Substitutional diffusion and Kirkendall effect in binary crystalline solids containing discrete vacancy sources and sinks

We investigate vacancy-mediated diffusion in a binary substitutional alloy by explicitly accounting for discrete vacancy sources and sinks. The regions between sources and sinks are treated as binary crystals with a perfect lattice structure containing a dilute concentration of vacancies. The sources and sinks are assumed ideal, maintaining an equilibrium vacancy concentration in their immediate vicinity. Diffusion within the perfect lattice is described with a diffusion-coefficient matrix determined by kinetic Monte Carlo simulations for a binary, thermodynamically ideal alloy in which the components have different vacancy-exchange frequencies. Continuum simulations are performed for diffusion couples with discrete grain boundaries acting as vacancy sources and sinks. Effective grain coarsening due to the Kirkendall effect is observed even in the absence of Gibbs-Thomson driving forces. As in standard ternary systems, uphill diffusion is observed. We also find that the drift of the lattice frame of reference as a result of the Kirkendall effect increases with the source/sink density. Upon increasing the density of vacancy sources and sinks, we recover the conventional treatment of substitutional diffusion, which assumes a dense and uniform distribution of vacancy sources and sinks that maintain an equilibrium vacancy concentration throughout the solid. The inverse Kirkendall effect, where the slower component segregates at grain boundaries acting as vacancy sinks, is also observed in the simulations.     

Competition of K and F sinks during void formation

The kinetics of void evolution in a binary alloy with two competing types of vacancy sinks is considered. In the course of interdiffusion, the competition of F and K sinks manifests itself in the competition of Frenkel voiding and Kirkendall shift of a lattice. The equations for the void-growth rate are obtained, depending on the dislocation-sink strengths. The maximal void sizes which can be reached during individual growth stage are found for two cases: (1) for a homogenous quenched alloy and (2) for a diffusion zone during interdiffusion process.     

Effects of coherency stress and vacancy sources/sinks on interdiffusion across coherent multilayer interfaces – Part I: Theory

A phase field model corresponding to vacancy-mediated interdiffusion in coherent multilayers with completely miscible constituents is developed to explore the effects of several factors on interdiffusion across coherent multilayer interfaces, such as: (1) the dependence of diffusion potentials and mobilities on coherency stress; (2) the dependence of diffusion potentials and mobilities on composition; (3) the elastic constant inhomogeneity resulting from a inhomogeneous composition distribution; and (4) the properties of vacancy sources/sinks. The Gibbs free energy of the system consists of chemical and elastic energies. The gradient energy is neglected as the multilayers under consideration can be chemically well approximated by an ideal substitutional solution model. Elastic energy is a function of the stress-free strain and inhomogeneous elastic moduli distributions, while the stress is solved by anisotropic phase field microelasticity theory. The diffusion potentials are obtained straightforwardly as functional derivatives of the free energy with respect to composition and are in keeping with previous derivations that involved many mathematical manipulations or quite advanced theories. The diffusion mobilities are affected by the stress through modification of the vacancy formation and migration energies. Two limiting cases of vacancy sources/sinks are taken into account: ideal vacancy sources/sinks are uniformly and densely distributed, or not present at all, so the vacancy concentration is in equilibrium all the times, as determined by the local stress and composition in the former case, but deviates from the equilibrium concentration in the latter. The model can be conveniently extended to consider the non-ideal activity of vacancy sources/sinks by introducing a general kinetic relation for the vacancy creation rate.     

Effects of coherency stress and vacancy sources/sinks on interdiffusion across coherent multilayer interfaces – Part II: Interface sharpening and intermixing rate

Vacancy-mediated interdiffusion in coherent Mo/V and Cu/Ni multilayers is simulated to evaluate the effects of coherency stress and vacancy sources/sinks on interface sharpening and the intermixing rate, using the phase field model developed in a previous paper for two limiting cases: ideal vacancy sources/sinks densely distributed or not present at all. Interface sharpening stems from a large diffusion coefficient asymmetry across the interface, which in turn originates from the large difference in vacancy formation and migration energies between the two constituent layers. Remarkable sharpening is found in Mo/V multilayers either with dense or without vacancy sources/sinks, but only in Cu/Ni with a high density of sources/sinks. Sharpening is suppressed by coherency stress in Cu/Ni regardless of the existence of vacancy sources/sinks, but only promoted in Mo/V with a high density of sources/sinks. The intermixing rate is suppressed in Mo/V by the introduction of a high density of vacancy sources/sinks that are parallel or perpendicular to the interfaces, or uniformly distributed in all orientations, but only promoted in Cu/Ni by the introduction of vacancy sources/sinks that are parallel to the interfaces. The intermixing rate is promoted in Mo/V by coherency stress regardless of the existence of vacancy sources/sinks, but promoted in Cu/Ni by coherency stress only when the vacancy sources/sinks are parallel to the interface or not present at all. The effects of that part of coherency stress induced by the mismatch in atomic volumes on interface sharpening and the intermixing rate are opposite to, but dominant over, those of the stress induced by lattice creation/annihilation in vacancy sources/sinks.     

Diffusion in multi-component systems with no or dense sources and sinks for vacancies

General equations for the multi-component diffusion in crystalline systems are derived in the framework of Onsager's non-equilibrium thermodynamics. The aim is to provide explicit equations needed for computer-modeling of, e.g., diffusion-controlled phase transformations, but avoiding the usual simplifying assumptions, such as the independence of the fluxes of different atomic species. An additional difficulty is introduced into the problem by the fact that the concentration of vacancies, which mediate diffusion of substitutional atoms, may follow different rules, depending on whether there is a sufficient density of sources and sinks to keep the local vacancy concentration in equilibrium. We treat two limiting cases, one where the total vacancy number is conserved (that is, there are no available sources and sinks) and one where the vacancy concentration is kept in equilibrium (the case of dense sources and sinks). First, the diffusive fluxes of all components and of vacancies are expressed by the well known Onsager relation. The kinetic coefficients L_ik from the Onsager relation are derived by means of an extremal thermodynamic principle with respect to the atomic mobilities of individual components and taking into account the constraint amongst fluxes resulting from the vacancy diffusion mechanism. In the case where a dense network of sources and sinks of vacancies is active, the number of lattice positions is not necessarily conserved in every region of the specimen. This means, that the second Fick law, derived for the local conservation of lattice positions, is not applicable in this case. Using mass conservation considerations, the second Fick law is modified to account for this effect. The lattice may shrink or expand for two reasons—either due to the generation or annihilation of vacancies or to the change of the molar volume connected with the change of chemical composition. The deformation of the system is expressed quantitatively by strain rates. Finally, the equations for the system evolution are expressed in both the lattice-fixed and in the laboratory-fixed frames of reference.     

空位对Cu/Sn无铅焊点界面元素扩散的影响

界面柯肯达尔空洞形成的过程伴随着空位的形成与扩散,对空位行为的研究有利于深入理解界面扩散和空洞形成过程. 运用分子动力学方法模拟Cu/Cu₃Sn界面上空位对扩散的影响,计算空位形成能、扩散势垒及空位扩散激活能. 结果表明,相同条件下含空位的模型发生扩散的几率要高于不含空位的模型. 另外,计算表明铜晶体的空位形成能大于Cu₃Sn晶体中铜空位的形成能;Cu₃Sn晶格中不同晶位的Cu空位(Cu1空位和Cu2空位)的形成能比较接近,但均小于锡的空位形成能. 此外,对Cu/Cu₃Sn界面的空位扩散势垒及空位扩散激活能的计算结果表明,Sn原子的空位扩散激活能高于Cu原子.     

Thermodynamic effects on the kinetics of vacancy-generating processes

The inhibiting effect of vacancies on the very process in which they are generated is considered from a thermodynamic viewpoint. Examples of such processes treated here in some detail are grain growth and pore dissolution. It is shown that these processes are inhibited due to vacancy generation. A particular scenario discussed implies intermittent “locking”. After a period of uninhibited kinetics the process comes to a halt due to a thermodynamic back force “locking” it. It can only re-start once the vacancies produced are removed by diffusion. This repetitive cycle leads to an overall reduction in the rate of the kinetic process in question. Specific predictions with regard to grain growth in fine-grained (particularly nanocrystalline) materials and void dissolution kinetics in sintering are made. A third example considered is vacancy drag on a moving individual grain boundary. The magnitude of the drag is re-assessed by taking into account the Gibbs free energy of the vacancies generated.     

Effects of residual S on Kirkendall void formation at Cu/Sn–3.5Ag solder joints

Solder joints of Cu/Sn–3.5Ag were prepared using Cu foil or electroplated Cu films with or without SPS additive. With a high level of SPS in the Cu electroplating bath, voids tended to localize at the Cu/Cu₃Sn interface during subsequent aging at 150 °C, which was highly detrimental to the drop impact resistance of the solder joints. In situ Auger electron spectroscopy of fractured joints revealed S segregation on the Cu/Cu₃Sn interface and void surfaces, suggesting that segregation of S to the Cu/Cu₃Sn interface lowered interface energy and thereby the free energy barrier for Kirkendall void nucleation. Once nucleated, voids can grow by local tensile stress, originating from residual stress in the film and/or the Kirkendall effect. Vacancy annihilation at the Cu/Cu₃Sn interface can induce tensile stress which drives the Kirkendall void growth.     

Semigrand Canonical and Kinetic Monte Carlo simulations of binary B2-ordered nano-films with triple defects

Equilibrium atomic configurations and the kinetics of “order–order” and surface segregation processes in B2-ordering stoichiometric A-50 at.%B binary thin films are investigated by means of Semigrand Canonical Monte Carlo (SGCMC) and Kinetic Monte Carlo (KMC) simulations. The (100)-oriented films are modeled with an Ising-type Hamiltonian with previously evaluated pair-interaction energy parameters yielding the effect of “triple-defect disordering”. The SGCMC simulations provide equilibrium vacancy concentration and atomic configuration in the films with B-atom termination of both free surfaces achieved at high temperatures by the generation of an antiphase boundary. Despite strong vacancy surface segregation, the thermodynamic activation energy for their formation inside the films is the same as in the bulk material. KMC simulations implemented with the SGCMC-determined equilibrium vacancy concentration reveal very slow relaxation of the films towards equilibrium configuration. The B-termination of the (100) free surfaces is produced by A-atom diffusion inwards into the films mediated by vacancies segregating on surfaces.     

Void growth by dislocation emission

Laser shock experiments conducted at an energy density of 61 MJ/m² revealed void initiation and growth at stress application times of approximately 10 ns. It is shown that void growth cannot be accomplished by vacancy diffusion under these conditions, even taking into account shock heating. An alternative, dislocation-emission-based mechanism, is proposed for void growth. The shear stresses are highest at 45° to the void surface and decay with increasing distance from the surface. Two mechanisms accounting for the generation of geometrically necessary dislocations required for void growth are proposed: prismatic and shear loops. A criterion for the emission of a dislocation from the surface of a void under remote tension is formulated, analogous to Rice and Thomson’s criterion for crack blunting by dislocation emission from the crack tip. The critical stress is calculated for the emission of a single dislocation and a dislocation pair for any size of initial void. It is shown that the critical stress for dislocation emission decreases with increasing void size. Dislocations with a wider core are more easily emitted than dislocations with a narrow core.     

Depletion and Voids Formation in the Substrate During High Temperature Oxidation of Ni–Cr Alloys

A numerical model to treat the kinetics of vacancy annihilation at the metal/oxide interface but also in the bulk metal has been implemented. This was done using EKINOX, which is a mesoscopic scale 1D-code that simulates oxide growth kinetics with explicit calculation of vacancy fluxes. Calculations were performed for high temperature Ni–Cr alloys oxidation forming a single chromia scale. The kinetic parameters used to describe the diffusion in the alloy were directly derived from an atomistic model. Our results showed that the Cr depletion profile can be strongly affected by the cold work state of the alloy. In fact, the oversaturation of vacancies is directly linked to the efficiency of the sinks which is proportional to the density of dislocations. The resulting vacancy profile highlights a supersaturation of vacancy within the metal. Based on the classical nucleation theory, the possibility and the rate of void formation are discussed.     

Vacancy-driven stress relaxation in layers

The role of vacancies together with their generally non-ideal sources and sinks has been investigated previously in the context of multi-component diffusion. This concept is applied to a thin layer on a substrate subjected to an initial eigenstress state. The surface of the layer is assumed to act as an ideal source and sink for vacancies. The interface may act either as an ideal or as no source and sink for vacancies. Non-ideal sources and sinks are supposed in the bulk. The formation of a vacancy site fraction profile across the layer due to active sources and sinks and due to diffusion is studied, provoking a relaxation of the residual eigenstress state. A set of two nonlinear partial differential equations for both the vacancy site fraction and the eigenstress distributions is developed. The problem is reformulated in dimension-free quantities. The relation between the diffusivity and the activity of sources and sinks for vacancies in the bulk is characterized by a vacancy intensity parameter k. Numerical solutions are presented, and the influence of nonlinear terms in the evolution equations is evaluated. The simulations provide the evolution of the vacancy site fraction and of the hydrostatic stress profiles in the layer. The development of a film of newly deposited or removed atoms at the layer surface and of the total thickness of the layer is also calculated. The results of simulations are discussed.     

Microstructure evolution in Cu pillar/eutectic SnPb solder system during isothermal annealing

The reaction between Cu pillar and eutectic SnPb solder during isothermal annealing was studied systematically. Intermetallic compounds (IMCs), such as Cu₆Sn₅ and Cu₃Sn, were formed in between Cu and SnThe parabolic rate law was observed on IMC formation, which indicated that the growth of IMCs was controlled by atomic diffusion (a diffusion-limited process). Annealing at 165 °C for 160 h decreased the growth rate of Cu₆Sn₅, and at the same time increased the growth rate of Cu₃Sn. This was when Sn in solder was exhausted completely. The activation energies for the growth of Cu₃Sn and Cu₆Sn₅ were measured to be 1.77 eV and 0.72 eV, respectively. The Kirkendall void that formed at the interface between Cu pillar and solder obeyed the parabolic rate law. The growth rate of the Kirkendall void increased when the Sn in solder was consumed in its entirety.     

Modeling of kinetics of diffusive phase transformation in binary systems with multiple stoichiometric phases

The thermodynamic extremal principle is used for the treatment of the evolution of a binary system under the assumption that all phases in the system are nearly stoichiometric with no sources and sinks for vacancies in the bulk. The interfaces between the individual phases are assumed to act as ideal sources and sinks for vacancies, and to have an infinite mobility. Furthermore, it is assumed that several phases are nucleated in the contact plane of the diffusion couple at the beginning of the computer experiment. Then, it is shown that the number of newly nucleated phases determines the maximum number of polyfurcations (i.e., branching of a single configuration into several distinct configurations) of the initial contact (Kirkendall) plane. The model is demonstrated on a hypothetical binary system with four stoichiometric phases. The inverse problem, namely, the determination of the tracer diffusion coefficients in newly nucleated phases from the thicknesses of new phases and the positions of polyfurcated Kirkendall planes, is treated too. This article was presented at the Multicomponent-Multiphase Diffusion Symposium in Honor of Mysore A. Dayananda, which was held during TMS 2006, the 135th Annual Meeting and Exhibition, March 12–16, 2006, in San Antonio, TX. The symposium was organized by Yongho Sohn of University of Central Florida, Carelyn E. Campbell of National Institute of Standards and Technology, Richard D. Sisson, Jr., of Worcester Polytechnic Institute, and John E. Morral of Ohio State University.     

Oxidation of Simple and Pt-Modified Aluminide Diffusion Coatings on Ni-Base Superalloys—II. Oxide Scale Failure

The spallation behavior of oxides formed during isothermal oxidation at 1050°C of one simple (PWA73) and three Pt-modified (RT22, SS82A and MDC150L) aluminide diffusion coatings on the same Ni-base, single-crystalline superalloy (CMSX-4) was investigated by scanning electron microscopy (SEM) and transmission-electron microscopy (TEM). It was found that the main spallation mechanism was the formation of large Kirkendall voids at the oxide–coating interface. It is believed that the void formation was caused by counter-current flow of vacancies to the diffusion of Ni away from the interface as Al is consumed by the oxide. The magnitude of the vacancy current was determined by the oxidation rate. The properties of the void-formation mechanism are discussed in view of previous data on the microstructure of the oxide scales.     

Self-similar growth of a compound layer in thin-film binary diffusion couples

The diffusion controlled growth of a compound phase A_nB between two thin films of material A and B is studied with the nonlinear Kirkendall effect included. This growth process is important in electronic materials processing and in synthesis of high-temperature materials using multilayer films. Previous models of the growth rate do not solve the diffusion equation, and thus do not fully utilize the predictive capability. This paper describes a self-similar transformation that reduces the nonlinear, time-dependent diffusion equation with two free boundaries into a nonlinear ordinary differential equation, which is solved numerically by a shooting method. It is found that the intrinsic diffusion coefficients of A and B in A_nB can be determined from the positions of the interfaces without using the concentration profile. This provides a simpler method for measuring intrinsic diffusion coefficients. An asymptotic solution valid for small concentration gradients is derived and agrees with the numerical results.     

Phase-field simulation of void migration in a temperature gradient

A phase-field model simulating vacancy diffusion in a solid with a strong vacancy mobility inhomogeneity is presented. The model is used to study void migration via bulk and surface diffusion in a temperature gradient. The simulations demonstrate that voids migrate up the temperature gradient, and the migration velocity varies inversely with the void size, in agreement with theory. It is also shown that the current model has the capability to investigate the effects of surface diffusion, temperature gradient and vacancy concentration on the void migration velocity. An interesting potential application of the model is to study the kinetics of void migration and the formation of a central hole in nuclear fuels.     

The discovery and acceptance of the Kirkendall Effect: The result of a short research career

In the 1940s, it was a common belief that atomic diffusion took place via a direct exchange or ring mechanism that indicated the equality of diffusion of binary elements in metals and alloys. However, Ernest Kirkendall first observed inequality in the diffusion of copper and zinc in interdiffusion between brass and copper. This article reports how Kirkendall discovered the effect, now known as the Kirkendall Effect, in his short research career. Editor's Note: A hypertext-enhanced version of this article can be found at http://www.tms.org/pubs/journals/JOM/9706/Nakajima-9706.html. Also some of the artwork employed here was photographically reproduced from existing publications. As a result, the quality of the images is sometimes less than ideal.     

Analytical treatment of diffusion during precipitate growth in multicomponent systems

We propose an approximate growth rate equation that takes into account both cross-diffusion and high supersaturations for modeling precipitation in multicomponent systems. We then apply it to an Fe-alloy in which interstitial C atoms diffuse much faster than substitutional solutes, and predict a spontaneous transition from slow growth under ortho-equilibrium to fast growth under the non-partitioning local equilibrium condition. The transition is caused by the decrease in the Gibbs–Thomson effect as the growing particle becomes larger. The results agree with DICTRA simulations where full diffusion fields are calculated.     

Interpretation of viscous deformation of Zr-based bulk metallic glass alloys based on Nabarro-Herring creep model

Superplastic-like viscous deformation of bulk metallic glass alloys around the glass transition temperature (Tg) was analyzed based on the Nabarro-Herring creep model, a classical creep model, where the diffusional motion of atoms or vacancies through the lattice (atomic configuration) is considered. The amorphous matrix of bulk metallic glasses that has a randomly-packed atomic configuration was assumed to behave in a manner similar to the grain boundary in polycrystalline metals so as to approximate the diffusivity of the major constituent element. In spite of rough approximation of the parameters in the Nabarro-Herring creep equation, a reasonable value of the diffusion path (d) could be obtained from the experimentally-obtained metal flow data, including the steady state stress and the strain rate. Due to the absence of vacancy sources such as grain boundaries in homogeneous metallic glasses, the diffusion path, which, in polycrystalline materials, generally is the average distance between vacancy sources such as grain boundaries, was considered in this work as the average distance between tunneling centers in bulk metallic glass alloys. The calculated diffusion path was comparable to the density of tunneling centers around Tg, proposed by M. H. Cohen and G. S. Grest based on free volume theory. The calculated diffusion path showed monotonous decrease with temperature over Tg for Zr-based bulk metallic glass alloys. Based on this analysis, a schematic model for viscous deformation of bulk metallic glass was proposed.