Rugotaxis: Droplet motion without external energy supply

Abstract

Nano-patterned substrates offer possibilities for controlling the motion of fluids without external energy supply in novel technologies in microfluidics, coatings,etc. Here, we report on the rugotaxial motion of droplets on wrinkled substrates with gradient in the wavelength of the wrinkles by exploring a broad range of parameters, such as amplitude of the wrinkles, substrate wettability, droplet size and wavelength gradient. Adopting a theoretical and molecular dynamics approach, we determine the Cassie–Baxter and Wenzel states of the droplets, investigate the efficiency of rugotaxis as a function of different parameters, and discuss additional effects, such as pinning. We find that shallow wrinkles characterised by small wavelength gradients, and moderate adhesion of the droplet to the substrate favour the rugotaxis motion with growing droplet size, when pinning is avoided. We also find that the driving force in rugotaxis is the gain in interfacial energy between the droplet and the substrate as the droplet enters regions of denser wrinkles (smaller wavelengths of the wrinkles).

Unidirectional Droplet Propulsion onto Gradient Brushes without External Energy Supply

“…These differences are generally considered small, especially compared to other in silico experiments. 13,24 Moreover, we see that the maximum shifts to higher values of the attraction strength ε db as the size of the droplet increases. For example, the maximum velocity is observed when ε db = 0.4 ϵ for droplets with N = 2000 beads and for ε db = 0.9 ϵ for droplets of 16 000 beads (Figure 5d).…”

“…From our earlier studies, we have seen that the minimization of the interfacial energy U db between the droplet and the substrate is the driving force for the durotaxis motion in the case of substrates with stiffness gradient 13 or wrinkled substrates with a gradient in the wavelength characterizing the wrinkles. 24 This has also been found in the case of a nanoflake on substrates with stiffness gradient. 16 Moreover, we have argued that the efficiency of the motion depends on the rate of change of the interfacial energy along the gradient direction.…”

“…This is crucial, as the nanoscale motion of nano-objects is usually controlled by tiny effects at the interface between the droplet and the substrate resulting from the molecular interactions between the two, which only a method with molecular-scale resolution can capture. As in the case of durotaxis 13 and rugotaxis 24 motions, we find that the motion is caused by a gradient in the droplet− substrate interfacial energy, which translates into an effective force that drives the droplet toward the stiffer, flatter parts of the substrate. Moreover, we find that the efficiency of the durotaxis motion for brush substrates is higher for moderate values of the grafting density and droplet adhesion to the substrate as well as for smaller droplets and longer brush chains.…”

“…55 Our analysis has also indicated that the durotaxis motion on brush substrates is driven by a corresponding gradient in the interfacial energy between the droplet and the substrate, in line with previous findings in the context of various other substrate designs. 13,16,24 Moreover, we have found that the origin of the steady increase of the interfacial energy is related to the state of the brush surface, which appears as a "rougher" profile in the softer parts of the substrate and a flatter interface in the stiffer regions. This translates into a larger number of contacts between the droplet and the substrate in the stiffer parts and, hence, a more negative interfacial energy along the direction of the stiffness gradient.…”

“…33 In contrast, motion caused by temperature gradient (thermotaxis), 34 electrical current, 35−38 charge, 39−41 or even simple stretch, 42 would require external energy supply, 19 as, also, in the case of chemically driven droplets, 43,44 droplets on vibrated substrates, 45−48 or wettability ratchets. 49−52 Inspired by our previous work with specific substrate designs that lead to the durotaxis 13 and rugotaxis 24 motion of nanodroplets as motivated by the corresponding experiments, 15,25 here, we propose a new design for the substrate, which is capable of sustaining the droplet motion. We consider a polymer brush, consisting of polymer chains grafted onto a flat, solid surface, and the stiffness gradient is introduced to the brush substrate by varying the stiffness of the polymer chains, which in practice amounts to tuning their persistence length.…”