The apparent ’five-fold’ nature of large T2 (AI6Li3Cu) crystals

LargeT ₂ (Al₆Li₃Cu) crystals which display apparent five-fold symmetry have been studied using several different electron microscopy techniques. Electron microprobe X-ray analysis was used to determine the phases present in two different alloys cast with bulk compositions close to stoi-chiometricT ₂. Convergent beam electron diffraction (CBED) of these crystals indicates that they do not display five-fold rotational symmetry, and TEM images and selected area diffraction from thin foil specimens indicate a microcrystalline or twinned structure is responsible for the apparent five-fold symmetry ofT ₂. Microdiffraction patterns obtained from the thinnest regions of the crystals are consistent with a cubic structure having a lattice parameter of ≈0.40 nm.     

Mechanism of Al2CuLi (T 1) nucleation and growth

Conventional strain contrast transmission electron microscopy (CTEM) and high-resolution transmission electron microscopy (HRTEM) were performed to establish the nucleation and growth mechanism of Al₂CuLi (T₁) precipitates in an Al-Li-Cu alloy. It is shown that the growth mechanism ofT ₁ precipitate plates occurs by the diffusional glide of growth ledges composed of b = 1/6〈112〉 partial dislocations on 111 matrix planes and that the growth ledges migrate by the ledge-kink mechanism, as previously suggested by Cassadaet al. 1 for this system.T ₁ plate nucleation is modeled as the dissociation of a perfect b = 1/2〈110〉 matrix dislocation in the vicinity of a dislocation jog. The coordinated dissociation of the dislocation line segments on each side of the sessile jog provides the displacement necessary for the formation of a new hexagonal plate or plate ledge. Strain contrast analysis of the Burgers vector of plate edges and the edges of growth ledges indicates the stacking of partial dislocations is of mixed displacement. Formerly Graduate Student, Department of Materials Science, University of Virginia,     

Crystallographic relationships of the

In a transmission electron microscopic study of rapidly as well as slowly solidified alloys in the composition range Al₇Cr-Al₄Cr, three mutually related structural variants of Al₄Cr, in ad- dition to the well-known monoclinic Ф-Al₄₅Cr₇, were found. The hexagonal μ-Al₄Cr is iso- structural with μ-Al₄Mn (P6₃/mmc), with a_μ = 2.00 nm and c_μ = 2.47 nm. The two orthorhombic ε- and ε′-Al₄Cr phases, one being B-centered and the other primitive, havea ≈ √3a_μ = 3.46 nm,b ≈ aμ = 2.00 nm, andc ≈ c_μ/2 = 1.24 nm. These three Al₄Cr phases are not only themselves structurally connected, but are also closely related both to the Al-Cr icosahedral quasicrystal found earlier and to the Al-Cr decagonal quasicrystal found for the first time in the present investigation in rapidly solidified Al-Cr alloys with 18 to 22 at. pct Cr.     

Convergent beam electron diffraction study of Al3Zr in Al-Zr AND Al-Li-Zr alloys

Point and space group analysis of metastable and equilibrium A1₃Zr precipitates in Al-Zr and Al-Li-Zr alloys was performed by convergent beam electron diffraction (CBED). The results obtained for the metastable Al₃Zr showed it possesses a cubic Pm3m (No. 221) space group with a lattice parameter of 0.408 nm. The equilibrium Al₃Zr (β) phase was determined to be body-centered tetragonal having a space group I4/mmm (No. 139) and lattice constants of a = 0.401 nm and c = 1.73 nm. In the lithium containing alloys, it was shown that if lithium incorporation into the metastable Al₃Zr particles occurred, it did not alter its structure. X-ray microanalysis of extracted metastable Al₃Zr precipitates was conducted using energy dispersive X-ray spectrometry (EDS), and the composition of these particles was determined to be ≈ 60 wt%Al and ≈ 40 wt%Zr. Since this composition is significantly different from stoichiometric Al₃Zr (48 wt%Al, 52 wt%Zr), it suggests that a metastable phase field exists in the α + β phase region of the Al-Zr phase diagram.     

Intermetallic phases formed during DC-casting of an Al−0.25 Wt Pct Fe−0.13 Wt Pct Si alloy

The Al−Fe and Al−Fe−Si particles formed during DC-casting of an Al-0.25 wt pct Fe-0.13 wt pct Si alloy have been examined. The particles were analyzed by transmission electron microscopy (TEM) and energy dispersive spectroscopy of X-rays (EDS). Crystal faults were studied by high resolution electron microscopy (HREM). Samples for electron microscopy were taken at various positions in the ingot,i.e., with different local cooling rates during solidification. At a cooling rate of 6 to 8 K/s the dominating phases were bcc α-AlFeSi and bct Al _m Fe. The space group of bcc α-AlFeSi was verified to be Im3. Superstructure reflections from Al _m Fe were caused by faults on {110}-planes. At a cooling rate of 1 K/s the dominating phases were monoclinic Al₃Fe and the incommensurate structure Al _x Fe. In Al₃Fe, stacking faults on {001} were frequently observed. The structure of Al _x Fe is probably related to Al₆Fe. Some amounts of other phases were detected. For EDS-analysis, extracted particles mounted on holey carbon films were examined. Extracted particles were obtained by dissolving aluminum samples in butanol. Accurate compositions of various Al−Fe−Si phases were determined by EDS-analysis of extracted crystals.     

Intermetallic phases formed during DC-casting

The Al-Fe and Al-Fe-Si particles formed during DC-casting of an Al-0.25 wt pct Fe-0.13 wt pct Si alloy have been examined. The particles were analyzed by transmission electron microscopy (TEM) and energy dispersive spectroscopy of X-rays (EDS). Crystal faults were studied by high resolution electron microscopy (HREM). Samples for electron microscopy were taken at various positions in the ingot,i.e., with different local cooling rates during solidification. At a cooling rate of 6 to 8 K/s the dominating phases were bcc α-AlFeSi and bct Al _m Fe. The space group of bcc α-AlFeSi was verified to be Im3. Superstructure reflections from Al _m Fe were caused by faults on {110}-planes. At a cooling rate of 1 K/s the dominating phases were monoclinic Al₃Fe and the incommensurate structure Al _x Fe. In Al₃Fe, stacking faults on {001} were frequently observed. The structure of Al _x Fe is probably related to Al₆Fe. Some amounts of other phases were detected. For EDS-analysis, extracted particles mounted on holey carbon films were examined. Extracted particles were obtained by dissolving aluminum samples in butanol. Accurate compositions of various Al-Fe-Si phases were determined by EDS-analysis of extracted crystals.     

Microstructural characterization of self-propagating high-temperature synthesis/ dynamically compacted and hot-pressed titanium carbides

Titanium-Carbide produced by combustion synthesis followed by rapid densification in a high-speed forging machine was characterized by optical microscopy, scanning electron microscopy, and transmission electron microscopy (TEM). The density of the combustion synthesized/dynamically compacted TiC reached values greater than 96 pct of theoretical density, based on TiC_0.9, while commercially produced hot-pressed TiC typically exceeded 99 pct of theoretical density. The higher density of the hot-pressed TiC was found to be attributable to a large volume fraction of heavy element containing inclusions. The microstructure of both TiCs consists of equiaxed TiC grains with some porosity located both at grain boundaries and within the grain interiors. In both cases, self-propagating high-temperature synthesis (SHS)/dynamically compacted (DC) and hot-pressed, the TiC is ordered cubic (NaCl-structure,B ₁; Space Group Fm3m) with a lattice parameter of ≈0.4310 nm, indicative of a slightly carbon deficient structure; stoichiometric TiC has a lattice parameter of 0.4320 nm. For the most part, the grains were free of dislocations, although some dislocation dipoles were found associated with the voids within the grain interiors. In one SHS/DC specimen, a new, complex Ti-Al-O(C) phase was observed. The structure could not be matched with any previously published phases but is believed to be hexagonal, with a c-axis/a-axis ratio of ≈6.6, similar to the AlCTi₂ phase which has a point group 6 mmm. In all other SHS/DC TiC samples, the grains and grain boundaries were devoid of any second-phase particles. The hot-pressed TiC exhibited a greater degree of porosity than the SHS/densified specimens and a large concentration of second-phase particles at grain boundaries and within grains. The structure and composition of these second-phase particles were determined by con-vergent beam electron diffraction (CBED) and X-ray microanalysis. This paper is based on a presentation made in the symposium “Reaction Synthesis of Materials” presented during the TMS Annual Meeting, New Orleans, LA, February 17–21, 1991, under the auspices of the TMS Powder Metallurgy Committee.     

Effects of welding and weld heat-affected zone simulation on the microstructure and mechanical behavior of a 2195 aluminum-lithium alloy

The microstructures, tensile properties, and fatigue properties of a 2195-T8 Al-Li alloy subjected to a weld heat-affected zone (HAZ) simulation and gas-tungsten-arc (GTA) welding using a 4043 filler metal, with and without a postweld heat treatment, were studied. The principal strengthening precipitate in the T8 base alloy was the T ₁ (Al₂CuLi) phase. The HAZ simulation resulted in the dissolution of T ₁ precipitates and the formation of T _B(Al₇Cu₄Li) phase, Guinier-Preston (G–P) zones, and δ′ (Al₃Li) particles. When the HAZ simulation was conducted at the highest temperature of 600 °C, microcracks and voids also formed along the grain boundaries (GBs). In the specimens welded with the 4043 alloy, T (AlLiSi) phase was found to form in the fusion zone (FZ). An elongated T _Bphase and microcracks were observed to occur along the GBs in the HAZ close to the FZ interface. The T ₁ phase was not observed in the HAZ. The postweld heat treatment resulted in the spheroidization of primary T phase and the precipitation of small secondary T particles in the FZ, the dissolution of T _B phase, and the reprecipitation of the T ₁ phase in the HAZ. Both the HAZ simulation and welding gave rise to a considerable decrease in the hardness, tensile properties, and fatigue strength. The hardness in the FZ was lower than that in the HAZ. Although the postweld heat treatment improved both the hardness and tensile properties due to the reprecipitation of T ₁ phase in the HAZ and a smaller interparticle spacing in the FZ, no increase in the fatigue strength was observed because of the presence of microcracks in the HAZ.     

Structural parameters of the low-temperature metastable form of the carbide W2C

A sufficiently complete spectrum of superstructure reflections was obtained by the x-ray diffraction analysis of a monocrystal to unequivocally determine, for the first time, the ordered structure of W₂C annealed for a long time at a temperature below that of its eutectoidal decomposition. It was shown that W₂C of eutectoidal composition in a metastable state has a rhombic ordered structure with the lattice parameters a=4.719 ± ± 0.003 nm × 10=c₀, b=6.017 ± 0.003 nm × 10 ≈ 2a₀, c=5.181 ± 0.003 nm × 10 ≈ √3a₀ (a₀, c₀ ≡ a_HCP, c_HCP). The specimens were obtained from tungsten alloys containing 30.5 and 35 at. % C, prepared by are melting followed by annealing at ∼2700°C (1 h) and stepwise cooling to 850°C (the total annealing time at temperatures below 1200°C was 538 h). Translated from Poroshkovaya Metallurgiya, Nos. 3/4(412), pp. 46–53, March–April, 2000.     

Decagonal quasicrystal and related crystalline phases in slowly solidified Al-Co alloys

A two-dimensional decagonal quasicrystal (DQC) has been found in binary Al-Co alloys in the composition range Al₁₁Co₄ to Al₁₀Co₄ after slow solidification from the melt. The crystalline phase found most frequently coexisting with this DQC is two new structure variants, oneC-centered and the other primitive, of the monoclinic layer compound Al₁₃Co₄. Theira andc parameters are roughly τ² times (τ = (1 + √5)/2 = 1.6180…) larger than the corresponding ones in Al₁₃Co₄. Moreover, a new orthorhombic phase Al₃Co or Al₁₁Co₄ (Pnmn, a = 1.25 nm,b = 0.81 nm, andc = 1.46 nm) has also been found. As shown by the characteristic electron diffraction patterns (EDPs), these crystalline phases can be considered as approximate structures of the DQC. The lattice relationship between these phases has been discussed.     

Discussion of “effects of tensile stress on microstructural change of eutectoid Zn-AI alloy”

In a recent contribution,[1] Zhu and Orozco presented a phase transformation of the ternary alloy Zn-20.2 wt pct Al-1.8 wt pct Cu, studied under tensile stress by using X-ray diffraction and scanning electron microscopy techniques. The authors report the existence of three phases in the alloy at room temperature after furnace cooling,α,ε, and a newη _′ ^T instead of the zinc-rich solid solutionη, as appears in the phase diagrams. The reported parameters for this hcp metastable phase are^[1,2] a = 0.2663 andc = 0.4873 nm; these values are close to the parameters of pure zinc,^[3] witha = 0.2664 nm andc = 0.4946 nm. The difference betweenη _′ ^T and zinc in thea parameter is around 0.03 pct, and it is 1.47 pet for thec parameter. When zinc is saturated with aluminum in the Zn-AI alloys, thea parameter shrinks^[3] to 0.2660 nm. It is possible to see that the value ofa of theη _′ ^T phase lies in-between the values of pure zinc and zinc-aluminum solid solution. The solubility of Al and Cu in Zn^[4] at 100 °C is 0.3 wt pct Cu and 0.06 wt pct Al. The covalent radius of Cu (0.117 nm) is smaller than the covalent radius of Al (0.118 nm) and Zn (0.125 nm), so the introduction of Cu in the zinc structure can result in a reduction of thec parameter. These values suggest that the metastable phaseη _′ ^T could be the hcp zincrich solid solution with low aluminum and copper contents. The article of Zhu and Goodwin,^[5] cited by Zhu and Orozco in their Reference 14, is related not to the eutectoid alloy, as they argue, but to an alloy with 27 wt pct Al, and no reports about the transformation ofε intoT′ were found. The presence of the metastable e phase (CuZn₄, sometimes called CuZn₅) at room temperature and its transformation to the stable phaseT′ (rhombohedral intermetallic phase, Al₄Cu₃Zn) have been observed by other authors.^[6,7] Y.H. ZHU and E. OROZCO:Metall Mater. Trans. A, 1995, vol. 26A, pp. 2611-15.     

The effects of silicon on the reaction between solid iron and liquid 55 wt pct Al−Zn baths

The effect of various silicon levels on the reaction between iron panels and Al-Zn-Si liquid baths during hot dipping at 610°C was studied. Five different baths were used: 55Al−0.7Si−Zn, 55Al−1.7Si−Zn, 55Al−3.0Si−Zn, 55Al−5.0Si−Zn, and 55Al−6.88Si−Zn (in wt pct). The phases which formed as a result of this reaction were identified as Fe₂Al5 and FeAl3 (binary Fe−Al phases with less than 2 wt pct Si and Zn in solution),T1, T2, T4, T8, andT _5H (ternary Fe−Al−Si phases), andT _5C (a quaternary Fe−Al−Si−Zn phase). Compositional variations through the reaction zone were determined. The phase sequence in the reaction zone of the panel dipped for 3600 seconds in the 1.7 wt pct Si bath was iron panel/(Fe₂Al₅+T ₁)/FeAl₃/(T _5H+T _5C)/overlay. In the panel dipped for 1800 seconds in the 3.0 wt pct Si bath the reaction zone consisted of iron panel/Fe₂Al₅/(Fe₂Al₅+T ₁)/T ₁/FeAl₃/(FeAl₃+T ₂)/T _5H/overlay. In the panel dipped for 3600 seconds in the 6.88 wt pct Si bath the phase sequence was iron panel/Fe₂Al₅/(Fe₂Al₅+T1)/(T1+FeAl3)/(T1+T2)/T2/T8/T4/overlay. The growth kinetics of the reaction zone were also studied. A minimum growth rate for the reaction zone which formed from a reaction between the iron panel and molten Al−Zn−Si bath was found in the 3.0 wt pct Si bath. The growth kinetics of the reaction layers were found to be diffusion controlled in the 0.7, 1.7, and 6.88 wt pct Si baths, and interface controlled in the 3.0 and 5.0 wt pct Si baths. The presence of the interface between theT2/T5H, Fe2Al5/T ₁, orT ₁/FeAl₃ phases is believed responsible for the interface controlled growth kinetics exhibited in the 3.0 and 5.0 wt pct Si baths.     

The identification of hydrogen trapping states in an Al-Li-Cu-Zr alloy using thermal desorption spectroscopy

Thermal desorption spectroscopy (TDS) was utilized to identify several metallurgical states in an Al -2Li - 2Cu-0.1Zr (wt pct) alloy, which trap absorbed hydrogen. Six distinct metallurgical desorption states for hydrogen were observed for tempers varying from the T3 to peakaged condition. Lower energy thermal desorption states were correlated with interstitial sites, lithium in solid solution, and δ′ (Al₃Li) precipitates. These states have trap-binding energies ≤25.2 kJ/mol. Under the charging conditions utilized, approximately 4 pct of the total (e.g., trapped and lattice) hydrogen content was associated with interstitial sites, consistent with the view that the intrinsic lattice solubility of hydrogen in aluminum is very low. In contrast, dislocations, grain boundaries, and T₁ (Al₂CuLi) particles were found to be higher energy-trap states with trap-binding energies ≥31.7 kJ/mol. Approximately 78 pct of all absorbed hydrogen occupied these states. Moreover, greater than 13 pct of the available trap sites at grain boundaries were occupied. Such a high hydrogen coverage at grain-boundary sites supports the notion that hydrogen contributes to grain-boundary environmental cracking in Al-Li-Cu-Zr alloys. Also, it points out the error in assuming that hydrogen cannot play a major role in cracking of Al-based alloys due to the low lattice solubility.     

Phase transitions in the magnetic superconductor (Dy<Subscript>0.8</Subscript>Y<Subscript>0.2</Subscript>)Rh<Subscript>4</Subscript>B<Subscript>4</Subscript>

A new (Dy_0.8Y_0.2)Rh₄B₄ superconductor (the superconducting transition temperature is T _c ≈ 5.5 K), which has an inherent magnetic subsystem whose properties are determined by the crystal structure of the superconductor, is synthesized at a high pressure (∼8 GPa) and t ≈ 1800°. The magnetic sublattice of the (Dy_0.8Y_0.2)Rh₄B₄ compound is found to substantially affect its superconducting properties and, in a number of cases, to lead to their anomalous variations, namely, to the absence of the traditional Meissner effect and an anomalously abrupt increase in magnetic induction B _k2 (upper critical field) upon a transition of the magnetic subsystem into the antiferromagnetic state. Upon cooling from 250 to 1.6 K, the (Dy_0.8Y_0.2)Rh₄B₄ compound undergoes a number of phase transformations, namely, a paramagnet-ferrimagnet transition at a Curie temperature T _C ≈ 30 K, a superconducting transition at T _c ≈ 5.5 K against the background of a ferrimagnetic order, and a ferrimagnet-antiferromagnet transition (the Neel temperature is T _N ≈ 2.8 K) in the retained superconducting state.     

Electron Diffraction Study of α- and α T -AlFeSi

Electron diffraction patterns from a-AIFeSi and α _T -AlFeSi are compared. Unusual Kikuchi band patterns are found in both a and α _T . Structure factor calculations can be used to explain these patterns. Additional higher order Laue zone (HOLZ) rings are seen in α _T . Reflections located at n/9[521] are seen in some zero order Laue zones (ZOLZ) in α _T . These additional reflections are responsible for the additional HOLZ rings found in α _T . Most atomic positions in a and α _T are similar. The specific structural feature which causes the additional reflections in α _T is not known. Lattice imaging and image calculation will be necessary to determine the exact structure of α _T .     

Evolution of Grain Boundary Precipitates in an Al-Cu-Li Alloy During Aging

The grain boundary microstructure of Al-Cu-Li alloy AA2050 was investigated for different isothermal aging times to rationalize intergranular corrosion (IGC) characteristics. In the underaged condition, the dominant grain boundary precipitates are fine T₁ (Al₂CuLi). Extended aging revealed that grain boundaries were decorated by large T₁ precipitates and S′ phase (Al₂CuMg), with S′ growth not dimensionally constrained. Such a transition in the precipitate type at grain boundaries is a unique feature of the Al-Cu-Li system.     

Microstructural characteristics of Ni-Sb eutectic alloys under substantial undercooling and containerless solidification conditions

Both Ni-36 wt pct Sb and Ni-52.8 wt pct Sb eutectic alloys were highly undercooled and rapidly solidified with the glass-fluxing method and drop-tube technique. Bulk samples of Ni-36 pct Sb and Ni-52.8 pct Sb eutectic alloys were undercooled by up to 225 K (0.16 T _E ) and 218 K (0.16 T _E ), respectively, with the glass-fluxing method. A transition from lamellar eutectic to anomalous eutectic was revealed beyond a critical undercooling ΔT ₁*, which was complete at an undercooling of ΔT ₂*. For Ni-36 pct Sb, ΔT ₁*≈60 K and ΔT ₂*≈218 K; for Ni-52.8 pct Sb, ΔT ₁*≈40 K and ΔT ₂*≈139 K. Under a drop-tube containerless solidification condition, the eutectic microstructures of these two eutectic alloys also exhibit such a “lamellar eutectic-anomalous eutectic” morphology transition. Meanwhile, a kind of spherical anomalous eutectic grain was found in a Ni-36 pct Sb eutectic alloy processed by the drop-tube technique, which was ascribed to the good spatial symmetry of the temperature field and concentration field caused by a reduced gravity condition during free fall. During the rapid solidification of a Ni-52.8 pct Sb eutectic alloy, surface nucleation dominates the nucleation event, even when the undercooling is relatively large. Theoretical calculations on the basis of the current eutectic growth and dendritic growth models reveal that γ-Ni₅Sb₂ dendritic growth displaces eutectic growth at large undercoolings in these two eutectic alloys. The tendency of independent nucleation of the two eutectic phases and their cooperative dendrite growth are responsible for the lamellar eutectic-anomalous eutectic microstructural transition.     

Structure and stability of a new ternary hexagonal phase in Al<Subscript>3</Subscript> Ti-based Al-Ti-V alloys

A novel hexagonal phase (designated H₍₂₎) has been detected as a major constituent of both Al₆₂Ti₁₀V₂₈ and Al₅₅Ti₁₀V₃₅ alloys, following chill casting and after homogenization at 1523 K. The phase H₂ has an ordered hexagonal crystal structure (space group P6₃/mmc, α=0.558±0.001 and c=0.450±0.001 nm), similar to that of α ₂-Ti₃Al, and an atomic composition of 54±1 Al−11±1 Ti−35±1 V in the chill-cast Al₅₅Ti₁₀V₃₅ alloy. A fine-scale, duplex lamellar structure, developed within the ordered H₂ phase in the solid state, was composed of parallel-sided multivariants of ξ-Ti₅Al₁₁ phase, formed parallel to (0001)_{H 2}. The orientation relationship between constituent phases was of the form

F-type icosahedral phase and a related cubic phase in the Al-Rh-Cu system

An F-type icosahedral phase and a related cubic phase (composition of Al_66.1Rh_21.5Cu_12.3, lattice constant a=1.5380(2) nm, and space group of Fm3) were observed in the Al₆₃Rh_18.5Cu_18.5 alloy by transmission electron microscopy (TEM). The structure of the Al-Rh-Cu cubic phase was determined by single-crystal X-ray analysis. A high-resolution electron microscopic image of the Al-Rh-Cu cubic phase is presented together with a simulated image. The structure of the cubic phase can be described by two types of atom clusters, which have outer shells with icosahedral symmetry. It is suggested that the structure of the Al-Rh-Cu cubic phase is helpful for understanding the structure of the i-Al-Rh-Cu F-type icosahedral quasicrystal.     

The niobium (columbium)-platinum constitution diagram

The Nb-Pt system was investigated over the entire composition range by metallography and X-ray diffraction analysis. The solubility limits of terminal and intermediate phases and solidus temperatures were determined. α-Nb dissolves ≈12 at. pct Pt at 2040 °C and ≈5 at. pct Pt at 1150 °C; α-Pt dissolves ≈20 at. pct Nb at 2000 °C and ≈ 18 at. pct Nb at temperatures below 1700 °C. The presence of six intermediate phases, Nb₃Pt (Cr₃O, A15 or β-W type), σ(≈Nb₂Pt, β-U type), Nb_1−xPt_1+x (AuCd type), α′-Pt (undetermined structure), NbPt₂ (MoPt₂ type), α-NbPt₃ (TiCu₃ type), and β-NbPt₃ (β-NbPt₃ type) was confirmed. The phase NbPt₃ melts congruently at ≈2040 °C, and σ forms peritectically at ≈1800 °C. By analogy with related systems, the high-temperature phase α′-Pt is probably an extension of and isomorphous with α-Pt solid solution. Eight three-phase reactions are described, the mean atomic volumes are given, and crystal chemical relationships among the six homologous T₅-T₁₀ systems (T₅ = V, Nb, Ta; T₆ = Pd, Pt) are discussed.