Turnbull Equation Research Articles

Fast crystallization below the glass transition temperature in hyperquenched systems.

Many phase change materials (PCMs) are found to crystallize without exhibiting a glass transition endotherm upon reheating. In this paper, we review experimental evidence revealing that these PCMs and likely other hyperquenched molecular and metallic systems can crystallize from the glassy state when reheated at a standard rate. Among these evidences, PCMs annealed below the glass transition temperature Tg exhibit slower crystallization kinetics despite an increase in the number of sub-critical nuclei that should promote the crystallization speed. Flash calorimetry uncovers the glass transition endotherm hidden by crystallization and reveals a distinct change in kinetics when crystallization switches from the glassy to the supercooled liquid state. The resulting Tg value also rationalizes the presence of the pre-Tg relaxation exotherm ubiquitous of hyperquenched systems. Finally, the shift in crystallization temperature during annealing exhibits a non-exponential decay that is characteristic of structural relaxation in the glass. Modeling using a modified Turnbull equation for nucleation rate supports the existence of sub-Tg fast crystallization and emphasizes the benefit of a fragile-to-strong transition for PCM applications due to a reduction in crystallization at low temperature (improved data retention) and increasing its speed at high temperature (faster computing).

The melting behavior of trinitrotoluene nanoconfined in controlled pore glasses

The depression of the melting temperature and heat of fusion of trinitrotoluene (TNT) confined in the nanoscale pores of controlled pore glasses (CPG) were studied by differential scanning calorimetry. 8, 12, 16, 35, and 70 nm pore size CPG were used in the experiment. Both the melting temperature and the heat of fusion of confined nanocrystals decreased with decreasing pore size, which is consistent with previous studies on other materials. When plotting the melting temperature depression as a function of reciprocal pore diameter, an excellent linear fit could be applied to the experimental data points. From the slope of this linear fit, the solid–liquid interface energy of TNT was calculated according to the Gibbs–Thomson equation and found equal to 22.1 ± 0.4 mJ m−2. This is in reasonable agreement with the values calculated from the empirical Turnbull equation and the liquid layer model.

Solid–liquid equilibria, thermochemical and microstructural studies of binary organic monotectic and eutectic alloy

Abstract The phase diagram of 1,4-dibromobenzene (DBB) with pyrogallol (PG) shows the formation of a monotectic and a eutectic alloys at 0.12 and 0.99 mol fractions of DBB, respectively. The phase equilibrium shows the large miscibility gap region with the upper consolute temperature 159.0 °C at 0.55 mol fraction of DBB. Growth kinetics of pure compounds and their monotectic and eutectic at different undercooling (ΔT) obey Hillig–Turnbull's equation: v = u (ΔT)n. Thermodynamic parameters such as enthalpy of mixing, entropy of fusion, interfacial energy, roughness parameters and excess thermodynamic functions were computed based on enthalpy of fusion values obtained from DSC studies. The Cahn wetting condition is applicable for monotectic alloy. The optical microphotographs of binary alloys show lamellar and dendritic features.

Synthesis and characterization of novel binary organic monotectic and eutectic alloys

The phase diagram of biphenyl – 3-hydroxybenzaldehyde (BP–HB) shows the formation of a eutectic with HB-0.935mole% BP and 4-bromochlorobenzene –m-aminophenol (BCB–AP) show the formation of a monotectic (AP-0.150mole% BCB) and a eutectic (AP-0.996mole% BCB). The BCB–AP shows large miscibility gap where the consolute temperature is 12.5°C above the monotectic horizontal. Growth kinetics of pure components, the eutectics and the monotectic were studied by measuring the rate of movement of the solid–liquid interface which suggests the applicability of Hillig–Turnbull's equation. The thermodynamic functions such as heat of mixing, entropy of fusion, roughness parameter, interfacial energy and excess thermodynamic functions were calculated from the enthalpy of fusion values, determined by the DSC method. The relation between interfacial energies of monotectic system satisfies the Cahn's wetting condition. The optical microphotograph of monotectic shows lamellar structure whereas eutectics show broken lamellar and lamellar pearlite structures.

Solid–liquid equilibrium and thermochemical studies of organic analogue of metal–nonmetal system: Succinonitrile–pentachloronitrobenzene

The phase diagram of an organic analogue of a metal–nonmetal system, involving succinonitrile–pentachloronitrobenzene, shows the formation of a eutectic and a monotectic. The two immiscible liquid phases are in equilibrium with a single liquid phase and the consolute temperature being 53.5 °C above the monotectic horizontal. The phase equilibrium study confirms the alloy composition of monotectic and eutectic at 0.150 and 0.985 mol fractions of succinonitrile, respectively. The solidification behaviour shows the validity of Hilling–Turnbull equation. The thermal properties such as heat of mixing, entropy of fusion, roughness parameter, interfacial energy, grain boundary energy and excess thermodynamic functions for parent components, monotectic and eutectic have been studied using their enthalpy of fusion values. The effects of solid–liquid interfacial energy on morphological change of monotectic have also been discussed. The microstructure of monotectic shows the lamellar growth along with droplets, however, eutectic infers the vertical growth of lamella.

Phase-field simulations of α → γ precipitations and transition to massive transformation in the Ti–Al alloy

A phase-field model for the solid–solid α → γ transition of Ti–Al binary alloys is presented based on analytical Gibbs free energies and couplings to the thermodynamical database ThermoCalc. The equilibrium values recover the α + γ phase boundaries. Morphological transitions from diffusive to massive (partitionless) growth are observed on increasing the initial mole fraction of aluminum. Temporal evolution of the interface shows a t behavior for diffusive and a linear behavior for massive growth, which is in accordance with theoretical predictions. An estimate of the interfacial mobility of Ti–Al based on the Burke–Turnbull equation is calculated. The expression of the mobility follows an Arrhenius law. Using the derived interfacial mobility, the calculated interfacial velocities of the massive transformation are in quantitative agreement with those observed in experiments.

Détermination sur lame du statut d’HER2 : bilan en France

Based on the random solution model and stoichiometric intermetallic compound Gibbs free energy model, the driving forces for the crystallization from the undercooled liquid of Cu-Zr binary alloys were evaluated by using Turnbull and Thompson-Spaepen (TS) approximate equations, respectively. Using Davies-Uhlmann kinetic formulations within the frame of continuous nucleation theory (CNT), two groups of time-temperature transformation (TTT) curves of eight compositions of the binary system were calculated, and thereby the corresponding critical cooling rates were obtained. It is shown that both groups of the calculated values for the glass-forming ability (GFA) are in good agreement with the experimental data. The critical cooling rates evaluated by using TS equation are in good agreement with the experimental data compared with that evaluated by using Turnbull equation.

Prediction of the glass forming ability in Cu–Zr binary and Cu–Zr–Ti ternary alloys

By combining CALPHAD technique with kinetic approach, we evaluated the glass forming ability(GFA) of nine compositions of Cu–Zr and thirteen of Cu–Zr–Ti alloys in terms of critical cooling rate and fragility. The driving forces for crystallization from the undercooled liquid alloys were calculated by using Turnbull and Thompson–Spaepen (TS) Gibbs free energy approximate equations, respectively. Time–temperature-transformation (TTT) curves of these alloys were obtained with Davies–Uhlmann kinetic equations based on classical nucleation theory. With Turnbull and TS equations, the critical cooling rates are calculated to be in the range of 9.78 × 10 3–8.23 × 10 5 K/s and 4.32 × 10 2–3.63 × 10 4 K/s, respectively, for Cu–Zr alloys, and 1.38 × 10 2–7.34 × 10 5 K/s and 0.64–1.36 × 10 4 K/s, respectively, for Cu–Zr–Ti alloys. The calculated fragility parameters are in the range of 6.32–9.3 for Cu–Zr binary and 5.7–8.24 ternary alloys, respectively. The comparison of the calculated results to the experimental data indicates that the cooling rate obtained with TS equation is more valid than that with Turnbull equation, and that obtained by other researchers for the evaluation of GFA. The fragility be used as parameter to evaluate relatively the GFA in an alloy system. This work suggests that the combined thermodynamic and kinetic modeling may provide an effective method for the prediction of GFA in Cu–Zr and Cu–Zr–Ti alloys.

Prediction of the Glass-Forming Ability of Cu-Zr Binary Alloys

Based on the random solution model and stoichiometric intermetallic compound Gibbs free energy model, the driving forces for the crystallization from the undercooled liquid of Cu-Zr binary alloys were evaluated by using Turnbull and Thompson-Spaepen (TS) approximate equations, respectively. Using Davies-Uhlmann kinetic formulations within the frame of continuous nucleation theory (CNT), two groups of time-temperature transformation (TTT) curves of eight compositions of the binary system were calculated, and thereby the corresponding critical cooling rates were obtained. It is shown that both groups of the calculated values for the glass-forming ability (GFA) are in good agreement with the experimental data. The critical cooling rates evaluated by using TS equation are in good agreement with the experimental data compared with that evaluated by using Turnbull equation.

Some physicochemical studies on organic eutectics

The phase diagrams of phenothiazine with each of m-nitrobenzoic acid ( m-NBA) and m-dinitrobenzene ( m-DNB) have been studied by thaw-melt method. These materials have been characterized by X-ray diffraction. Growth behavior of the parent components, eutectic and charge transfer complex (CTC) studied by measuring the rate of movement of the growth front in a capillary suggests the applicability of Hillig–Turnbull equation for the system. Microstructure and electrical conductivities of congruent melting complexes and eutectics have been determined. The low electrical conductivities of these materials have been due to weak interaction and mixed stacking of donor and acceptor. Excess thermodynamics functions of the charge-transfer (CT) materials and eutectics have been determined.

Some physicochemical studies on binary organic eutectic and monotectic alloys: p-dibromobenzene- m-aminophenol system

The phase diagram of p-dibromobenzene- m-aminophenol system, determined by the thaw–melt method, shows the formation of a eutectic at 86.8 °C and a monotectic at 120.0 °C at 0.9899 and 0.0496 mole fractions of p-dibromobenzene, respectively. Growth data for the pure components, the eutectic and the monotectic, determined by measuring the rate of movement of solid–liquid interface in a capillary at different undercoolings (Δ T) suggest the applicability of Hillig–Turnbull equation, v= u (Δ T) n , where u and n are constants depending on the nature of materials involved. The values of enthalpy of fusion of the pure components, the eutectic and the monotectic were determined and from these values, the enthalpy of mixing, entropy of fusion, Jackson's roughness parameter and excess thermodynamic functions were calculated. Optical microphotographs of the pure and binary materials show their characteristic features.

Ionic conductivity, glass transition, and local free volume in poly(ethylene oxide) electrolytes: Single and mixed ion conductors

Ethylene oxide based polymer electrolytes which exhibit single-ion (cation Li+ or anion ClO4−) and mixed ion (from the dissociation of LiClO4 salt) conduction were studied by employing positron lifetime annihilation spectroscopy (PALS) and conductivity (σ) measurements in the temperature range between 170 and 370 K. We present experimental evidence for the validity of (i) the linear expansion of local free volume from PALS, (ii) the Vogel–Tammann–Fulcher (VTF) law for σ, and (iii) the Cohen–Turnbull equation that relates σ to the local free volume. These were found to be valid in the temperature range above the end (or freezing) temperature of the glass transition, Tge≈1.06TgPALS≈1.18TgDSC [TgPALS and TgDSC are the Tg’s from PALS and differential scanning calorimetry (DSC), respectively]. From VTF fits to σ we obtained a Vogel temperature of T0≈TgDSC and pseudoactivations energies of B=3.7–5.7 kJ/mol. These parameters disagree with many data published in the literature but are in perfect agreement with the free volume experiments. Moreover, we found T0=TgPALS−(20–28) K and TgDSC=TgPALS−(25–27) K. Indications for the existence of two relaxation processes near Tg were observed in the free volume expansion curves, which were attributed to the motion of free polymer segments and those interacting with ions. The discrepancy between TgDSC and TgPALS can be attributed to the two-phase microseparation of the polymer electrolytes; DSC responds mainly to the polymer segments in the ion-depleted regions while PALS responds to the polymer segments in the ion-rich regions. From the Cohen–Turnbull plots the critical hole volume required for an elementary jump of an ion was estimated to be γν*≈1 nm3 and was found to be independent of the type of ion. This shows that each type of ionic conductivity is associated with the same segmental mobility. Below TgePALS the conductivity is larger than expected from the (extrapolated) VTF law, but smaller than displayed in the frozen-in free volume.

Kinetics of the Coesite-Quartz Transition: Application to the Exhumation of Ultrahigh-Pressure Rocks

The kinetics of the quartz–coesite phase transition has been studied in situ by X-ray diffraction in the 2·1–3·2 GPa, 500–1010°C pressure–temperature range. Analysis of the data within Cahn's model of nucleation and growth at grain boundaries reveals that the prograde and retrograde reactions have different kinetics. The quartz → coesite transformation is one order of magnitude faster than coesite → quartz. Both reactions are characterized by high nucleation rates, so that the overall reaction kinetics is controlled by crystal growth processes. For the coesite → quartz transformation, growth rates are extrapolated using Turnbull's equation with an activation energy for the transition of 163 ± 23 kJ/mol. This kinetic law is combined with an ‘inclusion in a host’ elastic model to study the contribution of kinetics in coesite preservation. This numerical modelling shows that above 400°C retrograde transformation of coesite to quartz is mainly controlled by the ‘pressure vessel’ effect of the host phase, whereas reaction kinetics is the controlling factor at lower temperatures. The influence of the shape of the P–T path and the exhumation rate upon the retrogression of coesite to quartz are investigated to use the percentage of unretrogressed coesite inclusions to constrain P–T–t paths.

Solidification and thermal behaviour of binary organic eutectic and monotectic; succinonitrile–pyrene system

Transparent binary alloy models are important in metallurgical and materials science, as phase transformations can be observed during solidification. This communication concerns the solidification and thermal studies of succinonitrile (SCN)–pyrene (PY) system, which is an organic analogue of a metal–nonmetal-type system. Phase diagram of the SCN–PY system, determined by the thaw-melt method shows the formation of a monotectic and a eutectic at 143.3°C and 55.3°C with 0.025 and 0.744 mole fractions of SCN, respectively. The critical solution temperature of the system lies 48.7°C above the monotectic temperature. The growth velocity ( v) data at different undercoolings obtained from the capillary method, obey the Hillig–Turnbull equation, v= u(Δ T) n . The heats of fusion of the binary as well as single materials were obtained from the DSC (Mettler DSC-4000 system) from which the entropy of fusion, enthalpy of mixing, Jackson's roughness parameter, excess thermodynamic functions, interfacial energy and radius of the critical nucleus were calculated. The optical microphotographs of the eutectic and monotectic show their characteristic features.

Physical chemistry of binary organic eutectic and monotectic alloys; 1,2,4,5-tetrachlorobenzene–α-naphthol and TCB–resorcinol systems

Phase diagrams of 1,2,4,5-tetrachlorobenzene (TCB) with each of α-naphthol and resorcinol determined by the thaw–melt method, show the formation a simple eutectic in the former case and a eutectic and a monotectic with large miscibility gap in the latter system. The growth behaviour of the pure components, the eutectics and the monotectic, studied by measuring the rate of movement of the solid–liquid interface in a capillary at different undercoolings (Δ T), suggests that the growth data obey the Hillig–Turnbull equation, v= u, (Δ T) n , where v is the linear velocity of crystallization and u and n are constants depending on the nature of materials involved. Using the values of enthalpy of fusion, determined by the DSC method (Mettler DSC-4000 system), entropy of fusion, heat of mixing, roughness parameter, excess thermodynamic functions and interfacial energies were calculated. The microstructures of the eutectics and the monotectic show their characteristic features.

Phase Diagram and Growth Behaviour of Durene–resorcinol System

Phase diagram of durene–resorcinol system, determined by the thaw-melt method, shows the formation of a monotectic (0.109 mole fraction of durene) and an eutectic (0.964 mole fraction of durene) with a large liquid miscibility gap in the region from 0.109 to 0.964 mole fraction of durene. The eutectic, monotectic and consolute temperatures are 78.4, 107.8 and 165.0°C, respectively. The growth behaviour studied by measuring the linear velocity of crystallization (v) in a capillary at different undercoolings (ΔT) suggests that the data obey the Hillig–Turnbull equation,v=u(ΔT)n, where u and n are constants depending on the nature of materials involved. From the values of enthalpy of fusion of the pure components, the eutectic and the monotectic determined by the DSC method using Mettler DSC-4000 system, entropy of fusion, enthalpy of mixing, Jackson’sroughness parameter, size of the critical nucleus interfacial energy and excess thermodynamic functions were calculated. The microstructures of the eutectic, and the monotectic, determined by the Leitz Laborlux D optical microscope show their characteristic features.

Physical chemistry of binary organic eutectic and monotectic alloys; 1,2,4,5-tetrachlorobenzene−β-naphthol and 1,2,4,5-tetramethylbenzene-succinonitrile systems

Phase diagrams of 1,2,4,5-tetrachlorobenzene–β-naphthol and 1,2,4,5-tetramethylbenzene–succinonitrile systems which are organic analogues of a nonmetal–nonmetal and a nonmetal–metal system, respectively, show the formation of a simple eutectic (melting point 103.7°C) with 0.71 mole fraction of β-naphthol in the former case and a monotectic (melting point 76.0°C) with 0.07 mole fraction of succinonitrile and a eutectic (melting point 52.5°C) with 0.97 mole fraction of succinonitrile in the latter case. The growth behaviour of the pure components, the eutectics and the monotectic studied by measuring the rate of movement of the solid–liquid interface in a capillary, suggests that the data obey the Hillig–Turnbull equation, v= u(Δ T) n , where v is the growth velocity, Δ T is the undercooling and u and n are constants depending on the nature of the materials involved. From the values of enthalpy of fusion determined by the DSC method using Mettler DSC-4000 system, entropy of fusion, interfacial energy, enthalpy of mixing and excess thermodynamic functions were calculated. The optical microphotographs of pure components and polyphase materials show their characteristic features.

Physical chemistry of binary organic eutectic and monotectic alloys; durene–pyrogallol system

The phase-diagram of durene–pyrogallol system, determined by the thaw-melt method, shows the formation of a eutectic at 79.6°C and a monotectic at 127.7°C containing 0.95 and 0.05 mol fractions of durene, respectively. Growth data for the pure components, the eutectic and the monotectic, determined by measuring the rate of movement of solid–liquid interface in a capillary at different undercoolings (Δ T) suggest the applicability of Hillig–Turnbull equation, v= u(Δ T) n , where u and n are constants depending on the nature of materials involved. The values of enthalpy of fusion of the pure components, the eutectic and the monotectic were determined and from these values, the enthalpy of mixing, size of the critical nucleus, interfacial energy and excess thermodynamic functions were calculated. While optical microphotographs of the pure components showed a faceted microstructure, those of eutectic and monotectic show their characteristic features.

Crystallization kinetics of palm stearin in blends with sesame seed oil

AbstractThis study investigates the crystallization kinetics of palm stearin (PS), a palm oil fraction, in blends with sesame seed oil. The results indicate that the crystallization behavior of PS in sesame oil is mainly associated with the crystallization of tripalmitin. Therefore, crystallization of blends of 26, 42, 60, and 80% (wt/vol) PS in sesame oil was described by equations developed for simpler systems (e.g., Fisher and Turnbull equation). The isothermal crystallization, melting profile, and fitting of the kinetics of nucleation to the Fisher and Turnbull equation showed that the 26, 42, and 60% PS/sesame oil blends crystallized mainly in the β1′ polymorph state. In contrast, the 80% blend crystallized in two different polymorph states (i.e., β1′ at T≼307.6 K and β1 at T≽308.2 K). The data indicated that, in spite of the higher concentration of PS in the 80% PS/sesame oil system, crystallization in the β1 state required more free energy for nucleation (δGc) than β1′ crystallization in the 26, 42, and 60% PS/sesame oil. At the low cooling rate used (1 K/min) it was observed that, for a particular PS blend, the higher the effective supercooling the higher the viscosity of the oil phase and the smaller the induction time of crystallization (Ti). Additionally, the β1′ crystals from PS, developed at the highest effective supercooling investigated, were smaller than the β1 crystals obtained at lower effective supercooling.

Some Physicochemical Studies on Organic Eutectics and Molecular Complex: Urea – p-nitrophenol System

The phase diagram of urea–p-nitrophenol system, in the form of a temperature-composition curve, shows the formation of a 1: 1 molecular complex surrounded by two eutectics containing 0.216 and 0.777 mole fraction of p-nitrophenol. Data on growth velocity (v), obtained by measuring the rate of movement of the interface at different undercoolings (ΔT), suggest that they obey the Hillig–Turnbull equation, i.e., v = u (ΔT)n, where u and n are constants depending on the nature of materials involved. From the heat of fusion values, determined by the differential scanning calorimetry (DSC) method, heat of mixing, entropy of fusion, roughness parameter, interfacial energy, radius of the critical nucleus, and the excess thermodynamic functions were calculated. While the x-ray diffraction data show that the eutectics are not mechanical mixtures of the components under investigation, the microstructural investigations give their characteristic features.