Coupled annealing temperature and layer thickness effect on strengthening mechanisms of Ti/Ni multilayer thin films

A systematic study was performed on mechanical and microstructural properties of Ti/Ni multilayers with layer thickness from 200 nm to 6 nm and annealing temperature from room temperature to 500 °C. Based on the observed hardness evolution, a coupled layer-thickness and annealing-temperature dependent strengthening mechanism map is proposed. For as-deposited films, the deformation behavior follows the traditional trend of dislocation mediated strengthening to grain boundary mediated softening with decreasing layer thickness. For annealed films, grain boundary relaxation is considered to be the initial strengthening mechanism with higher activation temperature required for thicker layers. Under further annealing, solid solution hardening, intermetallic precipitation hardening, and fully intermixed alloy structure continue to strengthen the thin layered films, while recrystallization and grain-growth lead to the eventual softening of thick layered films. For the films with intermediate layer thickness, a strong orientation dependent hardness behavior is exhibited under high temperature annealing due to mechanism switch from grain growth softening to intermetallic precipitation hardening when changing the loading orientation from perpendicular to parallel to the layer interfaces.     

Thickness effects in polycrystalline thin films: Surface constraint versus interior constraint

The uniaxial tension behavior of polycrystalline thin films, in which all grain boundaries (GBs) are penetrable by dislocations, is investigated by two-dimensional discrete dislocation dynamics (DDD) method with a penetrable dislocation-GB interaction model. In order to study thickness effect on the tensile strength of thin films with and without surface treatment, three types of thin films are comparatively considered, including the thin films without surface treatment, with surface passivation layers (SPLs) of nanometer thickness and with surface grain refinement zones (SGRZs) consisting of nano-sized grains. Our results show that thickness effects and their underlying dislocation mechanisms are quite distinct among different types of thin films. The thicker thin films without surface treatment are stronger than the thinner ones; however, opposite thickness effects are captured in the thin films with SPLs or SGRZs. Moreover, the underlying dislocation mechanisms of the same thickness effects of thin films with SPLs and SGRZs are different. In the thin films with SPLs, the thickness effect is caused by the sharp increase of dislocation density near the film-passivation interface, while it is mainly due to the sharp decrease of dislocation density within the refined surface grains of the thin films with SGRZs. No matter in what type of thin films, thickness effect gradually disappears when the number of grains in the thickness direction is large enough. Our analysis reveals that general mechanism of those thickness effects lies in the competition between the exterior surface-constraint and interior GB-constraint on gliding dislocations.     

Size effects in indentation response of thin films at the nanoscale: A molecular dynamics study

The indentation response of Ni thin films of thicknesses in the nanoscale was studied using molecular dynamics simulations with embedded atom method (EAM) interatomic potentials. A series of simulations were performed in films in the [1 1 1] orientation with thicknesses varying from 4 to 12.8 nm. The study included both single crystal films and films containing low angle grain boundaries perpendicular to the film surface. The simulation results for single crystal films show that as film thickness decreases larger forces are required for similar indentation depths but the contact stress necessary to emit the first dislocation under the indenter is nearly independent of film thickness. The low angle grain boundaries can act as dislocation sources under indentation. The mechanism of preferred dislocation emission from these boundaries operates at stresses that are lower as the film thickness increases and is not active for the thinnest films tested. These results are interpreted in terms of a simple model.     

Tensile response of passivated films with climb-assisted dislocation glide

The tensile response of single crystal films passivated on two sides is analysed using climb enabled discrete dislocation plasticity. Plastic deformation is modelled through the motion of edge dislocations in an elastic solid with a lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation incorporated through a set of constitutive rules. The dislocation motion in the films is by glide-only or by climb-assisted glide whereas in the surface passivation layers dislocation motion occurs by glide-only and penalized by a friction stress. For realistic values of the friction stress, the size dependence of the flow strength of the oxidised films was mainly a geometrical effect resulting from the fact that the ratio of the oxide layer thickness to film thickness increases with decreasing film thickness. However, if the passivation layer was modelled as impenetrable, i.e. an infinite friction stress, the plastic hardening rate of the films increases with decreasing film thickness even for geometrically self-similar specimens. This size dependence is an intrinsic material size effect that occurs because the dislocation pile-up lengths become on the order of the film thickness. Counter-intuitively, the films have a higher flow strength when dislocation motion is driven by climb-assisted glide compared to the case when dislocation motion is glide-only. This occurs because dislocation climb breaks up the dislocation pile-ups that aid dislocations to penetrate the passivation layers. The results also show that the Bauschinger effect in passivated thin films is stronger when dislocation motion is climb-assisted compared to films wherein dislocation motion is by glide-only.     

A continuum model for size-dependent deformation of elastic films of nano-scale thickness

Ultra-thin elastic films of nano-scale thickness with an arbitrary geometry and edge boundary conditions are analyzed. An analytical model is proposed to study the size-dependent mechanical response of the film based on continuum surface elasticity. By using the transfer-matrix method along with an asymptotic expansion technique of small parameter, closed-form solutions for the mechanical field in the film is presented in terms of the displacements on the mid-plane. The asymptotic expansion terminates after a few terms and exact solutions are obtained. The mid-plane displacements are governed by three two-dimensional equations, and the associated edge boundary conditions can be prescribed on average. Solving the two-dimensional boundary value problem yields the three-dimensional response of the film. The solution is exact throughout the interior of the film with the exception of a thin boundary layer having an order of thickness as the film in accordance with the Saint-Venant’s principle.     

Nanoindentation of polymeric thin films with an interfacial force microscope

The mechanical properties of interphase regions at bi-material interfaces can be quite different from the surrounding bulk materials. For composite materials, this interphase region is usually thin but plays an important role in their overall mechanical properties. Nanoindentation has become a commonly used experimental technique for measuring the mechanical properties of materials, especially when one of the dimensions is small. However, the extraction of reduced elastic modulus from the nanoindentation of thin films on substrates can pose challenges due to the influence of the substrate. In this study, the nanoindentation of thin films on substrates has been examined with a view to extracting the reduced modulus of thin polymer films.Thin films of (3-aminopropyl)triethoxysilane (C₉H₂₃NO₃Si, γ-APS) were deposited on silicon. An interfacial force microscope (IFM) was used to indent the γ-APS films. The effect of the substrate was studied by considering two very different thicknesses ( and ). The nanoindentation data were analyzed via contact mechanics theories and a finite element analysis that incorporated surface interactions. The analyses showed that nanoindentation experiments can provide reliable values of film modulus when the film is very different from the substrate. It was found that the commonly used rule of thumb that the indentation depth should be less than 10% of the thickness did not eliminate substrate effects for a wide range of material combinations. Instead, it is proposed that the contact radius should be less than 10% of the thickness so that contact mechanics theories for monolithic materials can be used without considering the presence of the substrate. The modulus of γ-APS polymer films and the surface energy between the tungsten tip of the IFM and γ-APS films were extracted and were related to their cure. A completely cured thick γ-APS film had a reduced modulus of . This value falls in the usual range for polymers due to the amorphous nature of the γ-APS films.     

Fracture mechanics analysis of thin coatings under plane-strain indentation

A combined experimental/analytical work is carried out to elucidate the fracture resistance of a thin, hard coating bonded to a semi-infinite substrate due to indentation by a cylindrical surface. The bending of the coating under the softer substrate induces concentrated tensile stress regions at the lower and upper surfaces of the coating, from which cracks may ensue. The evolution of such damage in a model transparent system (glass/polycarbonate) is viewed in situ from below and from the side of the specimen. The critical load needed to initiate a crack on the lower coating surface generally increase proportionally to the coatings thickness, d. An interesting departure from this trend occurs for thin coatings, where the fracture load, although marred by a large scatter, increases somewhat with decreasing d. The fracture data for the upper coating surface are limited to relatively thick coatings due to the recurrence of premature failure from the coating edges. The behavior in this range is similar to that for the lower surface crack, albeit with an order of magnitude greater fracture resistance.A fracture mechanics analysis in conjunction with FEM is performed to elucidate the stress intensity factors responsible for crack propagation. A crack normal to the coating surface is assumed to emanate either from the lower or upper surface of the coating. A major feature of the solution is the occurrence of a bending-induced compression stress field over a region ahead of the crack tip. This effect, which become more dominant as the ratio between the contact length and the coating thickness is increased, tends to delay the onset of crack propagation, especially for the lower surface crack. Consequently, in applications associated with large indenters, thin and/or tough coatings and stiff substrates, cracking from the upper coating surface may precede that from the lower surface. An interesting feature of this crack shielding mechanism is that when the coating surface contains a distribution of flaws rather than a single crack, small flaws in this population may be more detrimental than large ones. Incorporation of these aspects into the analysis leads to a good correlation with the test results. In the special case of line loading, which constitutes a lower bound for the critical loads, a closed-form, approximate solution for the stress intensity factors or the critical loads are obtained.Plane-strain indentation, although less common than spherical indentation, allows for characterizing the fracture resistance of opaque films through observation from the specimen edge. This approach is not easily implemented to thin films (i.e., less than about a hundred microns), however.     

The effect of out-of-plane restraint on plane-stress solutions near straight boundaries

The existence of out-of-plane displacement restraint where thin photoelastic plates are bonded to relaively rigid boundaries or inclusions will induce transverse stress components whose presence can cause the in-plane stresses to deviate significantly from the plane-stress solution. The extended generalized planestress formulation, which was developed in a previous paper to study the effect of out-of-plane restraint on the through-the-thickness averages of the in-plane stresses and displacements in thin plates, is applied here to obtain correction factors by which the photoelastically determined in-plane stress components at a straight restrained boundary can be multiplied to recover the desired twodimensional solution. Cases of mechanical and thermal loadings which vary sinusoidally in the direction parallel to the restrained boundary are treated. The stress-field alterations due to out-of-lane restraint are shown to depend strongly on Poisson's ratiov and the ratioL/H of the half-wavelength of the load variation to the plate thickness, and can be minimized by choosing a photoelastic material with the smallest values ofv andH possible. The simple asymptotic form of the stress-field alterations for large values ofL/H has enabled us to prescribe simple correction factors in this range. Finally, it is observed that, although the normal stress component which is parallel to the restrained boundary is altered to a greater degree than are the other in-plane stress components, the effect on this stress component penetrates into the plate a distance of only about one plate thickness, whereas the effect on the other components penetrates a distance of approximately one half-wavelength of the load variation.     

Size-dependent response of ultra-thin films with surface effects

A modified continuum model of elastic films with nano-scale thickness is proposed by incorporating surface elasticity into the conventional nonlinear Von Karman plate theory. By using Hamilton’s principle, the governing equations and boundary conditions of the ultra-thin film including surface effects are derived within the Kirchhoff’s assumption, where the effects of non-zero normal stress and large deflection are taken into account simultaneously. The present model is then applied to studying the bending, buckling and free vibration of simply supported micro/nano-scale thin films in-plane strains and explicit exact solutions can be obtained for these three cases. The size-dependent mechanical behavior of the thin film due to surface effects is well elucidated in the obtained solutions.     

Discrete dislocation dynamics simulations to interpret plasticity size and surface effects in freestanding FCC thin films

Strong size effects have been experimentally observed when microstructural features approach the geometric dimensions of the sample. In this work experimental investigations and discrete dislocation analyses of plastic deformation in metallic thin films have been performed. Columnar grains representative of the film microstructure are here considered. Simulations are based on the assumptions that sources are scarcely available in geometrically confined systems and nucleation sites are mainly located at grain boundaries. Especially, we investigated the influence on the mesoscopic constitutive response of the two characteristic length scales, i.e., film thickness and grain size. The simulated plastic response qualitatively reproduces the experimentally observed size effects while the main deformation mechanisms appear to be in agreement with TEM analyses of tested samples. A new interpretation of size scale plasticity is here proposed based on the probability of activating grain boundary dislocation sources. Moreover, the key role of a parameter such as the grain aspect ratio is highlighted. Finally, the unloading behavior has been investigated and a strong size dependent Bauschinger effect has been found. An interpretation of these phenomena is proposed based on the analysis of the back stress distribution within the samples.     

A theory of grain boundaries that accounts automatically for grain misorientation and grain-boundary orientation

This work is an attempt to answer the question:

Is there a physically natural method of characterizing the possible interactions between the slip systems of two grains that meet at a grain boundary—a method that could form the basis for the formulation of grain-boundary conditions?

Here we give a positive answer to this question based on the notion of a Burgers vector as described by a tensor field G on the grain boundary [Gurtin, M.E., Needleman, A., 2005. Boundary conditions in small-deformation single-crystal plasticity that account for the Burgers vector. J. Mech. Phys. Solids 53, 1-31]. We show that the magnitude of G can be expressed in terms of two types of moduli: inter-grain moduli that characterize slip-system interactions between the two grains; intra-grain moduli that for each grain characterize interactions between any two slip systems of that grain.We base the theory on microscopic force balances derived using the principle of virtual power, a version of the second law in the form of a free-energy imbalance, and thermodynamically compatible constitutive relations dependent on G and its rate. The resulting microscopic force balances represent flow rules for the grain boundary; and what is most important, these flow rules account automatically—via the intra- and inter-grain moduli—for the relative misorientation of the grains and the orientation of the grain boundary relative to those grains.     

Investigation of microstructure evolution during self-annealing in thin Cu films by combining mesoscale level set and ab initio modeling

Microstructure evolution in thin Cu films during room temperature self-annealing is investigated by means of a mesoscale level set model. The model is formulated such that the relative, or collective, influence of anisotropic grain boundary energy, mobility and heterogeneously distributed stored energy can be investigated. Density functional theory (DFT) calculations are performed in the present work to provide the variation of grain boundary energy for different grain boundary configurations. The stability of the predominant (111) fiber texture in the as-deposited state is studied as well as the stability of some special low-Σ grain boundaries. Further, the numerical model allows tracing of the grain size distribution and occurrence of abnormal grain growth during self-annealing. It is found that abnormal grain growth depends mainly on the presence of stored energy variations, whereas anisotropic grain boundary energy or mobility is insufficient to trigger any abnormal growth in the model. However, texture dependent grain boundary properties, mobility in particular, contribute to an increased content of low-Σ boundaries in the annealed microstructure. The increased presence of such boundaries is also promoted by stored energy variations. In addition, if the stored energy variations are sufficient the coexisting (111) and (001) texture components in the as-deposited state will evolve into a (001) dominated texture during annealing. Further, it is found that whereas stored energy variations promote the stability of the (001) texture component, anisotropic grain boundary energy and mobility tend to work the other way and stabilize the (111) component at the expense of (001) grains.     

Large-scale spanwise periodicity in a turbulent boundary layer induced by highly ordered and directional surface roughness

The effect of converging–diverging riblet-type surface roughness (riblets arranged in a ‘herringbone’ pattern) are investigated experimentally in a zero pressure gradient turbulent boundary layer. For this initial parametric investigation three different parameters of the surface roughness are analysed in detail; the converging–diverging riblet yaw angle α, the streamwise fetch or development length over the rough surface F_x and the viscous-scaled riblet height h⁺. It is observed that this highly directional surface roughness pattern induces a large-scale spanwise periodicity onto the boundary layer, resulting in a pronounced spanwise modification of the boundary layer thickness. Hot-wire measurements reveal that above the diverging region, the local mean velocity increases while the turbulent intensity decreases, resulting in a thinner overall boundary layer thickness in these locations. The opposite situation occurs over the converging region, where the local mean velocity is decreased and the turbulent intensity increases, producing a locally thicker boundary layer. Increasing the converging–diverging angle or the viscous-scaled riblet height results in stronger spanwise perturbations. For the strongest convergent–divergent angle, the spanwise variation of the boundary layer thickness between the diverging and converging region is almost a factor of two. Such a large variation is remarkable considering that the riblet height is only 1% of the unperturbed boundary layer thickness. Increasing the fetch seems to cause the perturbations to grow further from the surface, while the overall strength of the induced high and low speed regions remain relatively unaltered. Further analysis of the pre-multiplied energy spectra suggests that the surface roughness has modified or redistributed the largest scale energetic structures.     

Fracture in Microscale SU-8 Polymer Thin Films

In this work, the fracture of spin coated SU-8 epoxy thin films was investigated under mode I loading using in situ optical experiments on specimens with double edge notched tensile geometry. A method was developed to fabricate 3 μm thick SU-8 films with tapered Chevron type notches using a combination of electron beam and ultra-violet lithography techniques. Subsequently, through speckle patterning under tensile loading, the local deformation fields around the crack tip were extracted using digital image correlation. Since the notches were tapered through the thickness, a crack nucleated from them and grew stably until it spanned the entire thickness before propagating unstably leading to catastrophic failure. As SU-8 underwent brittle fracture with no evidence of a large process zone, the critical energy release rate, J _{I C} was computed from deformation fields, and was found to be 106.6 ± 12.03 J /m ². As the film thickness was small compared to lateral dimensions, assuming plane stress conditions, the critical stress intensity factor was calculated as 0.57 ± 0.03 MPa\(\sqrt {m}\). Furthermore, to assess the validity of the experimental method, a finite element simulation on the exact specimen geometry was conducted with experimentally evaluated far field displacement boundary conditions. The strain fields and J-integral value obtained from the simulation were in good agreement with the experimental results, implying the validity of the in situ experimental method proposed given the challenges of small scale specimens. Furthermore, using fractography and optical imaging it was confirmed that the unstable crack propagation started once the crack front reached full thickness, thereby providing sharp crack at the time of failure, which is necessary for brittle materials for valid fracture toughness experiment. It is expected that the proposed methods of specimen preparation and fracture experiments on microscale polymer thin films can be used on other materials.     

Distribution of dislocation source length and the size dependent yield strength in freestanding thin films

A method is proposed to estimate the size-dependent yield strength of columnar-grained freestanding thin films. The estimate relies on assuming a distribution of the size of Frank-Read sources, which is then translated into a log-normal distribution of the source strength, depending on film thickness, grain size and theoretical strength of the material, augmented with a single fit parameter. Two-dimensional discrete dislocation plasticity (DDP) simulations are carried out for two sets of Cu films and the fit parameter is determined from independent experiments. Subsequent DDP predictions of the stress-strain curves in comparison with the corresponding experimental data show excellent agreement of initial yield strength and hardening rate for films of varying film thickness and grain size. Having thus demonstrated the power of the proposed strength distribution, it is shown that the mode of this distribution governs the most effective source strength. This is then used to suggest a method to estimate the yield strength of thin films as a function of film thickness and grain size. Simple maps are presented that are in very good agreement with recent experimental results for Cu thin films.     

Effect of intrinsic stress gradient on the effective mode-I fracture toughness of amorphous diamond-like carbon films for MEMS

The influence of intrinsic stress gradient on the mode-I fracture of thin films with various thicknesses fabricated for Microelectromechanical Systems (MEMS) was investigated. The material system employed in this study was hydrogen-free tetrahedral amorphous diamond-like carbon (ta-C). Uniform gauge microscale specimens with thicknesses 0.5, 1, 2.2, and 3 μm, containing mathematically sharp edge pre-cracks were tested under mode-I loading in fixed grip configuration. The effective opening mode fracture toughness, as calculated from boundary force measurements, was 4.25±0.7 MPa√m for 0.5-μm thick specimens, 4.4±0.4 MPa√m for 1-μm specimens, 3.74±0.3 MPa√m for 2.2-μm specimens, and 3.06±0.17 MPa√m for 3-μm specimens. Thus, the apparent fracture toughness decreased with increasing film thickness. Local elastic property measurements showed no substantial change as a function of film thickness, which provided evidence for the stability of the sp²/sp³ carbon binding stoichiometry in films of different thicknesses. Detailed experiments and finite element analysis pointed out that the dependence of the effective fracture toughness on specimen thickness was due to the intrinsic stress gradient developed during fabrication and post-process annealing. This stress gradient is usually unaccounted for in mode-I fracture experiments with thin films. Thicker films, fabricated from multiple thin layers, underwent annealing for extended times, which resulted in a stress gradient across their thickness. This stress gradient caused an out-of-plane curvature upon film release from its substrate and, thus, combined bending and tensile mode-I loading at the crack tip under in-plane forces. Since the bending component cannot be isolated from the applied boundary force measurements, its contribution, becoming important for thick films, remains unaccounted for in the calculation of the critical stress intensity factor, thus resulting in reduced apparent fracture toughness that varies with film thickness and curvature. It was concluded that in the presence of a stress gradient, accounting only for the average intrinsic stresses could lead in an overestimate of the fracture resistance of a brittle film. Under these considerations the material fracture toughness of ta-C, as determined from specimens with negligible curvature, is K_IC=4.4±0.4 MPa√m.     

Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.     

齿向修形对滤波减速器润滑性能的影响分析

综合考虑了滤波减速器齿向修形参数、真实齿面粗糙度和瞬态效应等因素,建立了轮齿混合润滑数学模型,数值计算了不同修形参数值对应不同啮合点的最大压力和中心膜厚,分析了齿面粗糙度和转速对润滑性能的影响.结果表明:修形参数r和Ry均存在一个优化范围,使得轮齿表面最大油膜压力显著降低,边缘效应弱化,而中心膜厚则随着r和Ry的增大而逐渐增大;未修形轮齿边缘油膜压力受粗糙度的影响而急剧增大,边缘效应更加显著,修形后轮齿的边缘效应得到了明显改善,因此,轮齿修形也因粗糙表面的存在而显得更加重要;随着转速逐渐降低,轮齿表面的平均油膜厚度逐渐变小,接触比逐渐增大,轮齿表面由弹流润滑逐渐转为混合润滑,最后演变为边界润滑.     

Parallel transport and defects on nematic shells

Nematic shells are thin films of nematic liquid crystal deposited on the boundary of colloidal particles, where liquid crystal molecules may freely glide, while remaining tangent to the surface substrate. The surface nematic order is described here by an appropriate tensor field Q, which vanishes wherever a defect occurs in the molecular order. We show how the classical concept of parallel transport on a manifold introduced by Levi-Civita can be adapted to this setting to define the topological charge m of a defect. We arrive at a simple formula to compute m from a generic representation of Q. In a number of separate appendices, we revisit in a unified language several, apparently disparate applications of Levi-Civita’s parallel transport.     

Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation

Nanoindentation experiments have shown that microstructural inhomogeneities across the surface of gold thin films lead to position-dependent nanoindentation behavior [Phys. Rev. B (2002), to be submitted]. The rationale for such behavior was based on the availability of dislocation sources at the grain boundary for initiating plasticity. In order to verify or refute this theory, a computational approach has been pursued. Here, a simulation study of the initial stages of indentation using the embedded atom method (EAM) is presented. First, the principles of the EAM are given, and a comparison is made between atomistic simulations and continuum models for elastic deformation. Then, the mechanism of dislocation nucleation in single crystalline gold is analyzed, and the effects of elastic anisotropy are considered. Finally, a systematic study of the indentation response in the proximity of a high angle, high sigma (low symmetry) grain boundary is presented; indentation behavior is simulated for varying indenter positions relative to the boundary. The results indicate that high angle grain boundaries are a ready source of dislocations in indentation-induced deformation.