PHYSICA ELSEVIER Physica A 224 (1996) 403-411

Kinematic theory for scale-invariant patterns in acicular martensites

M a d a n R a o a, S u r a j i t S e n g u p t a b a Institute of Mathematical Sciences, Madras 600113, India

b Material Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, India

Abstract

We present a kinematic theory which explains the emergence of scale-invariant patterns in acicular martensites which occur, for example, in Fe-C and Fe-Ni alloys. Scale-invariant structures emerge naturally as a consequence of competition between the average tip-velocity, v, and the rate of nucleation, I, of the martensite grains. Martensite growth is analyzed in terms of fixed points of a well-defined renormalization group. It is shown that the stationary probability distribution of the martensite grain size P( / ) , is governed by two fixed points - an unstable (noncritical) fixed point at v/ I = 0, characterized by a Gamma distribution and a stable (critical) fixed point at v/1 = oo, characterized by a I_,6vy distribution, P(I) ~ l -a . A universal crossover function describes the SPD at intermediate values of v/l . The analysis may also be used to compute the tip-velocity from optical micrograph pictures of martensite grains.

Scale-invariant patterns [ 1 ] are common in nature. A variety of systems, like the

human lung [2] , cracks in brittle materials [3] , distribution of galaxies in the uni- verse [4] and martensitic microstructures [5] , show scale invariance over reasonably large length scales. However, in most cases, the origin of this scale invariance is not fully understood, In this paper, we furnish an explanation for the origin of scale-invariant mi-

crostructures in acicular martensites [6,8] as a straight-forward example of a crossover process involving a flow towards a "self-organized critical" (SOC) fixed point of a

renormalisation group (RG) transformation. The reader will find a detailed account of this work in Ref. [9].

The name "martensite" has been used to denote metastable configurations arising from structural transitions in a wide variety of systems [6,8] ranging from metals to polymeric

crystals and crystalline membranes [ 10]. To be specific, we therefore specialize to the well-studied example of acicular martensites occurring in Fe -C and Fe-Ni systems. Pure Fe exists in three crystalline phases [6] at atmospheric pressures, denoted by a

0378-4371/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 l ( 9 5 ) 0 0 3 4 9 - 5

404 M. Rao, S. Sengupta/Physica A 224 (1996) 403-411

(ferrite),), (austenite) and 8. The room temperature structure of Fe is BCC (a ) , which when heated to a temperature of 910 ° C, undergoes a structural change to the FCC (T) phase. Further heating to 1400 ° C transforms it back to BCC (8). Fast quenching can preempt the equilibrium solid state transformation T ~ ~ by the formation of a martensite. Alloying elements, e.g. C and Ni, are known to facilitate the formation of this metastable phase. We shall now discuss the kinetic and structural features of such martensites.

Upon quenching from the 9' phase to temperatures below the martensite-start tem- perature Tins, nuclei of the transformed phase grow within the austenite. The nucleation rate is independent of temperature (athermal) for martensites formed typically in Fe-C and Fe-Ni alloys. The dependence of nucleation rates on time is difficult to determine, but it is suggested that this is essentially a constant. This implies that the nucleation rates per unit volume of the untransformed material decreases with time [6]. Nucleation events for martensites are often localised close to defects in the parent lattice which presumably lowers elastic energy barriers.

Once nucleated, these martensite grains grow as lens-shaped plates with a constant radial velocity [ 11 ]. The martensitic plates in iron-based systems are known to grow extremely fast (,~ 105 cm s-I ) ~ and consequently there is very little diffusion of atoms. The elastic energy cost of transforming a finite region of the austenite phase (FCC) to BCC is large - this is crucial in determining the morphology of the plates. The solid minimises this elastic energy by forming lens-shaped plates with a well-defined substructure composed of alternating twinned variants of the BCC structure [8,12]. The plates have a high aspect ratio, and in two-dimensional sections appear as thin needles (hence the name acicular). The orientation of the crystallographic axes of the BCC crystallites within these plates relative to those of the parent matrix follow strict relationships. The plates themselves grow in such a way that their orientation (the habit plane) is fixed relative to the austenite. Finally, the martensite plates stop on colliding with other plates or with the austenite grain boundary, due to large elastic energy barriers. The growth thus proceeds without coalescence or breakup. A singular feature of this transformation is that the total nucleation time is finite, and so there is always some fraction of untransformed material (viz., "retained austenite").

Before elaborating on our Kinematic Theory describing the emergence of scale in- variance, it is convenient to picturise a simple kinematic model which incorporates the entire phenomenology concerning the structure and kinetics of single plates. We em- phasize that this model only serves as an arena to test the conclusions arising from our Kinematic Theory. The model depends only on kinematic parameters like the tip- velocity of the martensite plate and the rate of nucleation of the martensite grains. These parameters would in principle come from a microscopic kinetic theory of the martensite tranformation. One of the results of our work, is a novel method of determination of

I The tip-velocity of the plates in Fe-based systems, is of the order of the velocity of formation of mechanical twins (umklapp transformation). In such systems, the velocities are almost independent of the temperature of quench [81.

M. Rao, S. Sengupta /Phys ica A 224 (1996) 403-411 405

these parameters.

For convenience, our model is restricted to two dimensions (generalization to three

dimensions is straightforward). The transformed 2D austenite region is bounded by a 1D grain boundary, taken to be a square of size L (our results are insensitive to the shape of

the boundary as long as the number of "plates" is large). Following a quench below Tms, the austenite region becomes unstable to the production of critical size martensite nuclei. It is reasonable to assume that the spatial distribution of the nuclei is homogeneous (in the absence of correlated defects) and uncorrelated (in the absence of autocatalysis).

The detailed substructure of the grain is irrelevant to our study; the high aspect ratio then allows us to represent the lens-shaped grain as a straight line. Our model should

be viewed as being coarse-grained over length scales corresponding to the width of the martensite grains. This provides a small length scale cut-off, e.

Thus at time t = 0, we start with p nuclei, uniformly distributed within the square of

size L. Once nucleated, the points grow out as lines whose lengths grow with constant velocity v. The tips of these lines grow out with slopes 4-1, with equal probability (thus the habit "plane" is along the [11] direction). As these lines grow, more nuclei are

created at a rate I (rate of production of nuclei per unit area). The nucleation events

are uniformly distributed in time - thus p nuclei are born at each instant until a total nucleation time tN, so that finally a total of N lines are nucleated. A tip of a line stops when it hits other lines or the boundary. The length of a line, l, is defined as the euclidean distance between the two tips that have stopped due to a collision with other

lines or the boundary. After a time t > tN, when all the lines have stopped growing, we ask for the stationary probability distribution (SPD) of line lengths, P(1).

There are three time scales which describe the entire physics, tv = (NIL 2) - l / 2v - l ,

tl = L - 2 1 - 1 and tlv. Taking tl as our unit of time, one can construct the dimensionless

variables i',, = tv/tj and tN = tN/tl. It is easy to see that there are two extreme geometrical limits. When ?L71 --* 0, nucleation of N grains occurs in one single burst. The numerically computed SPD peaks at the characterictic length scale cx (NIL 2) -1/2

and is described by the gamma distribution,

a # x p. - I e - a x P(x) = (1)

r(~)

Here x = I / (N /L 2)-U2 and the values a = 1.64d:0.02 and/x = 4.52+0.1 (Fig. 1). The

lower bound for the exponent/x can be argued to be 3. The simulated microstructure in this limit (Fig. 1 (inset)), shows grains of comparable sizes.

The other geometric limit is obtained in the limit ~-1 __, o~. In this limit, subsequent nuclei partition space into smaller and smaller fragments, leading to a scale-invariant SPD! From our numerics, we obtain an extremely good fit to a stable Lrvy distribution with a power-law tail,

P(x) ,~, x -a, (2)

with a ~ 1.51 4- 0.03 (Fig. 2). Note that the upper cutoff on the length is L, and the lower cutoff is determined by the total number of grains N, so that all moments of

406 M. Rao, S. Sengupta/Physica A 224 (1996) 403-411

0.4

0.3

0.2

0.1

0.0

0 O

o % A []

0

t2x

2 4 6 8 10 1/10

Fig. 1. P(l) as a function of dimensionless length for N = 500 ([-]), 1000 (A),2000 (o),5000 (o) lines at the F-fixed point. (Inset) Typical configuration obtained for N = 1000 lines at the F-fixed point. The orientation of the "habit-plane" ([I1]) is along the diagonals.

the distribution function are well defined. Scaling arguments can then be used to obtain

analytically [9] the result a = 3/2. The hierarchy of length scales implied by the scale invariant SPD is readily apparent in Fig. 2 (inset).

Having evaluated the SPD at the two extremes, we now investigate the distribution for

intermediate values of ?v. For this purpose, it is convenient to define a renormalization

group (RG) decimation in time. We argue that the gamma distribution, Eq. (1), is

an unstable, noncritical fixed point of the RG, while the I_~vy distribution, Eq. (2) ,

is a stable, critical fixed point. Intermediate values of ?v are connected by a smooth

crossover. Imagine p martensite nuclei forming at each time in a discrete time sequence

{ts, t l , t2 . . . . . ti . . . . . t f } , ( t i - - t i - I = At and t f - - ts = tlv). The rate o f nucleation is thus I = p l A t . Define a sequence, Psi : {P~l, Pl2 . . . . . P/-l,i . . . . }, where Pi- l , i is the probability distribution of lengths of those lines that have stopped between ti-1 and ti.

Decimating over time, we construct the next sequence Pc2 : {Ps2, P24 . . . . . P i - 2 , i . . . . } ,

where Pi-2.i is now the cumulative probability distribution of lengths of those lines that have stopped between t i -2 and ti. This procedure of decimation can be continued till

one obtains the final sequence Psf : {Ps f } . There are clearly only two fixed point dis- tributions (we have also checked this numerically) - one, a trivial fixed point (F-fixed point) at ?~71 = 0, and the other at ?~-l ~ c~ (L-fixed point).

100

10

M. Rao, S. Sengupta/Physica A 224 (1996) 403-411

. . . . . . . . I . . . . . . . . I

407

1000

1

t ~

0.1

0.01

0.001

\ :

0 .0001 . . . . . . . . i . . . . . . . . t . . . . . . . 0.001 0.01 0.1

1

Fig. 2. P(l) as a function of length l for N = 9000 (o) lines at the L-fixed point. The solid line shows a plot of 1-3/2 for comparison (see text). For l < 0.1, we have chosen a linear scale, while for l > 0.1 the scale is logarithmic. (Inset) Typical configuration obtained for N = 1000 lines at the L-fixed point.

We have seen numerically, that the SPD for intermediate values of ?~ can be obtained

as a smooth crossover from the F-fixed point. This implies that apart from F and L,

there are no other fixed points o f the above RG. This is best seen in the crossover of

the various moments of the normalised P ( I ) , ~'(n) = ( In ) . We propose and confirm

the following kinematic scaling ansatz over the entire range of values of ?N and ?~:

¢(n) = ~o(n) tNO(n) fn(t~(n)/ tv), (3)

which for our model implies

~(n) = ~o(n)N-O(n) fn(VtN N~(n)~ (4) \ L N ]"

The scaling exponent O( n) = n/2 and the nonuniversal metric factor ~o( n) = a -n F(I.t + n ) / F ( l z ) can be trivially evaluated at the F-fixed point. The exponent v(n) is the

crossover exponent and fn(Y) (where y = (vtN/LN) N ~(n) is the scaling variable) is the universal crossover function (with the norrnalisation fn (0 ) = 1 to get the correct scaling for the gamma distribution at y = 0). Using simple scaling arguments [9] ,

408 M. Rao, S. Sengupta/Physica A 224 (1996) 403-411

r-

Z

i

10 3i

10

0*%=5 ck

o** n=4

)*** n=3

n=2 ~01~ 0 ~

n=]

l o - 1 1 o _ . . . . . . . . ' . . . . . . . . . . . . . . . ' . . . . . . . . . . 10 0 Y

Fig. 3. Kinematic scaling of the crossover of the first 5 moments N n/2 ( ( n ) / ( o ( n ) as a function of the scaling variable y = ( v t N / L N ) N v(n). The data sets for N = 500 (E]) , 1000 ( A ) , 2 0 0 0 (o ) , 5 0 0 0 (o ) , 9 0 0 0 ( . ) collapse on one curve for each value of n.

the crossover exponent for n = 1 can be shown to be v ( l ) = 2 and for n > 1 to be v(n> 1 ) = l + 3 n / ( 4 n - 2 ) .

In Fig. 3, we plot the crossover function fn(Y) by numerically determining the value of v(n) which leads to a collapse of our data for various values of v and N. We find that the best collapse of our data are obtained by v(1) = 1.92, v(2) = 1.80, v(3) = 1.79, v(4) = 1.82 and v(5) = 1.86 (these values are accurate upto 10%).

This is remarkably consistent with our prediction for v(n)! Our analysis shows that the F-fixed point (associated with Eq. (1) ) , is an unstable, noncritical fixed point of our RG, and governs the behaviour of the P(l) over the entire range of values of the scaling variable y. The RG flows drive the system to the stable, critical, L-fixed point (associated with Eq. (2) ) .

The most significant outcome of our theory, is a novel method of determining the kinematic parameters in acicular martensites. Our results suggest that the final optical micrograph of the martensitic plates has in it, an imprint of its kinetics. Given an optical micrograph, or rather, for the moment, a computer generated picture (such as the ones shown in Figs. la and 2a), it is possible to determine the kinematic parameters (scaling variable y) once the crossover functions and the exponents are known. The quantity vtN can be extracted fairly easily by computing the various moments of the SPD and reading off the value of y. An independent determination of one of the kinematic variables (v or

M. Rao, S. Sengupta/Physica A 224 (1996) 403--411 409

tN) will then yield the other. A simple static measurement thus enables a determination of the growth velocity, which is a difficult dynamic quantity to determine experimentally.

Although our theory holds for real martensites, the computed scaling function and the exponents are relevant only to the 2D model. To compare with real systems, accurate 3D simulations of growing 2D discs are required. The other important structural factor which may have some influence on our results, is "autocatalysis" [ 6,8], which describes the tendency of martensite plates to nucleate close to already existing plates, and near defect centers thereby building up correlations in the positions of martensite nuclei and producing "truss-like" microstructures. We expect that keeping such correlations in the nucleation centres will not change our qualitative picture.

We end this paper by commenting on the connection of our Kinematic Scaling theory to the growing field of "self-organized criticality" (SOC) [15] especially in the context of SOC as observed in martensites [ 16]. Power-law distributions, arising from self- organization are ubiquitous in nature [ 17-19]. Apart from a few simple cases [ 15], we do not understand the dynamics leading to such stationary, scale-invariant configurations. Two features, however, seem crucial for the emergence of scale-invariant behaviour. One is a time span long enough to allow for self-organization (dissipation), and the second is a coupling to an input flow which makes the system open (externally driven). Our system is both open (if viewed as a gas of interacting rods) and driven. With some imagination, one can also identify "avalanches" in our system analogous to those present in the more conventional SOC models in the following way. If one defines the composite process of birth, growth and death of a plate as an "event", then a correlated sequence of events can be identified with an "avalanche". In the limit of T~ -1 ~ 0 (i.e. F-fixed point), the events are uncorrelated. However when ?~-1 ~ oo (i.e. L-fixed point), the growth of plates becomes highly correlated because plates which have formed earlier partition space, thereby limiting the size of the subsequent plates. The "size" of the avalanche (correlated events) grows as one approaches the l_,6vy fixed point, reminiscent of avalanches seen in other dynamical systems evolving towards criticality. Thus our system can be classified as an SOC system, where distributions become critical on being given enough time tN to organize. Such SOC behaviour has been observed in acoustic emission experiments [ 16] in certain thermoelastic martensites, like Cu-Zn- AI, but the relation of these experiments to our theory is unclear. However, if one assumes that the amplitude A and duration ~- of acoustic emission signals is related to the size of the martensite grains L, then the distribution of A and 7- will be a power- law at the Lefy fixed point. Sethna et al. [20] claim that the presence of quenched disorder is crucial for the appearance of power law distributions in martensites. They study the hysteretic response of the random field Ising model to a time varying uniform magnetic field. Their numerical results suggest that for values of the random field above a critical threshold, the reversal of magnetization is dominated by avalanches of all possible sizes. The distribution of avalanche size and duration obeys a power law. They claim that this mimics qualitatively, the martensitic transition in thermoelastic reversible martensites as in Cu-Zn-AI, where the strength and duration of acoustic emission signals were observed to have power law distribution [ 16]. At the level of their theory, only

410 M. Rao, S. Sengupta/Physica A 224 (1996) 403-411

qualitative comparisons can be made with real martensite kinetics, since it lacks detailed information of structure and kinetics. On the other hand, our work shows that specific morphology (acicularity) and kinematics (nucleation and growth), in the absence of any explicit quenched disorder (pinning centres), is sufficient to generate power law distributions. Is it likely that disorder (which must be present in generic martensites) would completely modify our picture?

There are two ways in which disorder can influence the kinematics of martensites - one is by introducing correlations between nucleation events and the other by pinning the growing martensite plates. The effect of correlations has already been discussed. The effect of random pinning on the Kinematic Scaling is minor, as we argue below. Thus plates will stop not only when they encounter other plates (or the austenite grain boundary), but also when they come close to a random pinning centre. This implies that the net effect of pinning centers in our theory, is the creation of voids within the austenite which are inaccessible to the growing martensite plates. This is manifestly equivalent to a change in shape of the austenite (which is now a multiply connected region). Our conclusions are independent of the details of the shape of the austenite boundary, in the limit of a large number of plates. This is true as long as the Hausdorff dimension of the austenite is equal to the spatial dimension. For a very high concentration of random pinning centers, it is easy to see, that the stationary probablity distribution of the plate size would be sharply peaked at the mean spacing between the pinning centers and the L6vy fixed point would be inaccessible for all values of the kinematic parameters.

This paper is in part based on work done in collaboration with H.K. Sahu.

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