Coupled Faxen relations for non-uniform slip Janus spheres

Abstract

A non-uniform surface slip can cause a symmetry breaking in the geometry of an otherwise homogeneous spherical particle to give rise to an anisotropic hydrodynamic resistance to the particle. Here, we develop a more general theoretical framework capable of decoding the surface-pattern-dependent hydrodynamic features for single heterogeneous spheres having arbitrary non-uniform slip length distributions in small variations, especially for those of weakly stick–slip or slip–slip Janus spheres in either the two-faced or striped type. Utilizing the Lorentz reciprocal theorem in conjunction with surface spherical harmonic expansion, we derive a new coupled set of Faxen formulas for the hydrodynamic force and torque on a non-uniform slip sphere by expressing impacts of slip anisotropy in terms of surface dipole and quadrupole without solving detailed flow fields. Our results reveal not only how various additional forces/torques arise from surface dipole and quadrupole, but also that it is the anti-symmetric dipole responsible for distinctive force-rotation/torque-translation coupling. These features are very distinct from those of no-slip or uniform-slip particles, possibly spurring new means to characterize or sort Janus particles in microfluidic experiments. In addition, the coupled Faxen relations with surface moment contributions reported here may infer potential changes in the collective nature of hydrodynamic interactions between non-uniform slip spheres. Furthermore, the present framework can also be readily applied to heterogeneous self-propelled squirmers whose swimming velocities are sensitive to slip anisotropy.

Anisotropic stresslet and rheology of stick–slip Janus spheres

“…To use (2.25) for computing the stresslet of a stick–slip sphere, it is more convenient to expand the spatially varying part of the slip length in terms of surface harmonics (Premlata & Wei 2021). As to concerning how many terms are required to faithfully describe impacts of slip anisotropy on the stresslet, we need an additional rule, which is guided by general considerations, for constructing an anisotropic stresslet.…”

“…However, it is not clear how both the stresslet and torque are determined by the stick–slip pattern of the sphere. In particular, if the surface of the sphere is not equally partitioned by stick and slip parts, it will actually comprise both surface dipole and quadrupole contributions (Premlata & Wei 2021). We recently showed that these surface moments can have rather distinct impacts on the force and torque on a non-uniform slip sphere (Premlata & Wei 2021).…”

“…In particular, if the surface of the sphere is not equally partitioned by stick and slip parts, it will actually comprise both surface dipole and quadrupole contributions (Premlata & Wei 2021). We recently showed that these surface moments can have rather distinct impacts on the force and torque on a non-uniform slip sphere (Premlata & Wei 2021). This raises a question: How do distinct surface moments determine the anisotropic nature of the stresslet on a non-uniform slip sphere?…”

“…However, such an analogy may depend on the surface pattern of a Janus sphere. If the sphere is precisely half-faced, the surface pattern is antisymmetric, represented by a surface dipole (Premlata & Wei 2021). While an analogy to a doublet can be made by letting the two touched spheres be of different sizes, no correspondence to a two-faced Janus sphere can be found if the doublet becomes perfectly symmetric by letting the two spheres be of equal size.…”

“…While an analogy to a doublet can be made by letting the two touched spheres be of different sizes, no correspondence to a two-faced Janus sphere can be found if the doublet becomes perfectly symmetric by letting the two spheres be of equal size. In fact, a symmetric doublet or spheroid may be a better analogous model for a stick–slip–stick/slip–stick–slip Janus of striped type whose surface pattern is symmetric represented by a surface quadrupole (Premlata & Wei 2021). Because the two Janus sphere types mentioned above represent distinct symmetries in geometry, the corresponding stresslets should behave differently.…”