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 ΘR – cos ΘA), where γ is the interfacial tension, R is the radius of the particle, L is the diameter of the contact line, k = 24/π3 is a geometrical constant, and ΘR and ΘA 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.
Highlights
Young’s law states that the contact angle between a liquid−air interface and an ideal solid is given by[1] cos ΘY = γSA − γ γSL (1)where ΘY is Young’s contact angle, and γSA, γSL, and γ are the solid−air, solid−liquid, and liquid−air interfacial tensions, respectively
To gain further insights into the implications of the capillary torque, we consider two special cases: (1) when the particle rotates about its static equilibrium position and (2) when the particle is surrounded by a small liquid meniscus on a flat surface
Recent studies have argued that rotation is a relevant factor that needs to be considered when removing particles from surfaces by liquid−air interfaces.[24,25]
Summary
Young’s law states that the contact angle between a liquid−air interface and an ideal solid is given by[1] cos ΘY. To rotate a particle relative to a liquid−air interface, the contact angle on the side that rolls out of the liquid must be equal to ΘR, whereas the contact angle on the side that rolls into the liquid must be equal to ΘA [Figure 1b]. Our aim is to calculate the torque required to rotate the particle about the x-axis that goes through its center (Figure 2).[18] When the particle rotates counterclockwise, the liquid−air interface recedes (advances) on the right (left) side of the axis of rotation. We expect it to be analogous to contact angle variation around a drop moving on a flat surface In both cases, there is relative motion between a solid and a liquid, with a receding contact angle on one side and an advancing contact angle on the opposite side. As the contact angle hysteresis increases, the capillary torque increases since a higher ΔΘ causes a more asymmetric interface
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call