Modeling capillary forces for large displacements

Abstract

Originally applied to the accurate, passive positioning of submillimetric devices, recent works proved capillary self-alignment as effective also for larger components and relatively large initial offsets. In this paper we describe an analytic quasi-static model of 1D capillary restoring forces that generalizes existing geometrical models and extends the validity to large displacements from equilibrium. The piece-wise nature of the model accounts for contact line unpinning singularities ensuing from large perturbations of the liquid meniscus and dewetting of the bounding surfaces. The superior accuracy of the generalized model across the extended displacement range, and particularly beyond the elastic regime as compared to purely elastic models, is supported by finite element simulations and recent experimental evidence. Limits of the model are discussed in relation to the aspect ratio of the meniscus, contact angle hysteresis, tilting and self-alignment dynamics.

In-Plane Mode Dynamics of Capillary Self-Alignment

“…The second number α is a reduced viscosity given by 4ντ/h 2 , where ν is the kinematic viscosity, h is the gap, and τ is the characteristic time. τ is determined by (k/m) 1/2 , where k is the gradient of the capillary force along the shift DOF (i.e., the lateral stiffness of the meniscus in the full elastic regime 45 ). For the case illustrated in Figure 2, m̃= 2.88 and α = 0.32.…”

“…21 The non-constancy of h(x,y,θ) was implemented for physical consistency with (1) time scale separation, because of the dominance of vertical over lateral capillary forces for this system, 3 (2) the tendency of the liquid bridge to assume the geometry locally closest to a section of a sphere, 21,23 and (3) conservation of the meniscus volume, which upon contact line unpinning(s) induces an increase of h to partially compensate for the decrease of wet pad surface(s). 45 After setting h, the restoring capillary force and torque were computed by the method of virtual works 47 implemented using central finite differences. Figure 3.…”

“…1,24,26 The latter develops a geometric quasi-static approach which, while consistently supported by experimental data, 25 captures only partially the intrinsically dynamic nature of the process. 45 The liquid bridge can in fact be deformed along six degrees of freedom (DOFs), which are associated with the three translational and three rotational DOFs of the top floating component (Figure 1). Each type of deformation (or mode) is opposed by a corresponding restoring capillary force 25 (for translational modes, i.e., shifts and lift) or torque 27 (for rotational modes, i.e., twist and tilts) tending, with the possible exception of the tilt modes, 1 to bring the system back to its configuration of global equilibrium.…”

A Combined Experimental and Numerical Modeling Study of the Deformation and Rupture of Axisymmetric Liquid Bridges under Coaxial Stretching

“…We note, however, that contact line motions on more general solid substrates can significantly affect the statics and dynamics of liquid bridge 37 , including the bridge deformation and rupture patterns 19,38 . Further studies on this rich topic, including accurate modeling of the mechanics of contact line pinning and unpinning 39 would be worthy extension of our work.…”

Self-assembly of micro/nanosystems across scales and interfaces

“…Still, capillary self-alignment works also for relatively large component offsets from receptors [31]. An analytical model including partial wettability of the receptor surface can account for it [32]. Notably, with my colleagues I evidenced a dependency of the lateral selfalignment dynamics on the surface energy of components [33], and the coupled dynamics of in-plane translational and rotational modes under specific offset conditions [34].…”