Growth Kinetics Equation Research Articles

Size-controlled synthesis of copper oxide nanoparticles with a supercritical hydrothermal synthesis method

Supercritical hydrothermal synthesis (SCHS) is a promising technique for the preparation of copper oxide nanoparticles with fast reaction rate, green efficiency and low cost. In this paper, nanoparticles of copper oxide ranging from 53.0 to 94.0 nm were synthesized by supercritical hydrothermal method. The effects of different precursor concentrations, alkali addition, reaction pressure, reaction temperature and reaction time on the particle size, morphology and particle uniformity of copper oxide nanoparticles were also investigated, and the mechanism of each parameter was analyzed. The growth kinetic equation of copper oxide nanoparticles under supercritical hydrothermal conditions was established by analyzing the influence laws of temperature and reaction time on the grain size of the products. Further, the growth activation energy and the growth mechanism controlled by the grain surface process was obtained. The formation mechanism of copper oxide nanoparticles in rod form was analyzed by TEM results.

Grain growth kinetics and mechanical properties of spark plasma sintered Fe-15Cr-2W powders

In recent years, spark plasma sintering (SPS) technique has emerged as an alternative full density consolidation process as rapid sintering / densification is achieved with restricted grain growth as well as better purification and activation of particle surfaces. In the present investigation, consolidation of mechanically alloyed powder of nominal composition Fe-15Cr-2W at two pressures (50 MPa and 60 MPa) and three sintering temperatures (950 °C, 1025 °C, and 1100 °C) was studied. The consolidated samples were characterized and tested for density, grain size / distribution, hardness, and compressive strength. Consolidated samples were further annealed at 1100 °C for different holding periods (0, 60, and 120 mins) and at different temperatures of 950, 1100, and 1250 °C at a fixed holding period (60 min) to study the grain growth kinetics. The different stages in the displacement profiles during SPS were discussed in depth in view of the densification of the alloys. Pressure was found to be dominant factor in terms of densification at lower sintering temperature (i.e. 950 °C) while at higher temperature (≥1000 °C), temperature activated diffusion was found to be dominant. The grain growth exponent (n) and activation energy (Q) determined for the alloy sintered at 1025 °C and 50 MPa via classical grain growth kinetic equation were 7.34 and 214.9 kJ/mol respectively. The deviation of grain growth exponent from its ideal value (2) has been discussed in light of various attributes of mechanical alloying.

Novel metallic Fe recovery from copper smelting slag by the deep reduction method with renewable biochar reducing agent: Phase transformation process and Fe particle growth optimization

The land stockpiling of copper smelting slag causes serious environmental pollution and resource waste. It contains a large amount of Fe, which is difficult to recover by traditional methods. In this paper, a novel metallic Fe recovery method was presented by deep reduction with renewable biochar as a reducing agent. It not only recovers metallic Fe but also relieves the pressure of iron ore with a sustainable and eco-friendly method. It was found that the metallization ratio variation trend was explored by the temperature, time, CaO content and C/O molar ratio increasing. When biochar was used as a reducing agent, the metallization ratios of Fe under different reduction conditions were better than the traditional reducing agent coal, which indicated that biochar could replace coal as a reducing agent. The phase transition and microstructure change during the deep reduction process were analyzed by XRD, XPS and SEM-EDS. The metallic Fe particles were formed from the copper smelting slag particle outside and spread to the inside gradually, distinguishing the slag phase and metal phase eventually. The growth kinetic equations of the two stages were obtained after calculation and simulation. There's a certain amount of truth to the predicted value because R2 is 0.9745 with experimental measurements. This study demonstrates that the deep reduction is a green and effective technology for copper smelting slag utilization.

Microstructure and thermal stability of nanocrystalline AZ31 Mg alloys reinforced by ultrafine vanadium particles

In this paper, the microstructure evolution and thermal stability of nanocrystalline (NC) AZ31-2.5 wt%VP magnesium matrix composites were studied, while the hardness was analyzed. The results showed that the NC AZ31-2.5 wt%VP still kept NC scale after annealing at 300 °C and 350 °C. Even at 400 °C and 450 °C, the grain size of the annealed composite only grew to 113 nm and 132 nm, which was still in ultrafine grain scale, illustrating excellent thermal stability of NC AZ31-2.5 wt%VP. Furthermore, the V particles, which was uniformly distributed in the Mg matrix, had no obvious change in dimension after annealing, while it also had no new phase formed. The stabilization mechanism was expounded and found that the pinning effect of V particles effectively inhibited the grain growth of Mg matrix at high temperature. In addition, the grain growth kinetics equation (D6−D06=kt) was utilized to explain the grain growth behavior of NC AZ31-2.5 wt%VP, and the activation energy (Ea) of grain growth was calculated to be 130.9 kJ/mol, which was much higher than that of pure Mg (92 kJ/mol). Eventually, the hardness of NC AZ31-2.5 wt%VP was contrastive studied and the hardness evolution curves at different temperatures were obtained. After annealing treatment was performed at 450 °C for 180min, the hardness remained at 80 HV, which was higher than that of as-cast AZ31 (53 HV).

Kinetics of precipitation for graphite particle in high nickel ductile iron

In order to prevent premature failure of high nickel ductile iron used for engine exhaust manifold due to thermal fatigue, the precipitation morphology, nucleation and growth mechanism of graphite particles in high-nickel ductile iron were systematically studied by optical and SEM microscopy and the growth kinetic equation of graphite particles was derived. The results show that the precipitation density and average size of graphite particles within the austenite grain of high-nickel ductile iron are 44.1 particles/mm2 and 2.2 µm, respectively, and the precipitation density and average size of graphite particles on the austenite grain boundaries are increased to 76.6 particles/mm2 and 17 µm, respectively. The main nucleation mechanism of graphite particles in high nickel austenitic ductile iron is grain boundary nucleation. The maximum nucleation rate temperature of graphite particles nucleated on grain boundary is 650–850 °C, the fastest precipitation temperature is close to 680 °C, and the time from the beginning to the end of the growth of graphite particles nucleated by grain boundary is about 3400 s. The average size of graphite particles precipitated by grain boundary nucleation can grow to grade 7 (15–30 µm) under the high temperature of 715–805 °C for a long time (over 3400 s), which is beneficial to the thermal fatigue property of high nickel ductile iron. The local temperature at manifold should not be higher than 800 °C under long times.

Effect of HF, H2SiF6, and Al3+ ions on the crystal growth process and morphology of α-type hemihydrate calcium sulfate in phosphoric acid solution

The reaction of calcium dihydrogen phosphate solution with sulfuric acid in the presence or absence of HF, H2SiF6, and Al3+ ions was used to simulate the crystal growth process of α-type hemihydrate calcium sulfate (α-HH) with 30% P2O5 at a reaction temperature of 95 °C. The changes in Ca2+ ion concentration in the solution were measured at different reaction times. Then, the growth rates and reaction steps of α-HH crystals were fitted using the crystal growth kinetic equation. The growth morphologies of α-HH crystals were characterized using scanning electron microscopy, X-ray diffractometry, X-ray photoelectron spectrometry, and differential scanning calorimetry, and the influence mechanism of HF, H2SiF6, and Al3+ ions on the crystal growth process of α-HH were deeply investigated. Results show that the addition of HF, H2SiF6, and Al3+ ions decreased the growth rate constants of α-HH. Without HF, H2SiF6 and Al3+ ions, the growth rate constants of α-HH was 0.7452 × 10−3 (mol·min−1·m−2). When 0.060 mol·L−1 HF was added, the growth rate constants of α-HH decreased to 0.4954 × 10−3 (mol·min−1·m−2). When 0.015 mol·L−1 Al3+ ions and 0.060 mol·L−1 HF were added, the growth rate constants of α-HH decreased to 0.4548 × 10−3 (mol·min−1·m−2). The inhibition of α-HH growth by HF, H2SiF6, and Al3+ ions can be ranked as follows: HF, H2SiF6, and Al3+ ions > HF and Al3+ ions > H2SiF6 > HF. The reaction steps of α-HH in phosphoric acid solution in the presence or absence of HF, H2SiF6, and Al3+ ions were close to 2, and the growth of α-HH crystals was mainly affected by surface diffusion, adsorption, and helical growth mechanism. HF, H2SiF6, and Al3+ ions inhibited the growth of α-HH crystals along the C-axis, resulting in a decrease in aspect ratio, and the morphology of α-HH crystals is short columnar. These impurity ions decreased the intensity of diffraction peaks and crystallinity of the (200), (310), and (400) crystal planes of the α-HH crystals.

Numerical simulation of precipitation kinetics in multicomponent alloys

• Partial off-diagonal diffusion, involved in the precipitation kinetics equation, improves computational efficiency meanwhile keeping computational accuracy. • The ability to handle sophisticated alloy chemistry and compositional sensitivity of dimensional properties of precipitates, by quantitative equations. • Voronoi construction visualizes the spatial distribution of precipitates in a characteristic cell. • The diffusion distance in precipitation kinetics equation is depicted in a more realistic way by Voronoi construction. A universal numerical model based on the particle size distribution (PSD) approach has been developed for the simulation of precipitation kinetics in multicomponent alloys during isothermal ageing. Nucleation was implemented utilizing the classical nucleation theory (CNT). Growth and coarsening were modeled by a single growth kinetics equation, which is constructed based on the interfacial diffusion flux balance and the capillarity effect. Only partial off-diagonal terms in the diffusion matrix (diffusion of individual components in the matrix) were taken into account in the calculations to minimize the computational cost while coupling with CALPHAD to extract thermodynamics equilibrium around the interface. A new feature of the model is the incorporation of a more realistic spatial site distribution via a Voronoi construction in the characteristic cell, for the purpose of modifying the diffusion distance. Computational predictions of the precipitate dimensions and the precipitation kinetics were compared with the atom probe tomography (APT) measurements on ternary Ni-Al-Cr alloys isothermally aged at 873 K. It is found that the temporal evolution of the dimensions and composition of the precipitates is well captured, as is the dependence on changes in the alloy composition. The new modification with Voronoi construction demonstrates that the overall precipitation kinetics depends on the density and the spatial site distribution of precipitates. The ability to handle sophisticated alloy chemistries by quantitative equations, the compositional sensitivity of microstructural characteristics emerging from the simulation results, and the ability to visualize the spatial distribution of precipitates make the work very promising for multicomponent alloy design and optimization.

Thermodynamic and kinetic analysis of ternary intermetallics formation in Al–Cu–La alloy

The thermodynamic and kinetic analysis of rare earth (RE) intermetallics formation in Al–Cu–La alloy was carried out in this study. The formation enthalpy, excess entropy, excess Gibbs free-energy and activity of each component of Al–Cu–La ternary alloy were calculated combining Miedema model with Toop model. The calculated results show that the formation enthalpy, excess entropy and excess Gibbs free-energy of Al–Cu–La ternary alloy are all negative in the whole alloy composition range. The activities of Al, Cu and La are all tiny in the central zone of composition triangle of Al–Cu–La ternary alloy, indicating that Al, Cu and La components are easy to form ternary intermetallics. Moreover, the parabolic growth kinetics equation of RE intermetallics was derived integrating the Fick's second law with the mass balance law. According to the parabolic growth rate, the thickness of the RE intermetallics increases in proportion to the square root of the diffusion time.

Preparation of α-Al2O3/NiAl multilayer coatings on GH3535 superalloy surface by pack cementation and subsequent in-situ oxidation

Tritium would be produced in the Thorium-based Molten Salt Reactor and then quickly penetrate through the structural material of GH3535 superalloy into environment at elevated temperatures. To suppress the permeation of tritium, the α-Al2O3/NiAl multilayer coatings were prepared on GH3535 by pack aluminizing (PCA) and subsequent low vacuum oxidation as tritium permeation barriers. Herein, the effects of PCA process parameters on the composition and structure of the Ni–Al coatings were characterized, simultaneously, the influence of oxidation temperature on the composition and structure of Al2O3 films were analyzed in detail. Experimental results demonstrated that the main phase of the Ni–Al layers is Ni2Al3 (or NiAl) doped with Al86Cr14. Significant amounts of Mo-aluminides were found on the surface of the samples aluminized at 1050 °C. The Ni–Al layers growth kinetic equation under 650–850 °C was empirically obtained as h = 7.07•104W1/2t1/2exp[-7.53•104/RT]. After low vacuum heat-treatment, the α-alumina films with a thickness of approximately 1 μm were formed at 1050 °C for 2 h, densely and non-porously. Meanwhile, the brittle Ni2Al3 in the transition layer was transformed into the tough NiAl, providing a self-healing capability as well as alleviating the thermal mismatch between the coating and the substrate.

Kinetic study of complicated anti-solvent and cooling crystallization of disodium 5′-ribonucleotide

Disodium 5′-ribonucleotide, which is composed of disodium 5′-inosine (IMP) and disodium 5′-guanosine (GMP), is an important food additive. The lack of kinetic studies of it causes a lack of clarity in understanding the complicated multi-solute crystallization of IMP + GMP in ethanol-water. In this work, process analytical technology tools were used to obtain the thermodynamics and kinetic data from the experiments, the kinetic parameters of anti-solvent and cooling crystallization were investigated. The crystal form of IMP + GMP mixed crystal was determined, which was consistent with the IMP whether crystallized from pure water or ethanol-water. The effects of different anti-solvent addition rates and cooling rates on the metastable zone widths were studied, and the opposite effect on metastable zone width was found. The modified exponential empirical function was developed to correlate nucleation and growth kinetic equations under different conditions. The kinetic data were well fitted with adjusted correlation coefficient (adj-R2 > 0.7), which is sufficient to provide a valid reference for process design and control.

Interface Behavior of Brazing between Zr-Cu Filler Metal and SiC Ceramic

The interface behavior of brazing between Zr-Cu filler metal and SiC ceramic was investigated. Based on the brazing experiment, the formation of brazing interface products was analyzed using OM, SEM, XRD and other methods. The stable chemical potential phase diagram was established to analyze the possible diffusion path of interface elements, and then the growth behavior of the interface reaction layer was studied by establishing relevant models. The results show that the interface reaction between the active element Zr and SiC ceramic is the main reason in the brazing process the interface products are mainly ZrC and Zr2Si and the possible diffusion path of elements in the product formation process is explained. The kinetic equation of interfacial reaction layer growth is established, and the diffusion constant (2.1479 μm·s1/2) and activation energy (42.65 kJ·mol−1) are obtained. The growth kinetics equation of interfacial reaction layer thickness with holding time at different brazing temperatures is obtained.

Corrosion Behavior of TC4 Titanium Alloys in Al–Li Alloy Melt

The influences of Li content on the corrosion behavior of TC4 (Ti6Al4V) titanium alloy were explored when the TC4 titanium alloy was immersed in Al–Li alloy melt containing 0%, 1%, and 2% lithium at 680 °C, 700 °C, and 720 °C for 0.5 h, 1 h, and 2 h. The structure and growth law of the diffusion reaction layer at solid–liquid interface were studied, and the growth kinetic equation of the diffusion reaction layer was established. In addition, Ti content in Al–Li alloy melt was determined and its dissolution rate was calculated. The results showed that with the increase of lithium content in the melt, the thickness of the diffusion reaction layer (DRL) between TC4 titanium alloy and the melt increased significantly, and the activation energies of the diffusion reaction obtained were 141.28 kJ·mol−1 in liquid Al, 86.62 kJ·mol−1 in liquid Al–1Li alloy, and 43.42 kJ·mol−1 in liquid Al–2Li alloy, respectively. The dissolution rate of Ti in Al–Li alloy melts increased with the increase of lithium content in melts. When the holding time reached 3 h in a TC4 crucible, the content of Ti dissolved in the Al–2Li alloy melt was 0.105 wt%.

Biodegradation of phenolic compounds using immobilized Pseudomonas aeruginosa on granular activated carbon: Effect of immobilization, kinetic study and microbial regeneration

Phenol and its derivatives are one of the major pollutants present in industrial effluents. In this work, Pseudomonas aeruginosa (ATCC 9027) has been effectively used to carry out biodegradation of phenol, 3-aminophenol, and catechol. In order to develop an industrially viable technology, P. aeruginosa has been immobilized on granular activated carbon (GAC). The biofilm thickness was found in the range 14 to 34 μm. The adsorbed P. aeruginosa on the GAC is found to be efficient for bio-degradation of the phenolic compounds and in addition they also helped in regeneration of the activated carbon systems which in turn reduce the operability cost. The extent of bio-regeneration varied between 33-66%. Haldane’s growth kinetic equation is found to be suitable to fit the specific growth rate data of P. aeruginosa.

Diffusion bonding of Q345 steel to zirconium using an aluminum interlayer

The microstructure, interface reactions and mechanical properties of the Q345 steel/Al/Zr bonding interface were investigated. The thicknesses of the reaction layer in both Al/Fe and Al/Zr interfaces are temperature-dependent, with a higher temperature yielding a larger thickness. Both experimental results and thermodynamic calculation confirmed that the diffusion transition regions near the Al/Fe and Al/Zr interfaces mainly consist of Al5Fe2 and Al3Zr with thin Al3Fe and Al3Zr2 transitional phases, respectively. The growth kinetic equations of Al5Fe2 and Al3Zr as a function of temperature were investigated. The fitted curves from the equations well agree with the experimental results. The hardness of Al5Fe2 is higher than Al3Zr and the base materials. The propagation of shear crack occurs within Al5Fe2. Optimized shear strength can be obtained by controlling the thickness of reaction layer. The grain growth and fracture mechanisms of Al5Fe2 and Al3Zr at the interfaces were also discussed based on electron back scattering diffraction (EBSD) analysis of both reaction layers and fracture morphology.

A probabilistic model for predicting grain growth processes in alloys

This paper deals with a possible generalization of the traditional grain growth kinetic functions. A novel probabilistic model devoted to the grain growth prediction in alloys is presented. It is based on the extension of the traditional power law kinetic equation proposed by Beck. The Beck’s isothermal kinetic equation is defined as D(t)=A*exp(-E/RT)*tm where D(t) is the grain size (diameter), t is the time, E is the activation energy, A is the preexponential factor, m is the time exponent, R is the universal gas constant and T is the absolute temperature. The novel probabilistic model proposed is based on the concept of the mixture of probability density functions. In this model it is assumed that during grain growth the activation energy (E) is not constant, but its distribution is determined by the initial grain morphology of alloys. More exactly speaking, it is supposed that the activation energy is a random variable which is characterized by the probability density function g(E) of the activation energy. By using appropriately selected probability density functions, different types of generalized grain growth kinetic equations can be established. Based on computer simulations, the application of the extended grain growth kinetic models is demonstrated by examples.

Investigation on the Interfacial IMCs Layer of Sn-58Bi/Cu Solder Joints

The interfacial intermetallic compounds (IMCs) layer of Sn-58Bi lead-free solder reacting with Cu substrate were explored at aging stage after soldering for 1day, 3days, 6days, 8days and 10days, respectively. Results show that Bi doesn’t react with Cu atoms or Sn atoms. The IMCs layer is composed of Cu6Sn5 phase and Cu3Sn phase. The IMCs layer evolves from scalloped shape to layer shape with addition of aging time, meanwhile, Bi phase in matrix grows bigger. The growth kinetic equation of IMCs layer is obtained and described as X=1.06+0.69t1/2, where 0.69μm/d1/2 is the growth coefficient. Compared with Sn-3Ag-0.5Cu lead-free solder, the growth coefficient of Sn-58Bi solder is much smaller. Accordingly, the growth of IMCs layer of Sn-58Bi solder is more slowly than that of Sn-3Ag-0.5 Cu solder.

A new kinetic model for growth based on simultaneous substrate and biomass limitation in solid-state fermentation using agar spheres as the model substrate

Substrate limitation is a major concern in solid-state fermentation, especially for bacteria, as they are not capable of penetrating into the substrate particles. The use of a model substrate, such as agar spheres, makes it possible to study the intra-particle transfer phenomena in solid-state fermentation, reduce the system complexity, and determine the cell dry weight (CDW). A semi-mechanistic model was used to predict the growth of Pseudomonas aeruginosa on agar spheres that contain starch as the only carbon source. The model was developed by introducing a new simultaneous substrate- and biomass-limited growth kinetic equation. The parameters of the growth kinetic were estimated by fitting the experimental data of both the CDW and cumulative CO2 production on agar spheres containing glucose. Statistical analysis (F-test) showed that the new growth kinetic equation was significantly better than the conventional substrate-limited growth kinetics. The model predictions were validated by the experimental data of growth, starch concentration and cumulative CO2 production (CCP) of Pseudomonas aeruginosa growing on agar spheres containing starch in aerated small packed-bed bioreactors. The model confirmed that at high initial starch concentration, free glucose and O2 were not the limiting factors, and bacterial growth is limited by other factors that are related to the amount of biomass in the biofilm.

English

In order to describe and predict the growth and expression of recombinant proteins by using a genetically modified Pichia pastoris, we developed a number of unstructured models based on growth kinetic equation, fed-batch mass balance and the assumptions of constant cell and protein yields. The growth of P. pastoris on both glycerol and methanol could be represented by Monod kinetic equation. A simple simulation methodology and developed models were shown to satisfactorily describe both growth and production of recombinant human serum albumin (rHSA) using a genetically modified P. pastoris MutS strain. The obtained parameters from curve fitting were reasonable and could be acceptable. Moreover, the same parameter sets obtained by the experiments indicated the rigidity and consistency of the developed models and fermentation approach of this study. With correlation coefficients (r2) exceeding 0.99, the models were able to simulate and predict the cell growth behavior and recombinant protein production by P. pastoris without requiring complex models.   Key words: Exponential feed, growth modeling, Monod kinetic equation, Pichia pastoris, recombinant human serum albumin.

On the dynamic compared to static grain growth rate in 3 mole% yttria-stabilized tetragonal zirconia polycrystals (3 Y-TZP)

The reason for the higher dynamic grain growth rate compared to static rate is considered with focus on the results by Nied and Wadsworth on 3 mole% yttria-stabilized zirconia (3 Y-TZP). Included is a review of the models and theories of the pertinent grain growth kinetics and on the concurrent grain boundary cavitation. It is concluded that the same physical mechanism governs both dynamic and static grain growth, and that the existing grain size is an important factor in both cases. It is further concluded that the major factor responsible for the higher dynamic grain growth rate is the pre-exponential in the Arrhenius-type grain growth kinetics equation, the entropy corresponding to the atomic diffusion being an important parameter. There exists insufficient information to ascertain the influence of grain boundary cavitation on the concurrent dynamic grain growth.

Grain growth kinetics for B2O3-doped ZnO ceramics

Abstract Grain growth kinetics in 0.1 to 2 mol % B2O3-added ZnO ceramics was studied by using a simplified phenomenological grain growth kinetics equation Gn = K0 · t · exp(-Q/RT) together with the physical properties of sintered samples. The samples, prepared by conventional ceramics processing techniques, were sintered at temperatures between 1050 to 1250 °C for 1, 2, 3, 5 and 10 hours in air. The kinetic grain growth exponent value (n) and the activation energy for the grain growth of the 0.1 mol % B2O3-doped ZnO ceramics were found to be 2.8 and 332 kJ/mol, respectively. By increasing B2O3 content to 1 mol %, the grain growth exponent value (n) and the activation energy decreased to 2 and 238 kJ/mol, respectively. The XRD study revealed the presence of a second phase, Zn3B2O6 formed when the B2O3 content was > 1 mol %. The formation of Zn3B2O6 phase gave rise to an increase of the grain growth kinetic exponent and the grain growth activation energy. The kinetic grain growth exponent value (n) and the activation energy for the grain growth of the 2 mol % B2O3-doped ZnO ceramics were found to be 3 and 307 kJ/mol, respectively. This can be attributed to the second particle drag (pinning) mechanism in the liquid phase sintering.