Curvature capillary migration of microspheres

Abstract

We address the question: How does capillarity propel microspheres along curvature gradients? For a particle on a fluid interface, there are two conditions that can apply at the three phase contact line: Either the contact line adopts an equilibrium contact angle, or it can be pinned by kinetic trapping, e.g. at chemical heterogeneities, asperities or other pinning sites on the particle surface. We formulate the curvature capillary energy for both scenarios for particles smaller than the capillary length and far from any pinning boundaries. The scale and range of the distortion made by the particle are set by the particle radius; we use singular perturbation methods to find the distortions and to rigorously evaluate the associated capillary energies. For particles with equilibrium contact angles, contrary to the literature, we find that the capillary energy is negligible, with the first contribution bounded to fourth order in the product of the particle radius and the deviatoric curvature. For pinned contact lines, we find curvature capillary energies that are finite, with a functional form investigated previously by us for disks and microcylinders on curved interfaces. In experiments, we show microsphere migrate along deterministic trajectories toward regions of maximum deviatoric curvature with curvature capillary energies ranging from 6 × 10 3 − 5 × 10 4 kBT . These data agree with the curvature capillary energy for the case of pinned contact lines. The underlying physics of this migration is a coupling of the interface deviatoric curvature with the quadrupolar mode of nanometric disturbances in the interface owing to the particle's contact line undulations. This work is an example of the major implications of nanometric roughness and contact line pinning for colloidal dynamics.

Curvature capillary repulsion

“…As a result, curvature can act as an "external field" to dictate particle behavior. In prior work, we have reported capillary curvature attraction in the limit of small particle radius to interface radius of curvature [9][10][11][12]. Here we report more complex behavior that emerges when this limit is relaxed.…”

“…We then motivate our discussion of curvature capillary repulsion. The phenomenon of capillary curvature attraction is now well established, having been reported for microcylinders [9], microdisks [10], and microspheres [11]. These studies were performed for particles on a host interface created by pinning a water interface around a micropost of height H m , radius R m , with interface slope at the micropost of − tan Ψ, where Ψ is roughly 15 • .…”

Reply to the Comments on “Curvature capillary migration of microspheres” by P. Galatola and A. Würger

“…We have studied microparticle migration on curved interfaces in experiment and derived expressions for the capillary energy E for particles of radius a for pinned contact lines [1] (E PIN ) and equilibrium contact lines (E EQ ) [2]. Our expression for E EQ differs from that derived by Würger 3.…”

“…

We have studied microparticle migration on curved fluid interfaces in experiment and derived an expression for the associated capillary energy E for two cases, i.e., pinned contact lines [1] and equilibrium contact lines [2], which differ from expressions derived by others in the literature. In this problem, a particle of radius a makes a disturbance in a large domain characterized by principal radii of curvature R1 and R2.

…”

Pinned down

“…But there is strong impetus to find out: as recently shown, nanoscale heterogeneities can cause a particle to migrate on a curved surface 10 . So if pinning could be understood, it could be harnessed to create new dynamics and new assembly pathways.…”