Parabolic Growth Constants Research Articles

Investigation on the temperature-dependent diffusion growth of intermetallic compounds in the Mg-Al-Zn system: Experiment and modeling

With the rapid development of Mg alloys, deeper understanding to the thermodynamic and diffusional kinetic behavior of intermetallic compounds (IMCs) is important for studying the effect of alloying elements to the microstructure evolution. Specially, a systematic quantitative investigation on the diffusional growth of IMCs is of great necessity. However, the works studying the elemental diffusion behaviors of multiple-element IMCs are rare in magnesium alloy systems. The current work takes the ternary Mg-Al-Zn system as research target, and combines the diffusion couple technique, phase stability diagrams, in-situ observation technique and numerical inverse method to investigate the temperature-dependent kinetic coefficients. The parabolic growth constant (PGC) and interdiffusion coefficients for Mg solid-solution phase and γ-Mg17Al12, β-Mg2Al3, ε-Mg23Al30, MgZn2, Mg2Zn3, τ-Mg32(Zn, Al)49 and ϕ-Mg5Zn2Al2 IMCs in the Mg-Al-Zn alloy system are determined. By comparing the current experimental with calculation results, the rate-controlling factor of the temperature-dependent diffusion growth of ϕ, τ and ε ternary IMCs in the Mg-Al-Zn system is further discussed in detail.

Effect of ultrafine microstructure on interdiffusion-driven phase transformations in Ni-Sn sandwich diffusion couples

Solid-state diffusion in materials is greatly influenced by microstructural features such as grain boundaries, dislocations, and second phase particles. However, a systematic investigation of structure-kinetics correlation during interdiffusion is largely missing. Herein, a novel sandwich diffusion couple approach was utilized to demonstrate the effects of microstructure on interdiffusion-driven phase formation in the Ni-Sn system. Pure Ni samples were prepared by cold rolling (CR) and spark plasma sintering (SPS) with different microstructures. Two sandwich diffusion couples were prepared– a) NiAM/Sn/NiCR and b) NiCR/Sn/NiSPS, for phase growth analysis at each interface after annealing at 200 °C for 96 h. The intermetallic phase Ni3Sn4 formed at the NiAM/Sn, NiCR/Sn, and NiSPS/Sn interfaces had the thickness of 6, 14, and 41 µm, respectively, consistent with larger parabolic growth constant for the NiSPS/Sn interfaces. The enhanced kinetics at the SPS interface could be attributed to the presence of ultrafine-grained (UFG) (∼320 nm) microstructure dominated by high-angle boundaries.

Low-temperature, short-time, wafer-level bonding for Cu/Sn/Cu solid-state-diffusion interconnect in 3-D integration

In this paper, Cu/Sn/Cu solid-state diffusion (SSD) under low temperature is proposed and investigated for three-dimensional (3-D) integration. Cu and Sn films were deposited by high-efficiency and low-cost physical vapor deposition to fabricate 40-μm-pitch daisy-chain structures. Subsequently, the Cu bump surface was treated with Ar (5% H2) plasma. The Cu/Sn/Cu structure was bonded face to face at 200 °C for 15 min The interfacial composition of the as-bonded dies comprised five layers, Cu/Cu3Sn/Cu6Sn5/Cu3Sn/Cu, with no Sn remaining and no overflow. After annealing at 200 °C for 15 min under N2 atmosphere, as the Cu6Sn5 completely transformed into Cu3Sn, the microstructure changed to stable three layers: Cu/Cu3Sn/Cu. Additionally, the average bonding shear strength reached 27.0 MPa, which is higher than that for conventional Cu/Sn SSD bonding. The measured bonding resistance value was maintained at the theoretical value. Moreover, the parabolic growth constant of Cu3Sn reached 1.86 × 10−15 m2/s. Our study demonstrates the feasibility of using Cu/Sn/Cu SSD for low-temperature, short-time, wafer-level bonding.

A new insight on the diffusion growth mechanism of intermetallic compounds in Al-Er system

The diffusion growth of intermetallic compounds in Al-Er alloys are closely related to the properties of the alloys. The current work aims at explaining the dominance of Al3Er in the Al-Er alloys precipitation phases and the interface thin layer phenomenon by diffusion couple technique, estimating the parabolic growth constant and diffusion activation energy of intermetallic compound in Al-Er diffusion couples to provide theoretical guidance for the design of new Al-Er alloys. In this work, Al-Er diffusion couples were successfully prepared by casting-cladding method in the atmosphere. The growth of Al-Er intermetallic compounds at diffusion couple interface during annealing were observed and recorded by High-Temperature Laser-Scanning Confocal Microscopy at 673, 698, 723 and 748 K respectively. The results show that the growth characteristics of Al-Er intermetallic compounds were accord with layer-terraced growth during annealing. The thickness of intermetallic compound was linear with the square root of time at experimental temperature. The intermetallic compound layer was composed of Al3Er and a very thin AlEr phase. The parabolic growth constants of Al3Er phase at 673, 698, 723 and 748 K were 1.017 × 10−14, 1.609 × 10−14, 3.111 × 10−14 and 4.76 × 10−14 respectively. The activation energy of Al3Er phase was (88.4 ± 5.3) kJ/mol and the pre-exponential factor was 7.126 × 10−8 m2/s.

On the Theory of Directional Solidification in the Presence of a Mushy Zone

A model is developed for the directional solidification of a binary melt with a two-phase zone (mushy zone), where the fraction of the liquid phase is described by a space–time scaling relation. Self-similar variables are introduced and the interphase boundary growth is inversely proportional to the square root of time. The mathematical model of the process is reformulated using self-similar variables. Exact self-similar solutions of heat-and-mass transfer equations are determined in the presence of two mobile phase-transition boundaries, namely, solid–mushy zone and mushy zone–liquid ones. The temperature and impurity concentration distributions in the solid phase, the mushy zone, and the melt are found as integral expressions. A decrease in the dimensionless cooled-boundary temperature leads to an increase in the solidification rate and the fraction of the liquid phase. The solidification rate, the parabolic growth constants, and the fraction of the liquid phase at the solid–mushy zone boundary are determined depending on the scaling parameter and the thermophysical constants of the solidifying melt. The positions of the solid–mushy zone and mushy zone–binary melt phase transition boundaries are found. The dependences of the solidification rate (inversely proportional to the square root of time) are analyzed. The scaling parameter significantly is shown to substantially affect the solidification rate and the fraction of the liquid phase in the phase transformation region. The developed model and the method of its solution can be generalized to the case of directional solidification of multicomponent melts in the presence of several phase transformation regions (e.g., main and cotectic two-phase zones during the solidification of three-component melts).

On the temperature-dependent diffusion growth of ϕ-Mg5Al2Zn2 ternary intermetallic compound in the Mg–Al–Zn system

The study on the temperature-dependent kinetic behavior of intermetallic compounds (IMCs) in Mg–Al–Zn (AZ) series alloy system is important since its close interrelation with the microstructure evolution under various alloying conditions as well as the T-dependent performance. However, there is a very lack of experimental study on the diffusion kinetics of the ternary IMCs in the AZ series alloy systems. The current work combines the diffusion couple technique and numerical inverse method to investigate the T-dependent kinetic coefficients, i.e., the parabolic growth constant (PGC) and interdiffusion coefficients, for ϕ-Mg5Al2Zn2 ternary IMC in the Mg–Al–Zn alloy system. The Arrhenius formula of both PGC and main interdiffusivities is obtained, i.e., $${}_{{}}^{\phi } k_{p} = 5.48 \times 10^{ - 5} \exp \left( {\frac{120010}{{RT}}} \right)$$ m2/s, $${}_{{}}^{\phi } \tilde{D}_{{{\text{MgMg}}}}^{Al} = 3.48 \times 10^{ - 10} \exp \left( {\frac{50420}{{RT}}} \right)$$ m2/s and $${}_{{}}^{\phi } \tilde{D}_{{{\text{ZnZn}}}}^{Al} = 2.87 \times 10^{ - 7} \exp \left( {\frac{91440}{{RT}}} \right)$$ m2/s, based on which a numerical reproduction of the experimentally determined IMC growth is performed for verification. By comparing the current experimental and calculation results, the rate-controlling factor of the temperature-dependent diffusion growth of ϕ ternary IMC in the Mg–Al–Zn system is further discussed.

Diffusion growth of ϕ ternary intermetallic compound in the Mg-Al-Zn alloy system: In-situ observation and modeling

Study on the diffusion growth of ternary intermetallic compounds in Mg-Al-Zn based light-weight alloys is important due to its close interrelation with alloy property. However, there is a very lack of existing data due to difficulties in both experimental and computational aspects. The current work aims at presenting the experimental observation on the diffusion growth behavior of ϕ phase at 360 °C as well as calculating its composition-dependent interdiffusion coefficients. We designed and successfully fabricated four Mg-τ ternary diffusion couples annealed at 360 °C for different times, where the diffusion path goes across the ϕ phase region and the diffusion growth of ternary intermetallic compound can be solely detected. In-situ observation of the time-dependent growth of ϕ phase was performed to accurately determine the parabolic growth constant. The experimental data were then subjected to a numerical inverse method to generate a set of self-consistent interdiffusivities of the ternary intermetallic compounds, which can reproduce the presently observed diffusion growth behavior of ϕ ternary intermetallic compound in Mg-τ diffusion couples.

An in-situ study on the diffusion growth of intermetallic compounds in the Al–Mg diffusion couple

Study on the diffusion growth of intermetallic compounds in the Al–Mg based light-weight metal materials is important due to its close interrelation with property. The current work aims at solving the existing problems of estimating interdiffusivities of intermetallic compounds, such as the early-stage effect on the accurate determination of parabolic growth constant and the simultaneous calculation of composition- and temperature-dependent interdiffusion coefficient. We present the first usage of High-Temperature Laser-Scanning Confocal Microscopy for effective in-situ observation of the time-dependent intermetallic compound growth in the Al–Mg diffusion couples. The method is verified by successfully recording the growth of Al3Mg2 and Al12Mg17 phases in diffusion couples annealed at 380 °C, 400 °C and 420 °C, and the accurate parabolic growth constant of intermetallic compound are estimated without early-stage effect. The experimental data are then subject to a numerical inverse methods to generate a set of self-consistent interdiffusivities, which can reproduce the observed diffusion growth behavior of intermetallic compounds in the Al–Mg diffusion couples. The applicability of current method to other alloy systems is also discussed.

Interdiffusion study of the topologically closed packed μ phase and the phase boundary compositions in the Fe–Mo system

Fe/Mo bulk diffusion couples are annealed at 1050–1200 °C to study the growth and diffusion behaviour of μ phase in the Fe–Mo system. Parabolic growth constants and integrated interdiffusion coefficients are determined in the μ phase. The activation energies for the growth of μ phase and for the integrated interdiffusion coefficients are estimated as 228±22 and 733±10 kJ/mol, respectively. The relatively high activation energy for interdiffusion in μ phase, determined at high homologous temperature of 0.8–0.9, indicates that μ phase grows by lattice diffusion. Phase boundary compositions, measured by EPMA, indicates the homogeneity range of μ phase to be ~3 at.% higher towards the Fe–rich side (i.e., 36 at.% Mo) as compared to the reported composition of 39–44 at.% Mo in the phase diagram.

SIMULATION MODEL OF THE GROWTH KINETICS OF Fe2B LAYERS WITH CONSIDERATION OF THE BORIDE INCUBATION TIME EFFECT

In this work, a mathematical model for simulating the thermochemical boronizing process is presented. The diffusion model used in this paper is based on Fick’s laws by solving the mass balance equation of the (FeB/Fe2B) interface. In this developed model, the effect of boride incubation times during the formation of Fe2B layers on Armco iron was considered. To demonstrate the validity of our calculations, the simulation results are compared with experimentally obtained data on borided Armco iron, which allowed us to verify the validity of the model. Therefore, a good concordance was observed when comparing the experimental parabolic growth constants taken from the literature with our simulated values of the parabolic growth constants from the present diffusion model. From this study, it has been found that the incubation time has a very important influence on the evolution of the kinetics of the boride layers.

Diffusion multiple study of Co-Ni-Ti system at 1073 K

Co-Ni-Ti system is a key system in many industrial applications. The thermodynamic and kinetic information of this system is crucial to alloy and process design. Although the system has been studied largely, there is still confusion about the stability of the ternary phase τ_Ti(Ni0.5Co0.5)3 phase, and only few data on the diffusion in the intermediate compounds as available. In this work, high throughput diffusion multiple approach was applied to investigate the isothermal section and interdiffusion in this system at 1073 K. The stability of the ternary phase τ_Ti(Ni0.5Co0.5)3 at 1073 K was confirmed based on detailed EPMA element mapping and XRD analysis. A scheme of the isothermal section of the Ni-Co-Ti system at 1073 K including the new findings is presented. The parabolic growth constant of intermediate phases of the binary Ni-Ti and Co-Ti systems at 1073 K were also calculated and the corresponding integrated interdiffusion coefficients were determined using integrated Wagner equations.

Mechanical performance and oxidation resistance of an ODS γ-TiAl alloy processed by spark plasma sintering and laser additive manufacturing

In this work, the influence of Y2O3 additions on the mechanical properties and oxidation resistance of a Ti-45Al-3Nb (at.%) alloy have been studied. In particular, the mechanical properties from 293 K to 1073 K and oxidation resistance at 1073 K of spark plasma sintered and direct metal deposited material have been examined. At room temperature, higher yield stress (+34%) and ultimate tensile strength (+14%) at reduced ductility (−17%) is observed for the oxide dispersion strengthened variant compared to its non-strengthened counterpart. The strengthened variant shows superior strength retention up to 1073 K. Strengthened direct metal deposited material shows similar deformation characteristics as sintered material but suffers from premature fracture due to residual porosity. The addition of Y2O3 increases the oxidation resistance of both sintered and direct metal deposited material. Parabolic growth constants are decreased by −49% and −75% in sintered and direct metal deposited material, respectively. In sintered material the dispersoid size shows only slight changes from 29 nm to 26 nm at 923 K after 987 h and to 32 nm at 1073 K after 924 h demonstrating the high stability of the added particles. TEM analysis reveals abundant grain boundary pinning by the particles contributing to microstructural stability. The results show the potential of oxide dispersion strengthening in titanium aluminides for conventional sintering as well as for additive manufacturing processing routes.

Microstructure, Growth Kinetics and Some Mechanical Properties of Boride Layers Produced on Pure Titanium by Molten-Salt Boriding

To modify the surface properties of pure titanium, boride layers had been fabricated by the boron molten-salt diffusion on pure titanium surfaces in the temperature range of 900-1100 °C for 5- to 30-h treatments. The results demonstrated that the boride layers were mainly composed of TiB whiskers and TiB2 layers without the rutile titanium oxide TiO2. Two diffusion models were introduced to model the growth kinetics of boride layers. The parabolic growth constants and the boron diffusion coefficients were obtained. The boron activation energies for TiB2 and TiB were 225.617 and 165.266 kJ mol−1, respectively. The surface microhardness of the borided titanium decreased with the increase in distance from the surface. The results of wear tests indicated that the wear properties had been improved significantly compared to the pure titanium under dry sliding conditions.

Growth kinetics and some mechanical properties of two-phase boride layers produced on commercially pure titanium during plasma paste boriding

In this study, the plasma paste boriding (PPB) was applied in order to produce the boride layers on commercially pure titanium (Cp-Ti). PPB process was carried out in 50% H2–50% Ar atmosphere under a reduced pressure at various parameters (temperature and duration). The boride layers consisted of two zones: a continuous TiB2 zone close to the surface and a zone with TiB whiskers below the first one. The growth of both zones obeyed the parabolic growth law. The model of growth kinetics of such layers was analyzed. The parabolic growth constants were calculated and the boron diffusion coefficients were determined. Activation energy values of 123.33 and 178.71kJmol−1 were obtained for TiB2 and TiB phases, respectively. The calculated value for TiB2 phase was lower compared to the other boriding processes. Microhardness profile was determined for the layer of the highest depth. The maximal microhardness was characteristic of TiB2 zone (2452HV). Nanomechanical properties (nanohardness and Young's modulus) were studied using the nanoindenter. TiB2 zone was characterized by the nanohardness of 2867.6HV and indentation modulus EIT=383GPa. Wear resistance of plasma paste borided layers was up to 9 times higher when comparing to the untreated Cp-Ti.

Investigation of the diffusion behavior in Sn-xAg-yCu/Cu solid state diffusion couples

The interfacial reactions and growth behavior of the Cu3Sn and Cu6Sn5 phases in Sn-3.0%Ag-0.5%Cu/Cu and Sn-3.5%Ag-0.7%Cu/Cu diffusion couples were investigated at 130 °C, 150 °C, 170 °C and 200 °C and compared with those in binary Sn/Cu diffusion couples using identical experimental conditions. The Ag solubility in the Cu3Sn and Cu6Sn5 phases was found to be quite small. The parabolic growth constants of Cu3Sn and Cu6Sn5 in Sn-xAg-yCu/Cu diffusion couples were calculated. A transient stage for the growth of Cu3Sn was observed. A pseudo-binary method was proposed and performed to obtain the integrated interdiffusion coefficients and activation energies for diffusion of the intermetallics with limited solubility in one of the elements. The growth of Cu3Sn in Sn-xAg-yCu/Cu diffusion couples was found to be a little slower than that in binary Sn/Cu diffusion couples. The growth rate of the Cu6Sn5 phase in ternary Sn-xAg-yCu/Cu diffusion couples is quite close to that in Sn/Cu diffusion couples. The Cu addition in the Sn-based end-member is expected to favor the growth of Cu6Sn5 providing instantly available Cu atoms for the formation of Cu6Sn5. The Ag addition is suggested to slightly decrease the thermodynamic activity of Sn and further decrease the diffusion of Sn. These interpretations were confirmed by a study of the diffusion behavior in Sn-3.5%Ag/Cu and Sn-1.0%Cu/Cu diffusion couples. The calculated activation energy for interdiffusion in the Cu3Sn and Cu6Sn5 phases in Sn-xAg-yCu/Cu diffusion couples are quite close to those obtained for binary Cu/Sn couples. The obtained conclusions are informative for the Ag-Cu-Sn solder design.

Characteristics of Reactive Ni3Sn4 Formation and Growth in Ni-Sn Interlayer Systems

The near-isothermal growth and formation of Ni3Sn4 intermetallic compounds (IMC) in Ni-Sn interlayer systems was studied in the solid state at 473 K (200 °C) and under solid–liquid conditions at 523 and 573 K (250 °C and 300 °C) from an initial state of a few seconds. Scalloped solid-state IMC formation was mainly driven by grain boundary diffusion of Ni through the IMC layer combined with the grain coarsening of the IMC layer. Under solid–liquid conditions, the formation of faceted and needle-shaped Ni3Sn4 grains as well as an atypical IMC growth behavior with similar parabolic growth constants for 523 K and 573 K (250 °C and 300 °C) was observed within the first 180 seconds of the holding time, and IMC growth occurred as an isothermal solidification from the Ni-saturated Sn melt. Due to the progressive densification of the IMC layer and the diffusion-controlled growth, the kinetics slowed down by approximately one order of magnitude after 180 seconds of annealing. The final stage was characterized by the formation of IMC islands ahead of the interfacial Ni3Sn4 layer. Needle-like IMC growth was effectively suppressed under combined solid-state and solid–liquid conditions. Textured Ni3Sn4 IMC formation at the Ni-Sn interface was approved with pole figure measurements. The activation energy Q for solid–liquid IMC formation was calculated as 43.3 kJ/mol, and processing maps for IMC growth and Sn consumption were derived as functions of temperature and time, respectively.

Growth kinetics of boride coatings formed at the surface AISI M2 during dehydrated paste pack boriding

The growth kinetics of the boride coatings (FeB and Fe2B) at the surface of AISI M2 high speed steels were studied in this work. Boriding thermochemical treatment was carried out by dehydrated paste pack at three different temperatures 1173, 1223, and 1273K and four exposure times 1, 3, 5, and 7h, respectively. The presence of FeB and Fe2B phases was identified by scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS) and X-ray diffraction method. In order to obtain the boron diffusion coefficients at the FeB/Fe2B boride coatings, a mathematical model based on the mass balance at the growing interfaces was proposed under certain assumptions. Likewise the parabolic growth constants and the boride incubation time were established as a function of the parameters η(T) and ε(T). The activation energy values estimated for the FeB and Fe2B layers were 233.42 and 211.89kJmol−1 respectively. A good agreement was obtained between the simulated values of boride layer thicknesses and the experimental results. Finally, empirical relationships of boride coating thickness as a function of boriding temperature and time are presented.

Interdiffusion and Reaction between Pure Magnesium and Aluminum Alloy 6061

Using solid-to-solid couples investigation, this study characterized the reaction products evolved and quantified the diffusion kinetics when pure Mg bonded to AA6061 is subjected to thermal treatment at 300°C for 720 hours, 350°C for 360 hours, and 400°C for 240 hours. Characterization techniques include optical microscopy, scanning electron microscopy with X-ray energy dispersive spectroscopy, and transmission electron microscopy. Parabolic growth constants were determined for γ-Mg17Al12, β-Mg2Al3, and the elusive ε-phase. Similarly, the average effective interdiffusion coefficients of major constituents were calculated for Mg (ss), γ-Mg17Al12, β-Mg2Al3, and AA6061. The activation energies and pre-exponential factors for both parabolic growth constant and average effective interdiffusion coefficients were computed using the Arrhenius relationship. The activation energy for growth of γ-Mg17Al12 was significantly higher than that for β-Mg2Al3 while the activation energy for interdiffusion of γ-Mg17Al12 was only slightly higher than that for β-Mg2Al3. Comparisons are made between the results of this study and those of diffusion studies between pure Mg and pure Al [1] to examine the influence of alloying additions in AA6061.

Early stage growth characteristics of Ag3Sn intermetallic compounds during solid–solid and solid–liquid reactions in the Ag–Sn interlayer system: Experiments and simulations

Transient Liquid Phase Bonding (TLPB) makes use of the isothermal solidification of a liquid interlayer through the formation of intermetallic compounds (IMC). This work investigates the Ag3Sn IMC formation in a thin, electroplated Ag–Sn interlayer system appropriate for TLPB – while covering as-coated conditions, growth initiation up to a few minutes of holding time – with experimental and numerical methods. Both solid-state and solid–liquid reactions at 200, 250 and 300°C, respectively, were covered under identical and near-isothermal conditions with a novel experimental setup. Morphological aspects like the lateral IMC grain size or the scallop roughness were proven to strongly depend on temperature. Growth kinetics were captured by power laws, growth exponents n were determined as 0.24, 0.36 and 0.29, parabolic growth constants k as 1.32⋅10−11, 4.04⋅10−10 and 7.24⋅10−10cm2/s at 200, 250 and 300°C, respectively, and the activation energy Q as 30.6kJ/mol. Predominant diffusion mechanisms, i.e. volume or GB diffusion, as well as coarsening effects were identified as temperature and time dependent at early stages. A mathematical model was developed based on the Arrhenius relationship including a growth correction function zIMC,corr to calculate and predict IMC growth as a function of temperature and time. The study was completed by multiphase-field simulations which targeted to mimic kinetic and morphological features of Ag3Sn IMC under various conditions. Good agreement between experiments and simulations was found for a limited set of parameters which allowed drawing conclusions about the dominant transport mechanisms for the reacting elements.

A kinetic model for estimating the boron activation energies in the FeB and Fe2B layers during the gas-boriding of Armco iron: Effect of boride incubation times

The present work deals with a simulation of the growth kinetics of boride layers grown on Armco iron substrate. The formed boride layers (FeB+Fe2B) are obtained by the gas-boriding in the temperature range of 1073–1273K during a time duration ranging from 80 to 240min. The used approach solves the mass balance equations at the two growing fronts: (FeB/Fe2B) and (Fe2B/Fe) under certain assumptions. To consider the effect of the incubation times for the borides formation, the temperature-dependent function Φ(T) was incorporated in the model. The following input data: (the boriding temperature, the treatment time, the upper and lower values of boron concentrations in FeB and Fe2B and the experimental parabolic growth constants) are needed to determine the boron activation energies in the FeB and Fe2B layers. The obtained values of boron activation energies were then compared with the values available in the literature. Finally, a good agreement was obtained between the simulated values of boride layers thicknesses and the experimental ones in the temperature range of 1073–1273K.