Camille Duprat

Wetting of flexible fibre arrays

Fibrous media are functional and versatile materials, as demonstrated by their ubiquity both in natural systems such as feathers and adhesive pads and in engineered systems from nanotextured surfaces to textile products, where they offer benefits in filtration, insulation, wetting and colouring. The elasticity and high aspect ratios of the fibres allow deformation under capillary forces, which cause mechanical damage, matting self-assembly or colour changes, with many industrial and ecological consequences. Attempts to understand these systems have mostly focused on the wetting of rigid fibres or on elastocapillary effects in planar geometries and on a fibre brush withdrawn from an infinite bath. Here we consider the frequently encountered case of a liquid drop deposited on a flexible fibre array and show that flexibility, fibre geometry and drop volume are the crucial parameters that are necessary to understand the various observations referred to above. We identify the conditions required for a drop to remain compact with minimal spreading or to cause a pair of elastic fibres to coalesce. We find that there is a critical volume of liquid, and, hence, a critical drop size, above which this coalescence does not occur. We also identify a drop size that maximizes liquid capture. For both wetting and deformation of the substrates, we present rules that are deduced from the geometric and material properties of the fibres and the volume of the drop. These ideas are applicable to a wide range of fibrous materials, as we illustrate with examples for feathers, beetle tarsi, sprays and microfabricated systems.

Dynamics of elastocapillary rise

We present the results of a combined experimental and theoretical investigation of the surface-tension-driven coalescence of flexible structures. Specifically, we consider the dynamics of the rise of a wetting liquid between flexible sheets that are clamped at their upper ends. As the elasticity of the sheets is progressively increased, we observe a systematic deviation from the classical diffusive-like behaviour: the time to reach equilibrium increases dramatically and the departure from classical rise occurs sooner, trends that we elucidate via scaling analyses. Three distinct temporal regimes are identified and subsequently explored by developing a theoretical model based on lubrication theory and the linear theory of plates. The resulting free-boundary problem is solved numerically and good agreement is obtained with experiments.

Wetting on two parallel fibers: drop to column transitions

While the shape and stability of drops on single cylindrical fibers have received vast attention, there are few studies that consider a drop sitting between two fibers, which is a first step toward understanding the wetting of larger fibrous networks. In this paper, we investigate experimentally the behavior of a finite volume of liquid on two parallel rigid fibers. The liquid wetting the fibers can adopt two distinct equilibrium shapes: a compact approximately hemispherical drop shape or a long liquid column of constant cross-section. These two morphologies depend on the inter-fiber distance, the liquid volume, the fiber radius and the liquid–fiber contact angle. We study the transitions between a drop shape and a column by incrementally varying the inter-fiber distance and find that the transition depends on the global geometry of the system as well as on the volume of liquid. For totally wetting drops, we identify the regions where the drops or columns prevail, and find that there is a region where both morphologies are stable, and the transitions from one state to the other are hysteretic. These switches in morphologies may be used to manipulate or transport liquid at small scales.

Evaporation of Drops on Two Parallel Fibers: Influence of the Liquid Morphology and Fiber Elasticity

We investigate experimentally the evaporation of liquid accumulated on a pair of parallel fibers, rigid or flexible. The liquid wetting the fibers can adopt two distinct morphologies: a compact drop shape, whose evaporation dynamics is similar to that of an isolated aerosol droplet, or a long liquid column of constant cross-section, whose evaporation dynamics depends upon the aspect ratio of the column. We thus find that the evaporation rate is constant for drops, while it increases strongly for columns as the interfiber distance decreases, and we propose a model to explain this behavior. When the fibers are flexible, the transition from drops to columns can be induced by the deformation of the fibers because of the capillary forces applied by the drop. Thus, we find that the evaporation rate increases with increasing flexibility. Furthermore, complex morphology transitions occur upon drying, which results in spreading of the drop as it evaporates.

Wetting of crossed fibers: Multiple steady states and symmetry breaking

We investigate the wetting properties of the simplest element of an array of random fibers: two rigid fibers crossing with an inclination angle and in contact with a droplet of a perfectly wetting liquid. We show experimentally that the liquid adopts different morphologies when the inclination angle is increased: a column shape, a mixed morphology state where a drop lies at the end of a column, or a drop centered at the node. An analytical model is provided that predicts the wetting length as well as the presence of a non-symmetric state in the mixed morphology regime. The model also highlights a symmetry breaking at the transition between the column state and the mixed morphology. The possibility to tune the morphology of the liquid could have important implications for drying processes.

Capillary stretching of fibers

We study the interaction of a finite volume of liquid with two parallel thin flexible fibers. A tension along the fibers is imposed and may be varied. We report two morphologies, i.e. two types of wet adhesion: a weak capillary adhesion, where a liquid drop bridges the fibers, and a strong elastocapillary adhesion where the liquid is spread between two collapsed fibers. We show that geometry, capillarity and stretching are the key parameters at play. We describe the collapse and detachment of the fibers as a function of two nondimensional parameters, arising from the geometry of the system and a balance between capillary and stretching energies. In addition, we show that the morphology, thus the capillary adhesion, can be controlled by changing the tension within the fibers.

Oscillations of confined fibres transported in microchannels

We investigate the trajectories of rigid fibres as they are transported in a pressure-driven flow, at low Reynolds number, in shallow Hele-Shaw cells. The transverse confinement and the resulting viscous friction on these elongated objects, as well as the lateral confinement (i.e. the presence of lateral walls), lead to complex fibre trajectories that we characterize with a combination of microfluidic experiments and simulations using modified Brinkman equations. We show that the transported fibre behaves as an oscillator for which we obtain and analyse a complete state diagram.

CHAPTER 6:Elastocapillarity

In this chapter, we study phenomena associated with capillary forces interacting with a soft, deformable body. This interplay of surface tension and elasticity occurs in a wide variety of situations, including ones that involve the deformation of sheets and fibers. We introduce the concept of surface tension and discuss typical phenomena, such as the wetting of soft substrates; capillary adhesion; surface-tension-induced bending, buckling or wrinkling of structures; and a prototype of capillary flow, namely imbibition. We provide quantitative models to explain the observations, including using dimensional analysis and order-of-magnitude estimates, and present many cases where the predictions of the models are compared with experimental results. These ideas permeate many fields, and we have tried to capture the spirit and beauty of the subject.

CHAPTER 2:Low-Reynolds-Number Flows

In this chapter, we provide a brief description of some of the main results of low-Reynolds-number hydrodynamics. In particular, we introduce the general subject by way of several example flows and provide derivations or explanations of some of the fluid dynamics themes that are used in later chapters of this book: channel flows, Darcy’s approximation, kinematic reversibility, integral equations, point forces, slender body theory, Jeffery orbits, etc. Specifically, we discuss general theoretical principles, and describe various problems involving the motion of rigid spherical particles, deformable particles, and elongated particles. The main goal of this chapter is to present the physical intuition and underlying mathematics of common low-Reynolds-number flow situations.