Avrami Exponent Research Articles

Jeziorny Method Should Be Avoided in Avrami Analysis of Nonisothermal Crystallization.

The Jeziorny method treats nonisothermal crystallization data by replacing the variable temperature (T) values with the corresponding values of time and substituting them into the isothermal Avrami plot, ln[-ln(1 - α)] vs. lnt. For isothermal data, the slope of this plot is the Avrami exponent, n and the intercept is the rate constant, kA. This does not hold for nonisothermal data. Theoretical analysis suggests that in the case of nonisothermal data the intercept cannot be interpreted as kA, and its "correction" by dividing over the temperature change rate β is devoid of any meaning. In turn, the slope cannot be interpreted as n. It is demonstrated that the slope changes with time and its value depends not only on n but also on the temperature, temperature range, and activation energy of crystallization. Generally, the value of the slope is likely to markedly exceed the n value. The theoretical results are confirmed by analysis of simulated data. Overall, the Jeziorny method as well as other techniques that substitute nonisothermal data into the isothermal Avrami plot should be avoided as invalid and useless for any reasonable Avrami analysis. It is noted that n can be estimated from the nonlinear plot of ln[-ln(1 - α)] vs. T.

Crystallization of LiNbO3 and NaNbO3 in niobiosilicate glass–ceramics

AbstractAlthough glass–ceramics have been widely explored for their thermal stability and mechanical properties, they also offer unique symmetry‐dependent properties such as piezoelectricity and pyroelectricity through controlled crystallization of a polar phase. This work examines crystallization of LiNbO3 in a 35SiO2–30Nb2O5–35Li2O mol% composition and crystallization of LiNbO3 and NaNbO3 in a 35SiO2–30Nb2O5–25Li2O–10Na2O mol% composition. Crystallization kinetics are examined using the Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory where the Avrami exponent, n, is calculated to be 1.0–1.5. Microscopical analysis shows dendritic morphology, which when combined with the JMAK analysis, suggests diffusion‐controlled one‐dimensional growth. Adding Na2O to the glass composition increases the inter‐diffusivity of ions which causes LiNbO3 to crystallize faster and lowers the activation energy of transformation from 1054 ± 217 kJ/mol in the ternary composition to 882 ± 212 kJ/mol. Time‐temperature‐transformation diagrams are presented which show that the temperature for maximum rate of transformation for LiNbO3 is ∼650°C and for NaNbO3 is ∼715°C.

Crystallization analysis and determination of Avrami exponents during isothermal annealing and the effect of cooling rate on the evolution of the atomic structure of Pd78Si16Cu6 alloy: A molecular dynamics simulation study

In this study, molecular dynamics simulations have been carried out to investigate the effect of cooling rate on the atomic structure of the Pd78Si16Cu6 alloy and the isothermal annealing process on the evolution of its glassy structure. It is observed that the atomic structure, glass transition temperature and crystallization temperature of the system are affected significantly by the cooling rate. While the liquid system cooled at slower cooling rates than 1 K/ps transitions to a crystalline structure, a glassy structure is observed for faster cooling rates. The crystallization kinetics of the annealing process are explained by the Johnson, Mehl and Avrami model using Honeycutt–Andersen crystalline-type bonded pairs. The findings show that the system transitions from a metastable phase to a stable crystalline phase during isothermal annealing of the amorphous Pd78Si16Cu6 alloy, and increasing the annealing time for lower temperature increases the thermal stability of the amorphous phase.

Effects of Pressure on Homogeneous Nucleation and Growth during Isothermal Solidification in Pure Al: A Molecular Dynamics Simulation Study

Effects of different pressures on the isothermal-solidification process of pure Al were studied by molecular dynamics (MD) simulation using the embedded-atom method (EAM). Al was first subjected to a rapid-cooling process, and then it was annealed under different pressures conditions. Mean first-passage times (MFPT) method, Johnson-Mehl-Avrami (JMA) law, and X-ray diffraction (XRD) simulation analysis method were used to qualify the solidification- kinetic processing. Nucleation rate, critical-nucleus size, Avrami exponent, growth exponent, and crystallite size were calculated. Results show that the nucleation rate increases as the pressure increases. The change of critical-nucleation size is not obvious as the pressure increases. With the pressure increasing, growth exponent decreases, indicative of decreased grain-growth rate. It was also found that with the pressure increasing, the Avrami exponent decreases, indicating that the increased pressure has an effect on growth modes during solidification, which changes from three-dimensional growth to one-dimensional growth. Results of XRD simulation shows that with pressure increasing, crystallite size decreases.

Isothermal Crystallization of Poly(ε–Caprolactone) PCL in Three‐Armed Copolymers

AbstractBranched copolymers are a special class of polymeric materials in which are reflected the combined effects of polymer segments and architectural constraints of the branched architecture. In this work, three‐armed graft copolymers, poly (hydroxyethyl methacrylate‐graft‐poly(caprolactone), [P(HEMA‐g‐PCL)]3, are synthesized by combination of reversible addition‐fragmentation chain‐transfer (RAFT) and ring opening polymerization (ROP) mechanisms in a one‐pot/one‐step protocol. The resulting macromolecules are characterized by 1H Nuclear Magnetic Resonance (NMR), size exclusion chromatography (SEC), differential scanning calorimetry (DSC), and thermogravimetric analysis (TGA). The preliminary results indicate the success of the methodology and are in agreement with literature reports. In addition, DSC isothermal crystallization are performed, and Avrami's theory is employed in order to obtain kinetics parameters of interest, such as the half‐life for the crystallization process (t1/2), the bulk crystallization constant (k), and the Avrami's exponent (n). Thermal analysis evidences a noticeable reduction in the melting temperature compared with linear PCL homopolymer. However, the presence of branches does not modify the values of the maximum degradation temperatures significantly. Finally, the synthesized copolymers adopt a two‐dimensional crystallization type (disk and cylindrical).

Study of Nonisothermal Crystallization Kinetics of Unstretched and Uniaxially Stretched Electroactive PVDF Composite Films

AbstractIncorporation of graphene‐based fillers such as graphene oxide (GO) and reduced graphene oxide (rGO) into polyvinylidene fluoride (PVDF) followed by uniaxial stretching can significantly improve their electroactive phase content as well as the dielectric properties. In this study, the nonisothermal crystallization kinetics of unstretched and uniaxially stretched PVDF‐GO and PVDF‐rGO composite films are investigated through differential scanning calorimetry (DSC) at different cooling rates. Avrami‐modified Jeziorney's model is used to understand their crystallization behavior which leads to the nucleation of electroactive crystalline phases in PVDF. Avrami's exponent “n” values in the range of 2–4 are obtained for nonisothermal crystallization of PVDF in different samples which reveal both two‐ and three‐dimensional growth of spherulites in the crystalline region. Kissinger's method is used to calculate the activation energy of crystallization and the results reveal accelerated crystallization of PVDF after addition of GO and rGO fillers followed by uniaxial stretching.

Anomalous Avrami index recorded during the non-isothermal crystallization of borotitanosilicate glass powders

We investigated the crystallization kinetics of 24.5CaO-24.5SrO-3.7B2O3-6TiO2-41.3SiO2 and 25CaO-25SrO-3.8B2O3-8TiO2-42.2SiO2 (mol%) borotitanosilicate glass powders (≤ 43µm), herein referred as T6 and T8, respectively, using differential scanning calorimetry (DSC - β = 2.5°C/min – 40°C/min) and scanning electron microscopy (SEM). The kinetic parameters were calculated through model-free and model-fitting approaches. Despite minor variations in the corresponding activation energy, it was found that the local Avrami exponents changed significantly, and multiple-step reactions govern the crystallization process (with n ≥ 2 for 0.05 ≤ α ≤ 0.95). These unexpected n values were attributed to the complexity of the crystallization process, where crystals grow individually and three-dimensionally in the early stages, thereafter tending to grow in a single direction (1D) after meeting each other.

Microstructural evaluation and mechanical properties of bulk nanostructured Al86Cu6Y6La2 (at.%) alloy produced by hot consolidation of amorphous melt-spun flakes

Amorphous melt- spun Al86Cu6Y6La2 (at.%) alloy is hot consolidated at T = 723 K and P = 700 MPa. The glassy ribbons show three devitrification steps which fcc-Al phase is precipitated during the first crystallization stage, and formation of intermetallic compounds (Al3Y, AlCu, AlCu3 and Al3Y5) take place in the next steps. The calculated Avrami exponent (n ∼ 2.6) shows the first crystallization reaction is governed by the diffusional growth process and increasing nucleation rate. Moreover, the activation energy calculated with different methods show that Ea for the primary crystallization are in the range of about 254- 267 KJ.mol−1. Amorphous ribbons exhibit a remarkably tensile strength of 910 ± 7 MPa. Combined rapid solidification and hot consolidation produce a bulk sample with high hardness (345.1 ± 12.3 HV), compressive strength (460 ± 10 MPa) and 4 pct plastic strain. These high mechanical properties are due to ultrafine microstructure induced by the rapid solidification process.

On the Effect of Nucleation Undercooling on Phase Transformation Kinetics

We carry out an extensive comparison between Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory of first-order phase transformation kinetics and phase-field (PF) results of a benchmark problem on nucleation. To address the stochasticity of the problem, several hundreds of simulations are performed to establish a comprehensive, statistically significant analysis of the coincidences and discrepancies between PF and JMAK transformation kinetics. We find that PF predictions are in excellent agreement with both classical nucleation theory and JMAK theory, as long as the original assumptions of the latter are appropriately reproduced—in particular, the constant nucleation and growth rates in an infinite domain. When deviating from these assumptions, PF results are at odds with JMAK theory. In particular, we observe that the size of the initial particle radius \(r_0\) relative to the critical nucleation radius \(r^*\) has a significant effect on the rate of transformation. While PF and JMAK agree when \(r_0\) is sufficiently higher than \(r^*\), the duration of initial transient growth stage of a particle, before it reaches a steady growth velocity, increases as \(r_0/r^*\rightarrow 1\). This incubation time has a significant effect on the overall kinetics, e.g., on the Avrami exponent of the multi-particle simulations. In contrast, for the considered conditions and parameters, the effect of interface curvature upon transformation kinetics—in particular negative curvature regions appearing during particle impingement, present in PF but absent in JMAK theory—appears to be minor compared to that of \(r_0/r^*\). We argue that rigorous benchmarking of phase-field models of stochastic processes (e.g., nucleation) needs sufficient statistical data in order to make rigorous comparisons against ground truth theories. In these problems, analysis of probability distributions is clearly preferable to a deterministic approach.

An environmentally benign and sustainable process for carbon recovery and efficient defluorination of spent carbon cathode

A systematic and green low-temperature sulfation roasting—water leaching strategy was put forward to achieve a very high fluorine removal rate of 97.82% for spent carbon cathode (SCC), which was believed as a hazardous solid waste. And the carbon could be recycled with a purity of 90.29 wt.% in the flaky microstructure. Thermodynamic analysis and the results of SEM, XRD and EDS indicate that most of the fluoride could convert into water-soluble sulfate at low temperature. And the highest fluorine removal rate could be obtained when <0.15 mm SCC particles were mixed with sulfuric acid at a liquid-to-solid ratio of 1:1, and then roasted at 300 °C for 0.5 h. The sulfate was removed to purify the carbon via water-leaching process. Avrami exponents and corresponding activation energy for the roasting and leaching process demonstrated that both processes are controlled by diffusion.

Nucleation of recrystallization in magnesium alloy grains of varied orientation and the impacts on texture evolution

Selection of appropriate thermomechanical processing parameters is crucial for control of the crystallographic texture and grain size distribution in wrought magnesium alloys, which in turn dictate their mechanical properties. In this work, the recrystallization behavior of binary Mg-Zn alloys are examined after warm rolling as a function of alloy content and heat treatment temperature. The Avrami exponent increased with increasing heat treatment temperature up to the MgZn solvus, while the strength of the recrystallization texture decreased. Neither the Avrami exponent nor the qualitative trends in texture evolution were sensitive to the Zn alloying content in the examined range. EBSD and transmission electron microscopy observations revealed that the changes in texture and recrystallization kinetics could be attributed to differences in the nucleation behavior of basal and off-basal orientations.

Evaluation of novel composition (SiO2–CaO–Na2O–P2O5, SrO–CuO) degradable amorphous silicate glasses properties and comprehensive characterization of the thermal features

The current work presents and discusses the findings of a comprehensive study on the structural, chemical and thermal properties of SrO and CuO incorporated SiO2–CaO–Na2O–P2O5 amorphous silicate glass with a novel composition. Here, fundamental features (experimental density, oxygen density, and hardness) of all glasses were determined and chemical as well as phase composition of the glasses was verified with XRF and XRD, respectively. Moreover, the thermal behavior (viscos flow and crystallization kinetics) of amorphous silicate glass was investigated by non-isothermal methods using DTA analysis. The activation energies of glass transition (Eg) were calculated in the range of 546–1115 kJ/mol by Kissinger method, whereas the activation energies of crystallization (Ec) were calculated in the range of 164–270 kJ/mol by three different methods (Kissinger, Ozawa, Yinnon and Uhlmann). Avrami exponent (n) values ranged from 1.17 to 3.28 demonstrated that amorphous silicate glasses have different crystallization mechanism. Working temperature, which is one of the parameters indicating glass stability, increased with the incorporation of Sr and Cu from 187 °C to 245 °C. The initial dissolution measurement has been applied to study the degradability behavior of Sr and Cu incorporated amorphous glasses in vitro. Quantitative evaluation of Si4+ (0.156–0.373 kV), Ca2+ (0.043–0.332 kV), Na+ (0.044–0.329 kV), P5+ (0.057–0.289 kV), Sr2+ (0.134–0.385 kV), and Cu2+ (0.090–0.203 kV) depending on the ion activation energy (Ea-ion) and ion concentration at different temperature values (24, 37 and 55 °C) was performed in contact with Tris-HCl solution by ICP-OES analysis. The results revealed that investigated glasses were degradable and incorporation of Sr and Cu affected the glass initial dissolution. Overall, investigated glasses are suitable for various application such as hot-working production, glass-ceramic manufacturing, and glass or glass-ceramic scaffolds fabrication, due to wide working temperature ranges and high crystallization tendencies of the developed glasses.

Transformation kinetics for low temperature post-deposition crystallization of TiO2 thin films prepared via atomic layer deposition from TDMAT and water

Background: We report on the fundamental crystallization kinetics of atomic layer deposited (ALD) TiO2 thin films undergoing a post-deposition anneal (PDA) at low temperatures to probe differences in the as-deposited film microstructure. Methods: The system of study is ALD TiO2 thin films prepared from tetrakis(dimethylamino)titanium(IV) (TDMAT) and water at 120 °C, 140 °C and 160 °C followed by ex situ low temperature annealing at temperatures ranging from 140 °C to 220 °C. All as-deposited TiO2 thin films are amorphous by X-ray diffraction (XRD). Post-deposition annealing (PDA) produces large grain anatase crystals, confirmed by XRD and top-view scanning electron microscopy (SEM). A detailed SEM study is performed to quantify the nucleation and growth kinetics by fitting microstructural data to the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation. Finally, a time-temperature-transformation (TTT) diagram is constructed to summarize the differences in crystallization behavior at different ALD deposition temperatures. Results and Conclusions: Fitting microstructural data to the JMAK equation reveals an Avrami exponent close to 3 with continuous nucleation, suggesting two-dimensional, plate-like crystal growth. Applying an Arrhenius relationship to the phase transformation data, the combined activation energy for nucleation and growth is found to be 1.40–1.58 eV atom-1. Nucleation rates are determined, and an Arrhenius relationship is used to calculate the critical Gibbs free energy for nucleation (~1.3-1.4 eV atom-1). As such, nucleation is the rate-limiting step for the amorphous to anatase phase transformation. ALD growth temperature is found to dictate film microstructure with lower deposition temperatures reducing the nucleation rate and leading to larger grain sizes irrespective of PDA conditions. The nucleation rate pre-exponential frequency factor increases with increasing deposition temperature, thereby increasing the likelihood for nucleation. Interestingly, it is this difference in the vibrational modes of the amorphous structure, as indicated by the variation in the nucleation rate pre-exponential frequency factor, that alters the phase transformation rates and not a change in the activation energies for the transformation.

Crystallization behaviors and kinetic of argon oxygen decarburization slag via various SiO2 content: Conditions for obtaining amorphous state

The crystallization behaviors and morphology of industrial AOD slag with various SiO2 levels were studied. The time-temperature-transformation (TTT) and continuous-cooling- transformation (CCT) diagrams of argon oxygen decarburization (AOD) slag were obtained using confocal laser scanning microscopy (CLSM). The results show that the main crystalline phases evolved from dicalcium silicate, merwinite, and bredigite to an amorphous state when the dosage of SiO2 increased to 15 wt% in the quenching slag. Crystallization behaviors of AOD slag follow the Johnson-Mehl-Avrami (JMA) crystallization kinetic model. The Avrami exponent and the activation energy increase to 2.382 and 83.68 kJ/mol, respectively in the 15 wt% SiO2 samples. Thus, the crystal size shrinks, and the crystallization capacity of slag decreases. The critical cooling rate reaches the minimum value of 2052.5±34 °C/min in the 15 wt% SiO2 samples. This research showed that AOD slag modified by SiO2 can enhance the potential of preparing supplementary cementitious materials for added-value applications.

Crystallization behavior of a series of Zr-based metallic glasses

A series of Zr-Al-Ni-Cu metallic glasses were elaborately designed by combination of multiple optimization strategies and their nonisothermal and isothermal crystallization kinetics were systematically investigated. The results show that the compositions of these metallic glasses are off-eutectic. They have similar crystallization behaviors including glass transition temperature, crystallization temperature, single-step crystallization process, crystallization activation energy and Avrami exponent. The glass forming ability of these metallic glasses is almost equivalent, but slightly decreases with increasing Zr content. The crystalline phases are hp-Zr6Al2Ni and t-CuZr2 in both nonisothermal and isothermal cases for the partially and fully crystallized samples. The content of hp-Zr6Al2Ni decreases with increasing Zr content, while opposite for that of t-CuZr2. The crystallization mechanism is the growth of particles with decreasing nucleation rate for the studied Zr-based metallic glasses. This work would shed light on discovering the quantitative correlation between the microstructure and the crystalline phases for the metallic glasses.

Effects of Cooling Rate on the Solidification Process of Pure Metal Al: Molecular Dynamics Simulations Based on the MFPT Method

Isothermal solidification process of pure metal Al was studied by molecular dynamics (MD) simulation using EAM potential. The effects of different cooling rates on the isothermal solidification process of metallic Al were studied. Al was first subjected to a rapid cooling process, and then it was annealing under isothermal conditions. The mean first-passage times (MFPT) method and Johnson-Mehl-Avrami (JMA) law were used to qualify the solidification kinetic processing, and the nucleation rate, critical nucleus size, Avrami exponent and growth exponent of grains were calculated. Results show that the nucleation rate and critical size decrease as the cooling rate increases. Also, an increase in the cooling rate leads to the increase of grain growth rate. At all investigated cooling rates, nucleation and growth processes are in the typical three-dimensional growth mode.

Physics of Directional Polymer Crystallization.

It has been proposed that the nonisothermal directional crystallization of a polymer driven by a moving sink has an exact analogy to an equivalent isothermal crystallization protocol. We show that this is substantially true because polymers are poor thermal conductors; thus, polymer crystallization occurs over a relatively narrow spatial regime, while the thermal gradients created by this freezing occur over a much broader scale. This separation of scales allows us to replace the crystallization process, which is spatially distributed, with an equivalent step. The temperature at this step, which corresponds to the desired equivalent isothermal crystallization temperature, scales linearly with sink velocity. However, a few metrics, such as the Avrami exponent characterizing the kinetics of crystallization are very different in the two cases. These findings provide new insights into the physics of these spatially varying crystallization protocols and should inspire new experiments to probe the underlying equivalences more deeply.

Thermal Analysis of the Mechanism and Kinetics Parameters of the Metastable Phases Precipitation in the Al-Mg-Si Alloy

In this experimental study, the mechanism and kinetic parameters of the metastable phases precipitation in the Al-Mg-Si alloy were determined thermally by differential scanning calorimetric (DSC) analysis. All samples were treated up to 550 °C at heating rates of 5, 10, 20 and 30 °C/min. The apparent activation energy (56.74 kJ/mol) and the Avrami exponent (0.99), were determined by DSC from the non-isothermal method, using the Ozawa, Boswell and Kissinger methods while those obtained by isothermal method using the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model were 51.04 kJ/mol and 1.18. The activation energies values indicate that the formation of the metastable phases was mainly controlled by the migration of Mg and Si. The values of n, are characteristics of a growth of plate after saturation of nucleation. The frequency factor (ko) calculated by the isothermal method is found to be 8.36×107 s-1.

Transient nucleation in the rapid crystallization of Mg65Zn30Ca5 metallic glass revealed by nanocalorimetry

Due to the limited heating ability of conventional calorimetry, nucleation during the rapid crystallization remains ambiguous. In this study, nanocalorimetry was used to ascertain the role of nucleation in the rapid crystallization of Mg65Zn30Ca5 (at.%) metallic glass based on the Johnson-Mehl-Avrami (JMA) model. The rapid quenching rate of nanocalorimetry facilitates the in situ preparation of the amorphous phase, and the number of nuclei preserved in the sample can be tuned. On heating, transient nucleation rather than steady-state nucleation is considered as the dominant mechanism for rapid crystallization. With an increase in quenching rate, more nuclei are produced in the sample, contributing to the enhanced transient nucleation upon reheating. A larger deviation of nucleation rate from the steady-state nucleation is thus caused, which is revealed by a higher Avrami exponent. With the increase in heating rate, transient nucleation is enhanced, and the transition from transient nucleation to steady-state nucleation is postponed.

Recrystallization kinetics of cold rolled Gum Metal

A wire of a β-Titanium alloy known as Gum Metal was cold rolled to a reduction of 80% and heat treated for times in the range 3-120 minutes at a fixed temperature of 740 °C to yield a set of partially recrystallized microstructures. The recrystallization course was evaluated by light optical microscopy and Electron Backscatter Diffraction, based on which the recrystallization kinetics was analysed using the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model. This analysis showed that the kinetics of this material does not follow the traditional JMAK behaviour but has two distinct stages with very different Avrami exponents.