Buckling Of Thin Film Research Articles

Nodal properties of eigenfunctions of a generalized buckling problem on balls

In this paper we are interested in the following fourth order eigenvalue problem coming from the buckling of thin films on liquid substrates: $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2 u+ \kappa ^2 u=-\lambda \Delta u &{}\text {in } u=\partial _r u= 0 &{}\text {on } \partial B_1, \end{array}\right. } \end{aligned}$$ where $$B_1$$ is the unit ball in $${\mathbb R}^N$$ . When $$\kappa > 0$$ is small, we show that the first eigenvalue is simple and the first eigenfunction, which gives the shape of the film for small displacements, is positive. However, when $$\kappa $$ increases, we establish that the first eigenvalue is not always simple and the first eigenfunction may change sign. More precisely, for any $$\kappa \in \mathopen ]0,+\infty \mathclose [$$ , we give the exact multiplicity of the first eigenvalue and the number of nodal regions of the first eigenfunction.

The effect of mechanical-driven volumetric change on instability patterns of bilayered soft solids.

If a soft solid is compressible, its volume changes with imposed loading. The extent of the volume change depends on its Poisson's ratio. Here, we study the effect of the mechanical-driven volumetric change on buckling and post-buckling behaviors of a hard thin film perfectly bound on a compliant substrate through the theoretical analysis and finite element method. Poisson's ratio of the substrate has been chosen to be in the range of -1 to 0.5, allowing its volume change during deformation. We find that Poisson's ratio cannot only shift the critical strain for the onset of buckling, but also affect the buckling modes. When Poisson's ratio of the substrate is close to -1, the surface instabilities of the thin film can be suppressed and delayed to large deformation. The present study demonstrates a new way to control surface instabilities of a bilayered system by changing Poisson's ratio of the material.

The Mexican hat effect on the delamination buckling of a compressed thin film

Because of the interaction between film and substrate, the film buckling stress can vary significantly, depending on the delamination geometry, the film and substrate mechanical properties. The Mexican hat effect indicates such interaction. An analytical method is presented, and related dimensional analysis shows that a single dimensionless parameter can effectively evaluate the effect.

Combined effect of pressure and geometric imperfection on buckling of stressed thin films on substrates

This paper deals with the combined effect of pressure and geometric imperfection on buckling of stressed thin films on a semi-infinite rigid substrate. Analytical approximate solutions are proposed by using the total potential energy of the system and the Rayleigh–Ritz’s method, and their stabilities are also determined. The present analytical approximate solutions agree very well with the numerical solutions obtained via the improved shooting method. The effect of pressure mismatch together with imperfection on the critical stress (above which the film buckles) and on the film-center deflection is characterized. It is found that, compared with the classical case of the Euler column buckling, the equilibrium solutions and critical stress of the film are strongly dependent on the linear combination of the pressure mismatch and imperfection amplitude.

Buckling of Elastic Thin Films on Compliant Substrates for Determination of Surface Parameters

The buckling of an elastic thin film on a compliant substrate under small perturbation in normal deflection and tangential displacement is studied with the effect of surface energy and interfacial slip. It is revealed that Poisson's ratio of the soft substrate and the interfacial slip can be neglected for film's thickness smaller than tens of nanometers. The effect of surface energy nevertheless renders the critical strain for the onset of wrinkling and the wave length dependent on the thickness highly nonlinearly. The potential of using buckling of thin films on compliant substrates for determination of surface parameters has been discussed.

The effect of interface adhesion on buckling and cracking of hard thin films

The physics behind the strain-released buckling patterns including telephone cords and straight-sided wrinkles with and without cracks, as experimentally observed in sputter-deposited Ti-Si-N thin films on Si substrates, is investigated with model-based simulations by varying the mechanical properties of the interface. Our calculations reveal that the location of the cracks depends on the normal stiffness, the interfacial toughness, and the normal strength of the cohesive interface. These properties determine the geometrical shape of the buckles such as width, wavelength, and deflection, and hence the local bending-induced tensile stresses. Buckling patterns with cracks at the apexes occur for low-stiffness interfaces as well as for high-stiffness interfaces with high toughness. On the other hand, cracks at the bottom of the buckles are more likely to occur for interfaces with high stiffness and low toughness. By using an elastic material model with a fracture criterion for brittle behavior, we demonstrate that the crack will follow the path where the bending-induced principal stress exceeds the flexural strength of the film.

Buckling analysis of stretchable ferroelectric thin film on elastomeric substrates

The thin stiff films on pre-stretched compliant substrates can form wrinkles, which can be controlled in micro and nanoscale systems to generate smart structures. Recently, buckled piezoelectric/ferroelectric nanoribbons have been reported to show an enhancement in the piezoelectric effect and stretchability, which can be applied in energy harvesting devices, sensors and memory devices instead of polymeric polyvinylidine fluoride (PVDF). This paper studies the buckling and post-buckling process of ferroelectric thin films bonded to the pre-stretched soft layer, which in turn lies on a rigid support. Nonlinear electromechanical equations for the buckling of thin piezoelectric plates are deduced and employed to model the ferroelectric film poled in the thickness direction. Two buckling modes are analyzed and discussed: partially de-adhered buckling and fully adhered buckling. Transition from one buckling mode to the other is predicted and the effect of piezoelectricity on the critical buckling condition of piezoelectric film is examined.

Edge-defined metric buckling of temperature-responsive hydrogel ribbons and rings

We report the temperature-responsive buckling of ribbon-like hydrogel thin films based on photo-crosslinkable copolymers of poly(N-isopropyl acrylamide) (PNIPAM) with pendent benzophenone units. Due to the finite resolution of the photo-lithographic patterning method—here dominated by the convergence of light from the illumination source—nominally homogeneous layers are always surrounded by a more highly swelling border with a lateral size scale similar to the film thickness. As one of the in-plane dimensions of the gel sheet is reduced to only several times its thickness, this edge effect serves as a sufficiently large imperfection to drive buckling of the otherwise metrically flat ribbon. Due to the symmetric geometry, the resulting geometry has essentially constant curvature of the mid-line of the ribbon, with a radius of curvature defined by its thickness and width. In this manner, we prepare thermally responsive overcurved rings that buckle into more highly coiled configurations as they swell, due to the mismatch in metric curvature relative to the geometric shape of the object. We consider the conditions for unfrustrated coiling of both circular and polygonal rings in terms of the number of self-crossings made upon wrapping the surface of a sphere, and demonstrate the fabrication of ‘self-cinching unknots’, i.e. coiled ribbons that can fasten themselves onto other objects upon deswelling, due to topological constraints.

Nonlinear analysis of compressed elastic thin films on elastic substrates: From wrinkling to buckle-delamination

Nonlinear buckling of elastic thin films on compliant substrates is studied by modeling and simulations to reveal the roles of pre-strain, elastic modulus ratio, and interfacial properties in morphological transition from wrinkles to buckle-delamination blisters. The model integrates an interfacial cohesive zone model with the Föppl–von Kármán plate theory and Green function method within the general framework of energy minimization. A kinetics approach is developed for numerical simulations. Subject to a uniaxial pre-strain, the numerical simulations confirm the analytically predicted critical conditions for onset of wrinkling and wrinkle-induced delamination, with which a phase diagram is constructed. It is found that, with increasing pre-strain, the equilibrium configuration evolves from flat to wrinkles, to concomitant wrinkles and buckle-delamination, and to an array of parallel straight blisters. The height and width of the buckle-delamination blisters can be approximately described by a set of scaling laws with respect to the pre-strain and interfacial toughness. Subject to an equi-biaxial pre-strain, the critical conditions are determined numerically to construct a similar phase diagram for the buckling modes. Moreover, by varying the pre-strain, modulus ratio, and interfacial toughness, a rich variety of equilibrium configurations are simulated, including straight blisters, and network blisters with or without wrinkles. These results provide considerable insight into diverse surface patterns in layered material systems as a result of the mechanical interactions between the film and the substrate through their interface, which suggests potential control parameters for designing specific surface patterns.

A validated energy approach for the post-buckling design of micro-fabricated thin film devices

The buckling of an elastic thin film is studied in the light of an energy minimization method. Specifically, a comprehensive treatment of the Rayleigh–Ritz method is presented. Detailed mechanical modelling, analytical and numerical derivation of stability criteria, physical interpretation of buckling shapes, numerical code implementation, and experimental validations of selected simulations are addressed.The thin film deflection is prescribed as a superposition of buckle functions to provide displacement field parameterizations involving trigonometric functions. An energy minimization procedure is applied to calculate the unknown coefficients to predict the buckling shape and amplitude. Critical buckling values representing the thresholds for instability transitions in the system are calculated from the eigenvalues of the Hessian of the potential energy.Comparison between simulation results and experimental measurements show the great potential of this method to predict thin film buckling. The validated model is exploited by derivation of a new design space for thin film fabrication where the post-buckling mechanics is controlled.

Redeposition of a straight-sided buckle under pressure.

The unilateral buckling of a stressed thin film on a substrate has been investigated theoretically in the framework of the Föppl-von Kármán theory of thin plates when an increasing overpressure is considered onto the upper free surface of the film. It is found that, depending on the initial stress in the film and overpressure, two scenarios of evolution may occur. The snap-through of the one-dimensional buckle leading to the full redeposition should take place for low values of the initial stress. When the initial stress exceeds a critical value, a partial redeposition of the buckle should proceed as the overpressure increases. A snap-through while the redeposition mechanism has taken place should also occur for higher values of the overpressure.

Effects of interface sliding on the formation of telephone cord buckles

Sliding at interface during thin film buckling was reported by recent atomistic simulations. A stability analysis under the Föppl-von Kármán plate theory is performed to investigate the effect of interface sliding on the transition of a straight-sided blister to the telephone cord buckle in biaxially compressed thin films on rigid substrates. It is shown that the critical stress and the wavelength of the telephone cord buckle significantly increase as the interface sliding is noticeable in comparison with the constant values derived from previous classic analysis without interface sliding. Our theoretic analysis is further validated by results from numerical simulations based on our recently developed continuum model of nonlinear buckling. The results indicate that the wavelength variation in the telephone cord buckle observed in experiments may be due to interface sliding.

A Simply Analytic Study of Buckled Thin Films on Compliant Substrates

Buckling of stiff thin films on compliant substrates enables many new applications, such as stretchable electronics. Song et al. [2008, “Buckling of a Stiff Thin Film on a Compliant Substrate in Large Deformation,” Int. J. Solids Struct., 45(10), pp. 3107–3121] developed a finite deformation theory to explain the buckled amplitude and wavelength very well. This theory not only accounts for finite geometry change, but also the finite strain and a nonlinear constitutive model for the substrate. To provide a better physical insight, this paper investigates those three effects, and shows that finite geometry change dominates in the finite deformation theory and the simplified analysis leads to results that agree well with experiments and the finite element method.

Facile Fabrication of Ordered Crystalline‐Semiconductor Microstructures on Compliant Substrates

Buckling and wrinkling of thin films on a compliant material has proved to be a resource in several applications, such as flexible electronics, thin‐film metrology and fabrication of tunable optical components. A versatile approach for the fabrication of two‐dimensional and linear arrays of buckled structures is demonstrated here using a stiff material, in the form of a nanomembrane, on a compliant substrate. The novelty of the fabrication process is that the substrates are strained by isotropic volume expansion in solvents. This work illustrates in detail the potential of our technology to fabricate ordered arrays of 3D structures on large‐area compliant substrates, with important implications for a large number of fields. Furthermore, this paper discusses the interesting interface chemistry and mechanics leading to controllable and reproducible fabrication of our 3D structures.

A simplified analysis on buckling of stressed and pressurized thin films on substrates

This paper is concerned with effect of mismatched pressure on buckling of stressed thin films on a semi-infinite rigid substrate. Analytical approximate solutions are established in explicit form by using total potential energy of the system and the Rayleigh–Ritz’s method, and their stabilities have also been determined. The dependences of the critical stress and film-center deflection on the mismatched pressure have been formulated. Results are compared with exact or numerical shooting solutions, and excellent agreements are observed for a large range of film-center deflection. These expressions are brief and can easily be used to derive the effects of various parameters on mechanical behavior of films.

Creation of freestanding wrinkled nano-films with desired deformation properties by controlling the surface morphology of a sacrificial layer

Various wrinkle patterns can be formed due to the buckling of a stiff thin film on a compliant substrate. However, most wrinkled films previously reported were fixed on a large deformable substrate and thereby the potential deformability of the film was mechanically constrained by the substrate. In this study, we developed a technique for forming various wrinkled structures on the surface of a sacrificial resin layer. Since the sacrificial layer can be subsequently removed with a solvent, freestanding wrinkled films are created using the sacrificial layer. We found that a wrinkled structure is formed on the surface of the layer by applying a compressive strain to the resin layer at the appropriate moment during the hardening process. The wrinkle pattern depends on the curing time and the timing of the straining in two in-plane orthogonal directions. In addition to conventional stripe and labyrinth patterns by simple uniaxial and equi-biaxial strains, respectively, it was found that independent biaxial strains induce interesting structures, such as an orthogonally ordered wrinkle pattern and a nonsymmetrical buckling structure, in which the stripe array produced by the first straining remains and many finer wrinkles appear in each stripe by the second straining in the orthogonal direction. We conducted tensile experiments for 300-nm-thick freestanding Cu films having these wrinkled structures. The wrinkled nano-films have a variety of mechanical properties: the stripe structure has extremely high deformability (more than 10% strain) and reversibility, the labyrinth structure shows planar isotropic deformation, and the nonsymmetrical buckling structure has an anisotropic modulus and strength. Finite element analysis on the wrinkle structures revealed that the local stress concentration dominates the fracture limits.

Inhomogeneous large deformation study of temperature-sensitive hydrogel

In this paper, inhomogeneous deformation of a temperature-sensitive hydrogel has been studied and analyzed under arbitrary geometric and boundary conditions. We present the governing equations and equilibrium conditions of an isothermal process based on the monophase gel field theory of hydrogel via a variational approach. We have adopted and implemented an explicit form of energy for temperature-sensitive hydrogel in a three-dimensional finite element method (FEM) using a user-supply subroutine in ABAQUS. For verification purpose, a few numerical results obtained by the proposed approach are compared with existing experimental data and analytical solutions. They are all in good agreement. We also provide several examples to show the possible applications of the proposed method to explain various complex phenomena, including the bifurcation, buckling of membrane, buckling of thin film on compliant substrate and the opening and closure of flowers.

Detailed modelling of delamination buckling of thin films under global tension

Tensile specimens of metal films on compliant substrates are widely used for determining interfacial properties. These properties are identified by the comparison of experimentally observed delamination buckling and a mathematical model which contains the interface properties as parameters. The current two-dimensional models for delamination buckling are not able to capture the complex stress and deformation states arising in the considered uniaxial tension test in a satisfying way. Therefore, three-dimensional models are developed in a multi-scale approach. It is shown that, for the considered uniaxial tension test, the buckling and associated delamination process are initiated and driven by interfacial shear in addition to compressive stresses in the film. The proposed model is able to reproduce all important experimentally observed phenomena, like cracking stress of the film, film strip curvature and formation of triangular buckles. Combined with experimental data, the developed computational model is found to be effective in determining interface strength properties.

Surface Effects on the Mechanical Behavior of Buckled Thin Film

The buckling of thin films with natural nonlinearity can provide a useful tool in many applications. In the present paper, the mechanical properties of controllable buckling of thin films are investigated by accounting for both geometric nonlinearity and surface effects at nanoscale. The effects of surface elasticity and residual surface tension on both static and dynamic behaviors of buckled thin films are discussed based on the surface-layer-based model. The dynamic design strategy for buckled thin films as interconnects in flexible electronics is proposed to avoid resonance in a given noise environment based on the above analysis. Further discussion shows that the thermal and piezoelectric effects on mechanical behavior of buckled thin film are equivalent to that of residual surface tension.

Buckle-driven delamination of hydrophobic micro-, nano-, and heterostructured membranes without a sacrificial layer

A fabrication method, based on thin film buckling, is demonstrated to form unique membranes that can be used for applications in optics and biosensing. This method should be applicable to a variety of material systems, which, along with its simplicity and compatibility with different film architectures, allows for widespread implementation.