Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: Analysis for obtaining film stress from non-local curvature information

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. Recently Huang, Rosakis and co-workers [Huang, Y., Ngo, D., Rosakis, A.J., 2005. Non-uniform, axisymmetric misfit strain: in thin films bonded on plate substrates/substrate systems: the relation between non-uniform film stresses and system curvatures. Acta Mech. Sin. 21, 362–370; Huang, Y., Rosakis A.J., 2005. Extension of Stoney’s Formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry. J. Mech. Phys. Solids 53, 2483–2500; Ngo, D., Huang, Y., Rosakis, A. J., Feng, X. 2006. Spatially non-uniform, isotropic misfit strain in thin films bonded on plate substrates: the relation between non-uniform film stresses and system curvatures. Thin Solid Films (in press)] established methods for film/substrate system subject to non-uniform misfit strain and temperature changes. The film stresses were found to depend non-locally on system curvatures (i.e., depend on the full-field curvatures). The existing methods, however, all assume uniform film thickness which is often violated in the thin film/substrate system. We extend these methods to arbitrarily non-uniform film thickness for the thin film/substrate system subject to non-uniform misfit strain. Remarkably the stress-curvature relation for uniform film thickness still holds if the film thickness is replaced by its local value at the point where the stress is evaluated. This result has been experimentally validated in Part II of this paper.     

Non-uniform, axisymmetric misfit strain： in thin films bonded on plate substrates/substrate systems： the relation between non-uniform film stresses and system curvatures

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric misfit strain distributions in the thin film, we derived relations between the film stresses and the misfit strain, and between the plate system’s curvatures and the misfit strain. These relations feature a ‘‘local’’ part which involves a direct dependence of the stress or curvature components on the misfit strain at the same point, and a ‘‘non-local’’ part which reflects the effect of misfit strain of other points on the location of scrutiny. Most notably, we also derived relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial non-uniformities. These relations also feature a ‘‘non-local’’ dependence on curvatures making a full-field measurement a necessity. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.     

Multi-layer thin films/substrate system subjected to non-uniform misfit strains

Current methodologies used for the inference of thin film stress through curvature measurement are strictly restricted to stress and curvature states that are assumed to remain uniform over the entire film/substrate system. These methodologies have recently been extended to a single layer of thin film deposited on a substrate subjected to the non-uniform misfit strain in the thin film. Such methodologies are further extended to multi-layer thin films deposited on a substrate in the present study. Each thin film may have its own non-uniform misfit strain. We derive relations between the stresses in each thin film and the change of system curvatures due to the deposition of each thin film. The interface shear stresses between the adjacent films and between the thin film and the substrate are also obtained from the system curvatures. This provides the basis for the experimental determination of thin film stresses in multi-layer thin films on a substrate.     

Extension of Stoney's formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric temperature distributions, we derive relations between the film stresses and temperature, and between the plate system's curvatures and the temperature. These relations featured a “local” part which involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “non-local” part which reflects the effect of temperature of other points on the location of scrutiny. Most notably, we also derive relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial non-uniformities. These relations also feature a “non-local” dependence on curvatures making full-field measurements of curvature a necessity for the correct inference of stress. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.     

Thermo-mechanical solution of film/substrate systems under local thermal load and application to laser lift-off of GaN/sapphire structures

Film/substrate structures may undergo a localized thermal load, which can induce stresses, deformation and defects. In this paper, we present the solutions of temperature and stresses in a film/substrate structure under a local thermal load on the film surface. Then, the generalized Stoney formula, which connects the curvature of deformation and the stress field is obtained. The present solution takes into account the non-uniformity of the temperature field both in the width and thickness directions of the film. The thermo-mechanical solution is applied to the analysis of the temperature distribution, stresses, and damage of a GaN/sapphire system during the laser lift-off (LLO) process. It is shown that the laser with the Gaussian distribution of energy density causes much smaller tensile stresses at the edge of the heated area in the film than the laser with the uniform distribution of energy density, and thus can avoid damage to the GaN films separated from the substrate.     

Residual stresses in thin film systems: Effects of lattice mismatch,thermal mismatch and interface dislocations

This paper explores the mechanisms of the residual stress generation in thin film systems with large lattice mismatch strain, aiming to underpin the key mechanism for the observed variation of residual stress with the film thickness. Thermal mismatch, lattice mismatch and interface misfit dislocations caused by the disparity of the material layers were investigated in detail. The study revealed that the thickness-dependence of the residual stresses found in experiments cannot be elucidated by thermal mismatch, lattice mismatch, or their coupled effect. Instead, the interface misfit dislocations play the key role, leading to the variation of residual stresses in the films of thickness ranging from 100 nm to 500 nm. The agreement between the theoretical analysis and experimental results indicates that the effect of misfit dislocation is far-reaching and that the elastic analysis of dislocation, resolved by the finite element method, is sensible in predicting the residual stress distribution. It was quantitatively confirmed that dislocation density has a significant effect on the overall film stresses, but dislocation distribution has a negligible influence. Since the lattice mismatch strain varies with temperature, it was finally confirmed that the critical dislocation density that leads to the measured residual stress variation with film thickness should be determined from the lattice mismatch strain at the deposition temperature.     

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．     

Cracking mechanisms in lithiated silicon thin film electrodes

The massive cracking of silicon thin film electrodes in lithium ion batteries is associated with the colossal volume changes that occur during lithiation and delithiation cycles. However, the underlying cracking mechanism or even whether fracture initiates during lithiation or delithiation is still unknown. Here, we model the stress generation in amorphous silicon thin films during lithium insertion, fully accounting for the effects of finite strains, plastic flow, and pressure-gradients on the diffusion of lithium. Our finite element analyses demonstrate that the fracture of lithiated silicon films occurs by a sequential cracking mechanism which is distinct from fracture induced by residual tension in conventional thin films. During early-stage lithiation, the expansion of the lithium-silicon subsurface layer bends the film near the edges, and generates a high tensile stress zone at a critical distance away within the lithium-free silicon. Fracture initiates at this high tension zone and creates new film edges, which in turn bend and generate high tensile stresses a further critical distance away. Under repeated lithiation and delithiation cycles, this sequential cracking mechanism creates silicon islands of uniform diameter, which scales with the film thickness. The predicted island sizes, as well as the abrupt mitigation of fracture below a critical film thickness due to the diminishing tensile stress zone, is quantitatively in good agreement with experiments.     

Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.     

Analysis of vertical cracking phenomena in tensile-strained epitaxial film on a substrate: Part I. Mathematical formulation

This paper presents an analysis of a single vertical crack and periodically distributed vertical cracks in an epitaxial film on a semi-infinite substrate where the cracks penetrate into the substrate. The film and substrate materials have different anisotropic elastic constants, necessitating Stroh formalism in the analysis. The misfit strain due to the lattice mismatch between the film and the substrate serves as the driving force for crack formation. The solution for a dislocation in an anisotropic trimaterial is used as a Green function, so that the cracks are modeled as the continuous distributions of dislocations to yield the singular integral equations of Cauchy-type. The Gauss–Chebyshev quadrature formula is adopted to solve the singular integral equations numerically. Energy arguments provide the critical condition for crack formation, at which the cracks are energetically favorable configurations, in terms of the ratio of the penetration depth into the substrate to the film thickness, the ratio of the spacing of the periodic cracks to the film thickness, and the generalized Dundurs parameters between the film and substrate materials.     

Wrinkling of a charged elastic film on a viscous layer

A thin metallic film deposited on a compliant polymeric substrate begins to wrinkle under compression induced in curing process and afterwards cooling of the system. The wrinkle mode depends upon the thin film elasticity, thickness, compressive strain, as well as mechanical properties of the compliant substrate. This paper presents a simple model to study the modulation of the wrinkle mode of thin metallic films bonded on viscous layers in external electric field. During the procedure, linear perturbation analysis was performed for determining the characteristic relation that governs the evolution of the plane-strain wrinkle of the thin films under varying conditions, i.e., the maximally unstable wrinkle mode as a function of the film surface charge, film elasticity and thickness, misfit strain, as well as thickness and viscosity of the viscous layer. It shows that, in proper electric field, thin film may wrinkle subjected to either compression or tension. Therefore, external electric field can be employed to modulate the wrinkle mode of thin films. The present results can be used as the theoretical basis for wrinkling analysis and mode modulation in surface metallic coatings, drying adhesives and paints, and microelectromechanical systems (MEMS), etc.     

Three-Dimensional Compositional Elastic Field in Cubic Anisotropic Solid Films on Compliant Substrates

A two-dimensional Laplace transform technique is used to find an analytical solution for the three-dimensional elastic field in solid films attached to a compliant substrate. The stress field arises from the misfit between the film and substrate and also from the non-uniform composition field in the film itself. Both the film and substrate are assumed to have cubic symmetry. The solution method can also be applied to materials with general orthotropic symmetry. Mathematics Subject Classifications (2000) 74F05, 74G10, 74K35, 74E10, 44A30     

Thermally induced deformations in a partially coated elastic layer

The problem of a thin film coated on an elastic layer and subject to a thermal variation is analytically investigated in the present work. The analysis is developed in order to assess the mechanical behaviour of a crystalline undulator designed for obtaining high emission radiations through channelling phenomenon. It consists in a plane silicon wafer alternately patterned with thin films in silicon nitride on both surfaces. The system adopts a periodic curvature as a result of the misfit strain due to the different thermal expansivities of the layer and the film. The problem is governed by an integral equation which can be reduced to a linear algebraic system by approximating the unknown interfacial shear stress via series expansion of Chebyshev polynomials.     

Buoyancy-affected laminar film condensation

The flow and heat transfer in a laminar condensate flim on an isothermal vertical plate is modelled mathematically. The strict Boussinesq approximation is adopted to account for buoyancy due to local temperature variations within the film. A similarity transformation reduces the governing boundary-layer type equations to a coupled set of ordinary differential equations and the resulting three-parameter twopoint boundary value problem is solved numerically for Prandtl numbers,Pr, ranging from 0.001 to 1000 and Jakob numbers,Ja, between 0.0001 and 1.5. The principal effects of the favourable buoyancy are to reduce the thickness of the condensate film and increase the film velocity at the smooth liquid-vapour interface, whereas the friction and heat transfer at the plate are enhanced. In accordance with the classical Nusselt theory, it is found that the temperature varies nearly linearly across the film. The computed similarity profiles for velocity reveal, however, substantial departures from the parabolic distribution assumed in the simplified Nusselt analysis.     

Study of the thermocapillary effect on a wavy falling film using a fiber optical thickness probe

The wave flow of a water film down a vertical plate with a 150×150 mm heater has been experimentally studied. The effect of the heat flux on the film flow leads to the formation of periodically flowing rivulets and thin film between them due to the action of thermocapillary forces in spanwise direction. The local film thickness between rivulets is measured by means of a noncontact fiber optical probe. As the heat flux grows, the average film thickness continuously decreases but upon reaching about 50% of the initial thickness, the film spontaneously breaks down. It is found that the decrease of the wave amplitude between rivulets is caused by the reduction of the local Reynolds number and is in a qualitative agreement with the laws of the hydrodynamics for the isothermal case. That is, no appreciable effect of streamwise thermocapillary forces on the wave amplitudes is detected. The experimental results are in good agreement with recently published data obtained by the capacitance method.     

非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法（DTM）,研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

Boundary film shear elastic modulus effect in hydrodynamic contact. Part I: theoretical analysis and typical solution

Boundary film shear elastic modulus effect is analyzed in a hydrodynamic contact. The contact is one-dimensional composed of two parallel plane surfaces, which are, respectively, rough rigid with rectangular micro projections in profile periodically distributed on the surface and ideally smooth rigid. The whole contact is consisted of cavitated area and hydrodynamic area. The hydrodynamic area consists of many micro Raleigh bearings which are discontinuously and periodically distributed in the contact. Analysis is thus carried out for a micro Raleigh bearing in this contact. The hydrodynamic contact in this micro Raleigh bearing consists of boundary film area and fluid film area which, respectively, occur in the outlet and inlet zones. In boundary film area, the film slips at the upper contact surface due to the limited shear stress capacity of the film–contact interface, while the film does not slip at the lower contact surface due to the shear stress capacity large enough at the film–contact interface. In boundary film area, the viscosity, density and shear elastic modulus of the film are varied across the film thickness due to the film–contact interactions, and their effective values are used in modeling, which depend on the film thickness. The analytical approach proposed by Zhang (J Mol Liq 128:60–64, 2006) and Zhang et al. (Int J Fluid Mech Res 30:542–557, 2003) is used for boundary film area. In fluid film area, the film does not slip at either of the contact surfaces, and the shear elastic modulus of the film is neglected. Conventional hydrodynamic analysis is used for fluid film area. The present paper presents the theoretical analysis and a typical solution. It is found that for the simulated case the boundary film shear elastic modulus effects on the mass flow through the contact, the overall film thickness of the contact and the carried load of the contact are negligible but the boundary film shear elastic modulus effect on the local film thickness of the contact may be significant when the boundary film thickness is on the 1 nm scale and the contact surfaces are elastic. In Part II will be presented detailed results showing boundary film shear elastic modulus effects in different operating conditions.

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Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique

An experimental investigation of wavy water film falling down a flat plate has been carried out using confocal chromatic sensoring technique to determine the instantaneous and statistical characteristics of the film. The experiments involved three parameters: liquid feed mode, Reynolds number and plate inclination angle. The present time–average film thickness data is compared with the previous experimental and theoretical results showing a good agreement. A new correlation for the average film thickness is suggested. Our results show that the liquid feed mode has a vital influence on the film thickness and that the film thickness increases with Reynolds number and decreased plate inclination angle. The root–mean–square value of the film thickness fluctuations depends non-monotonically on the film Reynolds number. The corresponding mechanisms are analyzed.     

Patterned bilayer plate microstructures subjected to thermal loading: Deformation and stresses

We studied the deformation of a series of gold/polysilicon patterned plate microstructures fabricated by surface micromachining. The patterned plate microstructures were subjected to a uniform temperature change from 100 °C to room temperature that was intended to induce linear and geometrically nonlinear deformation. We used interferometry to measure full-field deformed shapes of the microstructures. From these measurements we determined the spatially-averaged curvature of the deformed microstructures within individual lines and across the entire plate. The deformation response of the patterned plates can be broadly characterized in terms of the average curvature as a function of temperature change and exhibits linear and geometrically nonlinear behavior. We modeled the deformation response of the patterned plates using geometrically nonlinear plate theory with the finite element method. Good agreement was obtained between predictions and measurements for both local curvature variations across lines and for the evolution of curvature of the entire plate with temperature change. Using a generalized plane strain approach with the finite element method we also modeled the spatial dependence of the stress distribution in the lines and substrate. For thick plates, our results agree with those of previous studies, showing a decrease in the von Mises stress in the metal lines with decreasing linewidth. For thinner substrates, though, we find the behavior with linewidth is opposite and there is a critical substrate thickness (about 10 μm for the system in our study) where the behavior with linewidth changes. These results have important implications in the design of patterned structures for micro-electro-mechanical systems (MEMS) applications where films are of comparable thickness to the underlying substrate.     

Stability of a laminar film flow

The problem of the wave motion of a liquid layer was first investigated by Kapitsa [1, 2], who gave an approximate analysis of the free flow and flow in contact with gas stream, and evaluated the influence of the heat transfer processes on the flow. The problem of the stability of such a flow was studied in detail by Benjamin [3] and Yih [4, 5], These authors proposed seeking the solution of the resulting Orr-Sommerfeld equation in the form of a series in a small parameter and developed a corresponding method of successive approximations. As the small parameter [3–5], they made use of the product of the disturbance wave number and the Reynolds number. In these studies, the tangential stress on the free surface was taken equal to zero, and the fluid film was always considered essentially plane. At the same time, there are certain types of problems of considerable interest in which neither of these assumptions is satisfied. A good example might be the problem on the stability of the annular regime of two-phase flow in pipes and capillaries, when the basic stream of one fluid is separated from the pipe walls by an annular layer of another fluid. In this case, the interface has a finite radius of curvature and the tangential stress on the interface may be significantly different from zero.In the present paper, the problem of the flow stability of a fluid layer with respect to small disturbances of the boundary surface is considered with account for both the finite radius of curvature of the boundary surface and the nonzero hydrodynamic friction at the boundary. The film is assumed to be quite thin. This enables us, firstly, to consider the Reynolds number small, to use the general method of [5], and, second ly, to consider the film thickness sufficiently small in comparison with the radius of curvature of the substrate on which the film lies. Furthermore, for evaluating the stability of the laminar flow of the curved film we can use the results obtained for a plane film with account for the terms which depend on the curvature of the substrate.As a rule, previous studies have considered only one-dimensional disturbances of the boundary surface. In the present paper, in the first approximation, the stability is examined in relation to two-dimensional disturbances of this surface, corresponding to three-dimensional flow disturbances.As an example, the results obtained are applied to the investigation of the stability of the free flow of a layer of fluid over an inclined plane under the sole influence of gravity.