M. Durand and J. Heu, “Thermally driven orderdisorder transition in twodimensional soft cellular systems”, to appear in Phys. Rev. Lett. (2019).
Many systems, including biological tissues and foams, are made of highly packed units having high deformability but low compressibility. At two dimensions, these systems offer natural tesselations of plane with fixed density, in which transitions from ordered to disordered patterns are often observed, in both directions. Using a modified Cellular Potts Model algorithm that allows rapid thermalization of extensive systems, we numerically explore the orderdisorder transition of monodisperse, twodimensional cellular systems driven by thermal agitation. We show that the transition follows most of the predictions of KosterlitzThoulessHalperinNelsonYoung theory developed for melting of 2D solids, extending the validity of this theory to systems with manybody interactions. In particular, we show the existence of an intermediate hexatic phase, which preserves the orientational order of the regular hexagonal tiling, but looses its positional order. In addition to shed light on the structural changes observed in experimental systems, our study shows that soft cellular systems offer macroscopic systems in which KTHNY melting scenario can be explored. 
E. Babilio, F. Fabbrocino, M. Durand and F. Fraternalli, “On the design of 2D stiffest elastic networks with tetrakislike architecture”, Extreme Mechanics Letters 15, 5762 (2017).
We derive general conditions for the design of twodimensional stiffest elastic networks with tetrakislike (or "Union Jack"like) topology. Upon generalizing recent results for tetrakis structures composed of two different rod geometries (length and crosssectional area), we derive the elasticity tensor of a lattice with generalized tetrakis architecture, which is composed of three kinds of rods and generally exhibits anisotropic response. This study is accompanied by an experimental verification of the theoretical prediction for the longitudinal modulus of the lattice. In addition, the introduction of a third rod geometry allows to extend considerably the possible lattice geometries for isotropic, stiffest elastic lattices with tetrakislike topology. The potential of the analyzed structures as innovative metamaterials featuring extremely high elastic moduli vs. density ratios is highlighted. 
M. Durand and E. Guesnet, “An efficient Cellular Potts Model algorithm that forbids cell fragmentation”, Computer Physics Communications 208, 5463 (2016).
The Cellular Potts Model (CPM) is a lattice based modeling technique which is widely used for simulating cellular patterns such as foams or biological tissues. Despite its realism and generality, the standard Monte Carlo algorithm used in the scientific literature to evolve this model works on a limited range of simulation temperature in order to preserve the connectivity of the cells. We present a new algorithm in which cell fragmentation is forbidden for all simulation temperatures. This allows to significantly enhance realism of the simulated patterns. It also increases the computational efficiency compared with the standard CPM algorithm even at same simulation temperature, thanks to the time spared in not doing unrealistic moves. Moreover, this algorithm restores the detailed balance equation, ensuring that the longterm stage is independent of the chosen acceptance rate and chosen path in the temperature space. 
A. CallanJones, M. Durand, and J.B. Fournier, “Hydrodynamics of bilayer membranes with diffusing transmembrane proteins”, Soft Matter 12, 17911800 (2016).
We consider the hydrodynamics of lipid bilayers containing transmembrane proteins of arbitrary shape. This biologicallymotivated problem is relevant to the cell membrane, whose fluctuating dynamics play a key role in phenomena ranging from cell migration, intercellular transport, and cell communication. Using Onsager's variational principle, we derive the equations that govern the relaxation dynamics of the membrane shape, of the mass densities of the bilayer leaflets, and of the diffusing proteins' concentration. With our generic formalism, we obtain several results on membrane dynamics. We find that proteins that span the bilayer increase the intermonolayer friction coefficient. The renormalization, which can be significant, is in inverse proportion to the protein's mobility. Second, we find that asymmetric proteins couple to the membrane curvature and to the difference in monolayer densities. For practically all accessible membrane tensions ($\sigma> 10^{8}$ N/m) we show that the protein density is the slowest relaxing variable. Furthermore, its relaxation rate decreases at small wavelengths due to the coupling to curvature. We apply our formalism to the largescale diffusion of a concentrated protein patch. We find that the diffusion profile is not selfsimilar, owing to the wavevector dependence of the effective diffusion coefficient. 
M. Durand, “Statistical mechanics of twodimensional shuffled foams: physical foundations of the model”, European Physical Journal E 38, 137 (2015).
In a recent series of papers, a statistical model that accounts for correlations between topological and geometrical properties of a twodimensional shuffled foam has been proposed and compared with experimental and numerical data. Here, the various assumptions on which the model is based are exposed and justified: the equiprobability hypothesis of the foam configurations is argued. The range of correlations between bubbles is discussed, and the mean field approximation that is used in the model is detailed. The two selfconsistency equations associated with this mean field description can be interpreted as the conservation laws of number of sides and bubble curvature, respectively. Finally, the use of a "GrandCanonical" description, in which the foam constitutes a reservoir of sides and curvature, is justified. 
S. Gauchet, M. Durand, and D. Langevin, “Foam drainage: possible influence of a nonnewtonian surface shear viscosity”, Journal of Colloid and Interface Science 449, 373376 (2015).
We describe forced drainage experiments of foams made with model surfactant solutions with different surface rheology. We analyze the origin of two distinct drainage transitions reported in the literature, between regimes where the bubble surfaces are mobile or rigid. We propose that both transitions are related to the surface shear viscosity and to its shear thinning behavior. Shear thinning could also account for the huge discrepancies between measurements reported in the literature. The role of surface tension gradients, i.e. Marangoni effect, could not possibly explain the behavior observed with the different solutions. 
M. Durand, "Lowdensity cellular materials with optimal conductivity and bulk modulus", irevues http://hdl.handle.net/2042/15750 (2007).
abstract: Optimal upper bounds on the average conductivity and the bulk modulus of low density cellular materials are established. These bounds are tighter than the classical HashinShtrikman upper bounds, and can be applied to anisotropic materials. Furthermore, necessary and sufficient conditions on the microgeometry of the cellular materials to maximize (for a given solid volume fraction) either the average conductivity or the bulk modulus are derived. The conditions are found to be identical for both quantities, so that the bounds are attained simultaneously: a lowdensity cellular material with maximal average conductivity will have maximal bulk modulus as well. 

D. Langevin, A. StJalmes, M. Durand, and M. Safouane, "Les Mousses" , bulletin de la SFP, 134, 1116 (2002).
