Capillary Torque on a Particle Rotating at an Interface

Abstract

Small particles attach to liquid−fluid interfaces due to capillary forces. The influence of rotation on the capillary force is largely unexplored, despite being relevant whenever particles roll at a liquid− fluid interface or on a moist solid. Here, we demonstrate that due to contact angle hysteresis, a particle needs to overcome a resistive capillary torque to rotate at an interface. We derive a general model for the capillary torque on a spherical particle. The capillary torque is given by M = γRLk(cos Θ R − cos Θ A ), where γ is the interfacial tension, R is the radius of the particle, L is the diameter of the contact line, k = 24/π 3 is a geometrical constant, and Θ R and Θ A are the receding and advancing contact angles, respectively. The expression for the capillary torque (normalized by the radius of the particle) is equivalent to the expression for the friction force that a drop experiences when moving on a flat surface. Our theory predicts that capillary torque reduces the mobility of wet granular matter and prevents small (nano/micro) particles from rotating when they are in Brownian motion at an interface. Article pubs.acs.org/Langmuir

The Force Required to Detach a Rotating Particle from a Liquid–Fluid Interface

“…Because rotation causes a decrease in the detachment force, it may, at first, seem that inducing the rotation of particles could be a useful way to facilitate the detachment of particles from interfaces. However, this may not be economical from an energetic perspective because in order to rotate a particle against an interface, energy needs to be supplied to overcome resistive capillary torque (described in ref 19). Capillary torque is due to the fact that the liquid−fluid interface is not axisymmetric about the center of a rotating particle.…”

“…23−28 On one side of the rotational axis, the particle rolls out of the liquid, and therefore, the contact angle corresponds to the receding contact angle, whereas on the opposite side the contact angle corresponds to the advancing contact angle, Θ A . 19 In general, Θ A differs from Θ R due to chemical and topographical inhomogeneities on the particle and/or adaptation of the particle to the liquid. 29,30 Consequently, eq 1 does not hold for a rotating particle and should be modified.…”

“…For example, it has been shown that the contact angle around spherical and prismatic particles varies along their three-phase contact line − and that the contact angle around a Janus particle varies around the contact line when the particle is oriented such that the boundary between the lyophobic and lyophilic hemispheres lies at a finite angle to an interface . Another example where the contact angle does not take a single value is when a particle rotates against a liquid–fluid interface, as shown in Figure a . There are several instances when rotation against an interface may become relevant: when a particle rolls down a wet or lubricated substrate, , when a water drop removes contaminant particles from a substrate, when strong winds blow on soil or dust particles at the surface of a lake or river, and when a particle with an electric or magnetic dipole moment is placed at an interface in an electric or magnetic field, respectively. − On one side of the rotational axis, the particle rolls out of the liquid, and therefore, the contact angle corresponds to the receding contact angle, whereas on the opposite side the contact angle corresponds to the advancing contact angle, Θ A .…”

The behavior of capillary suspensions at diverse length scales: From single capillary bridges to bulk

“…Contact dynamic effects, such as contact angle hysteresis or contact line pinning, are certainly important when capillary bonds are stretched or compressed but can also arise under rotation, as shown in Figure 5 [60]. Naga et al recently showed that a particle situated at an interface between two phases can experience similar effects as a liquid droplet sliding across a surface or a sphere rotating above a surface.…”

The behavior of capillary suspensions at diverse length scales: from single capillary bridges to bulk

“…Contact dynamical effects, such as contact angle hysteresis or contact line pinning, are certainly important when capillary bonds are stretched or compressed, but can also arise under rotation, as shown in Figure 5. [57] Naga et al recently showed that a particle situated at an interface between two phases can experience similar effects as a liquid droplet sliding across a surface or a sphere rotating above a surface. The resulting force or torque forms a barrier similar to an activation energy that needs to be overcome in order to reach a state of unhindered sliding or rotation.…”