Rigid Stamp Research Articles

Effect of residual stress on replication fidelity with nanoimprint

Relief of residual stress in an imprinted polymer may affect the replication fidelity by leading to recovery. The level of stress induced in the polymer depends on the method of imprint. For example, a “soft” imprint with an elastomeric stamp uses capillary forces to fill the cavities whereas a “hard” imprint with a rigid stamp relies on external pressure. To study the effect of residual stress after imprint, both methods are applied with different imprint times to vary the level of residual stress, as the stress remaining relaxes with imprint time. To visualize the residual stress a temperature treatment is performed after imprint. This temperature treatment allows recovery within a convenient experimental time. A comparison of the shape of the imprinted structures before and after temperature treatment clearly shows that with a hard imprint at short imprint times a considerable amount of stress remains in the polymer, in particular when the residual layer is thin and the imprinted stamp structures are wide. With a soft imprint residual stress is not evident. Similar results are obtained with a thermoplastic material and a crosslinking material (SU-8); however, with the latter recovery remains limited due to a decrease of mobility during crosslinking. Residual stress is of major importance for the replication fidelity with short imprint processes.

Closed-form solution of the two-dimensional sliding frictional contact problem for an orthotropic medium

Abstract The two-dimensional frictional contact problem of a rigid stamp sliding over an orthotropic medium is considered. The coordinate system is chosen such that it is aligned with the principal axes of orthotropy. It is parallel and perpendicular to the contact surface which is located along the x 1 axis. It is assumed that the condition of Coulomb friction prevails on the contact area. The two-dimensional half-plane problem is formulated using the Fourier integral transform method and the analytical formulation of the contact problem is reduced to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure. The singular integral equation is solved analytically utilizing the Jacobi polynomials. With the application of the results to the crack initiation in an orthotropic medium in mind, the main emphasis in the study has been on the investigation of the singular nature of the stress state at either end of the flat stamp and on the determination of the contact stresses based on orthotropic material parameters. The present study provides the analytical solution of the contact stresses in terms of orthotropic material parameters, the coefficient of friction and the spatial coordinates. The strength of the singularities and the stress intensity factors at both ends of the stamp are also found in terms of the orthotropic material parameters and the coefficient of friction. This study reveals that orthotropic material parameters and the coefficient of friction have a great effect on the strength of stress singularities and distribution of the contact stresses. Adjusting these parameters will reduce these stresses that may have a bearing on the failure of the orthotropic medium. The results of this study will provide benchmark results for finite element analysts and stress engineers.

Receding contact problem for a coated layer and a half-plane loaded by a rigid cylindrical stamp

In this study, the frictionless contact problem between a coated layer and a half-plane is considered when they are pressed by a rigid cylindrical stamp. Upon loading, two unknown contact widths and contact pressures occur on the contact areas the advanced contact area between the stamp and the coating; and the receding contact area between the layer and the half-plane. This problem is reduced to two singular integral equations by using the Fourier transform and applying the boundary conditions of the problem. The numerical solution of the system of singular integral equations is obtained by applying the Gauss–Chebyshev integration formulas.

Contact problem for an orthotropic plane with a slit

An exact solution of a mixed boundary-value problem for an orthotropic plane with a slit on one of the principal directions of orthotropy is constructed in the case where an absolutely rigid stamp with a flat base is pressed into one face of the slit. Using the method of discontinuous solutions for the equations of elasticity theory, the problem stated is reduced to the Riemann problem for two functions, and a closed solution of this problem is constructed. Simple expressions for the basic mechanical characteristics, such as the opening of the slit, contact stresses under the die, and the intensity factor of breaking stresses at the end points of the slit are obtained. Solutions of similar problems for an orthotropic plane are found when the stamp reaches only one or both end points of the slit, as well as for a semi-infinite slit. A numerical analysis is carried out.

Two-dimensional contact mechanics problems involving inhomogeneously elastic solids split into three distinct layers

This paper investigates the frictionless two-dimensional contact problem of an inhomogeneously elastic material under a rigid punch. The inhomogeneous solid is deemed to comprise three distinct regions which represent a homogeneously elastic coating and substrate joined together by a functionally graded transition layer (interlayer) whose shear modulus depends exponentially on the vertical coordinate.We propose closed form solutions for the horizontal and vertical displacements of the solid which are analytic if the contact pressure is known exactly. These solutions are further used to derive a fast and efficient iterative algorithm from which the contact footprint resulting from the rigid stamp problem may be computed. A selection of numerical results are then presented using this method and it is found that our model compares well with those of other authors in the two particular limiting cases considered here. We then investigate the effects of material inhomogeneity and coating thickness on the cylindrical stamp problem and it is found that the maximum principal stress is highly dependent on the thickness and mechanical properties of the layer. In particular, it is found that the maximum principal stress that occurs in a material with a hard coating may be reduced by increasing the thickness of the transition layer whilst lower stresses are achieved in materials with soft coatings by decreasing interlayer thickness.

Frictional Hertzian contact between a laterally graded elastic medium and a rigid circular stamp

This study investigates the problem of sliding frictional contact between a laterally graded elastic medium and a rigid circular stamp. Analytical and computational methods are developed to evaluate the contact stresses. In the analytical formulation, spatial variation in the shear modulus of the graded medium is represented by an exponential function, and Poisson’s ratio is taken as a constant. Coulomb’s dry friction law is assumed to hold within the contact area. The two-dimensional plane elasticity problem is formulated utilizing Fourier transforms, and the resulting Cauchy-type singular integral equation of the second type is solved by applying an expansion–collocation technique. The finite element method is used in the computational analysis of the contact problem. In the finite element model, continuous variation of the shear modulus is taken into account by specifying this property at the centroid of each finite element. The finite element-based solution procedure is verified by making comparisons to the results obtained through the analytical method. Numerical results generated for the laterally graded medium with an exponential variation in the shear modulus illustrate the influences of lateral gradation and coefficient of friction upon the contact stress distributions. The capability of the proposed finite element method is further demonstrated by providing numerical results for a laterally graded medium whose shear modulus is represented by a power function.

Contact problem for regular hexagon weakened with full-strength hole

A problem of the plane elasticity theory is addressed for a doubly connected body with an external boundary of the regular hexagon shape and with a 6-fold symmetric hole at the center. It is assumed that all the six sides of the hexagon are subjected to uniform normal displacements via smooth rigid stamps, while the uniformly distributed normal stress is applied to the internal hole boundary. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili’s complex potentials and the shape of the hole contour are determined from the condition that the circumferential normal stress is constant along the hole contour. Numerical results are given and shown in relevant graphs.

Effects of the moving speed of a rigid stamp on contact behaviors of anisotropic materials based on real fundamental solutions

The paper studies contact problem of a rigid stamp moving at a constant speed over the surface of anisotropic materials. The solution method is based on Galilean transformation, Fourier transform and singular integral equation. The stated mixed boundary value problem is reduced to a Cauchy type singular integral equation based on real fundamental solutions, which is solved exactly in the case of a rigid flat or cylindrical stamp. Explicit expressions for various stresses are obtained in terms of elementary functions. In particular, explicit formula is derived to determine the unknown contact region for the cylindrical stamp. For a flat stamp, detailed calculations are provided to show the influences of dimensionless moving speed on the normal and in-plane stress. For a cylindrical stamp, the effects of dimensionless moving speed, the mechanical loading and the radius on the contact region, the normal and in-plane stress are analyzed in detail.

Finite Element Method Simulation of the Molding Process for Thermal Nano-Imprint Lithography

We made a numerical study on the deformation of a viscoelastic polymethyl methacrylene (PMMA) resist when a rigid SiO2 stamp with a rectangular line pattern is imprinted into the PMMA resist for thermal nano-imprint lithography (NIL). The stress distribution in the polymer resist during the molding process is calculated by a finite element method (FEM). Our simulation results reveal that the asymmetric von Mises stress is distributed over the polymer around the external line, which seems to be due to the squeezing flow under the flat space. The stress seems to be concentrated at the sidewall close to the centerline of the whole structure. Our simulation also reveals that a micro gap is formed between the replicated structure and the outer wall of the mold.

Explicit solution of the frictional contact problem of anisotropic materials indented by a moving stamp with a triangular or parabolic profile

The frictional contact problem of anisotropic materials under a moving rigid stamp is solved exactly. Inside the contact region, the Coulomb friction law is applied. Both Galilean transformation and Fourier transform are employed to get the appropriate fundamental solutions, which can lead to real solutions of physical quantities no matter whether the eigenvalues are real or complex. The complicated mixed boundary value problem is converted to singular integral equations of the second kind, which are solved analytically in terms of elementary functions for either a triangular or a parabolic stamp. Explicit formulae of surface stresses are obtained. Numerical analyses are performed in detail to reveal the surface damage mechanism. It is also found that in the frictionally moving contact problem, the friction coefficient has a more important role than the moving velocity.

Transfer Printing of Nanoplasmonic Devices onto Flexible Polymer Substrates from a Rigid Stamp

Plasmonic color filters and polarizers were produced using nanotransfer printing to create aluminium nanostructures, as small as 75nm, on a polycarbonate sheet measuring 10mm × 12mm. Plasmonic filters showed good agreement with simulations.

On one contact problem of plane elasticity for a doubly connected domain: application to a hexagon

The paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a regular hexagon boundary, and the internal boundary is the required full-strength hole including the origin of coordinates. Hexagon’s two vertices are laid at the axis Oy, and the middle points of its two opposite sides are laid at the axis Ox. This full-strength hole is cycle symmetric. It is assumed that to every link of the broken line of the outer boundary of the given body are applied absolutely smooth rigid stamps with rectilinear bases, which are under action of the force P that applies to their middle points. There is no friction between the surface of given elastic body and stamps. The unknown full-strength contour is free from outer actions. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body) and unknown parts of its boundary are determined under the condition that the tangential normal moment arising at it takes a constant value. Such holes are called full-strength holes. Numerical analysis are also performed and the corresponding graphs are constructed.

The Rayleigh wave field in mixed problems for a half-plane

The note develops an approximate approach to mixed boundary value problems in linear elasticity starting from an explicit asymptotic model for the Rayleigh surface wave. It is demonstrated that the original vector mixed problem may be reduced to a scalar problem for the Laplace equation. As an illustration, the steady-state motion of a rigid stamp is analysed. Comparison of asymptotic and exact results is presented.

Frictional contact problem of a rigid stamp and an elastic layer bonded to a homogeneous substrate

In this study, the frictional contact problem for a layer bonded to a homogeneous substrate is considered according to the theory of elasticity. The layer is indented by a rigid cylindrical stamp which is subjected to concentrated normal and tangential forces. The friction between the layer and the stamp is taken into account. The problem is reduced to a singular integral equation of the second kind in which the contact pressure function and the contact area are the unknown by using integral transform technique and the boundary conditions of the problem. The singular integral equation is solved numerically using both the Jacobi polynomials and the Gauss–Jacobi integration formula, considering equilibrium and consistency conditions. Numerical results for the contact pressures, the contact areas, the normal stresses, and the shear stresses are given, for both the frictional and the frictionless contacts.

The Influence of Grain Size and Grain Size Distribution on Sliding Frictional Contact in Laterally Graded Materials

The sliding frictional contact problem for a laterally graded half-plane has been considered. Two finite element (FE) models, in macro and micro scales have been developed to investigate the effective parameters in contact mechanics of laterally graded materials loaded by flat and triangular rigid stamps. In macro scale model, the laterally graded half-plane is discretized by piecewise homogeneous layers for which the material properties are specified at the centroids by Mori-Tanaka method. In micro scale model, functionally graded material (FGM) structure has been modeled as ideal solid quadrant particles which are spatially distributed in a homogeneous matrix. Boundary conditions and loading is the same in both models. The microstructure has modeled as rearrangement and sizes changing of particles are possible to provide the possibility of crack nucleation investigation in non-singular regions. Analyses and comparison of the results showed that micro and macro scale results are in very good agreement. Also, increasing the grains aspect ratio and using optimum distribution of grains decrease stress distribution roughness on the surface. Therefore, the possibility of surface cracking far from stamp’s edges decreased.

Reusable Stamps for Printing Sub‐100 nm Patterns of Functional Nanoparticles

The production of reusable stamps for the creation of high-resolution patterns of functional nanoparticles is demonstrated. A patterned stamp is cleaned by removing nanoparticles in the recessed regions of the stamp, using a UV-curable adhesive, before moving on to the next cycle of printing with the stamp. A high-resolution patterning technique using rigid stamps with a low surface energy and the repeated use of the reusable stamps for defining sub-100 nm scale features is demonstrated. Color patterning has also been performed using just one color per stamp, then by the use of one stamp for multiple colors. Detailed facts of importance to specialist readers are published as ”Supporting Information”. Such documents are peer-reviewed, but not copy-edited or typeset. They are made available as submitted by the authors. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

Molecular Scale Modeling of Polymer Imprint Nanolithography

We present the results of large-scale molecular dynamics simulations of two different nanolithographic processes, step-flash imprint lithography (SFIL), and hot embossing. We insert rigid stamps into an entangled bead-spring polymer melt above the glass transition temperature. After equilibration, the polymer is then hardened in one of two ways, depending on the specific process to be modeled. For SFIL, we cross-link the polymer chains by introducing bonds between neighboring beads. To model hot embossing, we instead cool the melt to below the glass transition temperature. We then study the ability of these methods to retain features by removing the stamps, both with a zero-stress removal process in which stamp atoms are instantaneously deleted from the system as well as a more physical process in which the stamp is pulled from the hardened polymer at fixed velocity. We find that it is necessary to coat the stamp with an antifriction coating to achieve clean removal of the stamp. We further find that a high density of cross-links is necessary for good feature retention in the SFIL process. The hot embossing process results in good feature retention at all length scales studied as long as coated, low surface energy stamps are used.

A solution method for the sub-surface stresses and local deflection of a semi-infinite inhomogeneous elastic medium

This paper proposes analytical Fourier series solutions (based on the Airy stress function) for the local deflection and subsurface stress field of a two-dimensional graded elastic solid loaded by a pre-determined pressure distribution. We present a selection of numerical results for a simple sinusoidal pressure which indicates how the inhomogeneity of the solid affects its behaviour. The model is then adapted and used to derive an iterative algorithm which may be used to solve for the contact half width and pressure induced from contact with a rigid punch. Finally, the contact of a rigid cylindrical stamp is studied and our results compared to those predicted by Hertzian theory. It is found that solids with a slowly varying elastic modulus produce results in good agreement with those of Hertz whilst more quickly varying elastic moduli which correspond to solids that become stiffer below the surface give rise to larger maximum pressures and stresses whilst the contact pressure is found to act over a smaller area.

A double receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, the axisymmetric problem of a frictionless double receding contact between a rigid stamp of axisymmetric profile, an elastic functionally graded layer and a homogeneous half space is considered. The graded layer is modelled as a nonhomogeneous medium with an isotropic stress-strain law. Assuming the double contact between the bodies to be frictionless, only compressive normal tractions can be transmitted in each contact area while the rest of the surface is free of tractions. Using an appropriate integral transform, the axisymmetric elasticity equations are converted analytically into a system of singular integral equations where the unknowns are the pressures and the radii of the receding contact area in the two contact zones. The global equilibrium conditions are supplemented to solve the problem. The singular integral equations are solved numerically using orthogonal Chebyshev polynomials. An iterative scheme based on the Newton–Raphson method is employed to obtain the receding contact radii and pressures that satisfy the equilibrium conditions. The main objectives of the paper are to study the effect of the nonhomogeneity parameter, the thickness of the graded layer and the magnitude of the applied load on the contact pressures, the radii of the receding contact zones and the indentation for the case of a spherical rigid punch.

Fabrication of Large Area, 70 nm Pitch Nanograting Patterns by Nanoimprint Lithography Using Flexible Polymer Stamp

Flexible polymer stamps are considered as an attractive alternative to rigid, brittle and expensive stamps made of inorganic materials because of their low cost and ease of fabrication. In this paper, we present a nanoimprint process to fabricate large area, high-resolution nanograting patterns using flexible polymer stamp made from fluoropolymer. The flexibility and low surface energy of polymer stamp provide a clean release without fracture or deformation of the stamp and of the replicated nanograting. Large-area, high-density nanograting patterns with good shape homogeneity and size uniformity have been successfully fabricated using the flexible polymer stamp with advantages of its good conformal contact and low adhesion. Using flexible polymer stamps can resolve many serious issues in NIL and therefore can bring it to real industrial applications.