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Solitons and wave number selection in first order diffusive phase transformations
Scripta METALLURGICA et MATERIALIA
Vol. 24, pp. 2263-2268, 1990 Printed in the U.S.A.
Pergamon Press plc All rights reserved
S o l i t o n s and Nave Number S e l e c t i o n in F i r s t Order D i f f u s i v e Phase Transformations
McMaster
University,
J.S. Kirkaldy Hamilton. Canada
L8S-4M1
(Received May I, 1990) (Revised September 19, 1990) In 1981 the author introduced to the readers of t h i s journal the idea t h a t s o l i t o n s or s o l i t a r y waves play a p a r t in w a v e number s e l e c t i o n in e u t e c t o i d ( i c ) and Widmanstatten reactions. Since t h a t time there have been a t l e a s t three micrographic examples offered in the l i t e r a t u r e p e r t a i n i n g to e u t e c t i c s and c e l l u l a r s o l i d i f i c a t i o n . The aim of t h i s c o n t r i b u t i o n i s to describe an analogous r o l e for s o l i t o n s in the thermal d e n d r i t e problem,to o f f e r a t h e o r e t i c a l s y n t h e s i s and to f u r t h e r e s t a b l i s h the connection between the s o l i t o n mechanism and d i s s i p a t i o n p r i n c i p l e s . I t i s argued t h a t these e n t i t i e s l i e w i t h i n a d i s t i n c t and fundamental c l a s s of s t o c h a s t i c o b j e c t s which includes v o r t i c e s and d i s l o c a t i o n s . Evidence for S o l i t o n s in Phase Transformations S o l i t o n s were f i r s t observed in ocean waves (I) as n o n - l i n e a r l o c a l i z e d s p a t i a l d i s l o c a t i o n s which migrate e s s e n t i a l l y n o n - d i s s i p a t i v e l ywithin the manifold so as by c r e a t i o n and mutual a n n i h i l a t i o n s t a b i l i z e the symmetry and d i s s i p a t i v e p r o p e n s i t i e s of the optimal waves. Quite o u t s i d e t h i s general context an analogous m a n i f e s t a t i o n was recognized by Chalmers (2) and Jackson and Hunt (3) in three-dimensional forced v e l o c i t y e u t e c t i c s as wave number d i s l o c a t i o n s , and these were designated as l a m e l l a r f a u l t s (Figs.1 and 2; Refs.4 and 5). They c o r r e c t l y surmised t h a t s t a b i l i t y of a l a m e l l a r a r r a y with a s u f f i c i e n t but not too high d e n s i t y of f a u l t s could be approximately s p e c i f i e d by i d e n t i f y i n g a wave number in which an i s o l a t e d f a u l t of random or s t o c h a s t i c o r i g i n was d i f f u s i v e l y s t a t i o n a r y ( F i g . l a ) . While i t i s not easy to uniquely c h a r a c t e r i z e the dynamics of an i s o l a t e d f a u l t , the authors (3) were able to convincingly demonstrate t h a t the s t a b i l i t y point possesses a l a r g e r wavelength than t h a t given by the conventional minimumin the undercooling or maximum in the f r o n t a l v e l o c i t y curve (now v e r i f i e d e x p e r i m e n t a l l y by Seetharamin and Trivedi (6)). Later Kirkaldy and Sharma (4) offered a supportive p e r t u r b a t i o n argument to the e f f e c t t h a t a l a m e l l a r f a u l t would be s t a t i o n a r y i n an a r r a y w h o s e average spacing corresponded approximately to the I n f l e c t i o n p o i n t of the s p a c i n g - v e l o c i t ycurve which i s e q u i v a l e n t to ~ times the spacing a t minlmum undercooling Xm(Fig.lb). I t was then emphasized t h a t t h i s corresponded, withln the approximations of the thermnkinetic model, to a maximal c o n f i g u r a t i o n in the entropy production r a t e ( d i s s i p a t i o n ) . The l a t e r expansion of a very p l a u s i b l e s t a b i l i z a t i o n conjecture due to Langer (7-9) led within the same context of prompt s o l i t o n motion to the i d e n t i c a l s t a b i l i t y point. I s o l a t e d Wldmanstatten and d e n d r i t e f i g u r e s i n v o l v i n g free t i p growth present a s p e c i a l problem to the a p p l i c a t i o n of d i s s i p a t i o n p r i n c i p l e s s i n c e , u n l i k e the e u t e c t i c or e u t e c t o i d problems, there I s not a w e l l - d e f i n e d volume to which the degenerate s e t of steady s t a t e c o n f l g u r a t i o n s can be assigned. Note t h a t in the s o l i d s t a t e nature often chooses the a l t e r n a t e discontinuous ( e u t e c t o i d - l l k e ) steady s t a t e mode in which an appropriate volume can be defined (10,11). It i s the strong s e l f - o r g a n i z i n gc h a r a c t e r i s t i c w i t h i n i l l - d e f i n e d boundary c o n d i t i o n s which makes such systems e s p e c i a l l y i n t e r e s t i n g f r o m the pattern-forming p o l n t - o f - v l e w and i n v i t e s the search for a mechanism. We surmised in 1980 (12) t h a t a way out of the d i f f i c u l t y wlth volUme d e f i n i t i o n in the Widmanstatten p l a t e problem would be provided by the e x i s t e n c e of surface s o l i t o n s as represented in Fig.3. I t was argued t h a t t h i s f l u c t u a t i o n i n i t i a t e d kink c o n f i g u r a t i o n analogous to e u t e c t i c l a m e i l a r f a u l t s would he s t a t i o n a r y a t the i n f l e c t i o n point of the t i p r a d i u s versus v e l o c i t y curve i r r e s p e c t i v e of any thermodynamic c r i t e r i o n . We a n t i c i p a t e d analogous s o J i t o n c o n f i g u r a t i o n s for Widmanstatten and dendritic needles (12). Venugopalan and the author (13) found the first evidence of the involvement of solitons in wave number selection in cellular solidification of succinonitrile-salol (Fig.4) and this observation has been very recently confirmed by Simon et al (14) in liquid crystal solidification. Furthermore, a kink alternative to the lamellar fault soliton mechanism in eutectics has been reported by Falvre et al (15). This was first recorded in Jackson's Bell Telephone
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instructional film ( 1 6 ) from w h i c h F i g . 5 is reproduced. Note that the regression towards stability I s a damped o s c i l l a t i o n c a r r y i n g t h e s p a c i n g t o a v a l u e g r e a t e r t h a n Xm a s i n d e x e d by the flattened front, both f e a t u r e s a s p r e d i c t e d by t h e w r i t e r ( 4 , 9 ) . It is generally agreed that such manifestations are of stochastic or nucleate origin. It is undoubtedly a truism of pattern recognition t h a t one c a n n o t r e c o g n i z e u n t i l one knows w h a t t o look for. The d e n d r i t e tip shape identified i n F i g . 6 (26) may a c t u a l l y represent the symmetric configuration of four locked and stationary solitons of t h e form conjectured in F i g . 3 from R e f . 1 2 . T h e s e we s u r m i s e a r e i n s t a t i s t i c a l "equilibrium" within a phase space of surface solitons at smaller scales of length whose distribution is sustained by thermal feed-back from the fluctuating side-branching as characterized experimentally by Dougherty et al {17) {The diffusion length D/v>>r, the tip radius}. Flne precipitate arrays can also be flt into this conceptual mold since Ostwald ripening provides the mechanism for combination and annihilation, as occurs in certain versions of the pattern-forming Liesegang Phenomenon (18,19). The P h a s e Change S o l i t o n
as Generic with Vortices
and Dislocations
The p a r a d i g m o f s o l i t o n s t r u c t u r e in irreversible systems is the vortex in fluid systems. It is a non-linear mobile line d i s t u r b a n c e w h i c h p o s s e s s e s o p p o s i t e s i g n s a nd d e f i n e s a n lnvariant (the circulation). Equal signs repel a nd o p p o s i t e signs attract, the latter defining a possible annihilation, Vortices through their interactions are not only the agent of change in flow pattern, ultimately subsuming turbulence, but at l ow d e n s i t i e s actually represent the elements of macroscopic pattern. Examples are the cellular loop vortices of the R a y l e i g h - B e n a r d and T a y l o r s p e c i e s a n d t h o s e which nucleate a nd grow a r o u n d obstacles to laminar flow. V o r t i c e s c a n be n u c l e a t e d a t s u r f a c e s o r c a n e x t e n d t h e i r d o m a i n by l e n g t h e n i n g and ( o r ) i n t e r s e c t i n g . At v e r y high densities they are a perfect thermodynamic entity defining classical ergodicity at the chaotlc state (20). They e n t e r t a i n dissipation or loss of available energy not only locally as part of their viscous structures and their interactions but also through the trend to a fine-grained turbulence at high driving forces. On a f l u i d s u r f a c e the analogous disturbances are highly localized; hence the designation, soliton or solitary wave. The s o l i d s t a t e a n a l o g u e o f t h e v o r t e x i s t h e d i s l o c a t i o n with its p a i r e d s i g n s and i n v a r i a n t B u r g e r s v e c t o r . Because of the highly slngular nature dissipation m u s t be a c h i e v e d v i a phonon r a d i a t i o n , annihilation a nd o t h e r i n t e r a c t i o n s , a nd m a c r o s c o p i c pattern appears only as semi-ordered arrays such as ladder structures a nd p e r s i s t e n t slip bands (21). The s o l i t o n s p e r t a i n i n g to the heterogeneous systems of interest here, while p o s s e s s i n g many o f t h e g e n e r a l f e a t u r e s of vortices or dislocations, are of a less perfect structure. Fo r e x a m p l e , t h e d i s p l a c e m e n t c o r r e s p o n d i n g t o a B u r g e r s v e c t o r i s n o t n e c e s s a r i l y a constant (compare Figs.1 and 5). Yet they are line or "point" disturbances, they have the capability of nucleation, c o m b i n a t i o n and a n n i h i l a t i o n , t h e y can o f t e n f r e e l y e x t e n d , as t h e y multiply or stretch they enter a thermodynamic manifold wherein the system approaches a chaotic state, and a t low d e n s i t i e s individuals can actually define the macroscopic pattern at the stationary states ( c o m p a r e low d e n s i t y v o r t i c e s a nd F i g . 6 ) . Solltons
as "Particles"
in a Thermodynamic M i l i e u
Shoutd the involvement of solitons in stabilization of irreversible pattern p r o v e t o be ubiquitous, a s we b e l i e v e , a complete u n d e r s t a n d i n g o f wave number s e l e c t i o n w i l l demand a n e x p l a n a t i o n a s t o why d y n a m i c n a t u r e possesses a general propensity for digitization or "particle" formation up t o the macroscopic scales. We b e l i e v e t h a t t h i s w i l l u l t i m a t e l y be traced back to the quantization of atomic ma s s in the same way t h a t critical point fluctuations in a digital sequence of magnitudes are related'to spin lattice fluctuations via the block-spin technique. I n d e e d W l l s o n (22) s e e s t h e p r o b l e m o f t u r b u l e n c e as lylng within this framework. By a n a l o g y it is reasonable to r e g a r d o b s e r v a b l e s o l i t o n s a s members o f a h i e r a r c h y o f dynamic entities within a driven or supersaturated system in which the "stable" pattern represents the largest scale (the correlation length is the analogous quantlty in crJtlcal point theory) and to attribute a form o f dynamic equilibrium subsuming thermodynamic optimality to the aged manifold. T h i s c o n c e p t i o n l e a d s us d i r e c t l y to the minimax dissipation principle for irreversible steady states. Further analogy suggests for this class of high temperature systems that macroscopic states m u s t be t h e r m o d y n a m i c a l l y smooth a t the stationary state despite the existence of macroscopic conflgurational variations through fluctuations a nd (or) solitons which are capable of destroying the smoothness. It follows therefore that the dissipation function as the theoretically accessible and sensitive thermodynamic detector for macroscopic eonfigurational changes such as accrue from s o l l t o n nucleation and motion must for a
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completely relaxed flne-grained smoothness of temperature and entropy in space and t i m e be stationary with respect to the order parameters describing the mean macroscopic configurational changes. Statlonarity implies a minimum or a maximum or both. Hence the minimax designation, An equivalent argument which subsumes flne grainedness has been based on the principle of maximum path probability (23). The minimax is not to be confounded with the prlnclple of minimum entropy production which trivially applies in various approximations to continuous irreversible systems. The supportive experimental record is broad (24,25). It Is to be appreclated that this theorem is notably unspeclflc as to the coherency or non-coherency of the macroscoplc eonflgurational changes which test the stability, and this is its great strength. The changes can be stochastically originated chaotic or soliton forms of low but non-zero density or Idealized global changes as defined by steady state degeneracy slnce the latter defines feasible end-points accruing from the motion of the more slngular forms. The extreme case of local coherency, suggesting overdetermination, is exhibited by the observed maximum scale soliton configuration at a three-dlmensional dendrite tip (Fig.6) brought to approxlmate statiouarlty in a fluctuating milieu caused by the side-branchlng manifold under the influence of anlsotropy of surface tension. Note by contrast that solitons can escape to infinity in the two-dimenslonal case (Fig.3) so overall fixed point solutions (Wldmanstatten figures) become feaslble. Beyond the foregolng general theorem we have offered two nucleate or soliton-dependent perturbation arguments which for a volume dlffuslon controlled eutectoid(ic) process identify the stabllity point as a maximum in the entropy production rate with respect to spacing or the Inflection point configuration in the veloclty-spacing curve. As already noted, the latter feature encouraged Kirkaldy and Sharma (4) to construct the perturbation argument operating on a single lamellar fault (Fig.l) whose statlonary configuration also corresponds to the same optimum. This ties In nlcely with the general considerations presented above. The other more elementary "half-spacing" nucleation argument is the most transparent (27). It may now be appreciated that just as in equllibrium thermodynamics with its various potentials and equivalent Le Chatelier principles, stationary state potentials must possess equivalent perturbation prlnclples, rhe fact that we have been able to establish such equivalences is the strongest theoretical evidence for the existence of the dissipation potential.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 18 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
J. Moser, American S c i e n t i s t , 67, 689 ( 1 9 7 9 ) . B. C h a l m e r s , P r i n c i p l e s o f S o l i d i f i c a t i o n , J o h n W i l e y & S o n s , New York, 1964. K.A. J a c k s o n and J . D . Hunt, T r a n s . N e t . Soc. AIME, 236, 1129 ( 1 9 6 6 ) . J . S . K l r k a l d y and R.C. Sharma, A c t a Met. 28, 1009 ( 1 9 8 0 ) . B.L. B r a m f l t t and A.R. M a r d e r , INS P r o c e e d i n g s , 1968, p . 4 3 . V. S e e t h a r a m a n and R. T r i v e d i , Met. T r a n s . 19A, 2955 ( 1 9 8 8 ) . J . S . L a n g e r , P h y s . Rev. L e t t s . 44, 1023 ( 1 9 8 0 ) . J . S . K l r k a l d y , S c r l p t a Met. 15, 1255 ( 1 9 8 1 ) . J . S . K l r k a l d y , P h y s . Rev. B30, 6889 ( 1 9 8 4 ) . N.S. Sulonen, A c t a Met. 12, 748 ( 1 9 6 4 ) . J . S h a p i r o and J . S . K l r k a l d y , A c t a N e t . 16, 579 ( 1 9 6 8 ) . J . S . K i r k a l d y , S c r l p t a N e t . , 14, 531 ( 1 9 8 0 ) . D. V e n u g o p a l a n and J . S . K l r k a l d y , A c t a N e t . 32, 893 ( 1 9 8 4 ) . A . J . Simon, J . B a c k h o f e r and A. L i b c h a b e r , P h y s . Rev. L e t t s . 61, 2574 ( 1 9 6 8 ) . G. F a l v r e , S. de C h e v e i g n e , G. Guthmann and P. K u r o w s k i , P r e p r J n t , 1989. Bell Telephone System e d u c a t i o n a l film, Eutectic Solidification, prepared by K.A. Jackson (1967). A. D o u g h e r t y , P . P . K a p l a n and J . P . G o l l u b , P h y s . Rev. L e t t s . 58, 1652 ( 1 9 8 7 ) . J . S . K i r k a l d y , S c r l p t a N e t . 20, 1571 ( 1 9 8 6 ) . Y. B r e c h e t and J . S . K i r k a l d y , J . Chem. P h y s . 90, 1499 ( 1 9 8 9 ) . L. O n s a g e r , Nuovo C l m e n t o , Vol. VI ( S u p p l e m e n t ) , 279 ( 1 9 4 9 ) . E. A l f a n t i s , In Advances in Phase T r a n s i t i o n s , E d s . , J . D . Embury and G.R. P u r d y , Pergamon P r e s s , O x f o r d , 1988, p . 2 6 1 . K.G. W i l s o n , S c i . Am. 241, 773 ( 1 9 7 5 ) . J . S . K l r k a l d y , Phys. Rev. A31, 3376 ( 1 9 8 5 ) . J . S . K i r k a l d y , Met. T r a n s . 16A, 1781 ( 1 9 8 5 ) . J.S. Klrkaldy, i n Advances I n Phase T r a n s i t i o n s , Eds., J.D. Embury and G.R. P u r d y , Pergamon P r e s s , O x f o r d , 1988, p . 2 6 3 . S . - C . Huang and M.E. G l i c k s m a n , A c t a M e t . 29, 701 ( 1 9 8 1 ) . J . S . K l r k a l d y , S c r i p t a Met. 2, 565 ( 1 9 6 8 ) .
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A I
j
Fig.1 (~) Schematic lamellar fault array. Growth d i r e c t i o n i s o u t o f t h e p a g e . (6} S c h e m a t i c vertical profiles t h r o u g h a l a m e l l a r f a u l t (a ] s t a t i o n a r y configuration (b] f a u l t r e j e c t e d (c) fault absorbed. L o c a t l o n B' a t t e m p t s t o a d v a n c e b e c a u s e o f t h e r e d u c e d l o c a l s p a c l n g , b u t l s stabilized by t h e t h r e e - d i m e n s i o n a l nature of the fieJds. A f t e r K 1 r k a l d y and S h a r m a ( 4 ) .
Fig.2.
Lamellar
faults
i n Fe-Fe3C e u t e c t o i d .
After Bramfitt and Narder(5]. V D
la)
Fig.3: 1~1 P l a n and e l e v a t i o n o f k i n k neal; W l d m a n s t a t t e n t i p . perturbation on Widmanstatten tip. After Kirkaldy(8).
[ 6 ) Schematic
vlew of klnk
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Fig.4: Klnk solitons in succinonltrlle-salol cellular solidification actlng to decrease average spacing as a consequence of an injected cell which is expanding. In frame d, which is much delayed from the sequence a,b,c, a second phalanx of solitons has been emitted wlth equalization nearly attained. After Venugopalan and Kirkaldy(13).
Fig.5: front
as
Tilt the
soliton spacing
in forced velocity CBr 4 has increased on the Jeft.
C2C16 eutectic, Note After Jackson(16).
the
flattening
o1: t h e
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Fig.6: Anlsotroplc aspects of dendritic t i p s t r ~ c t u r e s in s u c c i n o n i t r l J e . ( a ) , (b) and ( c ) c o r r e s p o n d t o t h e v i e w i n g a n g l e s 0 , 2 2 . 5 ° and 45 ° shown (d). The s i d e branches extend in the f o u r <100> d i r e c t i o n s ; however, d e n d r i t e becomes a body o f r e v o l u t i o n . Compare t h i s w i t h F l g . 3 ( a ) w h e r e i n i t I n s e t (d) corresponds to the intersection and l o c k i n g o f f o u r c r o s s e d p l a t e particularly the matching of proflles along dashed lines in elevation and solid lines in plan. A f t e r Huang and G l l c k s m a n ( 2 6 ) .
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The t h r e e v l e w s s c h e m a t i c a l l y in near the tip the can be s e e n t h a t solitons. Note the matching of
Vol. 24, pp. 2263-2268, 1990 Printed in the U.S.A.
Pergamon Press plc All rights reserved
S o l i t o n s and Nave Number S e l e c t i o n in F i r s t Order D i f f u s i v e Phase Transformations
McMaster
University,
J.S. Kirkaldy Hamilton. Canada
L8S-4M1
(Received May I, 1990) (Revised September 19, 1990) In 1981 the author introduced to the readers of t h i s journal the idea t h a t s o l i t o n s or s o l i t a r y waves play a p a r t in w a v e number s e l e c t i o n in e u t e c t o i d ( i c ) and Widmanstatten reactions. Since t h a t time there have been a t l e a s t three micrographic examples offered in the l i t e r a t u r e p e r t a i n i n g to e u t e c t i c s and c e l l u l a r s o l i d i f i c a t i o n . The aim of t h i s c o n t r i b u t i o n i s to describe an analogous r o l e for s o l i t o n s in the thermal d e n d r i t e problem,to o f f e r a t h e o r e t i c a l s y n t h e s i s and to f u r t h e r e s t a b l i s h the connection between the s o l i t o n mechanism and d i s s i p a t i o n p r i n c i p l e s . I t i s argued t h a t these e n t i t i e s l i e w i t h i n a d i s t i n c t and fundamental c l a s s of s t o c h a s t i c o b j e c t s which includes v o r t i c e s and d i s l o c a t i o n s . Evidence for S o l i t o n s in Phase Transformations S o l i t o n s were f i r s t observed in ocean waves (I) as n o n - l i n e a r l o c a l i z e d s p a t i a l d i s l o c a t i o n s which migrate e s s e n t i a l l y n o n - d i s s i p a t i v e l ywithin the manifold so as by c r e a t i o n and mutual a n n i h i l a t i o n s t a b i l i z e the symmetry and d i s s i p a t i v e p r o p e n s i t i e s of the optimal waves. Quite o u t s i d e t h i s general context an analogous m a n i f e s t a t i o n was recognized by Chalmers (2) and Jackson and Hunt (3) in three-dimensional forced v e l o c i t y e u t e c t i c s as wave number d i s l o c a t i o n s , and these were designated as l a m e l l a r f a u l t s (Figs.1 and 2; Refs.4 and 5). They c o r r e c t l y surmised t h a t s t a b i l i t y of a l a m e l l a r a r r a y with a s u f f i c i e n t but not too high d e n s i t y of f a u l t s could be approximately s p e c i f i e d by i d e n t i f y i n g a wave number in which an i s o l a t e d f a u l t of random or s t o c h a s t i c o r i g i n was d i f f u s i v e l y s t a t i o n a r y ( F i g . l a ) . While i t i s not easy to uniquely c h a r a c t e r i z e the dynamics of an i s o l a t e d f a u l t , the authors (3) were able to convincingly demonstrate t h a t the s t a b i l i t y point possesses a l a r g e r wavelength than t h a t given by the conventional minimumin the undercooling or maximum in the f r o n t a l v e l o c i t y curve (now v e r i f i e d e x p e r i m e n t a l l y by Seetharamin and Trivedi (6)). Later Kirkaldy and Sharma (4) offered a supportive p e r t u r b a t i o n argument to the e f f e c t t h a t a l a m e l l a r f a u l t would be s t a t i o n a r y i n an a r r a y w h o s e average spacing corresponded approximately to the I n f l e c t i o n p o i n t of the s p a c i n g - v e l o c i t ycurve which i s e q u i v a l e n t to ~ times the spacing a t minlmum undercooling Xm(Fig.lb). I t was then emphasized t h a t t h i s corresponded, withln the approximations of the thermnkinetic model, to a maximal c o n f i g u r a t i o n in the entropy production r a t e ( d i s s i p a t i o n ) . The l a t e r expansion of a very p l a u s i b l e s t a b i l i z a t i o n conjecture due to Langer (7-9) led within the same context of prompt s o l i t o n motion to the i d e n t i c a l s t a b i l i t y point. I s o l a t e d Wldmanstatten and d e n d r i t e f i g u r e s i n v o l v i n g free t i p growth present a s p e c i a l problem to the a p p l i c a t i o n of d i s s i p a t i o n p r i n c i p l e s s i n c e , u n l i k e the e u t e c t i c or e u t e c t o i d problems, there I s not a w e l l - d e f i n e d volume to which the degenerate s e t of steady s t a t e c o n f l g u r a t i o n s can be assigned. Note t h a t in the s o l i d s t a t e nature often chooses the a l t e r n a t e discontinuous ( e u t e c t o i d - l l k e ) steady s t a t e mode in which an appropriate volume can be defined (10,11). It i s the strong s e l f - o r g a n i z i n gc h a r a c t e r i s t i c w i t h i n i l l - d e f i n e d boundary c o n d i t i o n s which makes such systems e s p e c i a l l y i n t e r e s t i n g f r o m the pattern-forming p o l n t - o f - v l e w and i n v i t e s the search for a mechanism. We surmised in 1980 (12) t h a t a way out of the d i f f i c u l t y wlth volUme d e f i n i t i o n in the Widmanstatten p l a t e problem would be provided by the e x i s t e n c e of surface s o l i t o n s as represented in Fig.3. I t was argued t h a t t h i s f l u c t u a t i o n i n i t i a t e d kink c o n f i g u r a t i o n analogous to e u t e c t i c l a m e i l a r f a u l t s would he s t a t i o n a r y a t the i n f l e c t i o n point of the t i p r a d i u s versus v e l o c i t y curve i r r e s p e c t i v e of any thermodynamic c r i t e r i o n . We a n t i c i p a t e d analogous s o J i t o n c o n f i g u r a t i o n s for Widmanstatten and dendritic needles (12). Venugopalan and the author (13) found the first evidence of the involvement of solitons in wave number selection in cellular solidification of succinonitrile-salol (Fig.4) and this observation has been very recently confirmed by Simon et al (14) in liquid crystal solidification. Furthermore, a kink alternative to the lamellar fault soliton mechanism in eutectics has been reported by Falvre et al (15). This was first recorded in Jackson's Bell Telephone
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instructional film ( 1 6 ) from w h i c h F i g . 5 is reproduced. Note that the regression towards stability I s a damped o s c i l l a t i o n c a r r y i n g t h e s p a c i n g t o a v a l u e g r e a t e r t h a n Xm a s i n d e x e d by the flattened front, both f e a t u r e s a s p r e d i c t e d by t h e w r i t e r ( 4 , 9 ) . It is generally agreed that such manifestations are of stochastic or nucleate origin. It is undoubtedly a truism of pattern recognition t h a t one c a n n o t r e c o g n i z e u n t i l one knows w h a t t o look for. The d e n d r i t e tip shape identified i n F i g . 6 (26) may a c t u a l l y represent the symmetric configuration of four locked and stationary solitons of t h e form conjectured in F i g . 3 from R e f . 1 2 . T h e s e we s u r m i s e a r e i n s t a t i s t i c a l "equilibrium" within a phase space of surface solitons at smaller scales of length whose distribution is sustained by thermal feed-back from the fluctuating side-branching as characterized experimentally by Dougherty et al {17) {The diffusion length D/v>>r, the tip radius}. Flne precipitate arrays can also be flt into this conceptual mold since Ostwald ripening provides the mechanism for combination and annihilation, as occurs in certain versions of the pattern-forming Liesegang Phenomenon (18,19). The P h a s e Change S o l i t o n
as Generic with Vortices
and Dislocations
The p a r a d i g m o f s o l i t o n s t r u c t u r e in irreversible systems is the vortex in fluid systems. It is a non-linear mobile line d i s t u r b a n c e w h i c h p o s s e s s e s o p p o s i t e s i g n s a nd d e f i n e s a n lnvariant (the circulation). Equal signs repel a nd o p p o s i t e signs attract, the latter defining a possible annihilation, Vortices through their interactions are not only the agent of change in flow pattern, ultimately subsuming turbulence, but at l ow d e n s i t i e s actually represent the elements of macroscopic pattern. Examples are the cellular loop vortices of the R a y l e i g h - B e n a r d and T a y l o r s p e c i e s a n d t h o s e which nucleate a nd grow a r o u n d obstacles to laminar flow. V o r t i c e s c a n be n u c l e a t e d a t s u r f a c e s o r c a n e x t e n d t h e i r d o m a i n by l e n g t h e n i n g and ( o r ) i n t e r s e c t i n g . At v e r y high densities they are a perfect thermodynamic entity defining classical ergodicity at the chaotlc state (20). They e n t e r t a i n dissipation or loss of available energy not only locally as part of their viscous structures and their interactions but also through the trend to a fine-grained turbulence at high driving forces. On a f l u i d s u r f a c e the analogous disturbances are highly localized; hence the designation, soliton or solitary wave. The s o l i d s t a t e a n a l o g u e o f t h e v o r t e x i s t h e d i s l o c a t i o n with its p a i r e d s i g n s and i n v a r i a n t B u r g e r s v e c t o r . Because of the highly slngular nature dissipation m u s t be a c h i e v e d v i a phonon r a d i a t i o n , annihilation a nd o t h e r i n t e r a c t i o n s , a nd m a c r o s c o p i c pattern appears only as semi-ordered arrays such as ladder structures a nd p e r s i s t e n t slip bands (21). The s o l i t o n s p e r t a i n i n g to the heterogeneous systems of interest here, while p o s s e s s i n g many o f t h e g e n e r a l f e a t u r e s of vortices or dislocations, are of a less perfect structure. Fo r e x a m p l e , t h e d i s p l a c e m e n t c o r r e s p o n d i n g t o a B u r g e r s v e c t o r i s n o t n e c e s s a r i l y a constant (compare Figs.1 and 5). Yet they are line or "point" disturbances, they have the capability of nucleation, c o m b i n a t i o n and a n n i h i l a t i o n , t h e y can o f t e n f r e e l y e x t e n d , as t h e y multiply or stretch they enter a thermodynamic manifold wherein the system approaches a chaotic state, and a t low d e n s i t i e s individuals can actually define the macroscopic pattern at the stationary states ( c o m p a r e low d e n s i t y v o r t i c e s a nd F i g . 6 ) . Solltons
as "Particles"
in a Thermodynamic M i l i e u
Shoutd the involvement of solitons in stabilization of irreversible pattern p r o v e t o be ubiquitous, a s we b e l i e v e , a complete u n d e r s t a n d i n g o f wave number s e l e c t i o n w i l l demand a n e x p l a n a t i o n a s t o why d y n a m i c n a t u r e possesses a general propensity for digitization or "particle" formation up t o the macroscopic scales. We b e l i e v e t h a t t h i s w i l l u l t i m a t e l y be traced back to the quantization of atomic ma s s in the same way t h a t critical point fluctuations in a digital sequence of magnitudes are related'to spin lattice fluctuations via the block-spin technique. I n d e e d W l l s o n (22) s e e s t h e p r o b l e m o f t u r b u l e n c e as lylng within this framework. By a n a l o g y it is reasonable to r e g a r d o b s e r v a b l e s o l i t o n s a s members o f a h i e r a r c h y o f dynamic entities within a driven or supersaturated system in which the "stable" pattern represents the largest scale (the correlation length is the analogous quantlty in crJtlcal point theory) and to attribute a form o f dynamic equilibrium subsuming thermodynamic optimality to the aged manifold. T h i s c o n c e p t i o n l e a d s us d i r e c t l y to the minimax dissipation principle for irreversible steady states. Further analogy suggests for this class of high temperature systems that macroscopic states m u s t be t h e r m o d y n a m i c a l l y smooth a t the stationary state despite the existence of macroscopic conflgurational variations through fluctuations a nd (or) solitons which are capable of destroying the smoothness. It follows therefore that the dissipation function as the theoretically accessible and sensitive thermodynamic detector for macroscopic eonfigurational changes such as accrue from s o l l t o n nucleation and motion must for a
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completely relaxed flne-grained smoothness of temperature and entropy in space and t i m e be stationary with respect to the order parameters describing the mean macroscopic configurational changes. Statlonarity implies a minimum or a maximum or both. Hence the minimax designation, An equivalent argument which subsumes flne grainedness has been based on the principle of maximum path probability (23). The minimax is not to be confounded with the prlnclple of minimum entropy production which trivially applies in various approximations to continuous irreversible systems. The supportive experimental record is broad (24,25). It Is to be appreclated that this theorem is notably unspeclflc as to the coherency or non-coherency of the macroscoplc eonflgurational changes which test the stability, and this is its great strength. The changes can be stochastically originated chaotic or soliton forms of low but non-zero density or Idealized global changes as defined by steady state degeneracy slnce the latter defines feasible end-points accruing from the motion of the more slngular forms. The extreme case of local coherency, suggesting overdetermination, is exhibited by the observed maximum scale soliton configuration at a three-dlmensional dendrite tip (Fig.6) brought to approxlmate statiouarlty in a fluctuating milieu caused by the side-branchlng manifold under the influence of anlsotropy of surface tension. Note by contrast that solitons can escape to infinity in the two-dimenslonal case (Fig.3) so overall fixed point solutions (Wldmanstatten figures) become feaslble. Beyond the foregolng general theorem we have offered two nucleate or soliton-dependent perturbation arguments which for a volume dlffuslon controlled eutectoid(ic) process identify the stabllity point as a maximum in the entropy production rate with respect to spacing or the Inflection point configuration in the veloclty-spacing curve. As already noted, the latter feature encouraged Kirkaldy and Sharma (4) to construct the perturbation argument operating on a single lamellar fault (Fig.l) whose statlonary configuration also corresponds to the same optimum. This ties In nlcely with the general considerations presented above. The other more elementary "half-spacing" nucleation argument is the most transparent (27). It may now be appreciated that just as in equllibrium thermodynamics with its various potentials and equivalent Le Chatelier principles, stationary state potentials must possess equivalent perturbation prlnclples, rhe fact that we have been able to establish such equivalences is the strongest theoretical evidence for the existence of the dissipation potential.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 18 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
J. Moser, American S c i e n t i s t , 67, 689 ( 1 9 7 9 ) . B. C h a l m e r s , P r i n c i p l e s o f S o l i d i f i c a t i o n , J o h n W i l e y & S o n s , New York, 1964. K.A. J a c k s o n and J . D . Hunt, T r a n s . N e t . Soc. AIME, 236, 1129 ( 1 9 6 6 ) . J . S . K l r k a l d y and R.C. Sharma, A c t a Met. 28, 1009 ( 1 9 8 0 ) . B.L. B r a m f l t t and A.R. M a r d e r , INS P r o c e e d i n g s , 1968, p . 4 3 . V. S e e t h a r a m a n and R. T r i v e d i , Met. T r a n s . 19A, 2955 ( 1 9 8 8 ) . J . S . L a n g e r , P h y s . Rev. L e t t s . 44, 1023 ( 1 9 8 0 ) . J . S . K l r k a l d y , S c r l p t a Met. 15, 1255 ( 1 9 8 1 ) . J . S . K l r k a l d y , P h y s . Rev. B30, 6889 ( 1 9 8 4 ) . N.S. Sulonen, A c t a Met. 12, 748 ( 1 9 6 4 ) . J . S h a p i r o and J . S . K l r k a l d y , A c t a N e t . 16, 579 ( 1 9 6 8 ) . J . S . K i r k a l d y , S c r l p t a N e t . , 14, 531 ( 1 9 8 0 ) . D. V e n u g o p a l a n and J . S . K l r k a l d y , A c t a N e t . 32, 893 ( 1 9 8 4 ) . A . J . Simon, J . B a c k h o f e r and A. L i b c h a b e r , P h y s . Rev. L e t t s . 61, 2574 ( 1 9 6 8 ) . G. F a l v r e , S. de C h e v e i g n e , G. Guthmann and P. K u r o w s k i , P r e p r J n t , 1989. Bell Telephone System e d u c a t i o n a l film, Eutectic Solidification, prepared by K.A. Jackson (1967). A. D o u g h e r t y , P . P . K a p l a n and J . P . G o l l u b , P h y s . Rev. L e t t s . 58, 1652 ( 1 9 8 7 ) . J . S . K i r k a l d y , S c r l p t a N e t . 20, 1571 ( 1 9 8 6 ) . Y. B r e c h e t and J . S . K i r k a l d y , J . Chem. P h y s . 90, 1499 ( 1 9 8 9 ) . L. O n s a g e r , Nuovo C l m e n t o , Vol. VI ( S u p p l e m e n t ) , 279 ( 1 9 4 9 ) . E. A l f a n t i s , In Advances in Phase T r a n s i t i o n s , E d s . , J . D . Embury and G.R. P u r d y , Pergamon P r e s s , O x f o r d , 1988, p . 2 6 1 . K.G. W i l s o n , S c i . Am. 241, 773 ( 1 9 7 5 ) . J . S . K l r k a l d y , Phys. Rev. A31, 3376 ( 1 9 8 5 ) . J . S . K i r k a l d y , Met. T r a n s . 16A, 1781 ( 1 9 8 5 ) . J.S. Klrkaldy, i n Advances I n Phase T r a n s i t i o n s , Eds., J.D. Embury and G.R. P u r d y , Pergamon P r e s s , O x f o r d , 1988, p . 2 6 3 . S . - C . Huang and M.E. G l i c k s m a n , A c t a M e t . 29, 701 ( 1 9 8 1 ) . J . S . K l r k a l d y , S c r i p t a Met. 2, 565 ( 1 9 6 8 ) .
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A I
j
Fig.1 (~) Schematic lamellar fault array. Growth d i r e c t i o n i s o u t o f t h e p a g e . (6} S c h e m a t i c vertical profiles t h r o u g h a l a m e l l a r f a u l t (a ] s t a t i o n a r y configuration (b] f a u l t r e j e c t e d (c) fault absorbed. L o c a t l o n B' a t t e m p t s t o a d v a n c e b e c a u s e o f t h e r e d u c e d l o c a l s p a c l n g , b u t l s stabilized by t h e t h r e e - d i m e n s i o n a l nature of the fieJds. A f t e r K 1 r k a l d y and S h a r m a ( 4 ) .
Fig.2.
Lamellar
faults
i n Fe-Fe3C e u t e c t o i d .
After Bramfitt and Narder(5]. V D
la)
Fig.3: 1~1 P l a n and e l e v a t i o n o f k i n k neal; W l d m a n s t a t t e n t i p . perturbation on Widmanstatten tip. After Kirkaldy(8).
[ 6 ) Schematic
vlew of klnk
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Fig.4: Klnk solitons in succinonltrlle-salol cellular solidification actlng to decrease average spacing as a consequence of an injected cell which is expanding. In frame d, which is much delayed from the sequence a,b,c, a second phalanx of solitons has been emitted wlth equalization nearly attained. After Venugopalan and Kirkaldy(13).
Fig.5: front
as
Tilt the
soliton spacing
in forced velocity CBr 4 has increased on the Jeft.
C2C16 eutectic, Note After Jackson(16).
the
flattening
o1: t h e
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Fig.6: Anlsotroplc aspects of dendritic t i p s t r ~ c t u r e s in s u c c i n o n i t r l J e . ( a ) , (b) and ( c ) c o r r e s p o n d t o t h e v i e w i n g a n g l e s 0 , 2 2 . 5 ° and 45 ° shown (d). The s i d e branches extend in the f o u r <100> d i r e c t i o n s ; however, d e n d r i t e becomes a body o f r e v o l u t i o n . Compare t h i s w i t h F l g . 3 ( a ) w h e r e i n i t I n s e t (d) corresponds to the intersection and l o c k i n g o f f o u r c r o s s e d p l a t e particularly the matching of proflles along dashed lines in elevation and solid lines in plan. A f t e r Huang and G l l c k s m a n ( 2 6 ) .
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The t h r e e v l e w s s c h e m a t i c a l l y in near the tip the can be s e e n t h a t solitons. Note the matching of