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Deterministic chaos and eutectoid phase transformations
Scripta METALLURGICA et M A T E R I A L I A
Vol. 24, pp. 1 7 9 - 1 8 4 , 1990 P r i n t e d in the U . S . A .
P e r g a m o n P r e s s plc All rights reserved
DETERMINISTIC CHAOS AND EUTECTOID PHASE TRANSFORMATIONS
Institute
J.S. Klrkaldy for Materials Research, McNaster University, H a m i l t o n . O n t a r i o , C a n a d a L8S 4M1.
( R e c e i v e d A p r i l 18, 1989) ( R e v i s e d N o v e m b e r i0, 1989) The w i d e recognition that certain deterministic non-linear difference and differential equations w i t h few d e g r e e s o f f r e e d o m c a n d e f i n e p h a s e s p a c e t r a j e c t o r i e s wlllch diverge exponentially lnto chaotlc motlons Is having a p r o f o u n d I n f l u e n c e on a p p l i e d p h y s i c s ( 1 . 2 ) . While presenting deep and unsolved problems of interpretation within which the experts find macroscopic analogies to microscopic quantum phenomena ( 1 . 2 ) t h e r e s e a r c h i s p r o v i d i n g i m p o r t a n t I n s i g h t s Into turbulent o r c h a o t i c phenomena i n h y d r o d y n a m i c s , c h e m i s t r y , m e c h a n i c s a nd b i o l o g y , f i n d i n g common e l e m e n t s or universality o f t h e k i n d s a l r e a d y r e c o g n i z e d ill t h e t h e o r y of s e c o n d o r d e r phase t r a n s i t i o n s (3,4). H e r e w i t h we u n d e r t a k e t o show t h a t the t r a n s i t i o n of pearlite I n FeFe_C t o t u r b u l e n t o r d e n d r i t i c f o r m s i s n o t o n l y encompassed by t h e same e l e m e n t s o f u n i v e r s a l i t y bu~ a l s o suggests the nature of t h e c o n n e c t i o n between s p a t i a l and t e m p o r a l chaos and t h e i r thermodynamic l i n k s . Much o f the fundamental theory prescription In virtual time (1,2)
Zn+ I = (]+r)Zn(l-z n) ; and this subsumes and economies, a {2,5). This map Verhulst pertaining
centers
on
the
non-linear
"logistic"
l+r ~ 4
map o r i t e r a t l v e
(l)
apparently deterministic phenomena as wide-ranging as self-llmit~ng ecologies damped driven pendulum and the capillarity-controlled Liesegang phenomenon is derivable from the growth-death type dlffercntJal equation or "flow" of to average self-llmlting population densities {6,7)
dX d-t = ~ X ( 1 - X )
(2)
through conversion to the numerical integral (8)
xn+ 1 = x n + c ~ t x (1 x ) It
l s t o be n o t e d t h a t
gq.(2)
(3)
possesses the closed solution
(8)
X = Xo/[Xo+{l-Xo)eXp(- ~t)]
(4)
which is always asymptotic to X = I. While it might be expected that thls solution can always be approximated by numerical operation on Eq.3 for positive r = ~ ~t this is not the case. If we make the linear transformation X n
(5)
~ (l+r)z r
n
specifying r ~ 3 the iterates of z in Eq.! are confined to the unit interval s o c a n be i n t e r preted as probability sequences, nFlg.1 exhlbtts the well-known distribution of asymptotic iterates z~ o f Eq.1 a s a f u n c t i o n o f t i l e n o r m a l i z e d c o n t r o l p a r a m e t e r ~ = ( ] + r ) / 4 . The e a s i e s t way t o appreciate this result, including the bifurcations or period doubling cycles and the transition to chaos at u = 0.892. is to p r o g r a m E q . 1 on a h a n d c a l c u l a t o r a nd e x p l o r e t h e Iterates manually. Observe t h a t t h e zm I t e r a t e s up t o t h e f i r s t b i f u r c a t i o n correspond to the X = 1 asymptotes of Eq.4. If for comparison one runs the Iterates on form ( 3 ) i t b e c o m e s c l e a r that the source of the Instability is the finite i n c r e m e n t a l change in s i g n of t h e r i g h t hand term w h i c h a c c r u e s when ~ t e x c e e d s 2. T h a t I s t o s a y , t h e i n s t a b i l i t y i s on t h e f a c e o f i t due
to an arithmetic artifact which is exhibited if the time increment is excessively large. As we shall argue below the apparent artifact is accorded physical significance when ~t is selected as a characteristic relaxation tlme for the problem at hand, given ~.
179 0 0 3 6 - 9 7 4 8 / 9 0 $ 3 . 0 0 + .00 C o p y r i g h t (c) 1990 P e r g a m o n P r e s s
plc
180
CHAOS
Transition The reaction (9,10) Sl .
entropy production by volume d i f f u s i o n
v &F. ~
XcTE
.
.
Vol.
from L a m e l l a r P e a r l l t e
24, No.
I
to C h a o t i c Forms
rate p e r u n i t volume o f t h e i s o t h e r m a l l a m e l l a r Fe-Fe~C e u t e c t o i d h a s been a p p r o x i m a t e d in t e r m s o f s p a c i n g ~ and f r o n t v b l o c i t y v by
c .
x
.c
~F
%TE
2~De
1 4 ~
%aT
.
~
c (1- _ . c ) 2
(6)
where ~ i s the wall surface tension, D is the diffusion coefficient and TE i s t h e e u t e c t o t d temperature. Note t h a t f o r any f i n i t e system t h i s r e p r e s e n t s a d i s c r e t e s e r i e s o f s p a t i a l s t e a d y states. Also &F ~ & ~ 11 -
Is the p o s i t i v e l y 2~
2~)
171
defined driving
f r e e energy change p e r u n i t
volume and
2~TE
(8)
~c ~-F-o ~V~-----T Is the relative
critical or cut-off supersaturation
s p a c i n g , w i t h ~T t h e u n d e r c o o l i n g and ~B t h e l a t e n t
heat,
and t h e
and t h i s is related t o ~T by t h e p h a s e d i a g r a m ( F t g . 2 ) . The d i s c r e t e s p a t i a l s e r l e s o f f r e e b o u n d a r y s t e a d y s t a t e s d e f i n e d by g q . 6 p o s s e s s e s a maximum a t X = 3~ , and i t h a s b e e n a r g u e d and Is consistent with the experiments, that this represents the stability p o i n t o r maximum p r o b a b i l ity state (10-13) in a kinetic phase space of fluctuating lame]lar dispersion. Now t h e s p a t i a l series can be t r a n s f o r m e d t o a v i r t u a l t i m e s e r i e s o r I a m e l l a r s t a t e s v i a an a s s o c i a t i o n , which for a large experimental enclosure equates the locally determined dissipation t o i t s volume normalized global expression, vJz., v~___F = _ 1 dF T T dt c and unit
this is interval,
! d(~F) T dt
introduced we o b t a i n
d..~X = 14~X(1-X) 2 d~
i n t o Eq.6 v i a Eq.7.
(lO)
D e f i n i n g X = ~ b / ~ w h i c h maps t h e s p a n o f ~ t o t h e
(11a)
where the normalized time is ~ = t/(R2=/D). S t a r t i n g w i t h any s m a l l Xu t h e e x a c t i n t e g r a l o f G this, w h i c h r e p r e s e n t s t h e g l o b a l e n t r o p y a c c u m u l a t e d , a s y m p t o t i c a l l y r e a c h e s a maximum v a l u e o f X = 1 as before. B e c a u s e o f t h e s q u a r e t e r m t h e c o r r e s p o n d i n g map n e v e r o s c i l l a t e s . However, a r e v i e w of the d e r i v a t i o n of Eq.6 i n d i c a t e s t h a t a r e l a t i v e l y f l a t i n t e r f a c e was a s s u m e d and t h i s . together with t h e l i n e a r 0 1 b b s - T h o m p s o n e f f e c t and a p p r o x i m a t i o n f o r R , i s o n l y v a l i d n e a r t h e optimum ( 9 , 1 1 ) . Indeed, the expression is otherwise patently lncorrec~ for it defines a physic a l l y u n a t t a i n a b l e v e l o c i t y s t a t e i n t h e a p p r o a c h t o ~ /~. = 1 and an a p p a r e n t l y f e a s l b l e p h y s i c a l state with negative v and p o s i t i v e d i s s i p a t i o n forCx / ~ e x c e e d i n g u n i t y even though t h ~ s e a r e both configurattonally I m p o s s i b l e on t h e I s o t h e r m ( c C . C F i g . 2 ) . One t h u s c o n c l u d e s t h a t S i must vanish by c r o s s i n g the axis approachir~ llnearity r a t h e r t h a n by f o r m i n g a q u a d r a t i c minimum. This Implies that for A / A ~ 1, v ~ S, i s a f e a s i b l e c o n f i g u r a t i o n which a p p r o a c h e s the a x i s vertically, and f o r ~ ~ > 1 that theldlsslpation I s n e g a t i v e and v i s i m a g i n a r y , p h y s i c a l l y excluding such a stats on b o t h t h e r m o d y n a m i c and c o n f i f f u r a t l o n a l g r o u n d s . I f E q . l l a i s t o be modified so as t o v a n i s h l i n e a r l y we c a n r e p r e s e n t i t by Eq. i l a up t o i t s i n f l e c t i o n point at X = 2 / 3 and a p p r o x l m a t e i t by i t s l i n e a r c o n t i n u a t i o n dX ~ 14~ 8 9 d-~ "~'~(1-~X)
(]]b)
thereafter. Note t h a t t h i s changes s i g n a t X = 8 / 9 , which c o o r d i n a t e d e f i n e s the e v o l u t i o n a r y maximum In X, and t h i s s i g n change as b e f o r e w i l l be t h e d i r e c t source o f t e m p o r a l l y u n s t a b l e iterates of the corresponding difference equation. Indeed. a t t h e n o n - p h y s i c a l s i g n change o f
Vol.
Si ~ t tm~,
24,
dF/dt
No.
1
one
CHAOS
might
infer
181
a nucleation event, where classically F increases momentarily
in
Converting Eq.ll to the equivalent difference equation for numerical integration in form (3) corresponding to a c h a r a c t e r i s t i c time incr e m e n t c o n s i s t ~ m t w i t h E q . l O , ~ = ~ = 1. and g e n e r ating the a s y m p t o t i c i t e r a t e s X as a f u n c t i o n o f t41S y i e l d s F i g . 3 . I n t h e u su a l f a s h i o n o f t h e Feigenba.m route to chaos (2? this exhibits period doubling, and unstable iterates when the control parameter 14~ > 5.9. The spatial asymptote X~ = 1/3 intersects with the temporal iterates at the second bifurcation (14~ ~ 6.7) and enters the chaotic zone at 14~ ~ 6.9. We hypothesize that the intersection defines a morphological interaction and from Fig.3 thereby identifies the onset control parameter for chaos with an undercoollng of approximately 200OC in the Fe.Fe~C system. This is experimentally about right for onset of the instability which attends the- bainite reaction and suggests that a search for evidence of a bifurcation near this value of 141s and the optimum spacing might be useful. These considerations augment our successful "half-spaclng" perturbation argument (10,12} which operated upon the v(X) curve {cf. Eq.6). This states that virtual half-spaclng nucleation events stabilize the h = 3h dissipation optimum because they grow at the same symmetrlcally disposed sub-maxlmal velocitles~ if the half-spaclng < 3/2A. then this region has a velocity deficit and will be overgrown, if it is > 3/2 ~ then the region c will be incorporated with average motion toward the stability point. Accordin~ to Flg.3 the time series asymptotes correspond to half-spacings < 3/2 ~ before the bifurcation so they would be overgrown even if nucleated. Slightly within the first b~furcation, however, the physical branch of asymptot(.s gains the stabilizing status of half-spacing fluctuations and beyond an increasing propensity for influencing the morphology. Indeed, we see that the lower time series blfurcati-r~ is f.rced asymptoti(:aJly toward the optimum at X~ = 1/3 with increased driving force or supersaturation and this extends the evidence favouring stability at the "half-spaclng" and dissipation optimum over the velocity optimum. All of this suggests that the first bifurcation wlll be manifested in the approach to the secn.d bif'urc~ition by the onset of alternating nucleation and growth events, which is a falr conception of upper balnlte. Certainly the balnlte reaction has elements of both chaos and mechanism change (17). A. analogous instability has been observed in the closely related Liesegang phenomenon (5). On[: may a s k a s t o t h e s i g n i f i c a n c e oi t h e f a c t t h a t t h e i t e r a t e s d i v e r g e f o r 14~ :> 7 . 5 ( c f . Fig.3). We can interpret this as the point of failure of diffusion models. Conceivably it could influence the onset of the catastrophic lower temperature shear mechanisms. Discussion It is a n o r m a l r e a c t i o n t,) s u s p e c t s u c h p r o c e e d i n g s o f t h e o r e t i c a l illegality, much a s t h e early c~r~trLbutions in quantum theory appeared in such a light to almost all scientists. The writer f i n d s some s u c c o r i n t h e f o l l o w i n g q u a l i t a t i v e considerations. When we d e r i v e p o p u l a t i o n g r o w t h e q u a t i o n s l i k e E q s . 2 and 11 we do n o t h e s i t a t e to intuitively average without reference to the discreteness and extreme fluctuating character of the argument entities, which are thereby entered as probabilities. However, such character is implicitly subsumed w/thin the derived continuum non-linear equation, i t s b o u n d a r y c o n d i t i o n s a nd i t s s o l u t i o n s through the characteristic lengths which can then on o c c a s i o n he r e c o v e r e d when t h e e q u a t i o n a n d i t s s o l u t i o n a r e appropriately dlscretized, Explicit discretlzatlon in Eq.ll puts some lost information or negentropy hack into the system and here it was a normalized characteristic relaxation time ~ = ~ = I for spacing change. Thus upon sufficient amplification by the control parameter ~ the recovery of dlscretlzatlon effects in the form of cyclic and strange attractors (8), together with chaotic elements does not seem implausible. That such gratuitous regularities and irregularities have close experimental counterparts in the relevant domain is, however, surprising and profound. It is tile universality of the phenomena, challenging (:onventinnal physical perception, that provides a key to complete understanding. For example, Eqs.1 o r 3 as used in b i o l o g y (18) describe a self-excluding or competitive population, a nd w h a t a t f i r s t s i g h t a p p e a r s a s a weak analogy with the pattern of competing a nd s u r v i v i n g p e a r l i t e lamellae through overgrowth is hereby established as a thenrettcally determined universality(2,3). Note particularly that the temporal instabilities are spontaneously precipitated h e r e by i s o t h e r m a l p o s i t i v e f r e e e n e r g y increments (of. Eq.]O) which events have a conventional but equally conjectural status In nucleation theory. For this reason we may r e g a r d this development a s a d y n a m i c a l addendum t o nucleation theory. Practically it s t a t e s w h a t i s g e n e r a l l y known t o be t h e c a s e , t h a t s t a b l e
182
CHAOS
spatial patterns supersaturattons.
will
I~e
overcom,~
by
repeated
Vol.
nucleation
events
at
24, No.
sufficiently
I
high
The e x p e r t s have s u r m i s e d that the discipline of deterministic c h a o s J a c k s two e s s e n t i a l features, a bridge between c h a o t i c m o t i o n s in t h e t e m p o r a l and s p a t i a l domain (2) and a f i r m c o n n e c t i o n w i t h thermodynamics ( 1 , 2 ) . Both o f t h e s e a r e p r o v i d e d by t h e p e a r l l t e model whi ch can therefore be r e p r e s e n t e d a;~ a paradigm f o r s t r o n g l y d r i v e n ! l - r e v e r s i b l e p r o c e s s e s . The t h e r modyuamJc q u e s t i o n h~ls be.n explored fundamentally according to the conjecture that in conservative mechanical systems o f many d e g r e e s o f ~reedom the e q u i l i b r i u m state of ergodicity (time averages replaceable by phase space averaK.~;s) can be e q . a t e d w i t h s e l f - g e n e r a t e d d e t e r ministic chaos ( 2 ) . By c o m p a r i s o n , s t e a d y s t a t e i r r e v e r s i b l e patterns of present Interest admit phase space f l u c t u a t i o n s only, so a r e n o n - - e r g o d l e . The term seems a p t o n l y t'or a h a r d - - d r i v e n fully chaotic state where f o r the p r e s e n t case t h e t e m p o r a l i t e r a t e s i n t e r s e c t t h e phase space O|llilgHl#= and 1.h(;reby d(~stroy tit(; p a t t e r n . (:haos in i r l . e v e r . ~ i h ] e sy,~t.~?ms o r t tzrbulenc(~ c a . t h u s for synthetic p u r p o s e s be e q u a t e d w i t h an e r g o d J c s t a t e o f e q u i l i b r a t i o n w i t h the a v a i l a b l e en(;r~:y sourc.e r a t h e r t h a n w i t h t h e h e a t b a t h s i n k . JRe _ ereJ!ces 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16, 17. 18.
P. Berge, Y. Pomeau and C. V l d a l , L ' O r d r e dan l e Chaos. Hermann, P a r i s . 1984. A.G. S c h u s t e r , D e t e r m i n i s t i c Chaos, VCH V e r l a g s g e s e l l s c h a f t mbfl.. WeinheJm, F . R . G . , 1988. N.J. Felgenbaum, J. S l a t . Phys. 21. 669 ( 1 9 7 9 ) . K.G. W i l s o n . S c i . Am. 2_.4.1, 158 (19"/91. P r i v a t e c o m m u n i c a t i o n w i t h Dr. Y. B r e c h e t . P.F. V c r h n ] s t , Mere. Acad. Roy. B r u x e l l e s . 2_0, 1 (.1846). A..I. L o t k a . Elements o f M a t h e m a t i c a l BJoJo~ffx'. Dover PubJJcatJon.~. l l l C . , blew Y o r k , ] 956. M. P r u f e r , SlAM J. AppJ. ,~ath., 45, ;32 ( 1 9 8 5 ) . hi, H t l l e r t , 3erukonr.orets A n n . , 141.. 757 (19571. J.S. K l r k n l d y , S c r l p t a Net. 2. 565 ( 1 9 6 8 ) . J. S h a p i r o and J . S . K i r k a l d y , A c t s M e t . , 16. 579 ( 1 9 6 8 ) . M.P. P u l s and J . S . K i r k a l d y , Met. T r a n s . , 3, 2777 ( 1 9 7 2 ] . J . $ . K l r k a l d y . Phys. Rev. A 3~, 3376 ( 1 9 8 5 ) , C. Z e n e r , T r a n s . A[blE, 16__7. 550 ( 1 9 4 6 ) . J . S . LanF.er, Phys. Rev. L e f t s . . 4.4, 1023 ( 1 9 8 0 ) . J . S . K t r k a J d y , Phys. Rev. 8 , 3 0 . 30 ( 1 9 8 4 ) . A . J . Lee, G. S p a n o s , G . J . S h t f l e t and l t . l . A a r o n s u n , A c t s M e t . , 36, 1129 ( 1 9 8 8 ) R.M. Hay, N a t u r ' e 2 6 l . 459 ( 1 9 7 6 ) . Acknowledgem(;nt s
I am i n d e b t e d t o Dr. Hrt.ko Buchmayr ¢~f t h e I n s t i t u t e of Technology, (;raz, A u s t r i a , f e r h i s assistance in tile prep~tl.ation Of F i g . 3 and to Dr. Hub A a r o n s o n o f C a r n e g i e - M e l l o n U n i v e r s i t y f o r some h e l p f u l comments on t h e b a J n t t e r e a c t i o n .
Vol.
24,
No.
I
CHAOS
183
Z~
f
'~#,T~,:?,:. ~ :.~,:! ~-,-'~',
0 0,5
Flg. l.
I 0.6
-q
I 0.7
Asymptotic
0.!
'
1
iterates z® of the logistic map (1).
IO0( ~o
80< I--
c~
/
I D
"-.
AT
t
AC,.
I
0.8
CARBON
Fig.2.
"~I 1.6
~,,
Fe-Fe3C p h a s e d i a g r a m c o n t r o l J i n g
I 2.4
"
6 . 6 7--.-~
the pearlite
reaction.
184
CHAOS
Vol.
2,9 1,8
1,6 1,4 ,.,-- -
- - , , . ~ , ; : •,",~.*. - ... v'~,,...;.., :: ":,
.
8/g
6.8 ~
9.~ 9.4
2 / 3 . ~
~.~,~. ~.!~,--,%',,, ,.:,.
~,~',"~.~* ¢:,..,';':
j
~,s'Y'-i,'a"~' ,' ,'
",
,_ ".. "..
/:.',C-'" ~;',! :, ;'
.,.v',..,.~.~ .% .~.. ..,,',~.,.t . ~ ',.. ~.~, ,',,,,~,,, ~,' ;:.,,.
"--_
.~'~,I,~'...,, ~..~,.. --------f-
.1/3
" % ~,,~.~"c'r ,,;' : " ~;• ,'11.~" 41~,' ~,,
1t.2 6.9
I
5
Fig.3.
I
I
6
?
• ~;l
Asymptotic iterates of the pearllte dissipation equation.
24,
No.
I
Vol. 24, pp. 1 7 9 - 1 8 4 , 1990 P r i n t e d in the U . S . A .
P e r g a m o n P r e s s plc All rights reserved
DETERMINISTIC CHAOS AND EUTECTOID PHASE TRANSFORMATIONS
Institute
J.S. Klrkaldy for Materials Research, McNaster University, H a m i l t o n . O n t a r i o , C a n a d a L8S 4M1.
( R e c e i v e d A p r i l 18, 1989) ( R e v i s e d N o v e m b e r i0, 1989) The w i d e recognition that certain deterministic non-linear difference and differential equations w i t h few d e g r e e s o f f r e e d o m c a n d e f i n e p h a s e s p a c e t r a j e c t o r i e s wlllch diverge exponentially lnto chaotlc motlons Is having a p r o f o u n d I n f l u e n c e on a p p l i e d p h y s i c s ( 1 . 2 ) . While presenting deep and unsolved problems of interpretation within which the experts find macroscopic analogies to microscopic quantum phenomena ( 1 . 2 ) t h e r e s e a r c h i s p r o v i d i n g i m p o r t a n t I n s i g h t s Into turbulent o r c h a o t i c phenomena i n h y d r o d y n a m i c s , c h e m i s t r y , m e c h a n i c s a nd b i o l o g y , f i n d i n g common e l e m e n t s or universality o f t h e k i n d s a l r e a d y r e c o g n i z e d ill t h e t h e o r y of s e c o n d o r d e r phase t r a n s i t i o n s (3,4). H e r e w i t h we u n d e r t a k e t o show t h a t the t r a n s i t i o n of pearlite I n FeFe_C t o t u r b u l e n t o r d e n d r i t i c f o r m s i s n o t o n l y encompassed by t h e same e l e m e n t s o f u n i v e r s a l i t y bu~ a l s o suggests the nature of t h e c o n n e c t i o n between s p a t i a l and t e m p o r a l chaos and t h e i r thermodynamic l i n k s . Much o f the fundamental theory prescription In virtual time (1,2)
Zn+ I = (]+r)Zn(l-z n) ; and this subsumes and economies, a {2,5). This map Verhulst pertaining
centers
on
the
non-linear
"logistic"
l+r ~ 4
map o r i t e r a t l v e
(l)
apparently deterministic phenomena as wide-ranging as self-llmit~ng ecologies damped driven pendulum and the capillarity-controlled Liesegang phenomenon is derivable from the growth-death type dlffercntJal equation or "flow" of to average self-llmlting population densities {6,7)
dX d-t = ~ X ( 1 - X )
(2)
through conversion to the numerical integral (8)
xn+ 1 = x n + c ~ t x (1 x ) It
l s t o be n o t e d t h a t
gq.(2)
(3)
possesses the closed solution
(8)
X = Xo/[Xo+{l-Xo)eXp(- ~t)]
(4)
which is always asymptotic to X = I. While it might be expected that thls solution can always be approximated by numerical operation on Eq.3 for positive r = ~ ~t this is not the case. If we make the linear transformation X n
(5)
~ (l+r)z r
n
specifying r ~ 3 the iterates of z in Eq.! are confined to the unit interval s o c a n be i n t e r preted as probability sequences, nFlg.1 exhlbtts the well-known distribution of asymptotic iterates z~ o f Eq.1 a s a f u n c t i o n o f t i l e n o r m a l i z e d c o n t r o l p a r a m e t e r ~ = ( ] + r ) / 4 . The e a s i e s t way t o appreciate this result, including the bifurcations or period doubling cycles and the transition to chaos at u = 0.892. is to p r o g r a m E q . 1 on a h a n d c a l c u l a t o r a nd e x p l o r e t h e Iterates manually. Observe t h a t t h e zm I t e r a t e s up t o t h e f i r s t b i f u r c a t i o n correspond to the X = 1 asymptotes of Eq.4. If for comparison one runs the Iterates on form ( 3 ) i t b e c o m e s c l e a r that the source of the Instability is the finite i n c r e m e n t a l change in s i g n of t h e r i g h t hand term w h i c h a c c r u e s when ~ t e x c e e d s 2. T h a t I s t o s a y , t h e i n s t a b i l i t y i s on t h e f a c e o f i t due
to an arithmetic artifact which is exhibited if the time increment is excessively large. As we shall argue below the apparent artifact is accorded physical significance when ~t is selected as a characteristic relaxation tlme for the problem at hand, given ~.
179 0 0 3 6 - 9 7 4 8 / 9 0 $ 3 . 0 0 + .00 C o p y r i g h t (c) 1990 P e r g a m o n P r e s s
plc
180
CHAOS
Transition The reaction (9,10) Sl .
entropy production by volume d i f f u s i o n
v &F. ~
XcTE
.
.
Vol.
from L a m e l l a r P e a r l l t e
24, No.
I
to C h a o t i c Forms
rate p e r u n i t volume o f t h e i s o t h e r m a l l a m e l l a r Fe-Fe~C e u t e c t o i d h a s been a p p r o x i m a t e d in t e r m s o f s p a c i n g ~ and f r o n t v b l o c i t y v by
c .
x
.c
~F
%TE
2~De
1 4 ~
%aT
.
~
c (1- _ . c ) 2
(6)
where ~ i s the wall surface tension, D is the diffusion coefficient and TE i s t h e e u t e c t o t d temperature. Note t h a t f o r any f i n i t e system t h i s r e p r e s e n t s a d i s c r e t e s e r i e s o f s p a t i a l s t e a d y states. Also &F ~ & ~ 11 -
Is the p o s i t i v e l y 2~
2~)
171
defined driving
f r e e energy change p e r u n i t
volume and
2~TE
(8)
~c ~-F-o ~V~-----T Is the relative
critical or cut-off supersaturation
s p a c i n g , w i t h ~T t h e u n d e r c o o l i n g and ~B t h e l a t e n t
heat,
and t h e
and t h i s is related t o ~T by t h e p h a s e d i a g r a m ( F t g . 2 ) . The d i s c r e t e s p a t i a l s e r l e s o f f r e e b o u n d a r y s t e a d y s t a t e s d e f i n e d by g q . 6 p o s s e s s e s a maximum a t X = 3~ , and i t h a s b e e n a r g u e d and Is consistent with the experiments, that this represents the stability p o i n t o r maximum p r o b a b i l ity state (10-13) in a kinetic phase space of fluctuating lame]lar dispersion. Now t h e s p a t i a l series can be t r a n s f o r m e d t o a v i r t u a l t i m e s e r i e s o r I a m e l l a r s t a t e s v i a an a s s o c i a t i o n , which for a large experimental enclosure equates the locally determined dissipation t o i t s volume normalized global expression, vJz., v~___F = _ 1 dF T T dt c and unit
this is interval,
! d(~F) T dt
introduced we o b t a i n
d..~X = 14~X(1-X) 2 d~
i n t o Eq.6 v i a Eq.7.
(lO)
D e f i n i n g X = ~ b / ~ w h i c h maps t h e s p a n o f ~ t o t h e
(11a)
where the normalized time is ~ = t/(R2=/D). S t a r t i n g w i t h any s m a l l Xu t h e e x a c t i n t e g r a l o f G this, w h i c h r e p r e s e n t s t h e g l o b a l e n t r o p y a c c u m u l a t e d , a s y m p t o t i c a l l y r e a c h e s a maximum v a l u e o f X = 1 as before. B e c a u s e o f t h e s q u a r e t e r m t h e c o r r e s p o n d i n g map n e v e r o s c i l l a t e s . However, a r e v i e w of the d e r i v a t i o n of Eq.6 i n d i c a t e s t h a t a r e l a t i v e l y f l a t i n t e r f a c e was a s s u m e d and t h i s . together with t h e l i n e a r 0 1 b b s - T h o m p s o n e f f e c t and a p p r o x i m a t i o n f o r R , i s o n l y v a l i d n e a r t h e optimum ( 9 , 1 1 ) . Indeed, the expression is otherwise patently lncorrec~ for it defines a physic a l l y u n a t t a i n a b l e v e l o c i t y s t a t e i n t h e a p p r o a c h t o ~ /~. = 1 and an a p p a r e n t l y f e a s l b l e p h y s i c a l state with negative v and p o s i t i v e d i s s i p a t i o n forCx / ~ e x c e e d i n g u n i t y even though t h ~ s e a r e both configurattonally I m p o s s i b l e on t h e I s o t h e r m ( c C . C F i g . 2 ) . One t h u s c o n c l u d e s t h a t S i must vanish by c r o s s i n g the axis approachir~ llnearity r a t h e r t h a n by f o r m i n g a q u a d r a t i c minimum. This Implies that for A / A ~ 1, v ~ S, i s a f e a s i b l e c o n f i g u r a t i o n which a p p r o a c h e s the a x i s vertically, and f o r ~ ~ > 1 that theldlsslpation I s n e g a t i v e and v i s i m a g i n a r y , p h y s i c a l l y excluding such a stats on b o t h t h e r m o d y n a m i c and c o n f i f f u r a t l o n a l g r o u n d s . I f E q . l l a i s t o be modified so as t o v a n i s h l i n e a r l y we c a n r e p r e s e n t i t by Eq. i l a up t o i t s i n f l e c t i o n point at X = 2 / 3 and a p p r o x l m a t e i t by i t s l i n e a r c o n t i n u a t i o n dX ~ 14~ 8 9 d-~ "~'~(1-~X)
(]]b)
thereafter. Note t h a t t h i s changes s i g n a t X = 8 / 9 , which c o o r d i n a t e d e f i n e s the e v o l u t i o n a r y maximum In X, and t h i s s i g n change as b e f o r e w i l l be t h e d i r e c t source o f t e m p o r a l l y u n s t a b l e iterates of the corresponding difference equation. Indeed. a t t h e n o n - p h y s i c a l s i g n change o f
Vol.
Si ~ t tm~,
24,
dF/dt
No.
1
one
CHAOS
might
infer
181
a nucleation event, where classically F increases momentarily
in
Converting Eq.ll to the equivalent difference equation for numerical integration in form (3) corresponding to a c h a r a c t e r i s t i c time incr e m e n t c o n s i s t ~ m t w i t h E q . l O , ~ = ~ = 1. and g e n e r ating the a s y m p t o t i c i t e r a t e s X as a f u n c t i o n o f t41S y i e l d s F i g . 3 . I n t h e u su a l f a s h i o n o f t h e Feigenba.m route to chaos (2? this exhibits period doubling, and unstable iterates when the control parameter 14~ > 5.9. The spatial asymptote X~ = 1/3 intersects with the temporal iterates at the second bifurcation (14~ ~ 6.7) and enters the chaotic zone at 14~ ~ 6.9. We hypothesize that the intersection defines a morphological interaction and from Fig.3 thereby identifies the onset control parameter for chaos with an undercoollng of approximately 200OC in the Fe.Fe~C system. This is experimentally about right for onset of the instability which attends the- bainite reaction and suggests that a search for evidence of a bifurcation near this value of 141s and the optimum spacing might be useful. These considerations augment our successful "half-spaclng" perturbation argument (10,12} which operated upon the v(X) curve {cf. Eq.6). This states that virtual half-spaclng nucleation events stabilize the h = 3h dissipation optimum because they grow at the same symmetrlcally disposed sub-maxlmal velocitles~ if the half-spaclng < 3/2A. then this region has a velocity deficit and will be overgrown, if it is > 3/2 ~ then the region c will be incorporated with average motion toward the stability point. Accordin~ to Flg.3 the time series asymptotes correspond to half-spacings < 3/2 ~ before the bifurcation so they would be overgrown even if nucleated. Slightly within the first b~furcation, however, the physical branch of asymptot(.s gains the stabilizing status of half-spacing fluctuations and beyond an increasing propensity for influencing the morphology. Indeed, we see that the lower time series blfurcati-r~ is f.rced asymptoti(:aJly toward the optimum at X~ = 1/3 with increased driving force or supersaturation and this extends the evidence favouring stability at the "half-spaclng" and dissipation optimum over the velocity optimum. All of this suggests that the first bifurcation wlll be manifested in the approach to the secn.d bif'urc~ition by the onset of alternating nucleation and growth events, which is a falr conception of upper balnlte. Certainly the balnlte reaction has elements of both chaos and mechanism change (17). A. analogous instability has been observed in the closely related Liesegang phenomenon (5). On[: may a s k a s t o t h e s i g n i f i c a n c e oi t h e f a c t t h a t t h e i t e r a t e s d i v e r g e f o r 14~ :> 7 . 5 ( c f . Fig.3). We can interpret this as the point of failure of diffusion models. Conceivably it could influence the onset of the catastrophic lower temperature shear mechanisms. Discussion It is a n o r m a l r e a c t i o n t,) s u s p e c t s u c h p r o c e e d i n g s o f t h e o r e t i c a l illegality, much a s t h e early c~r~trLbutions in quantum theory appeared in such a light to almost all scientists. The writer f i n d s some s u c c o r i n t h e f o l l o w i n g q u a l i t a t i v e considerations. When we d e r i v e p o p u l a t i o n g r o w t h e q u a t i o n s l i k e E q s . 2 and 11 we do n o t h e s i t a t e to intuitively average without reference to the discreteness and extreme fluctuating character of the argument entities, which are thereby entered as probabilities. However, such character is implicitly subsumed w/thin the derived continuum non-linear equation, i t s b o u n d a r y c o n d i t i o n s a nd i t s s o l u t i o n s through the characteristic lengths which can then on o c c a s i o n he r e c o v e r e d when t h e e q u a t i o n a n d i t s s o l u t i o n a r e appropriately dlscretized, Explicit discretlzatlon in Eq.ll puts some lost information or negentropy hack into the system and here it was a normalized characteristic relaxation time ~ = ~ = I for spacing change. Thus upon sufficient amplification by the control parameter ~ the recovery of dlscretlzatlon effects in the form of cyclic and strange attractors (8), together with chaotic elements does not seem implausible. That such gratuitous regularities and irregularities have close experimental counterparts in the relevant domain is, however, surprising and profound. It is tile universality of the phenomena, challenging (:onventinnal physical perception, that provides a key to complete understanding. For example, Eqs.1 o r 3 as used in b i o l o g y (18) describe a self-excluding or competitive population, a nd w h a t a t f i r s t s i g h t a p p e a r s a s a weak analogy with the pattern of competing a nd s u r v i v i n g p e a r l i t e lamellae through overgrowth is hereby established as a thenrettcally determined universality(2,3). Note particularly that the temporal instabilities are spontaneously precipitated h e r e by i s o t h e r m a l p o s i t i v e f r e e e n e r g y increments (of. Eq.]O) which events have a conventional but equally conjectural status In nucleation theory. For this reason we may r e g a r d this development a s a d y n a m i c a l addendum t o nucleation theory. Practically it s t a t e s w h a t i s g e n e r a l l y known t o be t h e c a s e , t h a t s t a b l e
182
CHAOS
spatial patterns supersaturattons.
will
I~e
overcom,~
by
repeated
Vol.
nucleation
events
at
24, No.
sufficiently
I
high
The e x p e r t s have s u r m i s e d that the discipline of deterministic c h a o s J a c k s two e s s e n t i a l features, a bridge between c h a o t i c m o t i o n s in t h e t e m p o r a l and s p a t i a l domain (2) and a f i r m c o n n e c t i o n w i t h thermodynamics ( 1 , 2 ) . Both o f t h e s e a r e p r o v i d e d by t h e p e a r l l t e model whi ch can therefore be r e p r e s e n t e d a;~ a paradigm f o r s t r o n g l y d r i v e n ! l - r e v e r s i b l e p r o c e s s e s . The t h e r modyuamJc q u e s t i o n h~ls be.n explored fundamentally according to the conjecture that in conservative mechanical systems o f many d e g r e e s o f ~reedom the e q u i l i b r i u m state of ergodicity (time averages replaceable by phase space averaK.~;s) can be e q . a t e d w i t h s e l f - g e n e r a t e d d e t e r ministic chaos ( 2 ) . By c o m p a r i s o n , s t e a d y s t a t e i r r e v e r s i b l e patterns of present Interest admit phase space f l u c t u a t i o n s only, so a r e n o n - - e r g o d l e . The term seems a p t o n l y t'or a h a r d - - d r i v e n fully chaotic state where f o r the p r e s e n t case t h e t e m p o r a l i t e r a t e s i n t e r s e c t t h e phase space O|llilgHl#= and 1.h(;reby d(~stroy tit(; p a t t e r n . (:haos in i r l . e v e r . ~ i h ] e sy,~t.~?ms o r t tzrbulenc(~ c a . t h u s for synthetic p u r p o s e s be e q u a t e d w i t h an e r g o d J c s t a t e o f e q u i l i b r a t i o n w i t h the a v a i l a b l e en(;r~:y sourc.e r a t h e r t h a n w i t h t h e h e a t b a t h s i n k . JRe _ ereJ!ces 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16, 17. 18.
P. Berge, Y. Pomeau and C. V l d a l , L ' O r d r e dan l e Chaos. Hermann, P a r i s . 1984. A.G. S c h u s t e r , D e t e r m i n i s t i c Chaos, VCH V e r l a g s g e s e l l s c h a f t mbfl.. WeinheJm, F . R . G . , 1988. N.J. Felgenbaum, J. S l a t . Phys. 21. 669 ( 1 9 7 9 ) . K.G. W i l s o n . S c i . Am. 2_.4.1, 158 (19"/91. P r i v a t e c o m m u n i c a t i o n w i t h Dr. Y. B r e c h e t . P.F. V c r h n ] s t , Mere. Acad. Roy. B r u x e l l e s . 2_0, 1 (.1846). A..I. L o t k a . Elements o f M a t h e m a t i c a l BJoJo~ffx'. Dover PubJJcatJon.~. l l l C . , blew Y o r k , ] 956. M. P r u f e r , SlAM J. AppJ. ,~ath., 45, ;32 ( 1 9 8 5 ) . hi, H t l l e r t , 3erukonr.orets A n n . , 141.. 757 (19571. J.S. K l r k n l d y , S c r l p t a Net. 2. 565 ( 1 9 6 8 ) . J. S h a p i r o and J . S . K i r k a l d y , A c t s M e t . , 16. 579 ( 1 9 6 8 ) . M.P. P u l s and J . S . K i r k a l d y , Met. T r a n s . , 3, 2777 ( 1 9 7 2 ] . J . $ . K l r k a l d y . Phys. Rev. A 3~, 3376 ( 1 9 8 5 ) , C. Z e n e r , T r a n s . A[blE, 16__7. 550 ( 1 9 4 6 ) . J . S . LanF.er, Phys. Rev. L e f t s . . 4.4, 1023 ( 1 9 8 0 ) . J . S . K t r k a J d y , Phys. Rev. 8 , 3 0 . 30 ( 1 9 8 4 ) . A . J . Lee, G. S p a n o s , G . J . S h t f l e t and l t . l . A a r o n s u n , A c t s M e t . , 36, 1129 ( 1 9 8 8 ) R.M. Hay, N a t u r ' e 2 6 l . 459 ( 1 9 7 6 ) . Acknowledgem(;nt s
I am i n d e b t e d t o Dr. Hrt.ko Buchmayr ¢~f t h e I n s t i t u t e of Technology, (;raz, A u s t r i a , f e r h i s assistance in tile prep~tl.ation Of F i g . 3 and to Dr. Hub A a r o n s o n o f C a r n e g i e - M e l l o n U n i v e r s i t y f o r some h e l p f u l comments on t h e b a J n t t e r e a c t i o n .
Vol.
24,
No.
I
CHAOS
183
Z~
f
'~#,T~,:?,:. ~ :.~,:! ~-,-'~',
0 0,5
Flg. l.
I 0.6
-q
I 0.7
Asymptotic
0.!
'
1
iterates z® of the logistic map (1).
IO0( ~o
80< I--
c~
/
I D
"-.
AT
t
AC,.
I
0.8
CARBON
Fig.2.
"~I 1.6
~,,
Fe-Fe3C p h a s e d i a g r a m c o n t r o l J i n g
I 2.4
"
6 . 6 7--.-~
the pearlite
reaction.
184
CHAOS
Vol.
2,9 1,8
1,6 1,4 ,.,-- -
- - , , . ~ , ; : •,",~.*. - ... v'~,,...;.., :: ":,
.
8/g
6.8 ~
9.~ 9.4
2 / 3 . ~
~.~,~. ~.!~,--,%',,, ,.:,.
~,~',"~.~* ¢:,..,';':
j
~,s'Y'-i,'a"~' ,' ,'
",
,_ ".. "..
/:.',C-'" ~;',! :, ;'
.,.v',..,.~.~ .% .~.. ..,,',~.,.t . ~ ',.. ~.~, ,',,,,~,,, ~,' ;:.,,.
"--_
.~'~,I,~'...,, ~..~,.. --------f-
.1/3
" % ~,,~.~"c'r ,,;' : " ~;• ,'11.~" 41~,' ~,,
1t.2 6.9
I
5
Fig.3.
I
I
6
?
• ~;l
Asymptotic iterates of the pearllte dissipation equation.
24,
No.
I