A resolution of the interface phase problem in titanium alloys

A resolution of the interface phase problem in titanium alloys

~cta merall. Vol. 36, No. 1, pp. 125-141, 1988 Printed in Great Britain. All rights r~~er~cd A RESOLUTION PROBLEM oool-6160/88 $3.00 + 0.00 Copyrigh...

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~cta merall. Vol. 36, No. 1, pp. 125-141, 1988 Printed in Great Britain. All rights r~~er~cd

A RESOLUTION PROBLEM

oool-6160/88 $3.00 + 0.00 Copyright 0 1988 Pergamon Journals Ltd

OF THE INTERFACE PHASE IN TITANIUM ALLOYS

D. BANERJEE’, C. G. SHELTON, B. RALPH3 and J. C. WILLIAMS’ ‘Defence Metallurgical Research Laboratory, Hyderabad, India, 2Shell Research Liited, Thornton Research Centre, P.O. Box 1, Chester CHI 3SH, England, 3Department of Materials Technology, Brunei University of West London, Uxbridge, Middx UBS 3PH England and ‘Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A. (Received 12 September 1986; in reoisedform 2 April 1987) Ahstraet-Although the interface phase in titanium alloys has recently been shown to be an artefact caused by thin foil specimen preparation techniques, its formation nevertheless presents some uniquely complex and puzzling features. Therefore, the wide variety of results that have emerged on this transformation (since the interface phase was first observed over a decade ago) are reviewed in this paper, and at attempt is made to provide a consistent view-point for its formation. Rkum&Bien que l’on ait rkmrnent montre que la phase interfaciale dans les alliages de titane etait un artefact cause par les techniques d’amincissement des echantillons, sa formation pn%ente cependant des caracteres particulitrement complexes et mysmrieux. C’est pouquoi nous passons en revue dam cet article la grande varitte des resultats qui concement cette transformation, et nous essayons de foumir un point de vue logique sur sa formation. Zusammenfassung-Wenn such vor kuaem gexeigt worden ist, dal3 die Grendliichenphase in Titanlegierungen ein durch die elektronenmikroskopische Prgparation bedlingter Artefakt ist, so ist die Bildung dieser Phase doch wegen der komplexen und erstaunlichen Eigenschaften interessant. In dieser Arbeit werden die vielfaltigen Ergebnisse tiber diem. Umwandlung (diese GrenztXchenphase wurde vor iiber xehn Jahren xum erstmals beobachtet) xusammengestelt. Es wird versucht, daraus ein konsistentes Bild Uber das Entstehen dieser Phase xu entwickeln.

1. INTRODUffION

The formation of an anomalous product at the interface between a(hexagona1) and /?(b.c.c.) phases of various titanium alloys has, over the last decade, been the subject of a wide variety of studies. Initial characterization of this phase [l, 21 showed it to possess either a hexagonal or f.c.c. structure. Several theories were then advanced to explain the formation of this interfacial region, and included the notion that the f.c.c. phase was a hydrogen induced or hydride phase stabilized by stress, strain or composition gradients [3-51, or that it was an intermediate transition phase formed during the /3 to a transformation [6]. The hexagonal form of the interface phase, usually designated as Type 2a, was thought to form as a stress induced reaction [I, or to be a more stable form of the a phase than Burgers orientation related a [8] (Type la), whose formation could possibly be caused by compositional differences [2]. More recently, it has been shown that both f.c.c. and hexagonal forms of the interface phase owe their origin to the nature of the thinning process used to prepare foils for observation by transmission electron microscopy (TEM) [!3-121.With these observations, it has become clear that hydrogen certainly plays a major role in the formation of both the hexagonal and f.c.c. products, since neither of these phases were observed when thin foils were prepared by techniques

designed to minimize or eliminate hydrogen absorption by the samples. An attempt is now made, using recent results generated at the authors’ laboratories and a careful consideration of all experimental observation available in the literature, to account for various aspects of interface phase formation. No effort is made in this paper to examine possible effects of the interface phase on mechanical properties since, as will be seen, the interface phase in any form is unlikely to be present in bulk material, at least at hydrogen levels normally encountered in commercial titanium alloys. The discussion is presented in four sections; the first two describe the formation of the hexagonal and f.c.c. products respectively in electropolished foils, and the third summarizes the effect of various thinning techniques and parameters on the transformation. A final section provides a comprehensive and self consistent view-point on the nature of interface phase formation.

2. THE HEXAGONAL INTERFACE PHASE IN ELECTROPOLISHED FOILS

2.1. Structure and crystallography Considerable confusion has accompanied the identification of the structure of the a/B interfacial 125

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BANERJEE et al.: INTERFACE PHASE PROBLEM IN Ti ALLOYS

0.2um Fig. 1. Type 2 01in Burgers a plates in Ti-lOV-2Fe3Al isothermally aged at 700°C for 50min.

layer following the initial study of Rhodes and Will-

iams [2] on the precipitation of a phase in metastable /3 phase alloys containing large amounts of /I stabilizing solute and hence high volume fractions of /I. In this early work, which involved the comparison of computer generated diffraction patterns with the complex, ex~~rnen~lly observed diffraction effects, it was suggested that a hexagonal product existed (hereafter referred to as Type 2a) which was not Burgers related to the fl phase (the classical orientation relationship between hexagonal and b.c.c. phases IS]) but twin related to the Burgers a plates (referred to as Type la) on (1012) planes. Characteristic of the Type 2a phases were “arced’ reflections, the presence of which in a diffraction pattern was subsequently interpreted by numerous authors (see for example, Ref. [13]) to imply the existence of Type 2a without adequate analysis of the diffraction pattern. Margohn et 01. [7] also report a hexagonal (1011) twin related phase at a/j? interfaces, albeit in an a-@ alloy with relatively low quantities of /I stabilizers, Ti-6AldV. Rhodes and Williams [l], however, also identified a f.c.c. structure at a/p interfaces in Ti-6Al-4V. More recently, Banejee [14] pointed out that the diffraction patterns characterized by Rhodes and Williams [2] as belonging to a hexagonal phase could also arise from a f.c.c. structure, and indeed Hammond and his co-workers [IS, 161 have consistently analysed their diffraction patterns as originating from an f.c.c. phase. Banerjee and Williams [17] have recently attempted a clear identification of the c~s~llography of the interface product in the Ti-IOV-2Fe-3AI alloy, which might be expected to contain the hexagonal

interfacial product. These results are now presented in some detail as they are believed to be conciusive in nature. It was revealed early in this work that a major problem in analysing diffraction effects in solution treated and aged metastable fl alloys lay in that the a phase was fine enough to allow several different variants of this phase to contribute to a selected area diffraction pattern. An example of this problem is shown in the typical diffraction patterns of the Type 2a phase presented by Rhodes and Williams [2]. Appropriate ageing treatments were therefore chosen for the alloy Ti-lOV-2Fe3Al in the range SOO-700°C so that the a plates were large enough to analyse infidel plates by selected area diffraction. Figure 1 shows transmission electron micrographs of such a plates after ageing at 700°C for 50 min. After a suitably sized and orientated a plate had been selected, it was tilted about a [OOOl]direction of the Burgers orientated Type la, starting from a (1120) zone axis or about a (lOTO) direction starting from an [OOOl] zone axis, in order to explore fully the reciprocal lattice of the type la. Figure 2 shows diffraction patterns obtained by tilting about [OOOl]starting from the [1120] zone axis of Fig. 2. Three variants of a hexagonal phase are also present in addition to the Burgers a spots in Fig. 2a and these arise from the interface product. Two of these are twin, related to Burgers a on the two (1011) planes present in this zone. Note that these two variants are themselves twin related to each other about a common type 2a (IOTI) plane parallel to the (0001) plane of the Burgers a or (110) plane of the fl phase, The variants desctibed above are indicated as A and A’ {identifying the specific twinning planes) in the stereographic projection of Fig. 3. A dark field image from one of the variants is shown in Fig. 4. A third variant which is twin related to Burgers a on a (lOT2) plane is also present in the diffraction pattern of Fig. 2(a). In the entire investi~tion this was the sole example of a (1012) twin related Type 2a crystallography recorded. Tilting about the [OOOI]Burgers a direction by the angles indicated results in diffraction patterns from the interface phase alone, recorded in Figs 2(b) and (c). These show the existence of other Type 2a variants, again twin related to each other about the (lOT1) Type 2a plane parallel to (0001) Burgers Type la. Since the amount of tilt required to produce these patterns is known, it is possible to obtain the relation between these Type 2a variants and the parent Burgers a plate. The analysis indicates that each of the Type 2a variants of Figs 2(b) and (c) are twin related to Burgers a on the remaining four {1011) Burgers a planes, and are indicated by the leter B and B’, and C and C’ in Fig 3. Thus, hexagonal “Type 2a” exists in six variants formed by twinning on the six possible (lOf1) planes of the parent Burgers CLplate. Of these six variants, three pairs A and A’, B and B’, and C and c’ are

BANERJEE et al.:

INTERFACE

PHASE PROBLEM

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. 062

(a)

Fig. 2. Diffraction patterns from Burgers a and Type 2a obtained by tilted about a [OOOI]Burgers u direction from a [I 120] Burgers a zone axis (a) [I 1201 Burgers a zone axis with spots from Burgers a belonging to the rectangular grid marked “b” . This zone axis also contains two [I 120] zone axee from Type 2a marked “a” and “c” which are twin related to Burgers a on the [IOIl] planes present in this zone axis. Spots from a third [1120] variant of Type 2a (marked “d”) are twin related to Burgers a a (0172) plane present in this zone axis. (b) Tilted by 14” about [0002] Burgers a from the orientation of (a). Two twin related variants of Type 2a of the [1213] zone axis are present. (c) Tilted by 10” about [0002] Burgers a from the orientation of (a). Two twin related variants of Type 2a of the [OTl 1) zone axis are present.

127

BANERJEE ef al.:

128

oil0

c

ooill

IOil Type2a C’ 0 0001, “olil

olio

IIOfJ

CRYSTALLOGRAPHYAND TYPE

INTERFACE

DISTRIBUTION

OF

2a IN Ti-IOV-2Fe-3AL

Fig. 3. Stereographic projection showing the relationship of Type 201:to Burgers a. The Type 2a plates are twin related to Burgexs a on the 6 {lOTI} Burgers a planes and are indicated by the variants A, A’, B, B’, C, C’. The distribution of these variations are shown in the accompanying schematic.

themselves twin related to each other on a common Type 2a (lOT1) plane formed parallel to the (0001) of Burgers a which all six variants share. Dark field analysis shows these pairs to coexist as indicated in the schematic of Fig. 3. It may be noted that out of these six variants, the pair A and A’ of Type 2a remain Burgers related to fl since twinning on this particular set of {1011) Burgers a planes simply restores the Burgers orientation relationship. The other Type 2a variants, B, B’, C and C’ are no longer Burgers related to the /I phase. The same results can be obtained by tilting to the [I2131 zone axes of the parent Burgers a plate, each of which contains two different {1011) Burgers a planes in the zone. It is believed that the evidence presented here constitutes an unambiguous characterization of a hexagonal interface product and its crystallography at the a/b boundaries of electropolished thin foils for TEM. 2.2. Formation characteristics

Rhodes and Williams [2], in the only study of the formation of “Type 2a” in Ti-Mo and Ti-Mo-Al alloys, concluded that the Type 2a was a more stable form of the a phase than the Burgers Type la. They also obtained a T-T-T curve for the formation of Type 2a. Banejee and Williams [ 17] have been unable to reproduce these results in their study of the

PHASE PROBLEM

IN Ti ALLOYS

formation of Type 2a in Ti-lOV-2Fe-3Al. On the contrary, the extent of hexagonal interface phase formation appeared to be closely related to the width of the parent Burgers a plates rather than directly on the ageing parameters. Typical micrographs illustrating this phenomenon are shown in Fig. 4. These are all taken from the same foil. A distribution of a sizes exists for a given ageing treatment. The {lOT1) twins of which the Type 2a phase in this alloy consists appear to consume the entire a plate when the a plate sizes are small but are confined to the a//? interfaces when the plates are large. The twinned product is never observed within the a plates alone, but always seem to extend from the interfaces into the plates. Since the a plate sizes are naturally smaller at lower ageing temperatures, an indirect dependence would then exist for the formation of the twinned product. Moreover, if the entire Burgers a plate is twinned on { lOT1) planes, the resulting diffraction effects could lead to the interpretation that Type 2a alone exists for a given heat treatment unless a very careful diffraction analysis is carried out. However, as will be seen in a subsequent section, the formation of the { 1071) twins is an artefact introduced by jet electropolishing samples to electron transparency, and in ion-milled foil of Ti-lOV-2Fe-3Al and other metastable /I alloys only Burgers a plates are observed at all ageing temperatures. It is worth noting that no composition differences have been detected between Type 2a and Burgers a. 3. THE F.C.C. INTERFACE PHASE IN ELECI’ROPOLISHED SAMPLES 3.1. Structure and crystallography

Having conclusively shown that a hexagonal interface product may be observed, it is now necessary to demonstrate the existence of an f.c.c. phase. There seems to be an overall consensus in the literature regarding the crystallography of the f.c.c. phase. However, some details require clarification and discussion. The f.c.c. phase exists in two crystallographic forms which have been expressed by various authors in different ways, that is, an orientation relationship may be shown to exist between the f.c.c. phase and the a phase or the /I phase. The first of these crystallographic forms may be expressed as (lTO),,.//(lToo),//(lT2)b

~~~~l~,~,,//~~~~l,//t~ ql [11%,,//[1 Qw/[1TTl,. The f.c.c. phase may nucleate on the a side of the a//I interface or the /I side. An analysis of this crystallography reveals that if the f.c.c. phase were to have formed within the fl phase, the orientation relationship expressed above would give rise to 12 variants in what is known as the Pitsch [18] relationship out of which one of the variants would be

BANERJEE

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0.7&m -

Fig. 4. The effect of Burgers a plate size on the formation of Type 2a.

C~stallo~aphi~Ily related to the (I phase since the ,!? and a are themselves related by the Burgers orientation relationship. On the other hand, if the f.c.c. phase were to have formed within the tl phase, three variants could result, again only one of which would be c~st~lo~phi~lIy related to 8. In general, only one variant of this crystallographic form is observed, that which is uniquely related to both the E and the j? phases, so that it is not possible to decide whether the f.c.c. phase has formed from either a or B phases. However, only in one isolated and reproducible situation, in the ahoy Ti-6Al-OSZr-3.3Mo0.25Si, argon quenched after B solution treatment, have three variants of the f.c.c. phase been observed in the A.M.

36,1-1

orientation

The single variant of this ~latio~~p which is refated to both the a and fi phases [as indicated in relationship (l)] lies along the interface; the other two variants extended from the a//3 interface into the a phase as plates with their broad faces parallel to (I TOO),.All these features are shown in Fig. 5. This observation suggests that this crystallographic form of the f.c.c. phase originates from the K side of the a//? interface, but a selective crystallographic

130

BANERJEE et al.: INTERFACE PHASE PROBLEM IN Ti ALLOYS

0

i

0-

-,“(A

r\

-2

O’\

?-

- - -7 [OOll TYPE 1 fee b _ __;, 3 VARIANTS I

Fig. 5. (a) A diffraction pattern from tbe [0002] Burgers a orientation shows that three variants of (001) f.c.c.//(OOOl) a, [lTO] f.c.c.//[lTOO] a are present. These variants are in the (001) orientation and are marked A, B and C in the schematic. Spots from each f.c.c. variant are marked as V, while spots from the B phase are marked as “b“ and spots from the Burgers a phase as “a”. (b) A dark tieid micrograph of variant A shows that this variant of the f.c.c. phase lies parallel to the a/j interface along the (lTO0) plane. (c) A dark field micrograph of variant B which lies across the a laths parallel to (tOTO).(d) A dark field micrograph of variant C which lies across the a laths parallel to (OTlO).(e) A dark field micrograph of the j3 phase.

variant predominates which is also crystatiographicaliy orientated with respect to the fl phase. The second crystallographic form of the f.c.c. phase has the orientation relationship (I I Ur,,,.//GJoQ,//(~

1%

r~~~l~,,.~.//r~~~~l,/l~~~~l,. If the f.c.c. phase were to have nucleated on the 01side

of the a//I interface, this orientation ~lations~p would give rise to two variants which would be twin related about the (11 I),,~.~,plane parailel to (OOOl}, (20,211. This is precisely what is observed as indicated in Fig. 6. It may be concluded therefore that this c~s~llographic form of the f.c.c. phase has formed from the a phase as well. Finally, observations have been made of an ordered structure for the f.c.c. phase in a Ti-O.gNi-O.3Mo (Ti CODE 12) alloy. While the

BANERJEE

et al.:

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PHASE PROBLEM

IN Ti ALLOYS

t31

superlattik spots correspond to values of h and k odd and 1 even, or 001 where 1 is odd. The ordering is associated with the introduction of tetragonality into the structure. Thus, the lattice parameter of the f.c.c. phase in this alloy is 4.53& while that of the ordered structure corresponds to a = 4.1511 if indexed on the basis of a face-centred tetragonal lattice. Diffraction patterns illustrating the ordered structure are shown in Fig. 7. It is noted that this ordered structure is identical to that of the Y hydride in zirconium alloys [22]. In this alloy, plates of the same phase are also found within the a grains.

Each of the two ~stallo~phic forms of the f-cc. phase is associated with a distinctive morphology. The so-called “monolithic” morphology is associated with the f.c.c. phase whose orientation is described by

OITO A

&

111 TYPE 2 FCC

)At

]

~//~~~,~*_

0001

fYPE,

FCC

/‘,

\\

\’\ \,,C I

111 TYPE 2 FCC (MATRIX A

:

\

002

6URGERS a

C2iiO3 ZONE t

:

TYPE 2 FCC (TWIN)

f

AXIS

The monolithic phase has been so designated because it often appears to form an ahnost continuous film of a single crystal (variant) along the a/J interface, particularly when the other crystallographic form is absent. An example of this situation is seen in Fig. 11 of Ref. [2]. Occasionally, this morphology may be heavily dislocated or heavily twinned [6]. The second morphology has been referred to as a “striated” mo~hology, and refers to the f.c.c. phase in the orientation

FCC PHASE

(b)

Fig. 6. (a) Twin related variants of f.c.c. phase in the orientation (I 1~~~.~.//(~i)~; [lTOJ,.,//[ll20~. @) Dark field TEM from one of the variants.

orientation relationships of the interface phase are identical to that observed in other alloys, superlattice spots are observed in the reciprocal lattice for the specific heat treatments listed in Table 1. These

The striated appearance results from the present of fine, parallel twin-related plates which constitute the two variants of this o~entation relationship. Clear evidence for nucleation of these plates on basal stacking faults in the a phase has been obtained IS] and can be seen in Fig. 8. The striated and monolithic morphologies may exist independently or together. When they do exist together, the monolithic morphology appears to form directly adjacent to the a/@ interface, with the striated form next to it within the a phase. This feature, shown in Fig. g is most clearly observed with the a//? interface approximately parallel to the electron beam direction that is in a zone axis containing the [ 112ja//[ 1TOO],directions (i.e. when the interface plane is very nearly parallel to the electron beam-see for example Williams et al. [23]). No systematic work exists to evaluate the effect of heat-treatment or alloy composition on the distribution of the two mo~holo~es. It is worth noting the strong resemblance of the striated morphology to that of Type 2a and this fact has led to much initial confusion in the identification of two interface phase structures in early work [2,6,23]. Careful diffraction analysis, however, quite clearly reveals the structural

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INTERFACE PHASE PROBLEM IN Ti ALLOYS

Fig. 7. Diffraction patterns from an ordered f.c.t. interface phase in Ti-O.SNi4.3Mo. (a) [OOl],,,,//[OOOl]~. (b) 11lol,,t.//[lT@-%

difference in the two phases as would be evident by comparing Fig. 6 with Fig. 2, both taken from a [1120] zone axis of the Burgers a phase. 3.3. The effect of heat treatment 0nf.c.c. phase formation in electropolished foils It has been shown by Rhodes and Paton [6] that isothermal heat treatments have no effect on the width of the interface layer in Ti-6Al-4V. However, both Rhodes [6] and Shelton [12] have observed the cooling rate following b solution treatment to influence the thickness of the interface region as illustrated in Fig. 9. Table 1. The relationship of interface structure to heat treatment in Ti-O.ENi4.3Mo Heat treatment 676”C-30 min-AC 788”C-30 min-AC 9WC-30 min-AC + 67692-30 min-AC 9OO”C-30min-AC + 788”C-30 min-AC

structure ordered f.c.t. Not known ordered f.c.t. f.c.c.

Fig. 8. The distribution of the striated and monolithic morphologies relative to the a//? interface.

BANERJEE et al.:

INTERFACE

PHASE PROBLEM

IN Ti ALLOYS

I33

spanding, to, the first plasmon loss peak of titanium for the a phase and for titanium hydride. The values of the position of the first plasmon peak compare well to those Zahuec [27J. The energy loss spectra for a,

0001

0.1

001 tmfq

1

Rate (KS-‘)

Fig. 9. The effect of cooling rate on the width of the interfacial layer and /I phase.

At cooling rates ~0.03 KS-‘, Shelton [12] found only a heavily dislocated interfacial region. The nature of the dislocations in this region could not be

easily identified, due to their high density. However, this dislocated layer appeared to create a locally rotated region, which could be lit up in dark field. It is possible that the possibility of imaging in such dark field conditions has also led to the erroneous low interface phase at identification of magnifications. Large strains occur within the a plates in the region adjacent to the interface, as indicated by the blurring of HOLZ lines in convergent beam electron diffraction patterns. The f.c.c. phase appears at cooling rates lower than 0.028 KS-‘, and at this point the width of the interfacial region increases to peak at about 0.008 KS-‘. The width of the interfacial layer clearly follows the trend in the /I phase width as a function of cooling rate. Assuming that volume fractions of the phases are constant, this implies that the interfacial region width also follows the a phase width. In situ heating of foils in the electron microscope results in a dissolution of the f.c.c. phase at around 300°C. This is shown in Fig. 10, where the diffraction spots marked “f” belonging to the f.c.c. phase gradually disappear from the pattern as the temperature of the foil is raised. 3.4. Chemistry of the j2.c. phase A number of attempts to determine the chemistry of the f.c.c. phase have been reported [12,25,26] and invariably the composition has been observed to lie between that of the a and /? phases. Whether these results truly reflect the composition of the f.c.c. phase or simply that beam broadening of the electron probe positioned at the a/B interface results in a sampling of both the a and fl chemistry is open to conjecture. In view of suggestions that the f.c.c. phase is a hydride of titanium [3-S], Shelton [12] has used electron energy loss spectroscopy to probe the nature of both titanium hydrides and the f.c.c. interface phase. Figure 11 shows the energy loss spectra corre-

Fig. 10. The effect of in drtc heating in the TEM on the interface phase. (a) Zone axis [OOO2]Burgers K at room temperature. Three variants of the “monolithic*’ f.c.c. phase are present as in Fig. 7. Spots marked ‘5” belong to the /I phase and those marked “a“ to the a phase, as in Fig. 7. (b) Heated to 250°C and (c) heated to 300°C. (b) and (c) show that spots marked “i” from the f.c.c. phase lengthen and finally vanish. All diffraction patterns are from the same area of foil.

BANERTEE et ul.: INTERFACE PHASE PROBLEM IN Ti ALLOYS

134

Table 2. Comparison of the position of the tirst plasmon energy loss peaks for OL, fl and f.c.c. interface ohasc in TMAMV SYStrm

c.p. a-Ti y-Titanium hydride Jet-electropolished TiiAMV ; f.c.c. interface Only dislocated interface

the f.c.c. interface phase and fl are shown in Fig. 11. Comparison between the positions of the peaks shown in this figure are made in Table 2, and clearly demonstrate the similarity between the f.c.c. phase and titanium hydride. Similar experiments on the a//j interface region in ion beam thinned foils containing no f.c.c. interface phase have shown that these results cannot merely be attributed to the presence of an interface in the area of specimen being examined, nor are they likely to be a result of some sort of chemical averaging process. It may be noted that Brown and Stephens [28] have recently reported a small energy

Ti=lZ6_

I

__TiH:?92

u I,

lo 20 30 Elechm Gwgy Los/eV

40

(4 Ti-6Al-N I3=18.6_

0

10 Electron

20

30

Plasmon energy (eV)

Energy change from ol-Ti (eV)

17.6 f 0.4 19.2 ? 0.4

1.6

17.2 18.4 f+ 0.2 19.0 t 0.2 17.4 IO.2

-0.4 0.8 1.4 -0.2

loss peak below the first plasmon loss peak for both niobium and vanadium hydrides. Although this energy loss region was closely examined for such a peak, none was observed.

4. THE EFFECT OF FOIL PREPARATION TECHNIQUE ON THE FORMATION OF ‘IME INTERFACE PHASE

A major step towards the understanding of the origin of the interface phase was made when it was found that ion milled foils contained neither the f.c.c. nor the hexagonal interface phase [9, lo]. A careful investigation of the influence of various thinning parameters was then carried out [ 111. The results of these investigations are summarized in this section. The foil preparation technique may be divided into two stages: the first consists of mechanically thinning or grinding the sample down to obtain a thickness suitable for the next stage that of electropolishing or ion milling. The mechanical grinding step is accomplished in a flowing lubricant which is usually water. In one experiment, samples of Ti-6Al4V were ground through fixed reductions in thickness using different lubricants and the hydrogen contents of the sample measured before and after the grinding. Table 3 indicates that the hydrogen concentration of the samples increased in all cases, but by the least amount when carbon tetrachloride was used as the coolant. Foils of Ti-6A1-4V prepared by_mechanical grinding in water, kerosene, or carbon tetrachloride were then electropolished in a Fischione twin jet electropolisher in different electrolytic solutions of either sulphuric acid in methanol or an acid free solution of zinc chloride in methanol. No interface phase was observed in foils ground in carbon tetrachloride and polished in the zinc chloride and methanol solution. No interface phase was also observed when foils of Ti-6Al4V were ion milled from a starting thickness of 0.025mm after mechanical grinding in water; or

co

Energy Loss/eV

Table 3. The effect of lubricant/coolant used in the mechanical grinding step on Hydrogen levels in Ti-6AcQV Lubricant/coolant

Fig. 11. (a) Energy loss spectra corresponding to the first plasmon loss peak of the a phase and titanium hydride. (b) Energy loss spectra corresponding to the first plasmon loss peak of a, /3 and the f.c.c. phase.

None (bulk) Water Kerosellc Carbon tctrachloride

H content (ppm wt) 6-1s 172 109 65

BANERJEE et al.:

INTERFACE PHASE PROBLEM IN Ti ALLOYS

135

Table 4. The effect of thin foil oreuaration technique on the formation of the f.c.c. uhase in Ti-6AUV Alloy Thinninn technique

Lubricant

Starting thickness 0.1 mm

Sulphuric acid methanol

Zinc chloride methanol Ion mill

Water Kerosene CC& Water Kerosene ccl, Water

0.025 0.025 0.025 0.025 0.025 0.025 0.025

when foils mechanically ground in water were vacuum annealed prior to thinning in the acid free solution. These results are summarized in Tables 4 and 5. The effect of electropolishing time was evaluated for a Ti-6Al-3Mc+O.SZr-O.25Si alloy [I 7] by altering the starting thickness of the blank for electropolishing. It was observed that for samples thinned in the sulphuric acid and methanol solution, higher starting thickness (approx. 0.1 mm), and therefore longer electropolishing times, resulted in the formation of an f.c.c. interface phase. However, at lower starting thickness (0.025 mm) a predominantly hexagonal interface phase (of the same crystallography as observed in Ti-lOV-2Fe+3Al) was obtained. Micrographs of ion milled samples of these two alloys are shown in Fig. 12. These indicate that both the f.c.c. and hexagonal forms of interface phase are absent in ion milled foils, and also show the true interface structure that exists between the a and b phases of titanium alloys. Shelton [12] has evaluated the effects of electropolishing on the lattice parameters of the a and /I phases. Figure 13 compares the 0002, and 111, lines taken from thin foil samples prepared by jet electropolishing, ion beam thinning and that from conventional powder samples. The lines broaden and move to a lower Bragg angle on jet electropolishing, indicating strains in both phases, and a volume expansion of /I and to a smaller extent, of the a phase. This may be compared with the observations of Pittinato and Hanna [29] who have observed considerable X-ray line broadening in the #I phase with hydrogen absorption, and associated increases in the d spacings of both a and /I phases in transformed B Ti-6Al-W.

VT9

Ti-6-4

Ti-IO-Z-3

f.c.c. h.c.p. No ifo

f.c.c./h.c.p. f.c.c./h.c.p. f.c.c./h.c.p. f.c.c./h.c.p. f.c.c./h.c.p. No ifp No ifp

h.c.p. No ifu

5. DISCUSSION The work described in the previous section shows conclusively that two structures of the interface phase do exist, an f.c.c. and an hexagonal structure. Further, both of these structures are observed in electropolished foils and not in ion milled samples, and therefore originate from effects due to the jet electropolishing process. In this section, these ideas are expanded upon to arrive at a comprehensive viewpoint of interface phase formation which is consistent with all the experimental data to hand.

5.1. Origin of the interface phase The results presented in Section 4 indicate clearly that parameters which influence hydrogen absorption by samples during thin foil preparation also affect interface phase formation, whether f.c.c. or hexagonal. To reiterate: Samples mechanically ground in Ccl, (minimum hydrogen absorption) and then electropolished in acid free solution do not contain the interface phase. However, the same mechanical grinding procedure combined with electropolishing in an acidic medium (sulphuric acid and methanol) results in interface phase formation. Samples ground in water but vacuum annealed (to remove hydrogen) prior to electropolishing in an acid free solution do not contain interface phase: however, samples ground in water and directly electropolished in either acid free or acid containing media do contain the interface phase. Samples ion milled from the bulk contain neither f.c.c. nor hexagonal interface phase.

Table 5. The effect of mechanical grinding on interface

phase

formation in TMAl-4V

Thinning technique Condition prior to final thinning Mechanically ground in water Mechanically ground in water, vacuum heat-treat, slow cool Mechanically ground in water, vacuum heat-treat, again mechanically ground in water

Sulphuric acid methanol

Zinc chloride methanol

Ion mill

ifp

ifp

No ifp

ifp

No ifp

-

ifp

-

No ifp

I36

BANERJEE et al.: INTERFACE PHASE PROBLEM IN Ti ALLOYS

showingthe absenceof Type 20:and the “true” a/j interfacialstructure.(b) Ion milledfoil of TiiAC3Mo-1 SZr-O.25Sishowingthe absenceof f.c.c.phase. Fig. 12. (a) Ion milled foils of Ti-lOV-2F*3AI

Concluding then, that both the f.c.c. and hexagonal phase owe their origin to an increase in hydrogen concentration during foil thinning, it is possible to examine the interrelationship between the two structures. More factors may now be taken into account: 1. Only the hexagonal phase forms when the volume fraction of /I phase is high, that is, metastable /I alloys such as Yi-IOV-2%3Al. 2. The hexagonal phase can form in a + /I alloys such as Ti-6Al-O.SZr-3Mo-O.25Si

provided that el~tro~lis~ng times are small (all other factors being constant). 3. The hexagonal phase is twin related to the parent Burgers a on {ldf 1) and sometimes possibly {1072) planes, which are both common twinning planes in the a phase of titanium alloys. 4. The f&c. and hexagonal varieties of interface phase are found only in transformed /.I microstructures. In equiaxed a + jl mill annealed microstructures, only the dislocated interface product has been observed [9, 121.

Fig. 13. X-ray diffraction patterns showing the effect of electropolishing on the 0002# and 110, lines in a thin foil of Ti-6Al-W as compared to the bulk sample. (a) Jet electropolished thin foil. (b) Ion beam thinned thin foil. (c) Bulk powder sample.

BANERJEE

et al.:

INTERFACE

It is thought that this is due to either the chemical composition at the interface (which may also be altered by changing the cooling rate), or, more likely, the fact that the a/p interface in such mill annealed microstructure is no longer of the Widmanstatten plate type (i.e. both habit plane and orientation relationship are different). 5. Hydrogen absorption during the thinning process increases the lattice parameter of the p phase, and to a lesser extent the a phase. This creates strains in the two phases.

The crystallography of the hexagonal phase strongly suggests that it is a stress induced product. A mechanism by which the a phase can be stressed lies in the volume expansion of the /3 phase during electropolishing through hydrogen absorption. The expansion of the b lattice would place the a phase in compression, and this phase may then deform by twinning or slip. It appears that when the a plates sizes are small in relation to the surrounding 8, twinning is the favoured deformation mechanism. It may be noted that both the creation of stresses and the resulting reaction are taking place in a thin foil where modes of deformation as well as stress states may well be different from those expected in the bulk, due to stress relaxation. An f.c.c. interface phase is formed when the volume fraction of the alloy is low or when electropolishing times are high (that is when the hydrogen concentration of the sample is expected to be relatively high). An explanation of this feature of interface phase formation may be proffered if it is accepted that the f.c.c. phase is a hydride of titanium. The notion that the f.c.c. phase is a hydride of titanium has been expressed several times [3-S]. Previous objections to this idea lay in that the hydrogen content of the bulk alloys did not seem sufficient to permit the extent of f.c.c. formation that is usually observed [I]. However, with the recent realization that hydrogen concentration measured in thin foils is substantially greater than those present in the bulk [ 11,301 this objection no longer appears to have any validity. The lattice parameter of the f.c.c. phase, moreover, is not much different from that of the binary hydride, and the crystallography of the f.c.c. phase in relation to the a phase is similar to that observed for titanium hydrides [31-331. Finally, Shelton [12] provides through electron energy loss absorption direct evidence of the similarity between the f.c.c. phase and the hydride. Additionally, the existence of the tetragonal face centred ordered phase in the Ti-O.8Ni+3Mo alloy which is structurally isomorphous with the ordered y hydride formed in zirconium alloys may be noted [23]. It may therefore be concluded that sufficient evidence now exists to indicate that the f.c.c. phase is a hydride of titanium. In order then to understand the

PHASE PROBLEM

IN Ti ALLOYS

137

effects of /l phase volume fraction and hydrogen content on interface phase formation, the nature of ternary phase equilibria generated by a combination of the Ti-H system and a phase diagram of a typical a + /? titanium alloy must be examined. Figure 14(a) shows such an isothermal section. The three phase a + /I hydride field must originate from the eutectoid in the Ti-H binary system. Of relevance to the present discussion is the slope of the line bounding the two phase a + j phase field and the a + /I hydride field. Since the /l phase in the binary Ti-H system has a considerably larger solubility for hydrogen than the a phase, the line ab must slope away from the 3~+ fi binary as the volume fraction of /l increases (with increasing /l stabilizing additions). Similar effects are to be observed on increasing the temperature in a vertical section cc’ shown in Fig. 14(b). Thus, for a given hydrogen concentration, the chance of the alloy existing in the three phase a + /3 + hydride region are greater for low volume fraction /? alloys. Thus, alloys such as IMI 685, Ti-6AlAV and TL6Al--OSZr-3Mo-O.2SSi all form the f.c.c. interface phase (or hydride). Similarly, for a given composition, an increase in hydrogen concentration tends to favour f.c.c. phase formation. as has been observed for lower electropolishing times in the Ti-6Al-OSZr-3Mo-O.25Si alloy. On the other hand, when the /3 phase volume fraction is high or the hydrogen content is low, the alloy would tend to lie in the two phase a + p region. In this situation, hydrogen concentration is not increased sufficiently during foil preparation for hydride formation, but is nevertheless sufficient (as discussed earlier) for a strain induced twinning reaction of the a phase which constitutes the hexagonal interphase phase. It is likely, given the observation that the spheroidized /l particles in mill annealed microstructure show only a dislocated layer at the interface, and neither hexagonal nor f.c.c. products, that the formation of interface phase in /? transformed microstructure is due to either chemical composition at the interface, as suggested above, or to the crystallography of the Widmanstatten a/b interface, or possibly both. 5.2. On a mechanism for j1c.c. phase formation The origins of both the f.c.c. and hexagonal interface phases have been traced to hydrogen induced effects. It remains to discuss the unique distribution and morphology of the f.c.c. phase (Section 3). The distribution and morphology of the hydride is shown in Fig. 8. It is difficult to find any parallel to these features in titanium systems elsewhere in the literature. Consider the phase diagrams of Fig. 14. The addition of hydrogen during the foil preparation process results in a shift in alloy equilibrium from a two phase a + b region to a three phase

BANERJEE et al.:

138

INTERFACE

PHASE PROBLEM

IN Ti ALLOYS

a

L 1

oc+P Ti

b

t+VERTICAL SECTION AT C-C'

Fig. 14. (a) A typical isothermal section (schematic) which should be formed by a combination of the Ti-H system and a pseudo binary section of a Ti a + /? alloy. The 3-phase space a +/I + hydride is indicated for clarity by dotted lines. (b) A vertical section through a 2-phase a + /? region and the 3-phase a+fl+yregion.

a +/I + hydride region. This shift may be accomplished by any of the following reactions [33]: a + hydride

/? + hydride a -I- b + hydride. It may further be noted that the transformation occurs either during the foil preparation process at temperatures ranging from - 50 to - 30°C (the electropolishing temperature) or as the foil is raised to room temperature following electropolishing. The temperature at which the reaction proceeds thus rules out any compositional change of the substitutional atoms constituting the reactive phases and therefore the reaction a +/I -P hydride. Either of the reactions a + hydride or /? -+ hydride may be accomplished without long range diffusion by

a massive transformation or a shear mechanism. Both these mechanisms require diffusion of hydrogen atoms to attain the hydride composition. A simple calculation, however, shows that this is not a rate limiting process even at room temperature. In order to evaluate the feasibility of a massive transformation occurring at room temperature the relationship for the growth of a disordered interface migrating without any change across the interface may be used [34]: G = Do/6 exp(-AHJRT)

x (-AF/RT)

where G is the growth rate, Do the diffusion coefficient, 6 the jump distance associated with atomic transfer across the interface, Hb the activation energy for interphase diffusion and AF the free energy change associated with the transformation. In the alloy Ti--6Al4V, the slowest moving species is V. Therefore the interdiffusion values for Ti-V

BANERJEE et al.: INTERFACE PHASE PROBLEM IN Ti ALLOYS

139

independently of the /l to f.c.t. transformation in such alloys. It is more difficult to distinguish between a shear mechanism and a massive mode of transformation. Recent studies [37], though controversial, indicate that the morphology and crystallography of a product arising out of a massive transformation may parallel closely that arising out of a martensitic mode. With the current information available on the f.c.c. phase formation, it does not seem possible to distinguish between the two mechanisms. Suffice it to say here that an examination of atomic matching across the habit planes for the two crystallographic forms indicate that the misfits are small and of similar magnitudes [5]. Thus, the choice of both the observed habits and orientation relationships is consistent with a surface energy dictated criterion. For instance, the predominant variant of the orientation relationship

alloys are used [35]. D,, = 1.6 x 10e4 cm/s H, = 30 kcal/mol. If it is assumed that the activation energy for interphase diffusion is half that of volume diffusion and D, remains approximately the same [36] F for the transformation is given as hydride u to - 19.78 kcal/mol[37]. With 6 = 4.4 A the growth rate arrived at the 298 K is O.O14pm/s, and is clearly enough for the massive transformation to be a possible transformation for f.c.c. interface phase formation. Thus, the f.c.c. phase may form from either the a or #I phases either by a shear mechanism or as a massive transformation. The crystallographic analysis presented in Section 3 strongly suggests that the hydride forms from the a phase, and this suggestion is reinforced by the fact that the observed crystallography is identical to that observed in situations where the hydride forms within a grains [30-321. With respect to this suggestion, attention is drawn particularly to an investigatin into the thin foil martensite observed in titanium alloys. Pennock et al. [39] have formed an illuminating set of experiments illustrating the connection between hydrogen absorption during thin foil preparation and the formation of the f.c.c. structure from the B phase which they specifically identify with titanium hydride. While the orientation relationship they observe between the f.c.c. phase and /3 is similar to that observed for the second crystallographic form of the f.c.c. interface phase, the morphology of the internally twinned plate structure bears little resemblance to that of the striated f.c.c. phase. Thin foils of a + fl alloys containing the f.c.c. phase also show the striations within the /l phase (see Fig. 11 of Ref. [6]) which Pennock et al. clearly identify with the dislocation debris left behind as the unstable f.c.t. martensite reverts back to 8. Thus the formation of the f.c.c. phase occurs

(lTO)rc.c.//(lTOO),;

Pw,c.c.//Poo1lz

is one which lies along the a/p interface with its broad faces parallel to the a phase on one side and the /I phase on the other. Figure 15 indicates schematically the atomic matching across the a, /I and hydride phases parallel to these broad faces. The existence of more than one orientation relationship for a precipitate forming from a diffusionally controlled reaction in a given alloy system has been observed [40,41]. However, a similar observation does not seem to have been recorded for a martensitic transformation. The effect of cooling rate on the thickness of the interface phase shown in Fig. 9 is more difficult to explain. A variety of complex effects may influence the formation of the interface phase as a function of cooling rate: variations in composition [41], lattice parameter of the a and jl phase [12], transformation substructure within the a phase [42] (see 3.3) and the volume fraction of the fl and a phase [ 121may all play their part, and it is difficult to separate all these possible variables in the equation.

_---___--__-___ llTl @&TYPE

1 fee

Ill01 IpR : -z MISMATCH : +2.2%

4 fee “, ;

f

l

I

I

I

I

!

I I

I

II I

[liO] TYPE 1 fee PLANE NORflAL:[ii00~ I( tri21 P

Fig. 15. The atomic matching across (I lO),,,,

A :* MISMATCH: -3.5%

,gg

TYPE 1 fee

[Iii]d

(ITOO), and (112), planes.

BANERJEE et al.:

140

INTERFACE

PHASE PROBLEM

5.3. A note on the solubility of hydrogen in titanium alloys The true solubility of hydrogen in a titanium has been shown to be significantly dependent on the work of elastic and plastic defo~ation [44,45] required to accommodate the difference in specific volume between hydride and a titanium (17%). This feature, for instance, leads to a wide variety of interesting phenomena such as hysteresis in resistivity curves on thermally cycling the sample through the hydride solvus [44], and the opposing and contradictory effect of aluminium on hydride solubility [43,46]. While aluminium reduces the activity of hydrogen in E titanium, it also enhances the yield strength, thus increasing the work of deformation in hydride precipitation and therefore increasing the apparent solubility of hydrogen. The solubility of hydrogen in titanium alloys cannot therefore be defined in terms of composition and phase equilibria alone. Factors such as the stress state at any given location 1461 or the existence of a pre-existing favourable disi~ation structure 1301can pfay a significant role in influencing local solubility since they affect the work of accommodation. Banerjee and Arunachalam [S] have pointed out some features of semicoherent a/p interfaces and a phase substructure which favour hydride formation. However, if hydride is present at the a//l boundaries of commercial titanium alloys in bulk samples, it may be expected to have a signifi~nt effect on mechanical properties; the soiubility of hydrogen in commercial a Jfl alloys is, therefore, of some importance. The purpose of the preceding discussion is to highlight the difficulties of determining this solubility through microstructural means. In conventionally electro-thinned titanium alloys, the increase of hydrogen concentrations during thinning and subsequent hydride fo~ation would mask the true absence or presence of hydride due to the intrinsic hydrogen content of the alloy. While recent work in this area has shown the way to prepare thin foils without hydrogen absorption, it is still not possible to be certain that the presence of hydride in foils prepared in this manner does truly indicate that hydride is present in the bulk. The considerable relaxation of stresses normal to the surfaces of a thin foil may allow the precipitation of hydride at hydrogen levels which lie below the solubility limit in the bulk.

the hydrogen levels are low enough that these phases are only observed as thin foil artefacts induced by an increase in hydrogen concentration in the thin foil during preparation, possibly accompanied by stress relaxation effects. However, the exact m~hanism by which the f.c.c. phase forms remains to be identified. Acknowledgements-C.

G. S. gratefully acknowledges the assistance of Drs F. J. Rocca and D. Imeson of the Cavendish Laboratory, Cambridge, in the acquisition of the Electron Energy LOSSdata.

REFERENCES 1. C. G. Rhodes and J. C. Williams, h&e&I. Trans. A $A, 1670 11935). 2. C. G. Rhodes and J. C. Williams, Mernli. Trcms. A 6A, 2103 (1975). 3. I. W. Hall, Rand. J. merall. 8, 17 (1979). 4. P. Hallam and C. Hammond, Tifanium Science and Technology, p. 1435. T.M.S.-A.I.M.E.. New York (1980). 5. D. Bane+ and V. S. Arunachalam, Acra merall. 29, 1685 (1981). 6. C. 0. Rhodes and N. E. Paton, Me&l. Trans. A lOA, (1979). 7. H. Margolin, E. Levine and M. Young, Met& Trans. A 8A, 377 (1977). 8. W. G. Burgers, Physicu 1,561, (1934). 9. C. G. Shelton and B. Ralph, Proc. Co& The Metallurgy of Light Alloys, pp. 180-183. Inst. of Metallurgists Conf. No. 20. (1983). 10. D. Banerjee and J. C. Williams, S&pm metali. 17, 1125 (1983). 11. D. Banerjee. C. G. Rhodes and J. C. Williams, Proc. 5rh lnt. Con$ on Ti Alloys, Munich (1984). 12. C. 0. Shelton, Ph.D. thesis, University of Cambridge (1985). 13. A. Lasalmonie and M. Loubradou. J. Marer. Sci. 14. 2589 (1979). 14. D. Banerjee, Merall. Trans. A 13A, 681 (1982). 15. 0. C. Morgan and C. Hammond. Ti Science and Te&ofogy,-p. 1443. T.M.S.-A.I.M.E., New York (1980). -- 16. G. Isaac and C. Hamond, Pm. 5th fat. Co& on Ti Aflovs,

The morphoiogy,

distribution and crystallography of the hexagonal and face centred cubic “interface” phases in titanium alloys have been described. Sufficient evidence has been accumulated to show that the f.c.c. phase is a hydride and that both f.c.c. and hexagonal structures arise from hydrogen induced transformations. In current commercial alloys

Munich fl9841.

17. D. Banerjee and-J. C.’ Williams, Unpublished research, Carnegie Mellon University (1982). 18. W. Pitsch, Phil. Mag. 4, 577, (1979). 19. D. Banerjee, R. V. Krishnan and K. 1. Vasu, Metall. Trans. A llA,1095(1980). 20.M. J. Blackburn, The Science, Technology and Applica-

21. 22. 23. 24.

6. CONCLUSIONS

IN Ti ALLOYS

25. 26. 27. 28.

tion of Titanium, p. 640. Pergamon Press, London (1970). D. Banerjee, Unpublished research, Carnegie Mellon University (1983). G. C. Weatherly, Actu metali. 29, 501 (198I). A. J. Williams, R. W. Cahn and C. S. Barrett, Acta metal 2, 117 (1954). C. G. Rhodes and N. E. Paton, Metal. Tram A IOA, 1753 (1979). P. Chenu. C. Servant and P. Lacombe. Scriota merall. 13, 951 (i979). C. Chen and J. C. Williams, Proc. 38th Electron Microscopy Sociefy Meeting, p. 362, Claitors Publishing Division, Baton Rouge, Calif. N. Zaluzec, T. Schober, B. W. Veal and D. G. Westlake, Anulytical Electron Microscopy, p. 191. San Fran&o Press (1981). L. M. Brown and A. P. Stevens, Acta metall. 33. 827 (1985).

BANERJEE er ai.:

INTERFACE

29. G. F. Pittinato and W. D. Hanna, Metali. Truns. $2905 (1972). 30. D. G. Westlake and W. R. Grey, Appl. Phys. Len. 9, 3 (1966). 31. J. D. Boyd, Trans. A.S.M. 62, 977 (1969). 32. N. E. Paton and R. E. Sourling. Merail. Trans. A 7A, 1769 (1976). 33. I. W. Hall, h&tall. Trans. A 9A, 815 (1978). 34. A. Prince, Allby Phase Equilibria. Elsevier, London (1966). 35. P. G. Shewmon, Diffusion in Solids, p. 61. McGraw-Hill, New York (1963). 36. 01% Defect Data 11, 95 (1975). 37. H. I. Aaronson and A. R. Kinsman, Acra metall. 25,367 (1977). 38. H. H. Arita, K. Shimizu and Y. Ichinose, Met&. Trans. A 13A, 1329 (1982). 1

_

PHASE PROBLEM

IN Ti ALLOYS

141

39. G. Pennock, H. M. Flowers and D. R. F. West, Metallography 10, 43 (1977). 40. M. G. Hall, H. I. Aaronson and A. R. Kinsman, Surface Sci. 31, 257 (1972). 41. A. M. Sriramurthy, S. Banerjee and S. N. Tewari, Acta meralf. 30, 1231 (1982). 42. C. G. Rhodes, Proc. 5th Inr. Co& on Ti ARow, Munich

(1984). 43. D. Banerjee, D. Mukherjee, R. L. Saha and K. Bose, Metall. Trans. A 14A, 413 (1983). 44. N. E. Paton, B. S. Hickman and D. H. Leslie. MeraN. Trans. 2, 2791 (1971). 45. J. C. Williams, E&ct of Hydrogen on the Behoviour of Metals, p. 367. T.M.S.-A.I.M.E.. New York (1976). 46. J. L. Waisman, Metali. Trans. A SA, 1249 (1972). 47. T. B. Flanagan. N. B. Mason and H. K. Birnbaum, Scripta mefuii. 15, 109 (1981).