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CLASSICAL AND NONCLASSICAL MECHANISMS OF MARTENSITIC TRANSFORMATIONS
G. Olson, M. Cohen
To cite this version:
G. Olson, M. Cohen. CLASSICAL AND NONCLASSICAL MECHANISMS OF MARTEN- SITIC TRANSFORMATIONS. Journal de Physique Colloques, 1982, 43 (C4), pp.C4-75-C4-88.
�10.1051/jphyscol:1982405�. �jpa-00221949�
JOURNAL DE PHYSIQUE
Colloque C4, suppl6ment au n o 12, Tome 4 3 , d6cembre 1982 page C4-75
C L A S S I C A L A N D NONCLASSICAL M E C H A N I S M S O F M A R T E N S I T I C T R A N S F O R M A T I O N S
G.B. O l s o n a n d M. Cohen MIT, Cambridge, MA, U.S. A.
( A c c e p t e d 9 A u g u s t 1982)
Abstract.
-
Theoretical approaches t o the nucleation of martensi t i c transformations have considered two types of nucleation paths: ( a ) t h e c l a s s i c a l path, involving a nucleus of fixed s t r u c t u r e and increasing s i z e , and (b) nonclassical paths, involving a nucleus of varying s t r u c t u r e . A unified approach i s out1 ined which allows f o r the t h e o r e t i c a l treatment of homogeneous and heterogeneous nucleation of coherent and semicoherent martensitic transformations by both types of nucleation paths, and ident-i f i e s t h e circumstances favoring each. Nonclassical paths would be favored i n the special case of homogeneous coherent nucleation near a mechanical i nstabi 1 i t y ; however, q u a n t i t a t i v e treatment requires f u r t h e r know1 edge of the energetics of homogeneous l a t t i c e deformations and the s t r a i n - gradient energies which control d i f f u s e i n t e r f a c e s . Heterogeneous nucleation, which governs martensitic transformations i n bulk materials, favors t h e c l a s s i c a l path and involves r e l a t i v e l y sharp i n t e r f a c e s .
Classical heterogeneous nucleation i s adequate1 y t r e a t e d by 1 i near-el a s t i c continuum and dislocation descriptions. Defect-nucleus i n t e r a c t i o n leads t o " b a r r i e r l e s s " nucleation with k i n e t i c s controlled by i n t e r f a c i a l motion.
Details requiring f u r t h e r a t t e n t i o n a r e the microscopic mechanisms of martensitic i n t e r f a c i a l motion i n d i s c r e t e c r y s t a l s , and the precise nature of the nucleating defects, both those i n i t i a l l y present and autocatalytic- a1 l y generated.
Introduction.- Martensitic transformations have been defined a s a subset of d i f f u s i o n l e s s displacive phase transformations, with s u f f i c i e n t l y large l a t t i c e - d i s t o r t i v e shear displacements such t h a t t h e transformation kinetics and morpho- logy a r e dominated by s t r a i n energy ( 1 ) . I t then follows t h a t martensitic trans- formations a r e f i r s t - o r d e r s o l i d - s t a t e reactions occurring by nucleation and growth.
While t h e growth stage of f i r s t - o r d e r transformations generally takes place by the motion of i n t e r f a c e s which convert the parent phase t o t h e fully-formed transformation product, two types of paths have been considered f o r the case of nucleation. The "classical " nucleation path, analogous t o t h e growth stage, involves a nucleus of the same s t r u c t u r e and/or composition a s the f u l l y - f o y e d product, increasing in s i z e by the motion of i t s interface. A "nonclassical path involves a continuous change in structure/and/or composition in a f i n i t e region. The two types of paths are i l l u s t r a t e d in Figure 1 where composition, s t r u c t u r e , and s i z e a r e denoted by C , rl, and n. Path "a" depicts the c l a s s i c a l path, while paths "b" and "c" represent nonclassical paths. For t h e simple case of purely compositional transformations, nucleation via nonclassical paths has been t r e a t e d q u a n t i t a t i v e l y by Cahn and Hill i a r d ( 2 ) .
The current s t a t u s of the theory of martensitic nucleation by both c l a s s i c a l and nonclassical paths has been recently reviewed ( 3 ) . While the theory of nucleation via c l a s s i c a l paths i s now well -developed f o r both homogeneous and heterogeneous martensitic nucleation, speculation regarding t h e possible re1 evance
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982405
JOURNAL DE PHYSIQUE
S ~ z e
Composition,
--_____
structure
-.
: b'\ I
IIL)
Figure 1 Classical and nonclassical Figure 2 Free energy per u n i t volume, nucleation paths f o r f i r s t - o r d e r sol id- Ag, associated with homogeneous d i s t o r - s t a t e phase transformations. Diffusion- t i o n of the metastable parent l a t t i c e l e s s martensitic transformation involves (n=0) t o produce the martensitic product variables 0 ( s t r u c t u r e ) and n ( s i z e ) . (n=1) under thermodynamic conditions
corresponding t o To, Ms, and the
" i n s t a b i l i t y temperature" Ti.
of nonclassical nucleation paths has not y e t evolved a q u a n t i t a t i v e theory. We here o u t l i n e a unified approach t o the q u a n t i t a t i v e treatment of martensitic nucleation, which a1 lows f o r both c l a s s i c a l and nonclassical paths, and i d e n t i - f i e s t h e circumstances under which each mode of behavior will prevail.
Lattice-Deformation Energetics.- O f importance t o both the energy of a nonclassical nucleus and the i n t e r f a c i a l energy of a classical nucleus are the energetics of intermediate configurations i n t h e l a t t i c e deformation of the martensitic trans- formation. The f ree-energy change per u n i t volume, Ag
,
accompanying homogeneous d i s t o r t i o n of t h e parent phase t o t h e product s t r u c t u r e along a deformation path (defined by the s t r u c t u r a l parameter n ) i s depicted i n Figure 2, with n=0 and rl=l representing t h e parent and product s t r u c t u r e s . The value of Ag a t n=1 corresponds t o the chemical free-energy change of the transformation, dgch, and the maximum i n the curve represents a "homogeneous l a t t i c e deformation b a r r i e r " of height 4.When
ngch=O
( a s indicated f o r the temperatu e T-TO), the b a r r i e r i s designated@,.
When ngch<O, i s l e s s than 4,; and f o r Agc6 su;flciently negative, the condition
@=0, can ( i n p r i n c i p l e ) be reached whereby t h e parent l a t t i c e becomes mechanically unstable as i l l u s t r a t e d f o r an " i n s t a b i l i t y temperature", T=Tj, i n Figure 2. While some transforming materials show an anornolous temperature dependence of e l a s t i c constants such t h a t a shear constant extrapolates t o zero a t a f i n i t e T i tempera- t u r e ( 4 ) , many systems do not e x h i b i t such anomolies and then a T - temperature does not e x i s t . As discussed in ref. 3, however, any metastable l a t t i c e can be- come unstable i f t h e thermodynamic driving force i s made s u f f i c i e n t l y large, as f o r example by the application of s t r e s s . As indicated by the curve f o r T=Ms i n Figure 2, martensitic transformations i n bulk materials typical1 y nucleate under thermodynamic driving forces which a r e f a r removed from the conditions f o r general mechanical i n s t a b i l i t y of t h e parent l a t t i c e .
Recent progress has been made i n t h e quantitative estimation of the energy curves represented by Figure 2, based on higher-order e l a s t i c constants (5,6), interatomic p o t e n t i a l s ( 7 ) , and e l e c t r o n i c calculations (8-11 ) . The b a r r i e r height 4 i s s e n s i t i v e t o the nature of t h e l a t t i c e deformation path. Paths examined so f a r include "proportional s t r a i n " paths along intermediate configur- a t i o n s defined by the deformation $3, where B i s the t o t a l deformation r e l a t i n g parent and product l a t t i c e s , and "sequential shear" paths involving close-packed shear systems as proposed by Bogers and Burgers (12). Calculations to-date suggest
lower b a r r i e r s f o r proportional s t r a i n paths, with (Ivalues as low as , 10 t o lo3 2 J/mole f o r FCCraCC transformations. The lower b a r r i e r f o r t h e proportional s t r a i n path compared t o t h e sequential shear paths complicates t h e i n t e r r e l a t i o n of thermodynamic s t a b i l i t y (Agch) and t h e behavior of second-order e l a s t i c shear constants, such as C'=1/2 (Cll-C12) i n BCC a l l o y s . Analysis of load-path bi- furcations during homogeneous l a t t i c e deformation, as s t u d i e d by Milstein and Farber ( 7 ) , may c l a r i f y t h i s matter.
The energetic c a l c u l a t i o n s suggest t h a t ~ g ( q ) can be reasonably approximated by a fourth-order Landau expansion:
2 3 4
Ag = Aq -Bo +Crl , (1
1
where the c o e f f i c i e n t s A , B , and C are related t o t h e second, t h i r d , and fourth order e l a s t i c constants of the parent phase. The dependence of Ag on thermodynamic conditions can be approximated by expressing these c o e f f i c i e n t s as functions of Agch constrained t o give local minima a t r1=0 and q=1. The simplest approxima- t i o n , which i s most appropriate near a mechanical i n s t a b i l i t y , i s t o take t h e coef- f i c i e n t A ( c o n t r o l l i n g mechanical s t a b i l i t y of the parent l a t t i c e ) as a l i n e a r function of ~ g c h . This gives expressions of the form:
where ACJ?~ i s t h e c r i t i c a l Agch f o r i n s t a b i l i t y . The l a t t e r can be estimated from t h e c r i t i c a l driving force f o r i n s t a b i l i t y under s t r e s s a t T , giving B=
-3.08 ( 3 ) . Energies normalized t o bo can then be simply expresse8 as functions of t h e dimensionless parameter a , where a=O corresponds t o equal f r e e energies of t h e parent and product phases ( A ~ c ~ = o ) and %=I represents mechanical i n s t a b i l i t y of t h e parent phase.
Coherent Interfacial/Gradient Energy.- As discussed i n r e f . 3, and a l s o developed by Roitburd (1 3,14), the lower energy of intermediate configurations accompanying
t h e reduction of $ near a mechanical i n s t a b i l i t y w i l l promote diffuseness of a coherent m a r t e n s i t i c i n t e r f a c e . The behavior of t h e i n t e r f a c e i s then best described by adopting a gradient-energy formulation s i m i l a r t o t h a t o r i g i n a l l y applied t o d i f f u s e i n t e r f a c e s in diffusional (compositional ) transformations by Cahn and H i l l i a r d (15). The volume and surface energy terms of a c l a s s i c a l description a r e then replaced by a functional of t h e form:
where K. i s a gradient-energy c o e f f i c i e n t and Ag i s the local free-energy density:
When t h e i n t e r f a c e i s t h e i n v a r i a n t plane of an invariant-plane s t r a i n ( I P S ) l a t t i c e deformation, Ag corresponds t o t h e function of Figure 2 and equation 1 . For a non-IPS coherent i n t e r f a c e additional s t r a i n energy a r i s e s from d i s t o r t i o n s i n the i n t e r f a c i a l plane a s required by compatabil i t y of t h e deformation across t h e i n t e r f a c e . The gradient energy < ( 0 ~ ) 2 ' i s a t t r i b u t a b l e t o an harmonic many- body i n t e r a c t i o n ( 1 6 ) .
*The simple form adopted f o r equation 3 i s based on the assumption t h a t a l l defor- mation s t a t e s l i e along t h e path corresponding t o the v a r i a b l e q. This w i l l be
v a l i d f o r the simple cases considered here. More general cases a r e describable via the gradients of the complete s t r a i n tensor (1 3,14).
C 4 - 7 8 JOUKNAI. I)E PHYSIQUE
Adopting t h e approximation t h a t K i s independent o f n (and n e g l e c t i n g gradient-energy t e n s o f h i g h e r power than ( v Q ) ~ ) , we can e s t i m a t e a v a l u e f o r r from knowlege of t h e coherent s u r f a c e energy a t T0(3). For a f l a t coherent I P S i n t e r f a c e i n the z=O plane, t h e s u r f a c e energy i s :
m
uo = f [ n g i
x
(J-I-)~] dz ,-03
( 4 )
S o l v i n g t h e E u l e r equation f o r t h e v a r i a t i o n n ( x ) which minimizes a,, and assuming Ag i s o f t h e form o f e q u a t i o n 1, a s o l u t i o n f o r a. i s o b t a i n e d ( 3 ) which i d e n t i f i e s
K as:
and g i v e s an i n t e r f a c i a l w i d t h o f :
For values o f a few mJ/m 2 ( t y p i c a l f o r coherent i n t e r f a c e s ) , t h estimated 9, values discussed e a r l i e r g i v e d = 10-12 - l ~ J l ~ / m w i t h a
hoof
a few1.
By f i t t i n g : r t o t h e i n t e r f a c i a l behavior a t T , t h e gradient-energy f o r m u l a t i o n w i l l account f o r b o t h t h e c l a s s i c a l i n t e r f a c e a? a=o and t h e d i f f u s e - i n t e r f a c e behavior a t a = l .A problem o f r e l e v a n c e t o m a r t e n s i t i c n u c l e a t i o n i s t h a t o f a t h i n s l a b f m a r t e n s i t e centered on t h e z=0 plane where ~ = n
.
T h i s geometry i s i l l u s t r a t e d n F i g u r e 3. For a coherent I P S p a r t i c l e , t h e t o f a 1 free-energy f u n c t i o n a l o fi n t e r e s t i s now: ea
Boundary c o n d i t i o n s f o r t h e v a r i a t i o n n ( z ) a r e q(o) = no. q(m) = 0, and nl(m) = 0 w i t h
rl'<D
f o r z>O. A f i x e d t o t a l displacement across t h e p a r t i c l e correspondst o t h e c o n d i t i o n :
9
ndz = IJ ( 8 )0
where u i s t h e displacement parameter i d e n t i f i e d i n F i g u r e 3. The l a t t e r c o n d i t i o n d e f i n e s a f i x e d e f f e c t i v e semithickness o f t h e s l a b expressed as:
c = u/nO ( 9 )
The form o f t h e E u l e r e q u a t i o n f o r d e t e r m i n i n g t h e v a r i a t i o n n ( z ) which minimizes t h e f u n c t i o n a l F A , e q u a t i o n 7, s u b j e c t t o t h e c o n s t r a i n t o f equation 8 i s :
where A i s a m u l t i p l i e r i n t r o d u c e d by t h e f i x e d displacement c o n s t r a i n t . T h i s y i e l d s a d i f f e r e n t i a l e q u a t i o n f o r 0 ( z ) o f t h e f o n :
(11)
Clossicol Nonclossicol
I
heterogeneous homogeneousF i g u r e 3 Slab o f m a r t e n s i t e centered on z=O plane. C l a s s i c a l p a r t i c l e of semi thickness 2c w i t h sharp i n t e r f a c e produces displacement u . y ~ , where yT i s t h e t r a n s f o r n a t i o n s t r a i n . Dotted l i n e i l l u s t r a t e s - displacement f i e l d o f d i f f u s e i n t e r f a c e producing same t o t a l displacement.
F i g u r e 4 N u c l e a t i o n b a r r i e r AG* vs a f o r coherent m a r t e n s i t i c t r a n s f o r m a t i o n . S o l i d l i n e s d e p i c t behavior f o r c l a s s i c a l heterogeneous n u c l e a t i o n f o r v a r i o u s values o f c/6 corresponding approximately t o number o f 8 i s l o c a t i o n s i n n u c l e a t i n g defect. Dashed 1 i ne represents b a r r i e r f o r n o n c l a s s i c a l homogeneous n u c l e a t i o n . For a given nQ, t h e minimum energy v a r i a t i o n f o r n(z) corresponds t o t h e a d d i t i o n a l boundary c o n d r t i o n q (0)=0, whereby A=-Ago/nO.
S o l u t i o n o f e q u a t i o n 11 f o r n ( z ) as a f u n c t i o n o f q, and h i s g i v e n by the i n t e g r a t i o n :
S u b s t i t u t i o n o f v a r i a b l e s according t o e q u a t i o n 11 determines FA and u as:
The e f f e c t i v e semithickness c=u/n, can be used t o d e f i n e an i n t e r f a c i a l energy:
These r e l a t i o n s p r o v i d e t h e b a s i s f o r a t r e a t m e n t o f t h e energy o f a m a r t e n s i t i c nucleus, which a l l o w s f o r b o t h c l a s s i c a l and n o n c l a s s i c a l n u c l e a t i o n paths.
Coherent Nucleation.- The m a j o r c o n t r i b u t i o n t o t h e i n t e r f a c i a l energy o f a t h i n - p l a t e I P S p a r t i c l e a r i s e s from i t s broad i n t e r f a c e s as opposed t o t h e p l a t e p e r i p h e r y , w h i l e t h e e l a s t i c energy accommodating t h e t r a n s f o r m a t i o n s t r a i n r e s i d e s p r i m a r i l y i n t h e r e g i o n of t h e m a t r i x beyond t h e p e r i p h e r y . A c c o r d i n g l y
C4-80 JOURXAT, DE PHYSIQUE
t h e t o t a l energy o f a t h i n - p l a t e nucleus o f r a d i u s r, l y i n g i n t h e z=O plane, can be reasonably approximated by t h e energy o f a one-dimensional v a r i a t i o n n ( z ) w i t h i n a c y l i n d e r o f r a d i u s r combined w i t h t h e e l a s t i c energy associated w i t h t h e long-range s t r a i n f i e l d o f a p a r t i c l e o f semi t h i c k n e s s c producing t h e n e t d i s p l a c e - ment d e f i n e d by u. T h i s approximation w i l l f i t b e s t when t h e i n t e r f a c i a l w i d t h 6 i s much l e s s than r. R e l a t i n g t h e m a t r i x e l a s t i c constants t o t h e c o e f f i c i e n t A i n e q u a t i o n 1, t h e nucleus energy can then be expressed by:
where 5 i s an accomnodation f a c t o r equal t o w(2-v)/4(1-v) f o r t h e case o f a simple shear. The f u n c t i o n s FA and u can be evaluated from e q u a t i o m 13 and 14 u s i n g
equations 1 a ~ d 2 t o express Ag. The a c t i v a t i o n energy f o r homogeneous coherfnt,
,
n u c l e a t i o n AG can t h a n be e v a l u a t e d from a saddle p o i n t o f AG ( r , n , A ) a t ( r ,n,,X ) corresponding t o a maximum of AC w i t h r e s p e c t t o r and a minimum w i t h r e s p e c t t o
Q and A. The c o e f f i c i e n t s of equations 2 and 7 a r e most c o n v e n i e n t l y f i t t e d t o tRe c l a s s i c a l i n t e r f a c i a l parameters a t a=O by n o r m a l i z i n g energies ( p e r volume) w i t h r e s p e c t t o @, and distances w i t h r e s p e c t t o 6, (equation 6 ) .
I n s p e c t i o n o f these equations r e v e a l s c e r t a i n f e a t u r e s o f a coherent nucleus which p a r a l l e l t h e C a h n - H i l l i a r d d i f f u s i o n a l n u c l e a t i o n b e h a v i o r as discussed i n r e f . 3. Since FA must be n e g a t i v e f o r a c r i t i c a l nucleus, no* must be s u f f i c i e n t l y l a r g e t h a t Ago<O. As i n d i c a t e d by F i g u r e 2, t h i s c o n d i t i o n i s met by p r o g r e s s i v e l y s m a l l e r n as w 1 . While a c r i t i c a l nucleus a t low a w i l l r e q u i r e n o = l and t h e h i g h energies o f i n t e r m e d i a t e c o n f i g u r a t i o n s w i l l cause t h e i n t e r f a c e t o remain f a i r l y sharp, t h e l o w e r a l l o w a b l e values o f no and lower i n t e r m e d i a t e energies w i l l promote d i f f u s e i n t e r f a c e s o f lower energy w i t h F A 2 as a = l i s approached.
Since b o t h t e n s o f e q u a t i o n 16 vanish as A 4 a t a = l , AG f o r a n o n c l a s s i c a l nucleus (n,<l) vanishes a t a = l as d e p i c t e d by t h e dashed l i n e i n F i g u r e 4 t Thus, w h i l e c l a s s i c a l n u c l e a t i o n t h e o r y g e n e r a l l y p r e d i c t s a AG which i s t o o h i g h f o r thermal a c t i v a t i o n o f homogeneous n u c l e a t i o n i n most c i r c ~ m s t a n c e s , d e v i a t i o n s from c l a s s i c a l b e h a v i o r near i n i n s t a b i l i t y can reduce AG enough so t h a t t h e r m a l l y - a c t i v a t e d homogeneous coherent n u c l e a t i o n w i l l precede t h e a c t u a l i n s t a b i l i t y . T h i s r e s u l t o b t a i n s f o r t h e simple case o f an IPS l a t t i c e deformation. I t i s n o t
c l e a r t h a t t h e a d d i t i o n a l s t r a i n energy a s s o c i a t e d w i t h i n t e r f a c i a l d i s t o r t i o n s i n a non-IPS coherent t r a n s f o r m a t i o n w i l l vanish a t a=], b u t i t can be expected t h a t n o n c l a s s i c a l behavior w i l l g e n e r a l l y a l l o w a s u b s t a n t i a l r e d u c t i o n i n AG* f o r homogeneous coherent n u c l e a t i o n near a mechanical i n s t a b i l i t y .
I t i s o f course w e l l e s t a b l i s h e d t h a t m a r t e n s i t i c t r a n s f o r m a t i o n s t y p i c a l l y n u c l e a t e heterogeneously. The h i g h a l e v e l s necessary f o r homogeneous n u c l e a t i o n can o n l y be
F7sCt87
i f n u c l e a t i o n a t i m p e r f e c t i o n s i s e l i m i n a t e d . T h e o r e t i c a l p r e d i c t i o n s and experimental observations (1 9-23) i n d i c a t e t h a t t h e most comon n u c l e a t i o n s i t e s a r e l i n e a r d e f e c t s which can be modelled as groups o f d i s l o c a t i o n s . Heterogeneous n u c l e a t i o n o f t h e coherent nucleus considered here can be modelled by t r e a t i n g i t s i n t e r a c t i o n w i t h t h e s t r e s s f i e l d od o f t h e d e f e c t i n t h e z=O plane. The i n t e r a c t i o n energy f o r a ribbon-shaped nucleus- o f l e n g t h L, w i d t h a, and semi t h i c k n e s s c l y i n g along a d e f e c t p a r a l l e l t o t h e y a x i s i s then :i n t - 2 ~ t - r ~
J
od-qcdx (17)r0
* w i t h i n t h e approximation o f e q u a t i o n 16, AG* o f a c l a s s i c a l nucleus (n,=l) a l s o vanishes as A-O. This a r i s e s from n e g l e c t o f t h e s u r f a c e energy contribution o f t h e p l a t e p e r i p h e r y . A mqre r e a l i s t i c (nonvanishing) b e h a v i o r o f t h e c l a s s i c a l GG* near a = l i s o b t a i n e d by adding a 4.i;rco term t o e q u a t i o n 16, w i t h a d e f i n e d by e q u a t i o n 17. The n o n c l a s s i c a l AG* s t i l l vanishes a t 3=1.
where y~ i s the transformation s t r a i n and r o i s the defect core radius. The t o t a l nucleation energy change AG can be expressed by combining E i with the two terms of equation 16 modified f o r the ribbon shape. For a disyocation defect with a
-
x-1 s t r a i n f i e l d , equation 17 introduces a negative energy contribution proportional t o In a which substantial 1 y reduces AG*, promoting nucleation a t lower a values.As equation 17 i n d i c a t e s , the nucleus-defect i n t e r a c t i o n energy i s propor- tional t o qo and heterogeneous nucleation thus favors c l a s s i c a l nucleation, The range of a required f o r c l a s s i c a l heterogeneous nucleation can be estimated from a dislocation-dissociation model of heterogeneous nucleation (17) in which nuclei form a t a constant thickness determined by the number of d i s l o c a t i o m i n the nucleating defect. The c l a s s i c a l b a r r i e r AG* goes abruptly t o zero when the f a u l t energy associated with the d i s s o c i a t i o n becomes negative. For the simple coherent transformation considered here, t h i s occurs a t a c r i t i c a l driving force of
agch = -o,/c. Taking o,=4$66,/3 from equations 5 and 6 , the c r i t i c a l a f o r nucteation i s then:
The behavior predicted f o r various values of c/6
,
corresponding approximately t o t h e number of dislocations in the nucleating dgfect, i s shown by the s o l i d l i n e s in Figure 3. For defects of the s i z e which accounts f o r bulk nucleation in s t e e l s , % i s of the order of 0.1. In addition t o the strong i n t e r a c t i o n energy favoring a c l a s s i c a l qO=l nucleus, the occurrence of nucleation a t such low a values will favor sharp i n t e r f a c e behavior such t h a t no s i g n i f i c a n t deviation from a c l a s s i c a l nucleation path i s expected f o r heterogeneous coherent nucleation.To summarize the r e s u l t s depicted i n Figure 4, heterogeneous coherent nuclea- tion a t low a will favor nucleation via t h e c l a s s i c a l path. I f heterogeneous nucleationcan be suppressed, homogeneous coherent nucleation a t high a will then
favor a nonclassical path and the associated nucleation b a r r i e r i s expected t o reach a thermally-surmountable level before general l a t t i c e i n s t a b i l i t y can be reached.
Semicoherent Nucleation.- W i t h the exception of F C C d C P and some BCC+9R
transformations, the 1 a t t i c e deformation accompanying most martensi t i c transforma- t i o n s i s not s u f f i c i e n t l y c l o s e t o on IPS t o allow f u l l y coherent transformation.
Under the thermodynamic conditions where these transformations a r e known t o occur, the volume s t r a i n energy associated w i t h d i s t o r t i o n s in a f u l l y coherent i n t e r f a c e would exceed the chemical driving force, and so a martensi t i c p a r t i c l e can only be s t a b l e i n a semicoherent form with s u f f i c i e n t 1 y reduced s t r a i n energy.
Semicoherence i s achieved through a l a t t i c e - i n v a r i a n t deformation by s l i p o r twinning which a l t e r s the net transformation shape s t r a i n , generally toward an IPS. A necessary consequence of the l a t t i c e - i n v a r i a n t deformation i s a modification of the i n t e r f a c i a l s t r u c t u r e , introducing a short-range i n t e r f a c i a l e l a s t i c s t r a i n f i e l d (with range comparable t o the spacing of the twins o r s l i p dislocations i n the i n t e r f a c e ) which contributes a substantial increment t o the i n t e r f a c i a l energy. In c o n t r a s t t o the behavior of f u l l y coherent i n t e r f a c e s just discussed, the i n t e r f a c i a l energy of a semicoherent p a r t i c l e i n an anisotropic crystal will generally have contributions from e l a s t i c constants which do not vanish a t an i n s t a b i l i t y . Similarly, nonvanishing e l a s t i c constants will a l s o l i k e l y contribute t o the accommodation s t r a i n energy associated with the net transformation s t r a i n (shape change). Thus the i n t g r f a c i a l energy and s t r a i n energy will remain f i n i t e , and the nucleation b a r r i e r AG f o r homogeneous nucleation of a semlcoherent martensitic p a r t i c l e will not vanish a t a = l
.
Nonclassical d i f f u s e - i n t e r f a c e behavior should play only a secondary r o l e in semicoherent martensi t i c nucleation, as r e f l e c t e d by j u s t a moderate reduction in i n t e r f a c i a l energy accompanying a widening of the cores of i n t e r f a c i a l dislocations. Classical calculations indicateJOURNAI. DE I'IIYSIQUE
I i i z
Figure 5 Schematic i n t e r f a c i a l dislocation s t r u c t u r e f o r semicoherent F C C 4 C C transformation nucleus lying i n close-packed (1 11) plane.
t h a t homogeneous nucleation of semicoherent martensitic transformations i s highly improbable under the thermodynamic conditions where these transformations typic- a l l y occur ( 2 4 ) .
For the case of heterogeneous nucleation, the considerations previously discussed f o r coherent nucleation combined with the properties of semicoherent i n t e r f a c e s j u s t described make i t highly unlikely t h a t semicoherent nucleation will deviate s i g n i f i c a n t l y from the c l a s s i c a l nucleation path. The e s s e n t i a l features of heterogeneous c l a s s i c a l nucleation of semicoherent transformations a t d i s l o c a t i o n - l i k e l i n e a r defects a r e now well established, based on both specific dislocation-dissociation models (17,25,26) and general continuum-elastic calcula- tions (18,27,28). A general r e s u l t i s t h a t ( a ) AG* f o r heterogeneous nucleation will vanish a t a r e l a t i v e l y modest (compared t o t h a t f o r a = l ) c r i t i c a l driving force which depends on t h e defect s i z e , and (b) the c r i t i c a l condition wi 11 be preceded by the formation of s t a b l e s u b c r i t i c a l classical embryos a t very low driving forces.
The short-rangeinteraction with d i s l o c a t i o n s i s so strong t h a t t h e l a t t e r embryos may e x i s t above To (Agch>O). The prediction of pre-existing c l a s s i c a l embryos and t h e g r e a t e r i n t e r a c t i o n energy f o r c l a s s i c a l vs. nonclassical embryos discussed e a r l i e r reinforce the conclusion t h a t heterogeneous semicoherent martensitic nuclea- tion will follow a c l a s s i c a l r a t h e r than non-classical nucleation oath.
The d e t a i l e d behavior of AG* f o r heterogeneous nucleation i s s e n s i t i v e t o the c o n s t r a i n t s applied t o t h e nucleus. Dislocation-dissociation models f o r heterogeneous nucleation a r e based upon a d i s c r e t e - d i s l o c a t i o n description of nucleus i n t e r f a c i a l s t r u c t u r e , intended t o r e f l e c t discrete-crystal c o n s t r a i n t s on the i n t e r f a c i a l mobility and morphology of a microscopic p a r t i c l e . A s p e c i f i c nucleus model ( 2 5 )
f o r semicoherent FCC+BCC trans formation i s depicted in Figure 5. The transformation l a t t i c e deformation i s accomplished bya a homogeneols a r r a y of coherency dislocations (29) with approximate Burgers vectors 11211 and [121], each a r r a y producing an IPS on crystal planes of c l o s e s t (30). A6semicoherent i n t e r f a c e i s obtained by the superpositign of a perigdic array of (anticoherency) s l i p disloca- tions with Burgers vectors - [TIO] and - [Oll], accomplishing a l a t t i c e - i n v a r i a n t deformatign. The major comaonent of t h i transformation net shape s t r a i n a r i s e s from t h e - [I211 coherency dislocations. I f the nucleation of individual loops of these A f s ~ o c a t i o n s i s d i f f i c u l t a t small s i z e s , the habit of a microscopic p a r t i c l e will be confined t o the close-packed (111) plane. A fixed number of the l a t t e r dislocations can be derived from the dissociation of a group of e x i s t i n g dislocatiorr, as represented i n Figure 6a, allowing formation of a ribbon-shaped nucleus of width a and thickness 2c. I f t h e d i f f i c u l t y of forming additional a [I211 coherency dislocations i s removed a t l a r g e r p a r t i c l e s i z e s by an athermal Xgchanism ( t o be discussed), t h e kinetics of nucleation will be controlled by the
F i g u r e 6 Geometry o f nucleus and d e f e c t f o r heterogeneous n u c l e a t i o n , d e s c r i b i n g nacleus as ( a ) d i s c r e t e d i s l o c a t i o n a r r a y , and ( b ) continuum p a r t i c l e . widening o f t h e nucleus a t c o n s t a n t thickness. As described e a r l i e r f o r t h e case o f heterogeneous coherent n u c l e a t i o n by d i s s o c i a t i o n t h i s geometric n u c l e a t i o n p a t h has t h e c h a r a c t e r i s t i c t h a t t h e free-energy b a r r i e r AG* drops a b r u p t l y t o zero when t h e e f f e c t i v e f a u l t energy accompanying d i s s o c i a t i o n becomes negative.
Widening o f t h e nucleus i s then c o n t r o l l e d by t h e d i s c r e t e - c r y s t a l k i n e t i c processes governing t h e motion o f t h e a [ I 2 1 1 coherency d i s l o c a t i o n s and t h e
concomitent f o r m a t i o n o f t h e i n t e r f a c l g l d i s l o c a t i o n s o f F i g u r e 5. The thermodynam- i c c o n d i t i o n s f o r t h e f a u l t energy t o become n e g a t i v e can be expressed as a f u n c t i o n o f d e f e c t s i z e (as f o r t h e coherent case i n e q u a t i o n 18) v i a c l a s s i c a l n u c l e a t i o n t h e o r y (1 7,25). By t h i s d i s s o c i a t i o n mechanism, t h e n u c l e a t i o n k i n e t i c s a r e governed n o t by heterophase f l u c t u a t i o n s surmounting t h e b a r r i e r AG*, b u t r a t h e r by t h e microscopic processes o f i n t e r f a c i a l motion a t a pre-formed c l a s s i c a l embryo where AG*=O.
Numerical continuum e l a s t i c c a l c u l a t i o n s by Suezawa and Cook (18) a l s o p r e d i c t t h a t AG* can vanish f o r heterogeneous c l a s s i c a l n u c l e a t i o n a t a l i n e a r defect, b u t t h e behavior o f AG* d i f f e r s i n d e t a i l f r o m the above d i s l o c a t i o n - d i s s o c i a t i o n model due t o d i f f e r e n t c o n s t r a i n t s a p p l i e d t o t h e nucleus; nucleus t h i c k n e s s i s unconstrained i n t h e Suezawa-Cook treatment, b u t an o b l a t e - s p h e r o i d a l nucleus shape i s imposed. L i n g and Olson (27) have demonstrated w i t h s i m i l a r c a l c u l a t i o n s t h a t a ribbon-shaped nucleus has a l o w e r energy than an o b l a t e - spheriodal nucleus when forming i n t h e s t r a i n f i e l d o f a l i n e a r defect. B a r r i e r - l e s s n u c l e a t i o n o f a ribbon-shaped p a r t i c l e w i l l o c c u r even when AG* f o r t h e o b l a t e s p h e r o i d i s s t i l l f i n i t e . These r e s u l t s a r e supported by a simple a n a l y t i c a l t r e a t m e n t o f t h e problem by R o i t b u r d ( 2 8 ) .
The c a l c u l a t i o n s o f L i n g and Olson (27) demonstrate t h a t t h e d i s c r e t e -
d i s l o c a t i o n and c o n t i n u u m - e l a s t i c nucleus d e s c r i p t i o n s a r e e n e r g e t i c a l l y e q u i v a l e n t when i d e n t i c a l c o n s t r a i n t s a r e employed. T h i s was shown b computing t h e energy o f a continuum p a r t i c l e w i t h a n e t t r a n s f o m a t i o n s t r a i n
~ 7 .
e q u i v a l e n t t o t h a t o f t h e nucleus o f F i g u r e 5 i n t e r a c t i n g w i t h t h e same d e f e c t a s J i n F i g u r e 6. Using parameters a p p r o p r i a t e t o s t e e l s , the c a l c u l a t e d t o t a l free-energy change p e r l e n g t h , Ge, vs. nucleus width, a, a r e p l o t t e d i n F i g u r e 7 f o r t h r e e l e v e l s o f chemical d r i v i n g f o r c e . The curves a r e f o r v a r i o u s f i x e d nucleus thicknesses, w i t h t h e numbers i n d i c a t i n g the t h i c k n e s s i n number o f c r y s t a l planes. Thed o t t e d curves denoting a t h i c k n e s s o f 15 planes correspond t o a nucleus t h i c k n e s s equal t o t h e h e i g h t h o f t h e d e f e c t ( F i g u r e 6 ) as adopted i n t h e d i s l o c a t i o n - d i s s o c i a t i o n d e s c r i p t i o n (25). The computed energies i n c l u d e a s t r a i n - e n e r g y c o n t r i b u t i o n a r i s i n g from e l a s t i c d i s t o r t i o n s i n the i n t e r f a c e ( h a b i t ) plane
JOUllNAL DE PHYSIQUE
Figure 7 Total free-energy change per unit nucleus length for heterogeneous FCC-tBCC nucleation in s t e e l s a t three levels of driving force.
Numbers i d e n t i f y nucleus thickness in terms of crystal close-packed planes. Shear modulus, u ; FCC l a t t i c e parameter, ao. Dislocation defect i n mixed o r i e n t a t i o n .
imposed by the c o n s t r a i n t s of a rational (111) crystal-plane habit. The behavior of t h e dotted curves i s identical t o thatderived from the dislocation-dissociation model (15,25).
Comparison of the curves f o r d i f f e r e n t thicknesses v e r i f i e s t h a t the short- range i n t e r a c t i o n s favor the formation of a s u b c r i t i c a l embryo with a thickness equal t o the defect height a t low driving force, a s indicated by the arrow (a/h=lO) i n Figure 7a. Once an embryo of t h i s thickness i s e s t a b l i s h ~ d . th e d i f f i c u l t y of nucleating additional coherency dislocations will constrain t h e thickness so t h a t the embryo i s s t i l l s u b c r i t i c a l (with a/h=30) under the driving- force conditions of Figure 7b even though a t h i c k e r embryo would be s u p e r c r i t i c a l . For the conditions of Figure 7c, the embryo i n question becomes supercri t i c a l , with a negative f a u l t energy then driving the process of i n t e r f a c i a l motion.
A plausible dislocation pole mechanism f o r embryo thickening has been proposed based on the i n t e r s e c t i o n with f o r e s t dislocations when t h e conditions f o r u n r e s t r i c t - ed nucleus widening a r e met (26). Once thickening can readily occur, the p a r t i c l e can adopt a minimum-energy habit and morphology, thereby allowing the attendant reduction in s t r a i n energy t o a c c e l e r a t e the growth process. In s i t u electron microscopy observations of heterogeneous nucleation in thennoelastic Au-Cd (1 9) i n d i c a t e a change from the close-packed habit t o t h e macroscopic habit without evidence of a pole mechanism. Nucleation of additional coherency dislocations may i n t h i s case be stimulated by the f r e e surfaces of the thin-foil specimen.
The increasing separation of the curves of Figure 7c a t large a s i g n i f i e s an increasing thickening force which might ultimately drive coherency-dislocation generation under bulk conditions. However, c l a s s i c a l (1 i n e a r - e l a s t i c )
dislocation-nucleation calculations have not y e t accounted f o r such a process ( 2 6 ) . Defining the end product of nucleation a s an "operational nucleus", t h e nucleation k i n e t i c behavior discussed here can bedescribedwithin the framework of general nucleation theory, adopting a nucleation-rate equation of the form ( 3 ) :
A
where Ni i s the number of nucleation s i t e s of t h e i t h type, xi i s the f r a c t i o n of these s i t e s occupied ( a t a given i n s t a n t ) by operational nuclei, and Bi i s a frequency f a c t o r controlling the r a t e a t whish nuclei become operational. In c l a s s j c a l nucleation, the f r a c t i o n of s i t e s
x
occupied by nuclei of operational s i z e n i s expressed by:= Z exp ( - & G * / ~ T ) (20)
where Z i s a nonequilibrium f a c t o r (expressing the f r a c t i o n of c r i t i c a l nuclei which reach ;) and AG* i s a s previously defined. The frequency factor
8
i s the r a t e a t which these nuclei a r r i v e a t n , and i s expressed as:where Qm i s the activation energy control ling motion of the nucleus/matrix
i n t e r f a c e and v i s an associated attempt frequency. Tn diffusional transformations, Qm i s the a c t i v a t i o n energy f o r atomic diffusion, while f o r martensitic transforma- tions i t i s the a c t i v a t i o n energy f o r motion of a g l i s s i l e i n t e r f a c e .
The operational nucleus i n martensitic transformations has come to be defined by the point a t which rapid growth begins, i . e . the condition t h a t leads t o a macroscopically detectable event (31 )
.
The strong defect/nucleus i n t e r a c t i o n i n heterogeneous martens i t i c nucleation indicates t h a t operational nucleation wi 11 occur under t h e condition AG*=O whereby the nucleation k i n e t i c s (equation 19)C4-86 J O U R N A L U E PHYSIQUE
a r e controlled e n t i r e l y by i n t e r f a c i a l motion. Further progress i n our understand- ing of the k i n e t i c s of martensitic nucleation requires b e t t e r knowledge of the microscopic processes of i n t e r f a c i a l motion subject t o the constraints of the d i s c r e t e c r y s t a l .
The k i n e t i c s of nucleus-interfacial motion have been modelled i n terms of t h e generation of dislocations producing the l a t t i c e - i n v a r i a n t deformation (24,32) and t h e motion of coherency d i s l o c a t i o n s producing the l a t t i c e deformation (26).
Either mechanism can account f o r the experimental nucleation kinetics which show a l i n e a r dependance of a c t i v a t i o n energy on driving force. As discussed in
reference 3, no attempt t o account f o r t h i s behavior i n terms of a smoothly changing AG ( a s opposed t o Q,) has succeeded so f a r .
Nucleation k i n e t i c experiments on the coherent F C C 4 C P transformation i n Fe-Ni-Mn a1 loys (33), where t h e l a t t i c e - i n v a r i a n t deformation i s absent, indicate the same behavior a s does the semicoherent FCC+BCC transformation in the same a l l o y system. This finding suggests t h a t r a t e control i s by the l a t t i c e deformation i n both cases, and favors t h e coherency dislocation motion model. Recent experiments on the mobility of macroscopic martensi t i c i n t e r f a c e s in thermoelastic 0 CuAl N i a l l o y s reveal the same k i n e t i c behavior as thermally-activated dislocation motion (34).
Analogous t o the c l a s s i c a l and nonclassical paths which describe the history of a nucleus as an e n t i t y , the microscopic fluctuations governing i n t e r f a c i a l motion of a c l a s s i c a l nucleus may be of e i t h e r "heterophase" o r "homophase"
c h a r a c t e r ( 3 ) . Hence, a1 though the overall nucleus follows a classical path, nonclassical d i f f u s e - i n t e r f a c e concepts may have some relevance t o t h e i n t e r f a c i a l mobil i t y , p a r t i c u l a r l y i f widening of the i n t e r f a c i a l core s t r u c t u r e increases mobility as suggested by estimates of the coherent i n t e r f a c e P e i r l s b a r r i e r (14).
The macroscopic i n t e r f a c i a l mobi 1 i t y experiments (34) suggest an anomol ous
mobility increase a t low temperatures where e l a s t i c softening e f f e c t s a r e encounter- ed.
While observed heterogeneous nucleation s i t e s of low potency possess many c h a r a c t e r i s t i c s of the nucleation-site models (19,23), the most potent s i t e s which i n i t i a t e transformation a t the Ms temperature i n s e e l s a r e known t o
be sparsely d i s t r i b u t e d with a number density of -106 c ! f S (one per lOOw grain) and a r e therefore v i r t u a l l y unobservable. Efforts t o a r t i f i c i a l l y produce crystallographically well-defined potent s i t e s i n pre-strained synthetic Fe30Ni b i c r y s t a l s have been hampered by interference from an unusual isothermal a u t o c a t a l y t i c mode of transformation (formation of paired martensitic units o r " b u t t e r f l i e s " ) which obscures the i n i t i a l events (35). Earlier detection of i n i t i a l events by acoustic emission measurements might circumvent t h i s d i f f i c u l t y .
Although a u t o c a t a l y t i c nucleation often i n t e r f e r e s with investiyations directed a t the i n i t i a l nucleation events, the f a c t t h a t the majority of martensitic units a r e a u t o c a t a l y t i c a l l y generated makes t h i s an important phenomenon worthy of study in i t s own r i g h t . I t i s c l e a r t h a t the s t r e s s - a s s i s t e d and strain-induced modes of nucleation which normally account f o r transformation during deformation wi 11 make an important contribution t o
a u t o c a t a l y s i s , b u t the a u t o c a t a l y t i c nucleation accompanying rapid growth events i n "bursting" alloys i s suggestive of an overdriven process f o r which c l a s s i c a l nucleation k i n e t i c concepts may be inadequate. Autocatalytic nucleation may represent a f e r t i l e ground f o r exploration of possible nonclassical phenomena.
As a f i n a l remark, we have here focussed on nucleation mechanisms f o r transformations in bulk m ~ t e r i a l s . I t i s well established t h a t f r e e surfaces can provide a strong stimulating e f f e c t on martensitic nucleation. While i t i s evident t h a t surface image forces will reduce t h e s t r a i n energy of a c l a s s i c a l nucleus, a q u a n t i t a t i v e theory of surface nucleation has not y e t been developed.
This i s another area which i s r i p e f o r f u r t h e r investigation.
Closure.- T h i s survey o f t h e t h e o r e t i c a l p o s s i b i l i t i e s f o r martensi t i c n u c l e a t i o n s e t s some d e f i n i t e g u i d e l i n e s f o r t h e circumstances f a v o r i n g c l a s s i c a l and n o n c l a s s i c a l n u c l e a t i o n paths. Nonclassical paths w i l l be favored i n t h e case o f homogeneous coherent n u c l e a t i o n near a mechanical i n s t a b i 1 i t y , b u t t h i s wi 11 r e q u i r e t h e suppression of heterogeneous n u c l e a t i o n . This s p e c i a l c o n d i t i o n m i g h t be achieved e x p e r i m e n t a l l y i n t h e case o f small d e f e c t - f r e e metastable p a r t i c l e s p r e c i p i t a t e d i n a s o l i d m a t r i x , (as i n t h e case o f FCC i r o n i n copper, o r o f p a r t i a l l y s t a b i l i z e d z i r c o n i a ) which a r e known t o t r a n s f o r m a t q u i t e h i g h d r i v i n g f o r c e s . A more q u a n t i t a t i v e t r e a t m e n t o f n o n c l a s s i c a l n u c l e a t i o n would b e n e f i t from f u r t h e r c a l c u l a t i o n s o f t h e e n e r g e t i c s o f homogeneous l a t t i c e deformations and measurements o f h i g h e r - o r d e r e l a s t i c constants i n metastable a1 l o y s , as w e l l as t h e development o f s t r a i n - g r a d i e n t energies.
The heterogeneous n u c l e a t i o n which c o n t r o l s m a r t e n s i t i c t r a n s f o r m a t i o n s i n b u l k m a t e r i a l s w i l l f a v o r n u c l e a t i o n by t h e c l a s s i c a l path, p a r t i c u l a r l y i n semicoherent t r a n s f o r m a t i o n s . I n view o f t h e f a c t t h a t c l a s s i c a l behavior, i n v o l v i n g r e l a t i v e l y sharp i n t e r f a c e s , i s adequately d e s c r i b e d by l i n e a r - e l a s t i c continuum and d i s l o c a t i o n t h e o r i e s , t h e e s s e n t i a l f e a t u r e s o f heterogeneous m a r t e n s i t i c n u c l e a t i o n a r e considered t o be w e l l understood. Although t h e d e t a i l e d d i s c r e t e c r y s t a l c o n s t r a i n t s which a p p l y t o a m i c r o s c o p i c p a r t i c l e m e r i t f u r t h e r a t t e n t i o n , t h e o r e t i c a l and experimental evidence p o i n t t o n u c l e a t i o n - r a t e c o n t r o l by i n t e r f a c i a l motion. Consequently, a b e t t e r understanding o f t h e mechanism and k i n e t i c s o f heterogeneous m a r t e n s i t i c n u c l e a t i o n r e q u i r e s f u r t h e r i n v e s t i g a t i o n o f t h e complete process o f m a r t e n s i t i c i n t e r f a c i a l motion i n a d i s c r e t e c r y s t a l ; t h i s i n t e r f a c i a l m o t i o n may i n w l ve n o n c l a s s i c a l phenomena i n some circumstances
.
I s o l a t i o n o f t h e s p a r s e l y d i s t r i b u t e d h i g h l y p o t e n t n u c l e a t i o n s i t e s which i n i t i a t e t r a n s f o r m a t i o n i n t h e b u l k w i l l r e q u i r e t h e f a b r i c a t i o n o f s y n t h e t i c n u c l e a t i o n s i t e s i n c a r e f u l l y designed experiments guided by s p e c i f i c s i t e models. While e x i s t i n g models appear t o reasonably account f o r n u c l e a t i o n i n deformation-induced t r a n s f o r m a t i o n s , f u r t h e r work i s needed on t h e mechanisms and k i n e t i c s of a u t o c a t a l y t i c n u c l e a t i o n ( e s p e c i a l l y i n m a t e r i a l s exhi b i t i n g b u r s t i n g behavior) as w e l l as t h e s p e c i a l case o f m a r t e n s i t i c n u c l e a t i o n a t f r e e surfaces.
Acknowledgements
.-
T h i s research i s sponsored by t h e N a t i o n a l Science Foundation under Grant #DMR79-15196. The authors a r e a l s o indebted t o Dr. H.C. L i n g o f Western E l e c t r i c Engineering Research Labs f o r t h e e l a s t i c c a 7 c u l a t i o n s quoted i n F i g u r e 6, and t o Dr. J.W. Cahn o f t h e N a t i o n a l Bureau o f Standards f o r h e l p f u l discussions concerning r n n c l a s s i c a l n u c l e a t i o n behavior.i o n c l u s i o n s r e g a r d i n g n o n c l a s s i c a l behavior near a mechanical i n s t a b i l i t y a r e based on p r e l i m i n a r y numerical c a l c u l a t i o n s performed by M r . K. B a l l o u as an undergraduate research p r o j e c t .
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