Alexandru Serbanescu
email: alexe1serbanescu@yahoo.com

Adrian Diaconu

Cornel Ioana

Angela Digulescu
email: angela.digulescu@gipsa-lab.grenoble-inp.frcornel.ioana@gipsa-lab.grenoble-inp.fr

Between Two Engineering Ages: Of Information And Complex Systems, Profesional Research Is Based On A Team Work!

Keywords: Nonlinear complex systems, hinking in systems, cryptography based on chaos, signal processing based on recurrence plot ananlysis

From twenty years we promote the idea that engineering's systems (and not only!) are nonlinear ones and usually can have chaotically behaviors and their associated processes need special studies' methods. In this paper we present two directions from our research: cryptography based on chaos and a new method to analyze signals from nonlinear dynamical systems.

In its most general sense, engineering is turning an idea into a reality, creating and using tools to accomplish a task or fulfill a purpose. There is the appreciation that the story of engineering divides into five ages: gravity, heat, electromagnetism, information, and systems. The same specialists appreciate that we are in the age of information, which has turned now into the age of complex systems.

Complexity describes objects with many interconnected parts and complex systems are a subfield of computer science, called computational complexity. Speaking of the concepts: complicated and complex we can say that it is easy to distinguish one from the other only at the extremes, but there is a middle ground where the distinction becomes unclear and arbitrary. The interactions of interest are nonlinear and this non-linearity yields levels of organization and hierarchies. Complex systems exhibit several kinds of behavior such as: selforganization into patterns, chaotic behavior adaptive interaction.

Analyzing complexity, we can ask how systems can generate perpetual novelty using limited resources. And besides many answers two concepts become important: laws and states. For example, for a system, the course over time appears chaotic rather than deterministic. Though the partial differential equations describing chaotic systems are fully deterministic, the presumed guarantees of determinism: similar starting conditions yield similar state trajectories-no longer hold! We must take in account a systemic approach because a system is more than the sum of its parts. It may exhibit adaptive, dynamic, goal-seeking, selfpreserving, and sometimes evolutionary behavior. The least obvious part of the system, its function or purpose, is often the most crucial determinant of the system's behavior. Complex behaviors of systems often arise as the relative strengths of feedback loops shift, causing first one loop and then another to dominate behavior! What can we do? Today, many authors recommend: Listen to the "Wisdom of the Systems", stay humble-Stay a Learner! Working with systems, often reminds me of how incomplete my mental models are, how complex the world is, and how much I don't know! Sometimes, I think to defy the Disciplines because seeing systems whole requires even more than being "interdisciplinary". Even so, we will contemplate the inevitable gaps between what we know, what we do, and why things go wrong. But, as we explore modern ideas of systems complexity, we will see that the gaps are filled by risk. Risk is at the heart of major engineering questions. How do we know what is safe? How safe is safe enough?

The simplest way of minimizing the risk of failure of a physical 'hard' tool is to use a large safety factor, but some tools have become so very complex.

Engineers have started to work and manage the risks in layers or levels. Strictly specializations in IT&C, for example, pose new risks since the relationships between levels and, more importantly, between the ways of understanding of specialists in those levels, are not straightforward. Now, in the 21st century, we are entering into the age of systems with a potential for new risks through interdependencies we may not fully understand. For example, we now know that some (but not all) physical processes are chaotic, in the sense that, whilst they may appear to be reasonably simple, they are inherently difficult to predict. We have discovered that they may be very sensitive to very small differences in initial conditions and may contain points of instability where paths diverge. Energy, entropy, and exergy are examples of how the challenges of the 21st century require the engineering disciplines that are much better at integrating their expertise to find synergy. As a result, some engineers have begun to think differentlythey use what many call 'systems thinking' and an engineering team is an example of a specific soft system.

If we are to make and maintain highly reliable and sustainable complex systems, then we need more of our specialist engineers to be systems thinkers that can deal both with the detail and the big picturea synergy from the integration of reductionism and holism.

Speaking about our research activity, I can confess that it was influenced by scientific trends, by our professors, our colleges and in the last thirty years by our enthusiastic PhD students.

In the following sections we present, for example, two directions from our research: cryptography based on chaos and a new method to analyze signals from nonlinear dynamical systems.

I. IMAGE ENCRYPTION ALGORITHMS BASED ON GAMES' THEORY

Within the past few years a new trend in the field of digital images' cryptographic algorithms' designing branch has been revealed. A new-built and unique approach (targeting the fruitful conjunction between games' theory, namely their rules' design principles, and cryptography, be it either classical or chaosbased), which would crystallize in its early years, was identified. Among the games whose principles are used in the designing of cryptographic algorithm can be enumerated, e.g., Rubik's cube [START_REF] Diaconu | An improved secure image encryption algorithm based on Rubik's cube and digital chaotic cipher[END_REF], Sudoku [START_REF] Diaconu | An image encryption algorithm with a chaotic dynamical system based Sudoku Grid[END_REF], Chinese Chess [START_REF] Diaconu | Color image scrambling technique based on transposition of pixels between RGB channels using Knight's moving rules and digital chaotic map[END_REF] etc.

An extended improved version of an image encryption algorithm based on Rubik's cube principle and digital chaotic map is presented in [START_REF] Diaconu | An improved secure image encryption algorithm based on Rubik's cube and digital chaotic cipher[END_REF]. Basically, as shown in Fig. 1, through an iterative process, image's rows and columns are scrambled via cyclic rotations (either to the left or right, resp., up or down, with a random number of steps, as determined by the intrinsic properties of the image). Then, by the use of a chaotic map, image pixels are doubly ciphered.

Other ideas revealed within the previously mentioned paper were: (i) to relocate original images' pixels as dictated by the associated Sudoku gird (i.e., constructed with the same dimensions as of image to be scrambled and (or) ciphered), as Fig. 2 suggests [START_REF] Diaconu | An image encryption algorithm with a chaotic dynamical system based Sudoku Grid[END_REF]; (ii) to take advantage of Knight's moving pattern nonlinearity in order to change image pixels' position both spatially and between the RGB channels.

Within their research works, with the aid of standard analyses (e.g., statisticalhistogram, adjacent pixels' correlation coefficients; differential -NPCR and UACI indices; local and global entropy; key space and key sensitivity to small changes; strength to different types of attacks etc.) authors of the above mentioned papers have proved that the newly proposed digital images' encryption schemes have similar or better performances with those proposed by other scholars.

As easily it can be seen, all this was possible only through the combined efforts of an intricate network of enthusiastic scholars with common research interests.

II. NEW APPLICATIONS OF TRANSIENT SIGNAL PROCESSING

Transient signals are signals with short duration and broad spectrum which are often present in various industrial applications such as turbine regime changes, electrical arches generated in photovoltaic panels or in electric cables, etc. In order to ensure the safety of such systems, it is very important that these signals can be detected, located and characterized in order to prevent an eventual irreversible damage. This paper presents new tools derived from the concept of Recurrence Plot Analysis (RPA) and applied to some real applications related to power systems. The concept of RPA is presented in figure 3.

Our applications concern the detection, localization and characterization of the transient signals measured in hydrodynamic systems and in electrical systems, respectively. The applications for the hydrodynamic systems particularly refer to the detection and quantification of the water hammer effect and to the Rankine vortex rope detection, while the electrical systems applications refer to the detection, localization and characterization of partial discharges in high voltage cables and of electrical arcs that appear in photovoltaic panels, respectively. For the detection and localization part, we used the multi-dimensional representation and its corresponding recurrence matrix for the computation of the TDR -time distributed recurrence measure: ,

M il i TDR l R l M TDR l TDR l TDR l     1 * ( ) , 1, () ( ) 1 max( ( ))
The characterization of the transient signals was achieved through phase diagram representation in polar coordinates of the trajectory. It goes that the fundamental frequency of the transient signal is:

0 0 2 s n T f  (2)
where 0 n is the number of points from the phase diagram between the angles 3 / 4  and /4  and s f is the sampling frequency.

The characterization of the water hammer effect was realized using an ultrasonic sensing system placed on a pipe line. The acoustic effect induced to the sensors is detected using the recurrence plot analysis concept [START_REF] Ioana | Recent Advances in Non-stationary Signal Processing Based on the Concept of Recurrence Plot Analysis[END_REF] (equation ( 1)) and, then, the time of flight of the signals arrived at both sensors is computed (figure 4). Hereby, the pressure variation performed by the water hammer phenomenon into the pipe line is determined and compared with the variation measured by the pressure sensor placed in the pipe. An error under 2% was obtained, the recurrence plot analysis method proving to be a much better approach in comparison to state of the art methods in hydraulics or signal processing. The detection and characterization of a rotational vortex rope using acoustic sensors in active configuration. The recorded signals are investigated through the quantification of the diagonal lines of the recurrence matrix (equation ( 3)) and the exploit of their frequency content. Results confirm the existence of the rotational vortex rope and highlight its evolution. The quantification is obtained using the measure:

1 1 [ ] ( , ), 0, 1
Mn i i n i sd n D v v n M Mn         (3)
where ( , )

i i n D v v  is the point ( , ) ij corresponding
to the distance matrix.

The applications concerning the electrical systems applications refer to the detection, localization and characterization of the electrical partial discharges measured in high voltage cables and in photovoltaic panels, respectively. While the detection and the localization of the partial discharges are achieved through the exploration of the characteristics of the recurrence matrix building a well suited energy detector (equation ( 1)), their characterization is obtained using the multi-lag recurrence plot analysis, a new concept which uses the 2D phase space representation (figure 5).

The multi-lag recurrence plot analysis tools investigate the evolution of the signals in the phase space using the elliptic modeling and the polar coordinates representation (equation ( 3)) and highlights slight changes between the partial discharge and its reflexions.

III. CONCLUSIONS

Are our scientific activities in consensus with actual transitions from "the age of information to the age of (big) systems in engineering? Difficult to answer! But, from twenty years we promote the idea that engineering's systems are nonlinear ones and usually can have chaotically behaviors and their associated processes need special studies' methods. And these ideas are in consensus with the worldwide research trends in communications and information technology.

Figure 1 .

 1 Figure 1. Image scrambling using Rubik's cube principle. Here, I(i, j) represents image's pixels, β1 and β2 are two random integers (which subsequently will be part of the encryption key) and ´mod´ represents the modulo operator.

Figure 2 .

 2 Figure 2. Inter bit-level permutation: (a) input image's associated matrix, (b) the

Figure 3 .

 3 Figure 3. The concept of Recurrence Plot Analysis

Figure 4 .

 4 Figure 4. Water hammer detection and localization using the TDR measure

Figure 5 .

 5 Figure 5. The area evolution of the phase space representations of the normalized multi-path acoustic signal and their corresponding phase diagrams

ACKNOWLEDGMENT

The paper has been financially supported within the project entitled "Horizon 2020 -Doctoral and Postdoctoral Studies: Promoting the National Interest through Excellence, Competitiveness and Responsibility in the Field of Romanian Fundamental and Applied Scientific Research", contract number POSDRU/159/1.5/S/140106. This project is cofinanced by European Social Fund through Sectoral Operational Programme for Human Resources Development 2007-2013. Investing in people!


domain: phys nlin chim

text:
S Akamatsu
email: akamatsu@insp.jussieu.fr

M. S S Bottin-Rousseau

G Faivre

A theory of thin lamellar eutectic growth with anisotropic interphase boundaries

Keywords: Eutectic, directional solidification, microstructures, interphase boundaries

Rotating-I-esub3

We present a semiempirical theory of the effects of an orientation dependence of the surface free energy of interphase boundaries (interphase boundary anisotropy) on lamellar eutectic growth in thin samples. We show that, to a good approximation, thin lamellar eutectic patterns with a strong interphase boundary anisotropy travel along the growth front at such a velocity -or, equivalently, at such an inclination angle of the lamellae left behind in the solid -that the surface tension force of the interphase boundary is nearly parallel to the applied thermal gradient. This explains, among other things, the crystallographic locking of lamellar eutectic patterns that occurs in those eutectic grains, which have cusp singularities in the Wulff plot of the interphase boundary. Based on this theory, we show that the rotating directional solidication (RDS) method, by which a thin sample is rotated with respect to a fixed unidirectional thermal gradient, must yield eutectic lamellae whose trajectories are nearly homothetic to the 2D Wulff form of the interphase boundary. This opens up new possibilities for the experimental study of interphase boundary anisotropy in eutectic alloys.

Introduction

The important effect of an orientation dependence of the surface free energies of interfaces on the morphology of eutectic microstructures has long been recognized. However, our understanding of this effect is still fragmentary. Following Jackson and Hunt's landmark articles of 1966 [1,2], attention has been mostly focussed on the type (nonfaceted or faceted) of anisotropy exhibited by solid-liquid interfaces. The theory of eutectic solidification in alloys with nonfaceted solidliquid interfaces (nf/nf eutectics) was developed under the assumption of a complete isotropy of all interfaces. The famous regularity-i.e. space periodicity and steadiness-of lamellar eutectic growth patterns was established theoretically under this assumption. The full success of this "isotropic" theory in suitably chosen eutectic grains [3,4,5] should not eclipse the importance of crystallographic effects linked to interphase boundary anisotropy in other eutectic grains. Numerous metallic eutectics, notably Al-Al 2 Cu [6] or Al-Al 2 Cu-based ternary eutectics [7], form "epitaxial" eutectic grains (i.e. eutectic grains with orientation relationships between phases such that there exists low-energy orientations for the interphase boundary) upon directional solidification [8]. In epitaxial eutectic grains, lamellar planes are locked onto low-energy orientations of the interface boundary and are therefore generally steeply inclined with respect to the direction of growth, in contradiction with the isotropic theory. These well-known facts pose two as yet unsolved problems. First, what is the selection mechanism of the epitaxial eutectic grains during directional solidification? Second, what is the locking mechanism of eutectic lamellae in epitaxial eutectic grains? This study is dealing with the latter question and focuses on two-dimensional (thin) eutectics. The phenomenon of crystallographic locking of the lamellar plane has previously been observed in thin samples of carbon tetrabromide-hexachloroethane (CBr 4 -C 2 Cl 6 ) [2,9,10], which is a transparent analog of nf/nf metallic eutectics, and In-In 2 Bi [11]. Interestingly, crystallographic locking has been sought but could not be found in the transparent nf/nf succinonitrile-(D)camphor (SCN-DC) eutectic, probably because no epitaxial interphase orientation relationships exist in this alloy [12].

The explanations of crystallographic locking that have previously been proposed were unsatisfactory because they did not take into account the dynamical (i.e. out-of-equilibrium) nature of lamellar eutectic patterns [8]. In this article, we propose a dynamical theory of crystallographic locking, and, more generally, of interphase boundary anisotropy effects in lamellar eutectic growth. It is known that the steady-state (st) lamellar patterns of eutectics with some anisotropy of any (solid-liquid or interphase) interface must travel along the growth front at a con-stant velocity v st [10,[START_REF] Faivre | Solidification Processes and Microstructures : A Symposium in Honor of Prof[END_REF]. These "anisotropy-driven travelling " patterns leave behind in the solid interphase boundaries that are inclined with respect to the growth axis z at an angle φ st =arctan (v st /V), where V is the growth velocity. This in turn modifies the equilibrium between surface tension forces at the junctions between the three phases on the growth front, and thus the shape of the solid-liquid interfaces. In this article, we focus on systems where the anisotropy of the solidliquid interfaces is negligible compared to that of the interphase boundaries. Our central assertion is that, in such systems, the shape of the solid-liquid interfaces in the anisotropy-driven travelling lamellar patterns can be assumed to be mirror symmetric. In other words, φ st is such that the surface tension force σ of the interphase boundary is approximately aligned with z (Fig. 1). This assumption will be referred to as the "symmetric-pattern" or " σ z" approximation thereafter. We first present the experimental and theoretical arguments that support this assumption. Next we show that crystallographic locking can be coherently explained on the basis of this assumption. Finally, we study the trajectories followed by eutectic lamellae during a rotating directional solidification (RDS) experiment, i.e., when a thin sample is rotated with respect to a fixed unidirectional thermal gradient, and show that, in the symmetric-pattern approximation, these trajectories are homothetic to the 2D Wulff form of the interphase boundary in the sample plane. Experimental observations made by the RDS method will be presented in a forthcoming article.

Basic concepts

To avoid any obscurity, a few basic concepts should first of all be recalled. A eutectic grain (EG) is a portion of the solid, inside which the crystal-lattice orientation of either eutectic phase is uniform [8]. Morphology apart, it is a heterophase bicrystal defined by two sets of orientational variables, one describing the interphase orientation relationship and the other the orientation of the whole EG with respect to a frame of reference attached to the experimental setup, e.g. the xyz reference frame defined in Fig. 1 [START_REF] Sutton | Interfaces in Crystalline Materials[END_REF]. Let n be the unit normal vector and γ( n) the surface free energy of the interphase boundary in a given EG. The 3D Wulff plot of the interphase boundary in this EG is the surface described by r = γ, where r is the position vector and γ = γ( n) n. The corresponding Wulff form (form of minimum surface energy for a given volume) is described by r = ξ 3D , where ξ 3D = γ + |r|∇ a γ, ∇ a representing the angular part of the gradient operator [START_REF] Hoffmann | [END_REF]. Let l be the unit tangent to a trijunction line between the interphase boundary and the solid-liquid interfaces. The equation for the equilibrium between surface tensions (Young-Herring's equation) at the trijunctions reads

γ a + γ b + σ = 0 , (1)
where σ = ξ 3D × l, while γ a and γ b are the surface tension forces of the isotropic solid-liquid interfaces.

Let us now turn to thin-sample directional solidification. By "thin", it is meant that the confinement of the system by the sample walls imposes a pronounced 2D character to the solidification dynamics, so that the interfaces are constrained to remain essentially perpendicular to the sample plane [16]. Exceptions to this rule called "out-of-plane lamellar patterns" will be considered later on in this article. For now, we assume a strict bidimensionality, i.e. n is parallel to the sample plane, so that γ depends on a single variable, e.g. the inclination angle φ of the interphase boundary. The 2D Wulff plot of this boundary is the cross section of its 3D Wulff plot by the xz plane, while its 2D Wulff form is described by r = ξ, where ξ is the orthogonal projection of ξ 3D onto the xz plane. One has ξ = γ n -γ t and σ = γ t + γ n, where γ = dγ/dφ and t = -d n/dφ (Fig. 2). Introducing the angular deviation ψ = arctan (γ /γ) of σ from the interface boundary, and the inclination angle δ of σ, we write the σ z condition in the form of an equation for φ, δ(φ

) = φ + ψ(φ) = 0 . (2)
The solutions to Eq. 2 may be formally written

φ st = δ -1 (0) , (3)
provided it is accepted that δ -1 can be a multivalued function. In fact, an elementary calculation leads to dδ/dφ = γ(γ + γ )/(γ 2 + γ 2 ). Thus δ -1 is single-valued when the surface stiffness τ = γ + γ of the interphase boundary is positive at all φ. In this case, φ st is unique. Conversely, δ -1 is multivalued when the interface boundary has one, or several, orientation ranges in which τ is negative, in which case φ st may not be unique. It should be noted that τ(φ) > 0 is the Herring condition for an interface to be stable against a hill-and-valley instability [17]. This issue is further discussed below. We conclude this introductory section by noting that δ is the polar angle of ξ and that dξ/dφ = γ (γ +γ )/ γ 2 + γ 2 , where ξ = |ξ|. One therefore has

dξ ξdδ = dγ γdφ , (4)
which is the differential equation for the 2D Wulff form of the interphase boundary in polar coordinates, assuming γ(φ) and hence φ(δ) to be known.

The symmetric-pattern approximation

The symmetric-pattern assumption was primarily made on the basis of experimental observations that showed space periodic lamellar patterns travelling laterally at relatively large velocities and nevertheless exhibiting nearly mirror symmetric shapes of their growth fronts (see Section 5). This feature is in strong contrast with the spectacularly asymmetric growth front shapes of the so-called "tilted" lamellar patterns that are observed in off-eutectic isotropic systems at large values of the interlamellar spacing λ -specifically, above a threshold spacing value λ tilt ≈ 1.5λ JH , where λ JH is the Jackson-Hunt (JH) minimum-undercooling spacing [3,4]. A comparison between the growth-front shapes of the two types of travelling patterns clearly shows that, in a mirror symmetric travelling pattern, σ must be parallel to z, and not to the interphase boundaries, and that, therefore, the interphase boundaries must have a strong (i.e. such that γ /γ ∼ 1) capillary anisotropy (Fig. 3).

An analytical demonstration of the validity of the symmetric-pattern assumption should proceed as follows. The equations to be considered are those of the JH model of 2D eutectic directional solidification [2], which include a diffusion equation for the solute in the liquid and, at the solid-liquid interface, a mass conservation equation and two local-equilibrium (Gibbs-Thomson and Young-Herring) equations. As is well known, Jackson and Hunt solved these equations using a planar-front approximation in the λV/D << 1 limit, where D is the solute diffusion coefficient in the liquid, and found a continuous set of steady, space periodic, mirror-symmetric solutions over a wide λ-range at given V. The fact that the solutions have mirror symmetry follows from the invariance of the governing equations with respect to mirror reflexion about yz. This invariance is destroyed, in general, by the introduction of interfacial anisotropy in the local-equilibrium equations. Consequently, the space periodic solutions of the JH model with anisotropic interfaces do not have mirror symmetry and travel along x at a finite velocity v st . In other words, they are steady in a frame of reference travelling at v st along x. The differences in concentration field and form of the solid-liquid interface between anisotropic patterns and isotropic ones arise from the additional "lateraladvection" terms appearing in the diffusion and mass conservation equations when moving from a fixed to a travelling reference frame. Reproducing JH's calculation while taking the lateral-advection terms into consideration leads to the conclusion that the asymmetric components of the concentration field and of the form of the solid-liquid interface due to lateral advection are little-o of λV/D, and are therefore negligible. This, strictly speaking, does not mean that the symmetric-pattern approximation is accurate when λV/D << 1, but only that its accuracy cannot be evaluated using a planar-front approximation. However, it strongly suggests that the accuracy of the symmetric-pattern approximation increases as the contact angles (angles with respect to x at the trijunctions) of the solid-liquid interfaces decrease, everything else equal, and that the approximation becomes exact in the limit of vanishing contact angles. To sum up, numerical simulations will be necessary to evaluate the accuracy of the symmetric-pattern approximation as a function of, in particular, the contact angles of the solid-liquid interfaces. For now, based on experimental observations, the symmetric-pattern approximation can be safely assumed to be at least semiquantitatively valid for the determination of the anisotropy-driven travelling velocities of lamellar patterns.

Locked versus floating lamellar patterns

Let us consider a EG, in which the 2D Wulff plot of the interphase boundary has a trough with a pointed end (cusp) at φ = θ, where θ is the inclination angle of the corresponding facet in the Wulff form. It is assumed that this facet has a high mobility so that Young-Herring's equilibrium condition can be fulfilled at, and near, φ = θ. Equivalently, it may be assumed that the pointed end of the trough is in fact rounded within a very small angular range [18,19]. Anticipating on Section 6, we furthermore allow the EG to be rotated about y. Then θ represents the rotation angle of the EG measured from the position at which the facet is parallel to z. We may write γ(φ, θ) = γ 0 (φ -θ), where γ 0 (φ) is the Wulff plot of the interphase boundary at θ = 0 and reformulate Eq. 3 as

φ st (θ) = δ -1 0 (-θ) + θ , (5)
where δ -1 0 is the inverse function of δ 0 (φ) = φ + arctan γ 0 /γ 0 . For ease of reference, the discussion of Eq. 5 will use an elementary model of a Wulff plot with a pointed trough. In suitable surface energy units, the model is

γ 0 (φ) = 1 - 1 -cos (|φ| -Φ) 1 -cosΦ (6)
for |φ| ≤ Φ, and γ 0 (φ) = 1 for |φ| ≥ Φ. In this model, represents the relative depth of the trough and Φ its angular width. For illustration purposes, we can set Φ at some typical value for heterophase interfaces, e.g. 0.2 rad [START_REF] Sutton | Interfaces in Crystalline Materials[END_REF], and keep only as a parameter. The surface stiffness is constant and equal to 1 -/ s , where s = 1 -cosΦ, inside the trough, and is equal to unity ouside the trough.

Thus the interphase boundary is stable everywhere when < s and unstable in the orientation ranges corresponding to the flanks of the trough ("vicinal" ranges) when > s . Figure 4 shows the aspect of three key functions [the Wulff plot, the Wulff form, the φ st (θ) function] in this model at < s . The discontinuity of γ 0 occurring at the bottom of the trough gives rise to a facet in the Wulff form, as is well known. In the φ st (θ) curve, it gives rise to a line segment (c c) of slope unity. The interphase boundary is parallel to the facet for all θ belonging to the finite-width range (the "locking range") spanned by this segment. This is the phenomenon known as crystallographic locking.

In Fig. 4, the transition from the locked (c c) to a non-locked branch of states (e.g. pq) as a function of θ occurs continuously through a branch of stable intermediate states (cp). However, the measured values of reported for low-energy interphase interfaces [START_REF] Sutton | Interfaces in Crystalline Materials[END_REF] are generally above the instability threshold value ( s ≈ 0.02) according to the above model. Situations where > s must therefore be frequently observed. In this case, Figure 5 shows that two features arise: the locked branch overlaps the non-locked ones; the intermediate branches are unstable with respect to Herring's instability. Therefore, locked and non-locked lamellar patterns are metastable with respect to each other within a certain orientation range. Hysteresis and spatial coexistence of different lamellar patterns can in principle be observed. The transitions between the two types of patterns as a function of θ must be discontinuous.

Qualitatively speaking, the above considerations are independent of the model chosen. Therefore, two conclusions of general validity may be drawn: first, crystallographic locking is due to the existence of cusp-like singularities at the minima of the Wulff plot of the interphase boundary, and is therefore specific for epitaxial EGs; second, discontinuous transitions between locked and non-locked lamellar patterns should be observed when, and only when, the vicinal orientations of the cusps are unstable with respect to Herring's instability. These conclusions emphasize the similarity that exists between crystallographic locking of lamellar patterns and faceting of equilibrium forms. There is however a significant difference between the two phenomena, which we will now stress. As is well known, isothermal equilibrium shapes must be convex, and therefore cannot coincide with theoretical Wulff forms in, and around, unstable orientation ranges. The orientation ranges eliminated from ("forbidden" in) the equilibrium forms include not only unstable segments but also adjacent stable segments, such as segments ec and ep in Fig. 5a. No similar elimination needs to occur in lamellar eutectic growth. As far as we can see, φ st can take any value belonging to a stable branch of the φ st (θ) diagram. Hence possible hysteresis and coexistence effects, as mentioned above.

Finally, a remark about the dynamics of locked lamellar patterns should be made. Experimental observations, which will be reported in a future article, showed locked lamellar patterns with very irregular λ(x) distributions (i.e. spacing distributions along the growth front) that did not evolve over time. This is in strong contrast with isotropic lamellar patterns, which, as is well known, are steady only when their spacing distribution is uniform. More specifically, an isotropic lamellar pattern with a spatial modulation in λ relaxes toward uniformity through a process called phase-diffusion (or λ-diffusion) whose time constant varies as the square of the wavelength of the modulation [20,21]. This is a linear-response mechanism, which relies on the fact that all the features of the pattern, especially, the shape of the solid-liquid interface, vary smoothly with φ. It should be clear from the foregoing that, in locked lamellar patterns, σ, and hence the shape of the solid-liquid interface, vary discontinuously at φ = θ. Therefore, locked lamellar patterns -at least, those with hysteretic transitions between locked and non-locked patternsare deprived of a linear mechanism of response to spatial variations of λ, in agreement with experimental observations. This leads us to an important terminological remark. We propose to define locked lamellar patterns as those patterns, in which the λ-diffusion mechanism is blocked. Correlatively, those lamellar patterns, be they isotropic or anisotropic, in which λ-diffusion is effective should be grouped into one category, which, following Ref [10], we propose to call "floating" . [4]. The facts to note here are (i) the mirror symmetry (within experimental error) of the form of the growth fronts in travelling lamellar patterns, (ii) the low value of the contact angles in locked patterns compared to isotropic ones, and (iii) the breaking of the bidimensionality condition in the "out-of-plane" locked pattern of Fig. 6c. The first fact is key for the validity of the symmetric-pattern approach, as was stressed above. The second one is a confirmation that locked lamellar planes correspond to low-energy interphase boundaries. The third fact will now be commented.

Out-of-plane locked lamellar patterns

Out-of-plane locked patterns break the bidimensionality condition in spite of strong confinement effects. Clear examples of such patterns were identified in-situ through their optical contrast in CBr 4 -C 2 Cl 6 , as illustrated in Fig. 6c, and through post-mortem transverse metallographic sections in In-In 2 Bi [11]. These patterns had irregular λ(x) distributions indicating that they were locked onto facet planes corresponding to out-of-plane cusps of the 3D Wulff plot of the interphase boundary. Given that reflection boundary conditions are imposed on the concentration field at the sample walls, the question arises as to the geometry of out-of-plane locked patterns near these walls. It should be noted that the 3D extension of an out-of-plane locked pattern by symmetry about the sample walls is a chevron pattern, which suggests the presence of a localized defect line along the walls [22]. For our present purpose, however, the most important point is as follows. A thin eutectic system that displays an out-of-plane locked pattern also possesses, as a general rule, a 2D solution, either locked of floating, to the equations of growth. There may be conditions under which these two patterns are metastable with respect to each other.

Rotating directional solidification

A RDS experiment is schematically represented in Figure 7a. A thin sample is placed in a directional thermal gradient, and then rotated at an angular speed ω = dθ/dt about the y axis. A 2D reference frame XZ with the same origin Ω as the laboratory reference frame is attached to the rotating sample. We focus on the growing side (x > 0) of the solidification front. We neglect the undercooling of the growth front, which is thus equated with the isotherm at T E . We assume the sample to be a single eutectic grain and Ω to sit at a fixed distance R from the T Eisotherm. We seek a differential equation for RDS lamellar trajectories under the σ z assumption. More specifically, we assume that RDS lamellar trajectories adjust their inclination angle to the solution of Eq. 2 instantaneously. We thus neglect any effect that may arise from a non-uniformity of λ(x) or from the spatial V-ramp generated by the RDS. Let M be the position vector of a trijunction at a given time and x and ẑ be the unit translation vectors of the x and z axis (Fig. 7b). We set M = ρ(θ) x + Rẑ, where ρ(θ) is the unknown. An elementary calculation leads to: dM/dt = v st x + V ẑ, where V = ωρ and v st = ω(dρ/dt -R), and hence to: tanφ = -(dρ/dθ -R) ρ. Substituting -ψ for φ in accordance with the σ z assumption, one obtains:

dρ dθ -tan ψ(θ) ρ = R . (7)
When R = 0, Eq. 7 is identical to Eq. 4. Thus RDS lamellar trajectories are homothetic to the Wulff form of the interphase boundary. This conclusion is unambiguous when there is no forbidden orientation range in the Wulff plot of the interphase boundary. When such a range exists, all the stable branches of the theoretical Wulff form, including those that may be "forbidden" from the equilibrium shape, must be taken into consideration, as explained in Section 4. The theory presented here does not specify the RDS lamellar trajectories in the forbidden ranges. It only indicates that these trajectories are likely to present serial sharp edges, i.e. zig-zag shapes, due to the hysteretic nature of the transition between locked and floating lamellar patterns in theses ranges. We now assume that R 0, as generally occurs in the experiments. Then the solutions to Eq. 7 are in the form

ρ = ρ 0 F(θ) + RS (θ) , (8)
where ρ 0 = ρ (θ = 0),

F(θ) = exp θ 0 tan ψ(θ ) dθ , (9)
and

S (θ) = F(θ) θ 0 1/F(θ ) dθ . (10)
The function F(θ), which represents the Wulff form, is 2π-periodic, but S (θ) is not.

Thus RDS lamellar trajectories are spiraling either toward the center (R < 0) or the periphery (R > 0) of the sample with a separation distance between successive turnings of about 2πR. The growth pattern correspondingly undergoes a global lateral drift in the laboratory reference frame, so that the centre of the sample is a lamellar sink and its periphery a lamellar source when R < 0, and vice versa when R > 0. It should be stressed that this effect has nothing to do with the dynamics of the growth pattern itself, but arises from the translational motion undergone by the V-ramp when R 0 (Fig. 7b). Other remarks that will be useful in discussing the experimental results are: 1-RDS lamellar trajectories do not depend on the relation between θ and t; 2-The evolution with θ of an initially uniform spacing distribution is given by λ(θ)/λ(0) = W(θ). Any quasi-steady non-uniform spatial distribution of λ will undergo a periodic succession of dilatations and compressions but will not change shape during RDS -assuming that the variations of λ during a cycle remain inside the stability interval of the pattern [21];

3-The value of R can be calculated after the end of the experiments by fitting theoretical RDS lamellar trajectories onto the measured ones. In isotropic systems, RDS lamellar trajectories are involutes of the circle of radius R and centre ω and have an explicit intrinsic equation, namely, s = R (φφ 0 ) 2 /2, where s is the arc length of the trajectory and φ 0 is a constant, which facilitates the determination of R.

4-The Wulff plot of an interface boundary can be calculated from experimental φ(θ) data using Eq. 9 and the symmetric-pattern assumption without knowing R.

Conclusion

We have studied theoretically the steady-state dynamics of traveling lamellareutectic patterns during directional solidification of a thin nonfaceted eutectic alloy with a strong capillary anisotropy of the interphase boundaries. Two main conclusions have been reached. The first one is the validity of the so-called symmetric-pattern approximation, which states that, in a traveling lamellar-eutectic pattern, the shape of the solid-liquid interfaces is nearly mirror symmetric. The second one is the homothety between lamellar trajectories and Wulff forms of interphase boundaries during rotating directional solidification, which illustrates the relevance of that experimental method to semiquantitative studies of the capillary anisotropy of eutectic interphase boundaries. New insights have also been gained about the nature and dynamics of crystallographically locked lamellar patterns. These theoretical results will be compared with RDS experimental observations in a future article.

List of figure captions

Figure 6

 6 Figure6displays high-magnification views of three different types of lamellar patterns observed in different EGs during standard (as opposed to rotating) directional solidification in a 12µm-thick sample of CBr 4 -C 2 Cl 6[4]. The facts to note here are (i) the mirror symmetry (within experimental error) of the form of the growth fronts in travelling lamellar patterns, (ii) the low value of the contact angles in locked patterns compared to isotropic ones, and (iii) the breaking of the bidimensionality condition in the "out-of-plane" locked pattern of Fig.6c. The first fact is key for the validity of the symmetric-pattern approach, as was stressed above. The second one is a confirmation that locked lamellar planes correspond to low-energy interphase boundaries. The third fact will now be commented.Out-of-plane locked patterns break the bidimensionality condition in spite of strong confinement effects. Clear examples of such patterns were identified in-situ through their optical contrast in CBr 4 -C 2 Cl 6 , as illustrated in Fig.6c, and through post-mortem transverse metallographic sections in In-In 2 Bi[11]. These patterns had irregular λ(x) distributions indicating that they were locked onto facet planes

Figure 1 :

 1 Figure 1: Repeat units of lamellar eutectic patterns. a) Isotropic system. b) System with a strong interfacial anisotropy of the interphase boundary in the symmetric-pattern approximation. a, b: eutectic phases. L: liquid. z: growth direction parallel to the thermal gradient. y: normal to the sample plane. x: average direction of the solid-liquid interface. φ st : steady-state inclination angle. Other symbols: see the text.

Figure 2 :

 2 Figure 2: Wulff construction for an interphase boundary. WF: Wulff form. t and n: unit tangent and normal vectors to the interphase boundary, respectively. See the text for the other symbols.

Figure 3 :

 3 Figure 3: Sketches of experimentally observed travelling lamellar eutectic patterns. a) Anisotropydriven travelling patterns at λ < λ tilt . b) Tilted travelling patterns at λ > λ tilt in isotropic systems [3, 4].

Figure 4 :

 4 Figure 4: Properties of a shallow ( = 0.01) cusp as described by Eq. 6. (a) Blow-up of the vicinal regions of the Wulff plot (thin line) and Wulff form (thick line) in suitable surface energy and length units. For clarity the plot has been greatly elongated in the z direction. (b) Inclination angle of lamellae as a function of grain orientation. Points c and c : endpoints of the facet (φ = 0 ± ). Points p and p : limits of the trough (φ = ±Φ).

Figure 5 :

 5 Figure5: Properties of a deep ( = 0.05) cusp as described by Eq. 6. Same symbols as in Fig.4. Points e and e : edges of equilibrium facets. Dotted lines: unstable orientation ranges of the interphase interface.

Figure 6 :

 6 Figure 6: Real-time observation of lamellar eutectic patterns during standard thin-sample directional solidification of the CBr 4 -C 2 Cl 6 eutectic. Growth velocity: 0.5 µms -1 . Sample thickness: 12µm. The z axis is directed vertically upward. The melt is visible above the solid. The different micrographs correspond to different EGs and show the following types of lamellar patterns: a) 2D isotropic; b) 2D locked; c) out-of-plane. The angle of the lamellar plane and y in c) is 10 ± 3 o . Bar: 20 µm.

Figure 7 :

 7 Figure 7: a) RDS experiment (sketch). A thin sample placed on a cold (T -) and a hot (T + ) oven is rotated about the y axis. Ω: centre of rotation. θ: angle of rotation (θ > 0 refers to a clockwise rotation of the sample.) xz: laboratory reference frame. XZ: sample reference frame. Arrows: directions of motion of the solid-liquid interface in the sample reference frame. RDS lamellar trajectories have been schematically represented as semi-circles. b) Enlarged view of the solidification front. O: zero-growth rate point. M: instantaneous position of a trijunction. R, ρ: coordinates of M in the laboratory reference frame.

Acknowledgments

We are grateful to Mathis Plapp (Ecole Polytechnique, Palaiseau) for numerous fruitful discussions. One of the authors (M.S.) was supported by the Centre National d'Etudes Spatiales (CNES), France.


domain: phys.cond.cm-ms nlin chim

text:
S Akamatsu
email: akamatsu@insp.jussieu.fr

M. S S Bottin-Rousseau

M Sererefoğlu

G S Faivre

Lamellar eutectic growth with anisotropic interphase boundaries: Experimental study using the rotating directional solidification method

Keywords: Eutectic solidification, directional solidification, solidification microstructures, interphase boundaries

Lamellar eutectic growth with anisotropic interphase boundaries: Experimental study using the rotating directional solidification method

Introduction

The dramatic effects of a strong capillary anisotropy of interphase boundaries on eutectic directional-solidification patterns have long been noted [1,2], but have never been systematically studied. The property of the alloy that controls these effects is the Wulff plot of the interphase boundary in the reference frame of the laboratory. This Wulff plot varies from eutectic grain to eutectic grain, where the term "eutectic grain" (EG) designates both a given interphase orientation relationship -i.e. a heterophase bicrystal -and a given orientation of this bicrystal with respect to the directional-solidification setup. Careful experimentation requires a good crystallographic homogeneity of the samples, i.e. large-sized eutectic grains, and a small number of free orientational parameters. The use of thin-sample directional solidification, in addition to reducing the dimensionality of the solidification pattern, allows one to grow nearly perfect large-sized eutectic grains [3]. The use of a rotating directional solidification (RDS) stage whereby a thin sample is rotated with respect to a fixed unidirectional thermal gradient, moreover permits to vary a single orientational parameter in situ while keeping the other parameters constant.

In this article, we present experimental observations performed by RDS in the CBr 4 -C 2 Cl 6 and In-In 2 Bi lamellar eutectics. More specifically, we study the RDS lamellar patterns (i.e. the forms of the RDS lamellar trajectories) in a series of different eutectic grains in both alloys. Each RDS pattern turned out to be a reproducible feature of the EG to which it belonged, indicating that RDS patterns only depend on the characteristics of the Wulff plot of the interphase boundary. All the observed RDS patterns could be grouped into three categories with contrasting characteristics (quasi-isotropic, nearly-locked, locked) independently of the alloy. We show that all the observations can be simply explained by means of the socalled "symmetric-pattern approximation". This approximation will be discussed in detail in another article. It can be summed up as follows. Based on general principles, an interphase boundary anisotropy is expected to break the spatiotemporal symmetry (symmetry with respect to the growth axis z and time-invariance) distinctive of isotropic lamellar patterns. In other words, lamellar patterns should have asymmetric forms of their solid-liquid interfaces and travel laterally along the growth front (entailing that the lamellae left behind in the solid are inclined with respect to z) in the presence of a strong interphase boundary anisotropy. However, on closer examination, it can be seen that the steady-state travelling velocity v st of the patterns must be such that the solid-liquid interfaces remain nearly symmetric. The symmetric-pattern approximation consists of assuming that v st is such that the solid-liquid interfaces are exactly symmetric, or, equivalently, that the surface tension force σ of the interphase boundaries is parallel to z (Fig. 1) -see Ref. [4] for a definition of σ. As regards RDS experiments, it can be demonstrated that the symmetric-pattern approximation implies that the trajectories of the lamellae in the sample reference frame are homothetic to the 2D Wulff form of the interphase boundary in the EG under consideration. Thus, for instance, RDS lamellar trajectories should be almost circular in EGs where the interphase boundary anisotropy is negligible, and faceted in EGs where the Wulff plot of the interphase boundary has cusp singularities (epitaxial EGs). This is the only consistent explanation of the familiar phenomenon of crystallographic locking of lamellar patterns. The report also includes a detailed description of the as-yet unknown dynamical features of (crystallographically) locked lamellar patterns.

Experimental methods

Alloys and crucibles

The CBr 4 -C 2 Cl 6 alloy has a eutectic plateau at T E = 84.4 o C. The concentration of the eutectic liquid is C E = 11.6mol%C 2 Cl 6 . The eutectic phases are a fcc phase (α) and a bcc phase (β) with concentrations 8.8 and 18.5mol%C 2 Cl 6 , respectively [1,5]. The In-In 2 Bi alloy has a eutectic plateau (T E = 72.7 o C, C E = 22.2at%Bi) between a fct phase ( ) of concentration 11.5 at%Bi and the hexagonal In 2 Bi intermetallic compound [START_REF] Massalsky | Binary Alloys Phase Diagram[END_REF]. The value of the Jackson-Hunt (JH) constant K JH = λ JH V 0.5 at eutectic concentration, where λ JH is the minimumundercooling spacing [1], was estimated to be 13.8 ± 0.5µm 1.5 s -0.5 in CBr 4 -C 2 Cl 6 [START_REF] Akamatsu | [END_REF], and 7.07 ± 0.35µm 1.5 s -0.5 in In-In 2 Bi [8]. Both CBr 4 -C 2 Cl 6 and In-In 2 Bi undergo a eutectoid transformation between T E and room temperature rendering post-mortem orientation measurements by X-ray diffraction problematic. We used nominally eutectic (in fact, slightly hypereutectic) alloys. The In-In 2 Bi alloy was prepared from 99.999% pure indium and bismuth (Goodfellow), and the CBr 4 -C 2 Cl 6 alloy from sublimated compounds. Thin crucibles were made of two 0.3mm-thick glass plates separated by 12µm-thick plastic strips or 13µm-diameter tungsten wires. It has been checked that convection flows in the liquid were suppressed in samples of this thickness. The in-plane dimensions of the rectangular space available for solidification were of 15 × 30 mm 2 . The crucibles were filled with molten alloys by capillarity (CBr 4 -C 2 Cl 6 ) or by a vacuum suction method (In-In 2 Bi), and then sealed at room temperature. Residual impurities contained in our samples induced the formation of eutectic cells at pulling velocities higher than 5 µms -1 in both alloys. The solidification rates considered herein are well below this threshold value.

Method of observation

The growth front was observed in real-time in the y direction with an inverted optical microscope. The transparent CBr 4 -C 2 Cl 6 eutectic was observed in transmitted light. With this method, the contrast originates from the differences in refrative index of the different thermodynamic phases and reveals the interfaces. The In-In 2 Bi eutectic was observed in reflected light through the lower glass wall of the crucible [9]. The differences in reflectivity between the different types of surfaces of contact gave rise to a contrast, which, after numerical enhancement, made the -glass surfaces appear white, the In 2 Bi-glass surfaces black, and the liquid-glass surfaces grey. Perturbations (contrast effects making the -glass contacts appear grey instead of white, barbules formed at the rear of the growth front) due to complex wetting and segregation phenomena at the sample walls [8] can be seen in some places in the micrographs, but are of no importance for our present purpose. Images were recorded with a monochrome digital camera, and transferred to a PC for further analysis. A several millimeter-wide region of interest was regularly scanned using the motorized stage of the microscope.

The rotating directional solidification stage

A RDS stage was built based on a model introduced by Oswald et al.

[10] (Fig. 2). It comprises: (i) a thermal-gradient bench composed of two thermally regulated copper blocks delimiting a 5mm-wide observation slot; (ii) a rectangular housing sliding on top of the copper blocks along z; (iii) a 15 cm-diameter circular plate, pierced by a rectangular sample-holding cavity, inserted in the rectangular housing, and rotatable about the y axis. Small clamps mildly press the sample on the copper blocks to ensure both good thermal contact and smooth sliding motion. The position of the growth front along z is monitored over time to ensure that the stability of the temperature distribution is satisfactory. Slight deviations of the thermal field from unidirectionality due to the geometry of the setup occurred during RDS. However, the T E -isotherm remained rectilinear inside the region of interest, so that these deviations were equivalent to small variations of angular speed over time, and therefore did not affect the RDS lamellar trajectories. The translation speed is denoted by V T , and the angular speed by ω = dθ/dt, where t is the time and θ is the angle between z and the axis of the sample. The ω values used were at most 5×10 -4 rads -1 (≈ 0.3 rph). The thermal gradient at the solid-liquid interface was 100 ± 10 Kcm -1 .

Experimental protocol

The experiments were conducted as follows (Fig. 3). A solidified sample is inserted into the experimental setup in a tilted position and pushed toward the hot block until the solid is reduced to a small seed polycrystal (Fig. 3a). The sample is then submitted to a EG-selection process. At the end of this process, the sample is directionally solidified at a constant V T value until the center of rotation Ω approximately sits on the growth front (Fig. 3b). At this moment, which is taken as zero time point, the pulling is stopped, and a RDS-run at constant ω is started (Fig. 3c). The position R of the growth front with respect to Ω was less than 100µm in all the experiments. It could not be measured with accuracy at this stage of the experiment, but was determined later by a method which is explained below. The RDS-run is then pursued during periods which can cover several complete revolutions of the sample. Although the grown microstructure is destroyed by melting after a half turn, an uninterrupted recording of the growth front allows one to perform retroactively a mosaic reconstruction of the whole solidification process in a reference frame attached to the sample. These reconstructed microstructures are, of course, multiple-valued functions of θ for RDS-runs covering more than a revolution of the sample, and have to be represented by a set of reconstructions limited to one revolution (thereafter called "space-time maps") containing a radial cut.

Particular requirements were met during this study. The duration time of the pulling at constant V T was sufficiently long for a quasi steady lamellar pattern, which served as an initial condition for the subsequent RDS-run, to be formed. Out-of-plane locked lamellar patterns were identified in situ and eliminated from the study in the case of CBr 4 -C 2 Cl 6 samples. The EG-selection process was performed in either of the two ways: (i) an epitaxial EG that preexisted in the seed (and was probably formed by nucleation of one phase onto the other during the solid-solid phase transition occurring between room temperature and T E ) was allowed to expand laterally, or (ii) the preexisting EGs were overgrown through a process called lateral invasion [3], which led to the formation of a few millimeterslarge EGs with more or less random interphase orientation relationships thereafter called random EGs.

Experimental results

Several epitaxial and random EGs were studied by RDS in CBr 4 -C 2 Cl 6 and In-In 2 Bi. The space-time maps recorded in these EGs can be classified into a small number of categories (quasi-isotropic, nearly locked, locked) independently of the alloy concerned. Representative examples of each category are studied below. Most importantly, all of the studied space-time maps were essentially πperiodic and reproducible from turn to turn, regardless of their complexity. This establishes that the space-time map of a RDS experiment reveals a characteristic feature, called "RDS pattern", of the EG to which it belongs. As mentioned above, our working hypothesis is that the lamellar trajectories making up an RDS pattern are homothetic to the 2D Wulff form of the interphase boundary in the EG studied.

Quasi-isotropic RDS patterns

Quasi-isotropic RDS patterns -i.e. RDS patterns, in which lamellar trajectories remained nearly perpendicular to the growth front throughout the RDS-run -were typical of random EGs in both CBr 4 -C 2 Cl 6 and In-In 2 Bi (Fig. 4). Let us first of all consider the question of the time-evolution of the λ(x) distribution in these patterns. It is generally known that the dynamics of isotropic lamellar patterns (also called "floating" dynamics) is essentially controlled by one variable, the reduced spacing Λ = λ/λ JH = λV 1/2 /K JH . During standard directional solidification, this dynamics includes (i) a process of relaxation toward regularity called λ-diffusion, which occurs inside a certain stability interval, and (ii) instability mechanisms occurring at the bounds of this interval. These bounds are typically located at Λ 1 = 0.8 and Λ 2 = 2.2 at eutectic concentration, the former value corresponding to the onset of a lamella elimination instability and the latter to an oscillatory bifurcation [11,12]. During RDS, the solidification velocity varies according to V = ρω, where ρ is the position along the growth front measured from Ω. Thus, under a local-dynamics assumption which is generally valid in lamellar eutectic growth, the stability bounds of a lamellar pattern during RDS are located at the positions ρ i=1,2 given by Λ i = λ (ρ i ω) 1/2 /K JH . In other words, a lamellar pattern with a uniform spacing at t = 0 should remain stable in the region of the growth front extending from ρ 1 to ρ 2 and undergo the above-mentioned instabilities outside this region.

The experimental observations clearly support this conjecture. Large-scale modulations of λ inherited from the zero-time state were observed to remain unchanged throughout the RDS-runs in agreement with the fact that the λ-diffusion times over the ranges of the modulations were long compared to experiment durations. In Fig. 4a, for instance, λ ranged from 16.4 to 17.6 µm, V ranged from 0.58 to 0.93 µms -1 , and Λ ranged from 0.90 to 1.26. In Fig. 4b, λ ranged from 14.4 to 11.5 µm, V ranged from 0.35 to 0.54 µms -1 , and Λ ≈ 1.2 over the field of view. In both cases, the λ distribution fell well within the stability interval, and did not significantly evolve over time. Still in agreement with the above considerations, the centre of the sample functioned as a lamellar sink as a result of the inwardbound instrumental drift linked to the fact that Ω sat slightly above the growth front in these experiments (see below). In Figure 5, which comes from the same experimental run (RDS1) as Fig. 4a, a lamella elimination event is seen to occur at ρ close to the calculated value of ρ 1 (850µm). A central region of radius about ρ 1 /3 is occupied by a β single crystal, the solid-liquid interface of which assumes a stationary sigmoid shape going from the temperature of the β liquidus on the melting side of the sample to T E on the growing side. No perturbation linked to this local structure was observed beyond the central region.

A space-time map from Run RDS1 is shown in Figure 6. It can be seen that the RDS lamellar trajectories are not circles, but spirals approaching the central region of the sample. This is an instrumental effect due to the non-zero distance of the growth front to Ω in this experiment. In such a case, the growth front remains tangent to the circle of radius R and centre Ω throughout the experimental run, and the orthogonal trajectories to the growth front -which, in isotropic systems, represent the RDS lamellar trajectories -are involutes of this circle. Moreover, the direction of travel along the circle involutes is centripetal when R < 0, which was the case for Run RDS1, and centrifugal when R > 0. We performed leastsquare fittings of the equation of a circle involute to lamellar trajectories taking R as adjustable parameter. The best fit was obtained for R = -70 ± 10 µm for all the trajectories studied. A π-periodic smoothing of the fitting residuals gave practically identical results for two widely (≈ 50λ) apart lamellae and showed that the measured lamellar trajectories significantly deviated from the calculated ones by about 10 -2 rad (see Inset). Whether this deviation was due to weak interfacialanisotropy effects (including interphase boundary anisotropy and, possibly, solidliquid interface anisotropy effects) or some instrumental imperfection could not be ascertained. Similar results were obtained in other quasi-isotropic EGs. Thus, the experimental uncertainty (excluding the errors of unknown magnitude pertaining to the symmetric-pattern approximation) on the 2D Wulff plots determined by our experiments was at most of 1%. This is a minor error compared to the anisotropy effects studied in the remainder of this article. Altogether, these observations substantiate the earlier assumption of a quasi-isotropic lamellar growth in some random EGs of CBr 4 -C 2 Cl 6 and In-In 2 Bi [11,8].

Nearly locked RDS patterns

In some EGs, we observed smooth-looking oval lamellar trajectories, indicative of a floating dynamics coexisting with a pronounced minimum in the 2D Wulff plot. The λ(x) distributions underwent dilatation-compression cycles linked to the oval shape of the lamellar trajectories while remaining essentially uniform at each time point, confirming the presence of an effective λ-diffusion. These RDS patterns were therefore called "nearly locked". An example from a run performed in In-In 2 Bi (Run RDS2) is given in Figure 7. The analysis of this example proceeded as follows. The inclination angle φ of the lamellae at the growth front was measured point by point over a sufficiently broad θ range. Different lamellae yielded identical data within experimental error. The Wulff plot γ(φ) was calculated from the φ(θ) data by integrating the differential equation that relates these two quantities [4]. Various π-periodic smoothing functions were tentatively adjusted onto the γ(φ) data. A good fit was achieved with the function γ(φ) = 1c exp -(φ/w c ) 2 -2 cos (2φ) -4 cos (4φ), where c = 0.0507, w c = 0.195, 2 = 0.0854, 4 = 0.0221, the origin of φ being taken at the minimum of γ(φ) (Fig. 7). The 2D Wulff plot thus determined exhibits a relatively strong mostly twofold anisotropy and a trough with quantitative features (a depth of ≈ 5% and an angular aperture of ≈ 10 o ) comparable to those found for cusps corresponding to low-energy orientations of heterophase boundaries in various materials [START_REF] Sutton | Interfaces in Crystalline Materials[END_REF]. To sum up, Run RDS2 is a representative example of EGs with a strong non-singular (i. e. devoid of cusp-like minima and forbidden angular ranges) interphase boundary anisotropy.

Locked RDS patterns

The contrasting characteristics (travelling vs. steady, irregular vs. regular) of the locked and floating types of lamellar patterns in epitaxial and random EGs, respectively, are illustrated in Fig. 8. The two displayed micrographs were taken during standard (without rotation) directional solidification in thin CBr 4 -C 2 Cl 6 (Fig. 8a) and In-In 2 Bi (Fig. 8b) samples. The similarity between either type of lamellar pattern in the two alloys should also be noted.

Let us now turn to RDS patterns. Most epitaxial EGs exhibited RDS patterns indicative of deep cusps with forbidden vicinal angular ranges in the Wulff plots of the interphase boundary. The space-time maps observed in these EGs showed a great geometrical variety, as could be expected from the large number of orientational degrees of freedom of the EGs. Broadly speaking, however, they all consisted of a succession of different locking or floating angular ranges, separated by transition angular ranges, exhibiting, in general, sawtooth lamellar trajectories. Figure 9 shows an example coming from a run performed in In-In 2 Bi (Run RDS3), which comprised eleven revolutions of the sample that yielded eleven almost identical space-time maps. During this run, the instrumental drift was centrifugal so that, after a few turns, the whole solid originated from the central source and was therefore certain to be a single EG. Globally, Fig. 9 discloses three main angular ranges separated by transition angular ranges of variable widths. A closer examination reveals an unexpected complexity of the growth dynamics in each of them.

The following aspects of locked RDS patterns deserve special attention: 1-Crystallographic locking. Snapshots taken near the two endpoints of a locking angular range during a run called RDS4 are shown in Figure 10. Each of the two snapshots shows a very irregular, essentially time-independent λ(x) distribution. In this example, the local λ values ranged from about 5 to 30µm, and Λ from about 0.5 to 3, which means that most of the lamellae were well outside the stability range of isotropic lamellar patterns and remained nevertheless steady. Incidentally, the Λ values that have just been mentioned were calculated taking into account the fact that λ JH decreases as the contact angles decrease, and is thus smaller in locked than in floating lamellar patterns, everything else equal, which led us to λ JH = 10 ± 1µm in this example. Furthermore, Fig. 10 shows that the rectilinear trajectories of the locked interphase boundaries are perturbed by a sporadic formation of notches of amplitude usually much smaller than λ, which sometimes leads to branching events. This is not understood at all at present. The segments of interphase boundary extending between successive notches appear perfectly straight and parallel to each other within experimental error. During Run RDS4, their inclination angle was measured to be a linear function of θ with slope unity over a θ-range of more than 45 o . Thus, these segments were undoubtedly locked onto a fixed crystallographic direction.

2-Transition angular ranges. A view of sawtooth RDS lamellar trajectories observed in In-In 2 Bi (Run RDS5) is shown in Figure 11. These trajectories are composed of relatively long locked and unlocked segments separated by sharp edges, indicating that the angular range under consideration is a forbidden range for the interphase boundary. The unlocked segments do not seem to correspond to a definite floating state of the system, but are rather part of a nonlinear oscillatory mode of the lamellar pattern. This view is further supported by the periodicity in space and time assumed by the locking-unlocking process in some places. Solitary waves sweeping large portions of the growth front bring about an additional complexity. This complex dynamics should be studied by standard directional solidification in the future.

3-Multistability. A coexistence of distinct lamellar patterns along the growth front can be seen in each of the main angular ranges of Fig. 9. A definitive identification of the corresponding metastable branches of lamellar states would require metallographic data about the 3D orientation of the lamellar planes, which are not available. The following remarks are therefore somewhat conjectural. Of the branches displayed in Fig. 9, two (1a and 3a) are visibly floating, and the others locked. The floating patterns 1a and 3a could plausibly be occurrences of a single 2D nearly locked branch similar to the one studied in Section 3.2. The locked patterns 1b, 2a and 3b could also correspond to 2D branches. In contrast, the 2b locked pattern, which covers essentially the same broad (≈ 65 o ) angular range as 2a (and is moreover separated from 1a by a very discontinuous transition process), is likely to be an out-of-plane pattern. This illustrates the relatively frequent occurrence of metastability between different kinds of locked and floating lamellar patterns in RDS of epitaxial EGs.

Conclusion

This study has shown that the form of lamellar trajectories yielded by rotating directional solidification (RDS pattern) is a reproducible, eutectic-grain specific feature, which has a crystallographic origin. The hypothesis that a RDS patterns is approximately homothetic to the Wulff form of the interphase boundary in the sample plane was found to be compatible with most of the experimental observations. Numerical studies will be needed to evaluate the error committed using this approximation.

The link between 2D Wulff plots acquired by RDS and interphase orientation relationships in the EGs is not yet fully clarified. Three distinct categories (quasiisotropic, nearly locked, locked) of RDS patterns have been highlighted during this study. A locked RDS pattern unambiguously indicates an epitaxial EG, i.e., an EG where the interphase boundary has singular low-energy orientations. Quasiisotropic RDS patterns, which were observed in more or less randomly oriented EGs, are likely to originate from EGs with quasi-isotropic 3D Wulff plots. Nearly locked RDS patterns pose a somewhat more complex, and particularly interesting, question. There are basically two possibilities. A EG, in which a nearly locked RDS patterns is observed, may be epitaxial in the strict sense, but oriented in such a way that the singularities of the 3D Wulff plot are outside the sample planethe smooth trough in the 2D Wulff plot corresponding to a smooth region of the depression surrounding a 3D singularity. Or the EG itself may be only pseudoepitaxial, meaning that the surface tension minima of the interphase boundary are smooth. X-ray and EBSD studies would be necessary to clarify this point. In any case, the discovery of nearly locked RDS patterns during this study demonstrated that a steep inclination of the lamellae with respect to the growth direction is not necessarily associated with a dynamical locking (i.e. an absence of λ-relaxation mechanism), but can coexist with a floating dynamics. The inclined but nevertheless essentially regular lamellar patterns that have been observed in some Al-based eutectics [START_REF] Hecht | [END_REF] might belong to the nearly-locked category.

Figure 1: Repeat unit of a lamellar eutectic pattern in a system with a strong interfacial anisotropy of the interphase boundary according to the symmetric-pattern approximation. α, β: eutectic phases. L: liquid. z: direction of the thermal gradient (growth direction). y: normal to the sample plane. x: average direction of the solid-liquid interface. φ: inclination angle. σ: surface tension force of the interphase boundary. γ α and γ β : surface tension forces of the supposedly isotropic solid-liquid interfaces.

Figure 2 :

 2 Figure 2: Technical sketch of the RDS stage. P: sole plate fixed on the translation stage of the microscope. T-and T+: cold and hot blocks fixed on the sole plate. H: housing. W: sprocket wheel on the periphery of a rotatable circular plate. M1: translation motor. M2: rotation motor. S: worm screw. V T : translation speed. ω: angular speed. xyz: laboratory reference frame.

Figure 3 :

 3 Figure 3: Protocol for an RDS experimental run. (a) Seed crystal and eutectic-grain selection; (b) Standard directional solidification; (c) Start of RDS; (d) Long-duration RDS. Cross: centre of rotation (Ω). Empty rectangles: hot and cold blocks of the thermal bench.

Figure 4 :

 4 Figure 4: Snapshot taken during RDS experimental runs displaying quasi-isotropic RDS patterns. The centre of rotation was located outside the field of view, slightly above the growth front, at a distance of about 1.2 mm from the left edge of the images. a) CBr 4 -C 2 Cl 6 ; ω = 5.0 × 10 -4 rad s -1 (Run RDS1). The thicker lamellae are α and the thinner ones β. Inset: enlarged view (width: 100µm). b) In-In 2 Bi. ω = 2.5 × 10 -4 rad s -1 . The phase appears white and In 2 Bi black. Bars: 100µm.

Figure 5 :

 5 Figure 5: Snapshot of the central region during Run RDS1. Cross: calculated position of Ω. Bar: 100µm.

Figure 6 :

 6 Figure 6: Space-time map from Run RDS1. The direction of growth is counterclockwise. Straight line: radial cut. For the blank central area, see Fig. 5. The rare lamella termination events that occurred inside the stability range were due to extraneous particles being pushed along by the growth front. L1, L2: best-fitting circle involutes. Image size: 3.675 × 3.675mm 2 . Inset: Residual plots of the best-fitting circle involutes. Curves: smoothed curves through the residuals.

Figure 7 :

 7 Figure 7: Top: Space-time map over a π/2 angular range of a nearly locked lamellar pattern (Run RDS2). In-In 2 Bi eutectic. ω = 2.5 × 10 -4 rad s -1 . Arrow: growth direction. For easy readability, the image has been binarized and every other In 2 Bi lamella has been removed. Bar: 100 µm. Inset: φ(θ) function measured along the two emphasized lamellae. Bottom: Calculated 2D Wulff form and 2D Wulff plot of the interphase boundary. Symbols: point-by-point calculation. Continuous lines: smoothing function.

Figure 8 :

 8 Figure 8: Real-time observation of locked (left) and floating (right) lamellar eutectic patterns colliding at a eutectic-grain boundary during standard thin-sample directional solidification. The growth direction is directed vertically upward. a) CBr 4 -C 2 Cl 6 . Growth velocity: V=1.0 µms -1 . The thicker lamellae are α and the thinner ones β. b) In-In 2 Bi. V=0.5 µms -1 . The phase appears white and In 2 Bi black. Bars: 20 µm.

Figure 9 :

 9 Figure 9: Space-time map from Run RDS3. In-In 2 Bi eutectic. ω = 2.5 × 10 -4 rad s -1 . The direction of growth is clockwise. R 1 , R 2 , R 3 : main angular ranges. T 12 , T 23 , T 31 : transition angular ranges. 1a to 3b: different types of lamellar patterns. The radial cut is located in the T 23 range on the left-hand side of the map. Image size: 2.4 × 2.4 mm 2 .

Figure 10 :

 10 Figure 10: Top: Snapshots taken at the indicated θ-values within the same locking angular range. CBr 4 -C 2 Cl 6 (Run RDS4). ω = 5 × 10 -4 rad s -1 . Horizontal dimension: 600µm. Bottom: φ(θ) function. Symbols: measured values. Continuous line: best-fit linear function (slope 1± 0.007). The error bars are smaller than the symbol size.

Figure 11 :

 11 Figure 11: Transition from a locking to a floating angular range during a RDS run. In-In 2 Bi eutectic (Run RDS5). ω = 2.5 × 10 -4 rad s -1 . Top: Space-time map. Arrow: growth direction. Straight lines: positions of the growth front at the indicated θ-values. Vertical dimension: 340µm. Bottom: Snapshot taken at θ = 35 o . Horizontal dimension: 300µm.

Acknowledgments

We are greatly indebted to C. Picard, who drew the plans of, and built the RDS setup. We warmly thank P. Ott for her help in the preparation of the samples. We gratefully acknowledge numerous fruitful discussions with M. Plapp, U. Hecht, S. Rex and V. Witusiewicz. We also thank J. Van Suchtelen for communicating inspiring excerpts from an unpublished RDS investigation of lamellar eutectic growth. One of us (MS) was supported financially by the Centre National d'Etudes Spatiales, France.