- Email: [email protected]
Neutron scattering study of convection in a double-diffusive system
Physica 120B (1983) 376-382 North-Holland Publishing Company
§4.2. NON-EQUILIBRIUM SYSTEMS N E U T R O N S C A T T E R I N G S T U D Y O F C O N V E C T I O N IN A D O U B L E - D I F F U S I V E SYSTEM T. R I S T E and K. O T N E S Institute for Energy Technology, 2007 Kjeller, Norway
Invited paper Nonequilibrium systems with two or more order parameters and competing relaxation mechanisms display bifurcations and phase diagrams resembling equilibrium systems with multicritical properties. One example is a convecting nematic crystal in a vertical magnetic field. Our neutron scattering data give the time-averaged order parameters as a function of field (H) and temperature gradient (AT). We also investigate the time behaviour of the order parameters of different values of H and AT, and find steady, oscillatory and aperiodic states. We study the co-existence and transition between these states, and search for critical scattering. Some information is obtained on the evolution of the frequency spectrum and of the phase trajectory as AT is changed.
I. Introduction
O n e of t h e m o s t i m p o r t a n t d i s c o v e r i e s in s c i e n c e in r e c e n t y e a r s is t h e u n d e r s t a n d i n g of t h e e v o l u t i o n f r o m o r d e r l y to c h a o t i c t i m e b e h a v i o u r in p h e n o m e n a as different as b a c t e r i a p o p u l a t i o n a n d e l e c t r o n i c n o i s e [1]. T h e first t y p e of e x p e r i m e n t t h a t g a v e a c o n v i n c i n g d e m o n s t r a t i o n of a p e r i o d - d o u b l i n g r o u t e to c h a o s was a s t u d y of R a y l e i g h - B e n a r d c o n v e c t i o n [2]. It is p r o b a b l y fair to say t h a t h y d r o d y n a m i c s has n o t h a d t h e a t t e n t i o n t h a t it d e s e r v e s f r o m physicists. Convection p l a y s a m a j o r role in m a n y p h e n o m e n a in n a t u r e , in t h e f o r m a t i o n of clouds, in t h e c o n t i n e n t a l drift, in t h e activity of sunspots, etc. [3] The development from laminar, time-ind e p e n d e n t flow to i r r e g u l a r (chaotic, t u r b u l e n t ) , t i m e - d e p e n d e n t flow o c c u r s t h r o u g h o n e o r s e v e r a l r e g i m e s w h e r e the flow is t i m e p e r i o d i c . T h e different r e g i m e s a r e s e p a r a t e d by instabilities, also c a l l e d b i f u r c a t i o n s . It s h o u l d c o m e as no s u r p r i s e that t h e s e b i f u r c a t i o n s m u s t b e a r s o m e r e s e m b l a n c e to p h a s e transitions. L a n d a u s h o w e d t h a t t h e t r a n s i t i o n f r o m the q u i e s c e n t to the c o n v e c t i n g state, i.e. a t r a n s i t i o n b e t w e e n two different t i m e - i n d e p e n d e n t symm e t r i e s , has a m e a n - f i e l d c h a r a c t e r , like a
s e c o n d - o r d e r t r a n s i t i o n [4]. F u r t h e r m o r e , p e o p l e s t u d y i n g c o n v e c t i o n w e r e in fact t h e first to use t h e c o n c e p t of scaling. In c o n v e n t i o n a l p h a s e t r a n s i t i o n s t e m p e r a t u r e is a c o m m o n c o n t r o l p a r a m e t e r . In c o n v e c t i o n t h e t e m p e r a t u r e g r a d i e n t is an e q u a l l y c o m m o n c o n t r o l p a r a m e t e r . A s t h e vertical t e m p e r a t u r e g r a d i e n t ( A T ) of a l a y e r of liquid is i n c r e a s e d , c o n v e c t i o n sets in a b r u p t l y at a t h r e s h o l d ATe. F r o m t h e r e on t h e v e l o c i t y of t h e c o n v e c t i n g fluid (v) i n c r e a s e s a c c o r d i n g to v ~ ( A T - ATe) °.5. T h e velocity t h e r e f o r e p l a y s t h e role of an o r d e r parameter. T h e e x i s t e n c e of a t e m p e r a t u r e g r a d i e n t m e a n s that t h e r e is an e n e r g y flow t h r o u g h t h e system, a n d t h e p h e n o m e n o n is t h e r e f o r e that of a n o n e q u i l i b r i u m p h a s e t r a n s i t i o n . In e q u i l i b r i u m p h a s e t r a n s i t i o n s m u c h a t t e n t i o n is p r e s e n t l y b e i n g p a i d to multi-critical p h e n o m e n a , i.e. to s y s t e m s with c o m p e t i n g o r d e r p a r a m e t e r s a n d i n t e r a c t i o n s . In n o n e q u i l i b r i u m physics a corr e s p o n d i n g case is that of c o n v e c t i o n in a d o u b l e diffusive system. In such a case s t e a d y c o n v e c t i o n is p r e c e d e d by o s c i l l a t o r y (time p e r i o d i c ) conv e c t i o n [5], as shown s c h e m a t i c a l l y in fig. 1. W e shall in this p a p e r d e s c r i b e e x p e r i m e n t s a i m e d at u n d e r s t a n d i n g the o n s e t of the two t y p e s of flow a n d t h e r e g i m e of t h e i r c o e x i s t e n c e .
0378-4363/83/0000--4)000/$03.00 © 1983 N o r t h - H o l l a n d a n d Y a m a d a Science F o u n d a t i o n
T. Riste and K. Otnes / Neutron scattering and convection
or ~ r p,zmme~
R2
R,"
8
Fig. 1. Schematic picture of order parameter (velocity v, or
(v2)°~) versus Rayleigh number R, defined in the text. Superscripts o and s on Rc identify onsets of oscillatory and steady flow, respectively.
The physical mechanism for the oscillatory flow can be explained through the example of a layer of cold, fresh water on top of warm, salty water. If a droplet of salty water gets w a r m e r than its surrounding it will rise tO the fresh layer. Since salinity diffuses much slower than temperature, the droplet will rapidly be cooled but remain salty. The droplet, now heavier than its surrounding, will subsequently sink and overshoot its initial position, and the whole process repeats.
2. Theoretical survey
Consider an isotropic, pure liquid contained in a vessel and heated from below. T h e solution in the different flow regimes can be given in the space of the different p a r a m e t e r s R (Rayleigh number) and P (Prandti number) defined by
R =/3gI3AT/vK,
P = v/K
(1)
w h e r e / 3 is the volume thermal expansion, g the gravitational acceleration, 1 the vertical layer thickness of the liquid, A T the vertical temperature difference, u the kinematic viscosity and K the heat diffusivity. The calculated critical value, Re, for the onset of steady convection, for rigid boundaries and a high aspect (width-todepth) ratio, is 1708. In a brick-shaped, rectangular vessel and preference is for two-dimensional roils with axes parallel to the shorter horizontal side of the vessel. The effect of decreasing the aspect ratio is to reduce the
377
n u m b e r of rolls and to increase Rc [6]. P is expected to be of importance only for the higher instabilities. As mentioned above, the Landau theory predicts v oc ( R - Re) °5 for the velocity in the roll. Fluctuations are predicted to be of importance only within a narrow region ( R / R c - 1 ) < 10 7 and thus not likely to be seen [7, 8]. For an anisotropic liquid, i.e. a liquid crystal, eq. (1) is m o r e complicated. An effect is that ATe corresponding to Rc is reduced by a factor 10 3_ 10 -5 for a nematic liquid crystal. In this case molecular alignment is easily achieved by a magnetic field (H). For horizontal alignment steady convection is achieved when heating from below, for vectical alignment when heating from above. In both cases ,STc increases with increasing H. The width of the critical region is, for weak fields, predicted to be wider than for isotropic liquids, but still narrow ( < 10 4). G r a h a m [7] has given the following picture for fluctuations near Re: In the pretransitional region fluctuations involve transient convection cells of different sizes (or wave numbers, q). Near Rc there is a wave n u m b e r selection such that rolls of dimensions that adapt to the vessel b e c o m e dominant and survive the transition. These macroscopic, hydrodynamic fluctuations couple to inner, thermal fluctuations of the liquid. In liquid crystals orientational fluctuations of the relevant wave n u m b e r are abnormally abundant, which makes liquid crystals the best candidate for observing fluctuations near the convective instability. Orientational fluctuations are conveniently reduced by the application of an external magnetic field. In isotropic, pure liquids oscillatory convection only occurs at higher transitions on the route from laminar flow to turbulence, and is a nonlinear effect. In two-component or doublediffusive systems oscillatory convection precedes steady convection, see fig. 1, and appears already in linear theory. In this case the oscillatory flow corresponds to alternate clockwise and anticlockwise rotation of each convection roll. In an extensive series of papers da Costa, Knobloch, Proctor and Weiss [9] have treated the oscillatory and steady convection in a thermohaline
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72 Riste and K. Otnes / Neutron scattering and convection
and in a metallic liquid, the latter in a vertical magnetic field. O u r system is a nematic liquid crystal in a vertical magnetic field, for which L e k k e r k e r k e r gave the first linear theory. H e predicts R~ to increase and R~ to decrease with increasing H. H e r e superscripts o and s d e n o t e oscillatory and steady flow, respectively. V e l a r d e et al. [5] have c o n j e c t u r e d that the transitions c o n v e r g e at a tricritical point. In their t r e a t m e n t the transition to steady convection is an inverted bifurcation, i.e. the o r d e r p a r a m e t e r exists also for R < RL as for a first-order phase transition. D a C o s t a et al. find that the bifurcation changes f r o m an inverted to a n o r m a l o n e as the height (aspect ratio) of the vessel is increased (decreased). T h e y also investigate the stability of the oscillations along the oscillatory branch, from its start at B~ to its termination on the steady branch. W h e n the latter is supercritical, i.e. R~ is a n o r m a l bifurcation, the oscillatory branch starts and terminates via a H o p f bifurcation. A w a y f r o m these bifurcations the stability of the oscillations gets p o o r e r , or m a y be completely lost, d e p e n d i n g on the m o d e l they choose. Period doubling and aperiodity m a y occur outside of the stable regime. T h e d e v e l o p m e n t f r o m oscillatory to steady flow is conveniently described by phase space trajectories, as used in Hamilton-Jakobi mechanics [11]. Let a and ci d e n o t e the velocity amplitude of a roll and its derivative. F o r small amplitude oscillations (at R = R~) the system point then traces out a closed orbit, a limit cycle, in the phase space of coordinates a and &. This is shown in fig. 2a, and is equivalent to the case of an action-angle orbit of a simple p e n d u l u m . For large amplitudes, R = R 2 ( > R1), the nonlinearity of the system will distort the trajectory, as in fig. 2b. In phase space steady flow is a fixed point on the abscissa. In one of the models used in the calculations by da C o s t a et al. the system point will at R R 3 ( > R2) start to pay visits at fixed points c o r r e s p o n d i n g to opposite rotations of a roll, as shown in fig. 2c. A t a still higher value R = R4 the system m a y start to show a preference for one fixed point, see fig. 2d. T h e orbit is n o w time asymmetric, reflecting a p r e f e r e n c e =
o~
o;
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(e)
('b) d
('at)
Fig. 2. Phase space trajectories showing possible evolution of oscillatory flow at increasing Rayleigh numbers, see text. for o n e sense of rotation. T h e trajectory drawn has a split orbit, c o r r e s p o n d i n g to period doubling. If the oscillations are unstable the system point never retraces the same orbit and m o v e s on a strange attractor. T h e three attractors m e n t i o n e d , a fixed point, a limit cycle and a strange attractor, are topologically different and separated by bifurcations [5, 12]. A transition f r o m a fixed point to a limit cycle is a H o p f bifurcation. Successive H o p f bifurcations m a y lead to a strange attractor. T h e papers q u o t e d on oscillatory flow m a k e little or no reference to phase transitions. In a recent p a p e r Sz6pfalusy and T61 [13] find that for a h a r d - m o d e H o p f bifurcation the amplitude and the f r e q u e n c y c h a n g e as "O - (R - Re) °5 and (w ~o0) - (R - Re), respectively.
3. Experiments A sample of fully d e u t e r a t e d parazoxyanisole ( P A A ) was contained in a brickshaped alum i n i u m vessel of dimensions (3 x 30 x 30) m m 3. Of the long dimensions one is parallel to the scattering vector and the o t h e r to a vertical magnetic field. A vertical t e m p e r a t u r e difference (AT) was o b t a i n e d by setting the difference of the p o w e r fed to electrical heating elements at the top and b o t t o m of the vessel. T h e temp e r a t u r e at the midheight of the aluminium side wall was kept constant at 121 ° C, at which temp e r a t u r e the sample is nematic and susceptible to
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alignment by a field > 2 5 Oe. With this arrangement AT stayed constant within +0.01 ° for hours. The neutron wavelength was 1.25,~ -~. o The scattered intensity was recorded at Q - 1.8 A, at the maximum of the first liquid diffraction peak. This intensity is very sensitive to changes of the molecular alignment. It is a maximum for vertical alignment (H±Q) and a minimum for HHQ, i.e. horizontal [14]. In the given geometry there is one convection roll. Using the whole convection roll as the scattering volume, the
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379
experiment is sensitive to the relevant wavelength range of hydrodynamic fluctuations. They show up as intensity fluctuations in realtime recordings of the intensity. The timeaveraged intensity gives the order parameter for steady flow (v) or for oscillatory flow ((v2)°5). It can be shown that the intensity (i.e. its deviation from that of the fully aligned, flow-free state) is proportional to the order parameter [15]. With a vertical magnetic field the deviation is negative, hence we shall in the following invert the intensity. Fig. 3 gives the time-averaged intensity as a function of A T for H = 200 Oe. We observe two order-parameter curves. They are not part of a hysteresis loop, since we can come from one to the other through several channels. We suggest that they represent crossing branches of oscillatory and steady convection, indicated by fulldrawn and broken curves, respectively. At a stronger field, see fig. 4, the curves have either merged, or one has been preferred, over most of the gradient range. Each point on the curves is an average of several hours of counting. Examples of real-time recordings of the intensity are seen in fig. 5. In addition to raw data we give curves of the autocorrelation function c(~-)= AI(t)AI(t+ ~-). Here A I denotes the deviation from the time average, and the averaging is taken over the whole time series, c0-) in most cases shows whether we have an oscillatory or a steady state, but when oscillations get irregular or chaotic the distinction is not clear. For H < 200 Oe the oscillatory branch is less disturbed by the presence of the steady branch, and fig. 6 shows data for the order parameter and the corresponding frequency of oscillating states for H = 160Oe. The frequencies have been obtained by Fourier transform of the data. We have also processed some of these data to obtain the trajectories in phase space. In order to obtain the coordinates in the plot we have to do some smoothing of the data. This we do by means of the Fourier spectrum of I(t), which ensures that we do not smooth away the relevant oscillations. In fig. 7 we see the effect of increasing AT, i . e . R . Only short time records were used, to avoid the effects of gradient drifting.
T. Riste and K. Otnes / Neutron scattering and convection
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Fig. 6. O x c i l l a t i o n f r e q u e n c y v e r s u s o r d e r p a r a m e t e r , m e a s u r e d b y n e u t r o n i n t e n s i t y . A r r o w p o i n t s at z e r o o r d e r p a r a m e t e r . Solid line is a g u i d e to t h e e y e , a n d r e p r e s e n t s a m o d e l d i s c u s s e d in t h e text.
Fig. 7. P h a s e t r a j e c t o r i e s c o r r e s p o n d i n g to p o i n t s at (o = 5 × III 3s 1 a n d 1 0 × 10 -3s t of fig. 6, s h o w i n g i n c r e a s i n g c o m p l e x i t y as c o n t r o l p a r a m e t e r is i n c r e a s e d .
T. Riste and K. Otnes
Neutron scattering and convection
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Fig. 8. Order parameter, measured by neutron intensity, versus vertical temperature difference. The sample cell is incompletely filled, i.e. the top surface is free. A careful series of m e a s u r e m e n t s was also p e r f o r m e d with the same vessel filled to a height of 2 0 r a m , such that the top b o u n d a r y is not fixed. A p p a r e n t l y , for such a n a r r o w horizontal gap and with conducting side walls, the o r d e r p a r a m e t e r is not very m u c h d e p e n d e n t on the b o u n d a r y condition at the top surface. In fig. 8 we see c o m p e t i n g o r d e r p a r a m e t e r s over s o m e gradient region, although not as wide as in fig. 3. A notable difference is, however, a conspicuous intensity m a x i m u m near the onset of flow. This intensity m a x i m u m decreases linearly with H 2 and disappears at H = 235 Oe.
4. D i s c u s s i o n
F r o m o u r data it is evident that nonequilibrium phase transitions have o r d e r p a r a m e t e r curves that resemble those of equilibrium phase transitions, and n e u t r o n scattering is very well suited for measuring such curves. In the case of c o m p e t i n g o r d e r p a r a m e t e r s the interpretation of the curves is difficult, and the lines drawn in fig. 3 should be considered as tentative. T h e y suggest, however, an interference b e t w e e n states of different time symmetries. A t higher (fig. 4) and
381
lower applied fields the region of coexistence of o r d e r p a r a m e t e r s is narrower, and the interpretation easier. T h e identification of oscillatory states is easy when nonlinearities are weak. A t increasing control p a r a m e t e r (gradient) the oscillations b e c o m e spasmodic and almost invisible. A t the same time the f r e q u e n c y spectrum widens, as seen in fig. 6. A t h e o r y m e n t i o n e d a b o v e [13[, predicts a relation t o - " 9 2 between the frequency and the o r d e r p a r a m e t e r . If the oscillations b e h a v e as a soft m o d e in an equilibrium system, one would instead expect t o - " 9 [16]. T h e data of fig. 6 seem to f a v o u r the latter b e h a v i o u r in its initial part indicated by the line, suggesting a soft-mode H o p f bifurcation. T h e softening at increasing o r d e r p a r a m e t e r a n n o u n c e s the a p p r o a c h to steady flow. T h e transition f r o m the conductive to the oscillatory, convective state is, in a g r e e m e n t with the above, always f o u n d to be continuous. T h e oscillatory and steady convective states join in a m a n n e r that d e p e n d s on the applied magnetic field. In the range of coexisting order parameters, as e.g. in fig. 8, the junction is very similar to one of the possibilities suggested by da C o s t a et al. [9]. T h e width of the oscillatory flow regime decreases with increasing field, and it disappears a b o v e H - 250 Oe. T o g e t h e r with the threshold gradient this field possibly defines a tricritical point. T h e intensity m a x i m u m o b s e r v e d at the convection threshold for an incompletely filled cell, see fig. 8, looks very similar to critical scattering. T h e slope of the pretransitional tail corresponds to a critical index y = 1.5 + 0.3. T h e m a x i m u m is very narrow, - 0 . 1 °. O n its steep side we have not been able to clearly identify any point on the d o w n w a r d slope, which prevents us from drawing any definite conclusion as to the origin of the peak. If it is due to critical fluctuations, we must be able to explain why it is so much stronger in an incompletely filled cell. Invoking G r a h a m ' s picture [7], a plausible explanation could be the following: in a sample, whose aspect ratio is non-integral, fluctuations in the pretransitional region involve a wider range of wave n u m b e r s before the final q (i.e. cell dimension) is selected.
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T. Riste and K. Otnes / Neutron scattering and convection
The disappearance of the peak at H ~ 235 Oe may reflect the approach to a tricritical point and a reduction of the inner, orientational fluctuations at increasing field. Due to the steepness of the order parameter in the oscillatory regime, small changes (drifting) of the gradient give new phase trajectories. Fig. 7 shows, however, a growing complexity of the trajectory as the control (and order) parameter is increased. Our experiments have not answered all the questions raised in the theoretical survey. In particular we have not been able to give a coherent picture of the development of the phase diagram, and to explain the narrow stability range of the oscillations. Nevertheless the neutron method seems to offer some advantage over the heat-transport method used by the Orsay group [17]. Not only can we measure the order parameters better, but we can also measure spontaneous (i.e. not induced) oscillations. In retrospect we should have paid more attention to the geometry of the convection rolls. This is possible by tedious local intensity measurements [14]. Also we should have used shorter counting times in order to obtain more accurate information on the phase trajectories and their evolution as the control parameter is changed.
Acknowledgements We are much indebted to Professors Lekkerkerker and Velarde for valuable discussions.
References [1] Physics Today 34 (1981) 17. [2] A. Libchaber and J. Maurer, J. Phys. 41 (1980) C3. [3] See e . g . M . G . Velarde and C. Normand, Scientific American 243 (1980) 93. [4] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading Mass., 1959) Chap. XVII. [5] See e . g . M . G . Velarde, in Nonlinear Phenomena at Phase Transitions and Instabilities, ed. T. Riste (Plenum Press, New York and London, 1981) p. 205. [6] I. Cotton, Int. J. Heat & Mass Tr. 15 (1972) 665. [7] R. Graham, Phys. Rev. 10A (1974) 1762, and in Fluctuations, Instabilities and Phase Transitions, ed. T. Riste (Plenum Press, New York and London, 1975) p. 215. [8] J. Swift and P.C. Hohenberg, Phys. Rev. 15A (1977) 319. [9] E. Knobloch, N.O. Weiss and L.N. da Costa, J. Fluid Mech. 113 (1981) 153 and references therein. [10] H.N,W. Lekkerkerker, J. Phys. Lett. 38 (1977) L277. [11] H. Goldstein, Classical Mechanics (Addison-Wesley, Reading and London, 1959) p. 288. [12[ J.-P. Eckmann, Rev. Mod. Phys. 53 (1981) 643. [13] P. Sz6pfalusy and T. T61, Z. Phys. B43 (1981) 77. [14] More on the experimental method by T. Riste, K. Otnes and H. Bjerrum M~ller in Neutron Inelastic Scattering 1977 (Intern. Atomic Energy Agency, Vienna, 1978) p. 511. [ 15] J. Feder, private communication. [16] E.F. Steigmeier and H. Auderset, Solid State Commun. 12 (1973) 565. [17] E. Guyon, P. Pieranski and J. Salan, J. Fluid Mech. 93 (1979) 65.
§4.2. NON-EQUILIBRIUM SYSTEMS N E U T R O N S C A T T E R I N G S T U D Y O F C O N V E C T I O N IN A D O U B L E - D I F F U S I V E SYSTEM T. R I S T E and K. O T N E S Institute for Energy Technology, 2007 Kjeller, Norway
Invited paper Nonequilibrium systems with two or more order parameters and competing relaxation mechanisms display bifurcations and phase diagrams resembling equilibrium systems with multicritical properties. One example is a convecting nematic crystal in a vertical magnetic field. Our neutron scattering data give the time-averaged order parameters as a function of field (H) and temperature gradient (AT). We also investigate the time behaviour of the order parameters of different values of H and AT, and find steady, oscillatory and aperiodic states. We study the co-existence and transition between these states, and search for critical scattering. Some information is obtained on the evolution of the frequency spectrum and of the phase trajectory as AT is changed.
I. Introduction
O n e of t h e m o s t i m p o r t a n t d i s c o v e r i e s in s c i e n c e in r e c e n t y e a r s is t h e u n d e r s t a n d i n g of t h e e v o l u t i o n f r o m o r d e r l y to c h a o t i c t i m e b e h a v i o u r in p h e n o m e n a as different as b a c t e r i a p o p u l a t i o n a n d e l e c t r o n i c n o i s e [1]. T h e first t y p e of e x p e r i m e n t t h a t g a v e a c o n v i n c i n g d e m o n s t r a t i o n of a p e r i o d - d o u b l i n g r o u t e to c h a o s was a s t u d y of R a y l e i g h - B e n a r d c o n v e c t i o n [2]. It is p r o b a b l y fair to say t h a t h y d r o d y n a m i c s has n o t h a d t h e a t t e n t i o n t h a t it d e s e r v e s f r o m physicists. Convection p l a y s a m a j o r role in m a n y p h e n o m e n a in n a t u r e , in t h e f o r m a t i o n of clouds, in t h e c o n t i n e n t a l drift, in t h e activity of sunspots, etc. [3] The development from laminar, time-ind e p e n d e n t flow to i r r e g u l a r (chaotic, t u r b u l e n t ) , t i m e - d e p e n d e n t flow o c c u r s t h r o u g h o n e o r s e v e r a l r e g i m e s w h e r e the flow is t i m e p e r i o d i c . T h e different r e g i m e s a r e s e p a r a t e d by instabilities, also c a l l e d b i f u r c a t i o n s . It s h o u l d c o m e as no s u r p r i s e that t h e s e b i f u r c a t i o n s m u s t b e a r s o m e r e s e m b l a n c e to p h a s e transitions. L a n d a u s h o w e d t h a t t h e t r a n s i t i o n f r o m the q u i e s c e n t to the c o n v e c t i n g state, i.e. a t r a n s i t i o n b e t w e e n two different t i m e - i n d e p e n d e n t symm e t r i e s , has a m e a n - f i e l d c h a r a c t e r , like a
s e c o n d - o r d e r t r a n s i t i o n [4]. F u r t h e r m o r e , p e o p l e s t u d y i n g c o n v e c t i o n w e r e in fact t h e first to use t h e c o n c e p t of scaling. In c o n v e n t i o n a l p h a s e t r a n s i t i o n s t e m p e r a t u r e is a c o m m o n c o n t r o l p a r a m e t e r . In c o n v e c t i o n t h e t e m p e r a t u r e g r a d i e n t is an e q u a l l y c o m m o n c o n t r o l p a r a m e t e r . A s t h e vertical t e m p e r a t u r e g r a d i e n t ( A T ) of a l a y e r of liquid is i n c r e a s e d , c o n v e c t i o n sets in a b r u p t l y at a t h r e s h o l d ATe. F r o m t h e r e on t h e v e l o c i t y of t h e c o n v e c t i n g fluid (v) i n c r e a s e s a c c o r d i n g to v ~ ( A T - ATe) °.5. T h e velocity t h e r e f o r e p l a y s t h e role of an o r d e r parameter. T h e e x i s t e n c e of a t e m p e r a t u r e g r a d i e n t m e a n s that t h e r e is an e n e r g y flow t h r o u g h t h e system, a n d t h e p h e n o m e n o n is t h e r e f o r e that of a n o n e q u i l i b r i u m p h a s e t r a n s i t i o n . In e q u i l i b r i u m p h a s e t r a n s i t i o n s m u c h a t t e n t i o n is p r e s e n t l y b e i n g p a i d to multi-critical p h e n o m e n a , i.e. to s y s t e m s with c o m p e t i n g o r d e r p a r a m e t e r s a n d i n t e r a c t i o n s . In n o n e q u i l i b r i u m physics a corr e s p o n d i n g case is that of c o n v e c t i o n in a d o u b l e diffusive system. In such a case s t e a d y c o n v e c t i o n is p r e c e d e d by o s c i l l a t o r y (time p e r i o d i c ) conv e c t i o n [5], as shown s c h e m a t i c a l l y in fig. 1. W e shall in this p a p e r d e s c r i b e e x p e r i m e n t s a i m e d at u n d e r s t a n d i n g the o n s e t of the two t y p e s of flow a n d t h e r e g i m e of t h e i r c o e x i s t e n c e .
0378-4363/83/0000--4)000/$03.00 © 1983 N o r t h - H o l l a n d a n d Y a m a d a Science F o u n d a t i o n
T. Riste and K. Otnes / Neutron scattering and convection
or ~ r p,zmme~
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8
Fig. 1. Schematic picture of order parameter (velocity v, or
(v2)°~) versus Rayleigh number R, defined in the text. Superscripts o and s on Rc identify onsets of oscillatory and steady flow, respectively.
The physical mechanism for the oscillatory flow can be explained through the example of a layer of cold, fresh water on top of warm, salty water. If a droplet of salty water gets w a r m e r than its surrounding it will rise tO the fresh layer. Since salinity diffuses much slower than temperature, the droplet will rapidly be cooled but remain salty. The droplet, now heavier than its surrounding, will subsequently sink and overshoot its initial position, and the whole process repeats.
2. Theoretical survey
Consider an isotropic, pure liquid contained in a vessel and heated from below. T h e solution in the different flow regimes can be given in the space of the different p a r a m e t e r s R (Rayleigh number) and P (Prandti number) defined by
R =/3gI3AT/vK,
P = v/K
(1)
w h e r e / 3 is the volume thermal expansion, g the gravitational acceleration, 1 the vertical layer thickness of the liquid, A T the vertical temperature difference, u the kinematic viscosity and K the heat diffusivity. The calculated critical value, Re, for the onset of steady convection, for rigid boundaries and a high aspect (width-todepth) ratio, is 1708. In a brick-shaped, rectangular vessel and preference is for two-dimensional roils with axes parallel to the shorter horizontal side of the vessel. The effect of decreasing the aspect ratio is to reduce the
377
n u m b e r of rolls and to increase Rc [6]. P is expected to be of importance only for the higher instabilities. As mentioned above, the Landau theory predicts v oc ( R - Re) °5 for the velocity in the roll. Fluctuations are predicted to be of importance only within a narrow region ( R / R c - 1 ) < 10 7 and thus not likely to be seen [7, 8]. For an anisotropic liquid, i.e. a liquid crystal, eq. (1) is m o r e complicated. An effect is that ATe corresponding to Rc is reduced by a factor 10 3_ 10 -5 for a nematic liquid crystal. In this case molecular alignment is easily achieved by a magnetic field (H). For horizontal alignment steady convection is achieved when heating from below, for vectical alignment when heating from above. In both cases ,STc increases with increasing H. The width of the critical region is, for weak fields, predicted to be wider than for isotropic liquids, but still narrow ( < 10 4). G r a h a m [7] has given the following picture for fluctuations near Re: In the pretransitional region fluctuations involve transient convection cells of different sizes (or wave numbers, q). Near Rc there is a wave n u m b e r selection such that rolls of dimensions that adapt to the vessel b e c o m e dominant and survive the transition. These macroscopic, hydrodynamic fluctuations couple to inner, thermal fluctuations of the liquid. In liquid crystals orientational fluctuations of the relevant wave n u m b e r are abnormally abundant, which makes liquid crystals the best candidate for observing fluctuations near the convective instability. Orientational fluctuations are conveniently reduced by the application of an external magnetic field. In isotropic, pure liquids oscillatory convection only occurs at higher transitions on the route from laminar flow to turbulence, and is a nonlinear effect. In two-component or doublediffusive systems oscillatory convection precedes steady convection, see fig. 1, and appears already in linear theory. In this case the oscillatory flow corresponds to alternate clockwise and anticlockwise rotation of each convection roll. In an extensive series of papers da Costa, Knobloch, Proctor and Weiss [9] have treated the oscillatory and steady convection in a thermohaline
378
72 Riste and K. Otnes / Neutron scattering and convection
and in a metallic liquid, the latter in a vertical magnetic field. O u r system is a nematic liquid crystal in a vertical magnetic field, for which L e k k e r k e r k e r gave the first linear theory. H e predicts R~ to increase and R~ to decrease with increasing H. H e r e superscripts o and s d e n o t e oscillatory and steady flow, respectively. V e l a r d e et al. [5] have c o n j e c t u r e d that the transitions c o n v e r g e at a tricritical point. In their t r e a t m e n t the transition to steady convection is an inverted bifurcation, i.e. the o r d e r p a r a m e t e r exists also for R < RL as for a first-order phase transition. D a C o s t a et al. find that the bifurcation changes f r o m an inverted to a n o r m a l o n e as the height (aspect ratio) of the vessel is increased (decreased). T h e y also investigate the stability of the oscillations along the oscillatory branch, from its start at B~ to its termination on the steady branch. W h e n the latter is supercritical, i.e. R~ is a n o r m a l bifurcation, the oscillatory branch starts and terminates via a H o p f bifurcation. A w a y f r o m these bifurcations the stability of the oscillations gets p o o r e r , or m a y be completely lost, d e p e n d i n g on the m o d e l they choose. Period doubling and aperiodity m a y occur outside of the stable regime. T h e d e v e l o p m e n t f r o m oscillatory to steady flow is conveniently described by phase space trajectories, as used in Hamilton-Jakobi mechanics [11]. Let a and ci d e n o t e the velocity amplitude of a roll and its derivative. F o r small amplitude oscillations (at R = R~) the system point then traces out a closed orbit, a limit cycle, in the phase space of coordinates a and &. This is shown in fig. 2a, and is equivalent to the case of an action-angle orbit of a simple p e n d u l u m . For large amplitudes, R = R 2 ( > R1), the nonlinearity of the system will distort the trajectory, as in fig. 2b. In phase space steady flow is a fixed point on the abscissa. In one of the models used in the calculations by da C o s t a et al. the system point will at R R 3 ( > R2) start to pay visits at fixed points c o r r e s p o n d i n g to opposite rotations of a roll, as shown in fig. 2c. A t a still higher value R = R4 the system m a y start to show a preference for one fixed point, see fig. 2d. T h e orbit is n o w time asymmetric, reflecting a p r e f e r e n c e =
o~
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('b) d
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Fig. 2. Phase space trajectories showing possible evolution of oscillatory flow at increasing Rayleigh numbers, see text. for o n e sense of rotation. T h e trajectory drawn has a split orbit, c o r r e s p o n d i n g to period doubling. If the oscillations are unstable the system point never retraces the same orbit and m o v e s on a strange attractor. T h e three attractors m e n t i o n e d , a fixed point, a limit cycle and a strange attractor, are topologically different and separated by bifurcations [5, 12]. A transition f r o m a fixed point to a limit cycle is a H o p f bifurcation. Successive H o p f bifurcations m a y lead to a strange attractor. T h e papers q u o t e d on oscillatory flow m a k e little or no reference to phase transitions. In a recent p a p e r Sz6pfalusy and T61 [13] find that for a h a r d - m o d e H o p f bifurcation the amplitude and the f r e q u e n c y c h a n g e as "O - (R - Re) °5 and (w ~o0) - (R - Re), respectively.
3. Experiments A sample of fully d e u t e r a t e d parazoxyanisole ( P A A ) was contained in a brickshaped alum i n i u m vessel of dimensions (3 x 30 x 30) m m 3. Of the long dimensions one is parallel to the scattering vector and the o t h e r to a vertical magnetic field. A vertical t e m p e r a t u r e difference (AT) was o b t a i n e d by setting the difference of the p o w e r fed to electrical heating elements at the top and b o t t o m of the vessel. T h e temp e r a t u r e at the midheight of the aluminium side wall was kept constant at 121 ° C, at which temp e r a t u r e the sample is nematic and susceptible to
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alignment by a field > 2 5 Oe. With this arrangement AT stayed constant within +0.01 ° for hours. The neutron wavelength was 1.25,~ -~. o The scattered intensity was recorded at Q - 1.8 A, at the maximum of the first liquid diffraction peak. This intensity is very sensitive to changes of the molecular alignment. It is a maximum for vertical alignment (H±Q) and a minimum for HHQ, i.e. horizontal [14]. In the given geometry there is one convection roll. Using the whole convection roll as the scattering volume, the
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379
experiment is sensitive to the relevant wavelength range of hydrodynamic fluctuations. They show up as intensity fluctuations in realtime recordings of the intensity. The timeaveraged intensity gives the order parameter for steady flow (v) or for oscillatory flow ((v2)°5). It can be shown that the intensity (i.e. its deviation from that of the fully aligned, flow-free state) is proportional to the order parameter [15]. With a vertical magnetic field the deviation is negative, hence we shall in the following invert the intensity. Fig. 3 gives the time-averaged intensity as a function of A T for H = 200 Oe. We observe two order-parameter curves. They are not part of a hysteresis loop, since we can come from one to the other through several channels. We suggest that they represent crossing branches of oscillatory and steady convection, indicated by fulldrawn and broken curves, respectively. At a stronger field, see fig. 4, the curves have either merged, or one has been preferred, over most of the gradient range. Each point on the curves is an average of several hours of counting. Examples of real-time recordings of the intensity are seen in fig. 5. In addition to raw data we give curves of the autocorrelation function c(~-)= AI(t)AI(t+ ~-). Here A I denotes the deviation from the time average, and the averaging is taken over the whole time series, c0-) in most cases shows whether we have an oscillatory or a steady state, but when oscillations get irregular or chaotic the distinction is not clear. For H < 200 Oe the oscillatory branch is less disturbed by the presence of the steady branch, and fig. 6 shows data for the order parameter and the corresponding frequency of oscillating states for H = 160Oe. The frequencies have been obtained by Fourier transform of the data. We have also processed some of these data to obtain the trajectories in phase space. In order to obtain the coordinates in the plot we have to do some smoothing of the data. This we do by means of the Fourier spectrum of I(t), which ensures that we do not smooth away the relevant oscillations. In fig. 7 we see the effect of increasing AT, i . e . R . Only short time records were used, to avoid the effects of gradient drifting.
T. Riste and K. Otnes / Neutron scattering and convection
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Fig. 7. P h a s e t r a j e c t o r i e s c o r r e s p o n d i n g to p o i n t s at (o = 5 × III 3s 1 a n d 1 0 × 10 -3s t of fig. 6, s h o w i n g i n c r e a s i n g c o m p l e x i t y as c o n t r o l p a r a m e t e r is i n c r e a s e d .
T. Riste and K. Otnes
Neutron scattering and convection
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4. D i s c u s s i o n
F r o m o u r data it is evident that nonequilibrium phase transitions have o r d e r p a r a m e t e r curves that resemble those of equilibrium phase transitions, and n e u t r o n scattering is very well suited for measuring such curves. In the case of c o m p e t i n g o r d e r p a r a m e t e r s the interpretation of the curves is difficult, and the lines drawn in fig. 3 should be considered as tentative. T h e y suggest, however, an interference b e t w e e n states of different time symmetries. A t higher (fig. 4) and
381
lower applied fields the region of coexistence of o r d e r p a r a m e t e r s is narrower, and the interpretation easier. T h e identification of oscillatory states is easy when nonlinearities are weak. A t increasing control p a r a m e t e r (gradient) the oscillations b e c o m e spasmodic and almost invisible. A t the same time the f r e q u e n c y spectrum widens, as seen in fig. 6. A t h e o r y m e n t i o n e d a b o v e [13[, predicts a relation t o - " 9 2 between the frequency and the o r d e r p a r a m e t e r . If the oscillations b e h a v e as a soft m o d e in an equilibrium system, one would instead expect t o - " 9 [16]. T h e data of fig. 6 seem to f a v o u r the latter b e h a v i o u r in its initial part indicated by the line, suggesting a soft-mode H o p f bifurcation. T h e softening at increasing o r d e r p a r a m e t e r a n n o u n c e s the a p p r o a c h to steady flow. T h e transition f r o m the conductive to the oscillatory, convective state is, in a g r e e m e n t with the above, always f o u n d to be continuous. T h e oscillatory and steady convective states join in a m a n n e r that d e p e n d s on the applied magnetic field. In the range of coexisting order parameters, as e.g. in fig. 8, the junction is very similar to one of the possibilities suggested by da C o s t a et al. [9]. T h e width of the oscillatory flow regime decreases with increasing field, and it disappears a b o v e H - 250 Oe. T o g e t h e r with the threshold gradient this field possibly defines a tricritical point. T h e intensity m a x i m u m o b s e r v e d at the convection threshold for an incompletely filled cell, see fig. 8, looks very similar to critical scattering. T h e slope of the pretransitional tail corresponds to a critical index y = 1.5 + 0.3. T h e m a x i m u m is very narrow, - 0 . 1 °. O n its steep side we have not been able to clearly identify any point on the d o w n w a r d slope, which prevents us from drawing any definite conclusion as to the origin of the peak. If it is due to critical fluctuations, we must be able to explain why it is so much stronger in an incompletely filled cell. Invoking G r a h a m ' s picture [7], a plausible explanation could be the following: in a sample, whose aspect ratio is non-integral, fluctuations in the pretransitional region involve a wider range of wave n u m b e r s before the final q (i.e. cell dimension) is selected.
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T. Riste and K. Otnes / Neutron scattering and convection
The disappearance of the peak at H ~ 235 Oe may reflect the approach to a tricritical point and a reduction of the inner, orientational fluctuations at increasing field. Due to the steepness of the order parameter in the oscillatory regime, small changes (drifting) of the gradient give new phase trajectories. Fig. 7 shows, however, a growing complexity of the trajectory as the control (and order) parameter is increased. Our experiments have not answered all the questions raised in the theoretical survey. In particular we have not been able to give a coherent picture of the development of the phase diagram, and to explain the narrow stability range of the oscillations. Nevertheless the neutron method seems to offer some advantage over the heat-transport method used by the Orsay group [17]. Not only can we measure the order parameters better, but we can also measure spontaneous (i.e. not induced) oscillations. In retrospect we should have paid more attention to the geometry of the convection rolls. This is possible by tedious local intensity measurements [14]. Also we should have used shorter counting times in order to obtain more accurate information on the phase trajectories and their evolution as the control parameter is changed.
Acknowledgements We are much indebted to Professors Lekkerkerker and Velarde for valuable discussions.
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