Two-parameter asymptotic expansions for elliptic equations with small geometric perturbation and high contrast ratio

We consider the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirich- let boundary condition and transmission condition, subject to the small geometric perturbation and the high contrast ratio of the conductivity. We consider two types of perturbations: the first corresponds to a thin layer coating a fixed bounded domain and the second is the per perturbation of the interface. As the perturbation size tends to zero and the ratio of the conductivities in two subdomains tends to zero, the two-parameter asymptotic expansions on the fixed reference domain are derived to any order after the single parameter expansions are solved be- forehand. Our main tool is the asymptotic analysis based on the Taylor expansions for the properly extended solutions on fixed domains. The Neumann boundary condition and Robin boundary condition arise in two-parameter expansions, depending on the relation of the geometric perturbation size and the contrast ratio

Micro-mechanical characteristics and dimensional change of Cu-Sn interconnects due to growth of interfacial intermetallic compounds

Sn-based solder alloys are extensively used in electronic devices to form interconnects between different components to provide mechanical support and electrical path. The formation of a reliable solder interconnects fundamentally relies on the metallurgic reaction between the molten solder and solid pad metallization in reflowing. The resultant IMC layer at the solder/pad metallization interface can grow continuously during service or aging at an elevated temperature, uplifting the proportion of IMCs in the entire solder joint. However, the essential mechanical properties of interfacial IMC (i.e. Cu6Sn5, Cu3Sn) layers, such as Young s modulus and hardness, are drastically different in comparison with Sn-based solder and substrate. Therefore, the increasing fraction of interfacial IMCs in the solder joint can lead to significant deformation incompatibility under exterior load, which becomes an important reliability concern in the uses of solder joints for electronic interconnects. In the past decades, extensive research works were implemented and reported regarding the growth of interfacial IMC layers and its effect on the mechanical integrity of solder joints. But, the following fundamental issues in terms of mechanical and microstructural evolution in the uses of solder joints still remain unclear, demanding further research to elaborate: (1) The protrusion of IMCs: Though the growth of interfacial IMC layers along the diffusion direction in solder joints were studied extensively, the growth of IMCs perpendicular to the diffusion direction were reported in only a few papers without any further detailed investigation. This phenomena can crucially govern the long-term reliability of solder interconnects, in particular, in the applications that require a robust microstructural integrity from a solder joint. (2) Fracture behaviour of interfacial IMC layers: The fracture behaviour of interfacial IMC layers is a vital factor in determining the failure mechanism of solder joints, but this was scarcely investigated due to numerous challenges to enable a potential in-situ micro-scale tests. It is therefore highly imperative to carry out such study in order to reveal the fracture behaviour of interfacial IMC layers which can eventually provide better understanding of the influence of interfacial IMC layers on the mechanical integrity of solder joints. (3) Volume shrinkage: The volume shrinkage (or solder joint collapse) induced by the growth of interfacial IMC layers was frequently ascribed as one of the main causes of the degradation of mechanical reliability during aging due to the potentially resulted voids and residual stress at the solder/substrate interface. However, very few experimental works on the characterisation of such type of volume shrinkage can be found in literatures, primarily due to the difficulties of observing the small dimensional changes that can be encountered in the course of IMCs growth. (4) Residual stress: The residual stress within solder joints is another key factor that contributes to the failure of solder joints under external loads. However, the stress evolution in solder joints as aging progresses and the potential correlation between the residual stress and the growth of interfacial IMC layers is yet to be fully understood, as stress/strain status can fundamentally alter the course of total failure of a solder joint. (5) Crack initiation and propagation in solder joints: Modelling on the mechanical behaviour of solder joints is often undertaken primarily on the stress distribution within solder joints, for instance, under a given external loading. But there is lack of utilising numerical analysis to simulate the crack initiation and propagation within solder joints, thus the effect of interfacial IMC layers on the fracture behaviour of the solder joints can be elaborated in further details. In this thesis, the growth of interfacial IMCs in parallel and perpendicular to the interdiffusion direction in the Sn99Cu1/Cu solder joints after aging was investigated and followed by observation with SEM, with an intention of correlating the growth of IMCs along these two directions with aging durations based on the measured thickness of IMC layer and height of perpendicular IMCs. The mechanism of the protrusion of IMCs and the mutual effect between the growth of IMCs along these two directions was also discussed. The tensile fracture behaviour of interfacial Cu6Sn5 and Cu3Sn layers at the Sn99Cu1/Cu interface was characterised by implementing cantilever bending tests on micro Cu6Sn5 and Cu3Sn pillars prepared by focused ion beam (FIB). The fracture stress and strain were evaluated by finite element modelling using Abaqus. The tensile fracture mechanism of both Cu6Sn5 and Cu3Sn can then be proposed and discussed based on the observed fracture surface of the micro IMC pillars. The volume shrinkage of solder joints induced by the growth of interfacial IMC layers in parallel to the interdiffusion direction in solder joint was also studied by specifically designed specimens, to enable the collapse of the solder joint to be estimated by surface profiling with Zygo Newview after increased durations of aging. Finite element modelling was also carried out to understand the residual stress potentially induced due to the volume shrinkage. The volume shrinkage in solder joints is likely to be subjected to the constraint from both the attached solder and substrate, which can lead to the build-up of residual stress at the solder/Cu interface. Depth-controlled nanoindentation tests were therefore carried out in the Sn99Cu1 solder, interfacial Cu6Sn5 layer, Cu3Sn layer and Cu with Vickers indenter after aging. The residual stress was then evaluated in the correlation with aging durations, different interlayers and the locations in the solder joint. Finally, finite element models incorporated with factors that may contribute to the failure of solder joints, including microstructure of solder joints, residual stress and the fracture of interfacial IMC, were built using Abaqus to reveal the effect of these factors on the fracture behaviour of solder joints under applied load. The effect of growth of IMC layer during aging on the fracture behaviour was then discussed to provide a better understanding of the degradation of mechanical integrity of solder joints due to aging. The results from this thesis can facilitate the understanding of the influence of interfacial IMC layers on the mechanical behaviour of solder joints due to long-term exposure to high temperatures

Stress concentration for nonlinear insulated conductivity problem with adjacent inclusions

A high-contrast two-phase nonlinear composite material with adjacent inclusions of $m$-convex shapes is considered for $m>2$. The mathematical formulation consists of the insulated conductivity problem with $p$-Laplace operator in $\mathbb{R}^{d}$ for $p>1$ and $d\geq2$. The stress, which is the gradient of the solution, always blows up with respect to the distance $\varepsilon$ between two inclusions as $\varepsilon$ goes to zero. We first establish the pointwise upper bound on the gradient possessing the singularity of order $\varepsilon^{-\beta}$ with $\beta=(1-\alpha)/m$ for some $\alpha\geq0$, where $\alpha=0$ if $d=2$ and $\alpha>0$ if $d\geq3$. In particular, we give a quantitative description for the range of horizontal length of the narrow channel in the process of establishing the gradient estimates, which provides a clear understanding for the applied techniques and methods. For $d\geq2$, we further construct a supersolution to sharpen the upper bound with any $\beta>(d+m-2)/(m(p-1))$ when $p>d+m-1$. Finally, a subsolution is also constructed to show the almost optimality of the blow-up rate $\varepsilon^{-1/\max\{p-1,m\}}$ in the presence of curvilinear squares. This fact reveals a novel dichotomy phenomena that the singularity of the gradient is uniquely determined by one of the convexity parameter $m$ and the nonlinear exponent $p$ except for the critical case of $p=m+1$ in two dimensions