Void Migration Research Articles

Development of three-dimensional simulation method of the void migration caused by electromigration in the interconnect line with bamboo structure

Development of three-dimensional simulation method of the void migration caused by electromigration in the interconnect line with bamboo structure

Boundary Element Method for the dynamic evolution of intra-tow voids in dual-scale fibrous reinforcements using a Stokes–Darcy formulation

The Boundary Element Method (BEM) is implemented in the simulation of compression, displacement, migration and splitting of intra-tow voids in dual-scale fibrous reinforcements. The last three processes have not been simulated at mesoscopic scale in previous works due to the consideration of a constant pressure in the channels of the Representative Unitary Cell. In this work, both the channels and tows are modeled using the Stokes and Darcy equations, respectively, a pressure gradient is prescribed along the fluid motion, and full air compressibility is deemed, thereby allowing to consider these three processes. The void migration process from the weft towards the channel is analyzed in terms of the ratio between the average air migration velocity and the average liquid velocity, and of the normalized air rate from the weft towards the channel. According to BEM results, the bubble can migrate at both lower and higher velocities with respect to the liquid velocity, and the void removal out of the tows can occur after several stages of compression–displacement–migration–splitting; additionally, the bubble breaks up after several cycles of expansion and compression. BEM results also show that the liquid surface tension, pressure gradient and average channel pressure have important influence in the void migration process.

Phase-field study on geometry-dependent migration behavior of voids under temperature gradient in UO2 crystal matrix

In this work, a phase-field model is established to capture the void migration behavior under a temperature gradient within a crystal matrix, with an appropriate consideration of the surface diffusion mechanism and the vapor transport mechanism. The interfacial energy and the coupling between the vacancy concentration field and the crystal order parameter field are carefully modeled. Simulations are performed on UO2. The result shows that for small voids (with an area ≤ πμm2), the well-known characteristics of void migration, in consistence with the analytical model, can be recovered. The migration is manifested by a constant velocity and a minor change of the void shape. In contrast, for large voids (with an area of ∼10 μm2) initially in circular shapes, significant deformation of the void from a circular to cashew-like shape is observed. After long-time migration, the deformed void would split into smaller voids. The size-dependent behavior of void migration is due to the combined effect of the interfacial energy (which tends to keep the void in circular shape) and the surface diffusion flow (which tends to deform the void due to the nonuniform diffusion along the surface). Moreover, the initial shape of the void modifies the migration velocity and the time point when splitting occurs (for large voids) at the beginning of migration due to the shape relaxation of the void. However, it has a minor effect on the long-time migration. Our work reveals novel void migration behaviors in conditions where the surface-diffusion mechanism is dominant over the vapor transport mechanism; meanwhile, the size of the void lies at a mediate size range.

A review: applications of the phase field method in predicting microstructure and property evolution of irradiated nuclear materials

Complex microstructure changes occur in nuclear fuel and structural materials due to the extreme environments of intense irradiation and high temperature. This paper evaluates the role of the phase field method in predicting the microstructure evolution of irradiated nuclear materials and the impact on their mechanical, thermal, and magnetic properties. The paper starts with an overview of the important physical mechanisms of defect evolution and the significant gaps in simulating microstructure evolution in irradiated nuclear materials. Then, the phase field method is introduced as a powerful and predictive tool and its applications to microstructure and property evolution in irradiated nuclear materials are reviewed. The review shows that (1) Phase field models can correctly describe important phenomena such as spatial-dependent generation, migration, and recombination of defects, radiation-induced dissolution, the Soret effect, strong interfacial energy anisotropy, and elastic interaction; (2) The phase field method can qualitatively and quantitatively simulate two-dimensional and three-dimensional microstructure evolution, including radiation-induced segregation, second phase nucleation, void migration, void and gas bubble superlattice formation, interstitial loop evolution, hydrate formation, and grain growth, and (3) The Phase field method correctly predicts the relationships between microstructures and properties. The final section is dedicated to a discussion of the strengths and limitations of the phase field method, as applied to irradiation effects in nuclear materials.

GROUND FAILURE DUE TO GYPSUM DISSOLUTION

Gypsum in Hellas and Cyprus occurs in three different types: (a) bedded (mainly of Messinian age in Cyprus and Crete), (b) domes (mainly western Hellas and Crete), and (c) as bodies, fragments and cementing material in Triassic conglomerate formations (western Hellas). Ground failure caused by void migration to the surface, resulting from gypsum dissolution, is a common phenomenon in such areas, which are also found in other European countries (e.g., Italy, Spain, Switzerland, U.K., Lithuania, Latvia, Poland, Romania, Turkey, Ukraine and Russia). In this paper three different case studies of ground failure are presented: Cyprus, Crete (Viannos) and Corfu. Engineering geological, stratigraphical, geophysical, hydrogeological and hydrogeochemical studies of these areas, revealed the direct relationship between surface runoff, and ground water circulation with the rate of gypsum dissolution, the subsequent development of karst hollows, and the associated ground failure in urban and suburban environments. Two main models were defined, according to different mechanisms of gypsum dissolution. The first model is associated with the erosion activity of surface runoff, the second with the dissolving capacity of ground water. Risks to the urban and suburban environments were assessed, and guidelines as well as mitigation measures were proposed.

Electromigration: Void Dynamics

The detailed knowledge of the void evolution is helpful to understand and interpret traditional electromigration (EM) lifetime experiments. As void formation and subsequent evolution is highly dynamic, an in situ observation experiment was employed using an SEM allowing for void detection down to a few nanometers in diameter. For void nucleation, it has been suggested that due to the large anisotropy in modulus in Cu, incompatible grain boundaries can act as stress concentrators for both stress-induced and EM-induced voiding. In addition, during growth, the shape change and migration have been associated with the microstructure. To better understand void formation and evolution, we examined the Cu microstructure using EBSD during the experiment without interruption. We tested interconnects consisting of Cu-based damascene (M1) drift type structures, capped with SiCN. Void formation could be confirmed to be associated with features such as incompatible grain boundaries in some instances. In other cases, triple points and grain boundary/sidewall intersections were found as void nucleation sites. Void growth and migration is found strongly correlated with the driving current density. We show conditions under which void migration takes place or is suppressed. In particular, a critical current density of approximately 10 $\mbox{mA}/\mu\mbox{m}^{2}$ is observed, below which no void migration occurs. At current densities of 15 $\mbox{mA}/\mu\mbox{m}^{2}$ and above, all voids migrated at a net drift velocity proportional to the current density. Independent of migration, all voids are observed to grow at rates consistent with published data for a mixed near bamboo microstructure and interface diffusion.

Phase-Field-Crystal Model for Electromigration in Metal Interconnects.

We propose an atomistic model of electromigration (EM) in metals based on a recently developed phase-field-crystal (PFC) technique. By coupling the PFC model's atomic density field with an applied electric field through the EM effective charge parameter, EM is successfully captured on diffusive time scales. Our framework reproduces the well-established EM phenomena known as Black's equation and the Blech effect, and also naturally captures commonly observed phenomena such as void nucleation and migration in bulk crystals. A resistivity dipole field arising from electron scattering on void surfaces is shown to contribute significantly to void migration velocity. With an intrinsic time scale set by atomic diffusion rather than atomic oscillations or hopping events, as in conventional atomistic methods, our theoretical approach makes it possible to investigate EM-induced circuit failure at atomic spatial resolution and experimentally relevant time scales.

Morphological evolution and migration of void in bi-piezoelectric interface based on nonlocal phase field method

This paper reports the result of investigation into the morphological evolution and migration of void in bi-piezoelectric material interface by utilizing nonlocal phase field model and finite element method (FEM), where the small scale effect containing the long-range forces among atoms is considered. The nonlocal elastic strain energy and the nonlocal electric energy around the void are firstly calculated by the finite element method. Then based on the finite difference method (FDM), the thermodynamic equilibrium equation containing the surface energy and anisotropic diffusivity is solved to simulate the morphological evolution and migration of elliptical void in bi-piezoelectric films interface. Results show that the way of load condition plays a significant role in the evolution process, and the boundary of void's long axis gradually collapses toward the center of ellipse. In addition, the evolutionary speed of left boundary gradually decreases with scale effect coefficient growth. This work can provide references for the safety evaluation of piezoelectric materials in micro electro mechanical system.

A fitted finite element method for the numerical approximation of void electro-stress migration

Microelectronic circuits usually contain small voids or cracks, and if those defects are large enough to sever the line, they cause an open circuit. A fully practical finite element method for the temporal analysis of the migration of voids in the presence of surface diffusion, electric loading and elastic stress is presented. We simulate a bulk–interface coupled system, with a moving interface governed by a fourth-order geometric evolution equation and a bulk where the electric potential and the displacement field are computed. The method presented here follows a fitted approach, since the interface grid is part of the boundary of the bulk grid. A detailed analysis, in terms of experimental order of convergence (when the exact solution to the free boundary problem is known) and coupling operations (e.g., smoothing/remeshing of the grids, intersection between elements of the two grids), is carried out. A comparison with a previously introduced unfitted approach (where the two grids are totally independent) is also performed, along with several numerical simulations in order to test the accuracy of the methods.

Scale effect on migration and coalescence of void in piezoelectric film subjected to gradient stress and electric fields

In this paper, an analytical method is presented to solve the scale effect on migration motion and coalescence of void in piezoelectric solid subjected to gradient stress and electric filed. The piezoelectric solid is considered as an infinite orthotropic piezoelectric film corresponding to the scale of void. The surface diffusion of void in piezoelectric film is taken as the dominant mechanism for the migration motion of the void. Based on the non-local elastic theory considering the interact force among atoms, the influences of small-scale parameter, stress gradient, stress state, electric loading, and shape of void on the migration motion coalescence of void in piezoelectric film are described.

A model of two-phase resin and void flow during composites processing

The reduction of porosity during composites manufacturing remains a critical design goal for successful processing. High porosity composite parts exhibit reduced mechanical properties and increased design risk. A primary source of porosity is the formation and transport of trapped air bubbles during polymer resin flow through fibrous porous media reinforcement. A model experiment has been developed in which a flow cell comprised of two transparent parallel plates is injected with a simulated resin and air bubbles to investigate two-phase flow. The flow cell is designed to simulate flow behavior exhibited during Resin Transfer Molding (RTM) or Out-of-Autoclave (OOA) prepreg composites processing. A flow visualization setup captures the two-phase resin flow and void migration through the flow cell. A computational model is formulated to describe the two phase flow and validated with experimental results. The model material and process parameters can be optimized to maximize the relative void velocity with respect to the resin, thus achieving effective void migration and overall porosity reduction during processing. A parametric scaling analysis is conducted to identify and quantify the sensitivity of material and process parameters on two phase resin and void transport in composites processing.

Criteria for Predicting the Presence of Voids in Abandoned Subsurface Mine Workings in Zones of Intensive Weathering

Empirical criteria (K1–K2 > 0 and η–K3 > 0) for predicting the presence of voids in abandoned shallow underground workings in Donetsk city (Ukraine) located in the zone of intensive weathering are proposed. The voids exist when both inequalities K1–K2 > 0 and η–K3 > 0 are satisfied. The statistical analysis has established that the value of criterion K2 is inversely related to rock quality designation (RQD) and that a decrease in RQD leads to a decrease in K1–K2 and, hence, the probability of the void existence is decreased. It is also established that a statistical parameter η in the left-hand part of inequality η–K3 > 0 decreases according to logarithmic law with increasing standard linear intensity of weathering-induced fractures (P10). When the fracture intensity (P10) increases, the probability of the presence of a void is decreased. The proposed criteria confirm that a thick immediate roof and sandstone layers contribute to the existence of voids of unknown stability. On the other hand, the probability of existence of these voids decreases with increasing thickness of alluvium and total thickness of argillites and sandy shale layers in the rock mass which in turn suggests a decreased probability of void migration and surface subsidence in the future as this has likely already occurred. The use of the proposed criteria in engineering practice is discussed.

An unfitted finite element method for the numerical approximation of void electromigration

Microelectronic circuits usually contain small voids or cracks, and if those defects are large enough to sever the line, they cause an open circuit. We present a numerical method for investigating the migration of voids in the presence of both surface diffusion and electric loading. Our mathematical model involves a bulk-interface coupled system, with a moving interface governed by a fourth-order geometric evolution equation and a bulk where the electric potential is computed. Thanks to a novel approximation of the interface, equidistribution of its vertices is guaranteed, and no remeshing is necessary. In addition, the used curvature approximation means that our method is unconditionally stable for the evolution by surface diffusion only. Various examples are performed to demonstrate the accuracy of the method.

Optimization of curing cycle in carbon fiber-reinforced laminates: Void distribution and mechanical properties

A strategy is presented to optimize out-of-autoclave processing of quasi-isotropic carbon fiber-reinforced laminates. Square panels of 4.6 mm nominal thickness with very low porosity ( ⩽ 0.2 % ) were manufactured by compression molding at low pressure (0.2 MPa) by careful design of the temperature cycle to maximize the processing window. The mechanisms of void migration during processing were ascertained by means of X-ray microtomography and the effect of ply clustering on porosity and on void shape was explained. Finally, the effect of porosity and ply clustering on the compressive strength before and after impact was studied.

Evaluating hazards at salt cavern sites using multichannel analysis of surface waves

Subsidence resulting in sinkholes within dissolution mine fields in and around the city of Hutchinson, Kansas, USA (Figure 1) has been reported for more than 90 years. Salt dissolution and the resulting unpredictable upward migration of the dissolution voids can be observed both in proximity to well bores (dissolution mining, brine disposal, and oil wells; seismic shot holes, etc.) and in areas known to experience natural dissolution with no apparent anthropogenic influences. One potential outcome of the upward migration of these long-inactive dissolution voids in the city of Hutchinson is significant structural damage to facilities, infrastructure, or traffic areas. Monitoring the condition of a void's roof is key to determining and managing this risk. Drilling is expensive for this investigative application in part because of the number of potential targets. At a depth of approximately 120 m, most surface geophysical methods have neither the sensitivity nor resolution necessary to detect these voids, much less interrogate the roof.

Electromigration damage mechanics of lead-free solder joints under pulsed DC: A computational model

A numerical method for studying migration of voids driven by pulse electric current and thermal gradient in 95.5Sn–4Ag–0.5Cu (SAC405) solder joints is developed. The theoretical model involves coupling electron wind force, chemical potential, joule heating and stress gradient driving mass diffusion processes. Entropy based damage criteria is adopted to characterize the mass transportation mechanism, which utilize irreversible entropy production rate as a measure of material degradation. The pulse current induced EM damage results were compared with DC EM response under otherwise the same conditions. It is observed that increasing duty factor, current density and frequency leads to a faster damage accumulation. A mean time to failure (MTF) equation for solder joints under pulse current loadings is proposed which incorporates both thermomigration (TM) and EM damage. The failure mechanism is verified by our experimental results. It is observed that MTF is inversely proportional to rm, fp, and jn, where duty factor exponent m equals to 1.1, frequency exponent p equals to 1.43 and current density exponent is 1.96. This equation is effective only when pulse period is below thermal relaxation time, commonly in μs range.

Effect of direct current stressing to Cu–Cu bond interface imperfection for three dimensional integrated circuits

The ability to be used as both a glue layer and the interconnection line has put Cu metal interconnection as the ultimate goal for 3D-IC. However, the inherent properties of Cu-Cu bond interface that are not always perfect have raised some concerns. This work investigates the evolution of the Cu-Cu bond interface that had been subjected to prolonged electrical current stress. Interface evolutions were characterized by a combination of electrical current stressing and bond interface cross-sectional analysis. While interface improvement was observed in terms of interface void reduction after current stressing, early failures to the interconnection line adjacent to the bond interface were observed. Electromigration had driven void migration from the large bond interface area to the much smaller adjoining interconnect line. This potentially has a significant impact on the future of 3D-IC technology that utilizes Cu-Cu bonding. However, this problem can be mitigated by inserting a barrier layer in between the bond interface and the interconnect line to prevent the migration of the voids into the interconnect line.

Recrystallization-induced void migration in electroplated Cu films

During the annealing of electroplated Cu at 300–400 °C, voids nucleated at the film/substrate interface and migrated toward the film-free surface as the recrystallization proceeded. Voids were located at the recrystallized/unrecrystallized (R/UR) interface and served as markers of the interface motion. Kinetics of the R/UR interface migration showed t 1 2 dependence, suggesting that the recrystallization of the electroplated Cu was controlled by the long-range diffusion of atoms through high-diffusivity paths, such as grain boundaries and dislocation cores.

The underground cavity network of south-eastern Rome (Italy): an evolutionary geological model oriented to hazard assessment

There are many abandoned underground mines beneath the city of Rome, created in the Pleistocene pyroclastic deposits over thousands of years. These pose a serious public safety risk hence the study was undertaken to better assess the stability of the underground workings. The Villa de Sanctis public park was selected as a suitable test site, as in this area the cavity network can be inspected and the different steps of the progressive migration of voids toward the surface can be identified. In addition to an underground geological survey, geophysical investigations were undertaken to elucidate some key characteristics of the involved lithological units, including elastic stiffness and layer thickness. Recommendations are made for further studies of the accessible and inaccessible areas.

Modeling and simulation of voids and saturation in liquid composite molding processes

We propose modeling and simulation of the void formation and the tow saturation in liquid composite molding processes. The present model addresses the void formations in the macropore between fiber tows as well as in the micropore inside fiber tows. We also consider the bubble compression and tow saturation after voids are formed at the flow front. Finally, a void migration model is considered. The present models are validated by a comparison between simulation results and experimental data. The simulation results successfully represent important features of the void physics: the correlation of void formation at the flow front with capillary number, the advantage of vacuum application at air vents to reduce the void content inside tows, and the existence of partially saturated zone behind the global flow front. The representation of mold filling patterns by a degree of saturation shows the presence of a bubbly zone behind the flow front.