Patterning droplets with durotaxis

Abstract

Numerous cell types have shown a remarkable ability to detect and move along gradients in stiffness of an underlying substrate-a process known as durotaxis. The mechanisms underlying durotaxis are still unresolved, but generally believed to involve active sensing and locomotion. Here, we show that simple liquid droplets also undergo durotaxis. By modulating substrate stiffness, we obtain fine control of droplet position on soft, flat substrates. Unlike other control mechanisms, droplet durotaxis works without imposing chemical, thermal, electrical, or topographical gradients. We show that droplet durotaxis can be used to create large-scale droplet patterns and is potentially useful for many applications, such as microfluidics, thermal control, and microfabrication.droplet control | elasticity | soft matter | wetting | mechanosensing T he control of liquids on surfaces is essential for microfluidics (1), microfabrication (2), and coatings (3-5), to name but a few applications. Wetting is typically manipulated by controlling interfacial energies (6). Heterogeneous surface chemistries have been exploited to pattern (7,8) and transport droplets (3, 9). Gradients in temperature or electric potential can drive droplet motion (3, 9). Alternatively, surface topography can control the spreading of fluids. For example, isotropically rough surfaces can exhibit superhydrophobicity (10, 11), whereas anisotropic surfaces exhibit anisotropic spreading (12) and even directed droplet transport (13-15). Here, we introduce a method to control droplets on surfaces inspired by the biological phenomenon of durotaxis-the ability of many eukaryotic cell types to move along gradients in the stiffness of their extracellular matrix (16)(17)(18)(19). Although the current explanation of durotaxis involves active sensing of matrix stiffness and actomyosin-based motility (18), we show here that even simple liquid droplets display durotaxis. Furthermore, we show that durotaxis can be exploited to achieve largescale droplet patterning. A simple theory explains how drops move toward softer parts of a substrate, and quantitatively captures the droplet distribution on patterned surfaces. Droplet durotaxis is prominent on soft substrates, which are significantly deformed by liquid surface tension (20,21).The spreading of liquid droplets on stiff, flat surfaces is primarily described by the contact angle. In equilibrium, a small droplet takes the shape of a spherical cap with uniform contact angle θ determined by Young's law: γ LV cosθ = γ SV − γ SL . Here, indices L, S, and V of interfacial energies, γ, represent liquid, solid, and vapor, respectively (6). Spontaneous droplet motion typically occurs in two main cases. First, if the actual contact angle of a droplet differs from its equilibrium contact angle, the droplet will be driven to spread/contract until it reaches its equilibrium shape (6). Second, if there is a difference between the equilibrium contact angle on either side of a droplet, the droplet will be driven toward the more wetting sid...

Drops on soft solids: free energy and double transition of contact angles

“…Finally, the full shapes of the drops and the surface deflections are worked out for a case where the solid surface energies γ SV = γ SL . We find that the free energy is lower on softer substrates, and relate this to experiments of drop motion on substrate exhibiting a stiffness gradient (Style et al 2013b). …”

“…In these calculations, the droplet volume was kept constant. This allows for a direct comparison with recent experiments by Style et al (2013b) where droplets were found to move to softer regions on a substrate with a stiffness gradient. Our calculations, which predict a lower total energy F tot ≡ F el +F S +F γ for droplets on softer substrates, provide an explanation for this observation.…”

“…Elastic deformations take place over a length on the order of the elastocapillary length γ/E, where γ is the liquid surface tension and E the solid Young's modulus. Recent experiments have considered liquids on very soft elastomers with γ/E of the order of 1-100 microns and have reported many interesting features, such as the geometry near the contact line Jerison et al 2011;Style et al 2013a), compression of the solid (Marchand et al 2012a), evaporation and spreading dynamics (Carre et al 1996;Li et al 2007;Sokuler et al 2010), as well as migration of droplets on substrates with a stiffness gradient (Style et al 2013b).…”

Surface tension and contact with soft elastic solids

“…Recent experiments have illustrated that solid surface tension, Ç sv , can have a dominant role in the behaviour of soft materials [21][22][23][24][25][26][27] . Solid surface tension drives a rippling instability in soft, elongated structures 28 .…”

Equilibrium of droplets on a deformable substrate: Influence of disjoining pressure

“…There have been a number of experimental studies recently that have considered liquids on deformable substrates [8][9][10][11][12][13][14], where deformation of soft solids was investigated near the apparent contact line, but there exists a gap in theoretically understanding of the problem. This void is primarily due to the stress singularity present at the three phase contact line.…”