Evaporation of a thin film: diffusion of the vapour and Marangoni instabilities

The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffusion. The boundary condition at the interface couples transfer from the liquid to its vapour and diffusion flux. A non-local lubrication equation is obtained; this non-local nature comes from the Laplace equation associated with quasi-static diffusion. The linear stability of a flat film is studied in this general framework. The subsequent analysis is restricted to moderately thick films for which it is shown that evaporation is diffusion limited and that the gas phase is saturated in vapour in the vicinity of the interface. The stability depends only on two control parameters, the capillary and Marangoni numbers. The Marangoni effect is destabilising whereas capillarity and evaporation are stabilising processes. The results of the linear stability analysis are compared with the experiments of Poulard et al (2003) performed in a different geometry. In order to study the resulting patterns, the amplitude equation is obtained through a systematic multiple-scale expansion. The evaporation rate is needed and is computed perturbatively by solving the Laplace problem for the diffusion of vapour. The bifurcation from the flat state is found to be a supercritical transition. Moreover, it appears that the non-local nature of the diffusion problem unusually affects the amplitude equation

Predicting the mechanical properties of spring networks

The elastic response of mechanical, chemical, and biological systems is often modeled using a discrete arrangement of Hookean springs, either modeling finite material elements or even the molecular bonds of a system. However, to date, there is no direct derivation of the relation between discrete spring network, and a general elastic continuum. Furthermore, understanding the networks' mechanical response requires simulations that may be expensive computationally. Here we report a method to derive the exact elastic continuum model of any discrete network of springs, requiring network geometry and topology only. We identify and calculate the so-called "non-affine" displacements. Explicit comparison of our calculations to simulations of different crystalline and disordered configurations, shows we successfully capture the mechanics even of auxetic materials. Our method is valid for residually stressed systems with non-trivial geometries, is easily generalizable to other discrete models, and opens the possibility of a rational design of elastic systems

Statistics of power injection in a plate set into chaotic vibration

A vibrating plate is set into a chaotic state of wave turbulence by either a periodic or a random local forcing. Correlations between the forcing and the local velocity response of the plate at the forcing point are studied. Statistical models with fairly good agreement with the experiments are proposed for each forcing. Both distributions of injected power have a logarithmic cusp for zero power, while the tails are Gaussian for the periodic driving and exponential for the random one. The distributions of injected work over long time intervals are investigated in the framework of the fluctuation theorem, also known as the Gallavotti-Cohen theorem. It appears that the conclusions of the theorem are verified only for the periodic, deterministic forcing. Using independent estimates of the phase space contraction, this result is discussed in the light of available theoretical framework

Regulatory Role of Cell Division Rules on Tissue Growth Heterogeneity

The coordination of cell division and cell expansion are critical to normal development of tissues. In plants, cell wall mechanics and the there from arising cell shapes and mechanical stresses can regulate cell division and cell expansion and thereby tissue growth and morphology. Limited by experimental accessibility it remains unknown how cell division and expansion cooperatively affect tissue growth dynamics. Employing a cell-based two dimensional tissue simulation we investigate the regulatory role of a range of cell division rules on tissue growth dynamics and in particular on the spatial heterogeneity of growth. We find that random cell divisions only add noise to the growth and therefore increase growth heterogeneity, while cell divisions following the shortest new wall or along the direction of maximal mechanical stress reduce growth heterogeneity by actively enhancing the regulation of growth by mechanical stresses. Thus, we find that, beyond tissue geometry and topology, cell divisions affect the dynamics of growth, and that their signature is embedded in the statistics of tissue growth.Engineering and Applied Science

A Mechanical Model to Interpret Cell-Scale Indentation Experiments on Plant Tissues in Terms of Cell Wall Elasticity and Turgor Pressure

International audienceMorphogenesis in plants is directly linked to the mechanical elements of growing tissues, namely cell wall and inner cell pressure. Studies of these structural elements are now often performed using indentation methods such as atomic force microscopy. In these methods, a probe applies a force to the tissue surface at a subcellular scale and its displacement is monitored, yielding force-displacement curves that reflect tissue mechanics. However, the interpretation of these curves is challenging as they may depend not only on the cell probed, but also on neighboring cells, or even on the whole tissue. Here, we build a realistic three-dimensional model of the indentation of a flower bud using SOFA (Simulation Open Framework Architecture), in order to provide a framework for the analysis of force-displacement curves obtained experimentally. We find that the shape of indentation curves mostly depends on the ratio between cell pressure and wall modulus. Hysteresis in force-displacement curves can be accounted for by a viscoelastic behavior of the cell wall. We consider differences in elastic modulus between cell layers and we show that, according to the location of indentation and to the size of the probe, force-displacement curves are sensitive with different weights to the mechanical components of the two most external cell layers. Our results confirm most of the interpretations of previous experiments and provide a guide to future experimental work

Multiscale modelling and analysis of growth of plant tissues

How morphogenesis depends on cell properties is an active direction of research. Here, we focus on mechanical models of growing plant tissues, where microscopic (sub)cellular structure is taken into account. In order to establish links between microscopic and macroscopic tissue properties, we perform a multiscale analysis of a model of growing plant tissue with subcellular resolution. We use homogenization to rigorously derive the corresponding macroscopic tissue scale model. Tissue scale mechanical properties are computed from all microscopic structural and material properties, taking into account deformation by the growth field. We then consider case studies and numerically compare the detailed microscopic model and the tissue-scale model, both implemented using finite element method. We find that the macroscopic model can be used to efficiently make predictions about several configurations of interest. Our work will help making links between microscopic measurements and macroscopic observations in growing tissues

‘Ruban à godets’: an elastic model for ripples in plant leaves

The formation of ripples along the edge of plant leaves is studied using a model of an elastic strip with spontaneous curvature. The equations of equilibrium of the strip are established in an explicit form. A numerical method of solution is presented and carried out. Owing to the presence of geometric nonlinearities, several equilibrium configurations are found but we show that only one of them is physical. To our knowledge, this is the first investigation of ripples in plant leaves that is based on the equations of elasticity. To cite this article: B. Audoly, A. Boudaoud, C. R. Mecanique 330 (2002) 831–836. © 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS solids and structures / elastic rods / growth in biology Ruban à godets: un modèle élastique pour les fronces des feuillages Résumé On étudie la formation des fronces au bord des feuillages grâce à un modèle de bande élastique à courbure spontanée. Les équations d’équilibre de la bande sont établies explicitement. Une méthode numérique de résolution est présentée puis mise en œuvre

Mechanics of elastic ellipsoidal shells

Complicated structural features at a much smaller scale than overall structure size form during the deformation of elastic shells under mechanical loading. These features which can be seen by simple experiments in everyday life, as well as in biological and engineering systems, are associated with high energy density and evolve in intricate ways as the shell is further loaded deep into the nonlinear regime. The key challenge in understanding these features is interaction of physics and geometry that leads to a mechanical response which is very different from the response of solid objects. The formation of localized periodic structures in the crushing of a spherical shell, such as a ping pong ball, is well documented in the literature and studies show that spherical shells manifest periodic structures as polygons under point and plate loading. We studied ellipsoidal shells under plate and point indentation and results are presented here. For plate indentation, we present a new instability that is observed in the indentation of a highly ellipsoidal shell. In this phenomenon, above a critical indentation depth, the plate loses contact with the shell in a series of well-defined “blisters” aligned with the smaller radius of curvature. We used detailed numerical model to study this instability and explained it using scaling arguments. We characterized the onset of instability and showed relation between number of blisters and their sizes with indentation depth and geometry of shell. Our study showed that properties of blister are independent of elastic properties of shell itself and this suggests a novel method for simply determining the thickness of highly ellipsoidal shells