Streaming and the nucleation of martensite

Scripta

METALLURGICA

V o l . II, pp. 1 0 4 5 - 1 0 4 9 , 1977 Printed in t h e U n i t e d States

STREAMING AND THE NUCLEATION

Pergamon

Press,

Inc.

OF MARTENSITE

M. Suezawa* and H. E. Cook*t *Department of Metallurgy ~The Materials Research Laboratory and tDepartment of Mechanical Engineering University of Illinois at Urbana-Champaign Urbana, Illinois 61801 (Received J u l y 18, 1977) (Revised October 3, 1977) Introduction In their important study of the order-disorder transformation in CuAu thin films, Tong and Wayman (I) have interpreted the curious fluctuating contrast phenomenon called "streaming," first observed in the electron microscope by Pashley et. al. (2,3), as a dynamical precursor to the nucleation of the ordered l o w t e m p e r a t u r e phase. They and co-workers (4,5) have reported similar precursors for a variety of first-order phase transformations. The origin of the observed effects remains controversial (6,7), however, and the purpose of this note is to provide a more extensive theoretical analysis of the problem and to present a qualitative model for nucleation which can be extended to the more classical martensltic transformations found, for example, in steels. Three aspects of streaming will be investigated: i) the nature of the fluctuations involved, 2) the direction of streaming and 3) the source of the fluctuation frequency which, because it is observable by eye, is roughly I0 Hz. Several simplifying assumptions will be made. The surface energy between ordered embryos and the disordered matrix will be ignored and the strain interaction between locally ordered regions will not be considered. A more complete analysis, now in progress, shows that these two assumptions do not affect the streaming direction but, as expected, do affect the slze of the region. One finding is that the computed size of the locally transformed region shrinks to zero in the CuAu system for interfacial free energies exceeding 20 to 30 ergs/cm 2. The loll will be assumed to be infinitely thick which will not allow us to examine the important role which relaxations at the free surface appear to have on the observed effects (i). Interaction with Screw Dislocations As shown by Tong and Wayman (i) the transition from the disordered phase to the CuAu II structure is initiated by the formation of regions of CuAu I near screw dislocations. (The high density of screw dislocations, as opposed to edge, is apparently a result of preferential formation during the growth of the films.) It is necessary, therefore, that we calculate the local free energy of an ordered region near a screw dislocation which we will do in three steps. The free energy change per atom when the crystal orders in the stress free state is taken to be: f(~ in which T is the temperature,

S =-(k/Z) is the Bragg-Williams which is proportional

= arl2 + brl~ - TS,

(i)

~ is the order parameter and [(l+r~£n(l+y 0 + (i-r0Zn(l-r0],

(2)

entropy. The second term (b~ ~) results from the volume change on ordering to the square of the stress free strains (g) 8ij ~ (l12)(82%ij/Sn2)I~,

(3)

associated with the transition. If we take ~ to be the order parameter for a [i00] composition wave of wavelengthl = ~ where ~ is the lattice parameter (the [lO0] variant), the stress free strain matrix Is of the form

1045

1046

STREAMING AND N U C L E A T I O N OF M A R T E N S I T E

=

Vol.

(:°: 1

12

(4)

~2

0

II, No.

~2

in which the axes 1,2,3 are taken along the [i00], [010] and [001] directions, respectively. Near the transformation temperature, E l = -5.49 x lO -2 and c 2 = 1.14 x lO -2, obtained from reference (9). The value calculated for the coefficient (h) is -2.28 x 10-14ergs/atom, and the coefficient (a) was chosen such that the (metastable) transition temperature ( T ) from the disordered phase to O CuAu I, was 670°K, midway between the two equilibrium ordering transitions (8). Inclusion in eq. (1) of the free energy associated with coherency strains requires adding a term of the form b'r~. The value of b' will depend on the shape and orientation of the ordered region. For our purposes we take the region to be cylindrically shaped with the axis of the cylinder parallel to the [II0] direction. The value calculated for b' is 0.511 x 10 -14 erg/atom. Finally we must compute the elastic free energy of interaction between the cylindrical region and the screw dislocation. On adopting the cartesian coordinate system x, y, z along the {Ii0], [i~2] and [ill] directions, respectively, we have the followlng expression from reference (10) for the interaction between a screw dislocation along [ii0] and the ordered cylindrical region located at (y,z) : g(y,z;T~ = -(ag74[0G(y,z) (C2-C I) in which ~ I s and

the volume per atom, ~

~C¼~(C11-C12) 3 G(y,z) =

2V~ 3

(~/~)

(5)

is the order parameter for the low temperature phase at T o ,

V~y H y2 - ~ 3 (C44 - ~)

+ z yz + (C44 - 6) H

z2

(6)

where H = 2 C44 + C12 - CllThe parameters CII, C12 and C~4 are the usual Voigt elastic constants for a cubic crystal. Their values -- CII = 1.59 x 1012, C12 = 1.28 x 1012 and C44 = 0.458 x 1012 dyn/cm 2 -- were obtained by interpolating between the two pure components (ii) at temperature T . The inclusion of O g(y,z;I~ in Eq. (i) results in an additional contribution proportional to ~ which is negative and which depends on the location of the ordered region with respect to the dislocation. A contour map of equal free energies and local order parameter at i0 degrees below T is shown in Fig. 1 with the screw dislocation at the origin. The free energy change is negative°inside the 0. 832 enclosing contour for the values of rj shown and is positive outside the contour for any finite value of Ti. The extent of the region (~ 7 N) is of the order of the observed streaming distances, but we place little confidence in this value, because the calculated size will ~e significantly less if a surface free energy of 5 to i0 ergs/cm 2 was included. The llne connecting the origin to the region of maximum extent of each contour is seen in Fig. 1 to be along the [001] direction. This calculated result is in agreement with the marked [I00] direction of streaming observed for CuAu. The one sided nature of the interaction with the screw dislocation is in agreement with Tong and Wayman's calculation and with their experiment. The size of the region is shown in Fig. 2 as a family of curves for temperatures ranging from 700 to 660°K. The phase transition is seen to take place locally on the dislocation well above T o . Discussion It is instructive to examine the entire free energy curve as a function of Ti in considering the local phase transition and the nature of the fluctuations involved. The total free energy change &F = f(T~ + g(y,z;r~

(8)

is shown in Fig. 3 for three values of r(= [z 2 + y2]½) along the [001] direction at 660 °K. The curve for r = 7.29 a is at the boundary of the locally transformed region. Within this region,

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f l u ~ a t i o n s in ~ about its local equilibrium value of roughly 0.83 should be small. Outside of this region but adjacent to it the fluctuations that form will likely be heterophase in nature representing essentially perturbations in the boundary of the locally transformed cylindrical region as shown schematically in Fig. 4. It is important to note that the free energy associated with small fluctuations about Ti z 0 which we can denote as short range order is not strongly affected by the presence of the dislocation in the region outside of the contours in Fig. i. Thus the observed streaming is not likely associated with the local enhancement of short range order. The observed streaming, however, may not be a result of intrinsic heterophase fluctuations but may result from a coupling of the locally ordered regions to fluctuating stresses in the films. One possibility is thermal stresses induced by oscillations in the electron beam current. The period of the major oscillation in beam current amplitude is comparable to the period observed for streaming as shown by Fig. 4 of reference (4). Another observation which suggests that the effect is of extrinsic origin is that the periods are about the same for both diffusion controlled (CuAu) and diffusionless (Fe - 32 pct Ni) transitions. To observe streaming by an extrinsic mechanism it Is only necessary to have the intrinsic relaxation time (~i) less than the impressed frequency. For CuAu, ~ ~ ~2/~) is calculated to be 10 -2 sec. near To using the interdiffusion coefficient ~ from reference (12) and is small enough for the local order to couple to an extrinsic field at the observed frequency of ~ I0 Hz. Another general result of the studies (1,4,5) on CuAu is that streaming is enhanced between pairs of dislocations. If the two dislocations have Burgers vectors of opposite sign, then the same variant would be induced to form between them. A more interesting case in regard to martensite nucleation occurs when the two Burgers vectors are of the same sign because a pairing up of twin related variants is stimulated. An example of this is shown schematically in Fig. 5 for an arrangement of three pre-existing dislocations of the same sign. The alternating twin related variants interact favorably to reduce the elastic energy (13) when they occur periodically spaced along the [Ii0] type direction as shown in Flg. 5. Wlth sufficient supercooling, the arrangement shown should become a critical nucleus at which time twins are formed as growth progresses along the [Ii0] direction through the reduction of elastic free energy. We see no reason why a similar nucleation mechanism can not operate for diffusionless martensitic transformations. Growth would be associated with the propagation of a soliton-like transition front with the required twinning or slip for forming the invarlant plane strain being nucleated at a small distance, equal to the twin spacing, behind the front. We are using the word soliton (14-16) in the sense of a propagating solitary wave in which the released stored energy, described formally by an anharmonic Hamiltonlon coupling of the parent and product states of the transformation, Is absorbed by the dissipative processes associated with its propagation. The fall of dominos is a familiar solitonllke wave propagation. It is curious why the CuAu I phase should be a precursor to the formation of the CuAu II structure (i). The reason may be that the larger transformation strains for CuAu I result in the greater overall decrease in free energy when considering the local interaction with dislocations. The difference in the stress free strains between the two ordered structures is also important in understanding the stability of CuAu I with respect to CuAu II at low temperatures as shown by Tachiki and Teramoto (17). Finally we phenomenon provide an tions (18)

would like to emphasize that it would be a great interest to investigate the streaming near edge dislocations in CuAu as the effect should be larger. Moreover, it should important test of the mechanism as streaming is calculated to be along the [153] direcfor edge dislocations. Acknowledgements

We are indebted to C. M. Wayman for several helpful discussions concerning this problem and gratefully acknowledge financial support by the NSF under Grant No. DMR76-00255 (M.S. and H.E.C.) and Grant No. DRR76-01058 (H.E.C.) which is through the Materials Research Laboratory. References i. 2. 3. 4. 5.

H. D. A. I. H.

C. Tong and C. M. Wayman, Acta Met. 21, 1381 (1973). W. Pashley and A. E. Presland, Unpublished. M. Hunt and D. W. Pashley, J. Phys. Rad. 23, 846 (1962). Cornelis, R. Oshima, H. C. Tong and C. M. Wayman, Scripta Met. 8, 133 (1974). C. Tong and C. M. Wayman, Phys. Rev. Lett. 32, 1185 (1974).

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6. 7. 8. 9. i0. ii. 12. 13. 14. 15. 16. 17. 18.

STREAMING

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NUCLEATION

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E. I. Estrin, A. V. Suyazov and M. P. Usikov, Scripta Met. 9, 485 (1975). C. M. Wayman, I. Cornells, R. Oshims and H. C. Tong, Scrlpta Met. 9, 489 (1975). T. KaJltanl and H. E. Cook, to be published. W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, Pergamon Press, London (1967). J. P. Hirth and J. Lothe, Theory of Dislocations (McGraw-Hill Book Co., 1968). K. H. Hellege, Landolt-BSrnsteln Numerlcal Data and Functional Relationships in Science and Technology Group Iii Vols. 1 and 2, Sprlnger-Veriag (1966). M. R. Pinnels and J. E. Bennett, Met. Trans. !, 1989 (1972). A. G. Khachaturyan and G. A. Shatalov, Soviet Phys. - JETP 29, 557 (1969). A. Scott, F. Chu and D. MeLaughlin, Proc. IEEE 61, 1443 (1973). B. Horovltz, J. A. Krumhansl and Eytan Damany, Phys. Rev. Lett. 38, 778 (1977). T. Suzuki, Solid State Physics i, 203 (1976)(in Japanese). M. Tachiki and K. Teramoto, J. Phys. Chem. Sol. 27, 335 (1966). M. Suezawa and H. E. Cook, to be published.

~ [1111 A t

Fig. i. Calculated order parameter contours for locally transformed region of CuAu I near screw dlslocation at a temperature below the equilibrium transformation temperature but above the coherent transformation temperature.

Ol~(in

I

E1111

unit of5)

[1--i-2]

660oK

Fig. 2. Calculated extent of the locally transformed region shown as a family of curves as a function of temperatures.

]

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+°If I+0.10 r + lOa-

~+0.05

~

Fig. 3. Calculated local free energy as a function of local order parameter (~) at three different distances along [001] direction from dislocation.

o -0.05

-0.10 L 0

J

J 0.2

I

J 0.4

I q

0.6

0.8

1.0

Fig. 4. Schematic representation of fluctuations of the boundary for the locally transformed region.

I,,o,

Fig. 5. A schematic representation of a nucleation mechanism for the twln related transformation product from a suitable arrangement of screw dislocations having the same Burgers vector.

_

>,~,j

C5..+S

a

b ~Ik [I00)-type ? [010]-type