# Publications

## Preprints and manuscripts in preparation

Pellé A. & Durand M., “Deflection of a cantilever under an elastoplastic flow", submitted (2023).

Durand M., “Melting of two-dimensional soft cellular systems”, in preparation.

Spera G., Duclut C., Durand M. & Tailleur J., "Nematic Torques in Scalar Active Matter: when Fluctuations Favor Polar Order and Persistence", submitted (2023).

Saiseau R., Busson V. & Durand M., “Network emergence and reorganization in confined slime moulds”, submitted (2023).

## Published articles

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 self-organization of contractile activity, we investigate the dynamic patterns of cortex contraction in true slime mold Physarum polycephalum confined in ring-shaped 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.

Two different tensions can be defined for a fluid membrane: the internal tension , 𝛾 , conjugated to the real membrane area in the Hamiltonian, and the frame tension, 𝜏, conjugated to the projected (or frame) area. According to the standard statistical description of a membrane, the fluctuation spectrum is governed by 𝛾. However, using rotational invariance arguments, several studies argued that fluctuation spectrum must be governed by the frame tension 𝜏 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 one-dimensional membrane, for which arc length offers a natural parametrization, we give a new proof that the fluctuations are driven by 𝜏, 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.

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 stress-strain relationship and structural changes that take place in cellular systems.

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 cell-sorting 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 power-law 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.

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 Young-Dupré 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).

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 out-of-plane 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 two-dimensional 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.

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 order-disorder transition of monodisperse, two-dimensional cellular systems driven by thermal agitation. We show that the transition follows most of the predictions of Kosterlitz-Thouless-Halperin-Nelson-Young theory developed for melting of 2D solids, extending the validity of this theory to systems with many-body 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.

We derive general conditions for the design of two-dimensional stiffest elastic networks with tetrakis-like (or "Union Jack"-like) topology. Upon generalizing recent results for tetrakis structures composed of two different rod geometries (length and cross-sectional 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 tetrakis-like topology. The potential of the analyzed structures as innovative metamaterials featuring extremely high elastic moduli vs. density ratios is highlighted.

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 long-term stage is independent of the chosen acceptance rate and chosen path in the temperature space.

We consider the hydrodynamics of lipid bilayers containing transmembrane proteins of arbitrary shape. This biologically-motivated 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 (σ>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 large-scale diffusion of a concentrated protein patch. We find that the diffusion profile is not self-similar, owing to the wavevector dependence of the effective diffusion coefficient.

In a recent series of papers, a statistical model that accounts for correlations between topological and geometrical properties of a two-dimensional 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 self-consistency 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 "Grand-Canonical" description, in which the foam constitutes a reservoir of sides and curvature, is justified.

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.

Bubble monolayers are model systems for experiments and simulations of two-dimensional packing problems of deformable objects. We explore the relation between the distributions of the number of bubble sides (topology) and the bubble areas (geometry) in the low liquid fraction limit. We use a statistical model [M. Durand, Europhys. Lett. 90, 60002 (2010)] which takes into account Plateau laws. We predict the correlation between geometrical disorder (bubble size dispersity) and topological disorder (width of bubble side number distribution) over an extended range of bubble size dispersities. Extensive data sets arising from shuffled foam experiments, SURFACE EVOLVER simulations, and cellular Potts model simulations all collapse surprisingly well and coincide with the model predictions, even at extremely high size dispersity. At moderate size dispersity, we recover our earlier approximate predictions [M. Durand, J. Kafer, C. Quilliet, S. Cox, S. A. Talebi, and F. Graner, Phys. Rev. Lett. 107, 168304 (2011)]. At extremely low dispersity, when approaching the perfectly regular honeycomb pattern, we study how both geometrical and topological disorders vanish. We identify a crystallization mechanism and explore it quantitatively in the case of bidisperse foams. Due to the deformability of the bubbles, foams can crystallize over a larger range of size dispersities than hard disks. The model predicts that the crystallization transition occurs when the ratio of largest to smallest bubble radii is 1.4.

The rigidity of a network of elastic beams is closely related to its microstructure. We show both numerically and theoretically that there is a class of isotropic networks, which are stiffer than any other isotropic network of same density. The elastic moduli of these stiffest elastic networks are explicitly given.They constitute upper-bounds, which compete or improve the well-known Hashin–Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions. In particular, we present an optimal and isotropic three-dimensional structure which, to our knowledge, is the first one to be presented as such.

We propose an analytical model for the statistical mechanics of shuffled two-dimensional foams with moderate bubble size polydispersity. It predicts without any adjustable parameters the correlations between the number of sides n of the bubbles (topology) and their areas A (geometry) observed in experiments and numerical simulations of shuffled foams. Detailed statistics show that in shuffled cellular patterns n correlates better with A½ (as claimed by Desch and Feltham) than with A (as claimed by Lewis and widely assumed in the literature). At the level of the whole foam, standard deviations 𝛥n and 𝛥A are in proportion. Possible applications include correlations of the detailed distributions of n and A, three-dimensional foams, and biological tissues.

The methods of statistical mechanics are applied to two-dimensional foams under macroscopic agitation. A new variable —the total cell curvature— is introduced, which plays the role of energy in conventional statistical thermodynamics. The probability distribution of the number of sides for a cell of given area is derived. This expression allows to correlate the distribution of sides (“topological disorder”) to the distribution of sizes (“geometrical disorder”) in a foam. The model predictions agree well with available experimental data.

Networks of elastic beams can deform either by stretching or bending of their members. The primary mode of deformation (bending or stretching) crucially depends on the specific details of the network architecture. In order to shed light on the relationship between microscopic geometry and macroscopic mechanics, we characterize the structural features of networks which deform uniformly, through the stretching of the beams only. We provide a convenient set of geometrical criteria to identify such networks, and derive the values of their effective elastic moduli. The analysis of these criteria elucidates the variability of mechanical response of elastic networks. In particular, our study rationalizes the difference in mechanical behavior of cellular and fiber networks.

### M. Durand, "Low-density cellular materials with optimal conductivity and bulk modulus", i-revues http://hdl.handle.net/2042/15750 (2007).

Networks of elastic beams can deform either by stretching or bending of their members. The primary mode of deformation (bending or stretching) crucially depends on the specific details of the network architecture. In order to shed light on the relationship between microscopic geometry and macroscopic mechanics, we characterize the structural features of networks which deform uniformly, through the stretching of the beams only. We provide a convenient set of geometrical criteria to identify such networks, and derive the values of their effective elastic moduli. The analysis of these criteria elucidates the variability of mechanical response of elastic networks. In particular, our study rationalizes the difference in mechanical behavior of cellular and fiber networks.

The structure of pipe networks minimizing the total energy dissipation rate is studied analytically. Among all the possible pipe networks that can be built with a given total pipe volume (or pipe lateral surface area), the network which minimizes the dissipation rate is shown to be loopless. Furthermore, such an optimal network is shown to contain at most N−2 nodes in addition to the N sources plus sinks that it connects. These results are valid whether the possible locations for the additional nodes are chosen freely or from a set of nodes (such as points of a grid). Applications of these results to various physical situations and to the efficient computation of optimal pipe networks are also discussed

The elementary topological T1 process in a two-dimensional foam corresponds to the flip of one film with respect to the geometrical constraints, and is a process by which the structure of an out-of-equilibrium foam evolves. We study both experimentally and theoretically the T1 dynamics in a dry two-dimensional foam. The dynamics is controlled by the surface viscoelastic properties of the films (surface shear plus dilatational viscosity, μs+κ, and Gibbs elasticity ϵ), and is independent of the shear viscosity of the bulk liquid. Moreover, the dynamics of the T1 process provides a tool for measuring the surface rheological properties: we obtained ϵ=32±8mN/m and μs+κ=1.3±0.7mPa·m·s for sodium dodecyl sulfate, and ϵ=65±12mN/m and μs+κ=31±12mPa·m·s for bovine serum albumin, in good agreement with literature values.

We analyze the structure of networks minimizing the global resistance to flow (or dissipative energy) with respect to two different constraints: fixed total channel volume and fixed total channel surface area. First, we show that channels must be straight and have uniform cross-sectional areas in such optimal networks. We then establish a relation between the cross-sectional areas of adjoining channels at each junction. Indeed, this relation is a generalization of Murray's law, originally established in the context of local optimization. We establish a relation too between angles and cross-sectional areas of adjoining channels at each junction, which can be represented as a vectorial force balance equation, where the force weight depends on the channel cross-sectional area. A scaling law between the minimal resistance value and the total volume or surface area value is also derived from the analysis. Furthermore, we show that no more than three or four channels meet at each junction of optimal bidimensional networks, depending on the flow profile (e.g., Poiseuille-like or pluglike) and the considered constraint (fixed volume or surface area). In particular, we show that sources are directly connected to wells, without intermediate junctions, for minimal resistance networks preserving the total channel volume in case of plug flow regime. Finally, all these results are compared with the structure of natural networks.

We present an alternative derivation of upper-bounds for the bulk modulus of both two-dimensional and three-dimensional cellular materials. For two-dimensional materials, we recover exactly the expression of the Hashin-Shtrikman (HS) upper-bound in the low-density limit, while for three-dimensional materials we even improve the HS bound. Furthermore, we establish necessary and sufficient conditions on the cellular structure for maximizing the bulk modulus, for a given solid volume fraction. The conditions are found to be exactly those under which the electrical (or thermal) conductivity of the material reaches its maximal value as well. These results provide a set of straightforward criteria allowing us to address the design of optimized cellular materials, and shed light on recent studies of structures with both maximal bulk modulus and maximal conductivity. Finally, we discuss the specific case of spring networks, and analyze the compatibility of the criteria presented here with the geometrical constraints caused by minimization of surface energy in a real foam.

Many situations in physics, biology, and engineering consist of the transport of some physical quantity through a network of narrow channels. The ability of a network to transport such a quantity in every direction can be described by the average conductivity associated with it. When the flow through each channel is conserved and derives from a potential function, we show that there exists an upper bound of the average conductivity and explicitly give the expression for this upper bound as a function of the channel permeability and channel length distributions. Moreover, we express the necessary and sufficient conditions on the network structure to maximize the average conductivity. These conditions are found to be independent of the connectivity of the vertices.

The Lemlich law provides a simple estimate of the relative conductivity of a three-dimensional foam, as one third of its liquid fraction. This is based on an expression for the conductivity of a network of uniform wires (conducting lines). We show this to be an exact upper bound for the conductivity, orientationally averaged in the case of anisotropic systems. We discuss the dependance of conductivity on the geometry of the network structure and establish two necessary and sufficient conditions to maximize the conductivity. We note the connection between this problem and that of line-length minimization and also that between anisotropic conductivity and stress for a two-dimensional foam. These results are illustrated by various numerical simulations of network conductivities. The theorems presented in this paper may also be applied to the thermal conductivity and the permeability of a network.

Recent research related to foam drainage is surveyed with emphasis on the influence of interfacial rheology. Active research directions are highlighted and the possible impact of these studies on macroscopic rheology is indicated.

In this thesis manuscript, we report theoretical and experimental results on drainage of aqueous foams. Two experimental setups were built to study foam drainage, the first one based on conductivity measurements and the second one based on multiple light scattering. These two devices enabled us to highlight the effect of the physical chemistry on the drainage of foam, and thus to reconcile in a qualitative way the two existing models of foam drainage. This effect of physical chemistry corresponds to the important role played by the surface viscoelasticity parameters of the channels on the drainage. Moreover, the experimental results show that the bubble size also has an influence on the drainage regime. This result can be understood in terms of competition between the dissipation at the surface and the dissipation in the bulk. Measurements of the dynamical surface tension were also made with the pendant drop technique. We proved this way that the surface of the channels remains locally in thermodynamic equilibrium with the bulk. On the theoretical level, a new model for the drainage explicitly including the surface viscoelastic parameters and the bubble size is proposed. The dependence of the drainage with these parameters is qualitatively in agreement with the experimental results. Finally, the effect of the disorder has been studied by calculating the effective conductivity of periodic foams. The conclusion of this study is that the laws governing the local structure of a foam (Plateau's laws) naturally imply a uniform distribution and orientation of the channels in the foam.

We have investigated theoretically the effect of surface viscoelasticity on the drainage of an aqueous foam. Former theories consider that the flow in Plateau borders is either Poiseuille flow or plug-flow. In the last case, the dissipation is attributed to flow in the nodes connecting Plateau borders. Although we do not include this dissipation in our model, we obtain a drainage equation that includes terms equivalent to those of the earlier models. We show that when the water solubility of the surfactant stabilizing the foam is low, the control parameter M for the transition between the two regimes is the ratio , where μ is the bulk viscosity, Ds the surface diffusion coefficient, R the radius of curvature of the Plateau border and ɛ the surface elasticity. When the surfactant is more soluble M is rather related to the bulk diffusion coefficient. Within the frame of this approach and in view of the estimated M values, we show that the flow in Plateau borders is Poiseuille-like.

We have studied the drainage of foams made with viscous solutions. In this way, drainage is slowed down, and microgravity conditions are approached. The behaviour of mixtures of water and glycerol follows the theoretical predictions, but that of aqueous solutions of a polymer (polyethylene oxide) do not: drainage is much faster than for water-glycerol mixtures of the same bulk viscosity. The origin of this puzzling behavior is still not understood.

The electronic density of states (DOS) of 1D quasicrystals looks like that of a highly doped, p-type semiconductor: the Fermi level lies in the “impurity band”, which consists of localized states. At low temperatures, the conductivity of an electronic structure of this type is by variable-range hopping, as observed recently in i-Al–Pd–Re.

The velocity of gravity-driven flow through aqueous foams forced drainage has been determined by using electrical conductivity measurements in foams made with solutions of different surfactants. There is always a scaling behavior power law between the drainage velocity V and the imposed flow rate Q: V~Q^α, but the α coefficient varies between the different surfactant solutions and increases with surface viscosity. An explanation of this behavior will be given in terms of a transition between a node-dominated and a Plateau border-dominated viscous dissipation, for which theory predicts respectively α=1/3 and α=1/2.