Predicted Transition Temperature Research Articles

Effects of the surface energy and surface stress on the phase stability of spin crossover nano-objects: a thermodynamic approach.

Size-induced phase transformation at the nanoscale is a common phenomenon whose understanding is essential for potential applications. Here we investigate phase equilibria in thin films and nanoparticles of molecular spin crossover (SCO) materials. To calculate the size-temperature phase diagrams we have developed a new nano-thermodynamic core-shell model in which intermolecular interactions are described through the volume misfit between molecules of different spin states, while the contributions of surface energy and surface stress are explicitly included. Based on this model, we rationalize the emergence of previously-reported incomplete spin transitions and the shift of the transition temperature in finite size objects due to their large surface-to-volume ratio. The results reveal a competition between the elastic intermolecular interaction and the internal pressure induced by the surface stress. The predicted transition temperature of thin films of the SCO compound [Fe(pyrazine)][Ni(CN)4] follows a clear reciprocal relationship with respect to the film thickness and the transition behavior matches the available experimental data. Importantly, all input parameters of the present model are experimentally accessible physical quantities, thus providing a simple, yet powerful tool to analyze SCO properties in nano-scale objects.

Strong Electron–Phonon Coupling in 3D WN and Coexistence of Intrinsic Superconductivity and Topological Nodal Line in Its 2D Limit

The electron–phonon coupling (EPC) and superconductivity in three‐component‐fermion materials WN, WC, and TaN are systematically investigated using first‐principles calculations. The EPC strength in pristine or pressurized WC and TaN are found to be negligibly weak (<0.2), but can be enhanced significantly by electron doping, and the predicted transition temperatures (Tc) reach the values observed in experiments (≈4 K), demonstrating the vital role of charge doping in the formation of superconductivity. Surprisingly, pristine WN has strong EPC due to a synergistic effect of strong Fermi nesting and large deformation potential and behaves as a good superconductor with Tc of 31 K. Going down from 3D to 2D, WN thin film (i.e., monolayer W3N4) is also predicted to be an intrinsic superconductor with Tc of 11 K. Most importantly, monolayer W3N4 hosts nodal lines that are robust against spin–orbit coupling (SOC), close to the Fermi level and apart from other trivial bands, which are scarce in real materials. The coexistence of high‐transition‐temperature superconductivity and topological states in 3D and 2D WN render them promising platforms for realizing topological superconductivity.

First-Principles Study of Order–Disorder Transitions in Pseudobinary Clathrates

It has been recently demonstrated that the pseudoternary Ba8AlxGayGe46–x–y clathrate undergoes an order–disorder transition with increasing temperature that can be observed via site occupation factors (SOFs) and manifests itself, e.g., in electrical transport properties. Here, we generalize this result and analyze the characteristics of this order–disorder transition in the pseudobinary clathrates Ba8GaxGe46–x, Ba8GaxSi46–x, Ba8AlxGe46–x, and Ba8AlxSi46–x. To this end, we employ atomistic simulations that combine alloy cluster expansions trained against density functional theory calculations with Wang–Landau and ensemble Monte Carlo simulations. The simulations show that all four systems studied here display order–disorder transitions for at least some composition range. Based on an extensive literature survey, we also provide evidence for signatures of the transition in earlier experimental studies that to the best of our knowledge have hitherto not been related to such transitions. The predicted transition temperatures are lower for Ba8GaxGe46–x and Ba8GaxSi46–x than for Ba8AlxGe46–x and Ba8AlxSi46–x, although it appears that the simulations underestimate the transition temperatures for Ga-containing systems compared to the experiment. This nonetheless provides a sensible explanation for why the experimentally determined Al SOFs agree better with the simulated high-temperature disordered configurations, while the Ga SOFs more closely agree with the simulated ground-state configurations. As a result of stronger interactions, the SOFs vary substantially, especially near the stoichiometric 16:30 composition, providing an indication of why it has proved difficult to synthesize Ba8AlxGe46–x and Ba8AlxSi46–x samples at this ratio. The present study thereby yields detailed atomic-scale insights into the ordering in inorganic clathrates that, given the connection to transport properties established earlier, are not only useful from a fundamental perspective but also relevant for applications.

Micromagnetic Characterization of Operation-Induced Damage in Charpy Specimens of RPV Steels

The embrittlement of two types of nuclear pressure vessel steel, 15Kh2NMFA and A508 Cl.2, was studied using two different methods of magnetic nondestructive testing: micromagnetic multiparameter microstructure and stress analysis (3MA-X8) and magnetic adaptive testing (MAT). The microstructure and mechanical properties of reactor pressure vessel (RPV) materials are modified due to neutron irradiation; this material degradation can be characterized using magnetic methods. For the first time, the progressive change in material properties due to neutron irradiation was investigated on the same specimens, before and after neutron irradiation. A correlation was found between magnetic characteristics and neutron-irradiation-induced damage, regardless of the type of material or the applied measurement technique. The results of the individual micromagnetic measurements proved their suitability for characterizing the degradation of RPV steel caused by simulated operating conditions. A calibration/training procedure was applied on the merged outcome of both testing methods, producing excellent results in predicting transition temperature, yield strength, and mechanical hardness for both materials.

Estimation of transition temperature Tc for doped MgB2 superconductors by empirical models

In the present study, some empirical models namely linear, exponential and quadratic regressions are used for the evaluation of predicted transition temperature, Tc for the doped MgB2 superconductors by using the topological descriptors including electrical connectivity index and valence energy level connectivity index. The values of a correlation coefficient (r) and a coefficient of determination (R2) have been measured in these proposed models. The experimental dataset on the 46 different doped MgB2 superconducting transition temperature is compared with the predicted transition temperature estimated from these three different models. Some standard statistical error estimations such as Root Mean Square Error (RMSE) and Mean Bias Error (MBE) are also taken into account to evaluate the performance of these proposed models. However, a quadratic regression model provides a better estimate of the transition temperature as compared to all other models, where the error difference between the measured values and those calculated are minimal for RMSE and MBE.

Breaking time-reversal and translational symmetry at edges of d -wave superconductors: Microscopic theory and comparison with quasiclassical theory

We report results of a microscopic calculation of a second-order phase transition into a state breaking time-reversal and translational invariance along pair-breaking edges of $d$-wave superconductors. By solving a tight-binding model through exact diagonalization with the Bogoliubov-de~Gennes method, we find that such a state with current loops having a diameter of a few coherence lengths is energetically favorable below $T^*$ between 10%-20% of $T_{\mathrm{c}}$ of bulk superconductivity, depending on model parameters. This extends our previous studies of such a phase crystal within the quasiclassical theory of superconductivity, and shows that the instability is not qualitatively different when including a more realistic band structure and the fast oscillations on the scale of the Fermi wavelength. Effects of size quantization and Friedel oscillations are not detrimental. We also report on a comparison with quasiclassical theory with the Fermi surfaces extracted from the tight-binding models used in the microscopic calculation. There are quantitative differences in for instance the value of $T^*$ between the different models, but we can explain the predicted transition temperature within each model as due to the different spectral weights of zero-energy Andreev bound states and the resulting gain in free energy by breaking time-reversal and translational invariance below $T^*$.

Theoretical study of the thickness dependence of the metal-insulator transition in Bi2Sr2Co2O8 nanosheets

The bulk ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{Co}}_{2}{\mathrm{O}}_{8}$ crystal is an insulator at low temperature and experiences an insulator-metal transition at ${T}_{C}\ensuremath{\approx}60\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The new experiment showed that ${T}_{C}$ increases to 140 K for a mechanically exfoliated nanosheet of four blocks and is beyond room temperature for nanosheets thinner than four blocks. We show that the thickness-dependent insulator-metal transition observed in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{Co}}_{2}{\mathrm{O}}_{8}$ nanosheets can naturally be explained by the strongly correlated low-spin-state insulator (LS state) and intermediate-spin-low-spin-state metal (IS-LS state). In particular, the energy difference between the LS state and IS-LS state qualitatively reproduces the trend of the transition temperature with the nanosheet thickness. The predicted transition temperature of a nanosheet with three blocks is only slightly above room temperature, a result that can be used to check our proposed mechanism. Further experiments on the distinct magnetotransport properties and spin-fluctuation behaviors of the LS state and IS-LS state are also very helpful to resolve the issue. The weak interblock binding is also consistent with the layer-resolved partial densities of states.

Ab initio model for GaAs1−xNx chemical beam epitaxy using GaAs(100) surface stability over As2, H2, and N2

A model for the chemical beam epitaxy (CBE) of GaAs1−xNx was previously constructed on the basis of first-principles calculations of (100) surfaces of GaAs with As2 and H2 adsorptions and As/N substitution to simulate As2, H2, and N2 mixed gas conditions. We previously demonstrated that this model can be used to predict the temperature and pressure dependences of the growth behavior in metal–organic chemical vapor deposition (MOCVD). In this paper, we show that little modification is needed to transpose this model to CBE experiments. Our model allows us to predict transition temperatures at which Arrhenius regimes of N2 incorporation are changed. Additionally, an explanation of the trend of resulting regimes is given, which is based on the analysis of surface stability during growth at different temperatures.

Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system.

A superconductor is a material that can conduct electricity without resistance below a superconducting transition temperature, Tc. The highest Tc that has been achieved to date is in the copper oxide system: 133 kelvin at ambient pressure and 164 kelvin at high pressures. As the nature of superconductivity in these materials is still not fully understood (they are not conventional superconductors), the prospects for achieving still higher transition temperatures by this route are not clear. In contrast, the Bardeen-Cooper-Schrieffer theory of conventional superconductivity gives a guide for achieving high Tc with no theoretical upper bound--all that is needed is a favourable combination of high-frequency phonons, strong electron-phonon coupling, and a high density of states. These conditions can in principle be fulfilled for metallic hydrogen and covalent compounds dominated by hydrogen, as hydrogen atoms provide the necessary high-frequency phonon modes as well as the strong electron-phonon coupling. Numerous calculations support this idea and have predicted transition temperatures in the range 50-235 kelvin for many hydrides, but only a moderate Tc of 17 kelvin has been observed experimentally. Here we investigate sulfur hydride, where a Tc of 80 kelvin has been predicted. We find that this system transforms to a metal at a pressure of approximately 90 gigapascals. On cooling, we see signatures of superconductivity: a sharp drop of the resistivity to zero and a decrease of the transition temperature with magnetic field, with magnetic susceptibility measurements confirming a Tc of 203 kelvin. Moreover, a pronounced isotope shift of Tc in sulfur deuteride is suggestive of an electron-phonon mechanism of superconductivity that is consistent with the Bardeen-Cooper-Schrieffer scenario. We argue that the phase responsible for high-Tc superconductivity in this system is likely to be H3S, formed from H2S by decomposition under pressure. These findings raise hope for the prospects for achieving room-temperature superconductivity in other hydrogen-based materials.

Synthesis, crystal structure, and high-temperature phase transition of the novel plumbide Na2MgPb.

A hitherto unknown sodium magnesium plumbide, Na2MgPb, was synthesized by heating the constituent elements. Na2MgPb crystallizes in a hexagonal unit cell with the Li2CuAs-type structure (P63/mmc, Z = 2, a = 5.110(2) Å, c = 10.171(4) Å at 293 K). The compound furthermore displays polymorphism: high-temperature powder XRD measurements revealed that hexagonal Na2MgPb (dubbed the "α" phase) transforms to another hexagonal phase (β) which is existent at 493-553 K, and the β phase changes to a cubic structure (γ) at 533-633 K further. The molar volume of γ-Na2MgPb is approximately 9% and 13% smaller than the molar volumes of the α phase and the β phase, respectively (at 543 K). The electrical resistivity of Na2MgPb is 0.39 mΩ at 300 K; it rises with increasing temperature from 300 to 491 K, and then drops at 491 and 523 K. These abrupt changes in resistivity may be attributed to the α → β and β → γ phase transitions, respectively. To gain further insight into the structure of cubic γ-Na2MgPb, putative models with regular Heusler-type (Cu2MnAl-type) and inverse Heusler-type (Li2AgSb-type) arrangements were probed using first-principles computations based on density functional theory (DFT). These computations indicate that, for the cubic γ phase, an inverse Heusler-type structure is distinctly more stable than the alternative regular Heusler type (at 0 K); beyond that, ab initio thermochemistry was successfully used to verify the stability ordering (α-Na2MgPb being favorable at low temperature, γ-Na2MgPb at high temperature), albeit the theoretically predicted transition temperature of 900 K which is higher than observed in experiment.

Enhancement predicting accuracy for elastin-like polypeptides temperature transition by back propagation neural network.

Elastin-like polypeptides (ELPs) have been widely used to promote the development of a variety of smart biomaterials. Transition temperature is a key attribute of ELPs central to ELPs researches. Therefore, it is essential to establish predictive models of transition temperature that are both computationally efficient and reliable by employing simple parameters. Back propagation neural network (BPNN), a powerful feed-forward algorithm designed to solve problems with overwhelming complexity, has been successfully used in non-linear predictor model. In this study, two pH-sensitive ELPs were expressed, purified and determined for temperature transition across a range of pH. The pH, concentration and molecular weight (MW) as well as isoelectric point (PI) and pseudo amino acid (PseAA) of these two ELPs were adopted as input parameters. Support vector machine (SVM) and back propagation neural network (BPNN) were performed respectively. We selected Lamda (λ) value by training set and evaluated predictor model by jackknife test that combined with Uniform Design (UD). According to the results of BPNN and SVM, whose mean absolute error (MAE) of BPNN model jackknife test were 4.80 and 4.95 respectively, the predictive ability of BPNN is a minor improvement over SVM. Applying Mackay's data, MAE of BPNN jackknife test was 2.02, while the MAE between experimental and predicted transition temperature was 2.30 in Mackay's predictor model. Compared with Mackay predictor model, the enhancement in the accuracy indicates that the proposed BPNN method could play a complementary role to predict ELPs transition temperature.

A comparative first-principles study of martensitic phase transformations in TiPd2 and TiPd intermetallics

Martensitic phase transformations in TiPd2 and TiPd alloys are studied employing density-functional, first-principles calculations. We examine the transformation of tetragonal C11b TiPd2 to the low-temperature orthorhombic phase (C11b → oI6), and the transformation of cubic B2 TiPd under orthorhombic (B2→B19) and subsequent monoclinic transformations (B19→B19′) as the system is cooled. We employ a theoretical approach based on a phenomenological Landau theory of the structural phase transitions and a mean-field approximation for the free energy, utilizing first-principles calculations to obtain the deformation energy as a function of strains and to deduce parameters for constructing the free energy. The predicted transition temperature for the TiPd2 C11b → oI6 transition is in good agreement with reported experimental results. To investigate the TiPd B2→B19 transformation, we employ both the Cauchy–Born rule and a soft-mode-based approach, and elucidate the importance of the coupling between lattice distortion and atomic displacements (i.e. shuffling) in the formation of the final structure. The calculated B2→B19 transition temperature for TiPd alloy agrees well with the experimental results. We also find that there exists a very small but finite (0.0005 eV/atom) energy barrier of B19 TiPd under monoclinic deformation for B19→B19′ structural phase transformation.

Kinetic interactions between hydrogen and carbon monoxide oxidation over platinum

The heterogeneous chemistry coupling of H2 and CO over platinum was investigated experimentally and numerically for H2/CO/O2/N2 mixtures with overall lean equivalence ratios φ=0.13–0.26, H2:CO molar ratios 1:5–3:1, and a pressure of 5bar. Experiments were performed in an optically accessible channel-flow reactor at surface temperatures 510–827K and involved in situ Raman measurements of major gas-phase species concentrations and thermocouple measurements of surface temperatures. Emphasis was placed on the low temperature range 510–600K, whereby H2 inhibited the CO oxidation, and which was of particular relevance to gas turbine idling and part-load operation. Comparisons of measurements with 2-D simulations attested the aptness of the employed kinetic scheme, not only for H2/CO fuel mixtures but also for pure CO. Measured and predicted transition temperatures below which H2 inhibited CO oxidation agreed well with each other, showing a main dependence on the overall equivalence ratio (550±5K at φ=0.13 and 600±5K at φ=0.26) and a weaker dependence on the H2:CO ratio. Furthermore, this inhibition was non-monotonically dependent on the H2:CO ratio, becoming higher at a value of 1:1. The inhibiting kinetic effect of H2 was an outcome of the competition between H2 and CO/O2 for surface adsorption and, most importantly, of the competition between the adsorbed H(s) and CO(s) for surface-deficient O(s). Finally, transient simulations in practical catalytic channels revealed the interplay between kinetic and thermal effects. While at φ=0.13 the H2/CO reactive mixture exothermicity was insufficient to overtake the kinetic inhibition, at φ=0.26 catalytic ignition could still be achieved at temperatures well-below the transition temperature. The effect of H2:CO molar ratio on the light-off times was quite strong, suggesting care when designing syngas catalytic rectors with varying compositions.

Ab initio study of the high‐temperature phase transition in crystalline GeO2

Germanium dioxide (GeO2 ) takes two forms at ambient pressure: a thermodynamically stable rutile-type structure and a high-temperature quartz-type polymorph. Here, we investigate the phase stability at finite temperatures by ab initio phonon and thermochemical computations. We use gradient-corrected density-functional theory (PBE-GGA) and pay particular attention to the modeling of the "semicore" germanium 3d orbitals (ascribing them either to the core or to the valence region). The phase transition is predicted correctly in both cases, and computed heat capacities and entropies are in excellent agreement with thermochemical database values. Nonetheless, the computed formation energies of α-quartz-type GeO2 (and, consequently, the predicted transition temperatures) differ significantly depending on theoretical method. Remarkably, the simpler and cheaper computational approach produces seemingly better results, not worse. In our opinion, GeO2 is a nice test case that illustrates both possibilities and limitations of modern ab initio thermochemistry.

Analytical bond-order potential for Sn

An analytical bond-order potential based on Tersoff-Brenner model for Sn is developed using the Levenberg-Marquardt optimization algorithm with the first-principles calculations on fundamental physical properties of five configurations of Sn, and with the available experimental results. The potential is used to determine the crystal structures, binding energies, bond distances and strengths, and bulk modulus of the β-Sn and body-centered-tetragonal Sn, and to predict the changes in free energy of the α and β phase with temperature. Result indicate that the calculated basic properties of the phases and the predicted transition temperature from α to β phase are in good agreement with the reported experimental data. The proposed interatomic potential is short-ranged and quickly evaluated, and it could be used to distinguish covalent bond from metallic bond in the phases, so it is very useful for large-scale molecular dynamics simulations of Sn-based solder.

Predicting the temperature for the solid–solid phase transition in II–VI semiconductor alloys

At atmospheric pressure, many of the II–VI semiconducting alloys are known to undergo a zinc-blende to wurtzite solid–solid transition. Experimental values for these transitions temperatures have only been reported for two alloys. We show here that chemical potentials for one of the components in a solid solution with the other can be used to estimate the transition temperature. The non-ideal behavior of the solvent component is addressed via an activity coefficient which is determined using the quasi-chemical model. The chemical potentials for each case, zinc-blende and wurtzite are then taken to be equal at the transition temperature. Predicted transition temperatures are reported here for ZnS, CdS, ZnSe, ZnTe, and CdSe. In the case of ZnS and CdS these values agree with the experimental. For ZnSe, ZnTe, and CdSe the values are above their respective melting points. This result suggests that there is no complete bulk transition below the melting point for these three alloys but calculations predict that the two phases are nearly isoenergetic.

Hard-Wired Dopant Networks and the Prediction of High Transition Temperatures in Ceramic Superconductors

I review the multiple successes of the discrete hard-wired dopant network model ZZIP, and comment on the equally numerous failures of continuum models, in describing and predicting the properties of ceramic superconductors. The prediction of transition temperatures can be regarded in several ways, either as an exacting test of theory, or as a tool for identifying theoretical rules for defining new homology models. Popular “first principle” methods for predicting transition temperatures in conventional crystalline superconductors have failed for cuprate HTSC, as have parameterized models based on CuO2 planes (with or without apical oxygen). Following a path suggested by Bayesian probability, we find that the glassy, self-organized dopant-network percolative model is so successful that it defines a new homology class appropriate to ceramic superconductors. The reasons for this success in an exponentially complex (Non-Polynomial Complete, NPC) problem are discussed, and a critical comparison is made with previous polynomial (PC) theories. The predictions are successful for the superfamily of all ceramics, including new non-cuprates based on FeAs in place of CuO2.

An investigation into the low temperature thermal behaviour of vitamin E preparation USP using differential scanning calorimetry and low frequency dielectric analysis.

The thermal and dielectric responses of Vitamin E Preparation USP have been examined to further understand the melting and solidification of this material. A TA Instruments 2920 Differential Scanning Calorimeter was used to examine the thermal response of the sample at a range of scanning speeds. Isothermal dielectric studies were performed using a Novocontrol Dielectric Spectrometer over a range of temperatures down to -70 degrees C and a frequency range of 10(6)-10(-2) Hz. The differential scanning calorimetry (DSC) studies showed an anomalous response whereby at slow heating rates (2 degrees C min(-1)) a small exotherm followed immediately by an endotherm was observed. This response was considerably diminished in magnitude at higher rates (5 degrees C min(-1)) and was not observed at the fastest heating rate of 10 degrees C min(-1). No thermal events were seen on cooling the sample to -60 degrees C. It was suggested that the material formed a glass on cooling, with a predicted transition temperature of approximately -100 degrees C. Further studies using a liquid nitrogen cooling system indicated that the system did indeed exhibit a glass transition, albeit at a higher temperature than predicted (ca -63 degrees C). Low frequency dielectric analysis showed a clear relaxation peak in the loss component, from which the relaxation time could be calculated using the Havriliak-Negami model. The relationship between the relaxation time and the temperature was studied and was found to follow the Vogel-Tammann-Fulcher (VTF) modification of the Arrhenius equation. It is therefore concluded that Vitamin E Preparation USP is a glass-forming material that exhibits kinetically-hindered recrystallisation and melting behaviour. The study has also indicated that DSC and low frequency dielectric analysis may be powerful complementary tools in the study of the low temperature behaviour of pharmaceuticals.

Hard-Wired Dopant Networks and the Prediction of High Transition Temperatures in Ceramic Superconductors

I review the multiple successes of the discrete hard-wired dopant network model ZZIP, and comment on the equally numerous failures of continuum models, in describing and predicting the properties of ceramic superconductors. The prediction of transition temperatures can be regarded in several ways, either as an exacting test of theory, or as a tool for identifying theoretical rules for defining new homology models. Popular “first principle” methods for predicting transition temperatures in conventional crystalline superconductors have failed for cuprate HTSC, as have parameterized models based on planes (with or without apical oxygen). Following a path suggested by Bayesian probability, it was found that the glassy, self-organized dopant network percolative model is so successful that it defines a new homology class appropriate to ceramic superconductors. The reasons for this success in an exponentially complex (non-polynomial complete, NPC) problem are discussed, and a critical comparison is made with previous polynomial (PC) theories. The predictions are successful for the superfamily of all ceramics, including new non-cuprates based on FeAs in place of .

Density-functional study of the thermodynamic properties and the pressure–temperature phase diagram of Ti

The phonon spectra of $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\omega}$ Ti were studied using the supercell approach. The lattice vibrational energy was calculated in the quasiharmonic approximation using both first-principles phonon density of state and Debye model. The thermal electronic contribution to the free energy was evaluated from the integration over the electronic density of states. The Helmholtz energy was thus obtained by combining them with the 0 K total energy calculated within the framework of all-electron projector-augmented-wave method. The thermodynamic properties of $\ensuremath{\alpha}$ and $\ensuremath{\omega}$ Ti calculated by phonon and Debye model are very close to each other. The predicted enthalpy, entropy, bulk modulus, thermal expansion coefficient and heat capacity of $\ensuremath{\alpha}$ are in good agreement with experiments. By comparing with the experimental enthalpy of $\ensuremath{\beta}$, we found that the 0 K total energy calculated from bcc Ti is incorrect. This problem can be solved by shifting the total energy of $\ensuremath{\beta}$ down by $8\text{ }\text{kJ}\text{ }{\text{mol}}^{\ensuremath{-}1}$ to match the experimental value. With the Gibbs energy calculated from the Debye model as a function of pressure and temperature, the phase transformation conditions of $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\omega}$, $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\beta}$, and $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\omega}$ were identified. The predicted transition temperature between $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ at ambient pressure and the triple point are close to experiments. It was found that the entropy plays an important role in the $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\alpha}$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\beta}$ transitions, and the thermal electronic contribution to the Gibbs energy cannot be neglected for studying the $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\beta}$ transition. Our calculations also showed that zero-point energy is crucial to predict the transition pressure of Ti at low temperatures.