Preprints and manuscripts in preparation


Published articles
V. Busson, R. Saiseau & M. Durand, “Emergence of dynamic contractile patterns in slime mold confined in a ring geometry”, J. Phys. D: Appl. Phys. 55, 415401 (2022).
Coordination of cytoplasmic flows on large scales in space and time are at the root of many cellular processes, including growth, migration or division. These flows are driven by organized contractions of the actomyosin cortex. In order to elucidate the basic mechanisms at work in the selforganization of contractile activity, we investigate the dynamic patterns of cortex contraction in true slime mold Physarum polycephalum confined in ringshaped chambers of controlled geometrical dimensions. We make an exhaustive inventory of the different stable contractile patterns in the absence of migration and growth. We show that the primary frequency of the oscillations is independent of the ring perimeter, while the wavelength scales linearly with it. We discuss the consistence of these results with the existing models, shedding light on the possible feedback mechanisms leading to coordinated contractile activity. 
M. Durand, “Frame tension governs the thermal fluctuations of a fluid membrane: new evidence”, Soft Matter 18, 3891 (2022).
Two different tensions can be defined for a fluid membrane: the internal tension, $\gamma$, conjugated to the real membrane area in the Hamiltonian, and the frame tension, $\tau$, conjugated to the projected (or frame) area. According to the standard statistical description of a membrane, the fluctuation spectrum is governed by $\gamma$. However, using rotational invariance arguments, several studies argued that fluctuation spectrum must be governed by the frame tension $\tau$ instead. These studies disagree on the origin of the result obtained with the standard description yet: either a miscounting of configurations, quantified with the integration measure, or the use of a quadratic approximation of the Helfrich Hamiltonian. Analyzing the simplest case of a onedimensional membrane, for which arc length offers a natural parametrization, we give a new proof that the fluctuations are driven by $\tau$, and show that the origin of the issue with the standard description is a miscounting of membrane configurations. The origin itself of this miscounting depends on the thermodynamic ensemble in which calculations are made. 
F. Villemot & M. Durand, “Quasistatic rheology of soft cellular systems using Cellular Potts Model”, Phys. Rev. E 104, 055303 (2021).
Soft cellular systems, such as foams or biological tissues, exhibit highly complex rheological properties, even in the quasistatic regime, that numerical modeling can help to apprehend. We present a numerical implementation of quasistatic strain within the widely used cellular Potts model. The accuracy of the method is tested by simulating the quasistatic strain of 2D dry foams, both ordered and disordered. The implementation of quasistatic strain in CPM allows the investigation of sophisticated interplays between stressstrain relationship and structural changes that take place in cellular systems. 
Durand M., “Largescale simulations of biological cell sorting driven by differential adhesion follow a diffusionlimited coalescence of rounded clusters regime”, PLOS Computational Biology 17(8): e1008576 (2021).
Cell sorting, whereby a heterogeneous cell mixture segregates and forms distinct homogeneous tissues, is one of the main collective cell behaviors at work during development. Although differences in interfacial energies are recognized to be a possible driving source for cell sorting, no clear consensus has emerged on the kinetic law of cell sorting driven by differential adhesion. Using a modified Cellular Potts Model algorithm that allows for efficient simulations while preserving the connectivity of cells, we numerically explore cellsorting dynamics over very large scales in space and time. For a binary mixture of cells surrounded by a medium, increase of domain size follows a powerlaw with exponent $n = 1/4$ independently of the mixture ratio, revealing that the kinetics is dominated by the diffusion and coalescence of rounded domains. We compare these results with recent numerical studies on cell sorting, and discuss the importance of algorithmic differences as well as boundary conditions on the observed scaling. 
Durand M., “Mechanical approach to surface tension and capillary phenomena”, American Journal of Physics 89(3), 261266 (2021).
Many textbooks dealing with surface tension favor the thermodynamic approach (minimization of some thermodynamic potential such as free energy) over the mechanical approach (balance of forces) to describe capillary phenomena, stating that the latter is flawed and misleading. Yet, mechanical approach is more intuitive for students than free energy minimization, and does not require any knowledge of thermodynamics. In this paper we show that capillary phenomena can be unmistakably described using the mechanical approach, as long as the system on which the forces act is properly defined. After reminding the microscopic origin of a tangential tensile force at the interface, we derive the YoungDupré equation, emphasizing that this relation should be interpreted as an interface condition at the contact line, rather than a force balance equation. This correct interpretation avoids misidentification of capillary forces acting on a given system. Moreover, we show that a reliable method to correctly identify the acting forces is to define a control volume that does not embed any contact line on its surface. Finally, as an illustration of this method, we apply the mechanical approach in a variety of ways on a classic example: the derivation of the equilibrium height of capillary rise (Jurin's law). 
F. Villemot, A. Calmettes & M. Durand, “Thermal shape fluctuations of a twodimensional compressible droplet”, Soft Matter 16, 10358  10367 (2020).
Analysis of thermal capillary waves on the surface of a liquid usually assumes incompressibility of the bulk fluid. However, for droplets or bubbles with submicronic size, or for epithelial cells whose outofplane elongation can be modeled by an effective 2D bulk modulus, compressibility of the internal fluid must be taken into account for the characterization of their shape fluctuations. We present a theoretical analysis of the fluctuations of a twodimensional compressible droplet. Analytical expressions for area, perimeter and energy fluctuations are derived and compared with Cellular Potts Model (CPM) simulations. This comparison shows a very good agreement between theory and simulations, and offers a precise calibration method for CPM simulations. 
M. Durand and J. Heu, “Thermally driven orderdisorder transition in twodimensional soft cellular systems”, Phys. Rev. Lett. 123, 188001 (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).
