Interfacial Energy Density Research Articles

An energy-based failure criterion for delamination initiation in electronic packaging

The significance of interfacial delamination as a crucial failure mechanism in electronic packaging has been documented in many papers. A number of failure criteria have been used to solve the problems with a pre-crack at the interface. However, in real electronic packages, the size and location of the cracks or/and delamination cannot be predicted. It is not easy to use the traditional fracture criteria to deal with more complicated 3D delamination problems. The epoxy molding compound (EMC)/copper leadframe interface was selected in this study. A series of button shear tests were conducted to evaluate the interfacial adhesion between the EMC and copper. In each test, the failure load acting on the EMC of the button shear sample was measured at different shear angles and a finite element model was used to evaluate the stresses at the EMC/copper interface. In this paper, an energy-based failure criterion is proposed using both the interfacial distortional and hydrostatic strain energy densities as two failure parameters. Stresses were extracted from the numerical simulation in order to calculate the interfacial distortional strain energy density, U d, and the interfacial hydrostatic strain energy density, U h, related, respectively, to the shear and tensile modes. U d and U h were averaged within a selected region of the finite element model where it exhibits high interfacial strain energy density values.

Interfacial acidities, charge densities, potentials, and energies of carboxylic acid-functionalized silica/water interfaces determined by second harmonic generation.

Second-harmonic studies were carried out to determine the interfacial acidity, the potential, and the interfacial energy density of an acid-functionalized silica/water interface between pH 2 and 12. The interfacial potential changes over 3 orders of magnitude, from 10-2 mV to several tens of millivolts, and the interfacial energy density changes by 7 orders of magnitude, from less than 10-7 mJ/m2 to several millijoules per square meter. The methodology presented in this study provides quantitative thermodynamic information necessary for understanding and predicting how solvated species interact with functionalized organic adlayers at liquid/solid interfaces over a wide pH range.

AF-Biased CoFe Multilayer Films With FMR Frequency at 5 GHz and Beyond

Antiferromagnetic/ferromagnetic/antiferromagnetic (AF/F/AF) multilayers are investigated. The AF layer is 500-/spl Aring/-thick Ni/sub 50/Mn/sub 50/. Films are studied as a function of the thickness of the ferromagnetic layer (e/sub F/). Films with Ni/sub 80/Fe/sub 20/, Co/sub 50/Fe/sub 50/, and Co/sub 90/Fe/sub 10/ exhibit interfacial exchange coupling energy densities (J/sub ex/) of 0.22, 0.82, and 0.97 erg.cm/sup -2/ respectively. High J/sub ex/ ensures strong exchange coupling (H/sub ex/) and large uniaxial anisotropy fields (H/sub ua/) at 90/spl deg/ from the pinning direction. This promotes soft magnetism in CoFe films while they are not soft naturally. With e/sub F/=150 /spl Aring/,H/sub ua/ is of 570 and 755 Oe for Co/sub 50/Fe/sub 50/ and Co/sub 90/Fe/sub 10/, respectively. Thicknesses of /spl sim/0.3 /spl mu/m are also shown possible for practical use by means of multiple alternations (n /spl ges/ 20). Microwave characterizations confirm coherent reversible magnetization processes. Co/sub 50/Fe/sub 50/ multilayers are the best candidates with the highest /spl mu/' and the lowest /spl mu/'' at high frequency because an exceptional combination of a high 4/spl pi/M/sub s/ of 21 kG and a small damping factor of 0.015. ferromagnetic resonance frequencies of /spl sim/5 GHz are reported, and values of /spl sim/10 GHz are shown realistic, for the first time.

Continuum Limits of Discrete Systems without Convexity Hypotheses

We describe the variational limit of one-dimensional nearest-neighbour systems of interactions, under no structure hypotheses on the discrete energy densities. We show that the continuum limit is characterized by a bulk and a interfacial energy density, which can be explicitly computed from the discrete energies through operations of limit, scaling and regularization that highlight possible bulk oscillations and multiple cracking.

Microstructural evolution in two-phase solids: morphological stability of near-periodic particle arrangements

Microstructural evolution in binary alloys is studied numerically using a sharp interface model and boundary integral techniques together with a time stepping scheme. Two-dimensional matrix-inclusion type microstructures are considered with an isotropic interfacial energy density and cubically orthotropic elasticity, being distinct in the two phases. An assumption of local equilibrium is applied which provides the coupling of the diffusion and the elasticity problems via an interfacial jump condition. Special attention is paid to the verification of a phenomenon observed in an earlier work: energy considerations suggested the possibility of sustained inverse coarsening leading to equilibrium microstructures with large numbers of small particles. It is found that, while such stationary states exist, their domain of attraction is rather small; reduction of surface area is the dominant driving force in the process.

Interfacial Energies for Incoherent Inclusions

We study a variational problem describing an incoherent interface between a rigid inclusion and a linearly elastic matrix. The elastic material is allowed to slip relative to the inclusion along the interface, and the resulting mismatch is penalized by an interfacial energy term that depends on the surface gradient of the relative displacement. The competition between the elastic and interfacial energies induces a threshold effect when the interfacial energy density is non-smooth: small inclusions are coherent (no mismatch); sufficiently large inclusions are incoherent. We also show that the relaxation of the energy functional can be written as the sum of the bulk elastic energy functional and the tangential quasiconvex envelope of the interfacial energy functional.

The Energy of a One-Dimensional Structured Deformation

Starting from an assumed expression for the energy of a simple deformation, a representation formula for the energy of a one-dimensional structured deformation is obtained. If w and 0 denote the bulk and the interfacial energy density for a simple deformation, the corresponding densities for a structured deformation are determined by the lower semicontinuous and convex envelope of w and by the subadditive envelope of 0. This result holds under a specific type of convergence assumed for approximating sequences of simple deformations; as discussed briefly in the final remarks, other physically meaningful notions of convergence may lead to different expressions for the energy.

Incoherence at heterogeneous interfaces

To describe the small scale defect structure of incoherent interfaces in an elastic bi-crystal, we consider a multiscale variational model composed of two parts: a bulk energy density W and a nonconvex interfacial energy density ψ that depends on the microscopic lattice mismatch at the grain boundary and the relative crystallographic normal. Mismatch is described using a referential coherence tensor that measures the relative misfit at the grain boundary and serves to couple the bulk elastic deformation of the grains to small scale defect structures at the interface. Minimizing this energy yields predictions for the distribution of defect microstructures at the interface as a function of grain size, elastic stiffness, and lattice mismatch. Calculations for a heterophase (01) / (01) interface in a two-dimensional bi-crystal system are presented.

Coherent Solid-State Phase Transitions with Atomic Diffusion: A Thermomechanical Treatment

Using the framework of modern continuum thermomechanics, we develop sharp- and diffuse-interface theories for coherent solid-state phase transitions. These theories account for atomic diffusion and for deformation. Of essential importance in our formulation of the sharp-interface theory are a system of “configurational forces” and an associated “configurational force balance.” These forces, which are distinct from standard Newtonian forces, describe the intrinsic material structure of a body. The configurational balance, when restricted to the interface, leads to a generalization of the classical Gibbs–Thomson relation, a generalization that accounts for the orientation dependence of the interfacial energy density and also for a broad spectrum of dissipative transition kinetics. Our diffuse-interface theory involves nonstandard “microforces” and an associated “microforce balance.” These forces arise naturally from an interpretation of the atomic densities as macroscopic parameters that describe atomistic kinematics distinct from the motion of material particles. When supplemented by thermodynamically consistent constitutive relations, the microforce balance yields a generalization of the Cahn–Hilliard relation giving the chemical potentials as variational derivatives of the total free energy with respect to the atomic densities. A formal asymptotic analysis (thickness of the transition layer approaching zero) demonstrates the correspondence between versions of our theories specialized to the case of a single mobile species for situations in which the time scale for interface propagation is small compared to that for bulk diffusion. While the configurational force balance is redundant in the diffuse-interface theory, when integrated over the transition layer, the limit of this balance is the interfacial configurational force balance (i.e., generalized Gibbs–Thomson relation) of the sharp-interface theory.

Simulation of metallic impurity gettering in silicon by MeV ion implantation

A simple model for metallic impurity gettering by buried layers created by MeV ion implantation in silicon is presented. For our experimental conditions, the precipitation of supersaturated Fe at dislocation loops in silicon is not diffusion limited. Furthermore, a non-zero silicide-matrix interfacial energy density was required to fit our experimental data. In samples containing dislocations and boron-implanted layers, the dislocations trap more Fe than the boron, but the latter is more effective at reducing the concentration of ungettered Fe. Although we were able to simulate impurity distributions fairly well in many cases, the model is not free from adjustable parameters due to a lack of knowledge of detailed impurity–defect interactions and the silicide-matrix interfacial energy density.

On a Γ-limit of a family of anisotropic singular perturbations

We construct an example confirming that the interfacial energy density of the Γ-limit of a family of nonconvex functionals, cannot be computed, in general, by assuming that the local behavior of a sequence of vector-valued minimizers near the interface is unidirectional.

The equilibrium shape of an elastically inhomogeneous inclusion

A method is proposed for determining the equilibrium shape of an elastically inhomogeneous coherent precipitate in an anisotropic medium. Arbitrary eigenstrains and an isotropic interfacial energy density are taken into account. Use is made of Eshelby's notion of a “force on an interface” and boundary integral techniques are employed. Results for the plane strain case of one single inclusion and a regular array of inclusions with dilatational eigenstrains in an orthotropic matrix are presented. We find that the three factors: difference of elastic moduli in matrix and inclusion, interfacial energy and particle-particle interaction can affect the equilibrium shape in much the same way. The elastic inhomogeneity can cause strongly non-convex particle shapes and considerably affects the stability of fourfold symmetric equilibrium shapes.

Bulk and Interfacial Energy Densities for Structured Deformations of Continua

to provide a basis for the application of techniques in the calculus of variations to non-classical deformations of continua. These structured deformations of continua are suitable for describing deformations of materials whose kine- matics warrants analysis at both the macroscopic and microscopic levels. The motivation for this work lies in the study of equilibrium configura- tions of crystals with defects. In a defective crystal, the macroscopic defor- mation together with the referential (Bravais) lattice configuration do not suAce to describe fully the configuration of a deformed body; phenomena such as slips, vacancies, and dislocations may be present in the deformed (Bravais) lattice basis, thus preventing the use of the Cauchy-Born hypoth- esis, as described below. In a perfect crystal, it is postulated that the crystal lattice consists of identical atoms located at all positions vectors

Defect structures at thin film-substrate interfaces

We consider the loss of lattice coherence at a planar interface between a thin film and substrate. Coherence is determined locally at the interface by a relative deformation gradient at the interface. We construct an interfacial free energy density f xs such that a fully coherent interface is a global minimizer f xs , and there are symmetry related local minima of f xs corresponding to line defects at the interface. By considering both the interfacial energy and the elastic energy of the film, we calculate equilibrium states as a function of film thickness to predict both the loss of coherence and the geometric pattern of interfacial defects.

Linear Elastic Behavior of Dry Soap Foams

Linear elastic constants are computed for three dry foams that have crystal symmetry, bubbles with equal volume V, and films with uniform surface tension T. The Kelvin, Williams, and Weaire–Phelan foams contain one, two, and eight bubbles in the unit cell, respectively. All three foams have 14-sided bubbles, but these tetrakaidecahedra have different topology; the Weaire–Phelan foam also contains pentagonal dodecahedra. In addition to the bulk modulus for volume compression, we calculate two shear moduli for the Kelvin and Weaire–Phelan foams, which have cubic symmetry, and four shear moduli for the Williams foam, which has tetragonal symmetry. The Williams foam has five elastic constants, not six, because the stress remains isotropic for uniform expansion; this is not guaranteed by symmetry alone. The two shear moduli for the Weaire–Phelan foam differ by less than 5%. The other two foams exhibit much greater elastic anisotropy; their shear moduli differ by a factor of 2. An effective isotropic shear modulus G, which represents the response averaged over all orientations, is evaluated for each foam. Scaled by T/ V 1/3, Gis 0.8070, 0.7955, and 0.8684 for the Kelvin, Williams, and Weaire–Phelan foams, respectively. When extrapolated to the dry limit, the shear modulus data of Princen and Kiss (for concentrated oil-in-water emulsions with polydisperse drop-size distributions) fall within the range of our calculations. The Surface Evolver program, developed by Brakke, was used to compute minimal surfaces for the dry foams. Also reported for each undeformed foam are various geometric constants relating to interfacial energy density, cell edge length, and bubble pressure.

Use of Interfacial Energy Density in a Modified Bernoulli Equation for Characterization of Spraying Systems

Use of Interfacial Energy Density in a Modified Bernoulli Equation for Characterization of Spraying Systems

A continuum description of partially coherent interfaces

A continuum model of a two-phase crystal-crystal system is constructed in which the structure of the interface between the phases is determined by energy minimization, rather than by being specified a priori. The interfacial structure is parameterized by a variableĤ corresponding to the jump in the surface deformation gradient (or strain) at the interface, so that coherence is defined locally by the conditionĤ = 0. The energy of the system is taken to be the sum of the bulk and interfacial energies, where the interfacial energy densityf xs depends onĤ. In order to explore how the equilibrium interfacial structure depends on the functionf xs (Ĥ), a model system consisting of an elastic film on a rigid substrate is studied, and the interfacial energy density is taken to be nonconvex with a sharp minimum associated with coherence. In this case, it can be shown that the energy of the system is driven to its infimum by separating the interface into coherent and incoherent regions, which may be viewed as a continuum analog to a partially coherent interface. Further, this solution only appears above a certain critical thickness of the film, in agreement with misfit dislocation models of partially coherent interfaces.

Thermodynamics of wetting of solid surfaces by magnetizable fluids

Abstract The thermodynamic behavior of deformable three phase contact systems is linked to a flow network of stream and equipotential surfaces, and the system is shown to satisfy the Bernoulli equation. In ‘form transformations’, occuring in wetting phenomena, the pressure is an additional degree of freedom that can balance changes in interfacial energy densities. For magnetizable fluids the magnetic energy density facilitates control of the shape of the bulk fluid and of the films that may extend from it.

Role of interfacial thermal resistance and laser energy density during laser processing of copper-sapphire couples

The effect of the interfacial thermal resistance and the laser energy density on film morphology and the extent of evaporation was studied in the excimer laser processing of copper-sapphire couples. Copper films of 80 nm thickness were sputter-deposited on sapphire substrates and laser-irradiated with energy densities in the range of 0.2 to 3.5 J/cm2. The changes in film morphology and thickness as a function of laser energy density were analyzed by energy dispersive x-ray spectroscopy. Four regimes can be established as a function of the laser energy density. First, for low energy densities up to a critical value, the film is partially removed by thermal stresses. Second, as the energy density is increased above that critical value, larger portions of the film remain attached to the substrate. In this regime adhesion enhancement takes place. Third, a further increase in the energy density results in film evaporation. Finally, the decrease in the specific mass removal rate of copper is related to the formation of a laser generated plasma that shields the sample from the incoming radiation. In this last regime, an intermediate compound may form at the substrate surface. The data were correlated with results from a computer model of the heat flow during laser processing of metal-ceramic couples.

Application of thermodynamic potentials for electromagnetic interfacial systems

The application of thermodynamic potentials for interfacial systems, in the extensive and intensive formulation, is reviewed. Using the extensive formulation it is shown that for interfacial systems, comprising more than one subsystem, summation of different interfacial potentials may be necessary for determination of equilibrium values of unconstrained parameters. Using the intensive formulation it is shown that electromagnetic fields may be effective in changing chemical activity of polarizable materials. Interfacial energy density is shown to be a thermodynamic variable that embodies the energy exchange capability between bulk and interfaces. Finally, it is shown that at equilibrium the sum of interfacial energy density, mechanical pressure, and electromagnetic energy density is constant irrespective of the interface being flat or curved.