Faraday Discuss. Chern. SOC., 1984, 77, 18 1 - 188 Time-dependent Behaviour and Regularity of Dissipative Structures of Interfacial Dynamic Instabilities BY HARTMUT LINDE Akademie der Wissenschaften der DDR, Zentralinstitut fur physikalische Chemie, Rudower Chaussee 5, 1199 Berlin, German Democratic Republic Received 5 th December, 1983 Interfacial dynamic instabilities with self-amplifying and self-organizing convections driven by interfacial tension in a non-equilibrium two-phase fluid system show a surprising variety of dissipative structures. A complete investigation and description of them has to take into consideration at least four aspects. (1) First are the kinetic features of the convection behaviour and the deformation of the interface concerning stationary basic units of flow systems (b.u.f.) with the related deformations of the interface and different time-dependent behaviour (travelling quasi-stationary b.u.f. driven by long-range driving forces, relaxing oscillations with related autowave behaviour and classical mechanical waves).(2) The topological features include different spatial patterns, e.g. parallel, concentrically circular or spiral stripes, polygonal networks and hierarchically ordered structures. (3) The order- disorder features concern the size, shape and packing of the b.u.f. with respect to spatial regularity or spatial chaos. The time-dependent behaviour can be distinguished for harmonic, anharmonic and chaotic oscillations. (4) The driving faces (d.f.) and conditions for Marangoni instability I are heat-and/or mass-transfer and/or chemical reaction at the fluid interface.The resulting b.u.f., an interface-renewing flow, can behave ( a ) as stationary or quasistationary in travelling substructures, ( b ) as relaxing oscillations, e.g. travelling or spiral-shaped autowaves and ( c ) as classical longitudinal capillary waves. Marangoni instability 11, with the same d.f. as instability I, induces in thin-layer amplification of the differences in the thickness of the layer. Marangoni instability 111, with shear stress as the d.f. at a tenside- covered fluid interface, shows stationary or oscillatory hair needle-like or elliptical eddies jn the plane of the interface itself. Meniscus instability, resulting from the viscous pressure as the d.f. at a travelling meniscus, is an excellent example of amplification as well as of stationary spatial deformations of the meniscus-shaped interface in a determinate way and travelling spatial deformations of substructures, which are caused by a repeated stochastic process.~~ One of the most manyfold spectra of dissipative structures (d.~.)'-~ is caused by interfacial dynamic instabilities for the driving forces of heat- and/or mass-transfer and/or chemical reaction and/or shear stress in two-phase systems with fluid interfaces, if the systems have an internal feedback mechanism and exceed critical conditions. Stability theories use the Navier-Stokes equations, the laws governing the transport of matter and heat and some additional boundary conditions with respect to the usual hydrodynamics.The Gibbs-Marangoni effect is expressed by the stress balance at the interface where ( d u l a x ) , , is the derivative of tangential velocity in a direction normal to the interface in phase a or 6 (x = 0), pa& is the dynamic viscosity in phase a or 6, cr is the interfacial tension and x, y are the Cartesian coordinates perpendicular and 181182 DISSIPATIVE STRUCTURES OF MARANGONI INSTABILITIES tangential to the flat interface. Eqn (1) shows that a fluid interface responds to the difference in the interfacial tension, which is due to local adsorption of surface active agents (tensides) or to local heating, by interfacial convection from areas of lower tension to areas of higher tension, inducing the related shear stress p(du/dx) in both phases.The inversion of action and reaction can be observed in a system with shear stress, which leads to a difference in the interfacial tension at a tenside- covered interface. Secondly, we have to take into consideration the reaction of the interface to a space-dependent viscous pressure difference in both phases at the interface. For an originally flat interface the normal stress (pressure) balance at the interface is for x=O, where Pa& is the hydrostatic pressure at the interface in phase a or b, (du/dX),b is the derivative of the velocity in a direction normal to the interface in phase a or b (x = 0) and r l , r2 are the principal curvatures of the slightly deformed interface. The viscous pressure causes a deformation ( r , and r2 have finite values) even if the interfacial tension i s not space dependent [contrary to the Gibbs- Marangoni effect, eqn (l)].At least four different features play an important role in the investigations and therefore also in the description of the d.s. of interfacial dynamics: ( a ) kinetics, ( b ) topology, (c) order-disorder transitions and f d ) driving forces. KINETIC FEATURES OF THE CONVECTION AND DEFORMATION BEHAVIOUR The stationary state of the basic unit of flow (b.u.f.), shown in fig. 1, is due to interfacial convection enclosing circulating flows in both phases. Periodic reamplifi- cation and breakdown of the b.u.f. of interfacial convection (relaxation oscillation of the intensity) occurs. Travelling b.u.f. convection systems originate at a leading line or centre4 and behave as autowaves (no reflection at a wall, no interference but annihilation of colliding wavefronts).The periodic tangential or normal deforma- tions of the interface (with related convections of the adherent fluid layers) behave as classical travelling or standing waves with reflection and interference. Deforma- tions occur in the shape of the interface, which can be stationary or autowave-like travelling along the interface. To distinguish classical waves and autowaves, note that classical elastic waves of small amplitude can be described by Helmholz’s wave equation (a hyperbolic differential equation, invariant with time inversion): aLx a t 2 -= c2 AX where A is the Laplace operator and c is the velocity of the waves. For longitudinal waves, x is the characteristic parameter of the scalar field, e.g.density in sound waves or the concentration of surface-active agents, c,.,,,., at the interface in longi- tudinal (capillary) waves. Autowaves can be described by a system of parabolic differential equations (variant with time inversion) i = l , ..., n in which non-linear functions Fi(xi) are necessary, and where, in chemical systems, xi are the volume concentrations of autocatalysing or inhibiting chemical species,H. LINDE 183 Fig. 1. Basic unit of flow. Dj are the diffusion coefficients of the chemical species. xi, in interfacial dynamic systems, are the concentrations of surface-active agents, c ~ , ~ , ~ , , at the interface. Note that in the same interfacial system both classical elastic (capillary) waves and autowaves are possible, with variation of the direction of mass- or heat-transfer, respectively, of d a d a or - d Cs.a.a.d T' TOPOLOGICAL FEATURES OF TWO-DIMENSIONAL IMAGES OF BASIC UNITS OF FLOW OR OF INTERFACIAL DEFORMATION Straightly parallel stripes are sometimes interrupted by splitting or by unification of stripes [fig. 2( a ) ] or by the free ends of newly formed stripes [fig. 2( b ) ] . Concentri- cally circular [fig. 2(c)] or irregular parallel curvaceous stripes can be stationary or travelling. Spirals may exist with one [fig. 2 ( d ) ] or more arms [fig. 2(e)] moving radially whilst the centre is rotating. Polygonal networks of stationary or travelling roll cells also occur [fig. 2 ( f ) ] , Structures of higher order (structure hierarchy by substructuration) occur if two or more of the basic structures expanding from different leading centres col!ide.The collision areas form new structural elements of higher order [fig. 2(g)]. Finally, there may be circular, elliptical [fig. 2(h)] or hair needle-like [fig. 2(i)] eddies in the plane of the interface itself, i.e. another convection system rather than the above-mentioned b.u.f. ORDER-DISORDER FEATURES Order-disorder features consist of ( a ) spatial structures characterized by one or more wavelengths, which can be highly ordered with single- or multi-peak184 DISSIPATIVE STRUCTURES OF MARANGONI INSTABILITIES Fig. 2. ( a ) Splitting and unification of stripes; ( b ) free ends of newly formed stripes; ( c ) concentrically circular stripes; ( d ) spiral stripes with one arm; ( e ) spiral stripes with two arms ; (j) polygonal networks (of roll cells); (8) structure hierarchy by substructuration; ( h ) elliptical eddies in the plane of the interface; (i) hair needle-like eddies in the plane of the interface.H.LINDE 185 wavelengths distributions or disordered with wide distributions of wavelengths or sizes, ( b ) spatial structures characterized by the shapes of the topological features, ( c ) high-order systems with polygonal networks of regularly shaped and regularly packed stripes or polygons or disordered structures with irregularly shaped polygons and therefore also with irregularities in their packing, ( d ) time-dependent behaviour of oscillations: harmonic oscillations, or anharmonic oscillations, e.g. sawtooth oscillations or chaotic oscillations.DRIVING FORCES, CONDITIONS AND ESSENTIAL BEHAVIOUR DISSIPATIVE STRUCTURES OF MARANGONI INSTABILITY I Heat- and/ or mass-transfer of surface-active agents leads to Marangoni instability I and spontaneous interfacial convection. Both the stationary regime and classical wave-like oscillation (longitudinal capillary waves) are predicted from linear stability theory’ and observed in real physical systems.”* From an analysis including the thickness of the adjoining fluid layers6 conditions for the critical Marangoni numbers of both basic regimes are predicted; the influence of additional exchange of latent heat on the Marangoni numbers and on the kind of regime has also been analysed recently.’ In reality the ‘stationary’ regime is more complicated than its theoretical descrip- tion: i.e.the ‘stationary’ regime is expected to be time-independent like fig. 1 or fig. 2 ( f ) and to exhibit order-disorder features with a sharp single peak. At the lowest level of instability, roll cylinders or roll cells of the theoretically predicted smallest size show relaxation oscillation, which is the reason for the recently recognized autowaves with travelling parallel stripes, concentrically circular waves and even spiral leading centres. These autowaves, like autowaves in autocatalytic reaction4 systems, are expected to be propagated by a local driving force along the interface. Whether these travelling autowaves can form a stationary stable structure of higher order because of the structure of the collision area is, even in the case of travelling autowaves of the Belousov-Zhabotinski r e a ~ t i o n , ~ unknown.The next highest level of instability allows roll cylinders or polygonal cells as a network in stationary convection systems, which remain in place, especially in thin liquid layers. With a further increase in roll cell instability, characterized by an increase in the Marangoni number, convection systems again show travelling waves but with higher velocity than the above-mentioned pure autowaves. From the point of view of origination at a leading area’ and annihilation at a wall or at a collision area, they behave like autowaves; but their greater velocity is caused by a long-range interfacial tension gradient from a second-order convection system.Roll cells of first, second and higher order are observed to form a hierarchical system. An approach using non-linear terms’ was able to show the existence of second-order roll cells. From this behaviour we find selection of different kinetic regimes with different topological features: e.g. hexagonal patterns are preferred for stationary structures and tetragons ; ladder structures with different diameters between rungs and beams are preferred for travelling systems, if the movement is caused by hydrodynamic (or interfacial dynamic) shear stress. High velocity and high shear stress can cause chevron (herring-bone) patterns to be formed by deformation of the ladder structure. With increasing driving force the irregularities increase and lead to chaotic distributions of sizes and roll cells of irregular shapes: i.e.the first transition to chaotic behaviour in Marangoni instability.186 DISSIPATIVE STRUCTURES OF MARANGONI INSTABILITIES Roll cells of second and higher order can degenerate by relaxation oscillation, giving a wide range of diameters of chaoticaly spreading and oscillating cells (diameters decreasing and frequencies increasing with increasing driving force). Autowaves of roll cells of higher order have also been observed. The chaotic interaction of an ensemble of these stationary roll cells is characterized by a fast annihilation of every roll cell after or during the first period of oscillation by the consecutively very often nucleating adjoining cells. The chaotic features of spatial and time-dependent behaviour may be due to the stochastic behaviour of nucleation resulting in irregular anharmonic oscillations, which are not synchronized.There is a strong contrast between the three kinetic features of the 'stationary' regime (real stationary roll cells, structure hierarchy by travelling substructures and relaxation oscillations, which lead on the one hand to autowave behaviour and on the other to chaotic oscillations) and the classical oscillatory regime of the Marangoni instability. The latter is identical to undamped longitudinal capillary waves and can occur for travelling or standing classical waves with reflection at a wall and with interference. For lower supercritical driving forces, the regularity is very high and the topological features correspond to fig.2(a) or ( b ) . We observe with increasing driving force a transition from travelling straight waves (stripes) to standing waves (with mutually penetrating stripes) and from these one-modal waves to two-modal and even three-modal waves [with angles between the corresponding parallel stripes of 90" (two modal) or 60" (three modal)]. For very high driving forces the irregularities increase again and lead to chaotic behaviour. The experimental results are in good agreement with the predictions of stability and with the more detailed description obtained by computer simulation.'' The classical wave can be amplified as mentioned before by the driving forces caused by heat- and/ or mass-transfer and/ or chemical reaction at the interface: i.e. autowaves and classical waves are distinguished theoretically by the above- mentioned differential equations and the related kinetic behaviour concerning reflec- tion and interference and not by the kind of energy supplied by local sources at the interface or by the entire interface itself.MARANGONI INSTABILITY I1 ' Marangoni instability I1 results from the same driving forces as Marangoni instability I, but is connected with increases in the thickness differences in thin layers, i.e. increases in deformations normal to the interface. Thin layers, jets and droplets can be destabilized and broken into smaller compartments. MARANGONI INSTABILITY I I I ~ * - ' ~ Marangoni instability 111 results from shear stress in the liquid and/or the gas phase at a tenside-covered liquid/gas interface and is characterized by stationary (with small driving force, e.g.in systems where flow in the gas phase is the only cause of this dissipative structure) or oscillatory (with higher driving force) hair needle-like [fig. 2 ( i ) ] or elliptical eddiesi4 [fig. 2(h)] in the plane of the interface itself. This means that there is no surface renewal convection in this pattern of shear flow on the surface (which is, of course, accomplished by related convection in both boundary layers). As with the other dissipative structures, this instability occurs only if we exceed critical conditions, which are lower than the values of the critical conditions for transition from laminar to turbulent flow at a solid wall.direction - of f l o w Plate 1. ( a ) Moving meniscus between two plates; ( h ) first-order deformation of a moving meniscus; ( c ) second-order deformation of a moving meniscus; ( d ) fixed consecutive structure of first-order, X, and second-order, 0, meniscus deformations.[To face page 187H. LINDE 187 MEN I SC U S I NSTA B I LITY ’ This last example of surface instabilities results from the viscous pressure being the driving force, which operates at a travelling meniscus [plate l(a)] and is separate from the above-mentioned Marangoni instabilities. This condition obtains if a Newtonian or non-Newtonian liquid in the gap between the two plates is displaced by the penetration of air, forming a moving half-cylinder-shaped meniscus. The first regime of meniscus instability is characterized by a stationary deforma- tion of the meniscus in the shape of a waveline in the z direction [plate l(b)].The fixed consecutive structure is a system of equidistant stripes of smaller and larger thickness in both layers, remaining at the surface of both plates [fig. 2(a) and (b)], and whilst the ‘air fingers’ produce thin stripe-like layers the ‘liquid fingers’ produce thick layers stripe-like layers [plate l(b)]. The distance between these stripes (to avoid the expression wavelength, because we have to distinguish between classical waves and a variety of other wave-like phenomena) depends strictly on the viscosity, the velocity of the moving meniscus and the distance between the two plates (the radius of meniscus), so we observe regular parallel stripes (at small driving force) and a transition to a stripe system with smaller or larger distances if we change the plate distance during the experiment.16 The transition to narrower stripes is caused by origination of new stripes of fig.2(b), and the opposite transition is caused by the unification of stripes, fig. 2(a), i.e. in the first case there are new liquid fingers and in the second case there is unification of two liquid fingers [plate l(c)]. This instability is a good example of a chance process, if with increasing driving force by a second bifurcation additional deformations (of second order) [plate l(c) and ( d ) ] with smaller characteristic length than the first-order deformation of the meniscus occur. The second-order deformations originate at the top of the air fingers as small liquid fingers, and they travel to the base of the air finger and then disappear.The nucleation of this second-order deformation is the cause of its chance mechan- ism: from both its timing and its direction of travel the process has an event distribution of probability. Thus we can recognize another basic mechanism causing chaotic behaviour, which follows the condition ‘sensitive to initial conditions’.’’ The resulting fixed consecutive structure shows relatively regular first-order stripes and the irregular branches of the second-order stripes [plate l(d)]. ‘ H. Linde, Marangoni Instabilities, in Dynamics and Instability of Fluid interfaces, Lecture Notes in Physics no. I US, ed. T. S. Sorensen (Springer-Verlag, Berlin, 1979). * H. Linde, in Conuective Transport and Instability Phenomena, ed. J. Zierep and H. Oertel Jr (G. Braun-Verlag, Karlsruhe, 1982). H. Linde, Dissipative Strukturen der Grenzflachendynamik, in Fortschritte der experimentellen und theoretischen Biophysik Bd. 21, ed. E. Kahrig and H. Beaerdich (VEB Georg Thieme-Verlag, Leipzig, 1977). A. M. Zhabotinsky and A. N. Zaikin, J. Theor. Biol., 1973, 40, 45. L. E. Scriven and C . V. Sternling, AIChE J., 1959, 5, 514. H. Linde and J. Reichenbach, J. Colloid Interface Sci., 1981, 84, 433. G. Frenzel and H. Linde, Teor. Om. Khim. Tekhnol., 1983/84, in press. A. N. Zaikin and A. M. Zhabotinsky, Nature (London), 1970, 225, 535; A. N. Zaikin and A. L. Kawczynski, J. Non-equilib. Thermodyn., 1977, 2, 39. H. Wilke, Chem. Tech. (Leipzig), 1974, 26, 456; 1977, deposition system: Z. Angew. Math. Mech. (ZAMM), 1980, 9, 437. ‘ I C . Arcuri and D. W. De Bruijne, Roc. Vth Int. Congr. Surface Active Substances, Barcelona, 1968. ’ H. Linde and P. Schwartz, Chem. Tech. (Leipzig), 1974, 26, 455; 1977, deposition system. 10188 DISSIPATIVE STRUCTURES OF MARANGONI INSTABILITIES H. Linde and P. Friese, 2. Phys. Chem. (Leipzig), 1971, 247, 225. l 3 H. Linde and N. Shulewa, Mber. Dtsch. Akad. Wiss. Berlin, 1970, 12, 883. l4 H. Linde and P. Schwartz, Teor. Osn. Khim. TekhnoL, 1971,401. I’ L. Weh, Dissertation (Humboldt-Universitat, Berlin, 1972). H. Linde, Nova Acta Leopold., 1984, in press. J. A. Yorke and E. D. Yorke, in Hydrodynamic Instabilities and rhe Transition to Turbulence (Springer-Verlag, Berlin, 198 1).