Dissipative structures and non-equilibrium assisted self-assembly
Symposium organized by René Lefever and Guy Dewel, from 5 to 10 September 2000.
David ANDELMAN (Beverly Sackler Faculty of Exact Sciences, Tel-Aviv), Basile AUDOLY (Éсоlе normale supérieure, Paris), Paul CLAVIN (Université Aiх-Marseille-I), Patrick de KEPPER (C.N.R.S., Pessac), Guy DEWEL (Université libre de Bruxelles), Dominique DORMONT (C.E.A., Fontenay-aux-Roses), Nadia HASSANI (Université libre de Bruxelles), René LEFEVER (Université libre de Bruxelles), Stanislas LEIBLER (Princeton University, USA), Jean-Claude MICHEAU (Université Paul Sabatier, Toulouse), Alexander MIKHAŒLOV (Fritz-Haber Institut der Max-Planck Gesellschaft, Berlin), Stefan C. MULLER (Université de Magdeburg), Yves POMEAU (Université Paris-Sud), Didier ROUX (C.N.R.S., Pessac), Mustapha TLIDI (Université Libre de Bruxelles), Soersen TOXVAERD (Orsted Institute, Copenhagen), Qui IRAN-CONG (Kyoto Institute of Technology, Japon), Tomohiko YAMAGUCHI (N.I.M.C., Japan).
Two main ordering mechаnisms are distinguished in physics: the first one, often called self-assembly, governs the formation of mesoscopic structures via equilibrium phase transitions; the second one, generally called self-organization, produces macroscopic (dissipative) structures via non-equilibrium instabilities.
The difference between these mechanisms concerns the non-linearities involved and the role of energy dissipation. Self-assembly requires molecular interactions (hydrophobic and/or ionic interactions, hydrogen bonds, Van der Waals forces) but no energy dissipation. It is observed in conservative systems, the evolution of which tends to minimize a free energy functional. Self-organization, on the contrary, is associated with non-conservative, irreversible processes (chemistry, diffusion, heat transfers, viscous flows) which need to dissipate energy to produce so called dissipative structures. Because of this profound difference, studies concerning equilibrium and of non-equilibrium ordering phenomena have, by and large, evolved independently and formed separate research domains. The progresses accomplished in both domains have, however, brought into evidence point of views, methods and concepts, the convergence of which indicates a need for more dialogue, more unified and more pluridisсiplinary oriented approaches. The meeting was conceived and held within this perspective. Its principal topics were the following:
Surprisingly diverse equilibrium and non-equilibrium systems, belonging to widely separated areas of physics and chemistry, exhibit supramolecular organizations forming domain shapes and patterns, which possess similar morphological properties. The phenomenology underlying these behaviors was reviewed by D. Andelman. Discussing the formation of equilibrium modulational phases in two-and three-dimensional systems, the point of view he emphasized attributes their occurrence to competing interactions which stabilize states, characterized by periodic spatial variations of the pertinent order parameter. Examples reviewed in detail were magnetic garnet films and block copolymers. To a less extent, ferrofluids and organic monolayers on water/air interface (Langmuir rnonolayers) have also been mentioned. In magnetic systems, the magnetic wall energy competes with demagnetizing fields, causing a thin magnetic film to break spontaneously into spin « up », and spin « down » domains. In block copolymers, a phenomenon of microphase separation is known. Instead of the usual microphase separation occurring in incompatible binary mixtures of polymers, here a microphase separation takes place, because the two incompatible blocks are linked together on each polymer chain.
All these systems have in common a selection of a dominant, spatial, non-zero q-mode at thermodynamical equilibrium. As a function of system parameters, it is possible to construct a phase diagram showing phase transitions between different modulated phases: lamellar phases composed of one-dimensional sheets; hexagonal phases composed of close packed cylinders and cubic phases composed of spheres packed with cubic symmetry.
Chemical bistability, which has been described in isothermal open flow reactors since the late sixties, represents the simplest non-equilibrium order-disorder transition of chemical origin. At the meeting, J-С. Micheau reviewed its general phenomenology and reported the first example of а bistable kinetic behavior observed in an organic two-phase oil/water reacting system, namely, in the biphasic liquid-liquid alcaline hydrolysis of С-4 to С-8 ethyl alkanoates. Theoretical kinetic models of isomerization reactions exhibiting bistable behaviors were presented by S. Toxvaerd. Не discussed their relevance for explaining the emergence of biomolecular chirality and showed how molecular dynamics simulation methods, taking molecular interactions as well as Chemistry into account, could be used to study the kinetic regimes which such systems display during their temporal evolution.
(ii) Turing instability and related patterns
The non-equilibrium analogue of the modulational instability is the Turing instability. Best-known in reaction-diffusion systems, the latter has also been found in many other non-equilibrium systems, notably, in non-linear optical systems. In the language of Chemistry, the Turing instability arises when short range « autocatalytic » effects amplify spatial inhomogeneities loсally, but are unau e to trigger their spreading, due to the existence of counteracting long range « inhibitory » influences.
The development of open reactors enabling to control chemical systems distance with respect to thermodynamic equilibrium, strongly stimulates the experimental study of this instability. The core of these reactors consists of a hydrogel disc diffusively fed by flow-through reservoirs of initial reagents. The feed-geometries adopted by De Kepper are of two types: (a) two-side-fed reactors where the opposing sides of the gel are in contact with different subsets of reactants; (b) one-side-fed reactors where the reduced dimensionality of the reacting medium allows a good control of the patterns dimensionalitу itself. In the case of the chlorite-iodidemalonic acid reaction and of the chlorine dioxide-iodine-malonic acid reaction, the patterns observed with such reactors are in good qualitative (in some times even quantitative) agreement with the predictions of theoretical models and simulations.
Generally however, quantitative comparisons between experiment and thеory remain difficult in the case of reaction-diffusion systems. First, because experimental boundary conditions are delicate to model; second, because the huge algebraic complеxity of realistic chemical models entails demanding computational task. The modulational instabilities encountered in so called transverse non-linear optics, where the spatial coupling is provided by diffraction instead of diffusion, seem better suited for that purpose. Indeed, their control parameter, i.e. light, is easy to manipulate and realistic phenomenological models are much simpler than those proposed in chemistry. M. Tlidi illustrated this in discussing the occurrence of localized structures in a passive diffractive resonator: an optical cavity with plane mirrors filled by resonant two-level atoms without population inversion, and driven by a coherent plane-wave injected signal. Within the framework of a purely absorptive limit, he established that, depending upon the value of the wave number characterizing a modulational instability, two kinds of localized structures may appear, corresponding either to circular localized bright spots or to a localized single stripe. The analysis also revealed the existence of a « curvature instability » that may destroy the single localized stripe.
(iii) Influence of nonequilibrium constraints on self-assembly – Coupling of hydrodynamics and chemistry
There exists a growing research effort concerning the influence of non-equilibrium constraints (temperature gradients, concentration gradients, hydrodynamic flow, shear effects, chemical reactions, electro-magnetic fields) on the properties of phase transitions. Under the label non-equilibrium assisted self-assembly, it is essentially to this broad, very active, research domain that the meeting’s title refers. Its development is stimulated by needs coming from various horizons. Among them is the necessity to gain deeper physical insight into many ordering phenomena encountered in biology; materials science and engineering also provide powerful motivations. Preoccupations there are economical as well as fundamental; thеу are associated with the elaboration of incessantly more complex industrial processes, in which equilibrium and non-equilibrium effects are intimately coupled and need to be tightly controlled.
Exemplifying the equilibrium and flow properties of surfactant in solutions, D. Roux reported on the effect of shear on lyotropic lamellar phases. A new hydrodynamic instability has been observed in these systems, which Ieads to a phase of multi-lamellar vesicles (so called onions) compactly, packed in space.
A lyotropic lamellar phase is made of surfactant and water, and some time an additional hydrophobic component (oil). The symmetry of the phase is that of a smectic A phase. It is composed of a stack of surfactant bilayers, separated by water. Upon dilution with extra water, two main behaviors can be described: the dilution of the lamellar phase is limited, either by a phase transition with an isotropic liquid phase (micellar phase or sponge phase) or by a phase transition with another liquid crystalline phase. In order to understand the effect of flow on such phases, using rheophysics methods, the structure of lyotropic phases submitted to a pure shear was studied. Using a number of structural techniques under shear, such as scattering techniques (light scattering, neutron, x-ray) or dielectric measurements, it was possible to represent shear effects into a diagram, which can be considered as an out-of-equilibrium generalization of equilibrium phase diagrams. It describes the effect of shear as a succession of stationary states of orientation, separated by dynamic transitions. Indeed, while always staying in the one phase – thermodynamically, stable lamellar phase – the phase experiences a series of transitions modifying the orientation of the lamellae in space, respecting the direction of the shear. These different orientations correspond to an organization of spatial defects, which is different from one orientation to the other. It is a modification of what is named the « texture » of the phase. Consequently, it is not a phase transition but has to be considered as an instability. It is, however, different from the classical hydrodynamic instabilities observed in fluids submitted to shear, because it does not involve a length scale directly related to the size of the shear cells, but rather some microscopic length scale related to the intrinsic properties of the fluid. At very low shear rate, the phase is more or less oriented with the membrane parallel to the velocity however defects remain in the velocity direction, as in the vorticity direction.
At high shear rate the orientation is basically similar, but the defects in the velocity direction have disappeared. In the intermediate regime, a new and interesting orientation appears. The membranes are broken into pieces by the flow and the phase organizes itself into a phase of multi-lamellar vesicles, all of the same size. These vesicles have been named: « onions » because of their multilamellar structure. Onions fill up space and their size is a function of the shear rate. Mani other surfactant systems have now been studied and exhibit the same kind of behavior.
Нydrodynamic is associated with most classical works on nonequilibrium stability thеоry. The aspect especially emphasized in the meeting is the coupling of hуdrodynamic flows with Chemistry. G. Dewel discussed the dynamic properties of open packed bed tubular reactors, where fresh reactants are continuously supplied by feeding flows, maintaining a chemical non-equilibrium state. He showed that the concepts of convective and absolute instabilities must be used, in order to properly describe the development of spatio-temporal structures in systems where fluid particles are advected through downstream flow boundaries. Remarkablу, when the reference state is convectively unstable, localized perturbations are driven by the mean flow in such a waу that they grow in the moving reference frame, but dесау at any fixed location. In the absolute instability regime, on the contrary, localized perturbations grow at any fixed station. The effects of a simple plug flow on chemical instabilities breaking respectively the time (Hopf) and space (Turing), translation symmetry were also discussed. It was shown that the advection-spreading balance of localized perturbations provides a way to control the structures that can develop in a reacting flow Convectively unstable states are highly sensitive to small boundary perturbations; the latter can play the role of non-linear filter and noise amplifier. In this framework, the disсovery of a new pattern forming mechanism has been reported: a permanent, localized perturbation affecting a state that is convectively unstable with respect to a Hopf bifurcation, can generate a stationary space periodic chemical structure. Since it does not require a differential diffusion process, this mechanism is to be contrasted with the Turing instability: These theoretical prediction have been experimentally verified in the case of the ferroin-catalyzed Belousov-Zhabotinsky reaction in a tubular reactor. Numerical integrations of the Rovinsky-Zhabotinsky model describing this reaction are in good agreement with the experimental observations. This mechanism has now been observed in other systems, including exothermic reactions.
A different kind of chemical instability, that is induced by a differential flow of the key species, was also presented. The primary role of this differential flow is to spatially disengage the antagonists species (the activator and inhibitor) and to allow the generation of travelling waves. Since the conditions for which the differential flow induced instability arises are not very restrictive, it can be speculated that it could play an important role not only in chemistry, but also in physics and biology.
The coupling of chemistry with hydrodynamic flows was further investigated by P. Clavin, in connection with the genesis of cellular flames; and by B. Audoly, in connection with the mechanism of flame propagation in the presence of advection. The question whether hydrodynamic flows and/or mechanical deformations play a role in the variety of patterns (chevron, honeycomb, hexagonal. .. ) which can be observed when the methylene blue-oxygen reaction is embedded in a polyacrylamide gel, was discussed by S. Müller.
(iv) Cross–control of phase transitions and chemistry
Structural lability is characteristic of soft matter systems exhibiting weakly attractive molecular interactions. These condensed systems can exist in many different equilibrium states and easily undergo phase transitions. A. Mikhailov studied the non-equilibrium structures, which mаy result from the interplay between these structural phase transitions, chemical reactions and diffusion. Generally, the size of such a structure, which is determined by the range of molecules attractive interactions, is vert’ small; it can be a fraction of nanometer.
Non-organic reactions occurring in molecular monolayers adsorbed on metallic surfaces provide good models for studying the formation of non-equilibrium nanostructures. Such systems have the advantage of obeying chemical reaction schemes and physical interactions, which are relatively simple. Moreovеr, fast scanning tunneling microscopy (STM) can be used to monitor surface reactions in real time, with atomic spatial resolution. In the talk, the properties of stationary and traveling nanoscale structures in surface chemical reactions have been theoretically discussed. Special attention has been paid to stationary localized structures, which behave as self-organized chemical nanoreactors.
On the other hand, Q. Tran Cong demonstrated that the morphology of mesoscopic modulated phases, appearing in polymer blends undergoing phase separation, can be selectively controlled by photo-chemical cross-linking reactions The chemical reactions used for that purpose are the photodimerization of anthracene and the photoisomerization of trans stilbene. Notably, Tran Cong showed that: (1) stationary structure with various length scales can be obtained by varying the irradiation conditions used to chemiсally « freeze » the phase separation of polymer mixtures; (2) anisotropic phase separation of polymer mixtures can be generated by photo-crosslinking with linearly polarized light. This tends to suppress concentration variations in the direction parallel to that of the polarized light and leads, as a result, to the formation of lamellae perpendicular to the light polarization; (3) temporal modulation of the irradiation with frequencies in the 1Hz to 200Hz range, produces morphologies, the characteristic lengths of which have a much narrower band width than those obtained under constant irradiation. There exists an optimal frequency at which this band width passes through a minimum.
R. Lefever discussed the thermodynamics and kinetic conditions under which chemical reactions prevent the coarsening terminating spinodal decomposition and freeze the unmoving of binary mixtures at some early, pattern forming, stage of evolution. Under very general conditions, it was established that:
(1) pattern freezing phenomena can only occur in non-equilibrium systems, the level of dissipation of which exceeds a finite, non zero threshold value; (2) at least two independent chemical processes must take place; (3) chemistry must be destabilizing, which requires that at least one of these processes must be autocatalytic; (4) pattern formation is possible even outside of the spinodal region, i.e. without involving a phase separation phenomenon, in unsymmetrical mixtures where the potential energies between pairs of identical particles are sufficiently different. This latter condition replaces for non ideal chemically reacting binary mixtures, the unequal diffusion coefficients condition which governs the appearance of Turing patterns in the classical reaction-diffusion theory.
The response of soft matter and of soft mechanical systems to external stimulations is often reminiscent of living matter. Smart gels are examples of made to order materials, with novel combinations of properties. The trick in engineering such gels is to balance on a knife-edge, the forces controlling the expansion and contraction of the gel. This confers on them the ability to undergo tremendous volume changes via a phase transition, which small external stimuli may trigger. Because of this phenomenon, smart gels are often viewed as potential material for artificial muscle. Many fundamental physical and engineering hurdles remain to be surmounted in this quest. Progresses in this direction, however, exist as the oscillating gels studied by T. Yamagushi show. More precisely, a hydrogel made of the copolymer of N-isopropylacrylamide and ruthenium tris (2,2′-bipyridine) exhibits autonomous swelling-deswelling oscillations in а solution containing reactants of the Belousov-Zhabotinsky reaction. Here, the ruthenium complex works both as а catalyst of the BZ reaction aid as а transducer of chemical oscillations into volume oscillations. When а piece of gel is used whose size is small enough compared to the wavelength of traveling BZ patterns, the gel exhibits large amplitude oscillations up to 20% in volume. Yamagushi also presented an interesting example of non-equilibrium assisted self-assembly observed when silver salts of fatty acid are decomposed by heating to obtain silver nano-particles.
(v) Miscellaneous questions and challenging perspectives
Yves Pomeau devoted his talk to the equilibrium problem of self-gravitating systems, which he recently worked on in collaboration with Marie-Line Chabanol and Francis Corson. The Universe has long been known to show striking, large scale structures like galaxies and globular clusters within, that are held by gravitational forces. It is tempting to explain these structures as the result of the tendency to equilibrium of systems of point masses (the stars), interacting by gravitation, and more specifically to apply some version of the Gibbs-Boltzmann statistical theory. Unfortunately no equilibrium is possible in these systems: a detailed study of a well defined example shows that, for few point masses in a trap, the tendency to equilibrium is replaced by an evolution going on for ever and amounting to fill a phase space of infinite volume by some kind of diffusion. The extension of this idea to the many (N large) body problem, relevant for astrophysics, remains a major challenge: the tendency to aggregation shown in globular clusters should be an intermediate time trend, increasing the entropy by increasing the volume of phase space of particles on free orbits, although the masses in the cluster go to a more dеeply bound state.
The final result not yet reached by the observed globular cluster, being a collapse of a few bodies at the center. Evidently, several kinds of ordering phenomena analyzed in the meeting could participate at different levels in the redoubtably complex organizations of the living world. The challenges posed, and the perspectives offered, by naturally self-organizing states of matter are immense. We had some flavor of what physico-chemists are facing there, thanks to the brilliant talks given by D. Dormont, concerning the knowns and unknowns of prion diseases and by S. Leibler concerning microtubules and the sensitivity of cells temporal rhythms to thermal and/or external noises.