium state. This is confirmed by numerical simulations of evolving wrinkle patterns. By varying the residual stresses in the film, numerical simulations show that a variety of wrinkle patterns (e.g., orthogonal, parallel, zigzag, and checkerboard patterns) emerge as a result of the competition between material anisotropy and stress anisotropy.     

Dynamic theory for smectic A liquid crystals

A dynamic continuum theory is presented for smectic A liquid crystals in which the usual director n and unit layer normal a do not always necessarily coincide. Most previous dynamic continuum theories equate n with a; the theory developed in this article allows n and a to differ in non-equilibrium situations, work that has been motivated by the recent investigations by Auernhammer et al. (Rheol. Acta 39, 215–222, 2000; Phys. Rev. E 66, 061707, 2002) and Soddemann et al. (Eur. Phys. J. E 13, 141–151, 2004). The usual Oseen constraint () for smectics is not imposed upon the unit normal a. Permeation is also included. After a summary of the complete dynamic equations, an application is given via an example which shows that planar aligned layers of smectic A subjected to an arbitrary periodic disturbance are linearly stable.      

Study of size effects in thin films by means of a crystal plasticity theory based on DiFT

In a recent publication, we derived the mesoscale continuum theory of plasticity for multiple-slip systems of parallel edge dislocations, motivated by the statistical-based nonlocal continuum crystal plasticity theory for single-glide given by Yefimov et al. [2004b. A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity simulations. J. Mech. Phys. Solids 52, 279-300]. In this dislocation field theory (DiFT) the transport equations for both the total dislocation density and geometrically necessary dislocation (GND) density on each slip system were obtained from the Peach-Koehler interactions through both single and pair dislocation correlations. The effect of pair correlation interactions manifested itself in the form of a back stress in addition to the external shear and the self-consistent internal stress. We here present the study of size effects in single crystalline thin films with symmetric double slip using the novel continuum theory. Two boundary value problems are analyzed: (1) stress relaxation in thin films on substrates subject to thermal loading, and (2) simple shear in constrained films. In these problems, earlier discrete dislocation simulations had shown that size effects are born out of layers of dislocations developing near constrained interfaces. These boundary layers depend on slip orientations and applied loading but are insensitive to the film thickness. We investigate the stress response to changes in controlled parameters in both problems. Comparisons with previous discrete dislocation simulations are discussed.     

Modelling the wetting of a solid occlusion by a liquid film

We investigate in this work how the presence of an occlusion affects the dynamics of the wetting front of a liquid film draining down a vertical surface. This numerical study is developed in the context of the lubrication approximation. Through a parametric study, we show that depending on the asymptotic film thickness and the fluid properties, there exists a critical substrate contact angle below which separation of the contact line from the occlusion wall is observed which results in the appearance of a dry zone in the wake of the occlusion. In analogy with external aerodynamics, we also show that a sharp corner in the occlusion can induce this contact line separation. Our numerical results also highlight the importance of the occlusion wettability on the morphology of the wetting front suggesting a possible mechanism to control and mitigate the often undesirable fingering instability.     

A film flow model for analysing gravity-driven,thin wavy fluid films

To analyse the physics underlying gravity-driven runoff of thin wavy films, a film flow model is developed, and is solved with computational fluid dynamics. This model is based on the lubrication theory, and takes into account the gravitational, wall shear and surface tension forces. A key characteristic of the model is that it assumes only one computational cell over the film height, which enables studying film flow on larger computational domains. A main aim of this study is to perform a detailed validation of the numerical model. The film flow model is validated against several experiments of gravity-driven, thin fluid films on smooth surfaces. The time-averaged film thickness and the fluid speed profiles predicted by the model show very good agreement with experimental results. Similarly, the film flow model is able to predict the wave speeds with sufficient accuracy. The energy spectra of the waves, where higher frequency waves are present in film flows at higher Reynolds numbers, show an exponentially decaying trend at these high frequencies. The model performs better than the Nusselt equation for film flows, which under-predicts the time-averaged film thickness and over-predicts the time-averaged fluid speeds, even for flows at low Reynolds numbers. The film flow model is compared qualitatively for fingering behaviour. This model also allows to investigate film flows on large surfaces, which can be rough, curved and of complex geometrical shape.     

Unraveling surfactant transport on a thin liquid film

When a drop of insoluble surfactant is deposited on the surface of a thin liquid film, a radial flow is induced by the resulting surface tension gradient. It is difficult in practice to measure or visualize the evolution of the surfactant concentration and the corresponding surface tension field. In this contribution, we propose a numerical technique which allows, in theory, the reconstruction of the surfactant concentration and surface tension fields from the knowledge of the free surface velocity. The method also requires the knowledge of the equation of state relating the surfactant concentration to the surface tension. The proposed method is based on a reformulation of the lubrication approximation which then takes as an input the free surface velocity field. As a by-product, the film thickness is also reconstructed. We also show in this contribution, that the surface diffusion coefficient can also be estimated, in principle. The methodologies are successfully tested on ideal, synthetic data-sets but also on under-resolved, noisy, data-sets more representative of true experimental conditions. This contribution may help shed some light on the phenomena involved in surfactant transport.     

A methodology for determining mechanical properties of freestanding thin films and MEMS materials

We have developed a novel chip-level membrane deflection experiment particularly suited for the investigation of sub-micron thin films and microelectro-mechanical systems. The experiment consists of loading a fixed-fixed membrane with a line load applied at the middle of the span using a nanoindenter. A Mirau microscope interferometer is positioned below the membrane to observe its response in real time. This is accomplished through a micromachined wafer containing a window that exposes the bottom surface of the specimen. A combined atomic force microscope/nanoindenter incorporates the interferometer to allow continuous monitoring of the membrane deflection during both loading and unloading. As the nanoindenter engages and deflects the sample downward, fringes are formed and acquired by means of a CCD camera. Digital monochromatic images are obtained and stored at periodic intervals of time to map the strain field. Stresses and strains are computed independently without recourse to mathematical assumptions or numerical calibrations. Additionally, no restrictions on the material behavior are imposed in the interpretation of the data. In fact, inelastic mechanisms including strain gradient plasticity, piezo and shape memory effects can be characterized by this technique.The test methodology, data acquisition and reduction are illustrated by investigating the response of 1-μm thick gold membranes. A Young's modulus of , a yield stress of and a residual stress of are consistently measured. The post-yield behavior leading to fracture exhibits typical statistical variations associated to plasticity and microcrack initiation.     

A microbeam bending method for studying stress-strain relations for metal thin films on silicon substrates

We have developed a microbeam bending technique for determining elastic-plastic, stress-strain relations for thin metal films on silicon substrates. The method is similar to previous microbeam bending techniques, except that triangular silicon microbeams are used in place of rectangular beams. The triangular beam has the advantage that the entire film on the top surface of the beam is subjected to a uniform state of plane strain as the beam is deflected, unlike the standard rectangular geometry where the bending is concentrated at the support. To extract the average stress-strain relations for the film, we present a method of analysis that requires computation of the neutral plane for bending, which changes as the film deforms plastically. This method can be used to determine the elastic-plastic properties of thin metal films on silicon substrates up to strains of about 1%.Utilizing this technique, both yielding and strain hardening of Cu thin films on silicon substrates have been investigated. Copper films with dual crystallographic textures and different grain sizes, as well as others with strong 〈1 1 1〉 textures have been studied. Three strongly textured 〈1 1 1〉 films were studied to examine the effect of film thickness on the deformation properties of the film. These films show very high rates of work hardening, and an increase in the yield stress and work hardening rate with decreasing film thickness, consistent with current dislocation models.     

Theory of linear viscoelasticity of chiral liquid crystals

The governing equations of monodomain isothermal cholesteric liquid crystals subjected to small amplitude oscillatory rectilinear shear have been derived for three representative helix orientations. The imposition of oscillatory flow excites splay-bend-twist deformations when the helix is aligned along the flow direction, splay-bend deformations when the helix is along the vorticity gradient, and twist deformations when aligned along the velocity axis. The different nature of the excited elastic modes as well as the anisotropic viscosities are reflected in the anisotropy of the linear viscoelastic material functions for small amplitude rectilinear oscillatory shear. When the helix is aligned along the flow direction, cholesteric viscoelasticity is strongest, and exists in a relatively narrow band of intermediate frequencies. When the helix is aligned along the vorticity direction cholesteric viscoelasticity is significant in a relatively broad range of intermediate frequencies. Finally, when the helix is aligned along the velocity gradient direction, cholesteric viscoelasticity is relatively insignificant and only exists in a narrow band of frequencies. The cholesteric pitch controls the location of viscoelastic region on the frequency spectrum, but only when the helix is not oriented along the vorticity axis.     

A variational approach to the fracture of brittle thin films subject to out-of-plane loading

We address the problem of fracture in homogenous linear elastic thin films using a variational model. We restrict our attention to quasi-static problems assuming that kinetic effects are minimal. We focus on out-of-plane displacement of the film and investigate the effect of bending on fracture. Our analysis is based on a two-dimensional model where the thickness of the film does not need to be resolved. We derive this model through a formal asymptotic analysis. We present numerical simulations in a highly idealized setting for the purpose of verification, as well as more realistic micro-indentation experiments.     

A relationship between effective work of adhesion and peel force for thin hyperelastic films undergoing large deformation

A method to determine the effective work of adhesion for hyperelastic thin films undergoing large deformations is presented. Starting from energy balance equation a relationship between work of adhesion, the peel force, the peel angle, and the stretch is derived. Based on this relation a procedure to compute the energy of adhesion from peel tests is proposed. To this end the peel force as well as the engineering stress vs. engineering strain diagram for thin film is required. The derived relationship shows that the non-linearity of the stress-stain relation must be taken into account in computing the effective work of adhesion from the peel force. The processing of experimental data within the standard linear elasticity approach would lead to an overestimation of effective work of adhesion. The error would increase with a decrease of the peel angle.     

3D finite element modeling for instabilities in thin films on soft substrates

Spatial pattern formation in stiff thin films on compliant substrates is investigated based on a nonlinear 3D finite element model. Typical post-bifurcation patterns include 1D sinusoidal, checkerboard and herringbone shapes, with possible spatial modulations, boundary effects and localizations. The post-buckling behavior often leads to intricate response curves with several secondary bifurcations that were rarely studied and only in the case of periodic cells. The proposed finite element procedure allows accurately describing these bifurcation portraits by taking into account the effect of boundary conditions. It relies on the Asymptotic Numerical Method (ANM) that offers considerable advantages to get a robust path-following technique and to detect multiple bifurcations. The occurrence and evolution of sinusoidal, checkerboard and herringbone patterns will be highlighted.     

A new solution for the rotation-driven spreading of a thin fluid film

Lie groups are used to solve the equation governing the flow of a thin liquid film subject to centrifugal spreading and viscous resistance. A new implicit solution is found. It is shown how this relates to the previous known solutions for the spreading of an initially flat film, the steady state and a separable solution. New permissible forms for the film evolution are also studied, including solutions exhibiting finite time blow-up. Near the contact line, where the film height tends to zero, an approximate explicit solution is obtained which may be used to describe a film with any size contact angle.     

Micron-scale channel formation by the release and bond-back of pre-stressed thin films: A finite element analysis

Buckling of thin films on a rigid substrate during use or fabrication is a well-known but unwanted phenomenon. However, this phenomenon can also be exploited to generate well-controlled patterns at the micro and nano-scale. These patterned surfaces find various technological applications such as optical gratings or micro/nano-fluidic channels. In this article, we present a numerical model that accounts for the buckling-up of pre-strained thin films by a reduction of the interface toughness and the subsequent bond-back. Channels are formed whose dimensions can be controlled by tuning the film dimensions, film thickness and stiffness, the eigenstrain in the film and the cohesive interface energy between the film and the substrate. We will show how the buckling-up and draping back processes can be captured in terms of a limited set of dimensionless parameters, providing quantitative insight on how these parameters should be tuned to generate a specified channel geometry.     

Glass-modified stress waves for adhesion measurement of ultra thin films for device applications

Laser-generated stress wave profiles with rarefaction shocks (almost zero post-peak decay times) have been uncovered in different types of glasses and presented in this communication. The rise time of the pulses was found to increase with their amplitude, with values reaching as high as . This is in contrast to measurements in other brittle crystalline solids where pulses with rise times of and post-peak decay times of were recorded. The formation of rarefaction shock is attributed to the increased compressibility of glasses with increasing pressures. This was demonstrated using a one-dimensional nonlinear elastic wave propagation model in which the wave speed was taken as a function of particle velocity. The technological importance of these pulses in measuring the tensile strength of very thin film interfaces is demonstrated by using a previously developed laser spallation experiment in which a laser-generated compressive stress pulse in the substrate reflects into a tensile wave from the free surface of the film and pries off its interface at a threshold amplitude. Because of the rarefaction shock, glass-modified waves allow generation of substantially higher interfacial tensile stress amplitudes compared with those with finite post-peak decay profiles. Thus, for the first time, tensile strengths of very strong and ultra thin film interfaces can be measured. Results presented here indicate that interfaces of 185-nm-thick films, and with strengths as high as , can be measured. Thus, an important advance has been made that should allow material optimization of ultra thin layer systems that may form the basis of future MEMS-based microelectronic, mechanical and clinical devices.     

A new method for manufacture of thin film heat flux gauges

The manufacture of thin film gauges for measuring transient temperatures and heat fluxes is described. A new method of using ceramic substrates (ZrO₂) with two sintered platinum wires is described. Examples of static and dynamic calibrations are given. Sample measurements in a shock tunnel are presented. The gauges show good mechanical strength and sensitivity.This article was processed using Springer- Verlag T_EX Shock Waves macro package 1990.     

A simple technique for determining yield strength of thin films

A technique to measure the yield strength of thin films has been developed which combines experimental observations of deflection and plastic deformation with finite element predictions of stress. This technique relies on integrated circuit technology to build bridge and cross beam test structures with a range of dimensions. Each structure is deflected in increments of 1 μm until the structure no longer elastically recovers upon release. In tandem with experimentally verified numerical predictions of force and stress, the yield strength of the thin film can be bounded between the highest elastic stress result and the lowest plastic stress result. For our test material of copper, this method provides a yield strength between 2.80 and 3.09 GPa, a value significantly larger than that for bulk copper, but consistent with thin film theory.     

A computational model for the indentation and phase transformation of a martensitic thin film

We propose a computational model for a stress-induced martensitic phase transformation of a single-crystal thin film by indentation and its reverse transformation to austenite by heating. Our model utilizes a surface energy that allows sharp interfaces with finite energy and a penalty that forces the film to lie above the indenter and undergo a stress-induced austenite-to-martensite phase transformation. We introduce a method to nucleate the martensite-to-austenite phase transformation since in our model the film would otherwise remain in the martensitic phase in a local minimum of the energy.