## Elastic networks

Various elastic systems can be understood as networks of interconnected struts which deform by a combination of bending, stretching, twisting and shearing mechanisms. Examples include paper sheets, polymer gels, protein networks and cytoskeletal structures , crystal atomic lattices, granular materials, foams, wood, bones... Moreover, the pairwise interaction potentials used in standard elastic percolation models can also be identified with the strain energy of elastic struts. Despite extensive research, the connection between the mechanical properties of such networks on a macroscopic level and the description of their structures on a microscopic level has not been completely elucidated yet.

Interestingly, under identical loading conditions, some structures appear to deform primarily through the local stretching of the struts, while in other structures the elastic energy is stored via local bending (twisting and shearing contributions are usually neglected). For instance, "foam-like" cellular architectures tend to be bending-dominated, while fibrous architectures exhibit a rich mechanical behavior: computational studies of the two-dimensional elastic deformation of a network of cross-linked fibers have shown a transition from a nonaffine, bending-dominated regime to an affine, stretch-dominated regime with increasing density of fibers. Recent experimental studies, have confirmed this transition.

We performed an analytical study of the macroscopic properties of elastic networks of beams. Our analysis provides a convenient set of geometrical criteria to identify the networks which deform exclusively by extension or compression of their members. Values of the effective elastic moduli of such networks are also derived. These results shed light on the relationship between microscopic geometry and macroscopic mechanics of elastic networks. In particular, they explain the difference in mechanical behavior of cellular and fiber networks.

Interestingly, under identical loading conditions, some structures appear to deform primarily through the local stretching of the struts, while in other structures the elastic energy is stored via local bending (twisting and shearing contributions are usually neglected). For instance, "foam-like" cellular architectures tend to be bending-dominated, while fibrous architectures exhibit a rich mechanical behavior: computational studies of the two-dimensional elastic deformation of a network of cross-linked fibers have shown a transition from a nonaffine, bending-dominated regime to an affine, stretch-dominated regime with increasing density of fibers. Recent experimental studies, have confirmed this transition.

We performed an analytical study of the macroscopic properties of elastic networks of beams. Our analysis provides a convenient set of geometrical criteria to identify the networks which deform exclusively by extension or compression of their members. Values of the effective elastic moduli of such networks are also derived. These results shed light on the relationship between microscopic geometry and macroscopic mechanics of elastic networks. In particular, they explain the difference in mechanical behavior of cellular and fiber networks.

## Transport networks

Finding the most efficient transport network is an issue arising in a wide variety of contexts. One can cite, among others, the water, natural gas and power supply of a city, telecommunication networks, rail and road traffic, and more recently the design of labs-on-chips or microfluidic devices. Moreover, this problem also appears in theoretical works intending to describe the architecture of the vascular systems of living organisms. Generally speaking, consider a set of sources and sinks embedded in a two- or three-dimensional space, their respective number and locations being fixed. The flow rates into the network from each source, and out of the network through each sink, are also given. The problem consists in interconnecting the sources and sinks via possible intermediate junctions, referred to as additional nodes, in the most efficient way. That is, to minimize a cost function of general form $\sum_{k} w_k f(i_k)$ , where the summation is over all the links that constitute the network. $w_k$ is the "weight" associated with the $k^{th}$ link, and is some function of the flow rate $i_k$ carried by this link. Minimization of the cost function can be done over different optimization parameters and with different constraints.

We have studied the structure of pipe networks that minimize the total dissipation rate $U=\sum_{k}r_k i_k^2$ is studied, where the weight $r_k$ is the "flow resistance" of pipe $k$, defined as: $r_k=\rho ~ l_k/ s_k^m$ , $\rho$ being some positive constant, $l_k$ and $s_k$ the length and cross-sectional area of each pipe respectively, and $m$ a positive constant characterizing the flow profile. For most flows encountered in physics, $m \geq 1$ . We have characterized the geometrical and topological features of such networks.

We have studied the structure of pipe networks that minimize the total dissipation rate $U=\sum_{k}r_k i_k^2$ is studied, where the weight $r_k$ is the "flow resistance" of pipe $k$, defined as: $r_k=\rho ~ l_k/ s_k^m$ , $\rho$ being some positive constant, $l_k$ and $s_k$ the length and cross-sectional area of each pipe respectively, and $m$ a positive constant characterizing the flow profile. For most flows encountered in physics, $m \geq 1$ . We have characterized the geometrical and topological features of such networks.

## Correlations between topological and geometrical disorders in soft cellular systems

Cellular materials are interesting both as disordered media with well-defined structural elements, and as models for more complex systems such as biological tissues. Among them, foams are ubiquitous in our daily lives and in many industries. Bubble monolayers are much easier to observe and to study. Such quasi two-dimensional foams are characterised by their number of bubbles, \(N\), area distribution, $p(A)$, and number-of-sides distribution, $p(n)$. When the liquid fraction is very low (“dry” foams), their bubbles are polygonal, with shapes that are locally governed by Plateau’s laws. Each side is a thin liquid film with a uniform surface tension; its curvature is determined by the difference of pressure between the two bubbles it separates. Bubble size distribution and packing (or “topology”) are crucial in determining e.g. rheological properties or coarsening rate. When a foam is shuffled (either mechanically or thermally), $N$ and $p(A)$ remain fixed, but bubbles undergo “T1” neighbour changes, which induce a random exploration of the foam configurations.

In two-dimensional foams, we explore the relation between the distributions of bubble number-of-sides (topology) and bubble areas (geometry). We develop a statistical model which takes into account physical ingredients based on Plateau laws. The model predicts that the mean number of sides of a bubble with area $A$ within a foam sample with moderate size dispersity is given by:

\[\bar{n}(A) = 3\left(1+\dfrac{\sqrt{A}}{\langle \sqrt{A} \rangle} \right),\]

where $\langle . \rangle$ denotes the average over all bubbles in the foam. The model also relates the \textit{topological disorder} $ \Delta n / \langle n \rangle =\sqrt{\langle n^2 \rangle - \langle n \rangle^2}/\langle n \rangle$ to the (known) moments of the size distribution:

\[\left(\dfrac{\Delta n}{\langle n \rangle}\right)^2=\frac{ 1 }{4}\left(\langle A^{1/2} \rangle \langle A^{-1/2} \rangle+\langle A \rangle \langle A^{1/2} \rangle^{-2} -2 \right).\]

Extensive data sets arising from experiments and simulations all collapse surprisingly well on a straight line, even at extremely high values of geometrical disorder. At the other extreme, when approaching the perfectly regular honeycomb pattern, we identify and quantitatively discuss a crystallisation mechanism whereby topological disorder vanishes.

In two-dimensional foams, we explore the relation between the distributions of bubble number-of-sides (topology) and bubble areas (geometry). We develop a statistical model which takes into account physical ingredients based on Plateau laws. The model predicts that the mean number of sides of a bubble with area $A$ within a foam sample with moderate size dispersity is given by:

\[\bar{n}(A) = 3\left(1+\dfrac{\sqrt{A}}{\langle \sqrt{A} \rangle} \right),\]

where $\langle . \rangle$ denotes the average over all bubbles in the foam. The model also relates the \textit{topological disorder} $ \Delta n / \langle n \rangle =\sqrt{\langle n^2 \rangle - \langle n \rangle^2}/\langle n \rangle$ to the (known) moments of the size distribution:

\[\left(\dfrac{\Delta n}{\langle n \rangle}\right)^2=\frac{ 1 }{4}\left(\langle A^{1/2} \rangle \langle A^{-1/2} \rangle+\langle A \rangle \langle A^{1/2} \rangle^{-2} -2 \right).\]

Extensive data sets arising from experiments and simulations all collapse surprisingly well on a straight line, even at extremely high values of geometrical disorder. At the other extreme, when approaching the perfectly regular honeycomb pattern, we identify and quantitatively discuss a crystallisation mechanism whereby topological disorder vanishes.

## Order-disorder transition in 2d soft cellular systems

Under small strains, a foam behaves elastically, with stress proportional to strain and no bubble rearrangements. Above the yield strain, bubble rearrangements occur and the foam deforms plastically. We call T1 the elementary neighbor switching event. In a regular (hexagonal) foam, a T1 event corresponds to the creation of paired dislocations (where a dislocation itself is composed of one 5-sided and one 7-sided bubbles). When the applied shear strain increases, the number of dislocations keeps increasing, and the foam structure gets more and more disordered.

This behavior is reminiscent of that of a two-dimensional crystal submitted to an increasing temperature. This comparison suggests that strain acts as an effective temperature for the foam structure. To probe further this hypothesis, we studied numerically the order-disorder transition driven by an effective temperature in a soft cellular system. We highlighte the existence of an intermediate "hexatic" phase in which positional order vanishes but orientational order is preserved. However, we found that the transition does not follow the standard Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY) scenario observed in many two-dimensional solids, in which dislocations split into two separate disclinations as temperature keeps increasing, but instead follows a dislocation condensation scenario.

This behavior is reminiscent of that of a two-dimensional crystal submitted to an increasing temperature. This comparison suggests that strain acts as an effective temperature for the foam structure. To probe further this hypothesis, we studied numerically the order-disorder transition driven by an effective temperature in a soft cellular system. We highlighte the existence of an intermediate "hexatic" phase in which positional order vanishes but orientational order is preserved. However, we found that the transition does not follow the standard Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY) scenario observed in many two-dimensional solids, in which dislocations split into two separate disclinations as temperature keeps increasing, but instead follows a dislocation condensation scenario.

## Flows and contractile activity in *Physarum Polycephalum*

Throughout life, adequate adaptation and growth of vascular networks are essential for our well being. In many diseases,

We study experimentally the vascular morphogenesis and the synchronization of the contractile activity in

*i.e.*cancer, hypertension, macular degeneration, diabetic retinopathy, etc., vascular adaptation is impaired. Understanding how vascular networks form might help to develop better treatment strategies or maybe even ways to prevent these diseases. Vascular morphogenesis is partly determined by self organized actions of shear stresses generated by the circulating blood and mechanical stresses generated by the growing, deforming surrounding tissue. The core of this work is to study the role of mechanical self-organized processes in growing tissue for vascular morphogenesis and to identify the basic physical laws that shape the vascular networks.We study experimentally the vascular morphogenesis and the synchronization of the contractile activity in

*P. Polycephalum*To watch a time lapse movie of Physarum Polycephalum growing on a map of France, click here.

## Hydrodynamics of bilayer membranes hosting transmembrane proteins

Biological membranes are lipid bilayers forming the envelopes of plasma membranes, nuclei, organelles, tubules and transport vesicles within a cell. They are versatile structures, both fluid and elastic, that can change shape or topology in order to accomplish the cell functions. From the dynamical point of view, membranes can be viewed as a system of four fluid phases in contact: a pair of two-dimensional (2D) lipid phases and two three-dimensional (3D) aqueous phases. These phases, separated but strongly coupled, exhibit nontrivial multiphase flow behaviors.

The dynamics of bilayer membranes containing transmembrane proteins at a high concentration are especially challenging because the proteins form a fifth phase that effectively interdigitates two components of the multiphase flow. This situation corresponds to the actual biological one, where macromolecular crowding effects are ubiquitous and which are known to make molecules in cells behave in radically different ways than in artificial lipid vesicles

Simple bilayer membranes already have a complex hydrodynamic behavior due to their soft out-of-plane elasticity. In the early studies of membrane hydrodynamics the bilayer structure of the membrane was neglected. While this is a good approximation for tensionless membranes at length-scales much larger than microns, experiments and theoretical studies have shown that taking the bilayer structure into account is essential at shorter length-scales. This is chiefly due to the importance of the dissipation due to the intermonolayer friction. The latter is caused by the relative motion of the lipid tails occurring when the two monolayers have different velocities. Clearly, in the case of membranes hosting integral proteins, an increase of the intermonolayer friction is expected.

In this work, we studied the multiphase flow of an almost planar deformable bilayer membrane hosting diffusing transmembrane proteins of arbitrary shape. Using Onsager's variational principle, we derive the general equations describing the multiphase flow of the system. We obtain a simple formula for the correction to the intermonolayer friction caused by the presence of the proteins. This correction is shown to vary in inverse proportion to the protein's mobility. We calculate the relaxation modes of the dynamical equations coupling the membrane shape, the lipid density-difference and the protein density. We discuss the role of membrane tension and protein asymmetry, and we derive the wavevector-dependent effective diffusion coefficient of the proteins. Finally, we analyse the spreading of a concentrated spot of proteins and we discuss its anomalous diffusion.

The dynamics of bilayer membranes containing transmembrane proteins at a high concentration are especially challenging because the proteins form a fifth phase that effectively interdigitates two components of the multiphase flow. This situation corresponds to the actual biological one, where macromolecular crowding effects are ubiquitous and which are known to make molecules in cells behave in radically different ways than in artificial lipid vesicles

Simple bilayer membranes already have a complex hydrodynamic behavior due to their soft out-of-plane elasticity. In the early studies of membrane hydrodynamics the bilayer structure of the membrane was neglected. While this is a good approximation for tensionless membranes at length-scales much larger than microns, experiments and theoretical studies have shown that taking the bilayer structure into account is essential at shorter length-scales. This is chiefly due to the importance of the dissipation due to the intermonolayer friction. The latter is caused by the relative motion of the lipid tails occurring when the two monolayers have different velocities. Clearly, in the case of membranes hosting integral proteins, an increase of the intermonolayer friction is expected.

In this work, we studied the multiphase flow of an almost planar deformable bilayer membrane hosting diffusing transmembrane proteins of arbitrary shape. Using Onsager's variational principle, we derive the general equations describing the multiphase flow of the system. We obtain a simple formula for the correction to the intermonolayer friction caused by the presence of the proteins. This correction is shown to vary in inverse proportion to the protein's mobility. We calculate the relaxation modes of the dynamical equations coupling the membrane shape, the lipid density-difference and the protein density. We discuss the role of membrane tension and protein asymmetry, and we derive the wavevector-dependent effective diffusion coefficient of the proteins. Finally, we analyse the spreading of a concentrated spot of proteins and we discuss its anomalous diffusion.

## Thermal Fluctuations in soft cellular systems

We investigate the thermal fluctuations in a two-dimensional soft cellular system. Each interface between cellular domains can fluctuate, but the modes are not independent because of the incompressibility of the domains. The effect of the dispersity in domain sizes on the spectrum of thermal fluctuations is investigated. This study will allow us to calculate the specific heat of a 2D cellular system, and help us to identify the "soft spots" in such a system.

## Plasticity of foams

The macroscopic response of "molecular" systems and "mesoscopic" materials, such as foams, emulsions, pastes, and slurries, to applied stress and strain share common features: elastic at small strains and plastic at larger strains. In the elastic regime, stress is proportional to applied strain, and deformations are reversible. Above the yield stress or strain, an elastic deformation and plastic flow occur.

Among mesoscopic materials, special attention has been devoted to two-dimensional (2D) foams. There is a two-fold interest in studying the mechanical properties of such systems: a foam is a model for other soft cellular materials such as emulsion or biological tissues: these systems are constituted of highly deformable (yet almost incompressible) units (bubbles, drops, cells,. . . ), interacting through attractive adhesive interactions and soft steric repulsions. Interface energy is then key in the cohesion and the rigidity of soft cellular systems. It confers the rigidity of these systems, sometimes constituted solely of fluids (e.g. foams and emulsions).

Foams are also interesting as a model for atomic or molecular system, both for crystalline and amorphous solids. The pioneering work of Bragg revealed the mechanism of defect propagation in crystalline solids by observing and quantitatively describing the displacement and rearrangement of individual bubbles in a raft. Even today, recent results on nanoindentation confirm that the structural dynamics of bubble rearrangements reveals crucial properties of plasticity on the nanoscale. Bubble raft (i.e. a monolayer of bubbles lying on liquid surface) is the most common system used in experiments and simulations. One major advantage of bubble rafts is that their two-dimensional nature allows for easy imaging and tracking of all of the "particles" in the system. Another reason that bubble rafts have been so useful in the study of molecular systems is that, thanks to the nearly circular shape of bubbles in these systems, bubble interactions can be calculated which allows for a direct evaluation of bond strengths and deformation energies. Yet, alternative setups have been used, such as a dry foam in a Hele-Shaw cell. In that case the bubbles are highly compacted and have polygonal shapes. The structure is then very similar to those of biological cells in a confluent tissue.

In foams and other soft cellular systems, plastic behavior is commonly described in terms of the T1 event, which is the elementary topological change in the structure corresponding to a neighbor switching. In a crystalline foam, where bubbles are disposed in an hexagonal lattice, a T1 event can be identified with the creation of a paired dislocations (i.e. two pairs of 5- and 7-sided cells). From a mechanical point of view, the T1 process corresponds to a transition from one metastable configuration to another, after passing through an unstable configuration where four films meet at one junction. The spontaneous evolution from one four-fold junction to two three-fold junctions, which involves creation of a new film, is driven by minimization of the surface area. Various experimental and theoretical studies on the

we investigated theoretically and experimentally the effect of the viscoelastic parameters on the dynamics of the T1 process. Experiments in a two-dimensional foam showed that the relaxation time depends on the interfacial viscoelasticity of the films, but not on the shear viscosity of the bulk liquid. These results are corroborated by a model, which allows for an estimation of the Gibbs elasticity and the surface viscosity of the surfactants used to make the foam.

Numerous studies have been carried out on the plastic behavior of foams, either experimentally or numerically, and with various setups (bubble raft, Hele-Shaw cell,...). No clear consensus emerges on the existence of a power-law behavior for the probability of T1 events, or even on the value of the power-law exponent. Two factors may explain these variability of observed behaviors: i) the liquid fraction, which is high in a bubble raft where circular bubbles are in contact with the pool, and low in a dry foam with polygonal bubbles. Liquid fraction has an impact on the dissipation during relaxation, but it also modifies the bubble interactions. It is worth mentioning that in the dry foam limit, energy interactions are not pairwise anymore, what could lead to a peculiar plastic behavior; ii) the dissipation at the top and bottom boundaries: for bubble rafts, the bottom surface is in contact with the liquid, while in Hele-Shaw cells foam is in contact with the glass plate. The top surface is in contact either with air or a glass plate.

We are currently investigating numerically the mean energy cost a unique T1 event, and the

Among mesoscopic materials, special attention has been devoted to two-dimensional (2D) foams. There is a two-fold interest in studying the mechanical properties of such systems: a foam is a model for other soft cellular materials such as emulsion or biological tissues: these systems are constituted of highly deformable (yet almost incompressible) units (bubbles, drops, cells,. . . ), interacting through attractive adhesive interactions and soft steric repulsions. Interface energy is then key in the cohesion and the rigidity of soft cellular systems. It confers the rigidity of these systems, sometimes constituted solely of fluids (e.g. foams and emulsions).

Foams are also interesting as a model for atomic or molecular system, both for crystalline and amorphous solids. The pioneering work of Bragg revealed the mechanism of defect propagation in crystalline solids by observing and quantitatively describing the displacement and rearrangement of individual bubbles in a raft. Even today, recent results on nanoindentation confirm that the structural dynamics of bubble rearrangements reveals crucial properties of plasticity on the nanoscale. Bubble raft (i.e. a monolayer of bubbles lying on liquid surface) is the most common system used in experiments and simulations. One major advantage of bubble rafts is that their two-dimensional nature allows for easy imaging and tracking of all of the "particles" in the system. Another reason that bubble rafts have been so useful in the study of molecular systems is that, thanks to the nearly circular shape of bubbles in these systems, bubble interactions can be calculated which allows for a direct evaluation of bond strengths and deformation energies. Yet, alternative setups have been used, such as a dry foam in a Hele-Shaw cell. In that case the bubbles are highly compacted and have polygonal shapes. The structure is then very similar to those of biological cells in a confluent tissue.

In foams and other soft cellular systems, plastic behavior is commonly described in terms of the T1 event, which is the elementary topological change in the structure corresponding to a neighbor switching. In a crystalline foam, where bubbles are disposed in an hexagonal lattice, a T1 event can be identified with the creation of a paired dislocations (i.e. two pairs of 5- and 7-sided cells). From a mechanical point of view, the T1 process corresponds to a transition from one metastable configuration to another, after passing through an unstable configuration where four films meet at one junction. The spontaneous evolution from one four-fold junction to two three-fold junctions, which involves creation of a new film, is driven by minimization of the surface area. Various experimental and theoretical studies on the

*frequency*of rearrangement events in foams have been conducted, but little is known about the typical relaxation time associated with such events. Indeed, the dynamics of the relaxation processes is usually neglected in simulations of foams even though the rheological behavior of a foam obviously depends on this relaxation time. More generally, to study the evolution of the foam structure, it is necessary to understand the dynamics of the elementary relaxation process.we investigated theoretically and experimentally the effect of the viscoelastic parameters on the dynamics of the T1 process. Experiments in a two-dimensional foam showed that the relaxation time depends on the interfacial viscoelasticity of the films, but not on the shear viscosity of the bulk liquid. These results are corroborated by a model, which allows for an estimation of the Gibbs elasticity and the surface viscosity of the surfactants used to make the foam.

Numerous studies have been carried out on the plastic behavior of foams, either experimentally or numerically, and with various setups (bubble raft, Hele-Shaw cell,...). No clear consensus emerges on the existence of a power-law behavior for the probability of T1 events, or even on the value of the power-law exponent. Two factors may explain these variability of observed behaviors: i) the liquid fraction, which is high in a bubble raft where circular bubbles are in contact with the pool, and low in a dry foam with polygonal bubbles. Liquid fraction has an impact on the dissipation during relaxation, but it also modifies the bubble interactions. It is worth mentioning that in the dry foam limit, energy interactions are not pairwise anymore, what could lead to a peculiar plastic behavior; ii) the dissipation at the top and bottom boundaries: for bubble rafts, the bottom surface is in contact with the liquid, while in Hele-Shaw cells foam is in contact with the glass plate. The top surface is in contact either with air or a glass plate.

We are currently investigating numerically the mean energy cost a unique T1 event, and the

*interaction energy*between two T1s.## Statistical description of membranes and filaments

Two different tensions can be defined for an interface: the internal tension, $\gamma$, conjugated to the real interface area in the Hamiltonian, and the frame tension, $\tau$, conjugated to the projected area. According to the standard Monge description of an interface, fluctuation spectrum is governed by $\gamma$. Yet, using rotational invariance arguments, several recent studies argued that fluctuation spectrum is rather governed by $\tau$, and not the intrinsic tension. However, these studies disagree on which hypothesis used in the Monge model is at the origin of the discrepancy: either a miscounting of configurations, quantified with the integration measure, or the use of a quadratic approximation of the Hamiltonian used in the calculations. Analyzing the simplest case of a one-dimensional interface with zero bending modulus, for which the statistical description can be made without any approximation, we give a new proof that the interface fluctuations are driven by $\tau$, and show that the origin of the issue of with the Monge mondel is a miscounting of interface configurations.