Transport of free surface liquid films and drops by external r atchets and self-ratcheting mechanisms Uwe Thiele ∗ Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK Karin John † Laboratoire de Spectrom´etrie Physique UMR 5588, CNRS and Universit´e Joseph Fourier - Grenoble I, BP 87 - 38402 Saint-Martin-d’H` eres, France We discuss the usage of ratchet mechanisms to transport a continuous phase in several micro-fluidic settings. In particular, we study the transport of a dielectric liquid in a heterogeneous ratchet capacitor that is periodically switched on and off. The second system consists of drops on a solid substrate that are transported by different types of harmonic substrate vibrations. We argue that the latter can be seen as a self-ratcheting process and discuss analogies between the employed class of thin film equ ations and Fokker-Planck equations for transport of discrete objects in a ’particle ratchet’. PACS numbers: 05.60.-k, 68.15.+e, 47.61.-k, 47.20.Ma, 47.55.Dz, 68.08.-p arXiv:1001.2417v1[cond-mat.soft]14 Jan 2010 Most of the many studied ratchet mechanisms are discussed in the context of directed transport and filtering of discret e objects as, e.g., colloids or macromolecules [1]. Examples include colloidal particles, that move in a directed manner through an array of periodic asymmetric micropores when un- der the influence of an oscillating external pressure [2]. Su ch particles may as well be driven by a dielectric potential of sawtooth shape that is switched on and off periodically [3]. In many of those ’discrete’ ratchets the carrier fluid (or solve nt) does not or nearly not move on average, i.e., they can not be employed to transport a continuous phase. One exception is the motion of magnetic particles in ferrofluids under the infl u- ence of an oscillating magnetic field [4]. There the resultin g net motion of the particles is transmitted to the carrier liquid by a strong viscous coupling. I. INTRODUCTION has not yet been studied in detail theoretically) as the static ratchet profile of the substrate interacts with the time-per iodic motion of the convection rolls to produce the mean flow, i.e., the time-periodic ’switching’ is done by the system itself. The strength and direction of the mean flow depend on the thick- ness of the liquid layer and the applied temperature gradient across the layer. A third example are Leidenfrost drops that are placed on a hot surface with a ratchet-like topography. This induces a directed motion of the drops [7, 9]. The effect is not only observed for many liquids but as well for small blocks of dry ice [10]. In contrast to the case of particle ratchets, not many models exist for the ratchet-driven transport of a continuous phase. Reference [11] considers a channel flow that is induced by locally asymmetric periodic arrays of electrodes under an AC voltage [12], and proposes an electro-osmotic model. However, the basic concept of ratchet transport – that a lo- cally asymmetric but globally homogeneous system may in- duce global transport if it is kept out of equilibrium [5] – ap - plies equally well to pure continuous media. This has been employed recently to transport a liquid or a solid phase in set- tings that do not show any macroscopic gradient [6–8]. The local asymmetry can result, for instance, from a periodic but asymmetric external potential. The periodic asymmetric vari- ation needs to be on a small length scale compared to the sys- tem size. In a first example, drops that are either placed on an asym- metrically structured substrate or between two such substrates move on average if a transverse electric field is applied peri - odically [8]. A similar transport may be achieved by vibrat- ing the structured substrate in a tangential direction [8]. A second example is the triggering of a large scale mean flow in Marangoni-B´enard convection over a solid substrate with asymmetric grooves [6]. That is an interesting example (that A different concept was analysed in relation to the first ex- perimental example: a two-layer film confined between two parallel plates can be driven by periodically switching on and off an external potential that is of ratchet shape in space. In the case that the potential is an electrical one and the two plates form a (heterogeneous) capacitor one may call the setup a flashing ’(electro-)wettability ratchet’ [13, 14]. The con cept of wettability may be understood in a rather general way as it may include any effective interactions between the liquid- liquid free interface and the solid walls as long as one is able to apply them in a time-periodic, spatially periodic (but locally asymmetric) manner. Net transport of a continuous phase may not only be caused by an external ratchet potential, but as well by a self-ratcheting effect: An enlightening example are drops that are driven up an incline by harmonic substrate vibrations at a finite angle to the substrate normal [15, 16]. A simple ratchet-like mech- anism has been proposed: The vibration component that is orthogonal to the substrate strongly modulates the hydrostatic pressure and thereby induces a nonlinear response in the drop shape. That in turn determines the strongly nonlinear drop mobility. As a result the drop reacts in an asymmetric manner to the vibration component parallel to the substrate. This sym- metry breaking between back and forth motion leads to the $^{∗}$Electronic address: u.thiele@lboro.ac.uk; URL: http://www.uwethiele.de $^{†}$Electronic address: kjohn@spectro.ujf-grenoble.fr