Phase equilibria and transformations in Ti–(25–52) at.% Al alloys studied by electrical resistivity measurements

PII:

Acta mater. Vol. 46, No. 2, pp. 405±421, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00274-7

PHASE EQUILIBRIA AND TRANSFORMATIONS IN Ti±(25±52) at.% Al ALLOYS STUDIED BY ELECTRICAL RESISTIVITY MEASUREMENTS D. VEERARAGHAVAN, U. PILCHOWSKI, B. NATARAJAN and V. K. VASUDEVAN Department of Materials Science and Engineering, University of Cincinnati, Cincinnati, OH 452210012, U.S.A. (Received 21 May 1997; accepted 29 July 1997) AbstractÐPhase equilibria and transformations in Ti±(25±52) at.% Al alloys were studied by electrical resistivity measurements over a range of temperatures using a special device that was constructed for this purpose. The a2, a, b and g phases are observed to have distinctly di€erent resistivities and temperature dependencies, owing to which phase transitions could be monitored. The results show that the currently accepted Ti±Al phase diagram is, by and large, accurate, except for minor modi®cations being required to the phase boundaries in the composition range of Ti±(25±43)Al. Also, the a single-phase ®eld is found to extend to the 25Al composition, which points to the absence of a b + a2 t a peritectoid reaction. The room temperature electrical resistivity of stoichiometric a2Ti3Al and g-TiAl are 118 and 31 mO cm, respectively, i.e. show a di€erence of 87 mO cm. The changes in resistivity with temperature are also signi®cantly di€erent in that in a2 the resistivity saturates to a near-constant value near 7508C, whereas that of the g phase shows a linear and near-constant slope with temperature like most metallic materials. In order to explain these di€erences, the electrical resistivity of the a2 and g phases has been modeled by ®tting the data using the Bloch±Gruneisen formulation with certain simplifying assumptions. Good agreement between the calculated and experimental resistivity±temperature curves, and between calculated and experimental values of the residual resistivity and Debye temperature, were obtained. From the model, parameters such as the Fermi velocity, e€ective mass of a conduction electron, the number of electrons participating in conduction and electron mean free path have been calculated for the two phases. The calculations reveal that the mean free path is of the order of the lattice parameter in the case of a2, which leads to high resistivity and resistivity saturation. The resistivity of the a2 phase is also higher than that of the g phase due to the fact that the Fermi velocity of the electrons is lower, e€ective electron mass higher and fewer electrons participate in conduction. These factors, coupled with hybridization and localization e€ects, cause the di€erent electrical resistivity behavior of the two phases. # 1998 Acta Metallurgica Inc.

1. INTRODUCTION

The gamma titanium aluminides (g-TiAl) have been receiving much attention in recent years as structural materials for elevated temperature aerospace applications, owing to their unique combination of attractive properties such as lightness, superior strength, creep and oxidation resistance [1]. A number of promising alloys have been developed based on the Ti±48Al composition (compositions in the text refer to atom percent), with additions of elements such as V, Cr, Nb, Mn, etc., singly or in combination [2±6]. All of these alloys are twophase, with a microstructure composed of either fully lamellar grains of Ti3Al + TiAl (a2+g) or duplex primary g plus lamellar grains, depending on processing and heat treatment [4]. The mechanical properties, in turn, are found to depend sensitively on the volume fraction, morphology, size and distribution of these constituents [3±7].

Over the years, and more recently, several studies have been conducted of phase equilibria and transformations in the binary Ti±Al system, including the two-phase a2+g alloys [8±20]. Of particular signi®cance are the results of recent investigations showing decidedly that the (b + g) two-phase ®eld shown in previous Ti±Al phase diagrams [18] does not exist and instead the (a + g) phase ®eld extends up to the peritectic temperature, i.e. the a region exists up to 14508C and 50 at.% Al. Although there is general agreement now on the broad shape of the phase diagram and the phases present, some uncertainty still exists with regard to the positions of phase boundaries in the region between 25 and 50Al. A study of phase transformations has therefore been undertaken, utilizing in situ high temperature electrical resistivity measurements, to validate the currently accepted Ti±Al phase diagram [19]

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and to determine whether modi®cations are necessary. The results are reported herein and discussed. The technique of electrical resistivity lends itself quite gracefully for studying phase transformations in alloy systems. Large, measurable changes in resistivity accompany most solid state transformations and particularly during long range ordering. Signi®cant di€erences in the absolute values of the resistivity for di€erent phases, as well as signi®cant changes with composition and temperature are often observed. This allows transformation temperatures and kinetics of phase changes to be determined accurately. Resistance arises from electron scattering caused by deviation from perfect periodicity. The two most important scatterers in metals and alloys are phonons and impurities: the electron±phonon component rep(T) dominates the total resistivity r(T) at high temperatures, and the electron impurity component, r0, dominates at suciently low temperatures. Mooij [21] studied the electrical resistivity behavior of Ti-rich Ti±Al alloys with compositions in the range of 0±33Al and temperatures from RT to 0900 K. The addition of Al was found to lead to an increase in the absolute value of resistivity at low and high temperatures. Furthermore, a tendency for saturation of resistivity at high temperatures was observed, with this feature being most pronounced for compositions around the a2-Ti3Al phase. He proposed, based on qualitative arguments, that the electron scattering caused by s±d hybridization of orbitals was a signi®cant factor in determining the resistivity behavior, and that at high temperatures the reduction in the electron mean free path to values approaching the interatomic distance causes the resistivity to saturate at a certain value and not increase linearly as most metallic materials do [21]. In our initial work [22], the resistivity characteristics of near-stoichiometric a2-Ti3Al was found to

be in general agreement with the results of Mooij [21]. However, that of near-stoichiometric g-TiAl was found to follow the general behavior of metals in that the resistivity increased linearly with temperature and did not show a tendency to saturate. In contrast with pure metals, the absolute value of electrical resistivity of near-stoichiometric g-TiAl at room temperature (RT) is high, 040 mOcm, but this value is signi®cantly lower than that of near-stoichiometric a2-Ti3Al, whose electrical resistivity at RT is 0120 mOcm. The reasons for the di€erences in the absolute resistivity of these two phases and their temperature dependence are currently not satisfactorily understood. The resistivity of these two phases are modeled and the di€erences in their behavior explained on a fundamental physical basis.

2. EXPERIMENTAL DETAILS

2.1. Alloy compositions, processing and heat treatment A series of alloys ranging in composition from 25 to 52Al were prepared. All alloys, with the exception of the nominal 48Al composition, were prepared as 250 g cigars by arc-melting 99.999% pure Al and commercial purity Ti under an argon atmosphere. The cigars were turned over and remelted at least ®ve times to assure compositional homogeneity. Subsequently, they were hot isostatically pressed (HIPped) using the conditions of 12008C, 2 h, 105 MPa. The nominal Ti±48Al alloy was obtained as an ingot from the Duriron Company. Cut sections of the ingot were solutionized in the fully a phase ®eld (14008C for 4 h) in vacuum, furnace cooled, then canned and vacuum sealed in Ti± 6Al±4V. Subsequently, the sealed piece was heated to 13908C for 2 h and then extruded at an extrusion ratio of 6:1. The chemical compositions (%Al, C,

Table 1. Chemical compositions of alloys studied Actual Composition Aimed composition (at.%) Ti±25Al Ti±35Al Ti±40Al Ti±43Al Ti±45Al Ti±46Al Ti±46.5Al Ti±47.5Al Ti±48Al Ti±48.5Al Ti±50Al Ti±52Al

Ti(at.%)

Al(at.%)

Bal Bal Bal Bal Bal Bal Bal Bal Bal Bal Bal Bal

25.34 35.01 41.23 42.69 44.74 46.55 46.81 47.55 47.86 48.50 49.86 52.30

O(wt%)

N(wt%)

0.073 0.081 0.035 0.078 0.017 0.037 0.056 0.047 0.049 0.028 0.047 0.083

0.005 0.008 0.006 0.004 0.007 0.014 0.005 0.006 0.006 0.003 0.005 0.013

C(wt%) 0.0146 0.0118 0.0108 0.0091 0.0102 0.0119 0.0194 0.0160 0.0120 0.0192 0.0202 0.0180

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS

O, N) of all the alloys were determined by wet analysis after the above treatments had been imposed, and are shown in Table 1. As can be seen, the actual Al contents are near the aimed values; the C, O, and N contents in each of the alloys are comparable, with values on the average around 0.05, 0.005 and 0.015 wt%, respectively. The exceptions are the 46 and 52Al alloys which have a much higher N content (0.013±0.014 wt%) than the others. The former values fall in the range found in alloys used in studies by other investigators [9±20]. Prior to studies of the transformation behavior, it was considered important to keep sample to sample variations in microstructure in each alloy to a minimum. Consequently, cut sections of each of the cigars (with the exception of the 52Al alloy) were given an additional homogenization treatment in a vacuum furnace (evacuated to 5  10ÿ5 Pa) for 0.5 h at temperatures in the a-phase ®eld, followed by slow furnace cooling to 9008C and held there for 24 h before slow cooling to room temperature. The a-solutionizing temperatures, which were chosen based on the currently accepted Ti±Al phase diagram [19], Fig. 1, increased with increasing Al content in the alloy. The complete schedule is shown in Table 2. The only exception was the 52Al single-phase g alloy, which received a modi®ed treatment to avoid the a + g phase ®eld during the initial solutionizing, as shown in Table 2. 2.2. Starting microstructure Both pre- and post-experiment samples were electropolished and etched using Kroll's reagent and

407

Table 2. Heat treatment schedule imposed on the alloys Nominal composition (at.%) Ti±25Al Ti±35Al Ti±40Al Ti±43Al Ti±45Al Ti±46Al Ti±46.5Al Ti±47.5Al Ti±48Al Ti±48.5Al Ti±50Al Ti±52Al

Heat treatment schedule 11208C, 12008C, 12508C, 13258C, 13508C, 13608C, 13708C, 14008C, 14008C, 14008C, 14108C, 13008C,

0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 9008C, 24 h 4 FC to 258C 0.5 h, FC 4 10008C, 50 h 4 FC to 258C

the microstructures observed by optical (OM) and scanning electron microscopy (SEM) (JEOL-T220A and Cambridge Model 90B).The phases present were determined by X-ray di€raction using a Philips X'Pert y±y powder di€ractometer. The alloys studied can be divided into three classes depending on the Al content and phases present: (I) single-phase a2 (25 and 35Al), (II) twophase a2+g (40±48.5Al) and (III) near single and single-phase g (50, 52Al). Representative optical micrographs showing the microstructure of alloys following the homogenization treatment (Table 2) are shown in Fig. 2(a±f). Both the 25 and 35Al alloys are completely a2 [Fig. 2(a,b)], whereas the 40±48.5Al alloys show the characteristic a2+g lamellar structure that is produced by slow cooling from the a region [Fig. 2(c,d)]; the 50Al alloy is nearly single phase g, with a small volume fraction of lamellar a2+g grains [Fig. 2(e)], whereas the 52Al alloy is single-phase g [Fig. 2(f)]. X-ray di€raction patterns recorded from the samples essentially con®rmed the phase identi®cation as a function of Al content. The grain size is large in each case, with values exceeding 500 mm on the average. These were the starting microstructures prior to the experiments in the TERMS (Section 2.3). 2.3. Temperature and electrical resistivity measurement system (TERMS)

Fig. 1. Central portion of the Ti±Al phase diagram (after Ref. [19]).

A novel computer controlled high-speed Temperature and Electrical Resistivity Measurement System (TERMS) was designed and constructed for the purpose of studying phase transformations in alloys by means of in situ measurements under controlled isothermal, continuous heating and cooling conditions or a combination of these. A schematic diagram of the system is shown in Fig. 3. The system broadly consists of the following: (1) a sample chamber connected to a vacuum system capable of attaining a vacuum of 10ÿ4 Pa, (2) a heating assembly which includes a computer controlled d.c. power supply (KEPCO Model ATE

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VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS

6-100M operated in the current mode) and appropriate connections for heating the sample by its electrical resistance, (3) a computer triggered quenching device designed to admit a jet of puri®ed helium gas at a predetermined ¯ow rate at the sample while shutting o€ the power, (4) fast-response thermocouples (25 mm diameter S-type Pt/ Pt±Rh) and resistance probes (Ta wires) spotwelded to the sample, and (5) hardware and the software interfaces for the purposes of closed loop temperature control, and data acquisition, synchronization and analyses. A more detailed description of TERMS is given elsewhere [22]. Samples measuring 25 mm long  3.5 mm wide  (0.5±0.8) mm thick were sliced from the homogenized alloys. The sample assembly consisted of a thermocouple spot welded to the sample center, a pair of Ta wires at a calibrated distance spot welded on the sample on both sides of the thermocouple, and 25 mm thick Ta sheets spot welded to the sample ends, then connected through spot welded Ti sheets to copper clamps, which, in turn, are connected to the power supply via copper busbars. Since Pt±Pt/Rh reacts with TiAl, a very thin sheet of Ta was ®rst spot welded to the sample, upon which a fused thermocouple bead was spot welded. Examination of the sample after heating to high temperatures did not reveal any signi®cant reaction between the Ta and the sample. Also, experiments on metals with known transition temperatures established that the temperature measurements were not compromised by this procedure. Room temperature electrical resistivity measurements were ®rst made by mechanically placing the sample across calibrated knife edges, and connecting the ends to a precisely controlled current source. With a controlled current of 03A passing, the voltage drop across the knife edges was measured using an accurate digital multimeter; the resistivity was then obtained using the standard expression,

rˆ

    V A  I L

where A is the sample cross-sectional area, L is the sample length between the knife edges, V is the voltage drop across L and I is the current passed through the sample. These measurements were then repeated after spot welding the Ta wires and from the known resistivity, the length between the Ta wires was determined. The latter, together with the cross sectional area, served as inputs to the computer program for calculating resistivity during heating and cooling. After insertion of the sample assembly and chamber evacuation to 010ÿ4 Pa (10ÿ5 torr), the sample was programmatically heated to a number of di€erent temperatures

(depending on the alloy) at a rate of 058C/min and the resistivity changes with temperature continuously monitored, displayed and stored. Similarly, the resistivity±temperature±time relationships on cooling (vacuum or He quench) at di€erent rates were also recorded. Heating and cooling experiments were conducted on at least 15 samples of each alloy.

3. ELECTRICAL RESISTIVITY CHANGES AND TRANSFORMATIONS ON HEATING

3.1. Room temperature resistivity vs Al content The resistivity values at room temperature as a function of Al content are shown tabulated in Table 3 and plotted in Fig. 4. The room temperature resistivity of the stoichiometric a2 phase (Ti3Al, 25Al) is quite high, being 0118mOcm. The value for the single-phase a2 35Al alloy at room temperature is 0170 mOcm, which is considerably higher than that of the similarly homogenized fully a2 25Al alloy. The higher resistivity value for the 35Al alloy over that of the stoichiometric composition is most likely caused by a lower degree of long range order and enhanced electron scattering by excess Al atoms. The room temperature resistivities of the a2+g alloys are much lower than those of the fully a2 alloys, which clearly shows that the g phase has a lower resistivity than a2. Secondly, there is a continuous decrease in the room temperature resistivity with an increase in Al content between 41.23 and 50.0Al, with a minimum being attained at the latter composition. For instance, the value at 41.83Al is 087 mOcm, whereas that at 50.0Al is 031 mOcm, with values for Al contents intermediate to these lying in between. These changes are associated with a continuous increase in the volume fraction of the lower resistivity g phase with increasing Al content. Beyond the 50Al composition, the resistivity rises again in the singlephase g region to a value of 050 mOcm at the 52Al composition; this increase can be attributed to increase in the degree of disorder of g due to non-stoichiometry (excess Al) and also a higher interstitial content (O and N) as compared with the 50Al alloy, as can be seen in Table 1. 3.2. Ti±25Al and Ti±35Al (single-phase a2) alloys Both these alloys in the homogenized condition have a completely a2 structure [Fig. 2(a,b)]. A typical resistivity±temperature plot for the 25Al alloy is shown in Fig. 5. The resistivity increases rapidly from a value of 0118 mOcm at room temperature to 0153 mOcm at 2258C and more gradually to a value of 0170 mOcm at 4508C. Beyond 4508C, the

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409

Fig. 2. Optical micrographs showing microstructure of as-homogenized alloys: (a) Ti±25Al, (b) Ti±35Al, (c) Ti±42.69, (d) Ti±48.5Al, (e) Ti±50Al and (f) Ti±52Al.

increase in resistivity is even more gradual, with signs of levelling-o€ near 7508C. There is a continuous decrease in the slope of the resistivity±temperature curve over this entire temperature range. Beyond 7508C, the resistivity begins to rise again with a higher slope till about 10708C, at which point there is sharp rise to a maximum 010908C. The increase in resistivity between 750 and 10708C is presumably associated with a decrease in the degree of long range order in the a2 structure, whereas the sharp rise at 010708C marks the transformation of a2 to disordered a, this transformation being complete at 10908C. The sharp, almost discontinuous rise in resistivity suggests that the a2 4a transformation is a ®rst order process. The resistivity remains almost constant between 1090 and 011208C corresponding to the single phase a

region. Above 11208C, and up to 011508C, there is a sharp drop in the resistivity, associated with the transformation from a to b. Between 1150 and 13008C, there is a gradual decrease in the resistivity in the fully b region, this feature suggestive of a negative temperature coecient of resistivity. These various features observed in the resistivity±temperature curve establish that at the 25Al composition, the a2 4a transus, a-transus, a 4 b transus and btransus are near 1070, 1090, 1120 and 11508C, respectively. Each of these transition temperatures is about 308C lower than the corresponding temperatures reported in the currently accepted Ti±Al phase diagram [19]. In order to have con®dence in these transition temperatures, samples were heated to just below and above each of these temperatures, cooled in

410

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS

Fig. 3. Block diagram of the Temperature and Electrical Resistivity Measurement System (TERMS).

vacuum and the microstructures examined (Fig. 6). The microstructure of a sample heated to 11208C and cooled showed evidence for slight transformation to b [Fig. 6(a)], that heated to 11308C and cooled showed evidence for additional transformation to b [Fig. 6(b)] and the one heated to and cooled from 12308C showed that complete transformation to b had occurred [Fig. 6(c)]. It should be noted that the b formed at high temperatures

will transform to a and then a2 on cooling. Because of the Burgers orientation relation (parallelism between the close-packed planes and directions) between b and a, individual grains of the parent a2 grain are converted on cooling from b to a plates (a2 after ordering) in several orientations. That the formation of b has occurred is judged by noting the presence of this Widmanstatten type a2 morphology in the microstructure. Thus, correlating Fig. 6 with

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS Table 3. Room temperature resistivity vs at.% of Ti±(25±52)Al alloys Nominal alloy Ti±25Al Ti±35Al Ti±40Al Ti±43Al Ti±45Al Ti±46Al Ti±46.5Al Ti±47.5Al Ti±48Al Ti±48.5Al Ti±50Al Ti±52Al

at.% Al

Resistivity (mOcm)

25.34 35.01 41.23 42.69 44.74 46.55 46.81 47.55 47.86 48.50 49.86 52.30

118.0 170.0 87.00 70.00 63.00 58.00 52.00 48.00 44.00 37.00 31.00 50.00

the resistivity±temperature plot for the 25Al alloy in Fig. 5 establishes that the transition temperatures determined based on those at which distinctive changes in resistivity take place do indeed correspond to the onset of the phase transformations that were assumed to have occurred. The resistivity±temperature plot for the 35Al alloy on heating is also shown in Fig. 5. The starting microstructure of this alloy after the homogenization treatment is also completely a2 [Fig. 2(b)]. The changes in resistivity on heating is somewhat similar to that of the 25Al alloy. The resistivity increases rapidly from 170 mOcm at room temperature to about 206 mOcm at 4508C, with the slope of the resistivity±temperature curve continuously decreasing; these features are similar to those in the 25Al alloy. Beyond 4508C, the increase in resistivity is much more gradual, with clear signs of attaining an almost constant value of 210±208 mOcm between 540 and 9758C. Beyond 9758C, the resistivity begins to rise again to a maximum of 0213 mOcm at 011508C. This rise is presumably associated with a

Fig. 4. Room temperature electrical resistivity vs Al content for Ti±(25±52)Al alloys.

411

continuous decrease in the degree of long range order in the a2 structure, eventually leading to the formation of fully disordered a at 11508C. Between 1150 and 12408C, the resistivity remains almost constant in the fully a region, and then begins to drop sharply with further increase in temperature to 012808C due to the transformation from a to b. The transformation to b appears to be complete at 012808C, since the resistivity appears to level-o€ at temperatures beyond this value. These results indicate that the a-transus, a 4 b transition temperature and the b-transus at the 35Al composition are near 1150, 1240 and 12808C, respectively. The latter two temperatures correspond closely to those reported in the currently accepted Ti±Al phase diagram [19], whereas the former is about 408C lower. As in the case of the 25Al alloy, the samples of the 35Al alloy were heated to just below and above each of these transition temperatures, cooled in vacuum and the microstructures examined (Fig. 7). The microstructures of the sample cooled from 12408C showed evidence of very slight transformation to b [Fig. 7(a)], that cooled from 12508C showed evidence for additional transformation to b [Fig. 7(b)] and the one cooled from 12808C was completely b at temperature [Fig. 7(c)]. These results give further con®rmation for the assignation of phase transition/boundary temperatures from the resistivity±temperature curve for this alloy. 3.3. Ti±40±48.5Al (a2+g) alloys The alloy compositions containing 40±48.5Al fall within the two-phase a2+g phase ®eld at room temperature. These alloys were initially solutionized in the fully a region for 0.5 h, slow cooled to 9008C and held for 24 h, and then slow cooled to room temperature. This treatment led to a fully lamellar a2+g structure in each case, as shown by the representative examples in Fig. 2(c,d).

Fig. 5. Variation of resistivity with temperature of Ti± 25Al and Ti±35Al alloys.

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Fig. 6. Optical micrographs showing microstructures of Ti±25Al alloy cooled at 01008C/s from (a) 11208C, (b) 11308C and (c) 12308C.

Fig. 7. Optical micrographs showing microstructures of Ti±35Al alloy cooled at 01008C/s from (a) 12408C, (b) 12508C and (c) 12808C.

The changes in resistivity on heating the homogenized two-phase alloys are generally similar. Figure 8(a±c), with up to four di€erent regimes of behavior depending on Al content. In the ®rst regime, which covers the range from room temperature and 09508C in all the alloys, the resistivity increases nearly linearly with increasing temperature. The linear behavior is more prominent in the alloys containing e44.75Al [Fig. 8(b,c)], whereas in the 41.23 and 42.69Al alloys [Fig. 8(a)], which contain a larger volume fraction of a2, there seems to be a continuous decrease in the slope of the resistivity±temperature curve over this temperature range. The latter behavior is similar to that seen in the 25 and 35Al a2 alloys (Fig. 5). In the second regime, which begins above 09508C and ends at the a-

transus in all the alloys, the resistivity initially rises with a higher slope compared with that at lower temperatures, and then is followed by an even steeper rise near 11258C. It appears that the ®rst slope change in the resistivity±temperature curve between 950 and 11258C is associated with a decrease in the degree of long range order in the a2, whereas the second at 011258C is associated with complete disordering of a2 to a. Above 11258C, the resistivity rises even more sharply with increasing temperature as more and more g transforms to the higher resistivity a phase, and then begins to ¯atten out upon completion of this transformation, i.e. upon crossing the a-transus. It is clearly evident that the atransus temperature increases with increasing Al content, consistent with the phase diagram [19]. The

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS

Fig. 8. Variation of resistivity with temperature of (a) Ti± 41.23Al and Ti±42.69Al alloys, (b) Ti±44.75Al, Ti±46.55Al and Ti±46.81Al alloys, and (c) Ti±47.55Al, Ti±47.86Al and Ti-48.50Al alloys.

values for the 41.23 and 48.5Al alloys are 01250 and 14108C, respectively. Above the a-transus, in the fully a phase ®eld, Regime 3, the resistivity remains practically constant with further increase in temperature. In the case of the 41.23 and 42.69Al alloys [Fig. 8(a)], an apparent fourth regime is seen in the resistivity±temperature curve, the beginning of which near 1350 and 13708C, respectively, is marked by a slight increase in resistivity from the value at the plateau in Regime 3. This increase can

413

possibly be attributed to the transformation from a to b, since these temperatures are close to the a±b phase boundary in the currently accepted phase diagram [19]. Further, from Fig. 8(a), near 1375 and 14008C in the 41.23Al and 42.69Al alloys, respectively, the resistivity seems to level-o€, possibly corresponding to complete transformation of a to b. These two temperatures then mark the b-transus at these compositions. It should be noted that in the 25Al and 35Al alloys, the transformation from a to b was accompanied by a decrease in resistivity, unlike the behavior of the 41.23Al and 42.69Al alloys where there is an increase. These results suggest that the relative resistivities of the a and b phases at high temperatures depends on Al content and that there is a crossover point at a composition between 35Al and 41.23Al, below/above which the a phase has a higher/lower resistivity than the b phase. It can be seen from the curves for the 25Al and 35Al alloys, Fig. 5, that the net decrease of resistivity during the transformation of a 4 b is lower in the case of the 35Al alloy than in the 25Al alloy. Since the temperature coecient of resistance for the two alloys may be slightly di€erent due to the di€erences in composition of the a and b phases, there may be a crossover point in temperature and composition at which the absolute value of the resistivity for the two phases are the same. Possibly, this crossover occurs in the range of temperatures between 1250 and 13508C and compositions between 35Al and 43Al. Due to this e€ect of crossover of absolute values of resistivities occurring in the same temperature range as the possible a 4 b transformation for the alloys between compositions 35Al and 43Al, further experiments were performed to conclusively determine whether the change in resistivities at these temperatures mark a phase transformation or a crossover of absolute resistivity values. Several samples were heated in the TERMS to a range of temperatures just above and just below those at which the slight increase in resistivity is seen from the plateau corresponding to the a region and slow cooled. The corresponding microstructures, shown in Fig. 9(a± d), establish that the a 4 b transformation temperature is 01350 and 013608C, and that this transformation is complete at 01375 and 013958C, respectively, for the 41.23Al and 42.69Al compositions. These temperatures are about 20±308C lower than the a±b and b phase boundary temperatures at these compositions in the currently accepted phase diagram [19]. 3.4. Ti±50Al and Ti±52Al (near g and g alloys) The Ti±50Al alloy was initially solutionized in the a + g region for 0.5 h, slow cooled to 9008C, held there for 24 h and then slow cooled to room temperature. This treatment led to nearly fully g

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Fig. 9. Optical micrographs showing microstructures of Ti±41.23Al alloy cooled from (a) 13508C and (b) 13758C; and Ti±42.69Al alloy cooled from (c) 13608C and (d) 13958C.

structure with a few lamellar grains, as shown Fig. 2(e). The single phase 52Al alloy was initially homogenized at 13008C to avoid entering the two phase a + g ®eld, held for 24 h and slow cooled to room temperature. This treatment led to a fully g microstructure as shown by representative micrograph in Fig. 2(f). Resistivity±temperature plots obtained from homogenized samples of these alloys on heating are shown in Fig. 10. The room temperature resistivity of stoichiometric single phase g

composition, Ti±50Al, is, as shown in Table 3 and Fig. 4, signi®cantly lower than that for the 48.5Al composition, which had a fully lamellar a2+g structure at room temperature. This clearly shows that the g phase has a lower resistivity than a2. The changes in resistivity with temperature on heating the 50Al alloy, Fig. 10, are generally similar to those of the higher Al two-phase alloys [Fig. 8(c)], with the same three regimes of behavior being observed. The resistivity increases nearly linearly

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS

Fig. 10. Variation of resistivity with temperature of Ti± 50Al and Ti±52Al alloys.

with temperature to 010008C and then with a higher slope to temperatures near 12808C. Beyond 12808C, the resistivity increases more rapidly until 14108C, above which it appears to begin levelling o€. The rapid increase in resistivity at 12808C is associated with the entry of the alloy into the two

415

phase a + g phase ®eld and the levelling o€ at 14108C is due to approach of the single phase a boundary. The resistivity behavior of the 52Al alloy, Fig. 10, is similar to that of the 50Al alloy at lower temperatures, except that the absolute values are higher. A rapid increase in resistivity occurs near 13308C on entering the two phase a + g ®eld. In contrast with the 50Al alloy no levelling o€ of the resistivity at higher temperatures is seen in this case. It can be inferred that the alloy has not reached the single phase a ®eld until the maximum temperature attained in the experiment, which was 14258C. One important aspect in resistivity±temperature plots of the near-single (Ti±50Al) and single-phase (Ti±52Al) g alloys in comparison with those of the single phase a2 compositions (Fig. 5) is that there is no evidence for saturation of resistivity near 7508C. The resistivity steadily increases with a constant slope until the alloys enter the a + g two-phase ®eld. Thus, the resistivity±temperature behavior of these two intermetallic phases are signi®cantly di€erent.

Fig. 11. Central portion of Ti±Al phase diagram showing data from present work (solid triangles) and modi®ed phase boundaries (long dashed lines).

416

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS 4. IMPLICATIONS ON THE Ti±Al PHASE DIAGRAM

The use of electrical resistivity measurements on heating has proven to be a viable method for mapping out phase boundaries in the Ti±Al system. Large measurable changes in resistivity are associated with the a t g transformation. The changes in resistivity are less marked in the a2ta, and a t b transformations, but nevertheless suciently discernible that near-equilibrium transition temperatures can be determined. The results of the phase boundary determinations using these measurements are shown in the Ti±Al phase diagram as long dashed lines in Fig. 11. In this diagram, the solid triangles denote the location of the various phase boundaries (transition temperatures) as a function of Al content. As can be seen, at high Al contents (a2+g alloys), the values of the a/a + g phase boundary determined herein are excellent agreement with the currently accepted Ti±Al phase diagram [19], implying that this diagram is quite accurate in this regime. At the lower Al contents (25±43Al), the results indicate that modi®cations to the a2/a, a/a + b, and b/a + b phase boundaries are necessary, and these are also shown in Fig. 11. These modi®cations include a shift of these boundaries to lower temperatures over the range (25± 43Al) of composition. Apart from these changes, the most recent version of the Ti±Al phase diagram [19] is quite accurate. The transformation temperatures determined for the lower Al content alloys (25 and 35Al) are signi®cantly di€erent than those reported by Kainuma et al. [20]. Also, the resistivity±temperature data obtained in these alloys suggests that the a singlephase ®eld does extend to the 25Al composition, in contrast with their proposal of a b + a2ta peritectoid reaction around this composition and temperature of 11602108C. 5. ELECTRICAL RESISTIVITY MODELING

The results of electrical resistivity measurements have shown that the a2, a, b and g phases are characterized by distinctive resistivities and that both composition and temperature have dominant e€ects. Stoichiometric single-phase a2 has a much higher resistivity than stoichiometric single phase g at room temperature. The resistivities of the a and b phases at high temperatures are a function of alloy composition. Many models [21, 24±28] have been proposed to explain the resistivity behavior of such ordered, transition metal alloys. These models have been based upon correlation between a negative dr/dT and the high impurity resistivity. They rely on the fact that the normal theories of conduction break down when the electronic mean free path approaches the lattice parameter of the crystal. For instance, based on qualitative reasoning, the high resistivity and subsequent saturation at high

temperatures observed in the case of a2 alloys has been attributed to a small mean free path [21]. To be able to explain the resistivity behavior of the a2 and g phases, the electrical resistivity±temperature data obtained on heating the 25Al (Ti3Al) and 50Al (TiAl) alloys have been modeled using the Bloch±Gruneisen model [29, 30] and the results used to obtain material parameters. The Bloch± Gruneisen (B±G) model uses the Mattheisen's rule that allows for a simple summation of the two important electron scattering contributions: temperature independent (impurity) and temperature dependent (electron±phonon, electron±electron, elastic) scattering. In the simplest B±G model, of the temperature dependent scattering contributions, only the electron±phonon contribution is considered to be in®nitely greater than the other contributions to electrical resistivity. Hence, in this study only the electron±phonon contribution is modeled using certain simplifying assumptions: (a) The Fermi surface is spherical which implies that only s-shell electrons and d-electrons of low velocity participate in the conduction process, (b) The phonon can be de®ned by assuming a Debye phonon spectrum, described by the Debye temperature, YD, (c) The electron±phonon coupling is only through longitudinal phonons, and (d) No umklapp scattering processes occur, which implies that there is no electron±phonon contribution to resistivity at extremely low temperatures and all the electron scattering can be accounted for by the impurity scattering. The rationale for these assumptions can be understood from an application of vibration spectra theory to describing the phonon spectrum and hence the electron±phonon interaction [31]. For a lattice vibration of wave number k, in a metallic material, it can be shown using the pseudopotential theory, that the structure factor expanded in the displacements of the ion around its mean position gives rise to non-zero intensity at wave numbers other than exact lattice wave numbers. These intensities, that are caused by lattice distortion, give rise to nonzero structure factors at satellites to each of the lattice wave numbers and lie at wave numbers on either side of the lattice wave number with a displacement of k. These satellite points give rise to electron scattering. Similar calculations can be done and structure factors can be computed using lattice distortion e€ects and pseudopotential theory for defects such as faults, dislocations, impurities and vacancies. Ordering and ordering parameter e€ects can also be similarly accounted for. The consequent satellite points can be expressed in matrix form to describe the electron±phonon interaction. From an evaluation of vibration spectra, it can be established

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS

417

that for normal scattering processes, i.e. those processes for which the lattice wave number vanishes to zero, only longitudinal modes contribute. This is due to the fact that in the case of the transverse mode, the electrostatic coulomb screening e€ects e€ectively cancel the band structure energy e€ects and hence do not give rise to electron scattering. Assuming that scattering does not occur at nonzero lattice wave number at low temperatures, umklapp scattering, which can occur from both longitudinal and transverse phonons, can be neglected. Another justi®cation for assuming that scattering contributions are signi®cant only from longitudinal phonos is that the transverse contributions are as signi®cant only in the acoustical modes, which cannot be generated thermally. In the case of the optical modes, the interaction with transverse modes is very weak. The simplest analytical expression for the electrical resistivity due to electron±phonon contributions, rep(T), is given by the B±G model as [30]:

rep ˆ r0 T…T=YD †:4

YD…=T

dx 0

…ex

x5 ÿ 1†…1 ÿ eÿx †

…1†

where the terms have the following meanings: r' = rep/T, the high temperature limit of the slope of the resistivity temperature curve, which is constant and di€ers for each material YD is the Debye temperature in Kelvin, constant for each material, and T is the temperature in Kelvin. This is the temperature dependent scattering contribution to the resistivity. The integral in equation (1) has been numerically solved and values have been obtained for di€erent T/YD values [32]. The temperature independent scattering term is given by r0, the impurity resistivity at liquid helium temperature, 4.2 K. Therefore, the ideal resistivity by Mattheissen's rule is given by: rideal …T† ˆ r0 ‡ rep …T†

…2†

In the case of the a2-Ti3Al alloy, the resistivity can

Fig. 12. Experimental and calculated resistivity±temperature plots of (a) stoichiometric a2 Ti±25Al (Ti3Al) alloy, and (b) stoichiometric gTi±50A (TiAl) alloy.

be seen to begin saturating near 7508C. It has been established that in systems where room temperature resistivity is large (>50mOcm), resistivity saturation is pronounced. This e€ect, causing the breakdown of the classical Boltzmann theory, has been rationalized in terms of the Io€e±Regel criterion [33], namely, that the electron mean free path cannot be shorter than the interatomic distance. The high temperature resistivity at saturation, rsat, can be incorporated into the B±G formulation in such cases using the rationale that the limitation on the mean free path implies the existence of a minimum time between collisions, which leads to a parallel resistor formula that accounts for the resistivity saturation:

Table 4. Bloch±Gruneisen model solution for stoichiometric a2-Ti3Al and g-TiAl and comparison with experimental data r0(mOcm) Composition Ti±25Al Ti±50Al

YD(K)

Model ®t

Expt Data

Ref.

Model ®t

Expt. Data

Ref.

34.95 11.40

43.90 Ð

[34] [34]

414 533

450 575

[34] [34]

rsat (mOcm) 226.5 228.6

r' (mOcm/K) 0.838 0.067

D = (Zw2)/N 0.21 0.12

418

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS Table 5. Values of di€erent parameters used in calculations of the electronic mean free path of stoichiometric a2-Ti3Al and g-TiAl

Composition4

Ti±25Al stoichiometric a2

Parametersq 3

2.N(EF) (states/m -J) ltr r' (mOcm/K) hWFi (m/s)

Ti±50Al stoichiometric g

Reference

47

2.76  10 0.36 0.838 1.22  105

[35] [35] Model solution Calculated

1 1 1 ‡ ˆ r…T† rideal …T† rsat

…3†

where r(T) is the measured resistivity and rsat is the resistivity at which saturation occurs. The resistivity of stoichiometric g-TiAl at room temperature is typically 030 mOcm. Also, its behavior on heating is signi®cantly di€erent than a2Ti3Al in that the resistivity does not saturate at high temperatures and the slope of the resistivity± temperature curve is nearly constant throughout the g phase ®eld except at temperatures higher than 011508C, wherein disorder and the g 4 a transformation contributes to resistivity in addition to the temperature coecient of resistance. Hence, the B±G formulation used in the case of stoichiometric g-TiAl was of the form: r…T† ˆ rideal …T† ˆ r0 ‡ rep …T†

1.44  10 0.24 0.067 4.56  105

[36] Calculated from electronic speci®c heat data Model solution Calculated

quite well with the experimentally reported value of 43.9 mOcm [34]. The margin of error most likely arises from the di€erent experimental conditions of the earlier work, as the impurity resistivity is strongly dependent on the defect structure and impurities in the material, and hence thermal history. The Debye temperatures, YD, for the two phases are also in close agreement with the experimental values reported, thus lending validity to the model. Based on this model and the parameters obtained a number of useful physical quantities of the two intermetallic phases can be calculated using the nearly free electron model. An estimate of the electronic mean free path due to electron± phonon scattering can be obtained from the expression [30]: `ep ˆ

…4†

where the terms have the same meaning as de®ned previously. Figure 12(a,b) shows computed resistivity±temperature plots for stoichiometric a2-Ti3Al and gTiAl using the B±G model. The best ®t was achieved by minimizing the root mean square deviation, allowing the four parameters in equation (1) to ¯oat. The experimental curves for the two compositions are superimposed on the calculated curves as a series of points. As can be seen, excellent agreement between the experimental and calculated resistivity±temperature curves is obtained for both a2-Ti3Al and g-TiAl. The parameters, r0, r' = dr/dT, and YD for the two alloys, and rsat for a2-Ti3Al that best ®t the data, as well as experimentally determined values for r0 and YD [34] are listed in Table 4. The value of r0 of 34.0 mOcm, in the case of a2-Ti3Al agrees

Reference

47

hhv  Fi   YD 2pltr kB TG T

where the terms have the following meanings: G

  YD T

is de®ned by the integral in equation (1) hvFi is the velocity of electrons at the Fermi level h = h/2p, h being Planck's constant kB is Boltzmann's constant, and T is temperature in Kelvin ltr is the electron±phonon coupling constant similar to the transport constant in superconductivity theory. This is a characteristic dimensionless constant for any material and can be modeled mathematically using the pseudopotential approxi-

Table 6. Electronic mean free path of stoichiometric a2-Ti3Al and g-TiAl Alloy composition Ti±25Al Ti±50Al

…5†

Iep(T) (AÊ)

Iep(T = 300 K) (AÊ)

Iep(T = 1000 K) (AÊ)

4119.4/TG(YD/T) 23095.9/TG(YD/T)

15.28 85.67

4.16 23.30

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS

mation. The constant depends primarily on Fermi energy and the pseudopotential [31]. The Fermi velocity can be calculated from the expression [30]: r0 ˆ

6pkB ltr he  2 :2N…EF †hv2F i

…6†

where e is the electronic charge, and N(EF) the density of states at the Fermi level. The Fermi density of states has been calculated for both a2 and g by earlier researchers using models based on the nearly free electron theory and taking into consideration the d-orbital electrons and bonding e€ects in the two phases [35, 36]. The electron±phonon coupling constant, ltr, has been estimated from the electronic speci®c heat data [35] and the model ®t for the two alloys (Table 4) provided the value for r'. The various parameters used and the Fermi velocity for the two alloys are shown in Table 5. Using the values in Table 5, the electron mean free path due to phonon scattering has been obtained for a2-Ti3Al and g-TiAl from equation (5) and the results are summarized in Table 6. Table 6 shows that the temperature dependence of the electron mean free path, lep, for the a2 and g phases are vastly di€erent, which explains their di€erent temperature dependencies. The model predicts that the electron mean free paths for a2-Ti3Al at 300 and 1000 K are 15.28 and 4.16 AÊ, respectively, i.e. are close to its lattice dimensions (a = 5.78AÊ, c = 4.62 AÊ). The corresponding values for g-TiAl are much higher, being 85.67 and 23.30 AÊ, respectively. The latter values are also much larger than the lattice parameters (a = 4.005 AÊ, c = 4.07 AÊ) of stoichiometric g-TiAl. Experimental data show that the resistivity of a2 is high at room temperature (0300 K) and it saturates near 7508C (1023 K). The model predictions can be interpreted in the following manner. The usual theory of conductivity is generally expected to break down when the electronic mean free path, lep, approaches the lattice parameter, a. The same holds good when lep
419

total resistivity when lep0a. Weger and Mott [24] proposed an alternative mechanism based on a two band model, in which it is assumed that above a certain temperature the electron±phonon scattering energies may exceed the hybridization interaction, thereby decoupling the s- and d-states. Conduction then mainly occurs by s-electrons with a reduced rate of increase of phonon scattering with increasing temperature. In the case of g-TiAl, resistivity saturation is not observed as the mean free path is signi®cantly greater than the lattice parameter even at high temperatures. The reasons for the di€erences in the electronic mean free paths of a2-Ti3Al and g-TiAl are not entirely understood, although some explanations can be forwarded based on the parameters calculated in the present study. Di€erences in bonding and hybridization e€ects along with di€erences in localization of electrons can be contributing factors [37]. The conduction band electrons in a2Ti3Al are more closely bonded to the atom core than the electrons in g-TiAl. This is borne out by the fact that the ratio e€ective mass to the free electron mass is 5.6 in the case of a2-Ti3Al, whereas the corresponding ratio in the case of g-TiAl is 1.9. This implies that the electrons in the a2 phase move as if they weigh 5.6 times more than electrons free to participate in conduction processes as one would expect in normal metals. This di€erence in m*/m0 (e€ective mass/free electron mass) between a2-Ti3Al and g-TiAl can partially explain the di€erence in the absolute value of the electrical resistivities. Also, the ratio of the number of conduction electrons available for conduction compared with the number that should be theoretically available in the absence of localization of electrons around atom cores is 3.1% in the case of Ti3Al and 9.2% in the case of TiAl. This implies that localization e€ects are much stronger in Ti3Al than in TiAl. The increased localization may create tunneling e€ects that circumvent classical Boltzmann behavior and establish conduction paths that limit the electrical resistivity to a maximum value above a certain temperature. The ordering parameter may also signi®cantly in¯uence the electrical resistivity changes with temperature. A reduction in the Long Range Order (LRO) parameter with increase in temperature may cause the hybridized electrons to be delocalized, thereby enhancing conduction and causing the resistivity to saturate in a2-Ti3Al. This implies the density of states function actually changes with temperature and also the localization of electrons is less pronounced at high temperatures. These e€ects may not be as pronounced in g-TiAl, since the lower impurity resistivity indicates that scattering in the g phase is not as strong as in the a2 phase.

420

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS 6. CONCLUSIONS

REFERENCES

The major conclusions arising from this study of phase equilibria and transformations in Ti±(25±52) at.%Al alloys using electrical resistivity measurements are as follows:

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1. The a2, a, b and g phases are observed to have distinctly di€erent resistivities and temperature dependencies. 2. The currently accepted Ti±Al phase diagram is, by and large, accurate, except for minor modi®cations being required to the phase boundaries in the composition range of Ti±(25±43) Al. Also, the a single-phase ®eld is found to extend to the 25Al composition, which points to the absence of a b + a2 t a peritectoid reaction. 3. The room temperature electrical resistivity of stoichiometric a2-Ti3Al (118 mOcm) is much higher than that of and g-TiAl (31 mOcm). The temperature dependence of the resistivity is also quite di€erent in that in a2 the resistivity saturates to a near-constant value near 7508C, whereas the resistivity of the g phase shows a linear and near-constant slope with temperature like most metallic materials. 4. The electrical resistivity of the a2 and g phases has been modeled using the Bloch±Gruneisen formulation. Excellent agreement is obtained between the calculated and experimental resistivities and between parameters (r0, YD) derived from the model and experimental data. 5. From the B±G model and related theory, parameters such as the Fermi velocity, electron mean free path, e€ective mass of a conduction electron and the number of electrons participating in conduction have been calculated for the stoichiometric a2-Ti3Al and g-TiAl phases. The calculations reveal that the high resistivity and saturation of resistivity at high temperatures in the case of the a2 phase is caused by mean free paths being of the order of the lattice parameter. The resistivity of the a2 phase is also higher than that of the g phase due to the fact that the Fermi velocity of the electrons is lower, e€ective mass higher and fewer electrons participate in conduction. These factors, coupled with hybridization and localization e€ects, causes the di€erent electrical resistivity behavior of the two phases.

AcknowledgementsÐSupport for this research from the National Science Foundation (DMR-9224473, B. MacDonald, Program Monitor) and the Wright Laboratories, Materials Directorate/UES (S-269-000-002/ SUB AF, M. Mendiratta, Program Monitor), is deeply appreciated.

VEERARAGHAVAN et al.: PHASE EQUILIBRIA AND TRANSFORMATIONS 34. Collings E. W., in Applied Superconductivity, Metallurgy, and Physics of Titanium Alloys, Vol. 1. Plenum Press, New York. 1986, p. 224. 35. Collings, E. W., Ho, J. C. and Ja€ee, R. I., in Titanium: Science and Technology ed. R. I. Ja€ee and H. M. Burte, Plenum Press, New York 1973, p. 831.

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36. Woodward, C., McLaren, J. M. and Rao, S., J. Mater. Res., 1992, 7, 1735. 37. Rossiter, P. L., in The Electrical Resistivity of Metals and Alloys. Cambridge University Press, Cambridge, 1991.