Tetragonality of Fe-Ni-Ti Martensite

TETRAGONALITY

OF Fe-Ni-Ti

MARTENSITE

$1. $1. HAL.L* and P. G. WIXCHELL School of Materials Engineering. Purdue University, Lafayette, IN 47907. U.S.A. With Appendix by P. GUY ZYA Oratoire, 69 Calaire, France (Receid

8 June

1976)

Abstract-Precise X-ray 200,020. and 002 peak shapes and positions are reported for martensite formed from austenite single crystals containing 2, 4.5 and 6.5 at.:,; Ti with about 257; Ni. The parent austenite has been rapidly quenched and prcaged at 7lO’C before transformation to martensite and the distribution of intensity in and near the austenite peaks has been measured. In all alloys briefy preaged austenite forms tetragonal martensite with fairly sharp X-ray peaks. Martensite tetragonality increases rapidly with preaging time. In 4.5 and 6.5 at.?: Ti alloys even longer preaging treatments produce austenite with satellite X-ray peaks. Martensite subsequently formed from such austenites exhibits a superposition of the fairly sharp peak previously described and a greatly broadened Bragg peak which is positioned to signify reduced tetragonality. With increased preaging time. austenite satellites grow in intensity and the broad martensite peaks grow at the expense of the narrow peaks; these broad peaks sharpen slightly and shift toward still lower tetragonaiity. In the 2 at.?; Ti alloy even extensive preaging failed to produce satellite peaks in the austenite and subsequently formed martensit: failed to exhibit broad peaks. These results mean that the very small solute atom clusters first formed are transformed coherently from austenite to martensite and serve as defects which distort the lattice to produce significant martmsite tetragonality and probably also significant strain energy in the martensite. In contrast continued aging of the richer Ti austenites produces larger clusters which are sufficiently different in atomic scattering factor of lattice parameter to produce X-ray satellite peaks from austenite. The larger clusters become incoherent during martensite formation, cause defects in the martensite. and lose the cooperative elastic strain which supports tetragonality. The clusters are hypothesized to be Ni,Ti and the formation of the smaller clusters is modeled to explain the lattice strains and to describe the kinetics of the initial stage of their formation. Resume--On presents des mesures prtcises de la forme et de la position des pits 200. 020 et 002 de rayons X sur la martensite formee dans des monocristaux austtnitiques contenant 2, 4.5 et 6 at.“, Ti et environ 25% Ni. On a trempe rapidement et previeilli a 710°C. avant la transformation martensitique, la matrice austenitique et on a mesure la repartition de l’intensite dans et autour des pits de l’austenite. Dans tous les alliages ayant subi un court vieillissement pialable, l’austenite produit une martensite quadratique avec des pits de rayons X ausuz fins. La quadraticite de la martensite augmente rapidement avec le temps de vieillissement. Dans les alliages a 4.5 et 6,s at.?; Ti, des traitements de vieillissement prealable meme longs produisent une austenite avec des pits de rayons X satellites. La martensite form&e a partir de telles austtnites presente une superposition du pit fin dont nous venons de parler et d’un pit de Bragg tres elargi dont la position correspond a une diminution de la quadraticiti. Si l’on augmente le temps de vieillissement prealable l’intensite des satellites de l’austenite augmente et les pits martensitiques larges croissent aux d&pens des pits etroits; les pits largcs deviennent legtrement plus fins et se deplacent vers une quadraticiti encore plus faible. Dans I’alliage a 2. at.% Ti, on n’observe pas de pits satellites dans I’austenite, et la martensite qui se forme ensuits ne prtsente pas de pit elargi. Nos rtsultats montrent que les plus petits amas d’atomes de solute form& au debut se transforment de maniere coherente dausttnite en martensite et servent de defauts qui distordent le riseau, produisant ainsi une quadraticite notable de la martensite et probablement aussi une importante tnergie de deformation dans la martensite. Par contre, le vieillissement continu des austenites plus riches en Ti produit des amas plus grands qui different sufhsamment en facteurs atomiques ou en paramttres cristahins pour produire des pits X satellites. Les amas Ies plus grands deviennent incohtrents au tours de la transformation martensitique, produisent des defauts dans la martensite et relachent la deformation cooperative tlastique qui produit la quadraticite. On suppose que les amas sont NisTi et on propose un modtle de formation des petits amas pour expliquer les deformations du reseau et pour decrire la cinitique du stade initial de leur formation. Zusammenfassung-PrSzise Messungen von Form und Lage des Rontgenmaxima 200. 020 und 002 werden vorgelegt fur Martensit. gebildet aus Austeniteinkristallen mit etwa 3:; Ni und Gehalten von 2. 4.5 und 6.5 At.‘, Ti. Der Mutteraustenit wurde rasch abgeschreckt und vorgealtert bei 710-C. bevor die Umwandlung zu Martensit und die Verteilung der Intensitlt in und nahe bei den Austenitmaxima verfolgt wurde. In samtlichen Legierungen bildet kurz vorgealterter Austenit tetragonalen ?.tartensit mit ziemlich scharfen Rontgenmaxima. Martensit. der daraufhin von solchen Austeniten gebildet wurde. zeigt eine Uberlagerung des eben erwahnten ziemlich scharfen Maximums mit einem stark

736

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WINCHELL:

TETRAGONALITY

OF Fe-Ni-Ti

MARTENSITE

verbreiterten Braggreflex, dessen Lage reduzierte Tetragonalitat anzeigt. Mit ansteigender Voralterungsdauer nehmen die Austenitsatelliten an Intensitlt zu und die breiten Martensitmaxirna wachsen auf Kosten der engen Maxima. Diese breiten Maxima werden etwas enger und bewegen sich in Richtung noch kleinerer Tetragonalitat. In der Legierung mit 2 At.% Ti fanden sich selbst nach ausgiebiger Voralterung keine Satellitenmaxima im Austenit, der daraulhin gebildete Martensit zeigte keine breiten Maxima. Die Ergebnisse bedeuten, daB die zuerst gebildeten sehr kleinen Anhaufungen von Losungsatomen sich kohlrent vom Austenit zum Martensit umwandeln und als Defekte wirken, die das Gitter verzerren, so daD eine deutliche Tetragonalitat des Martensits und wahrscheinlich such eine betrlchthche Verzerrungsenergie im Martensit auftreten. Im Gegensatz dazu ftihrt Iangeres Altem der Ti-reicheren Austenite zu gr83eren Anhaufungen, die sich geniigend im Atomstreufaktor des Gitterparameters unterscheiden, urn Satellitenreflexe des Austenits hervorzurufen. Die groBeren Anhlufungen werden wahrend der Martengitbildung inkohlrent, erzeugen Defekte im Martensit und verlieren die kooperative elastische Verzerrung, welche die Tetragonalittit unterstiitzt. Vermutlich sind die Anhlufungen NisTi. Die Bildung der kleineren Anhlufungen wird mit einem Model1 beschrieben, urn die Gitterverzerrungen zu erklaren und urn die Kinetik des Anfangsstadiums ihrer Bildung zu beschreiben.

INTRODUCTION Although carbon and nitrogen bearing martensites have long been known to be tetragonal, only more recently has evidence of martensite tetragonality due to substitutionally dissolved solutes been reported. Two general discussions [ 1,2] give descriptions of some [33 arrangements of substitutional solute atoms in austenite such that the martensite will be tetragonal. One of these discussions [l] centers on arrangements of solute atoms characterized by the point symmetry of various locations in the unit cell and describes the change in this symmetry brought about by the martensitic transformation, i.e. by the Bain strain. Of primary interest among these points was the tetrahedral array of solutes which might characterize the atomic arrangement of Ni and Ti in y’ Ni,Ti. This arrangement received particular attention because observations of martensite tetragonality in Fe-25 Ni-(O-7)Ti (at.%) were available and showed that the tetragonality increased with the titanium level and was greatly reduced by tempering the martensite [4-71. Subsequently preaging of the austenite, i.e. aging prior to martensitic transformation, has been found to increase martensite tetragonality [S]. Since such preaging was already known [4,6] to produce austenite X-ray satellite peaks which monitor the size and number of precipitate (or preprecipitate) particles, the precipitation process and martensite tetragonality were empirically correlated. The presence of tetragonality in martensite formed from austenite which had not been preaged was ascribed to precipitates supposed to be present in the as quenched austenite [S, 63. Evidence for the presence of such precipitates was based on the observation that the size of the precipitates (judged from the separation of the satellites from their Bragg peak) extrapolated at zero time to a significantly large value of about sixteen unit cells. but no measurements of the concen* Now with the Argonne National Laboratory, Argonne, IL 60439, U.S.A. t Applied Research Laboratory, Div. of Bausch and Lomb, Rochester, NY, U.S.A. Model ARL-AMX.

tration of precipitate particles were reported. The appearance of initial precipitates of finite size is expected nucleation behavior, and hence extrapolation to a finite size at zero time does not necessitate the presence of such precipitates in the as-quenched austenite. Consequently. the presence or absence of precipitate particles in the as-quenched austenite must be regarded as an open question. The present report describes a careful restudy of the preaging of austenite and its effect on martensite tetragonality. Satellites developed during preaging of single-crystal austenite and the tetragonality of the martensite formed from it are accurately recorded and directly correlated. These studies are carried out for 2, 4.5 and 6.5 at.% titanium. They show that the relation of the tetragonality to the precipitate formation is more complex and interesting then previously indicated.

MATERIALS

AND PROCEDURES

Single crystals of Fe-Ni-Ti austenite were grown in cooperation with Dr. H. Harrison at the Central Crystal Growth Facility at Purdue University by a Bridgman technique. Yitria stabilized zirconia crucibles were charged with premelted rods and were enclosed in a tantalum susceptor. A steep temperature gradient was obtained using r.f. induction heating through a multitum coil terminating at a slotted cop per spoiler plate. Growth was most often successful at a rate of about 4 cmfir with a melt temperature of about 1600°C. Crystals were 1 cm in diameter and 11 cm long. They contained cells with misorientations of up to 5”. Each of three crystals were wrapped in iron-nickel foil and sealed in fused silica capsules and annealed for ten days at 1150°C. The samples were then analysed for nickel and titanium on a microprobe? which was calibrated using known Fe-Ni-Ti alloys. Results are shown in Table 1. The samples were essentially free of microsegregation of nickel or titanium and exhibited little marcosegregation. In particular no systematic variation in titanium content along the length of crystals could be detected.

HALL ASD WINCHELL: Table 1. Compositions Nominal

at.Oo Ni

x-2 ZM.5 13-6.5

30.3 + o.s* 26.1 + 0.5 23.0 + 0.6

TETR.-ZGONALITY OF Fe-Xi-Ti

of crystals at.?: Ti 9.01 * 0.03* 1.51 + 0.26 6.54 & 0.08

* The standard deviations of individual probe readings.

Ths crystals were oriented within 2’ by the standard Lau2 back-reflection X-ray technique and were cut parallsl to [lOO:,&and {1lo;,on a low-speed diamond saw to obtain slices 3.5 cm x I cm x 0.1 cm. The slices were chemically thinned to 0.03 cm and hand lapped to 0.025 cm thick. These slices were joule hzated to 11OO’C for 30 min in a vacuum of 10-j (lo-’ Torr) and quenched in vacuum in liquid metal (Pb Bi cutectic at 125 + 1O’C) at an experimentally verified cooling rate of 10°C s- ’ near 700°C. The quenchrd slices were diced by cutting along (001) or (ITO) into eight or nine specimens each about 25 mm’ in area. The details of the quenching apparatus and its calibration have been r2ported [9].* Preaging of the austenitic specimens was carried out at 71O’C & 5’C for times between 0.5 and 90 min. (The 3&2. alloy was aged for up to 360 min.) For preaging times of 3 min or less the specimen was wrapped in Fe-Ni foil which was spot welded. For preaging times longer than 3 min, the specimen and its foil w2re sealed in fused silica. The samples were hzated by placing them in a tube immersed in molten lead. For study of satellite formation in austenite, sequential aging treatments were carried out, and the przaging time reported is the total time at the preaging temperature. Following preaging the capsules were water-quenched and opened only after the specimen had cooled. If martensitic transformation was required. the specimen was cooled in liquid nitrogen (-196'C).Some martensitic specimens were postaged (tempered) inside Fe-Ni foils at 296 + 5’C for LIP to 88 h. All surfaces were metallographically prepared before X-ray diffraction. X-ray diffraction experiments were carried out on 4 circle the diffractometer previously described [IO. I I]. Cr K, radiation was detected by an energy dispersive detector+ with a resolution of 230 eV operating with associated single channel analyser with rsal-time correction for counting losses so that as-corrected losses were unimportant below 3 x 10”

* In that publication the cooling curve must be read upside down! t Nuclear Semiconductor, Menlo Park, Cc\. U.S.A. : Russian workers use the reverse symbols for 200 and 020. The choice is arbitrary. DSuch shifts averaged 0.032’0 and were never greater than 0.0% ‘0. We are indebted to Professor Herman Rubin of the Purdue University Statistics Dept. for pointing out this distribution to us and for supplying the regression analysis program.

MARTENSITE

737

cps. The 330 eV window is sufficiently wide so that some Bragg diffracted white radiation could bs supzrimposed on satellites near th2 Bragg peaks. This radiation was effectively removed on the short wavzlzngth side of the Bragg peak by the use of a vanadium filter at the receiving slit. The X-ray peaks were counted in every 0.01’0 steps for fixed time. During this time th2 sampls was rocked +Z.5’ about the “w” axis (axis of the difiractometer) and a range of x angles of + 1.7’ were received by the counter. (The x axis is normal to the diffractometer axis and at w = 0 makes an angle of 8 with the diffracted beam.) Thus, a projection of ths diffracted intensity onto a radial linz through a lattice point in reciprocal space was recorded. Essentially th2 same technique was ussd for rscording both the martensite and austenite peaks. The rocking was omitted when the reciprocal lattice location of the satellite peaks was dctermin2d. As explained previously [lo] the preferrsd orientation of martensite in austenite. which is dsscribed accurately enough for the present purposes by (lOlh,1~(11I), cl l~]~,~icl~O],. is sufficient to distinguish ZOO.020, and 002 martensite peaks by their orientation in the austenite. They are readily distinguished sines (OOZ), poles are near [OOl1r\ and (020) are very near : 110; + while (ZOOh, are five or more degrees farth2r from (110: +$ With this technique 200, 020. and 001 peaks were recorded without overlap and nith ample intensity so that the peak shape and position could be accurately determined. A disadvantag2 of the technique is that to record the martensite peaks th2 specimen surface must be tilted from the focussing condition to as much as 15’. To reduce the error so incurred. a pure iron polycrystal was run after each martensite peak without changing any angular settings. The peaks were shifted slightly3 to bring the observed iron peak to a standard position. Th2 peaks were recorded on punched paper tape and \vsre corrected for background and corrected for slowly varying functions due to angular dependence of atomic scattering factors and polarization. Corrrctsd data was computer plotted and analysed for peak position. center of gravity, integral width, and if necessary, was processed to separate two superimposed psaks (both due to the same Bragg diffraction) as described below. In several specimens two peaks were found to be superimposed, on2 with a half maximum width of about 2. degrees and one with a width of about 0.7 degrees. In order to separate these peaks their shapes were approximated by a Pearson tvpe VII distribution, 1; P(x)= [la7cJ~l

+ rn-’ 2 a-‘)7-’

(1)

for m = 1 this is Cauchy and for m = YZit is normal. m = 3 was found to fit the data adequately for all the symmetric peaks or peak components. A standard nonlinear regression program was suCc2Ssful in separating nearly all mixed peaks into their two components.

HALL

73s

ASD

WINCHELL:

TETRAGONALITY

OF Fe-Ni-Ti

MARTENSITE

a a 2x N I,

0.56-

‘0 c-

0.55

-

I - 0.02

0.537

I -0.01 h, ,,.=

I 0.00 2snyA“

I 0.02

I 0.01 (8-q

Fig. I. Intensity distribution about 200 reciprocal lattice point (rip) projecteJ on the h, h20 plane of 23 at.yO Ni 6.5 at.?b Ti austenite preaged 15 min at 71OC. Horizontal splitting is due to substructure in austenite. Vertical satellites are also split.

EXPERIMESTAL The positions,

RESULTS

shapes. and intensities

are reported

for satellite peaks in austenite and for 200, 020 and 002 peaks in martensite. The austenite satellite peaks are described first and in their case an additional feature, their location in reciprocal space, is also reported. Austenite satellite peaks of the 200 were measured I

I

I

0.50 -

and a sample distribution of intensity projected onto the 001 reciprocal lattice plane is shown in Fig. 1. This is from a sample of alloy 23-6.5 aged 15 min at 710°C. This distribution is atypical in that diffraction occurred from two subgrains and a doublepeaked smear of intensity is observed in the Bragg peak. This smear is at constant radius in reciprocal space and is caused by growth defects in the crystals. At much lower intensity and along a radial line in I

I

I

0’ -1

L

0;

0.57 -

i 2

0.56Colculatcd

ii

Position

&

.5 L

of

Rodiol

Sotollitcs

;: 5 4

0.55-

II b =- 0.54

-

0.53

0.53

I 0.54

I 0.55 h,

a” = *

I 0.56 [

rinO+;lnWUt]

I 0.57

I 0.50

(A-1)

Fie~ 2. Intensity distribution about 220 RLP projected on the h, hZO plane of 23 at.% Ni 6.5 at.?, Ti austenite preaged 35 min at 7lO’C. Smearing in IT0 is due to austenite substructure. Satellites appear along the 110 and not in the 100 or 010.

HALL

AND

WINCHELL:

TETRAGONALITY

reciprocal space are the satellites. They also are smeared circumferentially. The intensity contours of the satellite nearer the origin is shown in more detail because our setup resolves the low-angle satellite more clearly because C’filtering is effective here: thus radial satellites in the 100 direction in reciprocal space are present. In order to see whether satellites are radial or 100, the satellites near a 720 reciprocal lattice point were observed in projection on a 001 reciprocal lattice plane. The result in Fig. Z shows again that the Bragg intensity is smeared circumferentially in reciprocal space due to growth in defects in the austenite crystal. Also intensity is diffracted along the radial line parallel to the 110 reciprocal lattice vector. This latter distribution includes the expected positions of satellites if they were radial. Along the 100 and 010 reciprocal lattice directions a minimum of intensity is observed. Along these directions satellites should be seen if they were 100. Consequently satellites are not along 100 direction but rather along the radial directions in reciprocal space as previously observed by Dofield and Phillips in a CuNiCo single crystal [I?]. The general location of the satellites did not vary with aging and \+as similar for the 6.5 and 4.5 at.:,; Ti alloys but changes in intensity and radial distance from the reciprocal lattice point were observed. Satellites were never observed in the 30-2 alloy samples. The development of satellites during preaging at 7 IO’C was pbserved by projecting the difTracted intensity on the radial reciprocal lattice line by rocking the crystal about the “cc)”axis during each counting step. (The usual Al range of angles continued to be accepted which provided effective projection normal to the w direction.) The resulting intensity distribu-

TvETA

OF Fr-Ni-Ti Xl.\RTENSITE

739

tions are shown in Figs. 3(a) and (b) for the 6.5 and 4.5 at.‘, Ti alloys. The Bragg peak is truncated at !“/Aof its maximum intensity for presentation of the data. No satellites \vere observed in the 2 at.“; Ti alloy although care was taken to detect them and aging L+XScontinued for as long as four hours. WC conclude that they do not form in this alloy under the experimental conditions. The 6.j:0 Ti and 4.Y0 Ti alloys show satellite development during aging and the satellites are approximately symmetrically distributsd about the Bragg peak. (As discussed previous!> our experimental technique suppresses white radiation background better at the low-angle side.) The most prominent feature of these observations is that the intensity in the satellite peak is essentially zero in the as-quenched crystal and develops at a measurable rate with increased preaging. The relative intensities of the satellites and the Bragg peak was measured by integration of the curves in Fig. 3 including the complete Bragg maximum (counted to avoid losses) and the results are plotted in Fig. 4. Notice that the satellite intensity starts at zero for all three alloys and remains zero for the ?‘,OTi alloy but rises toward a limiting value of S”,, for 3.5?& Ti and I?,, for 6.jo0 Ti with a rise timz of about ten minutes. The position of the maximum in satellite intensity shifts from about 1.5’0 away from the 200 peak center to about 0.75’0 as aging time is increased from 0.5 to 90 min. According to these separations, de, the wavelength. Q, of the perturbing variation in the hO0 direction is for the hkl reciprocal lattice point [13]. Q = h tan @/[(II’ + k’ + l’)dB]

(2)

which yields an initial wavelength of about 15 unit cells. a number which is approximately doubIed dur-

ANGLE

Fig. 3(a). Satellite intensity near 200 RLP projected on the It00 direction preaged at 71o’C for the times shown. 36 at.% Ni 4.5 at.?< Ti.

740

HALL

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Aged

90

TETRAGONALITY OF Fe-Ni-Ti MARTENSITE

min.-

Fig. 3(b). Satellite intensity near 200 LRP projected on the hO0 direction preaged at 71O’C for the times shown. 23 at.% Ni 6.5 at.% Ti.

ing subsequent aging. Clearly, the wavelength is sub-

stantial in initial satellites, but the amount of intensity involved is not large and essentially no satellite producing particles are present in the as-quenched alloys. The satellite peak widths are also measurable and are related in the Discussion of Results to the number of wavelengths in a single precipitate. In summary, the development of satellites from satellite-free, as-quenched austenite has been observed for aging at 71O’C. No satellites form in 2 at.% Ti austenite. Satellites develop in 4.5 and 6.5 at.% Ti alloy in about 10 mm. They are separated from the 200 reciprocal lattice point (rlp) by about one degree 8 and lie on a radial line through the rlp. They are broad and increase in intensity until they have about ten percent of the intensity associated with the Bragg peak. The martensite 200, 020 and 002 peaks were observed in austenite single crystal specimens which

had been quenched and preaged and then transformed to martensite by cooling to - 196°C. All measurements were made at room temperature. Specimens cut along (100), were used to record the martensite 002 peaks. A’ typical arrangement of diffracted 002 intensity near an austenite 100 pole is shown in Fig. 5(a). The martensite 002 pole locations approximate those expected according to the Kurdjumov-Sachs orientation relationship. Specimens cut along (llO), were used to record the martensite 200 and 020 peaks. A sample distribution of intensity is shown in Fig. j(b). Almost always a clear separation was obtained with several maxima (four according to Kurdjumov-Sachs) within five degrees of the IlO., pole and several (again 4 according to KurdjumovSachs) at about ten degrees. The closer poles are called the 020 and those further away the 200. Although nearly all x, w maps such as Fig. 4(b) clearly distinguished the 200, and 020,,, their proximity to

PREAGING 124

,, (

0.14 0.12

-

0.10

-

IO

,

TIME

lmin)

30

60

90

I

I

I

240

I6.5

% Ti

lo)

4.5

% Ti

(a)

2 % Ti

0

I 2

I 4

I 6 JPREAGING

I 0

I IO TIME

I I2

(0) I 14

16

IminI’/

Fig. 3. Ratio of satellite intensity to total diffracted intensity vs preaging time at 71O’C.

HALL AND WINCHELL:

TETR.aGONALITY

OF Fe-Ni-Ti

MARTENSITE

741

Aw Fig. 5a. Martensite 002 pole figure near austenite (001). The 1004 and 010, zones are vertical and horizontal.

one another and the presence of significant substructure in the austenite crystals casts some doubt on their correct identification in all cases. The shapes and positions* of the 200, 020, and 002 martensite peaks are displayed for several preaging times and for alloys with 2.0, 4.5 and 6.5 at.% Ti in Fig. 6. The 200 peaks are shown in the first row, the 020 in the second row, and the 002 in the third row. The first column contains peaks for 2:; Ti martensite, the second column for 4.5% Ti, and the third column for 6.5% Ti. In each figure the individual peak profiles are tagged to indicate the preaging treatment time at 710°C imposed on the austenite before transformation to martensite. This figure is rich in information. The 2% Ti martensites shown in the first column exhibit what we shall call sharp peaks, i.e. peaks with a width at half maximum of about 0.75% As a result of preaging these peaks remain sharp but shift by small but easily measurable amounts. yiuch of this shift may have occurred already in the as-quenched austenite subsequently transformed to martensite. In order to evaluate the lattice parameter of cubic martensite, the “c” parameter, determined from the 002 peak position, is plotted in Fig. 7 against the “a” and “b” parameters from the 200 and 020 peak positions. The line representing c[(a + b)/2] is extrapolated to intersect c = (a + b)/2. To reduce the extrapolation, an additional point was obtained by tempering for 24 hr at 296°C martensite from as-quenched austenite. Such a treatment is already known to reduce

*The integrated intensity is the same for all peaks in . a given plot.

tetragonality in these martensites [5]. The slope of c vs (a + b)/2 is also of interest; it is approximately -3.3. The reference lattice parameter established in this way is compared to the parameters observed in preaged martensite and their relative differences are

l2-

a-

4-

","

-4-

-8-

Fig. 5b. Martensite 200, 020 pole figure near austenite (110). The li0, zone is vertical and the 001, zone is horizontal. The 200, poles are distinguished from the 020, poles by being about 6 degrees farther from tie llO,%

342

WALL ANDWI~C~ELL:

i= d

A p’

c

TETR~GG~~L~T~

OF

Fe-Vi-Ti

MARTENSITE

HALL

AND

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TETRAGONALITY OF Fe-Ni-Ti >tARTENSITE PREAGING

2.69

TIME

124

IO

30

loll

I

I

t

-A

743 (mini

60

90

I 4.5

0 0 Prcoqtd

2.80

0.5

min

6.5

%Ti

. .

200 020 002

4

BROAD PEAK

n

-A

_=r::::::::fi;

t

A

,I

!I

dA

“,

08

Y8

n

1

0 2.86

a

(A)

NARROW PEAK

I

I 4 JPREAGING

2.07 or

I 2

-

I

t

,

6

8

IO

TIME

(min)‘JZ

b (0) (i)

Fig. 7. Variation of “c” with “a” and “b” for Z at.“; Ti martensite. The c = a line is also plotted. AQ means as quenched and tempering is at 296°C.

called EL1,c12.es3 for the “a”, “b”. and “c” directions respectively. These so-called strains are shown in Fig. 8. This indicates “b” is larger than “a” by about 0.15’;. an amount which is independent of preaging treatment. Clearly “c” increases and .‘a” and “b” decrease rapidly with preaging time. These changes occur in martensite formed from austenite whose aging produces no satellites. The peak profiles for the 4.5 and 6.5 at.% Ti alloys are richer in information. These austenites do form satellites during aging and subsequently formed martensite exhibits more complicated diffraction peaks. Two different effects are most clearly shown in the 002 profiles. The shorter preaging times result in a rapid shift of a sharp peak. More prolonged aging

PREAGING

TIME

(min)

633

002

Fig. 9. Width of the ZOO,020 and 001 peaks in 4.5 and 6.5% Ti alloys. The broad peak and the narrow peak are each approximated by a Pearson VII distribution with exponent 3 in accomplishing the separation.

a much broader peak to develop at higher angles and be superimposed on the sharp peak. More prolonged preaging causes the sharp peak to decrease in intensity and the broad peak is present alone. These two peaks, approximated as Pearson Type VII (m = 3) distributions. were separated and the resulting breadths are shown in Fig. 9. The breadth of the narrow peak is 0.7’8 and does not change during aging and that of the broad peak decreases with increased preaging from 2.5 to 2.0’0 and the 003 peaks are about 10% broader than the 200 or 020. The fraction of the intensity converted from the narrow peak to the broad peak increases with preaging time with a rise time of about 5.7 min as shown in Fig. 10. causes

PREAGING

TIME

JPREAGING

TIME

(min)

0.01 (.I 1, c_

r

w

o\ypT-L -0.01 0

E2*020 I 2

I 4 JPREAGING

I 6

I 6 TIME

(A) I 1. IO’

20

hi”)“2

Fig. 5. Average strains in martensite with 2 at.% Ti as a function preaging austenite at 7 1O’C.

(min)

‘I

2

Fig. 10. The diffracted intensity converted to the broad martensite peak by preaging the parent austenite in 4.5 and 6.57; Ti alloys.

HALL

744

.WD

PREAGiNG

WINCHELL:

TIME

TETRAGONALITY

OF Fe-Ni-Ti

tminf

MARTENSITE

PREAGING 10

124

TIME

lmin)

~,03O~~~.96

2.94 0.02 r: w

rQ 2.92 u I*1 200

H

0.01

#

4.5 % Ti

PEAK MAXIMUM (0) (A) 020

2.90 -0.03

w

0

--I 2.80

2

4

6

JPREAG~NG TIME 4

2

0

6

JPREAGING

8

IO

(min)‘+

TIME

Fig. 11. The change in position of the 002 peaks in 4.5% Ti martensite formed from austenite preaged at 710°C as shown. AQ means as quenched in liquid metal. The observed peak position and the resolved broad and narrow peak positions are a11shown. The peak position of the sharp peaks changes more rapidly. The lattice parameter of the reference cubic martensite is determined by extrapolation aided by measurements of the quenched and tempered martensite as was previously demonstrated for the 2% Ti alloy. (The AcjAa values determined from these extrapolations are -3.1 for 4.5% Ti and -4.0 for the 6.17; Ti.) The 002 peaks are most clearly resoived into sharp and broad components. The strains corresponding to the positions of the .individual 002 peaks relative to those of the reference martensite are plotted as es3 in Figs. 11 and 12. The narrow peak dis-

PREAGING

TIM& (mid

Fig. 13. The change in position of the observed 200 and 020 peaks in 4.5% Ti martensite preaged at 710°C as shown. AQ means as quenched in liquid metal. The resolved broad and narrow peak positions are not shown but they nearly coincide with those plotted.

placement indicates a rapid increase in strain with rise time of 1.4 min for the 4.5% Ti and 0.1 min for the 6.5% Ti. In both alloys before a subsequent maximum strain is obtained in material contributing to the narrow peak. a broad peak arises at a lower strain from some of the ma~ensite, and change in position of the broad peak corresponds to a slight increase in strain followed by a more significant decrease. The positions of the maximum of the compound experimental peak are also shown in Figs. 11 and 12. Since for the 200 and 020 peaks the broad and narrow peaks have nearly the same positions, the separation of their broad and narrow peaks is both less certain and less important than is that separation for the 002. The relative shift in the experimental 200 and 020 peak maxima are shown in Figs. 13 and 14 for the

2.98

0.03

10 f mi&

PREAGlNG

TIME

fminf

2.96 2.86 2.94

”

0.02

4z

" 2.92

PEAK MAXIMUM @I

0

0

I

I

I

2

4

6

JPREAGING

TIME

6.5 % Ti 2 8

(A) 020 2.88 IO

(rni”l~~

Fig. 12. The change in position of the 002 peaks in 6.5% Ti martensite formed from austenite preaged at 710°C as shown. AQ means as quenched in liquid metal. RQ means brine quenched. The observed peak position and the resolved broad and narrow peak positions are all shown.

-0.03* 0

1 2

I 4

JPREAGING

1 6 TIME

2.80 I 8

10

(min)‘$

Fig. 14. The change in position of the observed 200 and 020 peaks in 6.5% Ti martensite preaged at 710°C as shown. AQ means as quenched in liquid metal. BQ means brine quenched The broad and narrow peak positions are not shown but they nearly coincide with those plotted

HALL

AND

WINCHELL:

TETRAGONALIN

Table 1. Approximate times of preaging for changes in experimental parameters AllO)

Austenite

Broad peak

200, 010

002

at.“‘, Ti

satellite

conversion

shift

shift

3 1.5 6.5

none

none 57* min 5.7* min

3.9 min

0.7 min

9.3 min

2.8

1.1 0.1

12.1

0.7

* Measured together. J.5 and 6.5% Ti alloys. The initial decrease occurs with a fall time of about 2.8 min for the 4.5”/, Ti

and OS min for the 6.5% Ti alloy. In almost all cases “b” is larger than “a” and the average relative excess is 0.2”, for the 4.57; Ti and 0.1% for the 6.5% Ti alloy. Mthough (b - a)/a changes from measurement to measurement (and in an occasional measurement becomes negative perhaps because of improper identification due to excessive austenite crystal substructure). their changes are not correlated with the preaging or post-aging (tempering) treatments which are effective in changing the individual “a”, “b” and “c” parameters. In summary the martensite lattice parameters as indicated by 200. 020 and CO2 peak positions shift fairly rapidly towards increased tetragonality. Decreased tetragonality and line broadening occurs later at a time roughly comparable to that requirement for satellite formation in the parent austenite. The various experimental times are summarized in Table 2 for ease of comparison.

DISCUSSION

OF RESULTS

The general pattern of the correlation of preaging of austenite and the tetragonality of its martensite is clear directly from the experimental observations. At short preaging times and even during the initial quench from the austenitizing temperature, changes occur in the austenite; these changes cause peak shifts producing tetragonality in the martensite. This occurs in all three titanium levels. Only in the two richer Ti alloys and only after substantial preaging times, satellite peaks start to appear in the austenite diffraction patterns and markedly broadened peaks start to appear in the martensite diffraction patterns. Continued preaging results in continued increase in austenite satellite intensity and conversion of the narrow martensite peaks to broad peaks whose location indicates reduced tetragonality. Even longer preaging

* .4ctually a small increase in peak width is observed in the Zoo Ti alloy and a similar increase may also be present in the other alloys but may be hidden by the breadth and asymmetry of the as quenched peaks. These peaks are shifting rapidly and they may have contributed intensity from various parts of the specimen whose quenching rates differ slightly. +-This expression is not very sensitive to AG/kras can be.readily seen by taking as zero whence r = 1.7 rn.

OF Fe-Ni-Ti

MARTENSITE

715

results in further reduction in martensite tetragonality and some sharpening of the broad peaks, but after 90 minutes preaging the broad peaks still have three times the width of the narrow shifted peaks observed at short preaging times. The changes at short times are in the martensite diffraction peaks and involve their rapid shift toward increased tetragonality with little broadening.* As a result of preaging treatments in this time range the austenite peaks are not noticeably changed. The time required for the 002 shift appears to be less than that for the 200,020 shift. In all three alloys the magnitude of the 002 peak shift is about twice as large as that of the 200, 020 shift, but as the Ti content increases a small increase in the maximum 002 shift relative to the 200, 020 shifts may be present. In the two higher Ti alloys the maximum shift may be determined by the competition between development of tetragonality and its depletion evidenced by the lower strain, broad peaks. In the 2% Ti alloy no such depletion is observed, and maximum tetragonality is maintained after long preaging. We hypothesize that “centers of tetragonality” are formed rapidly and then depleted by a second process. In considering the nature of the centers of tetragonality, we compare the observed shift times with the mean time of stay of a Ni atom. tD, of about 0.20 min [14]. Thus the formation of the centers which support tetragonality during preaging at 7 IO’C occurs in the time of 5-40 atomic jumps of nickel. If a single tetrahedral NisTi atom cluster were such a center of tetragonality, a first order analysis [lj] of the kinetics of cluster formation at fixed (non-diffusing) Ti atoms yields for the time required: 5 = 12/7r,[l

- e-“XTi(l

- e-l’ “r)]-‘,

where Xn is the atom fraction Ti and AG is the Ni-Ti binding energy. This expression becomes r z 5rD for the 2 at.“/, Ti alloy with AG/kT = 0.6 at 7lO’C (AG = 0.085 eV) which is our best estimate based on other data as indicated 1ater.t This rise time is close to that for the 2% Ti 002 peak shift in Table 2. Thus the peak shift is due to very small solute clustersperhaps Ni,Ti having only one Ti atom. The magnitude of the peak shifts for such clusters can also be estimated and compared to the measured values. Appendix A contains a calculation of the magnitude of the strains to be expected from a random collection of such centers. This appendix is essentially a real space elastic calculation of the mean strains due to a random collection of centers each approximated by a combination of forces. The combination of forces is selected to elastically distort the four atoms neighboring a tetrahedral site in martensite to their arrangement when they neighbored the corresponding tetrahedral site in austenite. The accumulated strain due to a random collection of such defects distributed throughout a finite body is calculated using the approximation that they are double forces with moment. The method follows the procedure of

716

HALL

AND

WINCHELL:

TETRAGONALITY

OF

Fe-Ni-Ti

MARTENSITE

vention is used. For a body with a residual stress Erhalby [I63 as developed and refined by Townsend [ 17, IS]. Eshalby [ 163 developed the basic pro- distribution oij in the centers and ~ii in the matrix, both energies must be added. A simplification can cedure and applied it to centers of dilation, and be made because experimentally (eii)’ < efl or lil or Townsend [ 171 showed how it could be generalized to double forces without moment. Guy [ 191 showed ei3 so that the second term of the right hand side that collections of opposed moments will not contrib of equation (2) can be omitted. Let v’ = volume fracute to the long range strain, Finally Townsend [ 181 tion centers and (I - L”) that of the matrix. Equilib pointed out that the near defect strains must be calcu- rium requires V’c’,, = -(l - V’) EYE and since lated from the single forces without the double force E,? = lt3 = lz3 must be assumed to be zero, approximation. E ” Essentially two features in this calculation are critiw= E;; + 6;;). 2(1 + v) V’ cal to the result: first, the elastic model of the defect as a tetrahedron of 4 atoms which remains rigid durUsing E = 16.7 x 10” N/m’ and F = 0.35 [ZO], for ing the Bain Strain applied to the surrounding a 4% Ti alloy with the strains observed after five matrix; second, the saturation density of these centers minutes; e’it z li2 2 - 0.012; ei3 f 0.028, one calwhich is essentially controlled by the Ti atom content culates 2200 J/mole (520 Cal/mole) in elastic energy. and the Ni-Ti binding energy. The first feature, the This is a high value and is an overestimate because elastic model of the defect is most sensitively tested the strains in the defects are certainly too large to by comparing the predicted ratio of Ac/Aa to that correctly use the low strain elastic moduli. However, experimentally observed at short aging times. The the calculation does show that ample energy is inpredicted value from equation (A.12) is -3.9 for volved to drop the M, by 100°C since that requires AG/kTvery large (near perfect short range order) and only about 500 J/mole [21]. IS -2.8 for AG/kT= 0.6. The experimental values for The subsequent increase of the M, for preaging the three alloys are -3.3, -3.1 and -4.0 in order times of more than 5 min appears to be coincidental of increasing titanium content. Such agreement for with and hence correlated with the decrease in tetrathe ratio of the axial strains is not unsatisfactory for gonality accompanying satellite growth and the develthis kind of calculation. opment of the broad martensite peaks. The processes The saturation strain levels are most sensitive to going on in this stage will be discussed next. the strengths of the centers, which are modeled as In contrast to the rapid shift of narrow martensite noted above and detailed in Appendix A, and the peaks which occurs at short preaging times, at longer strength of the Ni-Ti binding, i.e. AC/k?: If preaging times, satellites appear in the austenite and AG/kT= 0.6 is chosen at 71O”C, the observed satubroad martensite peaks appear with the narrow ration es3 strain level for the 2% Ti alloy is matched peaks. During more extended preaging, the satellites by the model. The other alloys do not exhibit a satugrow in intensity to a saturation level and the broad ration strain level and a satisfactory calculation canmartensite peaks grow rapidly at the expense of the not be made for them, but their binding energy must narrow ones. Still further preaging results in shifting be significantly higher than the 0.6 kT used for the the broad peaks toward lower tetragonality. None of 2% alloys. Values of kT to 2kT will allow the maxithe changes considered in this discussion is observed mum strain values which were observed in the 4.5% in the 2 at.% Ti alloy. Ti and 6.5% Ti alloys. These higher binding energies The experimental measurements of the austenite may well mean that several Ti atoms are involved satellites, in particular their location in reciprocal in one effective center in these alloys. Such a grouping space, their separation from the austenite peak, their is likely since, as will be discussed presently, precipishape. and their intensity. characterize the lattice pertation of Ni,Ti occurs in these alloys at longer times. turbations which cause them. Firstly, the satellites are Another interesting feature of this stage of the radial in reciprocal space, thus the perturbation of preaging process is the marked depression in M, the austenite lattice which causes them extends in observedC7-J. Preaging a 4 at.% Ti alloy at 71O’C three dimensions, i.e. it forms a ball [22,12]. for times up to 5 mm has been observed to decrease Secondly, the separation of the satellites from the the M, by as much as about 100°C. Further aging Bragg peak shows the repeat distance of perturbation, results in a rapid increase in M, to above its initial Q, in the initial balls is about 15 unit cells. Thirdly, (not preaged) value. The reason for this M, depression the breadth of the satellite peak itself characterizes is here postulated to be the extraordinary strain the number of repeating perturbations, K within the energy in the martensite containing defects. This ball according to [23,24] energy is estimated using isotropic elasticity as follows: Q V = i/fli COS e

--I-c(&

w = kEijcrij =

E ---EijCij

2(1 + v)

+ &&)‘,

(2)

where uij is the stress, eij is the strain, v is Poisson’s ratio and E is Young’s modulus. The summation con-

+

where i. is the X-ray wavelength, 6’ the Bragg angle, and /3i the integral satellite width. For all the satellites observed V= 1.0 _+ 0.07 and the variation from one treatment to another is not systematic. Fourthly. the

HALL AXD WINCHELL:

TETRAGGNALITY

ratio of the intensity in the satellite peaks to that in the Bragg peak yields the volume fraction occupied by the satellites [U, 151. and these numbers reach eight and twelve percent for the two higher titanium aiioys. In summary, these results show that although they are absent in as-quenched austenite, ball-shaped lattice perturbations form at initial size of 15 unit cells. They increase in number and size until they occupy S-12 percent of the volume. At longer times than those reported here, the first precipitates have been observed by transmission eiectron microscopy as spheres with the structure of “coherent” 7’ presumed to be Ni,Ti [S, 61. As a consequence, we suppose that the satellite measurements monitor the formation and growth of 7’ precipitate particles in the austenite matrix. By examining the coincident diffraction effects in the martensite, we also conclude these precipitates greatly affect the defect structure of the martensite. Few if any such precipitates are expected in the 2% Ti alloy because this austenite is probably not supersaturated at 7lO’C [Z]. In agreement with this, no satellites were observed in the 204 Ti alloy in spite of a careful search. In martensite formed from austenite containing such precipitates a very broad diffraction peak is always observed. This broad peak is developed in the presence of the narrow, high-tetragonality peak. The simultaneous appearance of both peaks means the precipitate process causing the broad peak is inhomogeneous. The broad 002 peak maximum is always shifted from that of the narrow peak toward fewer tetragonality. Hence the precipitation and associated peak broadening involves a depletion of the centers supporting tetragonality. The centers are evidently still forming because initialiy the broad peak shifts toward increased tetmgonality slightfy; however, with continued preaging it reverses this shift and the tetragonality producing centers are apparently substantially depleted. The change in the 200 and 020 peak positions are similar but the broad and narrow peaks have neariy coincident maxima. An explanation is that the precipitates affect the “a” and “b” parameters nearly as much as do the centers. This may account for the 4.5% Ti and 6.5% Ti alloys fall time of er, and cZ2 is longer than the rise time of ea3 in Table 2, but it does not account for the analogous observation in the 2:; Ti martensites! The particles forming in austenite during the preaging process are less effective in supporting tetragonality of the subsequently formed martensite but are much more effective in broadening the martensite diffraction peaks. Those two observations have the common explanation that these particles “tear loose” and become incoherent during martensitic transfo~ation and in the course of the transformation strain cause a very large amount of substructure in the martensite. However, because of their loss of coherence, they are unable to support tetragonality as effectively as the small coherently transformed centers hypothesized

OF Fe-Ni-Ti

MARTENSITE

717

previously. These small centers evidently produce no X-ray peak broadening and have no substructure. The loss or partial loss of coherency is essential to the reduction in tetragonality because according to elasticity theory no first order change in the average elastic strain should occur just due to a clustering of the centers causing the strain. This relaxation of the elastic strains also results in a large decrease in strain energy and the initial M, increases. Continued M, increase to temperatures higher than the as-quenched M, may be due in part to alloy depletion.

Thetetragonality of Fe-Ni-Ti martensite is supported by centers consisting of a very small number of solute atoms. These centers are formed during the quench from the austenitizing tem~rature (even at 10soC/s) and during short aging times. The centers are successfully modeled as tetrahedral NiJTi clusters which retain their regular tetrahedral shape during martensitic transformation. At longer aging times in alloys which are supersaturated at the aging temperature, coherent precipitates (or preprecipitates) are nucleated and grow. These are not as effective in supporting martensite tetragonality because they become incoherent during the martensitic transformation, but they are very effective in producing martensite substructure which causes marked X-ray line broadening in martensite. The pre~pitation (or preprecjpitation) process is inhomogeneously distributed in the martensite. The precipitates are postulated to be Ni3Ti 7’. Acknowledgements-This work was supported by the National Science Foundation under Grant GH 40510 and was initiated under ARPA IDL Program DTHC-0213 and NSF MRL Program GH 33574. Special thanks are due Dr. H. Harrison of Purdue Universitv* Crystal Growth . Laboratory.

REFEREiiCES I. P. G. WinchelI and G. R. Speich, iteta &ler. 18, 53 (1970). 2. J. W. Cahn and W. Rosenberg, Scriptn Mer. 5. IOI

(1971). 3. J. W. Cahn, Personal communication. 4. Y. Honnorat, G. Henry and J. Manenc, &fPm. scient. Revue MkraX LXIE. No. 6, 429 (1965). 5. Y. Honnorat, G. Henry, G. Mumy and J. &nenc, C. r. Acad. Sci. Paris. 260. Gr. 7, 2214 (1964). 6. J. K. Abraham, J. K. Jackson and L. Leonard. Frorrs. Am. Sot. iMetals 61, 233 (1968). 7. J. K. Abraham and J. S. Pascover. Trans. rll.CIE 245. 759 (1969). 8. R. E. Miner, Mernll. Trans. 2, 1259 (1971). 9. M. M. Hall, &Zerail. Trilns. 5, 955 (1974). 10. A. K. Sachdev and P. G. Winchell, Met&i. Tmns. (A) 6, 59 (1975). Ii. V. G. Veeraraghaven and P. G. Winchell. Metafi. Trans. (A) 6. 701 (1975). 12. W. Dorfield and V. Phillips. Actn Met. 18, 955 (1970). 13. V. Daniel and H. Lipson. Proc.R. Sot. (
745

HALL ASR WINCHELL:

TETRAGONALITY

14. C. J. Smith&s. Meruls Reference Book. 4th Edn., Vol.

II. D. 649. Butterworth, London (196%. \ M. M. Hall. PhD. Thesis. p. 140. Purdue University, Lafayette, IN. iMay 1975. J. D. Eshalby. Solid Srate Physics, Vol. 3, p. 79. Academic Press, New York (1956). J. R. Townsend, Acta .Ciet. 13, 325 (1965). J. R. Townsend, Phys. Rec. (B) 9, 4ooO (1974). P. M. Guy. Sf. S. Dissertation June 1971, School of Materials Engineering, Purdue University, Lafayette, IN. G. R, Speich and W. C. Leslie, detain. Trans. 4, 1873 (1973). M. M. Rao. R. J. Russell and P. G. Winchell, Trans. metal/. Sot. AIME 239, 634 (1967). W. Cochran and G. Rartha, Acta cr,vstallogr. 9, 941 (1956). M. Hillert, Se. D. Thesis, MIT, 1956. B. Warren, X-Ray D~~actio~, p. 253. Addison Wesley, Reading (1969). G. R. Speich, Trans. meraIl. Sot. AlME 227, 754 (1963). A. E. H. Love. The Mathematical Theory of Elasticity, 4th Edn. Dover, New York (1944). D. T. Keating and A. N. Goland, Acta Met. 15, 1805 (19671. I

is 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27.

APPENDlX A Tetragorral Lattice Distorrion of Fe-Ni-Ti Martensite by Subsritutiomd Solure Atom Clusrers Small clusters of substitutional atoms containing a few atoms each may act as centers of anisotropic strain which distort the crystal lattice. The total lattice strain due to a random collection of such strain centers is simpty the sum of the strains due to individual atom clusters and has strain components cij given by

The sum is carried out over alt possible orientations of the strain centers: n(” is the number per unit volume and S$) is the strenPth* of centers with the rth orientation. A less than cubs lattice distortion can result when there is a non-uniform distribution of the numbers of strain centers among the possible orientations. Shown in Fig. 15 is a four atom cluster of three nickel

OF Fe-Ni-Ti

MARTENSITE

and one titanium atoms which has been proposed by Wincbell and Speich [I] to support the tetragonal distortion of body centered Fe-Ni-Ti martensite. These atoms are assumed to retain their close packed regular tetrahedron arrangement during the austenite to martensite transformation and consequently are displaced from the average position in the body centered martensite lattice. Strong chemical interaction amone the Ti and Ni atoms is required to maintain this &placement which also displaces surrounding lattice atoms from their equilibrium positions. The number of clusters that may be formed is proportional to the Ti atom fraction, XTi, and Fig. 15 shows that there are three possible orientations for tetrahedral clusters. Then equation (Al) may be rewritten as

64.2) where Vis atomic volume andf”’ is the expected number of clusters with the slth orientation per Ti atom. The functionsp) are determined by the local arrangement of Ni and Ti atoms brought about by aging of the austenite prior to transformation to martensite. In order to arrive at the form of these functions, each Ti atom is assumed to act as a center for Ni atom clustering during aging of the austenite; the Ti atoms are assumed to be fixed Inon-diffusing) and to be randomlv distributed. The functibns X, and z2 are introduced to bescribe the probability that a Ni atom is a first or second nearest austenite neighbor to a Ti atom respectively. For dilute solute atom concentrations the expected number of K,Ti clusters per Ti atom with [OOt], orientation is f3’ r sx:,

(A.31

since at each Ti site there are eight ways of forming clusters with [OOl&, orientation and since the probability of forming the required triangle of Ni atoms is X:. This result is illustrated in Fig. I6 which shows one of the eight Ni,Ti clusters having [OOIJu orientation. Tne remaining seven clusters having @O1]u o~entation have as their centers the martensite lattice sites shown by i in the figure. The functionsj’” andy*’ represent the expected number of defects per Ti atom with [lOOI, and [OIO],, orientation, respectively, and are given by fit) =f(Z) =4x:

+ 4x: x1.

(A.41

Figure 17 shows that of the eight ways to form clusters with [lOO], orientation, four require triangles of Ni atoms whose probability of formation is XT and four require Ni triangles whose probability is XI X1. The functiohs $1’ represent the components of the lattice strain due to a single Ni,z cluster in a unit volume having the zth orientation. Symmetry considerations require that SC,‘:= s’?zz’ = St::, -_I...--

(ASa)

ryr M

Fig. 15. Tetrahedral four atom Ni,Ti cluster proposed to support Fe-Ni-Ti lattice distortions. The three possible orientations are indicated by the three 4-foId rotary inversion axes. *By F$’ is meant . the change in scj when one z defect . is Introduced. per umt volume.

Fig. 16. Ni,Ti cluster having [OOllM orientation inscribed ^ in T.c.c.lattice of austenite. At the Ti atom (0) eight . . clusters . 1,. are possible with LOOI], oT]entation as marLeO ny tif.

HALL

AUD

WINCHELL:

TEYRAGONALITY

OF Fe-Ni-Ti

749

MARTENSfTE

and Xz(tjr) = xVi[l

-i($)(l

- e-‘;)I,

(A.7b)

where _ = 1 _ eeLzXT’(l- e-“IrT).

(AS)

In these equations, XNi and XTi are the average nickel and titanium concentrations and T is a charactenstic time for the clustering process which is related to the time for nickel diffusion. rn. by:

(a)

(A.9) If the strains per individual tetrahedral Ni,Ti atom cluster per unit volume, Si, are known, the saturation strain values may be obtained from equations (A.6) by substituting for the functions X, and X2 their values evaluated for t p r:

(b) Fig. 17. NisTi clusters having [ IOO], orientation inscribed in f.c.c. lattice of austenite. There are eight clusters possible with [loo], orientation as shown. When present, four of the clusters have one of the three Ni atoms as a second neighbor in the austenite to the Ti (0) as shown in (a). The remaining clusters have Ni atoms which are all first neighbors in the austenite to the Ti atom as shown in (b). and that

Then equation (A.2) may be combined (A.3). (A.4) and (AS) to give 8XT, E,,

=

El2

E33

=

=

y

-

with equations

(A. IO)

x2=xNi[l-~~)].

(A.11)

and

For fixed alloy composition, the magnitude of the function I is determined only by the Ni-Ti binding energy AG. Experimentally determined saturation strain values may then be used to fix AG. For times t > 2r the ratio of the axial strains Ac/Aa calculated from equations (A.6) and (A.7) as 3(1 + Ss,/St,) + 2X2/X, - 4/7 3 + (1 + S,JS,,)(X,/X, + 17,‘14) (A.12)

x:

approaches a constant value. This ratio is most sensitive to the strain ratio S33/S,,. A comparison the results given by (A.12) with the slope of an experimental curve of lattice parameter c versus lattice parameter a is therefore a sensitive test of a particular model chosen to calculate the strains due to a single strain center. In order to determine the lattice strains due to a single Ni3Ti cluster per unit volume, Guy [19] proposed an elastic point defect model depicted in Fig. 18. Guy proposed

V

(ZS,, + S,,) + (2

x, = XNii_’

- I)&,]. (A.6b)

‘i where the now superfluous superscripts have been dropped. The lattice distortion due to the Ni,T strain centers is now expressed as a function of the Ti concentration, XTI, the strains due to a single strain center, S, and the degree of solute atom clustering existing in the austenite pnor to the transformation to martensite as expressed by the probability functions X, and X,. The degree of Ni-Ti clustering obtained as a result of austenite aging is most sensitive to the Ni-Ti binding energy AG. A first order analysis of the kinetics of the clustering process [ 15] gives the following expressions for the functions X, and X2 which describe the extent of the clustering and therefore the strain magnitudes in equations (.A.6) above: 1 - (1 - z)e-*”

X,(V)

= xxi

z

1

(A.7a)

-3

t

Fig. 18. Tetrahedral point defect in martensite. The chemical interaction forces on the nearest neighbor atoms are shown.

750

HALL

&?-ID

WINCHELL:

TET’RAGONALITY

that the NisTi cluster of atoms may be modeled by this p3rticuIar arrangement of point forces acting in a continuous and finite elastic medium. This arrangement of force pairs constitutes a pair of double forces with moments. Guy termed the two concentrated moments which are equal in magnitude and opposite in sign a “double-moment,” and showed that there is no average long range strain due to a uniform distribution of double moments. The totaf lattice strain can therefore be considered as being due only to the distribution of double forces. Consider, then, a distribution of n “double force defects” whose force per defect per unit volume is Fd = -Jai,

-Jai,

+ gai,,

(A.13)

where f and g are forces of interaction as shown in Fig. 1s and a is the body centered martensite lattice parameter. According to the procedure of Eshaiby [l&J as developed and refined by Townsend [ 173, these forces give the same long range displacement fields as applying the surface force F, along the surface limiting the defect bearing crystal. For the defect considered F, = - nfa(i, . i,& -n&i2 . i,)i, f nga(i$ * in&, (A.14) where i, is the normal to the crystal surface. Then the components of the long range stress per defect per unit volume, (rlj, inside the crystal containing these defects are Cl1 = -fa;o,,

= -fa; us3 = ga;a,,

OF Fe-Ni-Ti

MARTENSITE

while the magnitude of the atom displacements required to maintain the close packed arrangemennr determines the individual force magnitudes Townsend [ 181 pointed out that calculations for the displacements near a localized defect are improved by considering the displacements due to the single forces without the double force approximation. Followutg the method of Townsend, consider that the forces of interaction of magnitude f and g as shown in Fig. I8 are point forces Lvhich act at the centers of atomic volumes which have radii ‘c 3 a/4 where a is the body centered manensite lattice parameter. If these atomic volumes behave as “hard spheres” undergoing displacements without rotation, the elastic medium at point P(a/2 c L 3 a/4, 0, a/4) undergoes a displacement U such that li, = - (is. Following Love[26] the displacement U of a linear and isotropic elastic continuum at a point R due to a force F acting at the origin is U_

F(Lf3p) i+RF.R(isp) 1 Snp(A + 2~) R Snp(l + ly) F

(A.18)

where i, and p are the Lame constants. Therefore, the displacement of the elastic medium at point P due to the forces acting at each of the atoms 1, 2. 3 and 4 whose coordinates are (a/2, 0, a/4), (0, -a/2, -a/4). (-a 2. 0, a/4) and (0, a/2. -a/4) respectively has components

= at3 = eft = 0. (A.15)

The strain per defect per unit volume or “defect strength” is given by Hooke’s Law for elastic isotropy: and 2 + j3 us = -W.-.-+ 3 + 4fi

Using equations (A.9). (A.16) becomes

x (A .7a) and a S CUf f (f + 33 = 2~(31+ 2~)

&&I~

(A.17b)

where i. and ~1are the Lam6 constants. To determine the forces of interaction fand g, consider as a limiting case that the four atom defect shown in Fig. IS retains the close packed arrangement of the austenite after transformation of the lattice to martensite. The geometry of the transformation imposes conditions on the resulting atom dispIacements which determine the ratio g/f

g ;. -I- 3/l

1

nap j. + 2p

(15 f 445’

’ iraic

[f-g)(~)(~~-g(~j]. (A.19b)

Using Lame constants proposed by Keating and Goland [27] which have the values i. = 1.0784 x 10” dyn/cm’ and p = 0.9023 x 10” dyn/cm’. and requiring U1 = - U3 the ratio g/$is found to be 2.tO. The displacement which is just sufficient to produce a close packed arrangement of the four atoms of the defect is Li, = -0.067 a,,, where a,,, is the martensite lattice parameter. Using the lattice parameter of pure iron a, = 2.8664 A, f = 3.30 x lo-” dyn and g = 7.92 x lo-’ dyn. Equations (2.17) may now be evaiuated to give ssj = 131 .. x IO-“cm3 and S 11 = - 0.569 x lo-‘3cm3.