- Email: [email protected]
The nucleation of martensite in steel
THE NUCLEATION
OF MARTENSITE
IN STEEL
K. E. EASTERLING Department of Materials Technology, University of Lulea. S-951 57 Sweden
and A. R. ‘I-I&LEN Department of Applied Physics I. University of Technology. Lyngby. Denmark (Received 9 3ul.r 1975: in recked form 23 October
1975)
Abstract-Calculations of the total energy for transforming austenite to martensite in the form of thin ellipsoidal plates, fully coherent with the austenite, show that the process may be spontaneous in the presence of pre-existing dislocations. It is found that dislocations, or groups of dislocations, in the austenite are suitable sites for martensite nucleation in that their strain fields may interact favourably with the strain field associated with the Bain deformation thereby eliminating the energy barrier to nucleation. The driving force for twinning to occur virtually simultaneously with nucleation is large and when this happens energy is released for thickening and growth of the nucleus. It is also found that the strain energy of coherent plates of martensite. whether twinned or untwinned. is a function of their orientation in the austenite, although the lowest strain energy cases occur nevertheless over a relatively wide range of orientations. The proposed theory of dislocation-assisted nucleation of martensite is qualitatively able to account for the majority of experimental observations pertaining to martensite nucleation. R&r&--Le calcul de l’inergie totale pour la transformation de l’austinite en martensite sous forme de fines plaquettes ellipsoidales, parfaitement coherentes avec t’ausdnite, montre que ce processus peutCtre spontani si des dislocations preexistent. On a trouve que des dislocations ou des groupes de dislocations forment, dans I’austenite, des sites de germination convenablcs pour la reaction martensirique: en effet. leur champ de deformation peut agir favorablement avec le champ de deformation associt B la deformation de Bain et iliminer ainsi la barriere d’energie de germination. La force motrice pour que le maclage apparaisse pratiquement en meme temps que la germination est grande, et dans se cas de l’energie est liberee pour le grossissement et la croissance des germes. On trouve egalemcnt que l’bnergie ilastique de plaquettes de martensite cohirentes. maclees ou non, depend de leur orientation par rapport Q l’austenite, bien que Ies cas d’energie tlastique minimale se produisent dans un domaine relativement etendu &orientations. Cette thtorie de la germination de la martensite aidee par des dislocations, permet de rendre compte de la majorite des observations experimentales concernant la germination de la martensite. Z~~~~~-~rechnungen der Gesamtenergie fur die Umwandlung von Austenit in Martensit in Gestalt dlinner ellipsoidf~rmiger Platten. welche vollstandig kohlrent mit dem Austenit sind, zeigen. daB der Prozess in Gegenwart van Versetzungen spontan ablaufen kann. Es wird gefunden, dal3 im Austenit Versetzungen oder Versetzungsgruppen geeignete Orte fir Martensit-Keimbildung sind, indem deren Verzenungsfelder giinstig mit denjenigen, mit der Bain-Verformung zusammenhangenden wechselwirken. Dabei wird die Energieschwelle der Keimbildung aufgehoben. Die treibende Kraft fur die etwa gleichzeitig mit der Keimbildung ablaufende Zwillingsbildung ist groB, und wenn diese ablIuft, wird Energie fur Dicken- und GroBenwachstum des Keimes frei. Es wurde ebenfails gefunden, dal3 die Verzerrungsenergie koharenter Martensitplatten. ob verzwillingt oder nicht, von deren Orientierung im Austenit abhtingt, wenn such die Fllle kleinster Verzerrungsenergie in einem relativ weiten Orientierungsbereich liegen. Die vorgelegte Theorie der versetzungsunterstitzten Martensit-Keimbildung beriicksichtigt qualitativ die meisten Beobachtungen zur Martensit-Keimbildung.
1. NTRoDwcrIoN The hardening of steels by quenching from high temperatures to obtain martensite is one of the oldest and most important metallurgical treatments known. It is an intriguing fact, however, that the mechanism by which martensite forms is still not clarified. A satisfactory theory of martensite nucleation in steels ought to: (a) show specifically how the diffusionless shear transformation nucleates from austenite, given the free energy difference between the austenite and
martensite phases at the MS temperature: (b) explain how, and what sort of heterogeneities assist nucieation and growth: (c) show the influence of the martensite embryo on the form and fine structure of the fullgrown plates. Inclusion of the latter requirement is justified on the basis that since martensite grows, without diffusion, at speeds approaching the velocity of sound [l J, there should be structural and crystallographic features common to both the nucleus and the full-grown plates [2].
334
EASTERLING
ASD
THijLfN:
THE NUCLEATIOX OF MARTENSITE IN STEEL
It has to be stated that theories of martensite nucleation have made disappointing progress since 1958 when Kaufman and Cohen published their excellent review of the subject [3]. The main reason for this is because of the experimental difficulties associated with studying nucleation, at least in steels. A single plate grows at such speed that it may span a complete grain within lo-’ set [l]. Nucleation sites appear to be randomly and fairly coarsely distributed in any given grain [3], and since electron microscopy is needed to study nucleation the chances of seeing any thing meaningful are extremely remote [4]. Electron microscopy and X-ray studies of transformed martensite have been useful, however, (see, e.g. the reviews of Christian [2] and Wayman [j]) and these results have clarified, e.g. the irrational and variable nature of the measured habit planes, the very fine (-20 A thick) twinned structure of typical steel martensites, and (more recently) the confirmation that the Bain distortion occurs (together with inhomogeneous shears) in ferrous martensite transformations [6]. The latter result in particular is an important one because until that piece of work the Bain deformation was considered a mathematical operation rather than a physical quantity. Indeed we shall take this result as the starting point of the present work. In 1924 Bain [7] showed how martensite can be generated from austenite by a homogeneous strain in which two of the cube axes of the f.c.c. lattice expand by about 12Y4and the other axis contracts by about 20?/,, (these amounts depend, of course, on the alloying constituents). Later it was shown that the so-called Bain deformation is the most efficient way of generating the martensite lattice in that it requires the minimum of atomic movement to do so [8].‘On the other hand, it has been found that the Bain deformation alone can not account for the observed invariant plane strain characteristics of fully grown martensite plates [2,5], and this led to the proposal, later confirmed experimentally, that in addition to the Bain deformation, heterogeneous shears (twinning or slip) were also needed. This form of transformation is mainly based on the criterion that the strain energy associated with the full-grown martensite plate (with respect to the surrounding austenite) be a minimum. As we shall now show, the same strain energy criterion appears to govern the fine structure and orientation of the critical martensite nucteus.
refers to the coherency strain due to the misfit between the two lattices and Eshcmis the free energy. difference, Ax per unit volume, between the austenite and martensite at the MS temperature. Assuming the martensite forms as a thin ellipsoid of volume V; with semi-thickness c and radius a, and that the transformation can be expressed as a simple shear, s, parallel to the plane of the ellipsoid, equation (1) may be written:
where ~1is the shear modulus of the austenite, v is Poisson’s ratio and V is the total volume of the nucleus. If v = l/3: 16x E101= 2nn=o-I” 3 (s/2)2pac2
(3)
In a Bain deformation of (1,12; 1,12; 0,80) with the principal strains parallel to the axes of the martensite ellipsoidal nucleus, in which c/a = 403, the coherency strain energy as calculated by Eshelby’s expression [93 yields a value of about 0,OSpV, or about 25 times the available chemical free energy (the last term in equation 3). In other words, even without the surface energy contribution the nucfeation barrier for critical growth is much too large for martensite to form spontaneously, assuming realistic values of AJ If the principal strains are changed, e.g. to (0,SO; 1,12; 1,12), the coherency energy is virtually unchanged. In other words, it makes no difference if the contraction is parallel to the small axis or not. The energy obtained is also fairly insensitive to the c/u ratio chosen. The question now arises, whether it is energetically more favourable for the ellipsoid to form at some othersorientation in the austensite, but still retaining the Bain deformation as the mode of transformation. Using a computer the energies for all possible orientations of the ellipsoid could be readily calculated. This was carried out using Eshelby’s (91 equation for the coherency energy due to a homogeneous shear, and by converting the principal strains of the transformation to the new coordinate system, as shown in Fig. 1. Since Eshelby’s treatment clearly distinguishes between shear and dilatational stresses, the respective energies could also be calculated independently of each other. Table 1 gives the lowest strain energy values resulting from these calculations, together with the shear 2. ENERGY CONSIDERATIONS and dilatational components of these energies, in Consider an ellipsoidal region of martensite, transterms of different c/u ratios (0.01; 0,03; 0.10). formed by a Bain deformation from the austenite, It is seen from the results given in Table 1 that such that the interface between the two lattices the minimum energies in the case of c/a = O,Ol, are remains fully coherent. The total energy change as- about 29 times less than for the case where the ellipsociated with the transformation is: soid is parallel to the cubic axes of the austensite. However, this is still ten times larger than the Ap S*0,= Erurf + Emai* + &hem (1) term in equation (3), and the transformation will not occur spontaneously by the Bain deformation, irreswhere EPUifcorresponds to the coherent interfacial energy between the austensite and martensite, Ertrain pective of the shape or orientation of the martensite
EASTERLIKG
ASD THijLEN:
THE PU’UCLEATION OF MARTEYXf’E
Ipu’ STEEL
335
Table 1. Examples of some of the lowest coherency energies of martensite ellipsoids transformed by a (1.12; 1.12; 0.80) Bain deformation
Fig. 1. (x), (y), (z) are principal axes of the Bain deformation, and (1).(2) (3) are the axes of the martensite ellipsoid. The ellipsoid’s orientation, together with its relevant coherency energy, can be plotted on a 001 stereographic projection, as in Fig. 2.
The total (shear + dilatational) minimum energy orientations are plotted on a 001 stereographic projection in Fig. 2 for the case in which c/a = 0.01. Note that the minimum coherency energy is not confined to a single orientation, but appears to extend over a range of orientations. This range is defined as a “minimum energy trough”, and is shown “crosshatched” in Fig. 2. An important result that emerges from the computed results, is that one of the components of either shear or dilatational energy may constitute a large proportion of the total strain energy. For instance, in the case where c,‘n = 0,Ol (see Table I), the dilatational component may be as low as 0,007 PV comprising in fact only about one third of the total energy. The possibility thus arises that if the Bain deformation could occur simultaneously in adjacent regions of the austenite in such a way that certain components of shear or dilatational energy compensate one another, a net reduction in total coherency energy should result.
3. TWINNED ellipsoid in the austenite. This result is hardly a surprise, since we know from experimental observations that the Bain deformation alone can not account for the form and orientation taken by martensite in practice.
MARTENSTTE
Consider an ellipsoidal martensite nucleus which has undergone the Bain deformation in adjacent regions of the austenite, such that after the uansformation the regions are twin related. Assume that the width of adjacent twins can be varied as shown in Fig. 3.
001
Fig. 2. Coherency energies of ellipsoidal martensite, transformed by a Bain deformation of the austenite, (in terms of 0.001 pV); c/a = 0.01. The “minimum energy trough’ (crosshatched) denotes the range of orientations over which the coherency energies are consistently low, although the actual energy minima describe a single line along the centre of the trough.
Fig. 3. A schematic diagram of a twinned, ellipsoidal nucleus of martensite. Alternate twins have different widths, I and II, and by adjusting these widths, as aefl as the orientation of the ellipsoid in the austenite, the strain energy can be changed (see Fig. 4).
EASTERLING
336
THijLeS:
ASD
THE NUCLEATION
Table 2. Net strains of twinned martensite ellipsoids (see Fig. 4) ‘:a:.:
c;
c;
3 ;
,
b 1
0.12
3.:2
-:.zc
2
0.12
3.2&
-C,l5
3
0.12
0.v
-0.12
4
0.12
i.00
-a.08
5
0.12
-0.22
-rl,:c
6
0.12
-c.ia
-:.:3
:‘LI-s jdinl
Let k be the proportion of thickness I twin (or twins) in the nucleus. By varying the relative thicknesses of adjacent twins, the energy can also be minimised (this treatment is also adopted in the crystallographic theories [2]). The changes in principal strains are related to k as follows: in the x direction, the net strain = E,, in the y direction, the net strain = E, + (1 - k)E3, in the z direction, the net strain = 6s + (1 - k)El. The net shear may now be calculated as a function of k, E,~,c3. e.g: net shear = kf3 + (1 - k)E1 - kc1 - (1 - k)E3 = (1 - 2k&,
- e3),
when ei = 0,12; Ez = 0,12;
63
=
-0,20. > c
OF MARTENSITE
IN STEEL
The energies of these new net strains were calculated for five cases (in addition to the pure Bain deformation), and these results are summarized in Fig. 4. The net strains used are shown in Table 2. Figure 4 shows that by adjustment of adjacent twin widths very low coherency energies, of the order of AL can be obtained. Furthermore, the lowest energy troughs may be very shallow and quite extensive, see, e.g. case 6. The latter result is particularly interesting bearing in mind the experimental observations that the habit planes of martensite tend to be irrational, and differ by several degrees at least in a given alloy (see, e.g. Ref. 2, p. 876). It should also be noted, however, that the orientations covered by the “minimum energy troughs” can vary substantially from case to case. It is seen from Fig. 4 that the decrease in energy due to twinning is largely independent of the c/n ratio, although the difference in energy between c/a = 0.01 and 0,l at the minimum is obviously important when considering nucleation. In other words. the results show that the critical nucleus is likely to be a thin, twinned ellipsoid, the orientation of which is not too critical provided it lies within the “minimum energy trough”. Equations (l-3) showed that the formation of a coherent martensite nucleus is a function of surface, coherency and chemical energy. For the process to /
0.024
C/U~O.l
0 016
21
40
Fig. 4. Coherency energies of ellipsoidal martensite, as a function of twinning configuration and orientation in the austenite. The graph plots the energy minima for c/a = 0.1; 0.03; 0.01. The stereo-graphic plots give energies for c/a = 0.01 only. See Table 2 for net strains used. Note that the lowest energy orientations of the martensite (denoted by cross-hatching in the stereographic plots) may vary markedly from case to case.
E;\STERLII\;G
ASD
THijLfiX:
THE NUCLEATIO?i
OF MARTENSITE
1s STEEL
337
be spontaneous. the latter term (which is negative) must outweigh the other terms. The minimum free energy barrier to nucleation can be estimated by differentiating equation (3). with respect to n and c. and by subsequent substitution this yields:
in the simple case where & = 0, and the dilatational energy is assumed zero. As seen from this equation, the energy barrier is extremely sensitive to the values chosen for c, &and s. It is also obvious that the critical size of nucleus is likewise dependent upon these parameters. and it can be shown that:
and &it =
16ap(siZ)’
T’
Typically, Af = 1,74 x IO9 erg/cm3 (for steel) and s = 0.20 (**net”shear in a plate). We can only guess at the surface energy of a coherent twinned nucleus, and estimate a value of 20 ergjcm’. Strictly speaking, equation (6) is not fully applicable to a twinned nucleus since the total energy should include the twin boundary energy as well as the interaction energy at twin junctions. Estimations of these contributions have been attempted and indicate that the energy is not affected significantly by such refinements. In Table 3, we compare the nucleation barrier and optimum nucleus size of a fully coherent nucleus (20 erg/cm) with a semi-coherent nucleus containing a grid of dislocations at the interface (surface energy = 2~erg~~rn2 [33). As may be expected, a change in order of magnitude in the value of G brings about a thousand fold decrease in the nucleation barrier. However, even a coherent nucleus (nucleation barrier = 14 eV) will not form spontaneously and as we know from experiments, the transformation is in fact heterogeneous [3]_ It is generally thought that martensite nucleates at defects in the austenite (e.g. groups of dislocations) which somehow help to reduce the energy for nucleation. We now consider this possibility in more detail. 4. MARTENSITE NUCLEATION AT DISLOCATIONS We will assume that the strain field associated with a dislocation could in certain cases provide a favourTable 3. Energy of formation of critical nuclei of martensite
Fig. 5. The formation of a thin martensite nucleus at a dislocation loop.
able interaction with the strain fieid of the nucleus, such chat one of the shear terms of the Bain deformation is neutralized. This interaction changes the total energy equation to:
Consider a dislocation loop, in which a thin martensite nucleus has formed, as illustrated in Fig. 5. The interaction energy between the nucleus and the disiocation can be expressed as: Ei”$ -
-j.f.sWa”
4
2 -r
‘-‘-
c
(8)
I-VU
where b is the burgers vector of the dislocation. Assuming v = l/3, EI”1= - 2 psx abc.
(9)
The above approach assumes that the martensite nucleus reacts in the most efficient way with a complete dislocation loop. A more likely event is that martensite forms at one part of dislocation, e.g. at a curved segment or at a group of dislocations. On this basis the interaction energy defined by equation (9) is likely to be a maximum. We also assume in this approach that the dislocation loop is not a structural part of the nucleus. i.e. it does not contribute to the surface energy. We may now write equation (7) in more complete form as: 16R E,, = Zrra’a + 7 (s/2)2 PC’
-
Ai4;a’c -
‘psxabc.
(10)
It should be noted that the value of s chosen for the strain energy and interaction terms depends on whether the nucleus is twinned or not. By summing the various contributions to the total energy, it is now possible to compute the energy of the martensite nucleus as a function of size and thickness, whether it is twinned or untwinned, if it forms with or without pre-existing dislocations, etc, The results are presented in Table 4. Note that two dislocation interaction energies are considered (the two final columns in the table), the maximum value (given by equation 9) and a partial interaction energy of 3006 of the maximum. Thus the important columns are the last three, i.e. total energy without help from dislocations, energy with partial help from dislocations and energy with maximum assistance from a
338
EASTERLING
AND
THijLLiEi:
THE NUCLEATION
OF MARTENSfTE
IX STEEL
Table 4. Energies of martensite nucleation (in terms of 10-‘Oerg) as based on equation (10) and Fig. 4
n Id
S.26 fZ6
96.5 rzco
-7.60 -750
65.86 6423
37,:s 5.35
1.26 125
15f.3 113oc
-12 -fZOO
116.96 10.0?0
75,s XT:_
0,144 5.4 160
-0.036 -3.8 -380
-1.35 -13.5 -135
0.119 2.66 -74
-0.33 -1,64
-i.Z
1000
0.0126 1.25 126
-119
-2::
100 1000
I,26 126
-7.60 -760
-26.9 -264
3.06 -27b
-0.84 -363
_Ti :I _~-f-3
-40.6 -406
:3 26 +2i4
2.76 -399
-2s.: -5::
1 10
100
~
100 1000
I
C
I
dislocation. The columns showing negative energies in these columns mean that in these cases the final two terms in equation (10) outweigh the first two terms thus implying that rapid growth should occur. Significantly, we can see that under certain conditions martensite can form spontaneously at dislocations. On the other hand martensite is extremely unlikely to form spontaneously without assistance from dislocations. In order to clarify the growth characteristics of the martensite nucleus, the results of Table 4 are presented in thr~~imens~oM1 form plotting energy changes in terms of nucleus size in Figs. 6 and 7. In both figures the full curves denote untwinned martensite and the broken curves denote twinned martensite. Figure 6 describes nucleation without the aid of
emi-thickness,
_..,-;*
a
Fig. 6. The energy of ellipsoidal martensite assuming no interaction with dislocations. The full curves denote untwinned martensite and the broken curves denote twinned martensite (all values taken from Table 4 and based on ~nimum energy co~gurations). The figureshows that marten&e can not form s~ntaneousIy, and that the minimum energy growth patb is uphill until it reaches a size of about 1OOOAdia.
Fig. 7. The energy of ellipsoidal martensite assuming some interaction with dislocations (i.e. 30% of masimum possible interaction only, see Table 4). The full curves denote energies of untwinned martensite and broken curves denote twinned martensite. The figure shows that the small nuclei of unhvinned martensite can form but they are unlikely to grow larger than lOO-ZOO Adia. On the other hand the growth path for twinned martensite is continuously downhill shouing that its formation and growth should be spontaneous in the presence of dislocations.
dislocations, and Fig. 7 concerns nucleation with the help of dislocations (i.e. the middle E,,, column denoting partial help only). As shown in Fig. 6, without dislocations the energy barrier is always positive, unless the nucleus is very large and tamned. Even if the nucleus is twinned, there is thus little chance that nuclei form which are able to grow spontaneously. The situation is quite different when dislocations help the nucteation process (Fig. 7). In this case, even the untwinned nuclei are able to form up to a certain size, although they then appear to run up against a nucleation barrier between a = IO and 100 A. On the other hand, twinned nuclei can et
EASTERLING
AZD
THijLgN:
THE NUCLEATION
and grow substantially after reaching a size of about 400-5OOA diameter. It could indeed be conjectured from Fig. 7 that the initial nucleus is untwinned, but when this is unable to grow further, twinning occurs releasing energy for rapid thickening and growth. In other words, the growth path changes somewhere between 10 and 100A and in doing so the nucleus grows rapidly and spontaneousIy. Following this criticai growth stage, it is unlikely that existing dislocations play any role in subsequent thickening and growth up to the full-size plate. 5. DISCUSSION The above calc~ations have shown that not only is martensite nucleation a heterogeneous process, but that the end product is likely to be finely twinned. The nucleation process depends on a favourable reaction occurring between the strain field of the Bain deformation and the strain field surrounding dislocations, or groups of dislocations in the austenite. Thus the dislocations acting as nucleation sites for the martensite have no need to possess some critical orientation or configuration. They do not strictly speaking need to be a structural part of the nucleus, any more than for instance the impurities that assist heterogeneous solidification. Both transformations, martensitic and solidification, borrow energy from existing heterogeneities, the former in the form of strain energy, the latter in the form of surface energy. In this respect the role of dislocations is somewhat similar to that of catalysts in chemical reactions. Since all practical steels contain disl~ations even after long, high temperature annealing, it follows that the MS temperature corresponding to the bulk formation of martensite is in practice highly reproducible. As shown by Fig. 2, the strain energy of the,transformation can be minimized significantly if the coherent ellipsoidal martensite forms within a certain, well defined band of orientations, defined in the figure as the minimum energy trough“. The same conditions apply to the formation of twinned nuclei, as shown by Fig. 4. We thus see the nucleation process as an interplay between choice of disIocation site and orientation of the martensite nucleus with which it reacts. Bulk transformations in steels need a number of nucleation sites per grain, which requires in turn that the conditions for nucleation cannot be too delicately balanced. On this basis it should not be surprising if martensite forms at orientations practically anywhere within a fairly broad low energy trough of the type shown in Fig. 4 (case 6) for instance. This would explain why the measured habit planes of martensite are usually irrational and can vary over several degrees in any given steel [2]. We have shown that twinned nuclei of martensite have a lower energy than untwinned nuclei (see, e.g. Figs. 4,? and 8). However, we envisage the first nuclei to be formed by the Bain deformation, and twinning then occurs to release energy for further thickening
OF MARTENSITE
IS STEEL
33Y
and growth. Referring to Fig. 7. sometime early in the process the nucleus growth path moves from the untwinned form to that of the twinned type. Indeed, still further strain energy reduction can be obtained if twins form on two or three systems instead of only one as assumed in the present calculations, reducing s and hence the strain energy, at the cost however of increasing surface energy (due to the larger number of twin interfaces). Such multi-finned examples are to be found in the literature [5]. As shown by Fig. 7. following the nucleation of twins, further growth and thickening of the martensite plates is no longer dependent upon the presence of existing dislocations in the austenite, and spontaneous growth occurs until the plate hits a grain boundary or some other barrier. Of course, the calculations made in the present paper are based primarily upon energy considerations and we have not considered twin growth mechanisms as such. It is thought [2] that the essential differences between lath martensite with its dislocation sub-structure. and fully twinned martensite is due to their different MS temperature. At higher MS temperatures it is possible that growth of the nucleus occurs by a dislocation mechanism, e.g. by creating dislocations at the interface of the nucleus, or by attracting existing dislocations to relieve the strain energy. Preliminary calculations show that both mechanisms are feasible [lo]. If this happens twinning becomes unnecessary as a means of further growth and thickening. In fact the essential differences between lath and twinned martensite may be linked with the ability of dislocations to play a role in the growth process.
6. THE PRESEilT THEORY Ill THE LIGHT OF PREVIOUS WORK (a) Co~~ri~~ with theories o~~rre~site cr~stu~logr~phy (for details of these theories, see Ref. 2) In general there is fairly good agreement between the two approaches concerning the predicted habit planes of martensite plates containing twins. For instance, Fig. 4 shows that for certain twin configurations, habit orientations roughly corresponding to {225}7 through to {3.iO.i5jy are to be expected. Kowever, on the assumption that full grown plates inherit their habit planes from those of the nuclei. our results very well explain the scatter in orientation about the energy minima, as measured experimentally ie.g. compare our Fig. 4, case 6, with Fig. XXII, 15 in Christian’s book 12-j). On the basis of our approach, martensite nucleation is dependent on obtaining both: a plate orientation in a minimum strain energy form and a suitable orientated dislocation with which the plate can interact. In practice this interaction is likely to be achieved by adjustments in orientation between the dislocation and plate, at least within the bounds of the low energy troughs of Fig. 4.
340
(b) Comparison
EASTERLING
AND
THijLiS:
THE NUCLEATION OF MARTENSITE IN STEEL
with experimental obsermtions
The present theory has revealed martensite nucleation to be a simple dislocation-assisted process. How well does this interpretation match up to experimental observations’? As mentioned above, the theory correctly predicts the scatter and irrational nature of habit planes, as well as the fine twinned structure of martensite. It also predicts that plates are nucleated heterogeneously and randomly, i.e. independently of grain boundaries, again in agreement with experiment [4]. It is further predicted that without dislocations present in the austenite, martensite could not form at all (except possibly at external surfaces where the lattice restraints are relaxed). The work of Easterling and Miekk-oja [ 111 showed that small (- 300 A), defectfree particles of austenitic iron in a copper matrix could not transform to martensite unless dislocations were forced through the particles. It was not necessary that dislocations remained inside the particles, only that dislocations were present to allow the necessary strain field interaction to occur. Experimentally it is observed that in athermal transformations in steel, nucleation sites appear to be fairly coarsely distributed [2-4]. Assuming reasonable figures for the ratio of number of nucleation sites to number of dislocations in a well annealed austenite, it can be estimated that only about one in ten dislocations nucleates martensite. On the basis of our theory, this is not a surprising result. The orientation relationship between the embryo and dislocation has to be just right if components of shear or dilatation are to be accommodated such as to bring the total strain energy below the available free energy. On the other hand, isothermal transformation appears to be a very efficient process in which the majority of dislocations are utilized as nucleation sites. Since isothermal transformations are thought to be influenced by thermal fluctuations [2], it can be surmised that thermal fluctuations could help dislocations to move in a way that enhances their interaction with the Bain deformation. It is not presently obvious, however, why thermal fluctuations should be more effective in some alloys than in others. The influence of plastic deformation on transformation to martensite is very interesting with respect to the present work. Experimentally it is found that small amounts of plastic deformation increase the MS temperature, while large amounts decrease MS [2]. Qualitatively it could be expected that increases in dislocation density by deformation likewise increases the number of nucleation sites, and that the suppression of MS by large deformations is due to the higher restraint of the deformed austenite. We have been unable to find any systematic work which plots the maximum increase in MS temperature as a function of deformation, although the indications are that this is fairly low (5-10% strain) in most steels [IZ]. It may be possible to put this on more quantitative grounds,
if it is assumed that the optimum deformation corresponding to the highest MS occurs when the mean spacing between dislocations is less than the critical plate size for martensite nucleation (400-5OOA in steels). Broadly speaking, this sort of dislocation density would probably be achieved at fairly low deformations in steels, in agreement with the above results. 7. CONCLUSIONS (1) Calculations of the total energy for transforming austenite to martensite in the form of thin ellipsoidal plates fully coherent with the austenite, show that at the MS temperature the process may be spontaneous in the presence of pre-existing dislocations. It is found that a dislocation, or group of dislocations in the austenite, are suitable sites for martensite nucleation in that their strain fields may interact favourably with the strain field of the Bain deformation, eliminating the energy barrier to nucleation. (2) It is found that martensite is most likely to be twinned. The driving force for twinning to occur virtually simultaneously with nucleation is large, and when this happens energy is released for thickening of the nucleus and further growth. It is shown that the size of nucleus when rapid thickening and growth occurs is about 400-5OOA dia. (3) Calculations show that the orientation of a coherent nucleus in the austenite, whether twinned or untwinned, significantly affects its strain energy. In certain twinning configurations, the lowest energies of the coherent ellipsoidal martensite nuclei occur over a fairly wide, but well defined range of orientations in the austenite, taking the form of a “low energy trough” in a stereographic projection. This could be an explanation of why measured habit planes of fully grown martensite plates are irrational and may vary over several degrees in a given alloy. (4) The proposed theory of dislocation-assisted nucleation of martensite is qualitatively able to account for the majority of experimental observations pertaining to martensite nucleation. Acknowledgements-We
should like to thank Prof Morris Cohen (M.I.T.) for some very helpful correspondence and guidance during the early stages of this work. We also thank Michael Hyman, tekn lit (University of Luleh) for critical comments on the manuscript. REFJ.XENCES 1. R. F. Bunshah and R. F. Mehl, Trans. AI&fME
1251 (1953). 2. J. W. Christian,
197,
The Theory of Transformationsin and Alloys. Pergamon P&s, Oxford (1965). 3. L. Kaufman and M. Cohen. Proa. Metal Phvs. 7. 165 Metals
”
.
(1958). 4. S. R. Pati and M. Cohen, Acta Met. 17, 189 (1969). 5. C. M. Wayman, Modern Difiaction and Imaging Techniques in Marerial Science, (Ed. S. Amelinckx et al.), p. 187. North-Holland, Amsterdam (1970). 6. K. Shimizu and Z. Nishiyama, Met. 7&s. 3, 1055 (1972).
EASTERLING
AND
THijLfN:
THE
VJCLE.-\TION
7. E. C. Bain, 7&s. rll.VME 70. 25 11924). 8. M. S. Wechsler, D. S. Lieberman and T. A. Read, ibid. 197, 1503 (1953). 9. J. W. Eshelby, Proc. R. Sot. (.-II 241. 3i6 (1957). IO. K. E. Easterlirg and A. R. ThiilCn, Coherency loss of
OF >lXRTENSITE
IN STEEL
341
a martensite nucleus. Chalmers University of Technology Int. Rept. (1971). It. K. E. Easterling and H. M. Miekk-oja. Acra Met. 15, 1133 (1967). 12. P. J. Kelly and J. Nutting. J.I.S.I. 197, 199 (1961).
OF MARTENSITE
IN STEEL
K. E. EASTERLING Department of Materials Technology, University of Lulea. S-951 57 Sweden
and A. R. ‘I-I&LEN Department of Applied Physics I. University of Technology. Lyngby. Denmark (Received 9 3ul.r 1975: in recked form 23 October
1975)
Abstract-Calculations of the total energy for transforming austenite to martensite in the form of thin ellipsoidal plates, fully coherent with the austenite, show that the process may be spontaneous in the presence of pre-existing dislocations. It is found that dislocations, or groups of dislocations, in the austenite are suitable sites for martensite nucleation in that their strain fields may interact favourably with the strain field associated with the Bain deformation thereby eliminating the energy barrier to nucleation. The driving force for twinning to occur virtually simultaneously with nucleation is large and when this happens energy is released for thickening and growth of the nucleus. It is also found that the strain energy of coherent plates of martensite. whether twinned or untwinned. is a function of their orientation in the austenite, although the lowest strain energy cases occur nevertheless over a relatively wide range of orientations. The proposed theory of dislocation-assisted nucleation of martensite is qualitatively able to account for the majority of experimental observations pertaining to martensite nucleation. R&r&--Le calcul de l’inergie totale pour la transformation de l’austinite en martensite sous forme de fines plaquettes ellipsoidales, parfaitement coherentes avec t’ausdnite, montre que ce processus peutCtre spontani si des dislocations preexistent. On a trouve que des dislocations ou des groupes de dislocations forment, dans I’austenite, des sites de germination convenablcs pour la reaction martensirique: en effet. leur champ de deformation peut agir favorablement avec le champ de deformation associt B la deformation de Bain et iliminer ainsi la barriere d’energie de germination. La force motrice pour que le maclage apparaisse pratiquement en meme temps que la germination est grande, et dans se cas de l’energie est liberee pour le grossissement et la croissance des germes. On trouve egalemcnt que l’bnergie ilastique de plaquettes de martensite cohirentes. maclees ou non, depend de leur orientation par rapport Q l’austenite, bien que Ies cas d’energie tlastique minimale se produisent dans un domaine relativement etendu &orientations. Cette thtorie de la germination de la martensite aidee par des dislocations, permet de rendre compte de la majorite des observations experimentales concernant la germination de la martensite. Z~~~~~-~rechnungen der Gesamtenergie fur die Umwandlung von Austenit in Martensit in Gestalt dlinner ellipsoidf~rmiger Platten. welche vollstandig kohlrent mit dem Austenit sind, zeigen. daB der Prozess in Gegenwart van Versetzungen spontan ablaufen kann. Es wird gefunden, dal3 im Austenit Versetzungen oder Versetzungsgruppen geeignete Orte fir Martensit-Keimbildung sind, indem deren Verzenungsfelder giinstig mit denjenigen, mit der Bain-Verformung zusammenhangenden wechselwirken. Dabei wird die Energieschwelle der Keimbildung aufgehoben. Die treibende Kraft fur die etwa gleichzeitig mit der Keimbildung ablaufende Zwillingsbildung ist groB, und wenn diese ablIuft, wird Energie fur Dicken- und GroBenwachstum des Keimes frei. Es wurde ebenfails gefunden, dal3 die Verzerrungsenergie koharenter Martensitplatten. ob verzwillingt oder nicht, von deren Orientierung im Austenit abhtingt, wenn such die Fllle kleinster Verzerrungsenergie in einem relativ weiten Orientierungsbereich liegen. Die vorgelegte Theorie der versetzungsunterstitzten Martensit-Keimbildung beriicksichtigt qualitativ die meisten Beobachtungen zur Martensit-Keimbildung.
1. NTRoDwcrIoN The hardening of steels by quenching from high temperatures to obtain martensite is one of the oldest and most important metallurgical treatments known. It is an intriguing fact, however, that the mechanism by which martensite forms is still not clarified. A satisfactory theory of martensite nucleation in steels ought to: (a) show specifically how the diffusionless shear transformation nucleates from austenite, given the free energy difference between the austenite and
martensite phases at the MS temperature: (b) explain how, and what sort of heterogeneities assist nucieation and growth: (c) show the influence of the martensite embryo on the form and fine structure of the fullgrown plates. Inclusion of the latter requirement is justified on the basis that since martensite grows, without diffusion, at speeds approaching the velocity of sound [l J, there should be structural and crystallographic features common to both the nucleus and the full-grown plates [2].
334
EASTERLING
ASD
THijLfN:
THE NUCLEATIOX OF MARTENSITE IN STEEL
It has to be stated that theories of martensite nucleation have made disappointing progress since 1958 when Kaufman and Cohen published their excellent review of the subject [3]. The main reason for this is because of the experimental difficulties associated with studying nucleation, at least in steels. A single plate grows at such speed that it may span a complete grain within lo-’ set [l]. Nucleation sites appear to be randomly and fairly coarsely distributed in any given grain [3], and since electron microscopy is needed to study nucleation the chances of seeing any thing meaningful are extremely remote [4]. Electron microscopy and X-ray studies of transformed martensite have been useful, however, (see, e.g. the reviews of Christian [2] and Wayman [j]) and these results have clarified, e.g. the irrational and variable nature of the measured habit planes, the very fine (-20 A thick) twinned structure of typical steel martensites, and (more recently) the confirmation that the Bain distortion occurs (together with inhomogeneous shears) in ferrous martensite transformations [6]. The latter result in particular is an important one because until that piece of work the Bain deformation was considered a mathematical operation rather than a physical quantity. Indeed we shall take this result as the starting point of the present work. In 1924 Bain [7] showed how martensite can be generated from austenite by a homogeneous strain in which two of the cube axes of the f.c.c. lattice expand by about 12Y4and the other axis contracts by about 20?/,, (these amounts depend, of course, on the alloying constituents). Later it was shown that the so-called Bain deformation is the most efficient way of generating the martensite lattice in that it requires the minimum of atomic movement to do so [8].‘On the other hand, it has been found that the Bain deformation alone can not account for the observed invariant plane strain characteristics of fully grown martensite plates [2,5], and this led to the proposal, later confirmed experimentally, that in addition to the Bain deformation, heterogeneous shears (twinning or slip) were also needed. This form of transformation is mainly based on the criterion that the strain energy associated with the full-grown martensite plate (with respect to the surrounding austenite) be a minimum. As we shall now show, the same strain energy criterion appears to govern the fine structure and orientation of the critical martensite nucteus.
refers to the coherency strain due to the misfit between the two lattices and Eshcmis the free energy. difference, Ax per unit volume, between the austenite and martensite at the MS temperature. Assuming the martensite forms as a thin ellipsoid of volume V; with semi-thickness c and radius a, and that the transformation can be expressed as a simple shear, s, parallel to the plane of the ellipsoid, equation (1) may be written:
where ~1is the shear modulus of the austenite, v is Poisson’s ratio and V is the total volume of the nucleus. If v = l/3: 16x E101= 2nn=o-I” 3 (s/2)2pac2
(3)
In a Bain deformation of (1,12; 1,12; 0,80) with the principal strains parallel to the axes of the martensite ellipsoidal nucleus, in which c/a = 403, the coherency strain energy as calculated by Eshelby’s expression [93 yields a value of about 0,OSpV, or about 25 times the available chemical free energy (the last term in equation 3). In other words, even without the surface energy contribution the nucfeation barrier for critical growth is much too large for martensite to form spontaneously, assuming realistic values of AJ If the principal strains are changed, e.g. to (0,SO; 1,12; 1,12), the coherency energy is virtually unchanged. In other words, it makes no difference if the contraction is parallel to the small axis or not. The energy obtained is also fairly insensitive to the c/u ratio chosen. The question now arises, whether it is energetically more favourable for the ellipsoid to form at some othersorientation in the austensite, but still retaining the Bain deformation as the mode of transformation. Using a computer the energies for all possible orientations of the ellipsoid could be readily calculated. This was carried out using Eshelby’s (91 equation for the coherency energy due to a homogeneous shear, and by converting the principal strains of the transformation to the new coordinate system, as shown in Fig. 1. Since Eshelby’s treatment clearly distinguishes between shear and dilatational stresses, the respective energies could also be calculated independently of each other. Table 1 gives the lowest strain energy values resulting from these calculations, together with the shear 2. ENERGY CONSIDERATIONS and dilatational components of these energies, in Consider an ellipsoidal region of martensite, transterms of different c/u ratios (0.01; 0,03; 0.10). formed by a Bain deformation from the austenite, It is seen from the results given in Table 1 that such that the interface between the two lattices the minimum energies in the case of c/a = O,Ol, are remains fully coherent. The total energy change as- about 29 times less than for the case where the ellipsociated with the transformation is: soid is parallel to the cubic axes of the austensite. However, this is still ten times larger than the Ap S*0,= Erurf + Emai* + &hem (1) term in equation (3), and the transformation will not occur spontaneously by the Bain deformation, irreswhere EPUifcorresponds to the coherent interfacial energy between the austensite and martensite, Ertrain pective of the shape or orientation of the martensite
EASTERLIKG
ASD THijLEN:
THE PU’UCLEATION OF MARTEYXf’E
Ipu’ STEEL
335
Table 1. Examples of some of the lowest coherency energies of martensite ellipsoids transformed by a (1.12; 1.12; 0.80) Bain deformation
Fig. 1. (x), (y), (z) are principal axes of the Bain deformation, and (1).(2) (3) are the axes of the martensite ellipsoid. The ellipsoid’s orientation, together with its relevant coherency energy, can be plotted on a 001 stereographic projection, as in Fig. 2.
The total (shear + dilatational) minimum energy orientations are plotted on a 001 stereographic projection in Fig. 2 for the case in which c/a = 0.01. Note that the minimum coherency energy is not confined to a single orientation, but appears to extend over a range of orientations. This range is defined as a “minimum energy trough”, and is shown “crosshatched” in Fig. 2. An important result that emerges from the computed results, is that one of the components of either shear or dilatational energy may constitute a large proportion of the total strain energy. For instance, in the case where c,‘n = 0,Ol (see Table I), the dilatational component may be as low as 0,007 PV comprising in fact only about one third of the total energy. The possibility thus arises that if the Bain deformation could occur simultaneously in adjacent regions of the austenite in such a way that certain components of shear or dilatational energy compensate one another, a net reduction in total coherency energy should result.
3. TWINNED ellipsoid in the austenite. This result is hardly a surprise, since we know from experimental observations that the Bain deformation alone can not account for the form and orientation taken by martensite in practice.
MARTENSTTE
Consider an ellipsoidal martensite nucleus which has undergone the Bain deformation in adjacent regions of the austenite, such that after the uansformation the regions are twin related. Assume that the width of adjacent twins can be varied as shown in Fig. 3.
001
Fig. 2. Coherency energies of ellipsoidal martensite, transformed by a Bain deformation of the austenite, (in terms of 0.001 pV); c/a = 0.01. The “minimum energy trough’ (crosshatched) denotes the range of orientations over which the coherency energies are consistently low, although the actual energy minima describe a single line along the centre of the trough.
Fig. 3. A schematic diagram of a twinned, ellipsoidal nucleus of martensite. Alternate twins have different widths, I and II, and by adjusting these widths, as aefl as the orientation of the ellipsoid in the austenite, the strain energy can be changed (see Fig. 4).
EASTERLING
336
THijLeS:
ASD
THE NUCLEATION
Table 2. Net strains of twinned martensite ellipsoids (see Fig. 4) ‘:a:.:
c;
c;
3 ;
,
b 1
0.12
3.:2
-:.zc
2
0.12
3.2&
-C,l5
3
0.12
0.v
-0.12
4
0.12
i.00
-a.08
5
0.12
-0.22
-rl,:c
6
0.12
-c.ia
-:.:3
:‘LI-s jdinl
Let k be the proportion of thickness I twin (or twins) in the nucleus. By varying the relative thicknesses of adjacent twins, the energy can also be minimised (this treatment is also adopted in the crystallographic theories [2]). The changes in principal strains are related to k as follows: in the x direction, the net strain = E,, in the y direction, the net strain = E, + (1 - k)E3, in the z direction, the net strain = 6s + (1 - k)El. The net shear may now be calculated as a function of k, E,~,c3. e.g: net shear = kf3 + (1 - k)E1 - kc1 - (1 - k)E3 = (1 - 2k&,
- e3),
when ei = 0,12; Ez = 0,12;
63
=
-0,20. > c
OF MARTENSITE
IN STEEL
The energies of these new net strains were calculated for five cases (in addition to the pure Bain deformation), and these results are summarized in Fig. 4. The net strains used are shown in Table 2. Figure 4 shows that by adjustment of adjacent twin widths very low coherency energies, of the order of AL can be obtained. Furthermore, the lowest energy troughs may be very shallow and quite extensive, see, e.g. case 6. The latter result is particularly interesting bearing in mind the experimental observations that the habit planes of martensite tend to be irrational, and differ by several degrees at least in a given alloy (see, e.g. Ref. 2, p. 876). It should also be noted, however, that the orientations covered by the “minimum energy troughs” can vary substantially from case to case. It is seen from Fig. 4 that the decrease in energy due to twinning is largely independent of the c/n ratio, although the difference in energy between c/a = 0.01 and 0,l at the minimum is obviously important when considering nucleation. In other words. the results show that the critical nucleus is likely to be a thin, twinned ellipsoid, the orientation of which is not too critical provided it lies within the “minimum energy trough”. Equations (l-3) showed that the formation of a coherent martensite nucleus is a function of surface, coherency and chemical energy. For the process to /
0.024
C/U~O.l
0 016
21
40
Fig. 4. Coherency energies of ellipsoidal martensite, as a function of twinning configuration and orientation in the austenite. The graph plots the energy minima for c/a = 0.1; 0.03; 0.01. The stereo-graphic plots give energies for c/a = 0.01 only. See Table 2 for net strains used. Note that the lowest energy orientations of the martensite (denoted by cross-hatching in the stereographic plots) may vary markedly from case to case.
E;\STERLII\;G
ASD
THijLfiX:
THE NUCLEATIO?i
OF MARTENSITE
1s STEEL
337
be spontaneous. the latter term (which is negative) must outweigh the other terms. The minimum free energy barrier to nucleation can be estimated by differentiating equation (3). with respect to n and c. and by subsequent substitution this yields:
in the simple case where & = 0, and the dilatational energy is assumed zero. As seen from this equation, the energy barrier is extremely sensitive to the values chosen for c, &and s. It is also obvious that the critical size of nucleus is likewise dependent upon these parameters. and it can be shown that:
and &it =
16ap(siZ)’
T’
Typically, Af = 1,74 x IO9 erg/cm3 (for steel) and s = 0.20 (**net”shear in a plate). We can only guess at the surface energy of a coherent twinned nucleus, and estimate a value of 20 ergjcm’. Strictly speaking, equation (6) is not fully applicable to a twinned nucleus since the total energy should include the twin boundary energy as well as the interaction energy at twin junctions. Estimations of these contributions have been attempted and indicate that the energy is not affected significantly by such refinements. In Table 3, we compare the nucleation barrier and optimum nucleus size of a fully coherent nucleus (20 erg/cm) with a semi-coherent nucleus containing a grid of dislocations at the interface (surface energy = 2~erg~~rn2 [33). As may be expected, a change in order of magnitude in the value of G brings about a thousand fold decrease in the nucleation barrier. However, even a coherent nucleus (nucleation barrier = 14 eV) will not form spontaneously and as we know from experiments, the transformation is in fact heterogeneous [3]_ It is generally thought that martensite nucleates at defects in the austenite (e.g. groups of dislocations) which somehow help to reduce the energy for nucleation. We now consider this possibility in more detail. 4. MARTENSITE NUCLEATION AT DISLOCATIONS We will assume that the strain field associated with a dislocation could in certain cases provide a favourTable 3. Energy of formation of critical nuclei of martensite
Fig. 5. The formation of a thin martensite nucleus at a dislocation loop.
able interaction with the strain fieid of the nucleus, such chat one of the shear terms of the Bain deformation is neutralized. This interaction changes the total energy equation to:
Consider a dislocation loop, in which a thin martensite nucleus has formed, as illustrated in Fig. 5. The interaction energy between the nucleus and the disiocation can be expressed as: Ei”$ -
-j.f.sWa”
4
2 -r
‘-‘-
c
(8)
I-VU
where b is the burgers vector of the dislocation. Assuming v = l/3, EI”1= - 2 psx abc.
(9)
The above approach assumes that the martensite nucleus reacts in the most efficient way with a complete dislocation loop. A more likely event is that martensite forms at one part of dislocation, e.g. at a curved segment or at a group of dislocations. On this basis the interaction energy defined by equation (9) is likely to be a maximum. We also assume in this approach that the dislocation loop is not a structural part of the nucleus. i.e. it does not contribute to the surface energy. We may now write equation (7) in more complete form as: 16R E,, = Zrra’a + 7 (s/2)2 PC’
-
Ai4;a’c -
‘psxabc.
(10)
It should be noted that the value of s chosen for the strain energy and interaction terms depends on whether the nucleus is twinned or not. By summing the various contributions to the total energy, it is now possible to compute the energy of the martensite nucleus as a function of size and thickness, whether it is twinned or untwinned, if it forms with or without pre-existing dislocations, etc, The results are presented in Table 4. Note that two dislocation interaction energies are considered (the two final columns in the table), the maximum value (given by equation 9) and a partial interaction energy of 3006 of the maximum. Thus the important columns are the last three, i.e. total energy without help from dislocations, energy with partial help from dislocations and energy with maximum assistance from a
338
EASTERLING
AND
THijLLiEi:
THE NUCLEATION
OF MARTENSfTE
IX STEEL
Table 4. Energies of martensite nucleation (in terms of 10-‘Oerg) as based on equation (10) and Fig. 4
n Id
S.26 fZ6
96.5 rzco
-7.60 -750
65.86 6423
37,:s 5.35
1.26 125
15f.3 113oc
-12 -fZOO
116.96 10.0?0
75,s XT:_
0,144 5.4 160
-0.036 -3.8 -380
-1.35 -13.5 -135
0.119 2.66 -74
-0.33 -1,64
-i.Z
1000
0.0126 1.25 126
-119
-2::
100 1000
I,26 126
-7.60 -760
-26.9 -264
3.06 -27b
-0.84 -363
_Ti :I _~-f-3
-40.6 -406
:3 26 +2i4
2.76 -399
-2s.: -5::
1 10
100
~
100 1000
I
C
I
dislocation. The columns showing negative energies in these columns mean that in these cases the final two terms in equation (10) outweigh the first two terms thus implying that rapid growth should occur. Significantly, we can see that under certain conditions martensite can form spontaneously at dislocations. On the other hand martensite is extremely unlikely to form spontaneously without assistance from dislocations. In order to clarify the growth characteristics of the martensite nucleus, the results of Table 4 are presented in thr~~imens~oM1 form plotting energy changes in terms of nucleus size in Figs. 6 and 7. In both figures the full curves denote untwinned martensite and the broken curves denote twinned martensite. Figure 6 describes nucleation without the aid of
emi-thickness,
_..,-;*
a
Fig. 6. The energy of ellipsoidal martensite assuming no interaction with dislocations. The full curves denote untwinned martensite and the broken curves denote twinned martensite (all values taken from Table 4 and based on ~nimum energy co~gurations). The figureshows that marten&e can not form s~ntaneousIy, and that the minimum energy growth patb is uphill until it reaches a size of about 1OOOAdia.
Fig. 7. The energy of ellipsoidal martensite assuming some interaction with dislocations (i.e. 30% of masimum possible interaction only, see Table 4). The full curves denote energies of untwinned martensite and broken curves denote twinned martensite. The figure shows that the small nuclei of unhvinned martensite can form but they are unlikely to grow larger than lOO-ZOO Adia. On the other hand the growth path for twinned martensite is continuously downhill shouing that its formation and growth should be spontaneous in the presence of dislocations.
dislocations, and Fig. 7 concerns nucleation with the help of dislocations (i.e. the middle E,,, column denoting partial help only). As shown in Fig. 6, without dislocations the energy barrier is always positive, unless the nucleus is very large and tamned. Even if the nucleus is twinned, there is thus little chance that nuclei form which are able to grow spontaneously. The situation is quite different when dislocations help the nucteation process (Fig. 7). In this case, even the untwinned nuclei are able to form up to a certain size, although they then appear to run up against a nucleation barrier between a = IO and 100 A. On the other hand, twinned nuclei can et
EASTERLING
AZD
THijLgN:
THE NUCLEATION
and grow substantially after reaching a size of about 400-5OOA diameter. It could indeed be conjectured from Fig. 7 that the initial nucleus is untwinned, but when this is unable to grow further, twinning occurs releasing energy for rapid thickening and growth. In other words, the growth path changes somewhere between 10 and 100A and in doing so the nucleus grows rapidly and spontaneousIy. Following this criticai growth stage, it is unlikely that existing dislocations play any role in subsequent thickening and growth up to the full-size plate. 5. DISCUSSION The above calc~ations have shown that not only is martensite nucleation a heterogeneous process, but that the end product is likely to be finely twinned. The nucleation process depends on a favourable reaction occurring between the strain field of the Bain deformation and the strain field surrounding dislocations, or groups of dislocations in the austenite. Thus the dislocations acting as nucleation sites for the martensite have no need to possess some critical orientation or configuration. They do not strictly speaking need to be a structural part of the nucleus, any more than for instance the impurities that assist heterogeneous solidification. Both transformations, martensitic and solidification, borrow energy from existing heterogeneities, the former in the form of strain energy, the latter in the form of surface energy. In this respect the role of dislocations is somewhat similar to that of catalysts in chemical reactions. Since all practical steels contain disl~ations even after long, high temperature annealing, it follows that the MS temperature corresponding to the bulk formation of martensite is in practice highly reproducible. As shown by Fig. 2, the strain energy of the,transformation can be minimized significantly if the coherent ellipsoidal martensite forms within a certain, well defined band of orientations, defined in the figure as the minimum energy trough“. The same conditions apply to the formation of twinned nuclei, as shown by Fig. 4. We thus see the nucleation process as an interplay between choice of disIocation site and orientation of the martensite nucleus with which it reacts. Bulk transformations in steels need a number of nucleation sites per grain, which requires in turn that the conditions for nucleation cannot be too delicately balanced. On this basis it should not be surprising if martensite forms at orientations practically anywhere within a fairly broad low energy trough of the type shown in Fig. 4 (case 6) for instance. This would explain why the measured habit planes of martensite are usually irrational and can vary over several degrees in any given steel [2]. We have shown that twinned nuclei of martensite have a lower energy than untwinned nuclei (see, e.g. Figs. 4,? and 8). However, we envisage the first nuclei to be formed by the Bain deformation, and twinning then occurs to release energy for further thickening
OF MARTENSITE
IS STEEL
33Y
and growth. Referring to Fig. 7. sometime early in the process the nucleus growth path moves from the untwinned form to that of the twinned type. Indeed, still further strain energy reduction can be obtained if twins form on two or three systems instead of only one as assumed in the present calculations, reducing s and hence the strain energy, at the cost however of increasing surface energy (due to the larger number of twin interfaces). Such multi-finned examples are to be found in the literature [5]. As shown by Fig. 7. following the nucleation of twins, further growth and thickening of the martensite plates is no longer dependent upon the presence of existing dislocations in the austenite, and spontaneous growth occurs until the plate hits a grain boundary or some other barrier. Of course, the calculations made in the present paper are based primarily upon energy considerations and we have not considered twin growth mechanisms as such. It is thought [2] that the essential differences between lath martensite with its dislocation sub-structure. and fully twinned martensite is due to their different MS temperature. At higher MS temperatures it is possible that growth of the nucleus occurs by a dislocation mechanism, e.g. by creating dislocations at the interface of the nucleus, or by attracting existing dislocations to relieve the strain energy. Preliminary calculations show that both mechanisms are feasible [lo]. If this happens twinning becomes unnecessary as a means of further growth and thickening. In fact the essential differences between lath and twinned martensite may be linked with the ability of dislocations to play a role in the growth process.
6. THE PRESEilT THEORY Ill THE LIGHT OF PREVIOUS WORK (a) Co~~ri~~ with theories o~~rre~site cr~stu~logr~phy (for details of these theories, see Ref. 2) In general there is fairly good agreement between the two approaches concerning the predicted habit planes of martensite plates containing twins. For instance, Fig. 4 shows that for certain twin configurations, habit orientations roughly corresponding to {225}7 through to {3.iO.i5jy are to be expected. Kowever, on the assumption that full grown plates inherit their habit planes from those of the nuclei. our results very well explain the scatter in orientation about the energy minima, as measured experimentally ie.g. compare our Fig. 4, case 6, with Fig. XXII, 15 in Christian’s book 12-j). On the basis of our approach, martensite nucleation is dependent on obtaining both: a plate orientation in a minimum strain energy form and a suitable orientated dislocation with which the plate can interact. In practice this interaction is likely to be achieved by adjustments in orientation between the dislocation and plate, at least within the bounds of the low energy troughs of Fig. 4.
340
(b) Comparison
EASTERLING
AND
THijLiS:
THE NUCLEATION OF MARTENSITE IN STEEL
with experimental obsermtions
The present theory has revealed martensite nucleation to be a simple dislocation-assisted process. How well does this interpretation match up to experimental observations’? As mentioned above, the theory correctly predicts the scatter and irrational nature of habit planes, as well as the fine twinned structure of martensite. It also predicts that plates are nucleated heterogeneously and randomly, i.e. independently of grain boundaries, again in agreement with experiment [4]. It is further predicted that without dislocations present in the austenite, martensite could not form at all (except possibly at external surfaces where the lattice restraints are relaxed). The work of Easterling and Miekk-oja [ 111 showed that small (- 300 A), defectfree particles of austenitic iron in a copper matrix could not transform to martensite unless dislocations were forced through the particles. It was not necessary that dislocations remained inside the particles, only that dislocations were present to allow the necessary strain field interaction to occur. Experimentally it is observed that in athermal transformations in steel, nucleation sites appear to be fairly coarsely distributed [2-4]. Assuming reasonable figures for the ratio of number of nucleation sites to number of dislocations in a well annealed austenite, it can be estimated that only about one in ten dislocations nucleates martensite. On the basis of our theory, this is not a surprising result. The orientation relationship between the embryo and dislocation has to be just right if components of shear or dilatation are to be accommodated such as to bring the total strain energy below the available free energy. On the other hand, isothermal transformation appears to be a very efficient process in which the majority of dislocations are utilized as nucleation sites. Since isothermal transformations are thought to be influenced by thermal fluctuations [2], it can be surmised that thermal fluctuations could help dislocations to move in a way that enhances their interaction with the Bain deformation. It is not presently obvious, however, why thermal fluctuations should be more effective in some alloys than in others. The influence of plastic deformation on transformation to martensite is very interesting with respect to the present work. Experimentally it is found that small amounts of plastic deformation increase the MS temperature, while large amounts decrease MS [2]. Qualitatively it could be expected that increases in dislocation density by deformation likewise increases the number of nucleation sites, and that the suppression of MS by large deformations is due to the higher restraint of the deformed austenite. We have been unable to find any systematic work which plots the maximum increase in MS temperature as a function of deformation, although the indications are that this is fairly low (5-10% strain) in most steels [IZ]. It may be possible to put this on more quantitative grounds,
if it is assumed that the optimum deformation corresponding to the highest MS occurs when the mean spacing between dislocations is less than the critical plate size for martensite nucleation (400-5OOA in steels). Broadly speaking, this sort of dislocation density would probably be achieved at fairly low deformations in steels, in agreement with the above results. 7. CONCLUSIONS (1) Calculations of the total energy for transforming austenite to martensite in the form of thin ellipsoidal plates fully coherent with the austenite, show that at the MS temperature the process may be spontaneous in the presence of pre-existing dislocations. It is found that a dislocation, or group of dislocations in the austenite, are suitable sites for martensite nucleation in that their strain fields may interact favourably with the strain field of the Bain deformation, eliminating the energy barrier to nucleation. (2) It is found that martensite is most likely to be twinned. The driving force for twinning to occur virtually simultaneously with nucleation is large, and when this happens energy is released for thickening of the nucleus and further growth. It is shown that the size of nucleus when rapid thickening and growth occurs is about 400-5OOA dia. (3) Calculations show that the orientation of a coherent nucleus in the austenite, whether twinned or untwinned, significantly affects its strain energy. In certain twinning configurations, the lowest energies of the coherent ellipsoidal martensite nuclei occur over a fairly wide, but well defined range of orientations in the austenite, taking the form of a “low energy trough” in a stereographic projection. This could be an explanation of why measured habit planes of fully grown martensite plates are irrational and may vary over several degrees in a given alloy. (4) The proposed theory of dislocation-assisted nucleation of martensite is qualitatively able to account for the majority of experimental observations pertaining to martensite nucleation. Acknowledgements-We
should like to thank Prof Morris Cohen (M.I.T.) for some very helpful correspondence and guidance during the early stages of this work. We also thank Michael Hyman, tekn lit (University of Luleh) for critical comments on the manuscript. REFJ.XENCES 1. R. F. Bunshah and R. F. Mehl, Trans. AI&fME
1251 (1953). 2. J. W. Christian,
197,
The Theory of Transformationsin and Alloys. Pergamon P&s, Oxford (1965). 3. L. Kaufman and M. Cohen. Proa. Metal Phvs. 7. 165 Metals
”
.
(1958). 4. S. R. Pati and M. Cohen, Acta Met. 17, 189 (1969). 5. C. M. Wayman, Modern Difiaction and Imaging Techniques in Marerial Science, (Ed. S. Amelinckx et al.), p. 187. North-Holland, Amsterdam (1970). 6. K. Shimizu and Z. Nishiyama, Met. 7&s. 3, 1055 (1972).
EASTERLING
AND
THijLfN:
THE
VJCLE.-\TION
7. E. C. Bain, 7&s. rll.VME 70. 25 11924). 8. M. S. Wechsler, D. S. Lieberman and T. A. Read, ibid. 197, 1503 (1953). 9. J. W. Eshelby, Proc. R. Sot. (.-II 241. 3i6 (1957). IO. K. E. Easterlirg and A. R. ThiilCn, Coherency loss of
OF >lXRTENSITE
IN STEEL
341
a martensite nucleus. Chalmers University of Technology Int. Rept. (1971). It. K. E. Easterling and H. M. Miekk-oja. Acra Met. 15, 1133 (1967). 12. P. J. Kelly and J. Nutting. J.I.S.I. 197, 199 (1961).