Theory and experiment of martensitic nucleation in ZrO2 containing ceramics and ferrous alloys

AC/U r,~~~al/. Vol. 33. No. IO. pp. 1827-1845. 1985 Printed in Great Britain. All rights reserved

OOOI-6160/85 [email protected]

Copyright 8 1985 Pergamon Press Ltd

THEORY AND EXPERIMENT OF MARTENSITIC NUCLEATION IN ZrO, CONTAINING CERAMICS AND FERROUS ALLOYS I-WE1 CHEN and Y-H. CHIAO Department of Nuclear Engineering and Department of Materials Science and Engineering, Massachusetts Institute of Technology. Cambridge, MA 02 139, U.S.A. (Received

1I

December

1984; in recised,form

24 March

1985)

Abstract-An in sifu TEM experiment on martensitic nucleation was performed using sub-micron ZrO, particles which were initially defect free. Extrinsic, spontaneous and heterogeneous nucleation was induced by introducing dislocation loops with a strong shear component and Hertzian contact stresses into selected particles. The critical characteristics of these defects which were required to trigger barrierless nucleation, in terms of the loop radius, Burgers vector, contact stress, and contact area were in good agreement with an analytic elasticity model. Along with dislocation walls, these defects which have a short range stress field can also account for intrinsic nucleation in ZrO, containing ceramics and ferrous alloys. The essential features of nucleating defects, which maybe classified into intrinsic or extrinsic ones depending on their origins, and short range or long range ones depending on their stress fields, are analyxed and contrasted in view of their experimental evidence. R&um&Nous avons elfectue une exptrience in siru par MET sur la germination de la martensite en utilisant des particules de ZrO, de taille inl&ieure au pm, initialement sans defauts. Nous avons induit la germination extrin&que, sponta& et h&t&g&e en introduisant des boucles de dislocation avec une forte composante de cisaillement et des contraintes de contact de Hertz dans des partlcules choisies. Les caract&stiques critiques des dCfauts n&cessaires pour declencher une germination sans bar&e, en tetmes de rayon des boucles. de vecteur de Burgers, de contrainte de contact et de surface de contact, tient en bon accord avec un mod&le Clastique analytique. Avec des parois de dislocations ces dCfauts qui ont un champ de contrainte a faibli port&e peuvent Cgalement rendre compte de la germination intrinsbqte darts des &amlquea et des alliages ferreux contenant de la x&one. Les caract&tlques principales des dCfauts de germination, que Ton peut classer en intrin&ques ou extrinsZques selon kurs otiglna, et en dtfauts I courte ou a longne port& selon leurs champs de contrainte, peuvent &t-e analys&s et distlngu6es exp&imentalement. Zaaammatfawmql-An anfangs defektfrcien ZrO,-Teilchen mit GriiBen unterhalb von I pm wurde die marten&is&n Kelmbildtmg im Durchstrahlungselektronenmikroskop in-situ untersucht. In einxelnen Teilchen wurde exttinsischa, spontane und heterogene Keimbildung erzeugt, indent Versetaungscbleifen mit starker Scberkomponente und Hettxsche Kontaktspannung en eingefiihrt wurden. Die fur eine Keimbildung ohne Schwelle, d.h. ohne Beriicksichtigung von Schleifenradius, Burgemvektor, Kontaktspannung und Kontaktlliiche. kritlschen EigenschaRen dieser Defekte stimmen mit einem analytischen Elastixitlitsmodell gut i&rein. l&se Defekte, die wie Versetxnngswiinde ein kutzrelchendes Spannungsfeld auhveisen, k&men such die intrinsische Keimbildung in ZrO, in Keramiken und Eisenlegierungen erklfuen. Die wesentlichen BlgenschaRen der die Keimbildung firdemden Defektc, die je nach Ursprung in intrinsische und extrinsische und je nach Spannungsfeld in solche mit kuner oder weiter Wechselwirkung eingeteilt werden kennen, werden anhand der experimentellen Ergebnisse analysiert.

1. B’lTRODUCTlON

The concept of heterogeneous martensitic nucleation is best illustrated by Fig. 1 which was originally due to Magee [I]. In this figure, it is shown that a martensitic nucleus, when formed homogeneously, inevitably runs into a very high nucleation barrier, AG*. However, if nucleation takes place near a defect, the elastic interaction between the shape strain of the nucleus and the stress field of the defect could result in a considerable lowering of the defect elastic energy, AC;. It is theoretically possible that, under certain circumstances, nucleation becomes barrierless because of the overall decrease of the total free energy of the system. Martensitic nucleations in Fe-Ni [2]

and ZrO, [3] are believed to proceed in the above manner. More specifically, three conditions have to be met before barrierless nucleation can realize. First, the coupling between the defect stress field and the transformation shape strain, which is dominated by shear, must be of an appropriate type to lead to a negative elastic interaction energy. Second, the magnitude of AC: must be sufficiently large to compensate AC* at the critical point. Third, the range of the defect stress field must be sufficiently long to cover a distance which is comparable to the critical size in homogeneous nucleation. Although these criteria will be further quantified later in this paper, we here show qualitatively, in Fig. 2, that a weak or a short ranged

182-f

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and

CHIAO:

MARTENSITIC NUCLEATION IN 210,

required for barrierless nucleation. In-situ TEM techniques were then developed to generate these defects in small and initially defect-free ZQ particles. As being observed, these particles suddenly underwent spontaneous martensitic transformation on the forbetween defect ond mation of certain critical nucleating defects. ThereEnergy fore, both the experiment and the theory demonstrated barrierless, heterogeneous nucleation. A Hetrogeneous nucleotion direct comparison betwen them furnishes quanAG M3l \ titative estimates of the critical characteristics of nucleating defects. F’reviously, heterogeneous, barrierless martensitic nucleation has been rationalized on the basis of a defect model due to Olson and Cohen [2]. Direct TEM observation of the nucleating defect envisioned Homogeneous nucleation in this model, however, has never been made, although numerous microscopy studies have already provided evidence which may be regarded as consis\ tent with and thus supportive of their model [ 1,4-71. We believe the electron microscopy observations Fig. 1. Schematicnucleusenergy curves for homogeneous nucleation with a barrier snd heterogeneousnucleation reported here are the fkst convincing proof of heterowithout a barrier. The latter is facilitated by a decreasing geneous, barrierless nucleation of martensitic transintcraqtion energy. formation to date. Such direct proof was possible because we employed very small particles, which, defect may not bring about barrierless nucleation. statistically, were free of any pre-existing nucleating Certain critical characteristics are thus required for defects. Thus the subsequent spontaneous transformation of these particles, on the introduction of defects to operate as nucleation sites. The main purpose of the present paper is twofold; certain characteristic defects, was unequivocally hetfirst, to experimentally provide direct observations of erogeneous in nature. Already, based on a statistical barrierless marten&k transformation nucleated at study of martensitic nucleation, we were able to draw, heterogeneities; second, to theoretically analyze the in an accompanying paper, an essential distinction critical characteristics which enable a specific defect between intrinsic and extrinsic marten&k nucleto act as a nucleating site. To make contact between ation, the former being one caused by pre-existing our theoretical and experimental findings, we place nucleating defects, while the latter was not [8]. We major focus on a simple-shear martensitic trans- now hold the view that the experimental observations formation as represented by the tetragonal to mono- reported here are examples of extrinsic nucleation, clinic transformation in ZrO,. Two types of defects, while the Olson-Cohen model and its corroborating a shear dislocation loop and a Hertzian contact, were experimental evidence are examples of intrinsic nuclefirst analyzed to obtain the critical characteristics ation. Although intrinsic nucleation is probably the

Energy

Energy

(01 Weok defect Fig. 2. Heterogeneous

nucleation

( b) Short-ranged

defect

with a barrier at (a) a weak defect, (b) a short range defect.

CHEN and CHIAO:

MARTENSITIC

NUCLEATION

dominant cause of martensitic transformation, especially in bulk materials due to pre-existing nucleating defects. it is extrinsic nucleation which must be sought to rigorously establish the proof of heterogeneous nucleation, as we did in the present study. The detailed characteristics of the nucleating defects which are responsible for intrinsic nucleation are somewhat different from the ones that we observed experimentally for extrinsic nucleation. In addition, we recall several previous studies of strain-induced

nucleation at tem~ratur~ above the normal Iw, temperature, which gave evidence of extrinsic nucfeation at defects generated by dislocation pile-ups at various geometrical discontinuities (9-141. Thus a broad spectrum of nucleating defects is theoretically and experimentally possible. In view of these possibilities, we also analyzed the defect characteristics of the Olson-Cohen type and the dislocation pile-ups, and compared them, broadly, with the experimental observations reported in the martensite literature. The major distinction between different kinds of defects, which became obvious from our theoretical analysis of their nu~~ting characteristics, is their range of the stress fietd. This mechanical ~stin~on is complementary to the statistica distinction mentioned earlier [8]. As a result of these deliberations, we attempt, at the end of this paper, a categorization of the major nucleating defects in terms of their ranges and their origins. A more comprehensive picture of martensitic nucleation hopefully will emerge from these viewpoints. The organization of the remainder of this paper is the following: In Section 2, we analyze two types of short-range defects, i.e. dislocation shear loops and Hertzian contacts. In Section 3, we analyze two types of long-range defects, i.e. dislocation walls and dislocation pile-ups. In Section 4, we report our in situ experiments on Zr02 and compare them with the theory of Sections 2 and 3. In Section 5, we further review the microscopy evidence of various nucleating defects by making distinctions on their stress fields and their origins. We finally categorize them along these lines.

2. NUCLEATINGDI%lXCR3 WITH A SHORT RANGESX’Rm FIELD 2.1. Shear d~~~catio~loop Theory of homogeneous martensitic nucleation predicts a thin disc shape for the martensitic nucleus at the critical stage [IS, 161. If the shape transformation strain is of the pure-shear type, representative of the tetragonal to monoclinic transformation in ZrOz, the shear distortion must lie parallel to the disc plane. To conform with the above configuration, our first example of the nucleating defect is thus chosen to be a shear dislocation loop. Heterogeneous nucfeation of a concentric coplanar martensitic disc therefrom is now analyzed.

IN ZrO,

1829

ST

w

ST 4

Fig. 3. Heterogeneous

nucleation at a shear dislocation loop.

In Fig. 3, the configuration of the nucleus and the loop is shown. Here, r, is the loop radius, b,, its Burgers vector, c the disk thickness and a the disc radius. The chemical driving force of martensitic t~sfo~ation is Ag& per unit volume, the transformation shape strain is Ss the in~~~a1 enerj$es of the disc surface and the disc edge are y ,, and yI respectively. Growth is along the radial direction at a constant disck thickness, c, which wili be later varied to obtain the optimal value that minimizes the nucleation barrier. Consider only the radial growth of the disc while letting thickness be a constant. The following equation describes the increase of the total free energy in this process

In the above, the left-hand side of the energy equation is cast in a form such that it can be readily interpreted as the equivalent stress exerted on the radial martensitic interface, from the combined action of chemical driving force, surface tensions, self stress due to the self energy arising from the shape strain, and the interaction between the disc and the loop. Barrierfess nucleation requires a negative equivalent stress at all radii. Frictional resistance to the motion of martensitic interface, if important, may also be incorporated as a positive term on the righthand side. The self stress due to the shape strain, a, can be evaluated by a similar radial differentiation of the elastic strain energy of the thin disc. We found the strain energy of the disc to be Knac2, where K

3x 2-v =jj l_v C

>

rs:.

Here p is the shear modulus and v the Poisson’s ratio. In the above, we used the Eshefby solution [I71 of an ellipsoidal thin disc of the same volume, as an approximation. By diffe~ntiation, we obtain the self

1830

CHEN and CHIAO:

MARTENSITJC

NUCLEATION

IN ZrO,

stress Kc

%lf =

z;;’

The strain due to the interaction of the shape strain and the dislocation stress field can be evaluated by a similar radial differentiation of the interaction energy Eb,. Noting that Ein,= -

where uD is the component

u. Srdv

(4)

of the dislocation stress

field aiding the shape strain and the integration is extended over the volume of the disc, we find, by differentiation

Fig. 4. A schematic diagram of the strength of a shear loop, A vs the thickness of a nucleus plate c, with the regime of barrierless nucleation indicated.

where

equation (12). A schematic plot of A vs c is shown in Fig. 4. The minimum A* can be solved analytically 1 2-v *L! G=(6) although its form is slightly too complicated to be 8 ( l-v > Or23 given here. However, since K * Ag&in martensitic is the average dislocation stress, over 6 = 0 to 2n, at transformation, the full analytic solution is simplitled a radial distance 4 which is assumed to be larger than considerably to become r, [I 81. From dislocation theory, uD is an inverse cubic field, hence equation (5) is re-written into the following form

+

_

3125K’ ?!I 729 A&

(13)

at

A CT*,=-

A

(13

(14)

where the amplitude of the u-r interaction is Referring back to the definition of A given by equa-

(8) tion (8), we find that the above condition

A

It is obvious from Figs 1 and 2 that the barrierless condition for nucleation is met if, at the inflection point, i.e.

a ;i;;

(

--

1 aAG

2nuc da

)

=

o

the equivalent stress on the martensitic remains negative

(9)

simply states that a critical amplitude of the dislocation loop field must be attained before the onset of barrierless nucleation in the coniiguration of Fig. 3. Under the same approximation, allowing radial growth at a constant thickness, the minimal (I** at which aG/&a = 0 at A = 0 can be found. These results may be used as the reference case of homogeneous nucleation. They are

interface *

(15)

(10) These parameters are not significantly different from Equation (9) gives simply

112

those of equation (14). Using the values of equation (15), and double-asterisks to denote quantities at the reference coordinate of homogeneous nucleation, it (11) can be readily verified that

where the asterisk is used to denote the inflection point. Substitution of u+ and equation (1) into equation (10) then gives

The critical condition of barrierless nucleation therefore corresponds to the smallest A which still satisfies

46656 2a:5=Agd=3125u::

(16)

at the reference coordinate of the critical nucleus of homogeneous nucleation. The last characteristic will be later used for comparison with other types of nucleating defects. In the above and throughout the present paper, we assume no thickness fluctuation along the nucleation path. Considering the simple shear nature of the

CHEN

and CHIAO:

MARTEN~ITIC

Fig. 5. Surface nucleation under a localized normal stress. transformation and the barrierless option of the nucleation, we believe it to be a reasonable assumption. However, relaxation of this assnmption should lower the critical characteristics required for the nucleating defects. 2.2. Hertzian

contact

The second case we will analyze is the nucleation from a region beneath a semi-infinite flat surface, on which a localized normal stress distribution is applied. Such a case may occur due to surface catalyzation, mismatch of thermal expansion between phases, and, in our experiment, Hertxian contact between two small ZtQ particles. A martensitic plate, with the aid of the normal stress, may nucleate from the surface, leaving a surface kink which will be determined by the transformation shape strain. In Fig. 5, the configuration of the surface region and the nucleus is shown in a side view. The normal stress P(y) on the surface, which extends between Y = t_ h/2, is assumed to be symmetric, i.e. P(j) = P( -y). The marten&i? plate is centered with the stress distribution extending for a width c along j and a length a along 3. We also follow the same nomenclature used in Section 2.1. Consider only the lateral growth of the nucleus in its length, while letting width c be a constant. The following equation describes the increase of the total free energy in this process. 1aAG --= c aa

%I

-Ag,,+c+u,,lf-ui,,G.

NUCLEATION

EN 210,

1831

tion is entirely equivalent to the one using be shape strain its]. The computational task, however, is considerably simplified as shown below. The presence of the free surface at x = 0 necessitates the consideration of image stresses of the dislocation array at x = a. Solution of this problem is remarkably simple, and can be constructed simply by introducing a second set of image dislocations at x = --a, with 6= -S, It should be noted that the resultant normal stress, due to these two sets of dislocation arrays, indeed vanishes at x = 0, but the shear stress does not. The complete solution should thus include another set of surface shear stress distributed symmetrically at x = 0. However, the latter symmetric interfacial shear stress exerts no net force on the dislocation array at x = a, and generates no normal displacement at x = 0. These features can be proved following the procedure of Ref. [ 181.Thus, for our purpose of computing the self stress and the interaction energy, only the two dislocation arrays are significant. The resultant configuration, including the normal displacement, isshown in Fig. 6. The self stress is computed from the mutual attraction between the two dislocation arrays. The computation is relatively strai~tfo~rd, involving double integration, and the result is given beiow “‘, ule’f= 2x(1 -v)

(g)ln(l+-$). c

(18)

The interaction energy of the present problem is the work done by the normal stress, with the normal displacement U(Y) kP I?,, = -

PU dy. s -h/2

(19)

(17)

The above equation is obviously cast in a similar way as equation (1) for the radial growth problem. The right-hand side of equation (17) is again the cquivalent stress exerted on the martensitic interface. The task, now, is to evaluate the self stress and the interaction. For this purpose, we introduce the following procedure which replaces the shape strain within the nucleus by an array of continuously distributed edge dislocations, at x = a between Y = f c/2, with a “Burgers” vector &pointing along -3, of a magnitude 6= 22, per unit distance along the array, Within the context of the theory of elasticity, the above construc-

~U’%ii~tiionr

+%iqid-body

Image shear traction (free surfocel

Fig. 6. A dislocation model of surface nucleation. Transformation stresses are given by continuum dislocations and by interfaeial shear tractions. Only dislocations result in normal displacmmnts and net force on each other. The rigid-body displacement is due to the extemal stress P to keep U (0, h/2) at zero.

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and CHIAO: MARTENSITIC NUCLEATION IN k&O,

As noted above, the normal displacement can be computed from the dislocation field, which, other than a rigid-body translation, is antisymmetric with respect to y. This is obvious from Fig. 6. Since P(Y) is symmetric, only the rigid-body displacement gives net contribution to the interaction energy. To determine the rigid-body displacement, we simply let the point at y = h/2 be fixed, thus

As before, equation (22) gives an a*, where the asterisk is used to denote the inflection point. Substitution of CI*into equation (23) then gives a condition for P. The critical condition of barrierless nucleation is finally determined by the minimal P satisfying equation (23). Due to the rather cumbersome form of CQ~and CJ;,~, an analytic solution of the above problem was not possible. Simplification is thus made since we note that, in most applications, c c h c a. We may therefore approximate the self stress and the interaction by

Differentiation with respect to x = a then gives rise to &4/$x, namely (strain),, to be evaluated at y = h/2, due to both sets of dislocations extended between y = &c/2 at x = * 0. The computation is again straightforward, involving one integration, and the result is given below

After rather involved analytical and numerical computation, the critical condition is tinally reduced to the following form P is the average normal stress over y = f h/2. We are now in a position to obtain the critical condition for barrierless nucleation. As before, it is obvious that the barrierless condition is met if, at the inflection point, i.e.

where F is a function shown in Fig. 7. In the limiting case of a* ti h2 + c2, a,, reduces to the aq2 form. The solution in this case becomes very (22) simple, which is given by

the equivalent stress on the marten&tic interface remains negative laAG
p*h*l=:

27 16n(I-v)(3-2v)x

(PMI)’

(27)

with

(23)

Under the same condition that equation (27) and (28) apply, we find, for homogeneous nucleation c**=;-=4h

Ag*

2x(1 -v)

(2%

At these reference coordinates of the critical nucleus, we obtain

Fig. 7. The critical condition of barrierless surface nucleation under a localized normal stress, of a strength P* as averaged over a distance h+.

The above equation is similar to equation (16). This set of solution corresponds to the branch of small Age*in Fig. 7. At large AgCh,a* approaches h in magnitude and the approximation of ue2 field is not valid. The solution in this case also becomes very simple and

CHEN and CHIAO:

corresponds

MARTENSITIC NUCLEATION IN ZrO,

to the other branch in Fig. 7

P*h*=0.863

--&- ), (v = 0.25). (psry C II .

(31)

It will be noted later that equation (3 1) resembles the result of Olson-Cohen model closely. If the pertinent interaction in the Hertzian contact problem indeed remains a short range one, i.e. if ulnl a a -*, the condition of heterogeneous barrierless nucleation as given by equation (30) is similar to the condition [equation (I 6)] required for the dislocation loop which also has a short range interaction of the type of din, aaT3. The above condition, which essentially states that 2 a:;

= Agch= lOui*,:

(32)

may be regarded as a general one for short range defects, and we propose to use it as a simple criterion to identify operational nucleating defects. It is interesting to note that the above condition is entirely independent of the interfacial energies. While the relation between as and Ag& is well-known in martensitic nucleation, and arises from the strain energy of a disc-shaped nucleus, the relation between a** mt and Ag& is new and is required to sufliciently tilt the balance of driving forces, at the size of a reference critical nucleus, in order to be in favor of barrierless nucleation. It appears that a relatively small ui+n:, of the order of 0.1 bg,, sufikea for the above purpose. This is because, at the top of nucleation barrier for homogeneous nucleation, different driving forces are already roughly in balance, as seen in Figs I and 2. Thus a significant, albeit not very large, interaction from the defect can cause very important effect.

following, we reanalyze their properties concerning martensitic nucleation to compare with nucleating defects with a short range stress field, The first is a dislocation wall, shown in Fig. 8(a), similar to the one considered by Olson and Cohen in their dissociation model of martensitic nucleation (21. However, we let the intensity of the strain field of the dislocation wall, characterized by its Burgers vector per unit height, &, be an arbitrary quantity which is not necessarily equal to Sr, as was assumed in the dissociation model. Only a continuous distribution of dislocations is treated. Thus, the defect shown in Fig. 8(a) is essentially an incomplete tilt boundary of a misorientation s,. Its height is denoted by h. Let a martensitic plate, of a length a and a constant width c, nucleate from the dislocation wall. The energy equation of interest is, as before, equation (I 7). As shown in Fig. 8(b), we replace the martensitic nucleus by two sets of continuous distributions of dislocations, with Burgers vector densities -S, and S, located at x = 0 and a, respectively, and extending between y = f c/2. The self stress and the interaction stress can again be easily evaluated from dislocation theory. The results are u

-

““-2n(l

“’

-v)

DEFECTSWITH A

LONG RANGE STEESS FIELD Two types of nucleating defects with a long range stress field have been considered in literature. In the

a

u”‘=27r(1-V)

c

In

-em----_

T

h

2 > a2+ h-C2 2 I ( )I

a-

(a)

FG,

11

;T :: II,f__________.L

h

L

h+C2 (

(34)

which is due to the two dislocation arrays, of the samesign,atx=Oandx=a. Equation (17) with u_,, and q,,, given by equations (33) and (34) at a2 > h2 + c2 can he closely approxi-

3rd

I

(33)

a2+ ct&b,

F: ‘ST

-

1

(f)ln(l+$) c

which is due to the two dislocation arrays, of the opposite sign, at x = 0 and x = a, and

()

3. NUCLEA~G

1833

1

PX

(b)

Fig. 8. (a) Heterogeneous nucleation at a dislocation wall. (b) A continuum dislocation model of (a).

1834

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and CHIAO:

MARTENSITIC

NUCLEATION

IN ZrO,

mated by the following equation I aAg

--= c

aa

-Agch+%+ ps:

c

2x(1-v)

00

- 2;::v)

(i)

(35)

N

N-t

N-Z X

since both the self stress and the interaction stress behave as an a - ’ function in the long range. The critical condition for barrierless nucleation, expressed as the minimal characteristics of the defect, is simply

Fig. 9. Heterogeneous nucleation at a dislocation pile-up. with c+ -2.

(37)

C

In this degenerate case, a* recedes toward infinity. Indeed, equations (36) and (37) simply state that the height of the dislocation wall needs to be adjusted such that its total dislocation content, 6; h+, is the same as that given in the dissociation model. When the above condition is met, there is a nearly complete cancellation between the two a-’ fields, due to the self stress and the interaction respectively. Defects with a stress field of a longer range than a-’ behave, in martensitic nucleation, in a quite different manner. The case of special interest is a stressed single pile-up of dislocations, as shown in Fig. 9. This problem was previously studied by Roitburd [19] using the saddlepoint method. Let the number of dislocations in the pile-up be N, their Burgers vector be b,, and the length of the pile-up be I. For the nucleation of a martensitic plate ahead of the pile-up, equation (17) applies again in this case. The self stress utir is again by equation (33), while the interaction stress uiimis of the same form as equation (5), where ii, is now the defect stress field at x = a, due to the pile-up. Using the stress field solution of a pile-up [18], it is found eimC=s,[($!y”-l].

is found that there exists a short range barrier due to the self stress, owing to uvlr going through a maximum at a/c = 0.505. At smaller a/c, u,,~ falls, to vanish at a/c = 0; at larger a/c, uwlrfalls according to c/a. This behavior is shown in Fig. 10. Also shown in Fig. 10 is the possible behavior of ui,, at different strengths. Rather than requiring the previous analysis such as that of equations (9) and (10) or (22) and (23), we find it obvious that, for barrierless nucleation, the condition to be met essentially simplifies, to a good approximation, to 90

a - 0.5oOk

because at both larger and smaller a/c, the equivalent stress is even more negative. Equation (40) reduces to NbJ’”

2 0.283 &c’~

M,Nb, n(l-v)l

2.5n(l - v) PS:

X (A&*-F)].

(41)

The minima1 value of the left-hand side which satisfies equation (41) at a certain c* is

N+b;/l*‘P

x

=

1-

= 0.566 2.5~ (1 - v)

(38)

Since I in a typical pile-up is much larger than the length of the martensitic nucleus at its critical state, only the near field solution is of immediate interest, i.e. din,

w

1

0 a

I/Z

l-2.5x(1

-~)($]}‘~f+y”

(42)

[ with c* = 2.5n(l -v)

&-

2711 . (43) 2.5n(l - V&T,,

(39) In the above, it was assumed that AgChis relatively

which is of the form of u-“* and thus of longer range than u,,,.. Due to the dominance of uvlr in the near field and the dominance of uinl in the far field, the analysis of the previous cases, exemplified by equations (9) and (10) or (22) and (23). when examined at the far field, needs to be re-examined for the present problem. It

small, which was the typical situation under which evidence for heterogeneous nucleation at dislocation pile-ups was reported. It is apparent that as the chemical driving force increases, the critical characteristic required of the pile-up as given by equation (42) decreases. In the meantime, the width at which the barrierless nucleation first proceeds increases. When c* approaches I+

CHEN and CHIAO:

MARTENSITIC NUCLEATION IN ZrO,

1835

form small dislocation loops. When these loops had grown to a critical size, spontaneous transformation was effected. Details of microscopy analysis will be given in Section 4.1.2. At higher density of ZrO, particles, a substantial portion of them were in contact or in clusters on the carbon film. The origin of such contact was essentially by chance, which was inevitable during the extraction replication transfer procedure. However, once a contact was made, strong contact stresses must arise by a mechanism which will be analyzed later in Section 4.2. Mechanically speaking, self 0.23 loaded He&an contacts were thus formed. Sub0 I 2 3 4 sequent TEM examination found such Hertzian con,C/O Fig. 10. The self stress of a nucleus plate. Also shown is uhl tacts to serve as effective nucleation sites and in-siru observation of martensitic nucleation could be made. due to a dislocation pileup of various strengths. Details of microscopy analysis will also be given in itself, the approximation of equation (39) ceases to be Section 4.1.2. Still another type of nucleating defects was manuvalid. Thus there is a lower limit of N*b$/l*‘12, even factured by forming bicrystals between ZrO, and at larger driving forces. For the case of c* 2 I+, the pile-up should be treated as a superdislocation for A&O, at high temperatures. This was accomplished which an analogous condition to equation (36) is by allowing A&O, vapor in contact with the Nb-Zr alloy during the high temperature internal oxidation more appropriate process. Heterogeneous precipitation of A&O, onto ZrO, particles was found, giving well-formed bicrystals of various shapes. The mismatch between thermal expansion of the two oxides, during subThe analysis of nucleating defects is now complete. sequent cooling to room temperature, generated sufficiently high internal stresses near the phase boundaries. TEM examination again found such 4. IN SITU TEM ORSERVATIONS OF stresses to serve as effective nucleation sites from HETEROGENEOUS NUCLEATION IN ZrO, which in situ observations of marmnsitic nucleation SMALL PARTICLES could be made. Details of microscopy analysis will be 4.1. Experimental given in Section 4.1.2. 4.1.1. Materials preparation. The statistical consid4.1.2. Microscopy on the nuckating &fects. (1) Dislocation loop. Zirconia particles obtained from eration of nucleating defects as given in the accompanying paper [S] makes obvious that experiments the above procedure were initially defect free, and designed for direct observations of heterogeneous thus featureless when examined under TEM. Due to nucleation must be performed using very small parion damage in the microscope operated at 200 kV, ticles. Our analysis [8] indicated that below 0.5 pm, however, diffraction contrast features started to ap ZrO, particles have virtually no probability of conpear inside particles a& a while. (That the ion taining any nucleating defects. In a previous study [3], damage was responsible for defect formation in this we manufactured small tetragonal ZrO, particles in a case was confirmed independently by other experiCu-Zr alloy, by internal oxidation, and verified that ments using thin gold crystals examined under similar conditions.) Under two beam orientations, these consuch particles remain tetragonal and thus metastable at 4.2“K. To allow for a higher formation tern- trast features were in the form of black-white lobes perature and a higher density of ZrQ particles, we or double arcs [photo l(a)]. For resolvable double selected a bigb purity Nb-Zr binary alloy for the arcs, reversing the g vector caused the image features present study. Using internal oxidation at 18OO”C!, to change their sixes. This inside/outside contrast tetragonal ZrOr particles up to a size of 0.5 pm were nature indicated that these features were small disobtained. The subsequent application of an extrao location loops. Size changes associated with reversing tion replication technique allowed the transfer of s and further dark field examination also corrobothese metastable particlea from the Nb matrix onto a rated this point. carbon substrate. Samples prepamd by this ptooedure Particles with such defects inside were rather unstable; changing the illumination intensity alone often were then examined under TEBI. At lower density of ZrOa particles, the majority of sufliced to trigger the spontaneous transformation them were isolated from eaob other on the carbon [photo l(b)]. The time available for imaging was thus film. These particles were selected under TEM for in severely limited and it was not feasible to perform a situ observations of the heterogeneous nucleation. complete series of imaging corresponding to different Essentially, radiation damage was implemented to g *b conditions in each individual particle. The nature I

A.M. ,3,10-E

I

1

1836

CHEN and CHIAO:

MARTENSITIC

NUCLEATION

IN ZQ

Photo 1. (a) Double arc strain contrast due to dislocation loops in a tetragonal Zro, particle. The projected ( 110) directions as indicated by black spikes were found to be parallel to the lines of no contrast across the dislocation loop. (Seetext.)(b) After the I + 8ntransformation,the sameparticlehad a twinned

structure. of the dislocation loops was, instead, deduced mainly mixed type. Tbe nature ofthese loops was determined from comparison of images obtained under dynam- by the standard (gb)s analysis. The results indicated ical two-beam conditions with computer simulated that they were interstitial loops. Very often, lines of images available in the literature. We use a distorted no contrast parallel to the projected (110) direction cubic fluorite indexing convention in the following. were observed bisecting the loops into double arm. As established by computer simulation, the angle Assuming the extinguished portions of the loops were (p, made between the diffraction vector g and the due to the mutual cancellation of two opposite edge black/white vector 1 depends essentially on the Bur- components, an analysis of those projected directions gers vector b and the loop normal n. Using an of no contrast also resulted in the same identific&on analytical expression derived by Wilkens et uZ.[ZO],4, of the loops as ones of a mixed type Iphoto l(a)]. for a series of ~rn~natio~ of b and II ranging from Loop sizes were measured from (g, 3g) dark field and to <112) were calculated for speciiic beam f g high order bright field images. The largest loops directions. Those computed CpIvalues were compared ever observed were of a size of 20 mn (Photo 2). (2) Herr&m conracf. Zro, particles dispersed on with the corresponding values measured from images obtained for that same imaging condition. Usually carbon films prepared by extraction replication found there are several different sets of b-n combinations occasional contacts between particles. This could which conform to the experimentally observed 4, range from two-particle contacts to densely packed value. Further elimination of some of these possibilities to make an unique identification of b and n requires the separate determination of either b or II. The procedure we used was to determine n by means of weak beam dark field or high order bright geld imaging for resolvable loops, assuming loops were all of the same nature irrespective of their sizes. It was found that under 2OOkV, the establishment of s8ccO.2nm-’ for weak beam imaging in 210, requires rather high order diffraction vectors in Bragg position, which were difficult to obtain given the time limitation imposed by the unstable nature of the particles. A compromising imaging condition for weak beam was usually set at (g,3g) orientation (Photo 2). A reasonably definitive determination of n turned out to be possible under such conditions. Following the above procedure, the Ioops were Photo 2. A (g, 3g) weak beam image of dislocation loops in a tetragonal ZrO, particle. found to be of b-(011), n=(Iil) type, i.e. of a

CHEN and CHIAO:

~ARTENSlTrC

NUCLEATION IN ZrO,

1837

Photo 3. (a) An edge-on view of two tetragonal Zr02 particles forming a Hertzian contact aa indicated by the arrow. (b) After the t -+ m transformation, both particles were twinned with apprekble twinning offsets at the particle surfaces.

clusters depending on the initial ZrQ precipitate density in the Nb matrix and the local replication condition. To set up frtvorable affection ~nd~tions in individ~ particles by Selected area diffraction and to simplify subsequent interpretation of martensitic nucleation, two-particle contacts were examined. Under dynamical conditions, strain contrast could be observed around the contact are8s [photo 3(a)]. Instability also existed in this case; contacted p8rticles showing prominent strain contrast were apt to transform spontaneously in a short time during electron illumination photo 3(b)]. Atthough SUlf8cCsof zro, p8rticles usually showed some extent of facetin& thecontact geometry w8s not simple. The angle of their depositions onto each other was essentially random and there were differences in their heights iodgsd in substrate catbon fihn. An estimate of the deformation and geometry in every contact ~8s therefore dif6cuh. In simple cases, when the contact geometry w8s highly symmetric, computer simulated strain contrast was compared with the electron micrographs [photo 4(a,b)]; the parameters which g8ve the best match were taken as the best estimates. Additional details of this procedure when applied to the contact problem 8re described in the Apendix. From the above, we estimated 8 typic81 contact width to be 4nm and the average cornpressive strain at the contact to be 0.01. In addition to two-p8rticle contacts found between ZrO,, bicrystals of ZrQ/A&O, were observed. Unlike contact boundaries between ZrOl which formed during replication, the phase boundary between Zr02 and Al,4 formed at high temperature and was fully equilibntted. Due to differential and, indeed, anisetropic them& expansion particularly from the A&O, phase, considerable thermal stresses arise within both particles 8t room temperature, With the phase boundaries usually being curved, even f8ceted, such therm81 stresses can accentuate to become easily

visible from their diffr8ctioncontrast faturea photo 5(8)]. Bicrystals of the latter type were also highly unst8bk. They ~sfo~~ ~n~~~ly tier short exposm to electron inaction photo S(b)]. A microcrack of a length of 1Omn which emanated from the intersection of a m-Z+ twin boundary and the ZrO,/Al,O, interphase boundary is clearly visible. 4.1.3. ikficrostructural feahdres during and after transformation. In each individual particle, transformation occurred in a bwst m8nner. Lattice correspondence between the tetr8gonal and the monoclinic phases was identified, [OOlLlf[OQl], and (IOO),11 (MO), by compating diffraction patterns taken from p8rticles oriented in
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a

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CHEN and CHIAO:

MARTENSITIC

NUCLEATION

IN ZrO,

a

Photo 5. (a) Strain contrast due to thermal stresses between tetragonal ZrO, and a comer A&O, particle (A). (b) After the ! -+ m transformation (100) twins formed in the monoclonic ZrO, particle. A microcrack as indicated by the arrow emanated from the twin intersection with the ZrOJA1201 boundary.

Photo 6. (a) A tetragonal ZrO, particle with the corresponding [OIO]diffraction pattern. Odd ordered diffraction spots in the g, row were forbidden. (b) After the r-m transformation, the same particle remained in the (010) orientation. Although &,),Il(g?oo),, 0, has rotated toward g,oo by 9”, constituting the Type C Lattice correspondence. (100) twins formed and were responsible for the extra diffraction spots disposed symmetrically with respect to the g,, row.

1839

1840

CHEN and CHIAO:

MARTENSITIC

NUCLEATION

IN ZrO,

the shape strain is essentially along f i plane [3]. Other material parameters used were p = 80 GPA,

v =0.25, &=0.154, y,, = 0.2 J/m’. At 2O”C,the net driving force A& is 4339.8J/mol, from which the shape independent in-plane distortion strain energy was already subtracted. The above value, or equivalently 2.04 x lO‘J/m’, is used for Ag&in our calculations. The above material parameters are identical to the ones used in our previous work [3,8,16]. Prediction of equation (13) gives r, = 10.3nm. This value coincides with our observation that no loops larger than 20.0 nm in diameter were retainable in tetragonal phase. Note that the calculated size of the loop is an upper bound, but the nature of a mixed-type loop also calls for a correction due to the resolutiott factor (5/a)‘” as compared with a pure shear loop. It appears that the two corrections are self-compensating to yield the excellent agreement noted above. We now m-examine the values of several parameters introduced in our theory and check our assump tions made therein. The parameter K was calculated to be 1.30 x 10sJ/m’, thus considerably larger than Ag&as we aheady assumed in deriving equation (13). The thickness of the plate, c*, was found to be 3.3 nm, while u+ is 18.1mn which is larger than rp These values arc smdler than the size of each twin variant remaining in the transformed particlea, as expected. The growth beyond the stage of (u+,c*) is governed by thermoelastic considerations and lies beyond the scope of the present paper. Transformed particles, due to both dislocation loops and Hertzian contacts, all exhibited conjugate twin variants. Note that these particles, as a whole, were not subject to cxtemal stresses or external constraint. We believe this to be the result of autocatalysis due to inclusion stresses diuing transformation. As illustrated in Fig. 11, a transforming particle containing a thin plate of martensite, which is still partially constrained by the tetmgonal matrix, must have shear stresses within and outside the thin plate of a magnitude -@r in a sense opposite to the shape strain. The above observation follows straightforwardly from Eshelby’s treatment of inclusion stresses and is essentially required by the equilibrium and compatibility considerations. Thus neighboring regions, above and beneath the partially formed martensitic plate, are assisted by these stresses to transform, nearly simultaneously, in a conjugate manner. This is believed to be the origin of twin variants in small particles. Of course, if additional external constraint were furnished on the particles themselves, deformation twins may also be introduced during post-transformation rearrangement. 4.2.2. Hertzian contact. To compare our observations on Hertzian contact as nucleating defects for barrierless nucleation with our theoretical predictions, we need to understand the physical origin of the contact stress. First consider two spherical spheres barely in contact. A decrease in the free

Fig. 11. A self-stressed transforming particle containing a nucleus which is still partially constrained before it reaches the free surfaces. The shear stresses inunediately above and beneath the plate are the same as the transformation stress and opposite to the shape strain.

surface area is possible if both spheres deform to form a flat contact between themselves. It is obvious that such processes cause a &crease in the surface energy and an increase in the elastic energy. Equivalently, we may regard that an attraction arises from the surface tension between two spheres, which is countered by a compressive stress across the contact. When the two processes are equilibrated, an equilibrium contact thus forma. To simplify the mechanical considerations and to avoid invoking the geometric &tails of the above problem, we illustrate in Fig. 12 an equivalent configuration which has a tapered contact; the attraction due to surface tensions is here replaced by an equivalent external load. Thus the upper particle with a tapered end, by indenting the flat substrate beneath, lowers the surface energies of the system by an amount of AE, = h ‘(2y, - yr)

(45)

where h is the length of the edge of the square contact, y, the surface energy, and yb the interfacial

yb

Pt2k$- 2 J (I-v)h

-d

Fig.

Yb)

12. Contact stresses due to surface energy reductions.

CHEN

and CHIAO: MARTENSITIC NUCLEATION IN 210~

energy of the contact. Meanwhile, inden~tion under a total load Q yields an average displa~m~nt W

f+f’ -v)Q. Gh The above form is known to be insensitive to the details of indentor geometry and load distributions for a broad class of contact problems [22]. Indeed, even for a rectangular contact of a fairly large aspect ratio, it remains applicable provided h is replaced by the square root of the rectangular area. Thus the elastic strain energy stored in indentation is

Equating AE, and AE, at equilibrium, we obtain the average pressure, P = Q/h2

Inasmuch as the above relation between P, h and surface energy reduction is essentially insensitive to the details of the contact geometry, it may be used as an approximate but general estimate for contact parameters. It is clear that the physical origin of the contact stress is the surface energy recovered by forming the contact. Indeed, this is exactly the opposite process of Griffith brittle fracture in which the elastic strain energy is expended to form two new fracture surfaces. It should be noted that ybabove need not be the same as the energy of grain boundaries of a particular kind. Since Hertzian contacts in our experiments were formed at low temperatures, relatively complete mlaxation of the interface structures, which is necessrvy for forming grain boundaries, was not accomplished. Large stresses therefore remained in the form of contact stresses. Indeed, for well-formed bicrystals with high quality grain boundaries, no sign&ant contact stress should be expected beyond the immediate vicinity of the cores of grain-boundary dislocations. By an image matching technique described in the Appendix, we estimated, for Photo 4(a), the contact width h = 4 nm, P = 2 GPa or an average strain of 0.01, and a contact length of 40 nm, The inferred surface energy reduction, 2y8- yb, is only 0.12 J/m’. The above value is obviously much lower than the surface energy reductions between clean surfaces and we11formed ~~u-~un~~~. Since our partides were extracted by an acid replica technique and the boundaries were formed at room temperature, y, should be relatively low while ybshould be relatively high. The inferred value is therefore not nmeasonable. On the other hand, the strain and the contact area estimated above did appear reasonabIe for the strain contrast and the lattice image distortion observed ex~~rnen~liy. Using the same estimates, we may examine the prediction of equation (261,or equivalently equation

1841

(27) and (3 1) for P* and h*. Referring to Fig. 7, we found the present @se fell very close to the intersection of the two branches there. Thus both equations gave essentially similar predictions. Using equation (31), we found P*h* = 10.4Pa.m. Let h*=4nm, then p* = 2.6 GPa. The above value is somewhat larger than the estimated contact pressure, as expected from the upper bound argument mentioned previously. The ag~ent between our experiment and theory thus appears satisfactory. The reason that equation (31) is applicable in the present case is mostly due to the very large Ag&in the ZrO, system. Since a* approaches h under such ~rcumstan~s, the short range contact stress field is being felt in nucleation. From equation (29), u* = 4.4 nm and is indeed approaching h. (We note here in passing that a* in this case is particularly small because of the surface relief of some of the strain energy of the nucleus.) The above trend will become even more prominent if h is larger and P is lower. It is inte~sting to note that equation (31) in the present limit bears a similar form as eqnation (36) which is the extension of the Olson-Cohen model [2]. As we shall discuss below, several possible origins of heterogeneous nucleation in ZrO, containing ceram& of current interest may share the same feature of the stress distribution found in Hertzian contacts. Thus they may be regarded as nucleation sites which are operationally equivalent to the dislocation wall described in Section 3. It is worthwhile to note that, in the accompanying paper [8], we obtained the potency distribution of the n&eating defects in ~*~on~ning ceramics and found it explicable in terms of the dislocation wall model. We note that, in all cases studied above, nucleation ensued only after electron illumination. Obviously, in addition to the contact stress field, beam heating or other electron irradiation effects aided nucleation. We interpret such observation as indicative of a small frictional barrier for the martensitic interface. From our recent experiment of the strain rate effect on transformation stress, we estimated an activation volume of the order of 2.6 (Burgers vector)‘, suggesting a Peierl-stress type barrier in ZrG, [23]. In literature, such frictional barrier was long speculated and termed “grass on the slope” by Kaufman and Cohen [15]. Essentially, it is a periodic ripple, of a wavelength of the lattice spacing, which is superimposed on the energy curves in Figs 1 and 2. Thus even though the energy curve is deemasing on the ~ntinu~ scale, it still has small barriers on the atomic scale which may require thermal activation. It is these barriers which held back spontaneous nucleation in small particles in the absence of electron illumination. 5. CLA!33FICA’MON OF NUCLEATING DEFECTS

Having established the direct evidence of hetero geneous nucleation by in situ microscopy and the

1842

CHEN and CHIAO: MARTENSITIC NUCLEATION IN ZrOz

theoretical model which identifies the critical characteristics of operating defects, we are in a position to quantitatively assess the overall picture of martensitic nucleation in terms of its nucleation sites. For this task, we use an approach which classifies nucleating defects by their origins and their stress fields. This approach is justified by two central observations made in our study. First, in the companion paper [cc], we conciuded, from a statisticai analysis, that preexisting nucleating defects tend to foilow a universal potency distribution. Nucleation from this origin is therefore statisticaliy predictable and may be IT+ garded intrinsic. For small particles falling below a size threshold, nucleating defects are statisticaily improbable to pm-exist at aii. Sticientiy potent defects must be generated by extrinsic means to facilitate nucleation. Likewise, when chemicai driving force is small, e.g. above the M, temperature of the buik material, aii the pm-existing defects are not sufiicientiy potent. Nudeation again must be fac& tated by more potent defects generated ex~i~y. In both cases, nude&ion may be regarded as extrinsic. Second, our analysis in Section II and III emphasisiag mechanical characteristics brought out one essential difference between various types of defects, namely the range of their stress fields. Criteria for bar&less nucleation among short range defects are very similar [see equation (3211,but they are different for long range defects [see equation (36, 42,44)]. Such difference in mechanical ebaracteristics is inherent in martensitic nucleation because the strain energy of a sheared nucleus causes a self-stress which has a range dependena of the 11-l type. Thus m&eating defects naturally ~stin~~ themselves from each other depending on whether their stress fieidsofthea-“typesatisfym~1orm~t.Dntwing on these two distinctions, we may classify nudeating defects as intrinsic/extrinsic and short ranged/long ranged. Experimental evidence suggesting nucleating defects of ail of the above types, in the literature of marten&c transformation of ferrous alloys and ZrO, ceramics, wiii now be reviewed in this light. intrinsic defects pre-exist in the crystal and are the remnant defect structures resulting from high temperature processing (including anneaiing). In the companion paper [8], we demonstrated that intrinsic defects have certain characteristic potency distribution which is explicable in terms of a defect configuration as a dislocation wail. This is the same type of configuration as envisioned in the Olson-Cohen model [2].As we discussed in Section 3, a more general dislocation wail of a diRerent content of Burgers vector for &.,,6, # S,, essentially behaves in the same way as evidenced by their critical characteristics, equation (36), which is of the same form as previously proposed in the dissociation model 121. Such defects tend to exist primarily as groups of interracial line defects. They are iikely to be “parasitically” attached to grain-boundaries and inter-

phase boundaries, where they can be partially stabilized by elastic interactions with those planar defects. It is noted that during high temperature processing, individual defects will tend to relax toward a state of lower energy (and thus lower po-

tency). Nonetheless, a certain small number of them can still be expected from accidental boundary motion and from differential thermal expansion at interphase boundaries during cooling. It is also noted that ~thou~ these defects, in a form analogous to an incomplete tilt boundary of Fig. 8, may be properly considered as extrinsic to the equilibrium interface structure, they are nevertheless the unavoidable residue defects resulting from high temperature processing and thus are intrinsic in ail materials. Their density, by necessity, is low and their statistics is predictable [8]. Indeed, nucleation due to intrinsic defects may be entirely suppressed in sufficiently smail particles, even at very large driving forces. In the companion paper [8], we estimated from nudeation statistics a uiticai size for defect free smali partides around O.Spm in Fe-Ni and ZrOs. Electron microscopy observations of marten&c embryos formed at internal interfaces are now quite numerous [I, 4-71. Although in no case has the defect structure been clearly resolved in these studies to provide definitive support for the above picture of intrinsic defects, the general features of these sites appear to be consistent with our expectation for nucleation at dislocation walls. In ferrous alloys, gram boundaries [4], twin boundaries [S], ~d~on~rna~ interfaces [6] and grain-boundary ~bid~/ma~x interfaces [7l have been reported to be the nucleation sites. In an YtO, doped ZrO, tetragonai polycrystai, subgrain boundaries were aiso observed to be the nucleation sites [24]. Thus nudeating defbcts of this type appeared quite general and probably are responsible for martensitic nucleation in bulk materials. We believe that nucleating defects in ail of the above cases are intrinsic in origin, having a long range stress field of the type of u-’ as discussed in Section 3. The critical characteristics of these defects, for barrierless nucleation, may thus be given by equation (32). It is obvious that the size of a critical defect decreases as the chemical driving force increases, hence only at smaller driving forces are these defmts larger and more resolvable by microscopy. However, large defects of higher potency are very rare statistically [8] and hard to find. For these reasons, direct microscopy confirmation of this mechanism has not been successful. Extrinsic defects, such as shear dislocation loops observed in our experiment, are generated at lower temperatures by external means. The shear loops in ZrO, observed by us are believed to be the result of radiation damage. Quenching and other treatment may also generate dislocation loops. In many cases, however, these loops are prismatic ones which are not as ef&ctive nucleating sites as shear loops for martensitic shear transformations. We believe this to be

CHEN and CHIAO:

~ARTE~S~IC

the main reason why a previous study on ion radiation damage in y-Fe particles embedded in a Cu matrix failed to indicate a definitive effect of prismatic loops on martensitic nucleation [25]. It should also be noted that the critical size required of the shear loops decreases rapidly with the driving force, as shown by equation (13). Thus extrinsic nucleation induced by a shear loop is considerably easier in ZrO, than in y-Fe at room temperature due to the higher driving force in the former material. (Their elastic constants are ~inciden~lly the same.) Provided prismatic dislocation loops can unfault or glide to acquire a shear component, they can eventually act as nucleating defects for barrierless nucleation at sufficiently large sixes. (Prismatic loops unfaulting or rotating to become mixed loops are prominent in anisotropic bee metals (Nb and MO [26]) and plausible in bstragonal ZrOl. Thus the very occurrence of mixed loops in tetragonalZ10~ is perhaps a result of its distorted cubic structure, since in a fully stabilized cubic zirconia only prismatic [I IO] loops were found following deviation from stochiometry [27&) It is likely that slightly over-a& semi-coherent precip itates, at their interface with the matrix, have such dislocation loops which initially formed to effect a partial loss of coherency. We have previously suggested this mechanism to be a possible cause for martensitic nucleation in over-aged MgO-partially stabilized zirconia [16]. Empirically, it is consistent with the fact that no incoherent precipitate in MgO-PSZ was ever observed to be tetragonal, yet free t-ZrO, particles are metastable. We therefore expect shear dislocation loops as nucleating defects to be either extrinsic or intrinsic, i.e. they can be either generated by extrinsic means or result from prior thermal history. Their stress field is always of a short range nature, and the critical requirement is given by equation (13), or, equivalently, by equations (t6.32). The other type of short range defects, namely Hertzlan contacts, was also found extrinsic in our experiments. It appears, however, that tensile or compressive stresses of a similarly localized nature may be generated by thermal expansion mismatch between different phases. For example, such stresses were observed in the Zr02-AlsOs bicrystal in our experiment and found to be effective nucleating sites of barrierless nucleation. In view of this possibility, they may also be intrinsic in origin. It is recognized that thermal stmsses at comers can be of various nature; their ,range dependence and their amplitudes are generally sensitive to the geometry and the size of the particle. Inasmuch as these stress fields are short range ones for the purpose of martensitic nucleation, equation (32) should serve as a valid criterion on their critical characteristics. In this respect, many nucleating defects of thii type may operate in a similar way. We note however that in cases of very large driving forces, which are believed to be present at least in A&O> containing ZrOz and possibly in MgO/CaO stabilized ZrOz, the critical nuclei are

NUCLEATION IN ZrQ,

1543

small (a* x h) and the interaction stresses between the contact stresses and the nuclei do not fall as a -2. Our analysis in Section 2.2 demonstrated that, in the latter case, the nucleation characteristic, equation (3 I), is essentially similar to the one of a dislocation wall, equation (36). The above similarity is also evident from comparing the second stress term of equation (21) and the stress term of equation (34) in the vicinity of a x h. Indeed, our experiments on ZrO, apparently fall into this latter regime. Thus for

a very large driving force Ag, and a not too small contact length h, the contact stress can be regarded, de facto, as a dislocation wall for the purpose of martensitic nucleation. In view of these possibilities, we expect particle-particle contacts and thermal stresses at corners to be either intrinsic or extrinsic nucleating defects, with stress fields of either a short range or a long range nature. As we already discussed in Section 4.2.2, nucleation statistics in several ZrOz containing ceramics as analyzed in the companion paper [8] is explicable in terms of a dislocation wall model. It is thus indicative that thii type of intrinsic nucleation is operational in such materials. Although the above three types of nucleating defects, according to our analysis, are believed to be at least occasionally intrinsic in their origins, the very long range defects as exemplified by a dislocation pile-up in Fig. 9 must arise extrinsically and are typically strain-induced. Numerous observations purporting to evidence heterogeneous nucleations at dislocation pile-ups were reported for plastically deformed austenite. Two particularly convincing studies which furnished in situ TEM observations of dislocation pile-ups and nucleation therefrom were made by Suzuki ei at. [9] and Brook et d [IO]. In general, under plastic defo~tio~ extrinsically introduced nucleation sites are plentiful, including shear-band intersections and their impingement with grain-boundaries [9-141. The shear bands may be mechanical twins, hcp s-marten&e, bundles of stacking faults, or simply slip bands. Not only being plentiful, nucleating defects generated in this manner are frequently large (of the size of 1 pm ot more) and thus more amenable to microscopy studies. Furthermore, nucleating sites of this type are frequently more potent than .those intrinsically available ones, at large N values. Thus martensitic nucleation can be induced thereby at temperatures above the A#,temperature. This latter feature is also an advantage for successful microscopy studies, since it precludes the interference of intrinsic defects in inducing the heterogeneous nucleation. To verify this possibility, we estimated that a pile-up length of Sprn and a total of 25 dislocations, at y ,, = 0.2 J/m’ and materials parameters representative of ferrous alloys, with a net driving force of the order of Agm = 450 J/mol, satisfies both equations (42 and 44) for barrier-less nucleation. Allowing for the shape-independent strain energy of the order of 500 J/m01 [Z], a total chemical driving force of 950 Jfmol is required, This value is smaller

1844

CHEN and CHIAO:

MARTENSITIC

than the chemical driving force at the M, temperature, which is 1100J/mol in Fe-Ni alloys. Thus, indeed strain-induced transformation may take place above the M, temperature as observed experimentally. The nucleating defects in this case are long-rang4 extrinsically introduced dislocation pileups or shear band intersections. 6. CLOSURE The above classification has incorporated all the nucleating defects which have found some experimentalsupportin studies of martenaiticnucleation in ferrous alloys and Zr02 containing ceramics. The overallpicturethus emergingfrom the presentexperiments and theory and the literaturereview portrays barrierlessheterogeneous nucleation of martensites from internal defects in all cases. These defects are either intrinsicor extrinsicin origin, and may have a long range or a short range stress field. Transformation, especially in bulk materials during cooling, is due to intrinsic defects which formed in prior higher tcmpera~urc processing, and by thermal or misfit stresses. Dislocation walls, contact/thermal stresses,and dislocation shear loops are in this category. Direct TEM observations of these defects are extremely difEcult for statistical and operational difficultiesdue to autocatalysis.Extrinsicnucleation, on the other hand, offers the best opportunity for direct TEM confirmation of heterogeneous nucleation. Using smallparticlesof Zr02, we have here, for the lirst time, provided the unambiguous identification of the nucleation sites and witnessed the spontaneous nucleation events therefrom,by generating defectsin selectedparticleswhich were initially defect-free.The previousTEM experimentsof Suzuki et al. [9] and Brooks et al. [lo] in 18-8 austenitic stainlesssteels which were in situ deformed above M, provided another example of extrinsic nucleation. In view of the direct microscopic confirmation of the heterogeneous mechanism in these specific experiments,and the overall consistency between the theoretical prediction and the experimental observations, including nucleation statistics [8], we conclude that the heterogeneous, barrierlessnucleation mechanism in martens& transformation is now firmlyestablished. Acknowfedtzemenf-We arc grateful for the financial support of this resear& from -U.S. Department of EnergY, under Grant No. DE-FGO2-84ER45154. Stimulating discussions during the course of this work were held wiyh Dr G. B. Olson of M.I.T. Nb-Zr alloys used in our experiments were kindly provided by Dr. C. T. Liu of Oak Ridge National Laboratory. APPENDIX

Simulorion of strain contrast of Herlzian contact From theory of elasticity [22,28], it is known that the deformation underneath a Hertzian contact is dependent on (a) the total contact load and (b) the load distribution at the

NUCLEATION

IN ZrOz

contact. The former essentially governs the far field while the latter is more important in the near field. For contact width as narrow as several nanometers in the present study, it would be unrealistic to try to resolve detailed futures in the near field from electron micrographs. Thus the actual load distribution cannot be specified from image matching. The major information obtained from the simulation of strain contrast pertains to the total contact load only. Howcvu, for a known geometry of the indentor, such as the one in Photo 4(a), the elasticity theory of Hertxian contacts also provides a relation between the contact width and the total load [22,28]. In this sense, both the average pressure and the contact width can be evcntuaIly infernal from image matching. Our simulation used the displacement field [22] of a point load onto a semi-infinite solid [221. Strain contrast was simulated using gm with a deviation parameter w = 1 and with both normal and anomalous absorption eoeflicknt set at 0.1. For ZrO, the g= extinction distance at 200 kV was calculated to be 13Omn. The thickness of the particle, measured by the projected width of the twinning plane after transformation, was around 65nm. The contact was extended for a length of 40 nm. The load which gave the best match was taken as the estimated total contact load. The geometry of photo 4(a) involves an indentor (the particle to the right) and a flat substrate (the particle to the left). The radius of curvature of the indentor was measured directly from the micrograph to be 4Onm. Using the elasticity solution of a circular cylinder of the same curvature indenting onto a flat surface [28], the contact width was obtained as a fundion of the total load. At the total load

estimated for photo 4a, the contact width obtained above was 4 nm. The average normal strain was 0.01. The simulated strain contrast is shown in Woto 4(b). While the general features of the simulated strain contrast compare rather favorably with the electron micrograph, there is a, slight difkrcncc between the aspect ratios of the contrast arcs in the two. The simulated contrast arcs protrudes into the particle less than the experimentally observed ones do. The above slight discrepancy can be attributed to anisotropy of tctragonal ZQ since only isotropic elasticity was used ln our simulation.

REFERENCES

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MARTENSITI~

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NUCLEATION

IN ZrG,

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