Morvan Ouisse
email: morvan.ouisse@femto-st.frmanuel.collet@ec-lyon.fr

Manuel Collet

Fabrizio Scarpa
email: f.scarpa@bristol.ac.uk

A piezo-shunted kirigami auxetic lattice for adaptive elastic wave filtering

Keywords: numbers: 46.40.-f, 77.84.Lf, 62.30.+d periodic structures, elastic wave propagation, band gaps, piezoelectric shunt, kirigami, auxetic structures

Tailoring the dynamical behavior of wave-guide structures can provide an efficient and physically elegant approach for optimizing mechanical components with regards to vibroacoustic propagation. Architectured materials as pyramidal core kirigami cells combined with smart systems may represent a promising way to improve the vibroacoustic quality of structural components. This paper describes the design and modelling of a pyramidal core with auxetic (negative Poisson's ratio) characteristics and distributed shunted piezoelectric patches that allow for wave propagation control. The core is produced using a kirigami technique, inspired by the cutting/folding processes of the ancient Japanese art. The kirigami structure has a pyramidal unit cell shape that creates an in-plane negative Poisson's ratio macroscopic behavior. This structure exhibits in-plane elastic properties (Young's and shear modulus) which are higher than the out-of-plane ones, and hence this lattice has very specific properties in terms of wave propagation that are investigated in this work. The short-circuited configuration is first analysed, before using negative capacitance and resistance as a shunt which provides impressive band gaps in the low frequency range. All configurations are investigated by using a full analysis of the Brillouin zone, rendering possible the deep understanding of the dynamical properties of the smart lattice. The results are presented in terms of dispersion and directivity diagrams, and the smart lattice shows quite interesting properties for the adaptive filtering of elastic waves at low frequencies bandwidths.

Introduction

Embedding smart materials in structures appears to be a very efficient way to provide new functionalities to users. In this work we focus on the dynamical behavior of twodimensional waveguides which are designed such that they can bring new properties in terms of elastic wave propagation, with a particular focus on the adaptivity of the system. Structures with smart mechanical behavior are most of the time designed for static and low frequency performances (see [START_REF] Preumont | Vibration control of structures : An introduction[END_REF][START_REF] Nelson | Active Control of Sound[END_REF][START_REF] Banks | Smart material structures Modeling Estimation and Control[END_REF] among many others). Recently, research activities have been developed to design new multi-functional structures that integrate electro-mechanical systems able to control their vibroacoustic behavior over a wide frequency band [START_REF] Thorp | Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches[END_REF][START_REF] Collet | Wave Motion Optimization in Periodically Distributed Shunted Piezocomposite Beam Structures[END_REF]. However, there is still a lack of studies in open literature related to the medium frequency (MF) optimization of the structural vibration. This paper is related in particular to this topic, with a focus on the ability of controlling elastic waves in auxetic lattices whose behavior is tuned with piezoelectric patches shunted with electric circuits. The work shown in this paper offers a contribution towards the solving of the challenges of the design and implementation of new classes of integrated smart metacomposites capable of improving the engineering performance in terms of mechanical and vibroacoustic behavior, as compared to strictly passive structures. We are interested here on the class of metacomposites that combine two different aspects of vibration control, namely the concept of periodic structures and the ability to change the mechanical behavior of structures using smart materials.

Recent research works about periodic structures are often presented as inspired by developments related to metamaterials, in particular within the context of light propagation. Many research efforts have been undertaken to investigate ways to design and construct photonic crystals exhibiting photonic band gaps that prevent light from propagating in certain directions at specific frequencies, or able to propagate light in anomalous and useful ways (i.e. negative refraction and artificial magnetism). These concepts can be extended to acoustic or elastic waves, defining the so-called phononic crystals which consists in periodic arrangement of the media in which the waves propagates in order to build interferences and/or diffraction effects that radicaly change the dispersion properties [START_REF] Yang | Ultrasound tunneling through 3D phononic crystals[END_REF]. The position of the Bragg band gap in the frequency domain is directly related to the spatial period of the crystal. This is interesting from the designer's point of view, but quite limited for practical applications in the audible frequency range in which the wavelengths of interest may be of dozens of centimetres or meters. An efficient way to eliminate this constraint consists in using locally resonant unit cells. In such approach a new frequency gap is opened around the resonance frequency, which may be tuned by changing the mass or stiffness of the resonator. [START_REF] Liu | Locally resonant sonic materials[END_REF] had demonstrated that a resonant sonic crystal including rubber-coated lead balls exhibits a low-frequency sonic band gap, and the resonance can provide a maximum impedance mismatch to shield against airborne sound. The same effect can be obtained using Helmoltz resonators as showed by Fang et al. [START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF][START_REF] Ambati | Surface resonant states and superlensing in acoustic metamaterials[END_REF] or Hu [START_REF] Hu | Two-dimensional sonic crystals with helmholtz resonators[END_REF]. Similar techniques can be applied to control propagation of elastic waves in structures [START_REF] Hussein | Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook[END_REF]. One of the ways to add a resonant system to the structure of interest without modifying substantially an original design consists in using piezoelectric patches, which are shunted with RL circuits [START_REF] Park | Vibration control of beams with negative capacitive shunting of interdigital electrode piezoceramics[END_REF]. The work of Hagood and Von Flotow [START_REF] Hagood | Damping of structural vibrations with piezoelectric materials and passive electrical networks[END_REF] should be mentioned as a precursor in this strategy, with practical design rules that still remain used today. The tunable characteristics of shunted piezo-patches allow the equivalent mechanical impedance of the structure to be tuned so that stop bands are generated over desired frequency ranges. The presence of a resistance in the shunt circuit generates a damped resonance of the electrical network. The resistance also allows the energy dissipation mechanism of shunted piezos to be exploited, which dampens the amplitude of vibrations also outside the stop bands. The original periodic shunting concept was numerically demonstrated on rods and fluid-loaded axisymmetric shells in [START_REF] Thorp | Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches[END_REF], before being extended to plates [START_REF] Casadei | Broadband vibration control through periodic arrays of resonant shunts: experimental investigation on plates[END_REF][START_REF] Spadoni | Vibration and wave propagation control of plates with periodic arrays of shunted piezoelectric patches[END_REF][START_REF] Chen | Wave propagation and attenuation in plates with periodic arrays of shunted piezo-patches[END_REF], where the Bloch theorem was used to predict the dispersion properties of the resulting periodic structure. This kind of approach inspired many recent contributions, and the research work for developing tools that can help engineers to design such systems is high [START_REF] Wang | Large low-frequency vibration attenuation induced by arrays of piezoelectric patches shunted with amplifier-resonator feedback circuits[END_REF][START_REF] Zhu | Experimental study of an adaptive elastic metamaterial controlled by electric circuits[END_REF]. However, the main limitation of this approach is the narrow-band effectiveness of the resonant circuits. An alternative originally proposed by Forward [START_REF] Forward | Electronic damping of vibrations in optical structures[END_REF] consists in using negative capacitance shunts. By tuning this capacitance in accordance with the effective capacitance of the embedded patch, the total impedance of the shunt circuit reduces to that of the remaining circuit, which opens the way to any arbitrary frequency-dependent effective impedance, that finally provides the desired behavior to the structure. Using this kind of approach for effective determination of the proper electrical impedance requires the use of optimization algorithms [START_REF] Livet | Structural multi-modal damping by optimizing shunted piezoelectric transducers[END_REF][START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF], hence allowing the frequency bands of interest to be extended in very large ways. Although the negative capacitance shunting strategy has been experimentally validated, it must be used with caution since it requires active elements that can destabilize the structure if improperly tuned [START_REF] Fukada | Sound isolation by piezoelectric polymer films connected to negative capacitance circuits[END_REF][START_REF] Kim | Broadband noise reduction of piezoelectric smart panel featuring negative-capacitive-converter shunt circuit[END_REF]. Efficiency band and stability can be improved by using specific parameters and circuit architecture [START_REF] Beck | Response-based tuning of a negative capacitance shunt for vibration control[END_REF]. This strategy may be followed to design new reconfigurable systems that can handle several functions with a single shunt circuit, allowing for example an effective change from wideband absorption to wideband reflection of elastic [START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF][START_REF] Tateo | Design variables for optimizing adaptive metacomposite made of shunted piezoelectric patches distribution[END_REF][START_REF] Collet | Adaptive metacomposites for vibroacoustic control applications[END_REF] or acoustic waves [START_REF] Collet | Semi-active optimization of 2D wave dispersion into shunted piezo-composite systems for controlling acoustic interaction[END_REF][START_REF] Zhang | Sound transmission loss of metamaterial thin plates with periodic subwavelength arrays of shunted piezoelectric patches[END_REF]. Electrical shunts may also reproduce physical phenomenon with analog components [START_REF] Lossouarn | Multimodal vibration damping of a beam with a periodic array of piezoelectric patches connected to a passive electrical network[END_REF][START_REF] Lossouarn | Multimodal coupling of periodic lattices and application to rod vibration damping with a piezoelectric network[END_REF]. These works all contribute to the development of what has been called programmable matter by Toffoli [START_REF] Toffoli | Programmable matter: Concepts and realization[END_REF] to refer to an ensemble of computing elements arranged in space, now extended to smart materials based on distributed piezoelectric actuators able to modify the inherent vibroacoustic properties of a given structure.

Innovative manufacturing techniques have been recently developed for the production of composite structures with complex shapes and functions [START_REF] Miura | Method of packaging and deployment of large membranes in space[END_REF][START_REF] Miura | Concept of the tension truss antenna[END_REF], and the kirigami process is one of them. kirigami is the ancient Japanese art of folding and cutting paper and has been applied to produce complex 3D cellular structures using modular moulding techniques [START_REF] Nojima | Development of newly designed ultra-light core structures[END_REF][START_REF] Saito | A cellular kirigami morphing wingbox concept[END_REF][START_REF] Hou | Graded conventional-auxetic kirigami sandwich structures: Flatwise compression and edgewise loading[END_REF], foldable structures [START_REF] Jianguo | Morphology analysis of a foldable kirigami structure based on miura origami[END_REF], robotic devices [START_REF] Sareh | Kirigami artificial muscles with complex biologically inspired morphologies[END_REF] and nanoscale artificial materials [START_REF] Hanakata | Highly stretchable mos2 kirigami[END_REF]. The interest reader may find a review of origami and kirigami engineering applications in [START_REF] Turner | A review of origami applications in mechanical engineering[END_REF]. Negative Poisson's ratio (or auxetic) solids have been extensively studied during the past three decades [START_REF] Lakes | Foam structures with a negative poisson's ratio[END_REF][START_REF] Evans | Molecular network design[END_REF]. The term auxetics covers a wide range of materials and structures exhibiting a negative Poisson's ratio. In cellular configurations, a negative Poisson's ratio can be achieved in re-entrant centre-symmetric (butterfly) honeycombs [START_REF] Gibson | The mechanics of three-dimensional cellular materials[END_REF][START_REF] Scarpa | Numerical and experimental uniaxial loading on in-plane auxetic honeycombs[END_REF], rotating rectangles and triangles [START_REF] Grima | Auxetic behaviour from connected different-sized squares and rectangles[END_REF], as well as arrow-head [START_REF] Larsen | Design and fabrication of compliant micromechanisms and structures with negative poisson's ratio[END_REF] and star-shaped configurations [START_REF] Grima | On the potential of connected stars as auxetic systems[END_REF]. The centre-symmetric auxetic configuration has also been considered as a basis for gradient cellular structures [START_REF] Lira | A gradient cellular core for aeroengine fan blades based on auxetic configurations[END_REF][START_REF] Prall | Properties of a chiral honeycomb with a poisson's ratio of -1[END_REF]. In a more general way, strategies allowing tailoring of mechanical properties [START_REF] Sigmund | Tailoring with prescribed elastic properties[END_REF] have reported for obtaining a Poisson's ratio up to -1 in 2D [START_REF] Cabras | Auxetic two-dimensional lattices with poisson's ratio arbitrarily close to-1[END_REF] or 3D [START_REF] Cabras | A class of auxetic three-dimensional lattices[END_REF]. All these configurations have been investigated in terms of manufacturing possibilities and mechanical performances, mainly in the static domain. However, new research activities in the areas of vibroacoustics of auxetic structures have been performed in recent years. The negative Poisson's ratio, which provides an unusual large volume deformation during loading induces tunable wave propagation directivities not commonly observed in classical systems. Recently, some kirigami auxetic cellular structures have also been deeply investigated in terms of wave propagation [START_REF] Scarpa | Kirigami auxetic pyramidal core: mechanical properties and wave propagation analysis in damped lattice[END_REF] and the concept has been pushed toward its limits with a null Poisson's ratio that induces negative stiffness regime under nonlinear deformation and high energy dissipation under cyclic loading [START_REF] Virk | Silicomb peek kirigami cellular structures: mechanical response and energy dissipation through zero and negative stiffness[END_REF]. Readers may refer to the recent monograph that summarizes major landmark works in auxetics research and development [START_REF] Lim | Auxetic materials and structures[END_REF].

Among the properties of interest for wave propagation in smart periodic structures, the directionality has been little investigated [START_REF] Scarpa | Kirigami auxetic pyramidal core: mechanical properties and wave propagation analysis in damped lattice[END_REF][START_REF] Wen | Directionality of wave propagation and attenuation in plates with resonant shunting arrays[END_REF]. However, the adaptive character of the electrical shunts may be a very efficient way to force the waves to follow a path of interest, for several purposes like isolation or energy harvesting. In this paper, we numerically investigate a strategy for controlling directionality properties of a kirigami auxetic lattice with the pyramidal topology described in [START_REF] Scarpa | Kirigami auxetic pyramidal core: mechanical properties and wave propagation analysis in damped lattice[END_REF]. Pyramidal lattices have been developed in recent years as lightweight cores for energy absorption [START_REF] Pingle | Collapse mechanism maps for a hollow pyramidal lattice[END_REF][START_REF] Hammetter | Pyramidal lattice structures for high strength and energy absorption[END_REF][START_REF] Yin | In-plane compression of hollow composite pyramidal lattice sandwich columns[END_REF] and unusual stretchable-bendable composites [START_REF] Yin | Stretch-bend-hybrid hierarchical composite pyramidal lattice cores[END_REF]. Concerning the kirigami pyramidal lattice of reference [START_REF] Scarpa | Kirigami auxetic pyramidal core: mechanical properties and wave propagation analysis in damped lattice[END_REF], the unusual deformation properties and its wave propagation characteristics are also due to the fact that the system is only partially auxetic [START_REF] Bilski | Tailoring poisson's ratio by introducing auxetic layers[END_REF], i.e., it exhibits auxetic behaviour only in the xy and yx planes. It is also worth noticing that a theoretical negative Poisson's ratio effect may be even induced in these structures if a geometrical perturbation of the lattice architecture is applied [START_REF] Aa Pozniak | Computer simulations of auxetic foams in two dimensions[END_REF]. Moreover, since 1987 it is known that it is not necessary to arrange auxetic structures by hand such structures, even in the isotropic case, can be obtained by a spontaneous self-organization of properly selected objects. Such objects can be hard, as in [START_REF] Wojciechowski | Constant thermodynamic tension monte carlo studies of elastic properties of a two-dimensional system of hard cyclic hexamers[END_REF] or soft [START_REF] Wojciechowski | Two-dimensional isotropic system with a negative poisson ratio[END_REF]. Obviously, depending on the shape of the objects forming them, the obtained structures can be auxetic or not, and can have various symmetries as illustrated, e.g. in [START_REF] Wojciechowski | Elastic properties of dense solid phases of hard cyclic pentamers and heptamers in two dimensions[END_REF].

The control is achieved through the use of shunted piezoelectric patches glued onto a periodical distribution of auxetic composite cells. This combination of both property induced by auxeticity and negative shunt circuit is analyzed in terms of wave propagation properties. This controlling capability is obtained by correctly tuning the parameters of the external circuit by which almost arbitrary effective structural impedance may be obtained. The particular bandwidth considered in this work (0 to 2.5 kHz) is typical of low frequency ranges excitations in most large scale airframe and transport structures. As it will be clear from the numerical results shown in the following paragraphs, the rich wave dispersion behaviour of this lattice in that particular bandwidth makes it particularly suitable as an active material vibration control platform for potential vibroacoustic applications at low frequencies. The paper is organized as follows: section 2 presents the structure of interest, section 3 recalls the methodology which has been applied here, section 4 provides the properties of the smart lattice when all piezoelectric patches are short-circuited, section 5 is dedicated to the analysis of the impact of a negative capacitance as shunt, section 6 considers the addition of a resistance in the shunt circuit, and conclusions are derived about the properties in terms of wave propagation of the smart kirigami auxetic lattice.

Description of the piezo-shunted kirigami auxetic lattice

In this section, the piezo-shunted kirigami auxetic lattice is described. Following the concept introduced in [START_REF] Scarpa | Kirigami auxetic pyramidal core: mechanical properties and wave propagation analysis in damped lattice[END_REF], the lattice is composed by a periodic distribution of kirigami -produced unit cells from kevlar/epoxy prepregs (Kevlar 49/914). Each cell includes shunted piezoelectric patches located on the base of the structure, as shown in figure 1. The mechanical properties of the host structure material are a Young's modulus of 30 GPa, Poisson's ratio of 0.4, loss factor of 10 -3 and density equal to 1600 kg.m -3 . The unit cell is a 120×120 mm 2 square, the legs are 17 mm width, 2.5 mm thick. The geometric configuration of the lattice was similar to the one shown in reference [START_REF] Scarpa | Kirigami auxetic pyramidal core: mechanical properties and wave propagation analysis in damped lattice[END_REF]. The in-plane Poisson's ratios ν xy = ν yx was -0.88, corresponding to a set of dimensionless parameters θ = 25 o and δ = 5.0. The piezoelectric patches are located at the bottom of the legs, they are 17 mm width, 25 mm long and 0.2 mm thick. The angle formed by the legs is such that the height of the central square is 25 mm. The piezoelectric ceramic is Lead Zirconate Titanate (PZT-4) [START_REF] Hooker | Properties of pzt-based piezoelectric properties of pzt-based piezoelectric ceramics between -150 and 250 • c[END_REF].

In the various configurations that will be investigated in this paper, the four patches may have an off/on shunt a pure negative capacitance (called Cneg) or a negative capacitance with resistance (called RCneg) circuit. The negative capacitance chosen is related to the cell instability point: an increase of the absolute value of the shunt capacitance results in a decrease of the effective cell stiffness until instability occurs. Figure 2 illustrates the change in the value of the first complex eigenfrequencies of the clamped-clamped cell when negative capacitance ratio α changes. This ratio is defined as C neg = -αC o where C o = 12.68 nF is the effective capacitance of the glued piezoelectric patch. The figure shows that one of the eigenfrequencies becomes imaginary for α = 1.61. This corresponds to the optimal point to tune the imaginary part of the electrical impedance [START_REF] Livet | Structural multi-modal damping by optimizing shunted piezoelectric transducers[END_REF][START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF][START_REF] Tateo | Design variables for optimizing adaptive metacomposite made of shunted piezoelectric patches distribution[END_REF][START_REF] Collet | Adaptive metacomposites for vibroacoustic control applications[END_REF][START_REF] Beck | Response-based tuning of a negative capacitance shunt for vibration control[END_REF].

The shunt circuit may also include an electrical resistance to impact the frequency efficiency of the system [START_REF] Beck | Response-based tuning of a negative capacitance shunt for vibration control[END_REF][START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF]. In the following computations, both R = 0 (pure C neg circuit) and R = 500Ω are used. Investigations presented in the paper will include results on the impact on the directivity of the lattice when dissipative elements are included in the shunt circuit [START_REF] Tateo | Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control[END_REF][START_REF] Tateo | Design variables for optimizing adaptive metacomposite made of shunted piezoelectric patches distribution[END_REF]. In order to distinguish the various configurations, the 4 branches of the unit cell are identified through cardinal directions (W, N, E, S). The E direction is aligned with direction 1 in figure 1, and corresponds to the reference axis for the angle ϕ that will be defined later. The N direction is aligned with direction 2 in figure 1. The lattice is obtained by repeating the unit cell in EW and NS directions, as illustrated in figure 3.

Methods and indicators for the dispersion analyses of the smart lattice

The dispersion diagrams are evaluated using the approach described in [START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF]. The computational method consists in reformulating the Floquet-Bloch theorem for computing waves dispersion for periodically smart distributed mechanical systems incorporating electronic components, damping effects or any frequency-dependent characteristics. The finite element approach used in this work is based on a fully 3D formulation of the periodic piezo elastodynamic problem introducing the wave number as the unknown eigenvalue of the the Floquet-Bloch shift cell operator [START_REF] Collet | A floquet-bloch decomposition of the elastodynamical equations: application to bi-dimensional wave's dispersion computation of damped mechanical system[END_REF]. The mesh size has been chosen to guarantee the convergence of the method in accordance to reference [START_REF] Collet | A floquet-bloch decomposition of the elastodynamical equations: application to bi-dimensional wave's dispersion computation of damped mechanical system[END_REF]. Simulations are performed on the first Brillouin zone of the lattice [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. The shape of this zone changes, depending on the symmetry of the problem, which may be broken when non identical shunt circuits are chosen in the various branches of the cell. In any case, the whole Brillouin zone is investigated to understand properly the properties of the system in terms of wave propagation. Following this objective, the frequency ω is fixed, together with the wave angle ϕ in the reciprocal space, and the wave number k is computed through a generalized eigenvalue problem. The problem being damped from the structural loss factor and by the piezoshunt coupling, k is complex. Its real part corresponds to propagative component, while its imaginary part is related to spatial attenuation. The wave number vector k has k cos ϕ and k sin ϕ components in the plane. Finally, ω is varied on the frequency range of interest, and θ is varied to cover the first Brillouin zone and the eigenvalue problem is solved for each configuration.

For each solution, the group velocity [START_REF] Maysenhölder | [END_REF] is computed by

C g n (ω, ϕ) = ∇ k ω = S e tot = I E tot ( 1
)
where : is the spatial and time average respectively on one cell and one period of time, S is the density of energy flow, I the mean intensity and e tot , E tot the total energy and its time average on a period (see [START_REF] Maysenhölder | [END_REF] for details). Due to the geometry and physical complexity of the lattice, the eigenvalue solver may return a very large number of solutions. Adequate filtering is necessary in order to determine the outputs of interest for the understanding of the physical phenomenon arising in this metacomposite. Hence two criteria are used for filtering the waves. The first one restricts the analysis only to wave numbers with real parts inside the first Brillouin zone, ie

Re(k) < τ BZ k BZ (ϕ) (2)
where k BZ (ϕ) is the length of the line investigated in the Brillouin zone at angle ϕ and τ is a threshold. Due to the shape of the Brillouin zone (square or triangle depending on the possible symmetry in the shunts),

k BZ (ϕ) = π L cos ϕ for ϕ ∈ 0 ; π 4 (3)
or

k BZ (ϕ) = π L cos(π/2 -ϕ) for ϕ ∈ π 4 ; π 2 ( 4
)
where L is the length of the square cell (120 mm for the considered case). τ should theoretically have a value of 1, but numerical issues may provide wave number with real parts slightly lower than k BZ (ϕ). This is the reason why in the numerical presented hereafter, a value of τ BZ = 0.99 has been used. The second criterion is related to the evanescent nature of the waves of interest. In the work described in this paper, both damping in the physical model of the structure and complex electrical impedance used to shunt the piezoelectric patch provide complex wave numbers as solutions of the eigenvalue problem [START_REF] Collet | A floquet-bloch decomposition of the elastodynamical equations: application to bi-dimensional wave's dispersion computation of damped mechanical system[END_REF]. The waves of interest for describing the dynamical behavior of the smart kirigami lattice being those that can propagate over several wavelengths, the highly evanescent waves may be removed from the dispersion diagrams. They are computed and correspond to important signature of the structure, in particular for the determination of the forced response of the system, but do not correspond to propagative effects of interest in this paper. The complexity of the wavenumber may be quantified by the ratio of imaginary part to its real part, hence only waves that verify

|Imag(k n )| |Real(k n )| ≤ τ e (5)
are considered in the analysis. Real(•) stands for real part and Imag(•) for imaginary part. In the results presented hereafter, a value of τ e = 0.8 has been used. It should be emphasized that this value is quite high since it corresponds to 96% of reduction in amplitude after a single wavelength. A lower value of τ e may be used to be more selective on the waves of interest. Finally, high-order evanescent waves appearing in the spectrum as quadruplets of eigenvalues (k n , kn , -k n , -kn ) are also removed from the analysis [START_REF] Maysenhölder | [END_REF].

Short-circuited reference configuration

4.1. Dispersion along Γ -X direction of the short-circuited reference configuration

The dispersion curves of the system along the Brillouin zone Γ -X direction (i.e for ϕ = 0) are shown in figure 4 for a closed circuit (ie, the piezo electrodes are in short circuit). The figure shows the real part of the non dimensional wavenumbers defined as k n (ω, ϕ) k BZ (ϕ) when no filtering is applied. As expected, numerous solutions are found, highlightning both the rich wave dispersion behaviour but also the complexity of the involved analysis.

After applying the three filters with τ BZ = 0.99 and τ e = 0.8, the dispersion diagram shown in figure 5 is obtained. It should be recalled that the calculations are performed by making ω varying in the frequency range of interest, which explains the fact that the dispersion curves do not reach the 0 and 1 values of the reduced wave number. A refined frequency step in the analysis would bring the dispersion curves to the borders of the domain. The links between the dots is done by tracking the shapes of the eigenvectors using a correlation criteria [START_REF] Collet | A floquet-bloch decomposition of the elastodynamical equations: application to bi-dimensional wave's dispersion computation of damped mechanical system[END_REF]. The results show the complexity of the vibroacoustic behavior of the system with three band gaps in the frequency range of interest, namely 0 to 2500 Hz. Between these band gaps, several waves are able to transport energy along the lattice. All dispersion diagrams presented hereafter use the same filters.

The modes occurring with the highest wavenumber at low frequencies correspond to an out-of-plane movement with propagation along the Γ -X axis, with the transverse legs moving creating a macroscopic flexural effect that can be compared to the first antisymmetric Lamb-wave mode (often called A 0 ) associated with the largest wavenumber in the low frequency band. An example is shown in figure 6, corresponding to the red box in figure 5. It should be emphasized that all shapes presented in the figures of this paper are associated to the propagation direction 1 (see figure 1 for the system of reference), ie EW or ϕ = 0. Moreover, the terminology used for the discrimination refers to this direction. In order to help readers to understand the behavior of the structure, in the corresponding figures, a color scale is used to show the distribution of modal kinetic energy

e n kin = 1 2 ρω 2 u n • u * n ( 6
)
where ρ is the density, u n is the shape of mode n and u * n its conjugate. In the same frequency range, in-plane shear motion of the cell can also be involved in the movement. This movement is illustrated in figure 7, corresponding to the yellow box highlighted in figure 5.

The last mode that is likely to occur in the same frequency range corresponds to the second flexural mode with anti-symetrical movements of NS and EW legs. It has the lowest wave number of the three described motions and may be interpreted as analogous to the second antisymmetric Lamb-wave mode (A 1 ) of an homogeneous plate with a very low cut-off frequency (for an elastic plate, the cut-off frequency is defined as the frequency above which higher modes like antisymmetric A 1 or symmetric S 1 appear). The associated shape is shown in figure 8, corresponding to the green box highlighted in figure 5. These three modes may participate to the response of the lattice up to 460 Hz. Between 460 and 650 Hz, a first band gap occurs, with a width to mean frequency ratio equals to 34%. Then, two branches appear, one corresponding to the second flexural mode (see figure 9, associated to cyan square in diagram 5) and to the first torsional mode which is shown in figure 10, corresponding to blue square highlighted in figure 5.

A second band gap then occurs between 950 and 1680 Hz, corresponding to a width to mean ratio equals to more than 55%, which is very high and may be efficiently used for practical filtering applications.

Still going up in frequency range brings the second torsional mode, illustrated in figure 11 and corresponding to purple square in figure 5. It may be emphasized that this mode generates electric charges only in the W and E piezoelectric patches, the N and S ones being almost unstrained. This point will explain some of the observations described in section 5. Close to the previous torsion dispersion line, the third flexural mode is observed, as seen in figure 12, corresponding to the gray box highlighted in figure 5. It involves strain mainly in N and S piezoelectric patches. The last band gap occurring in the frequency range of interest starts around 1950 Hz up to the upper analysis frequency of 2500 Hz.

While the piezoelectric patch is short-circuited, its impact on the dynamical behavior corresponds essentially to an added stiffness at the lower part of the legs. Before discussing the effect of the shunt circuit (see section 5), the behavior of the system in terms of directivity is addressed. A feature that is common to all the wave modes shown above is the fact that the modal deformations are mostly confined to bending and torsion of the ligaments, with global transverse flexural and shear modes. These wave modeshapes are compatible with the mechanical behaviour of the lattice, with the transverse stiffness (Young and shear modulus) significantly lower than the in-plane uniaxial and shear stiffness [START_REF] Scarpa | Kirigami auxetic pyramidal core: mechanical properties and wave propagation analysis in damped lattice[END_REF].

Directivity of the short-circuited reference configuration

By considering only the propagation along Γ -X (ϕ = 0), the directivity properties of the system can not be observed. A synthetic approach allowing the understanding of the directivity properties of the smart lattice consists in using the evanescence criteria which is computed for each configuration (ω, ϕ) as

Ind(ω, ϕ) = min 1, min n |Imag(k n )| |Real(k n )| , (7)
where n varies over all the computed waves. The saturation with unit value is used since all wave numbers with higher imaginary part than real part are spatially attenuated very rapidly. It should be emphasized that in this directivity analysis, no wave filtering is applied: all possible waves (among which out-of-plane, in-plane, shear movements) are provided by the calculation and included in the analysis. As a consequence, when the indicator Ind is close to 1, no wave can propagate in the system, whatever the excitation is: only highly evanescent waves can be involved in the movement of the structure.

Figure 13 shows the directivity diagram. In the frequency range of interest, the system exhibits a behavior with a reduced signature in terms of directivity: before 460 Hz, waves can propagate along any direction in the plane. The non-omnidirectional character of the lattice can be observed starting from the second band gap observed in figure 5: between 950 Hz and 1250 Hz, no wave can propagate along the ϕ = 0 direction, while the analysis performed for ϕ = 45 • shows that the waves are not filtered in this direction below 1250 Hz. Hence the system acts as a mechanical filter in this band. Above 1250 Hz (and below 2500 Hz which is the maximum frequency used in the analysis), the lattice is almost omnidirectional with the band gaps already commented for ϕ = 0.

Analysis of the shunted device: pure Cneg case

In this section, a pure negative capacitance (Cneg) shunt is used and the dispersion analysis is performed for two cases:

• case EW: only E and W patches are shunted (ie patches aligned with the ϕ = 0 direction, the NS patches are still short-circuited;

• case EWNS : all patches are shunted.

Dispersion along Γ -X direction of the lattice embedding a pure Cneg shunt

The figure 14 provides the dispersion diagrams of the three configurations (all patches short-circuited, EW shunts and EWNS shunts). As expected, the Cneg shunt has a strong impact on the dynamics of the lattice. In the low frequency range, the in-plane shear mode presented in figure 7 is not changed by the shunt since the piezoelectric patches are not strained (see yellow curve in the reference diagram of figure 5). A similar effect may be observed on the first torsional mode whose wavelength is long enough to generate almost no charge between the electrodes (see initial blue curve in figure 5 and associated mode shape in figure 10). All other branches are strongly affected by the shunt, whose main effect is to couple the various modes involving strain in the lowest part of the legs of the kirigami structure. The figure 15 illustrates this by showing the shape of the modified branch at 170 Hz, involving coupling between the out-of-plane shear mode (see figure 6) and the first bending mode shown in figure 8).

In the low frequency region, using only EW shunts provides almost the same effects as if the full EWNS shunts would have been used (this comment is only valid for waves propagating along Γ -X direction, see next section for other directions). The difference between EW and EWNS shunts effects can be observed on the second torsional mode and third flexural mode. The second torsional mode, as seen in purple curve in figure 5 and associated shape in figure 11), can be controlled only by NS shunts, since EW shunts are not strained while NS ones are located in the area where the strain energy is very high. This implies that the dispersion properties of this mode are impacted only if NS patches are shunted. On the opposite, controlling the third flexural mode (see gray box in figure 5 and shape in figure 12) requires EW shunts while NS shunts have almost no effect, as seen in figure 14. The most interesting feature of the shunted EWNS configuration lies in the fact that, despite the cancellation of the first band gap due to strong modes coupling, the second and third ones are combined, resulting in a large frequency band where no wave can propagate from 1100 Hz to more than 2500 Hz. Further analysis has shown that the upper limit of the band gap in Γ -X direction was 2900 Hz, corresponding to an impressive width to mean frequency ratio equals to 90%. The thin pass band below 2kHz for the EW configuration may be used for frequency filtering, providing that practical applications that would take advantage of this, involve deformations of the cell in coherence with the second torsional mode.

Directivity of the lattice embedding a pure Cneg shunt

The radical changes in the dispersion properties due to the negative capacitance shunt commented in the previous section imply strong evolutions of the directivity diagram, as illustrated in figures 16 for EW shunt and 17 for EWNS shunt. When only the E and W patches are shunted, the low frequency contents below 1000 Hz are highly dependent of ϕ, meaning that the directional character of the lattice is strong. Above his value, the band gaps are almost omnidirectional. The very selective character of the thin pass band just below 2 kHz is confirmed in a quite omnidirectional way.

This characteristic is more marked for the fully shunted configuration: in this case, starting from 1kHz up to more than 2.5 kHz, no wave can propagate in the system, meaning that a huge omnidirectional band gap has been obtained thanks to the Cneg piezo-shunt embedded on the kirigami lattice. Many practical applications may take advantage of this very large band gap.

Impact of resistance in the shunt circuit

The combination of the previous negative capacitance with a resistance in the shunt circuit adds further dissipation to the system. Hence only some little changes may be observed on the real part of the wave numbers, as shown in figure 18 for propagation along Γ -X. This figure is based on the plot shown in figure 14 on which the new computed wave numbers (including the resistance in the shunt) have been added.

On the other hand, when looking at the imaginary part of the wave numbers it is quite evident to observe that the resistance in the shunt increases the value of the damping by two orders of magnitude for almost all branches of the diagram compared with the pure Cneg case, as shown in figure 19, on which the non dimensional value of the imaginary part is obtained following the same strategy as for the real part. Similar trends as those described in the previous section are observed: depending on the correlation between the strain energy and piezoelectric patches' locations, the shunt has a variable degree of control of the modes. By using the full EWNS shunts one can achieve the ability to control all the waves in the frequency band of interest.

This added damping will clearly act on the directivity of the system, by smoothing the bounds of the gap, as it can be observed in figure 20, which shows the directivity diagram for the RCneg EW shunted lattice. Comparing these results with the ones in figures 13 and 16 it appears that in terms of directivity, the resistance somehow deteriorates the performances of the negative capacitance, by removing the small gaps in the low frequency range, and decreasing the value of the Ind criteria in the second gap, even if for almost all points the decrease remains high with Ind higher than 0.3, still corresponding to a relatively high spacial decay rate of the waves amplitudes.

Adding RCneg shunt on the S and N patches do not change fundamentally the behavior of the smart lattice. As shown in figure 21, the trends are similar to the previous case where only E and O patches were shunted. However it seems that a very thin path opened on the whole frequency range (ie. with no band gap) may be obtained for ϕ = π/4, while waves trying to propagate between 1 kHz and 2.5 kHz along other directions are highly damped, which could be of interest for some practical applications. Further investigations are however required to check the impact of the resistance value on this interesting behavior. Modifications of waves directivity by added dissipation into the system has already been oberved in [START_REF] Collet | A floquet-bloch decomposition of the elastodynamical equations: application to bi-dimensional wave's dispersion computation of damped mechanical system[END_REF]. In these two works, dissipation modifies the 2D propagation maps by creating specific frequency dependant directivity or as here opening a π/4 degree propagtion door on the whole frequency band of interest especially inside the bands gaps. This phenomenum can be use for filtering vibration even if the damping may induces a strong decay rate of the waves amplitudes.

Conclusions

The kirigami pyramidal auxetic active core described in this work has shown promising properties in terms of wave propagation, especially at low frequency badwidths. Various configurations have been investigated, corresponding to either short-circuited, EW or EWNS shunts. Shifting from one configuration to another in a practical implementation is very easy since only electrical shunts are involved. The behaviors observed are higly dependent of the configurations, opening the path to systems embedding devices with efficient adaptive filtering properties. Starting from a deep analysis of the short-circuited configuration, the effect of the shunts has been found to be coherent with the physical principles which are involved.

The short-circuited configuration exhibits a behavior with a small directivity signature. The non-omnidirectional character of the lattice can be observed starting from the second band gap, and it has been observed that the waves are not filtered if propagating at ϕ = 45 • below 1250 Hz. Several band gaps are observed in this configuration, but remain small in width compared with those that may be obtained with the shunts.

The negative capacitance shunt provides very interesting features since, despite the cancellation of the first band gap due to strong modes coupling, the upper band gaps are combined, resulting in a huge frequency band where no wave can propagate from 1100 Hz to 2900 Hz. Also, a very selective thin pass band below 2kHz for the EW configuration has been observed.

Adding a resistance in the shunt circuit somehow deteriorates the performances of the negative capacitance by removing the small gaps in the low frequency range. It provides however a quite interesting feature, namely the path at ϕ = 45 • which is extended to the whole frequency band of interest.

Figure 1 .

 1 Figure 1. The piezocomposite periodic auxetic cell

1 ]Figure 2 .

 12 Figure 2. Complex eigenfrequencies of clamped-clamped cell [rad.s -1 ] vs. negative capacitance ratio α: • real part, × imaginary part

Figure 3 .

 3 Figure 3. The piezocomposite lattice

Figure 4 .

 4 Figure 4. Dispersion analysis of the open circuit system along Γ -X (ϕ = 0) with no filtering: real part of non dimensional wavenumber vs. frequency [Hz].

Figure 5 .

 5 Figure 5. Dispersion analysis of the open circuit system along Γ -X (ϕ = 0) with filtering: real part of non dimensional wavenumber vs. frequency [Hz]. Square surroundings correspond to solutions whose shapes are shown in figures 6 to 12.

Figure 6 .

 6 Figure 6. Shape of mode occurring at 110 Hz, with Real(k n ) ≈ 0.7723 π L , corresponding to red square in figure 5. Color scale corresponds to modal kinetic energy density. Grey lines show the undeformed cell.

Figure 7 .

 7 Figure 7. Shape of mode occurring at 110 Hz, with Real(k n ) ≈ 0.1751 π L , corresponding to yellow square in figure 5. Color scale corresponds to modal kinetic energy density. Grey lines show the undeformed cell.

Figure 8 .

 8 Figure 8. Shape of mode occurring at 110 Hz, with Real(k n ) ≈ 0.1751 π L , corresponding to green square in figure 5. Color scale corresponds to modal kinetic energy density. Grey lines show the undeformed cell.

Figure 9 .

 9 Figure 9. Shape of mode occurring at 730 Hz, with Real(k n ) ≈ 0.9053 π L , corresponding to cyan square in figure 5. Color scale corresponds to modal kinetic energy density. Grey lines show the undeformed cell.

Figure 10 .

 10 Figure 10. Shape of mode occurring at 890 Hz, with Real(k n ) ≈ 0.686 π L , corresponding to blue square in figure 5. Color scale corresponds to modal kinetic energy density. Grey lines show the undeformed cell.

Figure 11 .

 11 Figure 11. Shape of mode occurring at 1870 Hz, with Real(k n ) ≈ 0.3678 π L , corresponding to purple square in figure 5. Color scale corresponds to modal kinetic energy density. Grey lines show the undeformed cell.

Figure 12 .

 12 Figure 12. Shape of mode occurring at 1870 Hz, with Real(k n ) ≈ 0.585678 π L , corresponding to grey square in figure 5. Color scale corresponds to modal kinetic energy density. Grey lines show the undeformed cell.

Figure 13 .

 13 Figure 13. Directivity diagram of the short-circuited system. Distance from the center is frequency. Color is Ind (blue: some waves can propagate in the lattice, red: band gap).

Figure 14 .

 14 Figure 14. Dispersion analysis along Γ -X (ϕ = 0) of the shunted system with pure negative capacitance (R = 0): real part of non dimensional wavenumber vs. frequency [Hz]. Short circuit (black • ), EW shunts (blue ), EWNS shunts (red ).

Figure 15 .

 15 Figure 15. Shape of mode occurring at 170 Hz, with Real(k n ) ≈ 0.3286 π L , for the EW pure Cneg shunt configuration. Color scale corresponds to modal kinetic energy density. Grey lines show the undeformed cell.

Figure 16 .

 16 Figure 16. Directivity diagram of the Cneg shunted lattice (only E and W patches are shunted here). Distance from the center is frequency. Color is Ind (blue: some waves can propagate in the lattice, red: band gap).

Figure 17 .

 17 Figure 17. Directivity diagram of the Cneg fully shunted lattice. Distance from the center is frequency. Color is Ind (blue: some waves can propagate in the lattice, red: band gap).

Figure 18 .

 18 Figure 18. Dispersion analysis along Γ -X (ϕ = 0) of the shunted system with resistance and negative capacitance (R = 500Ω): real part of non dimensional wavenumber vs. frequency [Hz]. Short circuit (black • ), EW Cneg (blue ), EWNS Cneg shunts (red ), EW RCneg shunts (pink ♦), EWNS RCneg shunts (cyan +).

Figure 19 .

 19 Figure 19. Dispersion analysis along Γ -X (ϕ = 0) of the shunted system with resistance and negative capacitance (R = 500 Ω): imaginary part of non dimensional wavenumber vs. frequency [Hz]. Short circuit (black • ), EW shunts (blue +), EWNS shunts (red ).

Figure 20 .

 20 Figure 20. Directivity diagram of the RCneg EW shunted lattice. Distance from the center is frequency. Color is Ind (blue: some waves can propagate in the lattice, red: band gap).

Figure 21 .

 21 Figure 21. Directivity diagram of the RCneg fully shunted lattice. Distance from the center is frequency. Color is Ind (blue: some waves can propagate in the lattice, red: band gap).

Acknowledgments

A part of this work was financed by The French National Research Agency under grant number ANR-12-JS09-008-COVIA. It has been performed in cooperation with the Labex ACTION program (ANR-11-LABX-0001-01). Fabrizio Scarpa thanks the support of the FP7-NMP3-LA-2010-246067 project for the logistics of the kirigami lattice design. The Authors would also like to thank the anonymous reviewers for the helpful suggestions.


Kristian Torbensen

Federico Rossi

Sandra Ristori

Ali Abou-Hassan

Chemical communication and dynamics of droplet emulsions in networks of

Introduction

The generation, propagation and reception of bio/chemical information between individual organisms are the keystones of many intelligent communicating systems. [START_REF] Strogatz | Sync: How Order Emerges From Chaos In the Universe, Nature, and Daily Life Hachette Books[END_REF] The transmission of a signal at the level of an individual generator-receptor couple is for example the basis for the bio/chemical signal transmission in neural synaptic communication.

2, 3 The propagation of a signal wave-front among different individual sources is then associated with the synchronization of the sources and the observation of cooperative phenomena. 4 Such cooperative effects are ubiquitous in nature. They are observed for example in the synchronous flashing within colonies of fireflies, in the contraction/expansion of heart muscles, in morphogenesis, calcium signalling dynamics, decentralized bacterial quorum sensing, etc. 1, 2 Specifically, in cell populations, a variety of dynamical behaviours have been reported, ranging from a single synchronization state to multiple synchronization states, which reflect different degrees of communication. 5 Among the mechanisms through which communication is achieved, the most characteristics for living systems are the (frequently negative, but sometime positive) feedback loops, closely related to the oscillation phenomena described in this review.

6, 7

At the extremities, synchronization in nature occurs, on the smallest scale, as simultaneously moving electrons in superconductors, and at the largest scale, as planetary systems hurled by gravitational synchrony. [START_REF] Strogatz | Sync: The Emerging Science of Spontaneous Order Penguin[END_REF] At the level of unicellular organisms, cooperation and synchronization occurs via chemical communication based on a messenger diffusion/reaction, which extends over a wide range of time and length-scales. 9, 10 Inside a cell, within trans-membrane protein machinery, chemical communication is restricted to the nanometer scale. However, in many other examples, chemical reagents are able to cross biological membranes and be transported to their target by diffusion over larger distances in an aqueous environment. For instance, the Please do not adjust margins Please do not adjust margins nervous system involves micron-scale chemical communication during synaptic communication from neurons to axons. 11 Diffusion of chemical reagents over even longer distances (hundreds of micrometers or more) in the extracellular solution is believed to be responsible of the largescale collective behaviours of colonies of unicellular organisms.

For trans-membrane communication over short or large distance a key role can also be assigned to specific lipids. For instance, phosphatidylinositol derivatives have been charged to promote cell migration by generating spontaneous waves which results in oscillatory dynamics. This example points out the relevance of the membrane composition, not only in terms of biomacromolecules embedded in the lipid matrix but also in terms of the very same lipids forming the bilayer skeleton.

12 Synchronized and cooperative behaviour are also widely exploited in our everyday life through information and communication technology (ICT). Inspired by biological paradigms, new technologies emerges, such as the bio/chemical-information and communication technology (bio/chemical-ICT), 11, 13, 14 which aims at extending the wellknown field of ICT, classically based on the transmission of electrical or electromagnetic signals, to the bio/chemical world of molecules. 15, 16 In a biomimetic approach, it proposes to carry chemical information into individual encapsulated microchemical reactors, and to propagate chemical information between them. Such naturally occurring information processes can provide a basis for innovative computational strategies. 14, 15, 17 Motivated by mimicking and understanding the collective behaviour and synchronicity involved in complex systems in living organisms, or by more applied objectives such as the field of ICT, the last ten years have witnessed extensive research in the field of chemical communication and dynamics of oscillatory droplets. To build reliable communicating systems capable of exhibiting dynamic and collective behaviour, several requirements must be fulfilled. First, the encapsulated chemical reaction should have a non-linear mechanism able to respond to external stimuli and, at the same time, generate chemical signals in a controllable way. The Belousov-Zhabotinsky (BZ) reaction, 18, 19 described in more detail in the following section, certainly matches these requisites. It is a well-known chemical oscillator, whose mechanism is effected by the interaction of activatory and inhibitory feedback loops, capable of generating periodic chemical signals. Second, once the signal emitter/receiver reaction is "stored" inside droplets, the interface of the droplets should be (semi) permeable to allow the propagation and transmission of the chemical information from one compartment to another, via chemical messengers, such as in living organisms. Various surfactants, phospholipids and polymers are usually employed to form such membrane-like interfaces. Third, there must be a reliable method of droplet generation capable of controlling the droplet size and dispersity, the chemical composition of the liquid/liquid interface between the droplets and the surrounding medium, as well as the spatial organization of the droplets. Here, microfluidic systems (vide infra) can offer unprecedented opportunities to control such parameters. As a consequence, an ideal experimental reaction-diffusion system that has been largely investigated in order to study chemical communication and the resulting dynamics, consists of microcompartments (emulsions, liposomes, polymersomes) produced using microfluidics, in which aqueous droplets containing the Belousov-Zhabotinsky (BZ) chemical oscillator are dispersed in oil or aqueous media.

Although biological and biochemical oscillators exist, [START_REF] Higgins | [END_REF][21][22] and can be used as a model of chemical information, interpretation of the autonomous and collective behaviour in such networks is onerous due to the complexity of the dynamics and the chemistry of the systems. In contrast, the chemistry of the BZreaction is well-understood; its rate constants are known, and the inter-droplet coupling between compartments is purely diffusive.

23

The aim of this manuscript is to provide a tutorial review to the field of chemical communication and dynamics in networks of compartments containing BZ micro-oscillators engineered using microfluidics. First the BZ-reaction is introduced, then the important role of microfluidics for generating communicative networks of micro-oscillators is discussed and finally different experimental efforts that have been achieved and reported recently in this field using microfluidics are reviewed For a more theoretical description of communicative networks of diffusively coupled oscillators, the reader can turn to these reports.

24-29

The Belousov-Zhabotinsky (BZ) chemical reaction as a system for chemical communication studies

Nowadays, the BZ reaction is probably the most studied oscillating chemical reaction; as stated by Anatol Zhabotinsky; [START_REF] Zhabotinsky | [END_REF] "the popularity of the BZ is mainly due to the fact that it is the simplest closed macroscopic system that can be maintained far from equilibrium by an internal source of free energy homogeneously distributed in space". This means that even in batch experiments, the BZ reaction can last for several hours without the need of supplying fresh reagents to the system; moreover, being outside of thermodynamic equilibrium, BZ can display several exotic dynamical regimes: periodic, aperiodic and chaotic oscillations, 31 autocatalysis and bistability, 32 Turing structures and pattern formation. 33, 34 For a more detailed review on the BZ reaction, we encourage the reader to consult these articles. 32, 35-38 Here, we briefly sketch the oscillatory mechanism, and highlight the most important features that make the BZ reaction a good system to be coupled with microfluidic for studying chemical communication. The BZ reaction is driven by the oxidation of an organic substrate, generally malonic acid, by bromate in an acidic solution in the presence of a catalytic species. The most used catalysts are redox active organo-metal complexes such as ferroin (a phenanthroline-iron(II) complex) and tris(2,2'bipyridyl)-ruthenium or metal ions such the redox couple Please do not adjust margins Please do not adjust margins . An example of the oscillating behavior is given in Figure 1A, where a spectrophotometric time-series show the changes in the concentration of the reduced form of the catalyst. In the specific case of ferroin, the two different oxidation states are reflected in a dramatic change of color, blue when oxidized and red when reduced. The simplest and oldest mechanism, which account for the oscillations of the BZ-system is known as FKN (proposed by Field, Korös and Noyes), 39 sketched in Figure 1B with ferroin as the catalyst. The scheme briefly consists of three main processes with three key species responsible for the activatory and inhibitory feedback loops that compete in the different stages of the reaction, giving rise to a periodic change in the concentrations of the reaction intermediates. Process A takes place when the concentration of bromide ions ) or inhibiting (Br 2 ) the oscillations. Occasionally, Ru-based catalysts are employed to render the reaction photosensitive. The photo-excited state of the Ruthenium biprydine complex (λ max = 450 nm), for example, is able to interact with bromate and oxygen to promote the formation of Br -, thus acting as a powerful oscillations inhibitor, externally controllable by light. 40 This feature is particularly useful to set initial and boundary conditions of BZbased oscillator networks.

As previously stated, the BZ reaction can produce several oscillation cycles (over a thousand for certain experimental parameters) even in batch conditions, thus in typical chemical communication experiments, the BZ can be easily confined with no need of feeding with fresh reactants; together with the "long life", another aspect that makes BZ appealing for such kind of experiments is the variety of intermediates produced during each cycle and their influence on the oscillations. In fact, when single oscillators are physically separated by a barrier or by a solvent immiscible with water, the intermediates can be used as messengers to deliver chemical information among different single units. By exploiting the properties of the intermediates, it is possible to choose the barriers (or the separating solvent) in order to have a specific type of coupling (global, inhibiting or activating) and the corresponding feedback on the oscillating mechanisms. Please do not adjust margins

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The compartmentalization of the BZ reaction in nano-and microreactors, microparticles, vesicles, liposomes, polymersomes, or aqueous microdroplets, has been used as a model system to investigate short and long-range chemical communication between microreactors arranged in various configurations, from linear arrays to network with intricate geometries and multiple connections. 9, 41-56

An important issue for controlling the communication dynamics in networks of chemical oscillators is the way the signal leaves one unit to reach the others. For example, messenger molecules can target a specific area or they can uniformly spread in the surrounding environment, depending on the dynamical behavior of the chemical oscillator inside a single compartment (see Figure 1C). One of the crucial parameter influencing the global dynamics of the BZ reaction is the medium homogeneity; in particular, in the absence of any macroscopic concentration gradients, the system oscillates uniformly in space (homogeneous oscillations). On the contrary, under the action of spatial gradients, the BZ behaves as a reaction-diffusion system generating fronts, pulses or waves. [57][58][START_REF] Field | Oscillations and Traveling Waves in Chemical Systems[END_REF] When the BZ system is confined and no stirring or advection is possible, the critical parameter responsible for the homogeneity of the system is the compartment size. In the case of droplets, the diameter d determines the time of diffusive mixing, t = d 2 /2D (with D being the diffusion coefficient of a BZ intermediate), that has to be compared with the BZ oscillation period, τ: when t < τ, bulk oscillations generate an isotropic distribution of the BZ intermediates in the inner aqueous phase, which, in turn, may cross the amphiphilic barriers in each point of the membrane; when t > τ, the coupling of the BZ with diffusion generates pulses or chemical waves, which are directional signals propagating along a straight line and can be controlled to deliver information to a precise target. Therefore, by taking advantage of the microfluidic techniques, it is relative simple to adjust the size of the compartments to obtain different ways of signal delivery. Although this review reports on chemical communication in BZ oscillating networks engineered using microfluidics, it is important also to highlight some results dealing with BZ and compartmentalization obtained by other approaches than microfluidics. Okano et al. 44, 48 studied the phase synchronization of a 2D array of BZ cell elements in a gel matrix mediated by external illumination. To model neurotransmission signalling, which is typically unidirectional, and occurs via discrete pulses of mass exchange in contrast to continuous mass diffusion, Horvath et al. [START_REF] Horvath | [END_REF] performed experiments with pulse-coupled BZ oscillators. The coupling strength between two oscillators was controlled by varying the concentration of the inhibitor specie bromide in one oscillator as a function of the oscillatory signal in the other. A similar method was employed by Weiner et al., 61 Holz et al. 62 well-defined size, dispersity and chemical composition, all of them acting as microreactors. The chemical communication between such microreactors can be obtained in various modalities: Either within the encapsulated entities, between the interior and the outside of the compartment (partition), or between neighbouring compartments. At the level of communication between neighbouring compartments the generation of intermediates that allow for the spatio-temporal propagation of chemical information is responsible for the production of collective behaviours, which can lead to coupling and synchronization. The spatio-temporal coupling between compartments can also be modulated and tuned using microfluidics, which provides not only a control over the dimensions and the boundaries of the compartments but also over their spatial organization and arrangements into different geometries including 1D and 2D networks. High control over the spatial resolution in manufacturing microfluidics 75, 76 enables for engineering assemblies of microcompartments of different dimensions, where the characteristic time of diffusion between these chemically coupled entities is less than the actual time of reaction, e.g., the oscillatory periodicity of the encapsulated BZ-reaction. Under these conditions, the collective behaviour of coupled microreactors is governed by the diffusive pathway between the microreactors, and can hence be tuned by engineering permeable-specific membranes encapsulating the microreactors, or introducing scavengers in the diffusive medium.

One of the first attempts to use microfluidics techniques in connection with the BZ reaction was made by the group of Steinbock, 77 who fabricated a patterned microfluidic device where BZ containing chambers were connected via a microchannel (see Figure 2), to constrain propagating wave signals in a well-defined path. By avoiding the typical wave broadening due to diffusion, with this system, they could

Please do not adjust margins Please do not adjust margins improve the study of spiral segmentation and pattern formation in heterogeneous media. An extensive general review on the potentiality of "active" droplets outside of equilibrium, from nano-to millimetre sized scale, has been recently published. 78 In this review, we focus on a biomimetic model of chemical communication proceeding via a reaction/diffusion pathway. In the following we describe different microfluidic methodologies that have been designed to elaborate biomimetic networks of BZ micro-oscillators using microfluidic platforms and we present a summary of the corresponding results. As the main theme of this review is the generation of networks of BZ microoscillators using microfluidics, we shall focus on the multi-scale approach used for the assembly of networks ranging from 1D to 2D.

BZ-oscillators stabilized by surfactant monolayers: from linear to planar networks

1-D arrays of BZ

A major contribution to the study of coupled compartmentalized oscillators came from the group of Epstein at Brandeis University. 43 They used, for the first time, a Tshaped microfluidic junction (Figure 3A) to generate BZ containing aqueous droplets separated by octane and stabilized by a surfactant. Droplets formed at the junction were collected in glass capillaries with diameters of 100-200 μm, which was rendered hydrophobic in order to obtain a linear network of BZ droplets. The authors showed the importance of the geometric configuration of the BZ containing droplets to control the dynamic behaviour of the network. For example coupling between droplets was only observed when the time of diffusion via the droplet spacing medium (which can be manipulated by the flow rate of the oil phase) was shorter than the period of the BZ oscillations. The oscillation dynamics was found to be rather sensitive to the concentration of the BZ component malonic acid (MA); at higher concentrations, anti-phase oscillation between neighbouring droplets was observed. For intermediate and smaller concentrations of MA, the system reached stationary Turing patterns, occurring after a few anti-phase oscillation periods or immediately, respectively. For these droplets, separated by a hydrophobic medium, nonpolar intermediates were thought to be responsible for the communication between the micro-compartments. In particular, the inhibitor Br 2 had the proper characteristic to diffuse fast in the nonpolar medium and facilitate the communication between two adjacent oscillators. The experimental findings were also in line with an inhibitory coupling type, as theoretically predicted few years earlier.

79

This was further confirmed by adding a scavenger for bromine to the organic phase; in this case the communication ceased to occur and the droplets oscillated eventually independently.

Analyses of the collective behaviour of the networks of oscillating droplets, generally based on space-time plots constructed from time-series of images of the droplet recorded over times, as shown in Figure 3B along with a simulation based on the FKN-model, led the authors to conclude that for the experimental conditions, the inhibitory coupling governed by bromine was dominant. In order to investigate these findings on a more quantitative basis by defining both the boundary and the initial conditions, Delgado et al. 49 exploited the photosensitivity of the catalyst Ru(bipy) 3

2+

. The production of bromide within the individual drops could be hence controlled, and hereby inhibiting on demand the oscillations within the drops exposed to light. After the removal of the illumination, the drops could recover their ability to oscillate synchronously, achieving an initial phase synchronization of an entire array of oscillators. Arrays of BZ micro oscillators in a perfluorinated oil (HFE 7500) containing the amphiphilic block copolymer PEG-PFPE were generated using the flow focusing microfluidic device schematized in Figure 3C and parked inside 100 μm diameter hydrophobically modified round glass capillary for manipulation and observation. Experimentally it was evidenced that such a drop with high bromide concentration positioned in the center of a 1D array of in-phase oscillators, emitted, upon ceasing illumination, a sufficient amount of bromide to cause neighbour drops to display out-of-phase coupling. Eventually, this caused the fraction of in-phase drops to decay monotonically, resulting in the entire array to oscillate out-of-phase. The perturbation of the in-phase oscillation was furthermore found to depend on the initial concentration of MA; higher concentrations resulted in weaker coupling and thus prolonging the time to attain anti-phase synchrony with a phase shift of exactly 180°. Numerical simulations based on an open system, i.e., the drops surveyed being positioned far from the boundaries, confirmed these findings. Restricting the number of oscillating drops by applying boundary conditions set by non-oscillating drops, and setting the surveyed drops in an initial synchronized state by illumination, the authors eventually observed the same general out-of-phase behaviour. However in this case, due to the restrictions introduced by the boundary conditions, the Please do not adjust margins Please do not adjust margins out-of-phase behaviour deviated slightly from the ideal 180° anti-phase synchrony. The role of the BZ intermediate bromine as the main messenger specie in the inhibitory coupling between BZ drops was also confirmed by Li et al. 51 who used the same microfluidic methodology. When the initial bromate concentration in the drops was increased, and the MA concentration was decreased, excitatory coupling leading to in-phase synchrony between neighbour drops was observed. Thus, under these conditions, the inter-droplet diffusion of the excitatory intermediate bromate dioxide radical plays a significant role. Based on finite element analysis, the authors suggested a theoretical model for the coupling strength between the BZ drops, as the ratio between the diffusive flux and the consumption of the inhibitor specie bromine. The transition from weak to strong coupling transforms the dynamical phase behaviour from out-of-phase to in-phase synchrony.

The possibility of cross-diffusion, i.e., the diffusion of one specie induced by a concentration gradient of another, has been investigated for other BZ species. These results suggest that a more complex diffusion-reaction pattern is occurring in the communication pathway between BZ droplets (see also Vanag et al. 82 for a complete overview on the effects of cross-diffusion on pattern formation). Moreover, cross-diffusion can also generate hydrodynamic motions, if the dimensions of the reactor allow for bulk movements, thus potentially affecting the information delivery among single oscillators. Reprinted from ref. 86 with kind permission of The European Physical Journal (EPJ). (B) Effect of membrane formation upon the phase coupling of chemical oscillators. The blue, red, and green traces represent the transmittance of the three droplets shown in the insets as a function of time. The black curve shows, on the same time axis, the distance of the 'blue' from the 'red' droplet (measured center-to-center). Clearly, the oscillations couple in phase as soon as the membrane is formed (jump in the black curve), but not before. Reproduced from ref. 45 with permission of the Royal Society of Chemistry . (C) The formation of travelling waves when bilayer membranes are formed between oscillator droplets. Each image is 5 s apart. The droplet diameter is 30 μm. Reprinted from ref. 86 with kind permission of The European Physical Journal (EPJ).

83-85

(A)

(B) (C) (A) (B) (C)
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In the previous examples, the BZ droplets were separated by a hydrophobic phase, and most frequently bromine was found to be the main messenger molecule diffusing between the microreactors promoting inhibitory coupling. What dynamical behaviour could result if BZ microreactors coated with a surfactant (present in the oil phase) were brought into close contact without coalescence? At the point of contact between two or more adjacent microreactors, as usually hypothesized a bilayer would be formed. This bilayer could give rise to new dynamic phenomena due to the interplay between the droplet network topologies and the type of coupling between the individual droplets. It was expected that mass transfer of less hydrophobic molecules, such as the BZ activator bromine dioxide radical, between the micoreactors would be facilitated. 4A). When drops were separated by even a small volume of oil, the coupling remained inhibitory, whereas drops in contact only separated by a monoolein bilayer exhibited excitatory in-phase synchrony, see Figure 4B and travelling waves could be observed in 2D arrays (Figure 4C). 53 with permission from the Royal Society of Chemistry. (B) Cuts of the snapshots of a BZ-doublet along the dotted lines (top) collected on a single time-plot (middle and bottom). One can notice the parabolic shape of the white stripe representing the chemical wave of the oxidized catalyst expanding from the center of the larger droplet (the black solid lines are guide to the eye) because in the center, the concentration of activator first exceeds the threshold for selfexcitation. The wave propagates outwards, passes through the facet between the droplets and propagates through the small drop. The long-time dynamics can be inferred from a stack containing multiple sequences of oscillations (bottom). Reproduced from ref. 53 with permission of the Royal Society of Chemistry.

So far, only networks of droplets with equal volumes have been considered. To extend the control over volumes to the level of individual compartments, Guzowski et al. 53 described recently an automated microfluidic platform for reproducible self-assembly of chemically communicating droplet networks with pre-designed number and type of the communicating compartments. The chip was obtained by direct milling in polycarbonate and comprising a triple T-junction. To control the flow in the chip, and to generate the droplets on-demand, external valves were used. Pairs of BZ light sensitive droplets were encapsulated in an oil shell (hexadecane) containing asolectin (a mixture of phospholipids from soy lecitin) and dispersed in fluorocarbon oil, see Figure 5A. At the contact point in the core, a thin lipid membrane was also formed between the droplets. To immobilize the core shell droplets into 1D linear networks, their buoyancy were exploited to trap them in engraved notches. The authors demonstrated how resonance between chemical oscillators depended on the relative volumes of the droplets; smaller drops eventually became in-phase synchronized with the larger ones, by an excitatory coupling mechanism, see Figure 5B.

From 1D to 2D networks

In order to observe new dynamic regimes, and reach a higher level of complexity, a higher level of assembly geometry of the oscillators is necessary. For this purpose, microfluidics have much to offer, and more complex synchronization patterns were observed for higher order (1.5D and 2D) droplet configurations. Toiya et al. 46 obtained 1.5D networks of surfactant-coated drops loaded with BZ by using the same microfluidic device and conditions previously described.

49

When the droplet diameter lied between 0.5d and d, where d is the diameter of the glass capillary, the droplet self-organized to form a zig-zag structure. To obtain 2D geometry, drops produced in the microfluidic device were transferred to a standard microscope slide. A variety of synchronous regimes were observed, including in-and anti-phase oscillations, stationary Turing patterns and more complex mixed patterns, including three-phase regimes. The original Turing mechanism for morphogenesis 87 was tested by Tompkins et al., 23 who employed 1-and 2D arrays of BZ drops suspended in oil. By tuning the coupling strength via drop size and MA concentration, the authors were able to generate the spatiotemporal chemical structures predicted by Turing. Due to the heterogeneity of the experimental setup, i.e., the diffusion of trigger molecules between BZ drops occurs through an oil phase, in contrast to the homogeneously coupled cells employed in Turing's prediction, an additional structure was found. In the continuity of the work of Toiya et al., 43 Delgado et al. 49 and Li et al., 51 where the emulsion systems suffered from limited abilities to define custom network geometries and controlled boundary conditions, Tompkins et al. 52 described the ability to create planar (close packed spheres) networks with Dirichlet (constant concentration) boundary conditions. This was obtained by physical trapping by confining photosensitive BZ drops generated using the flow focusing device between two glass

(A) (B)
Please do not adjust margins Please do not adjust margins walls, and by optically isolating a subset of the drops using programmable illumination. In order to create geometries not found within close-packed spheres, they developed new microfluidic manufacturing techniques capable of constructing custom planar geometries in PDMS. PDMS is highly permeable to nonpolar fluids and gases, but relatively impermeable to polar and charged species. To obtain networks with Neumann boundary conditions (no-flux boundary), containment patterns were obtained by Deep Reactive Ion Etching (DRIE) into silicon wafers and forcing the BZ emulsion into the cavities under a glass sheet (Figure 6A). The differences between the observed network dynamics for each boundary condition were described (example Figure 6B). Using light, the authors demonstrated the ability to control the initial conditions of the network and to cause individual oscillators within the network to undergo sustained period elongation or a one-time phase delay. Reprinted with permission from ref. 52 Copyright© 2015, American Institute of Physics.

As one of the main goals of building oscillatory networks is to approach biomimetic systems such as cells, and investigate their chemical communication, BZ droplets stabilized by 1,2dimyristoyl-sn-glycero-3-phosphocholine (DMPC) phospholipid layers at the interface, were generated using a coaxial flow microfluidic device by Rossi et al. 54 The BZ droplets in the oil phase (chloroform:cyclohexane), the number of BZ drops and their network geometry, was controlled by sliding the point of confluence between the exit of the microfluidic device, and a viscous aqueous solution of polymer (polyvinyl alcohol). The results showed how chemical coupling of BZ microemulsions, locally separated by a phospholipid layer, eventually transformed an initial out-of-phase synchrony into excitatory in-phase synchrony via trans-bilayer diffusion of bromous acid.

BZ-oscillators stabilized by surfactant bilayers:

The water in oil compartments previously reported are, despite the biomimetic approach, still far from water in water cell-like compartments found in nature. Giant phospholipid vesicles (or liposomes), where water droplets are compartmentalized by phospholipid membranes and dispersed in a water medium, provide a more reliable and promising artificial cell-like platform.88 With such, not only the mode of encapsulation, but also the permeability properties of the membrane and the inter-liposomes chemical communication will differ. Tomasi et al. 50 described, for the first time, the encapsulation of the BZ-reaction in liposomes, by employing a flow focusing microfluidic device (see figure 7A). This novel system provided the opportunity to study chemical communication between cell-like structures composed of a (DMPC) membrane, in a purely aqueous environment (see Figure 7B). This implied a high extent of partitioning of all the BZ species, some of them capable of crossing the phospholipid membrane, thus allowing for exploring the chemical communication pathways in a different environment with respect to the water-in-oil microdroplets. In this case, the microfluidic technique granted the best experimental method to obtain giant (100 -300 µm) liposomes in the presence of the acidic and oxidizing medium of the BZ reaction. Other techniques of liposome formation, such as the droplet transfer method, 89, 90 were effective for the formation of millimetre sized water-in-oil emulsions, but failed when droplets were dispersed in an aqueous medium. By using electrochemical techniques, it was possible to characterize the behaviour of the oxy-bromine intermediates at the interface between a hydrophilic and a hydrophobic medium, both in the presence and in the absence of a lipid membrane. 50, 91 The main outcome of this study was the unambiguous identification of brominated species as the messenger molecules between different compartments; this, together with numerical simulations, allowed to assess the activatorytype coupling, which was ensured by oxidation waves at the contact regions between different liposomes. A typical pulse transmission is depicted in Figure 7B. The liposomes generated by means of microfluidic techniques represent an interesting system to study the influence of the membranes on the communication dynamics in the network of chemical oscillators. In fact, to control the chemical communication, the membranes could be engineered to be selectively permeable to specific messenger molecules. 92 In practice, by intercalating different dopants into the DMPC lipid bilayers, it was possible to tune some of the membranes properties, such as lamellarity, permeability and local curvature. The specific effect on the membrane structure of dopants like cholesterol, myristic acid, sodium tetradecyl sulphate and tetradecylamine was studied by small angle X-ray scattering (SAXS), and the

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93

The introduction of these bilayer intercalating species brought about new dynamical regimes with respect to the pure DMPC membranes, where, for simple emulsions, it was found a mostly inhibitory-type coupling. It was found, that the bromine-scavenger cholesterol induced the formation of weakly phase-coupled patterns, while the anionic surfactant sodium dodecyl sulphate, promoted a strongly coupled system with the presence of phase-lock states between alternate oscillators. Reproduced from ref. 50 with permission of the Royal Society of Chemistry.

To the best of our knowledge, this is the only example of chemical communication between vesicles encapsulating the BZ oscillator. However, it is important to mention that the successful entrapment of the BZ reaction inside giant polymersomes was recently reported by Mercader et al.

55, 56

These systems can offer new possibilities to study the influence of the membrane composition on the global behaviour in networks of chemical oscillators.

Conclusions

Confined domains of the oscillatory Belousov-Zhabotinsky reaction arranged in various geometries, using microfluidics, have proven to be a significant and versatile tool for modelling bio-physical networks and communication pathways. The pertinent chemical species responsible for communication in these systems are, mainly, the inhibitory bromine and the excitatory bromine dioxide radical. However, other species, such as Br -and HBrO 2 can play an important role. Since their concentrations are governed by the experimental conditions, various coupling patterns can be achievable, i.e., the phase shift between different oscillators is tuneable. In this respect, membranes and surfactants confining the microreactors also play a significant role. In particular, they can promote preferential communication pathways or act as specific scavengers for messenger molecules, thus allowing a finetuning of the general dynamical properties of the networks of chemical oscillators. Droplet arrays separated by an organic medium represent a valuable tool for investigating the motion of chemical species through long distances, compared to the molecular scale, and for engineering coupled aqueous domains. Importantly, both the geometry and structure of compartmentalized oscillators can be finely tuned using microfluidic devices. The applications of such systems are not entirely defined yet, though they are considered as prototypes for many innovative and technically sophisticated devices.

Clearly, the most challenging configurations are those in which both the internal and external compartments (where signal emitters are produced and exchanged) encompass an aqueous medium. Semipermeable membranes, surrounding oil or water droplets, also represent a boundary region whose properties are still being investigated. In general, due to their "soft" nature, the means to study compartments separated by mono or multishell amphiphilic membranes, without perturbations, are limited and often indirect. Thus, only a broad range of physico-chemical, biological and computational methods can afford extensive understanding of communication within networks of chemically coupled oscillators confined in liquid domains. Microfluidics is a versatile platform, and offers exceptional technical control, for the production and organization of liquid droplets into different geometries. However, up to now all the geometries reported are 1D or 2D, with fixed positions of the droplets. Elaborating active geometries with different dimensions (1D to 3D), and ondemand oscillating networks would overwhelm the field of network dynamics, with new emergent patterns, new collective properties and advance of our knowledge on chemical communication.

Notes and references

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Figure 1 .

 1 Figure 1. (A) Spectrophotometric recording at λ = 510 nm (where ferroin has an absorption maximum) of a typical BZ reaction with malonic acid as organic substrate. Pictures on the left show the catalyst colour in the different oxidation states; (B) simplified sketch of the FKN mechanism. Different colours refer to the catalyst oxidation state during each stage of the reaction; (C) Signals directionality related with the type of dynamical behaviour of the BZ reaction.

Figure 2 .

 2 Figure 2. Oxidation wave traveling through a BZ system in a patterned microreactor. The channel has a height of 80 µm and a width of 50 µm. The circular portion of the reactor has a diameter of 1 mm, and the length of the reference bar (A) is 200 µm. Initial BZ concentrations are 1.50 M H2SO4, 0.15 M 1,4 cyclohexanedione, 0.10 M NaBrO3, and 3.1 mM ferroin. The snapshots are separated by 10 s. Reprinted from ref.77 . Copyright© 2004, American Chemical Society.

Figure 3 .

 3 Figure 3. (A) Schematic representation of the microfluidic device. Red droplets correspond to the reduced form of the catalyst (ferroin), blue droplets to the oxidized form (ferrin). Below: Snapshot of two capillaries with droplets. BZ droplets with convex surfaces are dark due to ferroin. Horizontal length of the frame and inner diameter (ID) of the capillary are 4.8 mm and 150 mm, respectively. Reproduced from ref. 43 Copyright© 2008 Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission. (B) Examples of space-time plots obtained by stacking consecutive images of the droplet arrays shown top-most in each image. (a) Anti-phase oscillations with spikes of oxidation of ferroin seen as light horizontal lines across BZ droplets and (b) stationary Turing structures with alternating oxidized and reduced states evolving from an initial oscillatory state. Horizontal lengths of the frame and the capillary ID are 4.8 mm and 150 mm, respectively; the total times are 5200 s (right) and 10 800 s (left), respectively. Patterns extend to the left and right of the segments shown. Reproduced from ref. 43 Copyright© 2008 Wiley-VCH Verlag GmbH & Co. KGaA. (C) Schematic drawing of the microfluidic PDMS device drop generator. At left, two different aqueous streams containing complementary reactants of the BZ solution are injected into the drop generator (light blue and red). The streams merge and coflow down a central channel, meeting two perpendicular oil flows that generate BZ droplets in a nozzle. The co-flow of the BZ solution is preserved immediately after the drops are formed, but complete mixing takes place in less than 1 s. A glass capillary (100 mm ID) was inserted in the PDMS chip a few millimetres downstream from the nozzle to collect the BZ droplets. Reproduced from ref.49

Figure 4 .

 4 Figure 4. (A) The contents of the BZ reaction are mixed on a microfluidic chip to prevent any pre-reaction. Seen through an optical 480/20nm notch filter, the transition from red colour to the blue colour of the BZ reaction is seen by the change in the brightness of the droplet. Correspondingly, red appears dark/black and blue appears bright/white in the grayscale images. (Scale bar: 100 μm.). Reprinted from ref.86 with kind permission of The European Physical Journal (EPJ). (B) Effect of membrane formation upon the phase coupling of chemical oscillators. The blue, red, and green traces represent the transmittance of the three droplets shown in the insets as a function of time. The black curve shows, on the same time axis, the distance of the 'blue' from the 'red' droplet (measured center-to-center). Clearly, the oscillations couple in phase as soon as the membrane is formed (jump in the black curve), but not before. Reproduced from ref.45 with permission of the Royal Society of Chemistry . (C) The formation of travelling waves when bilayer membranes are formed between oscillator droplets. Each image is 5 s apart. The droplet diameter is 30 μm. Reprinted from ref.86 with kind permission of The European Physical Journal (EPJ).

Figure 5 .

 5 Figure 5. (A) Layout of channels on the microfluidic chip. The fluorocarbon phase supplied via valve 1 pushes BZ components A and B. The 'refilling inlets' were closed during the operation of the valves. The opening angle of the inlet to the observation chamber was α ≈ 30°. This smooth widening prevented fragmentation of the larger drops. The notches at the sidewall of the chamber served as traps for the capsules, allowing for their immobilization during monitoring of the progress of reactions. The lengths of the channels were l1 = 17 mm and l2 = 9.5 mm. The widths of the inlet and outlet channels at all three Tjunctions were equal to w1 = 0.3 mm and their depths h1 = 0.3 mm. The width of the chamber was w2 = 2 mm, the depth was h2 = 2 mm, and the sizes of the notches were w3 = 0.3 mm by l3 = 2 mm. Reproduced from ref.53 with permission from the Royal Society of Chemistry. (B) Cuts of the snapshots of a BZ-doublet along the dotted lines (top) collected on a single time-plot (middle and bottom). One can notice the parabolic shape of the white stripe representing the chemical wave of the oxidized catalyst expanding from the center of the larger droplet (the black solid lines are guide to the eye) because in the center, the concentration of activator first exceeds the threshold for selfexcitation. The wave propagates outwards, passes through the facet between the droplets and propagates through the small drop. The long-time dynamics can be inferred from a stack containing multiple sequences of oscillations (bottom). Reproduced from ref.53 with permission of the Royal Society of Chemistry.

Figure 6 .

 6 Figure 6. (A) Planar networks with custom geometries. Typical results shown. (a) A polydimethylsiloxane (PDMS) device designed to create a ring network of seven with constant-concentration boundary conditions. The drops color coded as in (b). (b) A radial-phase-time plot of the ring of seven shown in (a). (c) An etched silicon device designed to create a ring network of seven with no-flux boundary conditions. The drops are color coded as in (d) and the white arrows indicate the order of oscillation. (d) A radial-phase-time plot of the ring of seven shown in (c). (e)-(h) Network patterns etched into silicon wafers imaged using an optical profilometer. (e) The ring network, (f) a six-star network, (g) a square lattice, and (h) a triangular lattice. Reprinted with permission from ref. 52 Copyright© 2015, American Institute of Physics. (B) Network patterns observed from ring networks of three and four members with constant-concentration and no-flux boundary conditions. Top: Images of the networks with the oscillating drops colour coded as below. All scale bars are 100 μm. Bottom: Radial-phasetime plots of the networks above. Left: Circular networks with constantconcentration boundary conditions implemented via optical isolation in closepacked drops. Only the colour coded drops are oscillatory; the other drops are optically inhibited. Middle: Circular networks with constant-concentration boundary conditions implemented in thin PDMS with optically inhibited drops in surrounding channels. The surrounding channels are not shown. Right: Circular networks with no-flux boundary conditions implemented in etched silicon.Reprinted with permission from ref.52 Copyright© 2015, American Institute of Physics.

Figure 7 .

 7 Figure 7. (A) Cartoon showing the coaxial microfluidic device and schematizing the encapsulation process. (B) Formation of the BZ/O/W double emulsions in the microfluidic device at the entrance (below left) of the collection capillary and (below right) at the exit of the collection capillary, time elapsed 80 s. (B) The pulse transmission across the touching liposomes after solvent evaporation. White arrows indicate the direction of pulse propagation. (Left): Image showing the space-time plot of liposomes 2, 5, 6 along the white bar. (B) Consecutive images showing a wave front propagating through three liposomes, along with the corresponding space-time plot.Reproduced from ref.50 with permission of the Royal Society of Chemistry.

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