Modification of the jogged-screw model for creep of γ-TiAl

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Acta mater. Vol. 47, No. 5, pp. 1399±1411, 1999 # 1999 Published by Elsevier Science Ltd On behalf of Acta Metallurgica Inc. All rights reserved Printed in Great Britain S1359-6454(99)00021-X 1359-6454/99 $20.00 + 0.00

MODIFICATION OF THE JOGGED-SCREW MODEL FOR CREEP OF g-TiAl G. B. VISWANATHAN1{, V. K. VASUDEVAN2 and M. J. MILLS1 Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, and 2Department of Materials Science and Engineering, University of Cincinnati, Cincinnati, OH 45221, U.S.A.

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(Received 5 November 1998; accepted 12 January 1999) AbstractÐDuring the high-temperature creep of the g-phase (L10 structure) of a ``near-gamma'' Ti±48Al microstructure, observations using transmission electron microscopy indicate that a=2h110Š or ``unit'' dislocation activity is a dominant deformation mode. These unit dislocations tend to be elongated along the screw orientation, and exhibit a large number of localized pinning points. Tilting experiments demonstrate that these pinning points are associated with jogs on the screw dislocations, suggesting that the joggedscrew model for creep should be appropriate in this case. However, it is shown that in its conventional formulation, the jogged-screw model is not capable of reproducing the measured creep response (i.e. stress exponents or absolute creep rates). Microscopic observations also demonstrate that a spectrum of jog heights are present, with some as large as 40 nm, based on present observations. A modi®cation of the jogged-screw model is proposed in which the average jog height is assumed to depend on stress. This modi®ed model results in good agreement between predicted and measured creep rates while using reasonable model parameters. Additional implications of the model and required experiments to further validate the model are also discussed. # 1999 Published by Elsevier Science Ltd on behalf of Acta Metallurgica Inc. All rights reserved.

1. INTRODUCTION

Intermetallic compounds based on gamma titanium aluminides (g-TiAl) are attractive due to their light weight (3.7±3.9 g/mm3), sti€ness (room temperature elastic modulus of 170 MPa), superior high-temperature strength (0600 MPa at 873 K) and excellent oxidation and creep resistance up to 9008C [1]. The potential as far as high-temperature structural applications of g-TiAi requires that the creep behavior of this compound be well understood. Though the single-phase alloys are of lesser engineering importance, the creep behavior of single- and twophase alloys is largely dependent on the g-phase since it is the primary participant in the deformation process. Signi®cant strides have been made in the last few years in understanding and modeling the lower temperature deformation behavior of these alloys, for example the yield strength anomaly which occurs in single-phase g-TiAl [2]. Widely reported values of creep activation energies equal to or greater than that for self-di€usion and stress exponents in the range 4±6 [2±8] both imply that the underlying mechanism of creep is related to climb-controlled recovery of dislocations. However, a distinct absence of subgrain formation at smaller strains (i.e. the secondary creep rate regime) [5, 6, 8] indicates that this power-law creep behavior is fundamentally di€erent from that found in pure {To whom all correspondence should be addressed.

metals [9]. There is at present no physically-based description of creep in single-phase g-TiAl. Previous studies have shown that ``unit'' a=2h110Š dislocations tend to dominate the microstructures following creep deformation in g-TiAl [4±6, 8]. Additional deformation modes have been reported in a number of investigations, including twinning [5, 6, 8, 10, 11, 13], dynamic recrystal-lization [5, 12], and grain boundary sliding [14]. The former deformation modes tend to become increasingly important at larger strains well beyond the minimum creep regime. Recent work indicates that for smaller grain size materials, grain boundary sliding may contribute to deformation even at smaller strains [14]. It has been suggested that strain hardening during primary creep in single-phase alloys may be due to the exhaustion of superdislocation motion caused by a combination of processes such as localized pinning, formation of Kear±Wilsdorf (KW) or roof barriers and faulted dipoles [6]. However, the importance of superdislocations to the creep process appears to be minimal based upon the present transmission electron microscopy (TEM) observations [6]. The purpose of this paper is in part to point out that the jogged-screw model, which has previously been used to describe creep primarily in pure metals and solid solution alloys [15], would appear to be particularly appropriate for the case of creep in gTiAl. Based on the premise that secondary or mini-

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mum creep rates are determined by the motion of jogged unit dislocations, it is shown that the jogged-screw model, in its original form, fails to accurately reproduce the macroscopically observed creep behavior. We then present a relatively minor, yet physically reasonable, modi®cation of the original jogged-screw model which produces much improved agreement with experiment. Following a section which brie¯y describes experimental details in Section 2, we present a synopsis of the available creep data and microstructural information related to creep in g-TiAl in Section 3, including a discussion of the possible source of jog formation. In Section 4, the predictions of the conventional jogged-screw model are compared with the experiment. New TEM observations of dislocation structures in creep-deformed, near-gamma Ti±48Al are then presented which clearly demonstrate that a=2h110Š ``unit'' dislocations contain jogs of varying heights. In Section 5, we present a modi®cation of the jogged-screw model which incorporates a stress dependence of the jog height, and the predictions of this modi®ed model are compared with experiment. Finally, in Section 6, some of the critical features and assumptions of the modi®ed model are examined, before presenting a summary.

3. CREEP BEHAVIOR AND DISLOCATION STRUCTURES IN g-TiAl

As is conventionally adopted in the literature, the creep response in g-TiAl may be characterized by the Dorn equation in which the steady state creep rate is expressed as e_ ˆ Asn exp…ÿQ=RT †, where the Q is the activation energy, s the applied stress, n the stress exponent, R the gas constant and T the creep temperature. Activation energy values in the range 300±600 kJ/mol and stress exponents in the range 4±6 have been reported [3±7, 16]. Given the fact that the activation energy for self-di€usion of Ti in TiAl is only 291 kJ/mol [16], the larger values of Q which have been reported in the literature are rather surprising. In a recent review of creep literature for single-phase g-TiAl by Parthasarathy et al. [17], it was concluded that creep in single-phase g-TiAl appears to be sensitive to a number of uncontrolled factors, such as interstitial impurity content, and details of heat treatment. However, stress exponent values of around 5 have been con-

2. EXPERIMENTAL PROCEDURES

For comparison with the calculated creep behavior based on the models described below, we will present both creep data and microstructural evidence based on an alloy of composition: AlÐ 47.86 at.%, OÐ0.116 at.%, NÐ0.016 at.%, CÐ 0.041 at.%, HÐ0.076 at.% and the balance titanium. Detailed description of the creep testing is available elsewhere [4, 8]. To brie¯y summarize the experimental procedures utilized in this previous study, prior to machining, cylindrical blanks of the forged alloy were heat treated at 1473 K in the (a ‡ g) two-phase region just above the eutectoid temperature and followed by a stabilization treatment at 1173 K for 6 h. This heat treatment yields a near-gamma microstructure for which the volume fraction of a2 is very small and the g grains are in an equiaxed morphology with an average grain size of 050 mm. Stress-increment tests were used to obtain the stress dependence of the creep rate and select monotonic tests were also performed to con®rm the results of the stress-increment tests [4, 8]. Thin foils for TEM observations were prepared from discs sectioned normal to the stress axis of the creep-tested samples. The foils were prepared by the twin-jet thinning technique using a solution consisting of 300 ml methanol, 180 ml butyl alcohol, 20 ml perchloric acid and 10 ml hydrochloric acid, at a voltage of 070 V, current of 20±30 mA and temperature of 233 K. Observations of the deformation structures were conducted in a Philips CM200 TEM operated at 200 kV.

Fig. 1. TEM micrograph showing overall dislocation structures in creep deformed g-TiAl: (a) T = 1041 K, s = 207 MPa; (b) T = 1088 K, s = 207 MPa.

VISWANATHAN et al.: JOGGED-SCREW MODEL FOR CREEP

sistently reported over a wide range of stress. In this study, the results from creep tests and microstructural study of a simple binary Ti±48Al alloy with an equiaxed, near-gamma microstructure are described in order to directly address the creep aspects of the g-phase, thus separating the issues that are related to other microstructures with lamellar constituent. Figure 1 shows the typical dislocation structures in a near-gamma Ti±48Al sample tested to the minimum creep regime at 1041 and 1088 K both at stresses of 207 MPa. The dislocations that are seen in the micrograph are all unit or 1=2h110Š type dislocations. In a number of previous studies, similar dislocation structures have been reported to dominate creep microstructures [6, 8]. The dislocations tend to be elongated in screw orientation, and appear to be pinned at many places along their length. The segments on either side of these pinning points are signi®cantly bowed. From observations such as that shown in Fig. 1, the average distance between these pinning points is estimated to be about 200 nm for these creep conditions. This type of dislocation morphology is in fact quite similar to that which has been observed following constant strain rate deformation of g-TiAl at lower temperatures [2, 18, 19]. In these previous works, such cusped con®gurations have been attributed primarily to two possible sources: (a) intrinsic pinning due to the presence of jogs along the screw dislocations [2, 19], and (b) extrinsic pinning due to ®ne-scale precipitates (e.g. oxides) [20±22]. Support for the former source of pinning lies in the fact that it provides a natural explanation for the yield strength anomaly observed between room temperature and about 800 K in this material [23]. Louchet and Viguier [24] have proposed the picture illustrated in Fig. 2 as the origin of pinning points along these unit dislocations. Since the core of the unit dislocation is considered to be quite compact, based on high resolution transmission electron microscopy (HRTEM) investigation of these dislocations in Al-rich alloys [25], a kink-pair along a screw dislocation can, in principle, form on a variety of planes. Assuming, for example, that two di€erent octahedral planes have a signi®cant

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Schmid factor, stable kink-pairs may be formed and expand on these two planes along the same dislocation line. A small jog on the screw dislocation would e€ectively be formed at the point of collision between the two migrating kinks. As the screw segments between the jogs continue to advance by a kink-pair process, the kinks would tend to collect at the jogs, which could cause the jogs to grow in height. It should also be noted that the jogs can move laterally along the dislocation line via glide, so that some interactions between jogs along the dislocation line may be expected. In the original Louchet and Viguier model [24], it was suggested that the dislocations which bowed between the jogs could lie on a variety of planes. More recent work by Sriram et al. [19] seems to indicate that the dislocation segments bowing between the jogs lie on parallel {111} planes, perhaps indicating that the resolved shear stress is a maximum for a particular {111} plane. It should be noted that this collidingkink-pair picture was developed based on constantstrain-rate experiments. Such a sequence of events has also been proposed recently by Lu and Hemker [6] as a result of their TEM analysis of creep-deformed Ti±52Al. Tilting experiments about the Burgers vector of dislocations such as that shown in Fig. 1 indicate that most of the bowed segments are in fact on parallel {111} planes, in agreement with the con®gurations reported by Sriram et al. [19], and thereby suggesting a similar possible origin for the pinned con®gurations. Direct evidence for the presence of jogs associated with these cusps is described below. 4. CONVENTIONAL JOGGED-SCREW MODEL

The direct observation of cusped screw dislocations during creep of g-TiAl, coupled with the absence of highly developed subgrain structures, suggests that the non-conservative motion of jogs along screw dislocations might indeed be an appropriate rate-limiting process during creep in these materials. We will now demonstrate that the original jogged-screw model, when applied to the gTiAl case, is not capable of reproducing the observed creep behavior when conventional

Fig. 2. Illustration of intrinsic formation of non-conservative jogs along screw dislocations.

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Fig. 3. Predicted axial creep rate vs axial stress using the conventional jogged-screw model (labeled as J-S Model with h ˆ b) for a temperature of 1088 K and two values of jog spacing, l. Also shown are predictions assuming a larger, constant value of jog height h ˆ 100b. Included for comparison are experimental data from the work of Viswanathan [4, 8].

assumptions are made concerning jog heights and the stress dependence of the dislocation density. The general concept that the motion of dislocations at high temperatures may be limited by the presence of jogs on screw dislocations originated with Mott [26], who proposed a quantitative form for the strain rate. This theory was subsequently developed further by Hirsch and Warrington [27], Friedel [28] and Barrett and Nix [15]. The velocity of the screw dislocation assuming that the jogs are all of the vacancy-producing type is given by [15, 29]      4pDs tOl vs ˆ ÿ1 …1† exp hkT h where Ds is the self-di€usion coecient, h the jog height, t the applied shear stress, O the atomic volume, l the average spacing between the jogs, k is Boltzman's constant, and T the temperature. If it is assumed that jogs of the vacancy producing/absorbing type are present in equal density, then the term sinh…tOl=hkT † replaces the exponential in equation (1). However, these two functional forms are practically indistinguishable for the range of parameter values of interest here. It is normally assumed that the jog height is of atomic dimensions (approximately equal to b), and that the jogs may arise due to the intersection of dislocations. We may reasonably make the assumption that the mobile screw dislocation density scales as predicted from the Taylor expression [30]   t 2 rs ˆ …2† aGb where t is the applied shear stress, G the shear modulus and a the Taylor factor. Once again, such a stress dependence is suggested by the bulk of the

data available in the literature [31]. Then, using the Orowan relation, the functional form of the conventional jogged-screw model can be seen in Fig. 3. We have assumed a temperature of 1041 K for which the di€usion coecient of g-TiAl is calculated to be 3.82  10ÿ19 m2/s, using the data from Kroll for a Ti±54Al alloy [16]. A shear modulus of 59 GPa has been used based on the estimates from Ref. [32]. A Burgers vector value of b = 2.832 AÊ has been considered. We have converted from shear to tensile values of stress and strain assuming e_ ˆ g_ =2 and t ˆ s=2. The value of a in the Taylor expression can be deduced from dislocation density measurements. Based on TEM observations in near-gamma Ti± 48Al crept at the stress of 207 MPa and temperature of 7688C, the dislocation densities presented in Table 1 have been measured. The sample used for these measurements was obtained following a creep strain of 0.015 corresponding to the minimum creep rate condition. At this strain level, the density was found to vary signi®cantly even within a given grain, with the density being higher near the grain boundaries and lower within the grain interiors. Both limiting values of the dislocation density are given in Table 1. These dislocation densities imply a-values (using the Orowan equation) of between about 0.7 and 3. Since the majority of the deformation volume within grains appear to have disloTable 1. Experimental values for r (dislocation density) and a (Taylor factor)

Grain interior Near grain boundary

Dislocation density, r (/cm2)

Taylor factor, a

0.028  1010 0.500  1010

2.93 0.72

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Fig. 4. (a) Stereogram showing the beam directions B used to obtain the weak-beam TEM images from Ti±48Al alloy deformed at 1041 K and 207 MPa; (b) g = 220; B1‰334Š, close to the glide plane ‰111Š, (c) g ˆ 111; B1‰143Š, close to [011], (d) g = 111; B1‰413Š, close to ‰101Š. (e) A schematic illustration of the tall jog con®gurations deduced from these observations. The approximate viewing directions used in (b) through (d) are indicated.

cation densities closer to the lower value in Table 1, this analysis would suggest that the value of a should be on the higher end of this range. An a value of 2 will therefore be used below. The sensitivity of the model to the choice of a is discussed in Section 6.

Several features of the predicted creep response based on the conventional model should be noted. First, the limiting stress exponent at lower stresses is 3. Second, at higher stresses, a continual increase in the value of the stress exponent is observed. Third, the form of the curves is sensitively depen-

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dent upon the value of l, the distance between jogs. It should be emphasized that the value of l is the only adjustable parameter in the model, and therefore determines the e€ective stress exponent for a given range of stress. A quantitative comparison of the model with experimental creep data for the near-gamma Ti±48Al alloy is also shown in Fig. 3. Clearly, the predicted strain rates using the jogged-screw model (h0d) are many orders of magnitude too large relative to the data, irrespective of the choice of l value. Particularly if the experimentally observed jog spacing is used (i.e. approx. 200 nm), then the predicted stress exponent is over an order of magnitude larger than that observed (about 6.0) over the relevant stress range. 5. POSSIBLE MODIFICATIONS TO THE JOGGEDSCREW MODEL

As mentioned above, a fundamental assumption of the conventional model is that the jog height is of atomic dimensions (i.e. equal to the Burgers vector), and invariant with stress. A possible source of the large disparity between the predicted and observed creep rates might be that the atomicheight jogs assumed in the conventional model are far too weak, and can be transported too readily, to account for the observed creep rates. For each vacancy emitted from a vacancy-producing jog, the dislocation may advance by the distance, a, given by [29] a ˆ O=…bh†:

…3†

In the conventional model, h0b, so that a0O=b2 . As h increases, a decreases, implying that more climb is required for a given advance of the dislocation. The work of Sriram et al. [19] indicates that jogs can have a spectrum of heights, ranging from 9 to 40 nm, in a Ti±52Al sample deformed under a constant strain rate of 1.7  10ÿ4/s at 573 K. They further propose that the origin of these larger jog heights is due to lateral motion of the jogs, which enables jogs to both annihilate and grow in height. Through this statistical process, larger jog heights may develop. Shown in Fig. 4 is a tilting experiment performed on a portion of a unit dislocation in a near-gamma Ti±48Al sample tested at 1041 K and 207 MPa to minimum creep regime, which proves the existence of jogs much larger than atomic dimensions. The sample orientations for this tilting procedure are indicated in the stereographic projection shown in Fig. 4(a). The dislocation under scrutiny here has b ˆ 1=2‰110Š that lies on the …111† glide plane and is pinned by jogs (marked by arrows). In Fig. 4(c), the dislocation is imaged very close to the glide plane (g ˆ 220; B1‰334Š, close to ‰111Š). Figure 4(c) and (d) were imaged by tilting symmetrically about the

Burgers vector direction by 0248 in both senses. In Fig. 4(c) the dislocation is imaged with (g ˆ 111; B1‰143Š, close to [011]) and in Fig. 4(d) the dislocation is imaged with (g ˆ 111; B1‰413Š, close to ‰101Š). The presence of large jogs at these pinning points is clearly revealed from these series of micrographs. For example, the cusped segments (with mixed character) on either side of the pinning point labeled as j1, are drawn apart when tilted in one direction [Fig. 4(d)], and cross over each other when tilted in the opposite direction [Fig. 4(c)]. The relationship between these three viewing directions and the apparent jog geometry is illustrated schematically in Fig. 4(e). A rough estimate of the jog height for j1 normal to the …111† glide plane is about 23 nm. The jog indicated as j2 appears to be even taller (about 40 nm). However, it also should be noted that many of the observed cusped con®gurations are associated with much shorter jogs since the jog height cannot be readily discriminated by similar tilting experiments. Another interesting feature of these jogs is that they appear to be aligned essentially normal to the glide plane. This can be deduced from Fig. 4(c) by noting that the jog has no apparent dimension when viewed perpendicular to the glide plane. This observation indicates that the glide plane for these jogs would be …112†. Since this is not a close-packed glide plane, it is expected that lateral movement of these jogs would be a dicult process. Having justi®ed the fact that some tall jogs do exist along unit dislocations following creep, it is interesting to consider whether such a modi®cation to the jogged-screw model can account for its overestimate of the observed creep rates. If a larger, constant value of h is assumed, then much better agreement in terms of the magnitude of the strain rate can be achieved. For example, also shown in Fig. 3 are plots of the predicted strain rates assuming that the jog height, h, in equation (1) is equal to 100b, which would be of the same magnitude as that observed in this work as well as that reported by Sriram et al. [19]. With this modi®cation, the observed strain rates in the stress range of the creep data are quite close to the observed values. However, it should be noted that the predicted stress exponent in the relevant stress range is now equal to 3, which is clearly inconsistent with the creep data shown for the near-gamma Ti±48Al, as well as the other reports in the literature. Note also that this stress exponent also extends to unreasonably large stress values. Clearly the incorporation of taller jog heights is required in order for the predicted strain rates to approach those which are observed. In addition, there is good reason to expect that the spectrum of jog heights should depend on the applied stress. Consider the jog con®gurations illustrated in Fig. 5, which have recently been discussed by Sriram et al. [19]. For short jogs [Fig. 5(a)], the non-conserva-

VISWANATHAN et al.: JOGGED-SCREW MODEL FOR CREEP

Fig. 5. Illustration of (a) short jogs being dragged by vacancy creation and (b) a tall jog acting as a dislocation source.

tive movement of the jog can occur, as assumed in the conventional model. A natural upper-bound on h is indicated in Fig. 5(b), where the jog is suciently tall that the oppositely signed, near-edge segments attached to the top and bottom of the jog can bypass each other. This maximum jog height, hd, can be approximated using the condition for breaking of a pure edge dipole:   Gb hd ˆ : …4† f8p…1 ÿ †tg These considerations then bound the possible dimensions of the jogs. If h hd , then the jog will no longer be dragged non-conservatively, but will instead act as a dislocation source. Overlapping dislocation segments, such as that illustrated in Fig. 5(b) and indicating the bypassing of the dipole, have been noted by several investigators following creep of gTiAl [6, 8]. Such a dislocation source can be envisioned to generate a new line length which initially has no jogs. Thus, the range of jog heights which are undergoing non-conservative motion should lie between b and hd. As exempli®ed in Fig. 4, we have in fact observed that jogs of a variety of heights are present following creep of the near-gamma Ti±48Al. It should be noted, however, that the measured jog heights are signi®cantly larger than would be predicted based on the direct application of equation (4). The value of hd obtained from equation (4) is only 10 nm, taking into account the Schmid factor of 0.41 for this dislocation as determined from experiment. However, the jog j1 in Fig. 4 has a measured height of about 023 nm. Since the adjacent, bowed segments are not overlapped, this particular jog appears to be less than the critical height (i.e. it is being dragged non-conservatively). On the other hand, the jog indicated as j2 in Fig. 4 would appear to be quite close to the critical bypass condition based on the fact that the near-edge segments are slightly extended in a ``hairpin'' or dipole con®guration. The height of this particular jog is about 40 nm. However, the value of hd obtained from equation (4) is only 10 nm, taking into account the

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Schmid factor of 0.41 for this dislocation as determined from experiment. Possible reasons why the value of hd is apparently underestimated by equation (4) are discussed later. For the moment, however, we will assume that the actual critical jog height for bypass is 4hd in order to be consistent with the present jog-height measurements. Therefore, we expect our actual jog heights to lie in a range between b and 4hd. We will nevertheless assume that the functional form provided by equation (4) is correct. We are presently undertaking a systematic study to determine the statistical distribution of jog heights as a function of stress following creep. The results of this study will be published elsewhere. However, in the absence of such detailed TEM measurements, we now make a rather crude assumption that the ``characteristic'' jog height on the ``average'' dislocation segment is a constant fraction of the critical jog height. For the present discussion, we will set the value of h in equation (1) equal to b*4hd, which yields the following, modi®ed expression for the jogged-screw model:        pDs t 2 tOl g_ ˆ exp ÿ1 : …5† aG 4bhd kT bbhd In this expression, we have once again assumed that the Taylor relation of equation (2) holds. The predicted strain rate as a function of stress using this modi®ed expression is shown in Fig. 6 for the same model parameters as were used in generating the curves for the conventional model. The only additional parameter is the value of b, which is assumed equal to 0.5. This ®nal assumption once again is a statement that the ``characteristic'' jog height is equal to half the critical jog height. Several features of these plots are worthy of note. First, the limiting stress exponent at lower stresses using the modi®ed model is 5, instead of the value of 3 given by the conventional model. This ``natural'' value of 5 also extends to much higher stresses for a given value of l. At higher stresses, the stress exponent increases gradually. However, the sensitivity to the value of l at higher stresses is not as large as in the conventional model. As can be seen in Fig. 6, quite remarkable agreement is obtained between the experimental creep data and the strain rates predicted from the modi®ed model. As mentioned above, preliminary estimates indicate that the observed jog spacing in samples following creep at 207 MPa and 7688C is of the order of 200 nm. Using this value, the predicted rates from the modi®ed model are within an order of magnitude of the experimental data. The stress exponents are also in excellent agreement. Choice of larger or smaller values of l results in poorer agreement with experiment, as illustrated in Fig. 6 for l = 20 nm.

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Fig. 6. Predicted axial creep rate vs axial stress using the conventional (h ˆ b) and modi®ed joggedscrew model, with model parameters appropriate for g-TiAl. Included for comparison are experimental data from the work of Viswanathan [4, 8].

Finally, shown in Fig. 7 is a comparison of predicted and measured creep rates for two temperatures (1041 and 1088 K) over a range of stresses. In this calculation, it is assumed that the model parameters a, b and l are constant with temperature. In fact the value of l deduced from observations [e.g. see Fig. 1(b)] is similar to that found at lower temperatures. The only temperature-dependent variables included in this calculation are di€usion coecient [16] and shear modulus [32]. It can be seen that reasonable agreement in absolute strain rates and stress exponents can be achieved at both temperatures. Several implications of the assumptions regarding the temperature dependence (or lack

thereof) of the model parameters is discussed in the next section. 6. DISCUSSION

This initial analysis of the modi®ed model suggests that it represents an improvement over the conventional jogged-screw model in several important respects relative to the creep of g-TiAl. First, reasonable stress exponent values are obtained, in the range of 5 or greater, assuming that the Taylor relationship holds for mobile dislocation density. In previous ®tting of the conventional model to experimental data by Barrett and Nix [15] and Seitho€ et al. [33], quite large values of the stress-dependence

Fig. 7. Comparison of predicted and measured creep rates for two temperatures (1041 and 1088 K) over a range of stresses, using the modi®ed jog-screw model. The model parameters used in these calculations are indicated.

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Fig. 8. Predicted creep rates as a function of stress for di€erent values of (a) the Taylor factor, a, and (b) the characteristic jog height, b.

for dislocation density must often be assumed in order to obtain agreement with experiment. While this assumption was experimentally justi®ed in the case of the work of Barrett and Nix on Fe±3% Si [115], many other measurements following creep [31, 34] point toward a signi®cantly lower stress dependence, similar to that given in the Taylor relation. The modi®ed model also accounts, at least in a crude way, for the experimental observation of ``tall'' jogs and ``hairpin'' con®gurations during deformation of g-TiAl. Finally, this initial comparison with experiment also indicates that these ``taller'' jogs, which require more di€usion for a given incremental advance of the dislocation, are important to consider in order to approach the creep rates observed in experiment. A number of the features and assumptions in the modi®ed model require additional consideration

and further development. The only ``adjustable'' parameters in the modi®ed model are the values of l, a and b, together with the assumption of the Taylor relation. All of these parameters can be measured or estimated from TEM analyses. The values utilized in this work have been justi®ed based upon a limited set of observations at one temperature and stress. The sensitivity of the model to the choice of a is shown in Fig. 8(a). We have estimated the value of a from dislocation density measurements to be in the range of 0.7±3, depending on the density variation in intragrain. Theoretical estimates of a usually yield the expectation that it should have a value between 0.1 and 1 [34]. However, such a choice of a would lead to an overestimation of the dislocation density in our case. As shown in Fig. 8(a), this will consequently result in overestimation of the strain rate at all

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stresses. A clear need is to obtain measurements of dislocation density as a function of both stress and temperature in order to validate the use of the Taylor equation, and con®rm the chosen value of a utilized here. The sensitivity of the model to the value of b is shown in Fig. 8(b). The observations presented in Fig. 4 clearly indicate that tall jogs can form during creep. This ®nding agrees with that of Sriram et al. [19] following deformation at lower temperatures under constant strain rate conditions. As discussed above, the expected ``limiting'' jog-height value predicted by equation (4) appears to underestimate the height of the taller jogs which have been observed. There are a number of possible reasons for this discrepancy. First, equation (4) is based on isotropic elasticity theory which is not strictly valid in this case since the anisotropy factor [35] is signi®cantly larger than 1 (02.8). Equation (4) also assumes in®nitely long, straight dislocations, while in our case the breaking of the dipole will occur over a relatively short segment length which is also constrained at one end. Thus, bowing forces will tend to oppose the applied stress. In utilizing the applied shear stress in equation (4), we also ignore the fact that there may be frictional forces on the near-edge dislocations due to Peierls e€ects or solute interaction, which would tend to increase the value of hd. It should also be noted that instead of using a single, characteristic jog height, a functional form could be chosen for the actual distribution of these heights. However, in the absence of such statistical information from TEM observations, this re®nement is presently not warranted. Clearly, systematic measurements following creep deformation at the relevant stresses and temperatures are required in order to re®ne the model with respect to the distribution of jog heights. This work is presently being undertaken. We now turn our attention to the possible temperature dependence of the model parameters. Based on the colliding-kink-pair picture of jog formation, theoretical calculations [29] which account for the rates of kink-pair formation and kink propagation suggest that the jog spacing should depend only on temperature according to   DF l ˆ 2a exp …6† 2kT where a is the atomic jump distance (approximately equal to b) and DF is the free energy for the formation of kink-pairs. If the value for DF in equation (6) is small relative to the activation energy for self-di€usion, then clearly l will not vary strongly with temperature, and the predicted activation energy for creep will be similar to that for self-di€usion. However, if the value of DF is large, then the predicted activation energy for creep will be signi®cantly smaller than that for self-di€usion,

which would be contrary to experimental evidence [4±6, 8]. It is interesting to note that in the work of Barrett and Nix [15] as applied to creep of a Fe±3% Si alloy, it was assumed that the value of l is indeed a constant with stress and temperature. To justify this assumption, they argued that the jogs are the result of dislocation intersections. Although the overall density increases with stress, it was still assumed that the jog spacing was constant with stress. Assuming for the present case a rather small value of DF relative to the activation energy for self-di€usion (e.g. DF = 0.1 eV), then l will not depend strongly on temperature, but the magnitude of l would be only of the order of 1.0 nm. However, this is an extremely small value for l as compared with the apparent jog spacing seen in Fig. 1. On the other hand, if equation (6) holds, then the observed jog spacing (approx. 200 nm) would imply a much larger value for DF equal to about 0.8 eV, which is about 1/3 the value of the activation energy for self-di€usion. This value of DF actually yields an activation energy for creep which is substantially less than that for self-di€usion. Since this is not the case, the validity of equation (6) must be brought into question for the case of g-TiAl. It should be noted that our present TEM observations at two di€erent temperatures (1041 and 1088 K) and the same applied stress (207 MPa) indicate that the jog spacings are actually quite similar. In addition, the jog-spacing measurements of Sriram et al. [19] following constant strain rate testing at much lower temperature indicate a similar value of l (about 100 nm) to that determined in this work. Thus, the temperature sensitivity of this parameter as suggested by equation (6) does not seem to be borne out by the presently available data. Due to the importance of l as a parameter in the model, it is suggested that the temperature and stress dependence of l be explored, perhaps using dynamical dislocation calculations based on the colliding-kink-pair picture of Fig. 2, and that additional experiments to determine these dependencies would also be extremely valuable. The distribution of jog heights may also be temperature dependent. Consider the jog con®guration shown in Fig. 2(b) in which the bowing dislocations on parallel glide planes cause a near-edge dipole to form. There are attractive elastic interaction forces for climb between the edge dislocations of the dipole. With increasing temperature, these interaction forces might tend to skew the jog-height distribution to smaller values (i.e. the value of b would become smaller). At a given applied stress value, this e€ect would then weaken the average strength of the jogs, which would increase the temperature sensitivity (i.e. increase the activation energy for creep) relative to a model without such a dependence.

VISWANATHAN et al.: JOGGED-SCREW MODEL FOR CREEP

Other factors which have been explicitly excluded in the present model are the e€ect of both friction and/or internal stresses. Signi®cant frictional forces on unit dislocations in g-TiAl have been deduced from post-mortem TEM analysis of dislocation con®gurations following constant-strain-rate deformation. The curved nature of dislocation segments between jogs, which have been seen in g-TiAl, suggests that some lattice (or possibly solute) frictional forces are operative. In a further re®nement of the present model, the applied shear stress, t, in equation (4) could be replaced by an e€ective stress, teff ˆ t ÿ tfric . However, from measurement of the radius of curvature of several cusped con®gurations, a value of the friction stress equal to about 13 MPa is obtained. Using the appropriate Schmid factor for the grain orientation and slip system of the dislocations analyzed, this corresponds to a tensile stress of about 32 MPa. This value represents only about 15% of the applied stress at which the samples were crept (i.e. 207 MPa). It should be recognized that this estimate probably represents a signi®cant overestimate of the friction stress which is actually operative at creep temperatures since the creep samples analyzed in this study were cooled under load. Thus, the measured friction stresses are actually relevant to frictional forces operative at room temperature. Complementary experiments in which samples are unloaded prior to cooling could help determine the actual friction stresses at higher temperature. Nevertheless, these estimates indicate that the friction forces are quite small relative to the applied stress under creep conditions, so that use of the applied shear stress in equation (5) is probably justi®ed. The in¯uence of inter-dislocation stresses could also be included using a similar e€ective stress approach. This modi®cation could in principle enable the model to account for the normal primary creep transients observed in these alloys. However, it is not clear that this would be appropriate in the case of these materials since the dislocation microstructure is relatively homogeneous, suggesting that the applied stress is supported primarily by individual dislocations rather than hard heterogeneities in the microstructure. This reasoning is similar to the case in solute strengthened alloys where it has been argued previously that the internal stresses should be relatively small [36]. As an alternative to an ``internal stress'' approach, it is intriguing to consider that the normal primary and extended tertiary creep transients which have been observed [4±8] may originate from the evolution of the jog height and spacing along dislocations as a function of strain. Considering for the present only the e€ect of jog height, a possible scenario for the evolution of a presumed jog-height distribution with strain is illustrated in Fig. 9(a). Note that these distributions are for illustrative purposes only and are not intended to imply variations

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Fig. 9. Schematic illustrations indicating (a) the possible evolution of the jog-height distribution as a function of strain. Nj(h) represents the number of jogs per unit length of a given height, h, and hd is the critical jog height for dipole bypass. (b) The corresponding variation of dislocation density, dislocation velocity and strain rate as a function of strain.

in the total number of jogs. At the beginning of deformation, intrinsic sources active upon loading would produce dislocations which have a preponderance of short jogs. Since these jogs are on average rather weak, then the strain rate would initially be large, as illustrated in Fig. 9(b). With increasing strain, the average jog height will increase, and the number of jogs of elementary height (i.e. equal to b) will decline. Consequently, the strain rate should decrease in this regime of strain. Eventually, an increasing number of jog heights larger than hd will develop, which could have two e€ects. First, these jogs will act as sources of new dislocation line length, so the total dislocation density would increase more rapidly once a large number of jogs have heights exceeding hd. Second, the new dislocation line length formed in the process will have a jog distribution once again skewed to heights close to b. This situation is illustrated in Fig. 9(a) as an increase in the value of Nj …h ˆ b† for curve ``C''. Both of these e€ects would tend to increase the overall strain rate since the total number of dislocations is increasing and their average velocity (at constant stress) is also increasing. In a recent analysis of creep for Ti±52Al by Lu and Hemker [6], they attempted to correlate the extended region of tertiary creep (in which the strain rate was increasing) with the measured increase in dislocation den-

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VISWANATHAN et al.: JOGGED-SCREW MODEL FOR CREEP

sity. Their work indicated that the increasing strain rate could not be fully accounted for on the basis of the increased dislocation density as a function of strain, and concluded that the average dislocation velocity must be increasing with strain as well. The qualitative scenario described here could o€er a possible explanation for this conclusion by Lu and Hemker [6]. The attractiveness of this proposed model is that essentially all of the key model parameters can be measured experimentally. Jog heights are perhaps the most dicult of the parameters to conclusively measure. While this has been done presently for a single stress and temperature, and previously for gTiAl under constant strain rate conditions by Sriram et al. [19], additional observations of this kind following creep deformation are clearly called for in order to con®rm the existence of tall jogs during creep deformation, and hopefully quantify the typical distribution of such heights. 7. CONCLUSIONS

. The creep behavior of a near-gamma microstructure in a Ti±48Al alloy has been investigated in this study. Minimum creep rates have been obtained from constant load creep tests at temperatures of 1041 and 1088 K in the stress range 107±240 MPa. Stress exponent values have been determined from stress increment tests to be about 6 at 1041 K, and about 5 at 1088 K. . TEM analysis indicates that the deformation microstructure at minimum creep is dominated by unit 1=2‰110Š dislocations, and that these dislocations are pinned by jogs of varying heights. Jog heights as large as 40 nm have been observed. A possible mechanism for the formation of tall jogs has been forwarded based upon earlier work aimed at understanding the ¯ow strength anomaly under constant strain-rate conditions [2, 19]. . Assuming that the Taylor relation holds for expressing the stress dependence of the dislocation density, the jogged-screw model in its conventional form (in which jog heights are assumed equal to the Burgers vector) predicts creep rates which are far too large relative to those observed in the near-gamma Ti±48Al. If the presence of tall jogs is incorporated into the conventional model, strain rates in reasonable agreement with experiment can be obtained, however, the predicted stress exponent of about 3 is signi®cantly less than that obtained from experiment. . An additional modi®cation of the original jogscrew model has been proposed which is based on the premise that there should exist an upper bound on the jog height which can be dragged by climb. If the jog height exceeds a critical value at

a given stress, then the jog instead acts as a dislocation source. Incorporating such a stress dependence to the jog-height distribution, and representing such a distribution by a single ``characteristic'' jog height, this modi®ed joggedscrew model is shown to yield remarkably good agreement with the measured creep response in the near-gamma Ti±48Al. The parameters of this modi®ed model have been ``calibrated'' based on TEM observations of dislocation structures following creep of the same material. . This modi®ed model produces a ``natural'' stress exponent of 5 at lower stresses, and increasing stress dependence at higher stresses. The critical model parameters can all be measured from TEM observations. Further re®nement of the proposed model is expected to result from additional, experimental measurement of dislocation density, jog-spacing distributions and jog-height distributions as a function of both stress and temperature. AcknowledgementsÐG.B.V. and M.J.M. acknowledge the National Science Foundation under grant DMR-9709029 with Bruce MacDonald as program manager for support of the TEM studies and model development. G.B.V. and V.K.V. are thankful to the State of Ohio Edison Materials Technology Center for the ®nancial support during the initial stages of this work under grant EMTEC/CT-29. REFERENCES 1. Kim, Y.-W., J. Metals, 1989, 41, 24. 2. Viguier, B., Hemker, K. J., Bonneville, J., Louchet, F. and Martin, J. L., Phil. Mag. A, 1995, 71, 1295. 3. Beddoes, J., Wallace, W. and Zho, L., Int. Mater. Rev., 1995, 40, 197. 4. Viswanathan, G. B. and Vasudevan, V. K., in Gamma Titanium Aluminides, ed. Y.-W. Kim, R. Wagner and M. Yamaguchi. TMS, Warrendale, Pennsylvania, 1995, pp. 967. 5. Ishikawa, Y. and Oikawa, H., Mater. Trans. JIM, 1994, 35, 336. 6. Lu, M. and Hemker, K. J., Acta mater., 1997, 45, 3573. 7. Hayes, R. W. and Martin, P. L., Acta metall. mater., 1995, 43, 2761. 8. Viswanathan, G. B., Ph.D. thesis, University of Cincinnatti, 1997. 9. Sherby, O. D., Klundt, R. H. and Miller, A. K., Metall. Trans., 1997, 8A, 843. 10. Sriram, S., Viswanathan, G. B. and Vasudevan, V. K., in Twinning in Advanced Materials, ed. M. H. Yoo and M. Wuttig. TMS, Warrendale, Pennsylvania, 1994, p. 423. 11. Skrotzki, B., Unal, M. and Eggeler, G., Scripta mater., 1998, 39, 1023. 12. Skrotzki, B., Rudolf, T. and Eggeler, G., Scripta mater., in press. 13. Jin, Z. and Bieler, T. R., Phil. Mag. A, 1995, 71, 925. 14. Ott, E. A. and Pollock, T. M., Metall. Mater. Trans., 1998, 29A, 965. 15. Barrett, C. R. and Nix, W. D., Acta metall., 1965, 13, 1247. 16. Kroll, S., Z. Metallk., 1992, 83, 591. 17. Parthasarathy, T. A., Mendriatta, M. G. and Dimiduk, D. M., Scripta mater., 1997, 37, 315.

VISWANATHAN et al.: JOGGED-SCREW MODEL FOR CREEP 18. Bird, N., Taylor, G. and Sun, Y. Q., MRS Symp. Proc., 1995, 364, 635. 19. Sriram, S., Dimiduk, D. M., Hazzledine, P. M. and Vasudevan, V. K., Phil. Mag. A, 1997, 76, 965. 20. Kad, B. and Fraser, H. L., Phil. Mag. A, 1994, 69, 689. 21. Haussler, D., Bartsch, M., Aindow, M., Jones, I. P. and Messerschmidt, U., Acta mater., in press. 22. Messerschmidt, U., Bartsch, M., Haussler, D., Aindow, M., Hattenhaurer, R. and Jones, I. P., MRS Symp. Proc., 1995, 364, 47. 23. Kawabata, T. S., Kanai, T. and Izumi, O., Acta metall., 1985, 33, 1355. 24. Louchet, F. and Viguier, B., Phil. Mag. A, 1995, 71, 1313. 25. Simmons, J. P., Mills, M. J. and Rao, S. I., MRS Symp. Proc., 1995, 364, 335. 26. Mott, N. F., in Creep and Fracture of Metals at High Temperatures, Proc. NPL Symp. HMSO, London, 1956, p. 21.

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27. Hirsch, P. B. and Warrington, D. H., Phil. Mag. A, 1961, 6, 715. 28. Friedel, J., Phil. Mag. A, 1956, 46, 1169. 29. Hirth, J. P. and Lothe, J., Theory of Dislocations. John Wiley, New York, 1982. 30. Taylor, G. I., Proc. R. Soc., 1934, A145, 362. 31. Mukherjee, A. K., Bird, J. E. and Dorn, J. E., Trans. Am. Soc. Metals, 1969, 62, 155. 32. Schafrik, R. E., Metall. Trans. A, 1977, 8, 1003. 33. Sietho€, H., Schroter, W. and Ahlborn, K., Acta metall., 1985, 33, 443. 34. Haasen, P., in Dislocation Dynamics, ed. A. R. Rosen®eld, G. T. Hahn, A. L. Bement Jr and R. I. Ja€ee. McGraw-Hill Inc., NY, 1968, p. 701. 35. Viguier, B. and Hemker, K. J., Phil. Mag. A, 1996, 73, 575. 36. Mills, M. J., Gibeling, J. C. and Nix, W. D., Acta metall., 1985, 33, 1503.