Order-disorder Phase Transformation Research Articles

Colloid and nanoparticle-driven phase behavior in weakly perturbed nematic liquid crystals

We study theoretically weakly colloidal particle or nano-particle perturbed nematic liquid crystal (LC) order. We consider cases where the particles are essentially homogeneously dispersed in LC matrix. A mesoscopic Landau-de Gennes approach is used where we focus on the temperature variations in the amplitude order parameter field for different particle volume concentrations ϕ. In the dilute regime, we describe the nematic order in terms of interface and volume amplitude order fields. This modeling suggests that the temperature-driven order-disorder crossover exhibits a two-step transformation. In the case that particles locally support LC order, on decreasing temperature firstly the interface order is established, followed by the build-up of volume order. The latter appears via the 1st order phase transition. On the other hand, the interface order growth is formed either gradually or via the 1st order interfacial transition. In the regime of relatively high values of ϕ the system can be effectively described in terms of a single amplitude order parameter field. In this case, the temperature-driven phase behaviour is qualitatively similar to the interface order temperature response in the diluted regime. We analytically express phase transition temperatures and determine critical points in the temperature-interface interaction phase space. Key novelties of the paper are the determination of conditions for particle-LC interface-driven discontinuous change of nematic ordering and nanoparticle-enabled critical point condition in pressure-driven order-disorder phase transformations. Experimental evidences supporting the predicted phase behavior are presented.

A High-Rate and Long-Life Sodium Metal Battery Based on a NaB3H8 ⋅ xNH3@NaB3H8 Composite Solid-State Electrolyte.

All-solid-state sodium metal batteries are promising for large-scale energy storage applications owing to their intrinsic safety and cost-effectiveness. However, they generally suffer from sodium dendrite growth or rapid capacity fading, especially at high rates, mainly due to poor wettability, sluggish ionic transport, or low interfacial stability of the solid electrolytes. Herein, we report a novel composite, NaB3H8 ⋅ xNH3@NaB3H8 (x<1), as a new class of solid electrolyte for high-rate batteries. NaB3H8 ⋅ xNH3@NaB3H8 is obtained from the sticky NaB3H8 ⋅ NH3 after removal of NH3 partially at room temperature. It delivers an ionic conductivity of 0.84 mS cm-1 at 25 °C and reaches 20.64 mS cm-1 at 45 °C after an order-disorder phase transformation. It also reveals a good capability of dendrite suppression and remarkable stability against sodium metal. These performances enable the all-solid-state Na//TiS2 battery with a high capacity of 232.4 mAh g-1 (97.2 % of theoretical capacity) and long-term cycling stability at 1 C. Notably, this battery shows superior long-life cycling stability even at 5 and 10 C, which has been rarely reported in all-solid-state sodium metal batteries. This work opens a new group of solid electrolytes, contributing to fast-charging or high-power-density sodium metal batteries.

Highly Coercive L10-Phase Dots Obtained through Low Temperature Annealing for Nano-Logic Magnetic Structures

Nano-logic magnetic structures are of great interest for spintronic applications. While the methods used for developing arrays of magnetic L10-phase dots are, in most cases, based on deposition followed by annealing at high temperatures, usually around 700 °C, we demonstrate here a technique where a much lower annealing temperature (i.e., 400 °C) is needed in order to promote fully the disorder–order phase transformation and achievement of highly coercive L10-phase dots. In order to develop building blocks based on arrays of L10-phase FePt dots for further spintronic applications, an engraving technique using electron beam lithography is employed. This paper describes the fabrication, as well as the morphological and magnetic characterization, of regularly placed FePt dots of various shapes, as pre-requisites for integration into nano-logic devices. As a proof of concept, regular arrays of FePt circular dots were devised and their structural characterization, using X-ray diffraction (XRD) and transmission electron microscopy (TEM), was performed. It has been shown that annealing at only 400 °C for 30 min proved the occurrence of the tetragonal L10 phase. Moreover, structural characterization showed that the disorder–order phase transformation was complete with only the L10 phase detected in high resolution TEM. The magnetic characterization provided more insight into the potential of such arrays of magnetic devices with convenient values of magnetic coercivity, remanent and saturation magnetization. These findings show good potential for developing regular arrays of uniformly shaped magnetic entities with encouraging magnetic performances in view of various applications.

Self-oscillatory instability of the driven phase front propagation induced by liberation of latent heat.

We theoretically address crystals exhibiting first-order phase transformations subjected to a steadily propagating temperature gradient. The latter drives a nonisothermal propagation of a phase front. We theoretically demonstrate that for the phase transformations of the displacive type, the phase front always steadily follows the isotherm. In contrast, in the case of the order-disorder or hybrid phase transformations in a crystal containing pinning defects, one finds a velocity of the isotherm, the first critical velocity, at which the steady front motion becomes unstable, and a stick-slip front propagation starts. Upon reaching the second critical velocity, the stick-slip behavior vanishes, and the motion becomes steady again. Our results enable one to determine the activation energy of the leading order-disorder process from the measurements of the driven motion of the phase front. In light of these results, we discuss experimental findings for PbTiO_{3} and NaNbO_{3}.

A study of order-disorder phase transformation in Ag50Pd50 alloy

In this study, the structure of Ag50Pd50 alloy was investigated with temperature dependent in-situ X-ray diffraction, residual resistometry, and differential scanning calorimetry (DSC) as well as differential thermal analysis (DTA) techniques. Isochronal annealing experiment was performed to determine the impact of ordering on the residual electrical resistivity. The residual resistivity curve shows a sharp minimum near 250 °C indicating the on-set of ordering. The in-situ X-ray diffraction data taken for a well annealed sample shows no super lattice reflections over the investigated temperature range but the lattice parameter determined from this data shows an abrupt decrease in its value at 250 °C showing an order-disorder phase transformation (ODPT). The diffraction pattern of a sample annealed at 230 °C shows splitting of the high angle 511 fundamental reflection due to the tetragonal distortion. The peak intensity and c/a ratios are found to be 1:1.94 and 1.0028, respectively. A similar phase change was also observed during a DSC experiment around 250 °C on this alloy. The existence of these anomalies may be attributed to L1o - FCC type order-disorder phase transformation at this temperature.

Intense luminescence in nanostructured germanate glass

The phosphors free of active dopants have been attracted much attention due to their interesting optical properties and potentials for construction of rare-earth free functional materials. In this letter, we report the construction of phosphor free of active dopants via nanocrystallization of glass. In a typical example, a single-phase Zn2GeO4 phosphor is fabricated via controllable crystallization of the germinate glass. Intense green luminescence with the central wavelength at 523 nm can be realized in nanostructured glass and it is estimated to be ∼150 times higher than that of the as-made glass. The physical mechanism is associated with the formation of Zn related interstitial defects during the disorder-order phase transformation in glass. The progress in this work indicates that the avenue of glass crystallization provides an effective strategy for the development of novel active photonic materials free of active dopants.

Persistence of variant selection in red gold alloys

Recently, a new thermally activated distortion with amplification (TADA) effect has been reported in red gold alloys caused by the A1 → L10 phase transformation. The macroscopic amplification is due to the persistence of variant selection nucleated under stress but growing in stress-free condition. In this work, we show that the TADA effect can generate a compressive force whose absolute value exceeds 40 MPa in bending and uniaxial loading. Complementary EBSD analyses indicate that the compressive force has no influence on the variant selection initiated under tension. This result confirms that the A1 → L10 disorder-order phase transformation is diffusive, thermodynamically of 1st order, displacive and crystallographically continuous. The TADA force may be exploited for actuators.

Role of Disordered Precursor in L10 Phase Formation in FePt-Based Nanocomposite Magnet

In order to prove the usefulness of having a structurally disordered precursor to the formation of FePt L10 phase and to facilitate the co-existence of exchange coupled hard and soft magnetic phases with optimized magnetic properties in various conditions of annealing, a Fe-Pt-Zr-B melt spun alloy has been synthesized and detailed structural and magnetic investigations have been undertaken to probe its phase evolution during annealing. The dynamics of formation of the hard magnetic L10 phase during the gradual disorder–order phase transformation has been monitored by using a complex combination of X-ray diffraction methods and 57Fe Mössbauer spectroscopy methods, over a wide range of annealing temperatures. Multiple phases co-existing in the annealed sample microstructures, observed in XRD, have been reconfirmed by the Mössbauer spectra analysis and, moreover, accurate quantitative data have been acquired in what concerns the relative abundance of each of the observed crystalline phases in every stage of annealing. It is shown that the formation of the hard magnetic phase, emerging from the chemically disordered precursor, is gradual and occurs via complex mechanisms, involving the presence of a disordered Fe-Zr-B-rich intergranular region which contributes to an increase in the abundance of the L10 phase for higher annealing temperatures. Magnetic measurements have confirmed the good performances of these alloys in terms of coercivity and remanence. These results contribute to the development of these alloys as the next generation of rare earth, free permanent magnets.

Magnetic Phase Coexistence and Hard-Soft Exchange Coupling in FePt Nanocomposite Magnets.

With the aim of demonstrating phase coexistence of two magnetic phases in an intermediate annealing regime and obtaining highly coercive FePt nanocomposite magnets, two alloys of slightly off-equiatomic composition of a binary Fe-Pt system were prepared by dynamic rotation switching and ball milling. The alloys, with a composition Fe53Pt47 and Fe55Pt45, were subsequently annealed at 400 °C and 550 °C and structurally and magnetically characterized by means of X-ray diffraction, 57Fe Mössbauer spectrometry and Superconducting Quantum Interference Device (SQUID) magnetometry measurements. Gradual disorder–order phase transformation and temperature-dependent evolution of the phase structure were monitored using X-ray diffraction of synchrotron radiation. It was shown that for annealing temperatures as low as 400 °C, a predominant, highly ordered L10 phase is formed in both alloys, coexisting with a cubic L12 soft magnetic FePt phase. The coexistence of the two phases is evidenced through all the investigating techniques that we employed. SQUID magnetometry hysteresis loops of samples annealed at 400 °C exhibit inflection points that witness the coexistence of the soft and hard magnetic phases and high values of coercivity and remanence are obtained. For the samples annealed at 500 °C, the hysteresis loops are continuous, without inflection points, witnessing complete exchange coupling of the hard and soft magnetic phases and further enhancement of the coercive field. Maximum energy products comparable with values of current permanent magnets are found for both samples for annealing temperatures as low as 500 °C. These findings demonstrate an interesting method to obtain rare earth-free permanent nanocomposite magnets with hard–soft exchange-coupled magnetic phases.

Mn-Induced Thermal Stability of L10 Phase in Fept Magnetic Nanoscale Ribbons.

Magnetic nanoscale materials exhibiting the L10 tetragonal phase such as FePt or ternary alloys derived from FePt show most promising magnetic properties as a novel class of rare earth free permanent magnets with high operating temperature. A granular alloy derived from binary FePt with low Pt content and the addition of Mn with the nominal composition Fe57Mn8Pt35 has been synthesized in the shape of melt-spun ribbons and subsequently annealed at 600 °C and 700 °C for promoting the formation of single phase, L10 tetragonal, hard magnetic phase. Proton-induced X-ray emission spectroscopy PIXE has been utilized for checking the compositional effect of Mn addition. Structural properties were analyzed using X-ray diffraction and diffractograms were analyzed using full profile Rietveld-type analysis with MAUD (Materials Analysis Using Diffraction) software. By using temperature-dependent synchrotron X-ray diffraction, the disorder–order phase transformation and the stability of the hard magnetic L10 phase were monitored over a large temperature range (50–800 °C). A large interval of structural stability of the L10 phase was observed and this stability was interpreted in terms of higher ordering of the L10 phase promoted by the Mn addition. It was moreover found that both crystal growth and unit cell expansion are inhibited, up to the highest temperature investigated (800 °C), proving thus that the Mn addition stabilizes the formed L10 structure further. Magnetic hysteresis loops confirmed structural data, revealing a strong coercive field for a sample wherein single phase, hard, magnetic tetragonal L10 exists. These findings open good perspectives for use as nanocomposite, rare earth free magnets, working in extreme operation conditions.

Effect of stoichiometry on the evolution of thermally annealed long-range ordering in Ni–Cr alloys

Ni-based alloys, such as alloys 690 and 625, are widely used in the nuclear industry as structural components, because of their desirable mechanical properties and resistance to stress corrosion cracking. However, in some high chromium alloys, a disorder-order phase transformation near 33 at% Cr, is known to decrease ductility and fracture toughness. In this study, the ordering transformation is investigated in Ni–Cr binary model alloys to better understand the effects of composition. Model alloys with different stoichiometries (Ni/Cr = 1.8, 2.0, 2.2, 2.4) were isothermally aged up to 10,000 h at three temperatures (373 °C, 418 °C, and 475 °C) and characterized by transmission electron microscopy (TEM), microhardness, and synchrotron-based X-ray diffraction (XRD). TEM results show the evolution of the Ni2Cr (MoPt2-type) ordered precipitates between 3000 h and 10,000 h with corresponding size of ∼10–20 nm. Microhardness testing results show that off-stoichiometry (Ni/Cr ≠ 2.0) alloys exhibit a smaller change with ordering compared to the stoichiometric (Ni/Cr = 2.0) alloy at all temperatures. XRD quantifies ordering induced lattice contraction in the matrix structure and the size of the ordered precipitates. No BCC Cr was detected by XRD or TEM during characterization in the range of 29.83–35.66 at% Cr after 10,000 h of aging, confirming that all of the hardening can be attributed to the development of Ni2Cr in alloys ranging from Ni/Cr of 1.8 to Ni/Cr of 2.4.

Effect of La addition on high-temperature order-disorder phase transformation in Fe − 18Ga alloy

The microstructure, internal friction (IF) behavior spectra, heat flow curves, and mechanical strength of Fe-18Ga-xLa (x = 0.05, 0.1, 0.2, 0.5, and 1.0 at.%) alloys were systematically analyzed in this investigation. In the IF spectra, a relaxation IF peak (Labeled as P1) and a peak related to order-disorder phase transition (Labeled as Ptr) are observed in the temperature range of 400 °C–690 °C. It is found trace La doping has a significant effect on the Ptr peak. With the increase of the La-doped content, the peak of Ptr gradually shifts to a lower temperature and the net peak height decreases. Combining with phase diagram, DSC, Grazing Incidence X-ray diffraction (GI-XRD) spectra and SEM, the mechanisms of the reduction of both the critical temperature and the strength of Ptr peak for the Fe-18Ga-xLa alloy are ascribed to the precipitation of LaGa2 phase, which reduces the Ga content gradually in the matrix. Further, the mechanism is verified in Fe-18.1Ga alloy and Fe-19.3Ga alloy. In addition, the trace La doping can significantly improve the mechanical strength of the Fe − 18Ga alloy.

Probing the temperature effects in the radiation stability of Nd2Zr2O7 pyrochlore under swift ion irradiation

The pyrochlore structure is of particular interest to the nuclear industry due to its potential for nuclear waste immobilization and as a host material for inert matrix fuels. Herein, the experimental results of the temperature and ion fluence dependence on the disordered ion tracks produced by the electronic excitation in the pyrochlore compound Nd2Zr2O7 are presented. The samples were exposed to extreme conditions of ion irradiation with 100 MeV 107Ag9+ ions at ion fluences ranging from 3 × 1011 ions/cm2 to 2 × 1013 ions/cm2 at temperatures of 300 K and 1000 K. The in-situ X-ray diffraction along with micro-Raman spectroscopic analysis on Nd2Zr2O7 samples irradiated at 300 K displays order-disorder phase transformation which is consistent with earlier reports. The direct-impact model is used to calculate the amorphous fraction which exists due to massive deposition of electronic-energy in a localized region, and the estimated value of track radius is 2.98 ± 0.56 nm. Furthermore, theoretical thermal spike simulations are performed to estimate the track radius (∼4.2 nm) which provides strong support for experimental results. To simulate the response of Nd2Zr2O7 under nuclear reactor environment, irradiation carried out at an elevated temperature that exhibits significant radiation tolerance. Its substantiality is revealed from comparative temperature dependent irradiation studies where a slow rate of amorphization is found at the elevated temperature because of the competition between defect production and thermal healing mechanism. Thus, obtained results strongly endorse the appositeness of Nd2Zr2O7 as inert matrix material for the fabrication of futuristic nuclear fuel.

Thermodynamic Origin of Solute-Enriched Stacking-Fault in Dilute Mg-Zn-Y Alloys

We investigate thermodynamic behaviors of dilute Mg-Zn-Y ternary alloys to form a unique solute-enriched stacking-fault (SESF), which is an intrinsic-II type stacking-fault (I2-SF) enriched by the Zn and Y atoms and represents the structural-unit of the long-period stacking/order (LPSO) phase. SESF in the hexagonal-close-packed (hcp) Mg matrix forms a local face-centered-cubic (fcc) environment, and hence our thermodynamic analysis is based on the Gibbs energy comparison between hcp and fcc phases over the Mg-Zn-Y ternary composition ranges, using the calculation of phase diagrams (CALPHAD) method aided by the first principles calculations. Segregation behaviors of solute Zn/Y atoms into the SESF are firstly estimated according to the Hillert's parallel tangent law, followed by the possible disorder-order phase transformation within the SESF using the multiple-sublattice model. We find that the Zn/Y co-segregations at the SESF provide a remarkable condition that the fcc layers become more stable than the hcp-Mg matrix. Besides, within the SESF, the following spinodal-like decomposition into the Mg-rich solid-solution and the Zn/Y-rich L12-type order phase causes a significant reduction of the total Gibbs energy of the system. These thermodynamic behaviors explain fairly well a phenomenological origin of the Zn-Y clustering with the L12-type short-range order, which is known to occur for the LPSO phases and also confirmed for the present SESF by electron microscopy experiments. Therefore, strong Zn-Y interactions even in dilute conditions play a key role to stabilize firmly the SESF in the Mg-Zn-Y alloys.

Annealing-Induced High Ordering and Coercivity in Novel L10 CoPt-Based Nanocomposite Magnets

A novel class of quaternary intermetallic alloys based on CoPt is investigated in view of their interesting magnetic properties induced by the presence of hard magnetic L10 phase. A Co48Pt28Ag6B18 alloy has been prepared by rapid solidification from the melt and subjected to various isothermal annealing procedures. The structure and magnetism of both as-cast and annealed samples as well as the phase evolution with temperature are investigated by means of thermal analysis, X-ray, and selected area electron diffraction, scanning and high-resolution electron microscopy, and magnetic measurements. The X-ray diffraction (XRD) analysis shows that both the as-cast alloy and the sample annealed at 400 °C (673 K) have a nanocrystalline structure where fcc CoPt phase predominates. Annealing at 473 °C promotes the formation of L10 phase triggered by the disorder-order phase transformation as documented in the differential scanning calorimetry results. The sample annealed at 670 °C (943 K) shows full formation of L10 CoPt as revealed by XRD. Magnetic measurements showed coercivity values ten times increased compared to the as-cast state. This confirms the full formation of L10 CoPt in the annealed samples. Moreover, detailed atomic resolution HREM images and SAED patterns show the occurrence of the rarely seen (003) superlattice peaks, which translated into a high ordering of the L10 CoPt superlattice. Such results spur more interest in finding novel classes of nanocomposite magnets based on L10 phase.

Preparation of magnetic Fe-Co alloy using texture controlled technique

Microstructure and texture controlled 48Fe-52Co at% alloy was prepared by repeated cold-rolling and recrystallization through heat treatment at temperatures below order-disorder phase transformation using Fe and Co foils. Furthermore, distortion of controlled texture orientated cubic crystal structured 48Fe-52Co at% alloy along the 〈0 0 1〉 direction was attempted by introducing nitrogen interstitially through plasma nitriding. The textural analysis suggested that the microstructure is composed of micron-sized grains randomly aligned along rolling direction (RD). Furthermore, from the results of orientation distribution function (ODF), the dominance γ-fiber was confirmed in as-rolled samples. However, in recrystallized sample, the intensity of rotated-cube orientation increased while the presence of γ-fiber decreased. Furthermore, a complete rotated-cube orientation was realized in nitrided texture. The appearance of the prominent X-ray diffraction peak (2 0 0) in all samples indicated the presence of rotated-cube orientation. However, the FWHM of (2 0 0) peak of the nitrided sample was broader than the rest. This was attributed to the diffusion of nitrogen into the matrix during plasma nitriding and internal nitriding was confirmed from the EDX analysis. However, further investigation is necessary to understand the behavior of nitrogen.Magnetic properties of as-rolled 48Fe-52Co at% alloy inferred from in-plane magnetic hysteresis loops was similar Fe foil. Furthermore, magnetic properties of recrystallized and nitrided 48Fe-52Co at% alloy were different from Fe-Co particles or crushed bulk Fe-Co alloy prepared by arc melting. However, the internal diffusion of nitrogen was not adequate enough to bring about a considerable change in the magnetic properties. We believe that considerable crystal distortion to induce change in the magnetic properties can be exerted by introducing interstitial carbon and the addition of third or fourth elements.

Calorimetric study and microstructure analysis of the order-disorder phase transformation in silicon steel built by SLM

Innovative Additive Manufacturing (AM) technologies like Selective Laser Melting (SLM) could prove to be efficient for the processing of brittle silicon steel (Fe-Si) with high silicon content. This research elucidates the effects of heat-treatment on the microstructure of SLM-built high silicon steel, with particular emphasis on the formation of ordered phases, which are known to cause undesired material embrittlement.Silicon steel with 6.9 wt % Si is produced by SLM and investigated performing Differential Scanning Calorimetry (DSC), microstructure analysis and hardness measurements. As-built Fe-Si parts are found to consist primarily of disordered A2 phase as the high cooling rates typical of SLM suppress the ordering phase transformations. It is shown how heat treatments can be applied to modify the state of ordering and morphology of the rapidly solidified microstructure. By shedding light on the phase-ordering mechanisms and the effect of heat treatments on microstructure of high-Si steel built by SLM, the present study paves the way towards the optimisation of the mechanical and magnetic properties of this alloy.

Sputtering pressure effects on microstructure and grain orientation distribution in FePt thin films

We use the magnetron sputtering method to fabricate FePt thin films on amorphous oxidized Si substrates by controlling the sputtering pressure and investigate the evolution of microstructure and grain orientation using X-ray diffraction and electron back-scatter diffraction, and discuss the effect of microstructure on magnetic properties of L10-FePt thin film. The experimental results show that the L10-FePt phase is formed when the sputtering pressure is 0.25 Pa, and the L10-FePt phase is not found when the sputtering pressure is higher. The higher sputtering pressure leads to poor crystallinity and smaller grain size in FePt thin films, which inhibit disorder–order phase transformation. Meanwhile, higher sputtering pressure results in a larger residual strain, because it increases the degree of local gain misorientation in FePt thin films. The inverse pole figure map and band contrast map reveal the grain growth orientation and microstructure of L10-FePt thin film from cross-sectional, indicating that the grain grows along different direction, and exists in the form of nanocolumnar crystal. The experimental results demonstrate that when there is no texture in L10-FePt thin film, the out-of-plane magneto anisotropy of L10-FePt thin film should be mainly contributed by the shape anisotropy.

The effects of ordered carbon vacancies on stability and thermo-mechanical properties of V8C7 compared with VC

The ordered non-stoichiometric V8C7 can form in the VCy carbides by the disorder–order phase transformation. The intrusion of ordered carbon vacancies can affect their stability, mechanical, thermal and electronic properties. The relatively thermodynamic stability and mechanical properties at high temperature for the ordered stoichiometric VC and non-stoichiometric V8C7 are investigated in this paper by first-principle calculations combined with the quasi-harmonic approximation. The difference between the properties of VC and V8C7 can be obtained. We find that the V8C7 is thermodynamic more stable than VC, but has weaker elastic heat resistance than VC. Moreover, the minimum thermal conductivity of VC is a little larger than V8C7 and a simple way is proposed to characterize the anisotropy of lattice thermal conductivity based on the Cahill’s model.

Raman Study of the Structural Distortion in the Ni1-xCoxTiO3 Solid Solution.

Raman spectra were collected on Ni1-xCoxTiO3 (0 ≤ x ≤ 1) ilmenite samples as a function of the temperature between 4 and 1200 K. An evident symmetry lowering from the prototype R3̅ symmetry is observed. The distortion was largest for the x = 0.40 and 0.50 samples and significantly diminished for small and large values of x. The distortion was preserved in the whole temperature range and, except for the x = 0.50 sample, did not show significant changes. Notably, between 25 and 69 K, distortion of the x = 0.40 sample is accompanied by ferromagnetic order. The direct cation-cation and O-mediated indirect interactions are discussed as mechanisms behind the distortion and magnetic order. A reversible order-disorder phase transformation, assigned to occur between the ilmenite and corundum phases, took place at 973 K in the x = 0.50 sample. Completion of the transformation took over 1 h. The role of the overlap of Co/Ni and Ti 3d orbitals through O octahedral faces for charge transfer is discussed.