Kinetics of martensitic type restacking transitions ...

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Kinetics of martensitic type restacking transitions: dynamic scaling, universal growth exponent and evolution of diffuse scattering B y S h a n k a r P r a s a d S h r e s t h a† a n d Dhananjai Pandey

School of Materials Science and Technology, Banaras Hindu University, Varanasi 221005, India Results of Monte Carlo simulations of the kinetics of 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H restacking transitions by a martensitic process using a one-dimensional kinetic Ising model are presented. It is shown that the evolution of diffuse scattering during these transitions can be calculated by using such a model. We show that the characteristic length scale L(t) grows as t1/2 in the late stages while the equal-time two-point correlation functions exhibit dynamic scaling. A simple diffuse scattering experiment is proposed to determine L(t) from single-crystal diffraction data for verifying the power law dependence of L(t) at the late stages of the restacking transitions.

1. Introduction Recent years have witnessed considerable interest in the study of the kinetics of phase separation and phase ordering using zero temperature low-dimensional stochastic dynamic models. Exact solutions for both conserved and non-conserved zero temperature dynamics have been obtained for one-dimensional systems (Amar & Family 1990; Privman 1992a, b; Gobron 1992; Majumdar & Sire 1993; Majumdar et al. 1994). For the conserved case (Privman 1992a; Majumdar et al. 1994), the zero temperature dynamics is irreversible (non-ergodic) and leads to frozen states whose structure strongly depends on the initial conditions. The domain walls in the frozen configurations are locally pinned. Domain coarsening in such conserved models can occur only in the limit of T → 0 and on a special time scale (Cornell et al. 1991). The growth exponent for such a coarsening process is found to be 1/3. Unlike the case of conserved dynamics, coarsening can occur even at 0 K for non-conserved dynamics with a growth exponent of 1/2 at late stages (Amar & Family 1990). The mechanism of domain growth at late stages involves pairwise annihilation of interfaces separating ordered domains (Amar & Family 1990). The interfacial motion is diffusional and resembles in certain respects diffusion-limited particle–antiparticle annihilation models (Amar & Family 1990; Privman 1992b). One of the most interesting consequences of power law growth for the characteristic length scale (linear size of the domain: L(t) ∼ tn ) is the scaling of the domain size † Permanent address: Tribhuvan University, Patan M. Campus, Physics Department, Patan Dhoka, Nepal. Proc. R. Soc. Lond. A (1997) 453, 1311–1330 Printed in Great Britain

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c 1997 The Royal Society

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S. P. Shrestha and D. Pandey

distribution function and equal-time two-point correlation function at late stages of coarsening. The dynamic scaling of the equal-time two-point correlation function is of considerable interest since it can be experimentally verified in scattering experiments (Gunton et al. 1983). The theoretical predictions for the non-conserved zero temperature dynamics in one dimension have been discussed in the context of binary reactions (Torney & McConnell 1983), exciton fusion kinetics (Parus & Kopelman 1989) and particle– antiparticle annihilation (Toussaint & Wilczek 1983). In recent years, there has been a growing realization that several three-dimensional problems can be effectively modelled in terms of one-dimensional systems (Kabra & Pandey 1988: Sheng & Zhang 1995; Shrestha & Pandey 1996; Yi & Canright 1997). The present investigation was undertaken to show the relevance of zero temperature non-conserved dynamics on one-dimensional Ising chains for studying the kinetics of martensitic type restacking transitions between close-packed layered materials using Monte Carlo techniques. It is shown that irrespective of the initial and the product phases involved in the restacking transition, the characteristic length scale shows a power law dependence on time with an exponenet of 0.5. We confirm the universality of this exponent for the characteristic length scale by using dynamic scaling of the equal-time twopoint correlation function. Unlike the theoretical study of Amar & Family (1990), which is restricted to nearest-neighbour interactions, we show that the growth exponent remains the same for competing interactions ranging up to second and third neighbours. It is proposed that the systems exhibiting martensitic type restacking transitions can serve as model systems for experimentally verifying the theoretically predicted dynamic scaling behaviour of pair correlation functions in one-dimensional stochastic dynamic systems. It is shown that the characteristic length scale for such a study can be conveniently determined from the FWHM of the single-crystal X-ray diffraction profiles at the late stages of the transition. We also show that such Ising models can be used to predict the evolution of diffuse scattering during restacking transitions due to non-random insertion of deformation faults, for which Pandey and his associates (Pandey 1976; Pandy et al. 1980a, b; Pandey & Lele 1986a, b; Kabra et al. 1986) had to take recourse to elaborate Markovian chain type calculations proposed by Hendricks & Teller (1942) and Wilson (1942) for random stacking faults.

2. Mapping of restacking transitions into one-dimensional Ising systems Martensitic type restacking transition between 2H and 3C, 2H and 4H, 2H and 6H, and 3C and 9R have been observed in materials like pure Co, Co–Ni alloy (Frey et al. 1979; Hitzenberger et al. 1988; Blaschko et al. 1988), ZnS (Pandey 1981; Pandey & Krishna 1982; Sebastian et al. 1982), fullerenes (Muto et al. 1993), CdI2 (Trigunayat & Verma 1976; Minagawa 1978) and Cu–Zn–Al alloys (Ahlers & Plegrina 1992). It is known that the restacking transitions commence with the appearance of characteristic diffuse streaks in the c∗ direction of reciprocal space on single-crystal diffraction patterns (Minagawa 1978; Pandey et al. 1980c; Pandey 1981; Pandey & Krishna 1982; Sebastian et al. 1982; Pandey & Lele 1986a). These diffuse streaks are restricted to reciprocal lattice rows with Miller–Bravais indices HK.L for which H − K 6= 0 mod 3 and arise due to non-random insertion of stacking faults on basal plane. Stacking faults involved in the martensitic type restacking transitions are of Proc. R. Soc. Lond. A (1997)


Kinetics of martensitic type restacking transitions

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deformation type (Pandey 1981; Sebastian et al. 1982; Pandey & Lele 1986a, b). Theory of diffraction from 3C and 2H crystals containing random distribution of deformation faults was developed by Paterson (1952) and Christian (1954). However, such theories cannot explain the evolution of diffuse scattering during the martensitic type restacking transitions since the distribution of faults is now non-random (Pandey 1976, 1981). The theory of diffraction from non-randomly faulted closepacked structures has been developed by Pandey and his associates (Pandey 1976; Pandey et al. 1980a, b; Pandey & Lele 1986a, b; Kabra et al. 1986). All these theories for random as well as non-random distribution of stacking faults are based on the assumption that the faults are introduced in a sequential manner from one end of the stack of layers towards the other, as in a typical random walk problem. Recently, Kabra & Pandey (1989, 1995) and Shrestha et al. (1996) have questioned the validity of such sequential models for the insertion of stacking faults during restacking transitions and have proposed Monte Carlo techniques as alternatives for the calculation of the evolution of diffuse scattering during restacking transitions involving non-random distribution of stacking faults. Geometrically, a deformation fault (DF) can be visualized to result by shearing of one part of the crystal past the other across the close-packed planes through partial slip vectors s of the form a/3h10¯ 10i, where a is the unit cell parameter in the basal plane. This is illustrated below for the 3C structure: Perfect 3C structure: 3C structure with DF:

. . . A B C A B C A B C. . . . . . A B C A B A B C A. . .

The dotted vertical line represents the slip plane across which the right-hand part has sheared past the left-hand part of the crystal. Shearing by vector +s causes a cyclic shift of layers from A to B, B to C, C to A above the slip plane while an anticyclic shift of layers from A to C, C to B, B to A results through the vector −s. Such deformation faults are known to result by the splitting of perfect dislocations of burgers vector b into two Shockley partial dislocations of burgers vectors b1 and b2 in low stacking fault energy (SFE) materials (Hirth & Lothe 1968). The pair of partial dislocations with burgers vectors b1 and b2 repel each other but when they separate out, a strip of deformation fault is generated whose positive SFE is responsible for holding these partials at some equilibrium separation (Hirth & Lothe 1968). In materials undergoing martensitic type restacking transitions, the energy of the deformation fault can become negative since the faulted region locally corresponds to the product phase configuration. The negative SFE concept has been justified by Olson & Cohen (1978) in the context of 3C to 2H transition theoretically. It has also been verified experimentally in Co (Hitzenberger et al. 1985). Once the SFE becomes negative, the two Shockley partials bounding the strip of deformation fault will shoot out to the crystal boundaries because of their mutual repulsion causing the entire layer to be covered by deformation faults. In the absence of any obstacles, the partial dislocations will move with the velocity of sound and as such the kinetics of spreading of deformation faults in the close-packed planes will be extremely fast. The thickening of such faulted local configurations in the c direction will, however, be stochastic in nature since it requires occurrence of faults on layers succeeding or preceding the faulted layer with certain minimum separation (Pandey et al. 1980b; Pandey & Lele 1986a, b). As a result of this, the kinetics of evolution of the product Proc. R. Soc. Lond. A (1997)


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phase regions in the c direction during restacking transitions will effectively be a one-dimensional process (Shrestha & Pandey 1996a, b). We shall now proceed to show that the kinetics of martensitic type restacking transitions can be studied by using spin flip dynamics on a one-dimensional Ising chain. In close-packed structures, any layer on top of another layer (say of A type) can be related to the preceding one either by vector +s or by −s depending on whether it is of B or C type respectively. This fact has been used to describe the close-packed stacking sequence of layers in terms of two state variables +s and −s. Following earlier workers (Yeomans 1987; Heine 1988), we assign Ising spin up (↑) for nearest-neighbour pairs of layers of type AB, BC, CA related through vectors +s and Ising spin down (↓) for the pairs BA, CB, AC related through vector −s. Thus, for example, 2H(AB,AB) and 3C(ABC,ABC) structures are Ising antiferromagnet and ferromagnet, respectively, in this notation. As shown by Kabra & Pandey (1988), the introduction of a deformation fault in such close packed structures is equivalent to a single spin flip as illustrated below taking the example of a 3C structure: Perfect 3C: Faulted 3C:

. . . A+ B+ C+ A+ B+ C+ A+ B . . . . . . A+ B+ C+ A− C+ A+ B+ C . . .

↑↑↑↑↑↑↑ ↑↑↑↓↑↑↑

Thus the kinetics of martensitic type restacking transitions between close packed structures can be studied by using non-conserved spin flip or Glauber dynamics (Glauber 1963). It may be mentioned that Ramasesha & Rao (1977) mapped the close-packed layer stacking sequence into a sequence of Ising spins by using h−k notation (for details of this notation, see Pandey & Krishna (1982, 1992)). This was in fact the first attempt to use the Ising model for the simulation of layer stackings. However, the single spinflip (Glauber) or spin-exchange (Kawasaki) dynamics in the h − k notation do not correspond to the introduction of deformation and layer displacement faults which are known to be involved in restacking transitions (see Pandey & Krishna 1982). Use of +s and −s vectors for mapping the ABC-sequence of layers into a sequence of Ising spins, on the other hand, correctly describes the introduction of deformation fault and layer displacement fault in terms of well-known Glauber and Kawasaki dynamics, as was first pointed out by Kabra & Pandey (1988). Consider a one-dimensional Ising chain with pairwise interactions given by the Hamiltonian, XX Jr si si±r , si = ±1. (2.1) H=− r

i

Here Jr are the interaction parameters between the ith and (i±r)th Ising spins which can take values si = ±1. The ground-state phase diagrams of this Hamiltonian for r = 2 and 3 are known (Yeomans 1987) and contain 2H, 3C, 4H structures for r = 2 and 2H, 3C, 4H, 6H and 9R structures for r = 3. The negative energy of a deformation fault, discussed earlier, in the spin language implies that the energy cost (∆H) of a single spin flip has become negative. For example, the energy cost of a single spin flip in an Ising antiferromagnet (2H) for r = 2 can be written as ∆H = J2 − J1 . For ∆H > 0, the 2H structure can evidently exist in the 3C phase field, albeit metastably. On the other hand, for ∆H < 0, the 2H structure is unstable with respect to the single spin flip process. Thus the ∆H = 0 condition fixes the boundary between the unstable and metastable regimes, and this is usually called the spinodal boundary (Gunton et al. 1983). The location Proc. R. Soc. Lond. A (1997)



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of the spinodal boundaries for various types of restacking transitions are given by the following equations:  for 2H to 3C,  J2 = J1    for 3C to 2H,  J2 = −J1 (2.2) for 2H to 4H,  J2 = J1 /2    J3 /J1 = J2 /J1 − 1 for 2H to 6H.  In the metastable regime (i.e. ∆H > 0), the SFE is positive for the most elementary process involving a single spin flip but can become negative for a critical size nucleus consisting of more than one stacking fault or spin flip. In the present work, we study the kinetics of restacking transition by martensitic process in the spinodal regime only. The details of kinetics for the metastable regime will be reported afterwards. We study the dynamics at 0 K since our one-dimensional Hamiltonian does not have a long-range interaction term. The role of temperature in this study is implicit through the choice of the interaction parameters Jr which decide whether the transition will proceed in the unstable or metastable way. In real physical systems, the temperature is expected to alter the interaction parameters Jr in such a way that the restacking transition evolves either by metastable or unstable dynamics.

3. Simulation procedure We have studied the kinetics of 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H martensitic type restacking transitions in the unstable regime by the Monte Carlo technique using zero temperature Glauber dynamics. For the simulation, we begin with an ensemble of antiferromagnetically coupled 10 000 spins for the 2H to 3C, 2H to 4H and 2H to 6H transitions. For the 3C to 2H transition, we start with an ensemble of ferromagnetically coupled spins. We use integer pseudo-random numbers [1,10 000] to select the spin sites randomly for trial spin flips. We then calculate the energy cost of the trial spin flip at the randomly selected site by using the Hamiltonian given by equation (2.1). For the 2H to 3C, 3C to 2H and 2H to 4H transitions, it is sufficient to go up to the next nearest neighbours (i.e. r = 2 in equation (2.1)) while for the 2H to 6H transition, we need to include the third neighbour interactions (i.e. r = 3) also. We have studied the dynamics of these four transitions in the spinodal (or unstable) regime where the energy cost of a single spin flip is negative, i.e. the spin flip can occur spontaneously. We have therefore chosen the following set of interaction parameters for the four transitions such that the initial configuration of the spins becomes unstable with respect to single spin flip dynamics: J1 = 1, J2 = 0.5 for 2H to 3C, J1 = −1, J2 = 0.5 for 3C to 2H, J1 = −0.5, J2 = −1 for 2H to 4H and J3 /J1 = −0.5, J2 /J1 = −0.5 for 2H to 6H. If the energy cost (∆H) of effecting single spin flip at the randomly selected site is found to be not more than 0, the spin flip process is carried through, or else a new site is selected by using the random number generator. This process is then continued. The unit of time in the simulation is the usual Monte Carlo step per spin (MCS/s), which is the time taken in N microtrials, where N is the total number of spins in the ensemble. For all the four restacking transitions, we have monitored the temporal evolution of layer stackings, equal-time pair correlation functions and excess energy (∆E) over ground state. For studying the evolution of the layer stackings and the pair correProc. R. Soc. Lond. A (1997)


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Table 1. Snapshot pictures of an ABC sequence of layers (Snapshots taken during (a) 2H to 3C, (b) 3C to 2H, (c) 2H to 4H and (d) 2H to 6H transitions.) (a) 2H–3C

(b) 3C–2H

t = 0.04 MCS/s BCBCBCABABABABABABAB ABCACACACACACACACACA CACACACACACBACBCBCBA BCBCBCBCBCBCBCBCBCBC BCBCBCBCBCBCBCBCBCBC BCBCBCBCBCBCBCABABAB t = 0.1 MCS/s ABCACACACACABCBCBCBC BCBCBACACACBACBCBACA CACACACACACACACBACBC BCBCBCBCBCBCBACACACA CACACBABABABABCABCBC BCABABABABABABABABAC t = 30 MCS/s BC BCABCABCABCABCABCA BCABCABCABCABCAB ACBA CBACBACBACBACBACBACB ACBACBACBACBACBACBAC ABCABCABCABCABCABCAB CAB ACBACBACBACBACBAC

t = 0.04 MCS/s BCABCBCABCBCABCBCACA ABCABCACABCABCABCABC BCABCBCABCABCABCABCA BCABCABCABCABCABABCA BCABCABCABCABCABCABC ABCABCABCABCABCABCAB t = 0.1 MCS/s CABCABCBCABABCABCABC BCABCBCABCABCABCABCA BCABCABCABCABCBCBCAC ABCABCABCABCABCABCAB CABCBCABCABCABCABCAB CBCABCABCABCABCABCAB t = 30 MCS/s BCBC ACACACACACACACAC A BABABABABABABABABAB ABABABABABABABAB CBCB CBCBCBCBCBCBCBCBCBCB ABABABABABA CACACACAC ACACACACAC BCBCBCBCBC

lation functions, we convert the Ising spins into the 0 A–B–C0 – sequence of layers. It was found necessary to average over 20 ensembles to get rid of statistical fluctuations in the pair correlation functions.

4. Results and discussion (a ) Evolution of stacking sequences From a study of the snapshot pictures of the evolution of the ABC sequences of layers at different times, we find that at early stages of transition (less than 1 MCS/s), islands of untransformed parent phase co-exist with islands of the product phase. As the transition progresses, the volume fraction of the parent phase decreases with a corresponding increase in the volume fraction of the product phase. Even when the volume fraction of the parent phase regions becomes zero, the entire ensemble does not correspond to one single domain of the product phase. Instead, the microstructure consists of domains of product phase which are fully interconnected to each other. This is illustrated in table 1a–d, which depict the evolution of the layer stackings for the 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H transitions at various instants of time. If the contiguous domains are related through shear vectors ±s, the domain wall is Proc. R. Soc. Lond. A (1997)



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Table 1. Cont. (c) 2H–4H

(d) 2H–6H

t = 0.04 MCS/s CBCBCBCBCBCBCBCBCBCB CBCBCBCBCBACACACACAC ACACACACACACACACBABA BABCACACACACACACACAC ABCBCBCBCBCBCBCBCBCB CBCBCBCBCBCBCBCBCBCB t = 0.1 MCS/s CBCBCBCBCBCBACACACAC ACACACACBABABABABABA BABABCACABCBCBACACAC ABCBCABABABABABABABA BABABABABABABACABACA BABABABABABABCACACAC t = 30 MCS/s BCACBCACBCAC ABACABAC A CBCACBCACBCACBCACBC ACBC BABCBABCBABCBABC BABCBAB ACABA BCBABCBA BCBABCBAB ACABACABACA CBCACBCACBCACBCACBCA

t = 0.04 MCS/s ACACACACACACACABCBCB CBCBCBCBCBCBCBCBCBCB ACACACACACABCBCABABC ACACACACABCABABABABA BABCACABCBCBCBCBCBCB CBCBCBCBCBCBCBCABABA t = 0.1 MCS/s CBCBCBCABABCACABCBCA BABABABABCABCBACACAC ACABCBCABABABABABABA BABABABABABACABCABCB CBCBCBCBCBCBCBCBCBCB ACACACACACACACACBABA t = 30 MCS/s CABCBA BCACBABCACBABC ACBABCACBABCAC ACBABC ACBABCACBABCACBABCAC BA CABCBACABCBAC BCABA CBCABACBCABACBCABACB CABACBCABACB ABCACBAB

expected to be of deformation type, as already pointed out in § 2. In such a situation, a single spin flip at the interface (or domain wall) can bring about the merger of the neighbouring domains as shown below for the 3C to 2H transition: Before spin flip:

+ − + − + + + − + − + A | B A {zB A B } C | A C{zA C A} domain-I domain-II

↑↓↑↓↑↑↑↓↑↓↑

After spin flip:

A+ B− A+ B− A+ B− A+ B− A+ B− A+ B

↑↓↑↓↑↓↑↓↑↓↑

We find that the domain growth brought about by the merger of contiguous domains in the manner shown above does not occur for any of the four types of restacking transitions after the formation of the fully interconnected microstructure since the domain wall configurations found in the fully interconnected microstructures do not correspond to deformation type. Furthermore, unlike the 2H to 3C and the 3C to 2H transitions, where all domain walls are of the same type, the domain walls observed for 2H to 4H and 2H to 6H transitions can be of different types even though none of these is of deformation type. Typical interconnected microstructures for the 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H transitions are depicted in table 1a–d respectively, for 30 MCS/s. The most common domain wall configurations observed at the late stages after the formation of fully interconnected microstructure are listed below for Proc. R. Soc. Lond. A (1997)


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each of the four transitions in terms of ABC-, Ising spin- and Ir,s (for definition see Pandey 1984) notations. 1. 2H to 3C transition ABC sequence + + + − A B C A C− B − A 2. 3C to 2H transition − + + − + − A+ 0 B 1 A0 B 1 C 1 B 0 C 1 B 0 3. 2H to 4H transition − − + − − + + A+ 1 B2 A3 C0 A1 C3 B0 C1 A2 4. 2H to 6H transition + + − − − − + + + − − B+ 0 C1 A2 B3 A4 C5 B0 A0 B1 C2 A3 C4 B5 − − − + + − − − + + + A+ 2 B3 A4 C5 B0 C1 A2 C4 B5 A0 B1 C2 A3

spin sequence ↑↑ ↑↓ ↓↓

Ir,s notation ···

↑↓ ↑↑ ↓↑↓

I11 (≡ I00 )

↑↓ ↓ ↑ ↓ ↓↑↑

I13 (≡ I31 )

↑↑↑ ↓ ↓↓ ↓ ↑↑↑↓↓ ↑↓↓ ↓ ↑↑ ↓ ↓↓↑↑↑

I00 (≡ I33 ) I24 (≡ I51 )

Before the formation of the fully interconnected microstructure, the introduction of spin flip leads either to the formation of islands of the product phase or layer by layer growth of the product phase domains. But after the formation of fully interconnected microstructure, the only sites where spin flips are favoured energetically correspond to the domain wall region and are underlined in the above list. The movement of domain walls brought about by the spin flip in the wall region is diffusive in nature (Amar & Family 1990; Majumdar et al. 1994) since the wall can move either to the left or to the right. Whenever two walls meet, they annihilate each other leading to the merger of domains. This implies that the domain growth after the formation of the interconnected structure takes place by pairwise diffusive annihilation of the domain walls (Amar & Family 1990). (b ) Excess energy over the ground state Figures 1a–d depict the variation of excess energy over the ground state (∆E) with time (t) for 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H martensitic transitions, respectively. It is evident from these figures that in all the cases, there is a precipitous fall in the excess energy over the ground state in the early stages of transition (less than 5 MCS/s). This is followed by a very slow decrease in ∆E. The initial precipitous fall in energy, before the crossover to a very slow decrease in energy, is due to the transformation of parent phase regions into the product phase islands. After the formation of fully interconnected microstructure, the energy cost of the movement of a wall, before its annihilation on meeting another wall, is invariably zero for all the transitions. Only when the walls meet and annihilate each other pairwise, there is a reduction in ∆E. This is responsible for the slow decrease in the energy after the crossover from the precipitous fall in ∆E. (c ) Evolution of pair correlations As explained later on (see § 5 a), the calculation of the evolution of diffuse scattering patterns during restacking transitions requires knowledge of equal-time pair correlation functions. The equal-time pair correlation functions P (m), Q(m) and R(m) are defined as the probabilities of finding A-A, B-B, C-C; A-B, B-C, C-A and A-C, C-B, B-A type of pairs of layers with m-layer separations, respectively (Kabra & Pandey 1988, 1995). It is essential to convert the simulated sequence of Ising spins Proc. R. Soc. Lond. A (1997)


Kinetics of martensitic type restacking transitions 20

20

∆E (arb units)

(a)

(b)

16

16

12

12

8

8

4

4

0

0

16

20 (c)

(d) 16

12 ∆E (arb units)

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12 8 8 4

0

4

6

12 t (MCS

18 s–1)

24

30

0

6

12 t (MCS

18

24

30

s–1)

Figure 1. Variation of excess energy over the ground state (∆E) with time (MCS/s) for (a) 2H to 3C, (b) 3C to 2H, (c) 2H to 4H and (d) 2H to 6H transitions.

at various instants of time into the corresponding ABC- sequence of layers to obtain these pair correlation functions which are defined in the ABC- notation only and not in terms of the Ising spins. For ordered structures, P (m), Q(m) and R(m) have fixed periodically varying values depicted in figure 2. Thus, for example, P (m) = 1 and 0 for all close-packed structures for m = 0 mod q and m = 1 mod q, respectively, where q takes values 2, 3, 4 and 6 corresponding to the 2H, 3C, 4H and 6H structures respectively. For m = 2 mod q, P (m) = 0, 1/2 and 1/3 for 3C, 4H and 6H structures. Similarly, P (m) for m = 3 mod q is zero for 4H and for m = 3, 4, 5 mod q for 6H are 1/3, 1/3 and zero. Before the commencement of the transition, P (m) (as also Q(m) and R(m)) corresponding to the parent phase repeat periodically over the entire ensemble size. With the introduction of deformation faults in the very early stages of transition, the correlation length of the parent phase decreases. This is illustrated in figure 3 which depicts a typical spatial variation of P (m) corresponding to a very early (t = 0.1 MCS/s) stage of transition. It is evident from this figure that in all four cases, P (m) converges to 1/3 beyond a certain value of m. Similarly Q(m) and R(m) also converge to 1/3 in all the transitions studied. This convergence of P (m), Q(m) and R(m) to 1/3 implies that the probability of finding the mth layer in A, B or C orientation has become equal, i.e. beyond this value of m there is no correlation among the stacking symbols Proc. R. Soc. Lond. A (1997)


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1 (b)

P(m)

(a)

0

(c)

(d)

P(m)

1

0

4

12

8

0

m

4

8

12

m

Figure 2. Variation of pair correlations (P (m)) with m for perfect (a) 2H, (b) 3C, (c) 4H and (d) 6H structures. The probability P (m) is defined only for integer values of m and the lines between the integer points are drawn as a guide to the eyes.

A, B and C. Thus the value of m at which pair correlation functions converge to 1/3 is the correlation length (ξ). It is evident from figure 3 that even at a very early stage (t = 0.1 MCS/s), the correlation length for the parent phase has been drastically reduced from its initial (at t = 0 MCS/s) value which extended up to the entire ensemble size (i.e. 10 000). With further passage of time, the parent phase correlation length decreases further until it eventually leads to a completely disordered arrangement of A, B and C type layers. The product phase correlations emerge from this disordered state later on. We depict in figure 3 typical spatial variation of P (m) after the emergence of the product phase correlations at t = 10, 3, 5 and 10 MCS/s for 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H transitions, respectively. The correlation length of the product phase is found to increase with increasing time. (d ) Growth exponents and dynamic scaling The spatial variation of P (m) for the product phase can be described by using an exponential function of the type given below for m = 0 mod q (q = 3, 2, 4, 6 for 3C, 2H, 4H and 6H respectively) for all the transitions: P (m) =

1 3

+ 23 e−m/L ,

(4.1)

where L is a characteristic length scale which is a measure of the correlation length ξ. We have determined L as a function of time by fitting the above function to the Proc. R. Soc. Lond. A (1997)


Kinetics of martensitic type restacking transitions 1.0

(a)

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(b)

P(m)

0.8 t = 0.1

0.6

t = 10

t = 0.1

t=3

t = 0.1

t = 10

0.4 0.2

0 1.0

(c)

(d)

P(m)

0.8 0.6

t = 0.1

t=5

0.4 0.2

0

20

0

40

80

0

20

0

40

80

Figure 3. Evolution of the pair correlation function P (m) at (a) t = 0.1 and 10 MCS/s for 2H to 3C, (b) t = 0.1 and 3 MCS/s for 3C to 2H, (c) t = 0.1 and 5 MCS/s for 2H to 4H and (d) t = 0.1 and 10 MCS/s for 2H to 6H transitions.

simulated data by using the least squares method. Figures 4a–d depict the quality of fits for m = 0 mod 3, 0 mod 2, 0 mod 4 and 0 mod 6 for 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H transitions, respectively. The characteristic length scale (L(t)), as determined from these fits, shows power law dependence on time: L(t) = ktn .

(4.2)

This is confirmed by the excellent straight line fits between log(L(t)) and log(t) as shown in the inset to figure 5. The values of the exponent (n) for 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H transitions are found to be 0.50 ± 0.02, 0.49 ± 0.02, 0.49 ± 0.04 and 0.50 ± 0.05, respectively. Thus, for all these transitions, the growth exponent is found to be nearly the same (n = 0.50), confirming the universality of the growth exponent. As pointed out in § 4 a, the domain growth after the formation of the fully interconnected microstructure occurs by the diffusive motion of the domain walls. For this type of mechanism, one indeed expects an exponent of 0.5. It has, however, been pointed out by Amar & Family (1990) that the equivalence between one-dimensional kinetic Ising models at zero temperature and diffusion annihilation in one dimension is only partial. While the exponent for the domain growth remains the same for both the models, the pre-exponent factor gets modified. For pure diffusive annihiProc. R. Soc. Lond. A (1997)


1322


1.0

time (MCS s–1) 10 30 60 90

(a)

P(m)

0.8 0.6

(b)

time (MCS s–1) 10 20 50 80

(d)

time (MCS s–1) 10 20 30 40

0.4 0.33 0.2

1.0

time (MCS s–1) 10 20 30 50

(c)

P(m)

0.8 0.6 0.4 0.33 0.2 0

40

80

120

160

200

0

40

80

m

120

160

200

m

Figure 4. Exponential fits to P (m) at four different times in the scaling regime for (a) 2H to 3C, (b) 3C to 2H, (c) 2H to 4H and (d) 2H to 6H transitions.

lation model in one √ dimension, the pre-exponent factor has been shown (Amar & Family 1990) to be 2 π (≈ 3.55) whereas the values obtained by us are 4.51 ± 0.33, 7.71 ± 0.57, 5.18 ± 0.69 and 4.17 ± 0.76 for 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H transitions respectively. Since the prediction of the pre-exponent factor by Amar & Family is for nearest-neighbour interactions, it seems that this may not hold good for interactions ranging to the second and third neighbours considered by us. Our results, however, clearly establish that the growth exponent remains 0.5 irrespective of the range of interactions. Figures 5a–d depict the variation of the pair correlation function P (m) as a function of a scaled variable m/L(t) at five different times, for m = 0 mod 3, m = 0 mod 2, m = 0 mod 4 and m = 0 mod 6 for 2H to 3C, 3C to 2H, 2H to 4H and 2H to 6H transitions, respectively. It is evident from this figure that in each case, the pair correlations P (m) at different times collapse into one single master curve in excellent conformity with dynamic scaling hypothesis (see Bray 1993).

5. Evolution of diffuse scattering As mentioned in § 2, the restacking transitions commence with the statistical insertion of stacking faults as revealed by the appearance of characteristic diffuse streaks Proc. R. Soc. Lond. A (1997)


Kinetics of martensitic type restacking transitions 102

10

10 log(t) 102

10 log(t) 102

0.6

L t (MCS s–1) 14 10 0.4 21 20 32 50 43 80 44 90 0.2

L t (MCS s–1) 25 10 34 20 48 40 53 50 59 60

102

(c)

0.8

102

(d)

log(L)

1.0

P(m)

10

log(L)

P(m)

0.8

102

(b)

log(L)

(a)

log(L)

1.0

1323

10

10

L t (MCS s–1) 16 10 0.4 27 30 36 50 41 70 46 90 0.2 0 1

10 log(t) 102

log(t) 102

10

0.6

L t (MCS s–1) 13 10 19 20 23 30 28 50 30 60 m/L(t)

2

3

0

1

m

2

3

Figure 5. Variation of P (m) with scaled variable m/L(t) for five different times, distinguished by different symbols, showing collapse of entire data on a single master curve in the scaling regime. Inset: Plot of ln(L(t)) versus ln(t) for (a) 2H to 3C, (b) 3C to 2H, (c) 2H to 4H and (d) 2H to 6H transitions confirming power law growth of characteristic length scale L(t).

on single-crystal X-ray diffraction patterns taken from partially transformed crystals. The model developed in this paper can also be used for the calculation of the evolution of diffuse scattering patterns, as explained below. (a ) General considerations Consider a stack of N close-packed layers numbered as j = 0 to N − 1. The resultant diffracted amplitude, G, from such a stack of layers can be written as G = (1/N )[F0 + F1 e−iφ + . . . + Fj e−ijφ + . . . + FN −1 e−i(N −1)φ ],

(5.1)

where Fj is the layer form factor for the jth layer and φ is the phase difference between rays diffracted from the origins of rhombic unit cells in the adjacent closepacked layers. The diffracted intensity is given by GG∗ =

∞ X 1 N −m ∗ −imφ + [Jm eimφ + Jm e ], N m=1 N 2

(5.2)

∗ i is the average spatial correlation function between the form where Jm = hFj Fj+m factor of a close-packed layer and the complex conjugate of the form factor for an∗ = hFj∗ Fj+m i = other layer which is m-layers apart (Wilson 1942). Furthermore, Jm ∗ hFj Fj+m i = Jm , where ∗ denotes the complex conjugation operation. Splitting Jm

Proc. R. Soc. Lond. A (1997)


1324


into real and imaginary parts, we can write 0 00 + iJm . Jm = Jm

(5.3)

Using the relationship given by equation (5.3) in equation (5.2) we get N −1 X 1 N −m 0 00 GG = +2 (Jm cos mφ − Jm sin mφ). 2 N N m=1 ∗

(5.4)

Choosing the origin on an A type site in a close-packed layer, the x, y coordinates of B and C sites in the mth layer with respect to the hexagonal unit cell can be written as ( 13 , − 13 ) and (− 13 , 13 ), respectively. The layer form factors (relative to the atomic scattering factor) for A, B, C type layers can thus be written as FA = 1,

FB = e2πi(H−K)/3 ,

FC = e−2πi(H−K)/3 .

(5.5)

where HK.L are the Miller–Bravais indices with respect to the hexagonal unit-cell. Assuming that the spacing between close-packed layers to be independent of faulting, we can write φ = 2πh3 /q, where h3 is a continuous variable along c∗ and q, as earlier, takes values 2, 3, 4 and 6 for the 2H, 3C, 4H and 6H structures. Defining wA , wB , wC as the probabilities of finding A, B, C type layers in any arbitrary region of the crystal and pA (m), pB (m), pC (m) as the probabilities of mth ∗ i can be expressed as layer being in A, B, C orientation, hFj Fj+m ∗ hFj Fj+m i = wA pA (m)FA FA∗ + wB pB (m)FB FB∗ + wC pC (m)FC FC∗

+wA pB (m)FA FB∗ + wB pC (m)FB FC∗ + wC pA (m)FC FA∗

+wA pC (m)FA FC∗ + wB pA (m)FB FA∗ + wC pB (m)FC FB∗ .

(5.6)

The equal-time pair correlation functions, P (m), Q(m) and R(m), by definition are therefore  P (m) = wA pA (m) + wB pB (m) + wC pC (m),   (5.7) Q(m) = wA pB (m) + wB pC (m) + wC pA (m),   R(m) = wA pC (m) + wB pA (m) + wC pB (m). Using P (m), Q(m) and R(m), equation (5.7) simplifies to ∗ i = P (m) + Q(m)e−iθ + R(m)eiθ , Jm = hFj Fj+m

(5.8)

where θ = 2π(H − K)/3. It is evident from equation (5.8) that reflections with H − 0 and imaginary K = 0 mod 3 will not be affected by faulting. Separating the real Jm 00 Jm parts of equation (5.8), we get 0 Jm = P (m) + [Q(m) + R(m)] cos θ, 00 Jm

= −[Q(m) − R(m)] sin θ.

(5.9) (5.10)

Substituting equations (5.9) and (5.10) in equation (5.4) one can express the diffracted intensity in terms of P (m), Q(m) and R(m). (b ) Calculation of diffuse scattering patterns during restacking transitions It is evident from the foregoing that the calculation of diffracted intensity along c∗ direction requires a knowledge of P (m), Q(m) and R(m) which can be obtained from our simulation studies, as already discussed in § 4. As pointed out earlier, P (m), Q(m) and R(m) converge to 1/3 beyond a certain value of m, say m0 for all the transitions Proc. R. Soc. Lond. A (1997)



1325

0 00 and Jm for m > m0 become zero as can be (see figure 3). As a result of this, Jm easily seen from equations (5.9) and (5.10) for reflections with H − K = 1 mod 3 and H −K = 2 mod 3. Therefore, the intensity expression given by equation (5.4) reduces to m0 X 1 N −m 0 00 +2 [Jm cos mφ − Jm sin mφ]. (5.11) I(h3 ) = 2 N N m=1

Substituting the values of P (m), Q(m) and R(m) for m 6 m0 corresponding to different intermediate states of transition, as obtained by simulations, we get the intensity distributions along c∗ which are given in figure 6. It is evident from these figures that all the restacking transitions commence with an appearance of diffuse scattering between the Bragg peaks as well as an initial broadening of the parent phase reflections. New reflections, characteristic of the product phase, appear from the diffuse scattering background at a later stage. The product phase reflections are initially considerably broad and shifted from their registered sites. As the transition advances, these reflections sharpen and approach the registered positions as can be seen from these figures. Figure 7 depicts the evolution of the peak position of the product phase as a function of time. It is evident from this figure that the peak position almost levels off after about 5 MCS/s on the formation of the fully interconnected microstructure. As shown in the next section the FWHM continues to decrease even though the peak positions have levelled off in the dynamic scaling regime. (c ) Proposed procedure for experimentally verifying the power law behaviour and dynamic scaling We shall now proceed to demonstrate a procedure for the determination of the characteristic length scale L(t) in a diffuse scattering experiment. For this we shall concentrate on the 2H to 3C transition only, since some qualitative comparison can be made between the theoretically predicted evolution of diffuse scattering patterns with those experimentally reported for ZnS (Sebastian et al. 1982; Shrestha & Pandey 1996c). Single-crystal X-ray diffraction studies have shown that the 2H to 3C transition in ZnS commences with a statistical insertion of deformation faults as revealed by the (i) broadening of 2H reflections at L = 0, ±1(mod 2) positions and (ii) appearance of diffuse streaks between these reflections parallel to the c∗ direction for all those reciprocal lattice rows for which H − K 6= 0 mod 3. With the progress of the transition, the 2H reflection at L = 1 mod 2 position becomes very broad with an almost flat top while the reflections at L = 0(mod 2) positions merge with the diffuse background. As the transition advances further, two distinct reflections approaching the 3C positions at L = ±2/3, ±4/3(mod 2) appear from the flat top L = ±1(mod 2) reflections of 2H. The intensity and sharpness of these new 3C reflections increases as the transition advances still further. It is easy to see from figure 6a, which depicts the theoretically predicted evolution of diffuse intensity distribution along c∗ during 2H to 3C martensitic type restacking transition, that the transition commences with an initial broadening of 2H reflections until the peak at h3 (L) = 1(mod 2) becomes flat-top while the other peak merges with the diffuse background. At a later stage, 3C reflections appear from the flat-top 2H reflection and approach the 3C positions at h3 (L) = ±2/3, ±4/3(mod 2) in perfect agreement with the experimental observations. In the scaling regime (t > 3 MCS/s), P (m) and Q(m) for m = 0, 1, 2 mod 3 reflecProc. R. Soc. Lond. A (1997)


1326

S. P. Shrestha and D. Pandey (b)

intensity (arb. units)

(a) 20 2H

3C

2H

3C

2H

2H

10

0

3C

2H

2/3

1 h3

4/3

5/4

3 2 1 0.7 0.5 0.4 0.3 0.2 0.1

IX VIII VII VI V IV III II I 0 2

IX VIII VII VI V IV III II I

1/3

2/3

4/3

5/4

2

6H

6H 2H/6H 6H

6H

2H

1 h3

(d)

(c) 20 2H intensity (arb. units)

2H

t (MCS s–1)

t (MCS s–1)

5 3 2 1 0.6 0.4 0.3 0.2 0.1 1/3

3C

4H

2H

4H

2H

2H

t (MCS s–1) t (MCS s–1)

10

0

20 10 3 2 1 0.6 0.3 0.1 1/2

1 h3

1/2

VIII VII VI V IV III II I 0 2

20 10 5 2 1 0.6 0.2 0.1

VIII VII VI V IV III II I

1/3

2/3

1 h3

4/3

5/4

2

Figure 6. Evolution of diffuse scattering at various instants of time along c∗ during (a) 2H to 3C, (b) 3C to 2H, (c) 2H to 4H and (d) 2H to 6H transitions. h3 is a continuous variable in reciprocal space along c∗ . Some peaks for t = 5, 3 and 2, 3, 10 and 20, 10 and 20 MCS/s in (a), (b), (c) and (d) respectively are truncated.

tions can be described by using the following functions: P (m) = P (m) = Q(m) = Q(m) =

1 3 1 3 1 3 1 3

+ 23 e−m/L

for m = 0 mod 3,

(5.12)

− 13 e−m/L

for m = 1, 2 mod 3,

(5.13)

for m = 0 mod 3,

(5.14)

for m = 1, 2 mod 3.

(5.15)

− +

1 −m/L e 3 1 −m/L e 6

For reflections with H − K = 1 mod 3, equations (5.9) and (5.10) simplify to 0 = 12 [3P (m) − 1] Jm

00 and Jm =

√ 3 [1 2

− P (m) − 2Q(m)].

(5.16)

Substituting from equations (5.12)–(5.15) in (5.16) and after simplifications, we get  0  Jm = e−m/L for m = 0 mod 3,  1 −m/L 0 (5.17) for m = 1, 2 mod 3, Jm = − 2 e   00 for all m. Jm = 0 Proc. R. Soc. Lond. A (1997)


Kinetics of martensitic type restacking transitions 4/3

5/3

(b)

peak position

(a)

1

4/3

2/3

1 0

9

18

27

0

36

5

10

15

20

25

5

10 15 t(MCSs–1)

20

25

5/3

7/4

(d)

(c) peak position

1327

6/4 4/3 5/4

1

1 0

5

10 15 t(MCSs–1)

20

25

0

Figure 7. Temporal evolution of the peak position of the (a) 3C reflection near h3 = 4/3, (b) 2H reflection near h3 = 1, (c) 4H reflection near h3 = 6/4, (d) 6H reflection near h3 = 4/3 positions.

For an infinite size crystal (N m), equation (5.4) reduces to (Warren 1969) I(h3 ) = GG∗ =

∞ 2 X 0 1 00 + [J cos(mπh3 ) − Jm sin(mπh3 )]. N N m=1 m

(5.18)

0 00 and Jm from equation (5.18) and then performing the summation Substituting for Jm over m, one obtains the following expression:

I(h3 ) =

1 − ρ cos(πh3 ) 1 3[1 − ρ3 cos(3πh3 )] − − , N (1 − 2ρ3 cos(3πh3 ) + ρ6 ) N (1 − 2ρ cos(πh3 ) + ρ2 ) N

(5.19)

where ρ = e−1/L(t) . The intensity expression given by equation (5.19) has L(t) as the only input parameter. The numerically computed intensity distributions using equation (5.19) are found to be in excellent agreement with those obtained directly by simulation for the scaling regime. This is illustrated in figure 8 for t = 5, 30, 50 and 70 MCS/s. Thus the diffuse scattering patterns at the late stages of the restacking transitions can be predicted directly from equation (5.19) without taking recourse to the usual elaborate Markovian chain type calculations (Pandey et al. 1980b; Pandey & Lele 1986a, b). The first term in equation (5.19) peaks at h3 = 0, 23 , 43 (mod 2) positions while the second term peaks at h3 = 0(mod 2) positions only. However, the contribution Proc. R. Soc. Lond. A (1997)


1328

S. P. Shrestha and D. Pandey 2H

3C

2H

3C

2H

60

intensity (arb. units)

simulation approach scaling function approach

40 t (MCS s–1) 70 IV 20

50 III 30 II 5

0 1/3

I 2/3 h3

1

Figure 8. Comparison of diffuse intensity distribution obtained using scaling function and simulation approach for four different times for the 2H to 3C transition.

of the first term around h3 = 0 mod 2 positions is cancelled by the second term leaving behind only the 3C reflections at h3 = 23 , 43 (mod 2) positions on the diffraction pattern. The FWHM of these 3C reflections is therefore decided by the first term only. With slight mathematical manipulation, it can be easily shown that FWHM ≈ 3/πL(t).

(5.20)

Thus one can determine L(t) from the FWHM of 3C reflections in a diffuse scattering experiment and verify its power law behaviour with respect to time.

6. Conclusion We have shown that the kinetics of the martensitic type restacking transitions can be modelled by using a one-dimensional zero temperature kinetic Ising model with Glauber dynamics. One can use this model to predict the evolution of diffuse scattering during such transitions. The correlation length at late stages is found to exhibit power law dependence on time with a universal growth exponent of 0.5 irrespective of the structure of the initial and product phases. It is shown that the characteristic length scale can be determined from the FWHM of product phase reflections on a diffraction pattern. Such a measurement can therefore be used to Proc. R. Soc. Lond. A (1997)



1329

(i) determine the growth exponents for the correlation length and (ii) verify the dynamic scaling behaviour of the equal-time pair correlation functions at late stages.

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Proc. R. Soc. Lond. A (1997)