History hydrodynamic torque transitions in oscillatory spinning of stick-slip Janus particles

We theoretically investigate the oscillatory spinning of an axisymmetric stick-slip Janus particle (SSJP) under the creeping flow condition. Solving the unsteady Stokes equation together with a matched asymptotic boundary layer theory, we find that such a particle can display unusual viscous torque responses in the high frequency regime depending on the Stokes boundary layer thickness δ, the slip length λ of the slip face, and the coverage of the stick face. Our analysis reveals that an SSJP will always experience a reduced Basset torque of 1/δ decay due to the presence of the slip face, with amplitude smaller than the no-slip counterpart irrespective of the value of λ. If the coverage of the stick face is sufficiently small, the reduced Basset torque can turn into a constant torque plateau due to prevailing slip effects at larger values of δ, representing a new history torque transition prior to the slip-stick transition at δ ∼ λ. All these features are markedly different from those for no-slip and uniform slip particles, providing not only distinctive fingerprints for Janus particles but also a new means for manipulating these particles.We theoretically investigate the oscillatory spinning of an axisymmetric stick-slip Janus particle (SSJP) under the creeping flow condition. Solving the unsteady Stokes equation together with a matched asymptotic boundary layer theory, we find that such a particle can display unusual viscous torque responses in the high frequency regime depending on the Stokes boundary layer thickness δ, the slip length λ of the slip face, and the coverage of the stick face. Our analysis reveals that an SSJP will always experience a reduced Basset torque of 1/δ decay due to the presence of the slip face, with amplitude smaller than the no-slip counterpart irrespective of the value of λ. If the coverage of the stick face is sufficiently small, the reduced Basset torque can turn into a constant torque plateau due to prevailing slip effects at larger values of δ, representing a new history torque transition prior to the slip-stick transition at δ ∼ λ. All these features are markedly different from those for no-slip and unifo...


I. INTRODUCTION
A Janus particle is a compartmentalized colloid of two faces having distinct properties. It is commonly made of hydrophilic and hydrophobic caps, termed stick-slip Janus particle (SSJP). Because of its surface polarity, such a particle can work as an active cargo for expediting transport or be used to aid in colloidal assembly. 1 In addition, because the mobility of an SSJP can be further adjusted by the stick-slip partition, this may offer a tunable means to realize more precise hydrodynamic manipulations. While there are a few studies on the hydrodynamics of SSJP, they are mainly focused on the steady situation. [2][3][4][5][6][7][8] The present work will be extended to the time-dependent scenario.
The present work is motivated by magnetically driven microrotors for generating rotational flows 9 or by the use of rotating magnetic beads for biosensing applications. 10 In the former, an SSJP could be a more efficient microrotor by having its slip portion faced down to the bottom wall to reduce drag. For the latter, a bead can be patched with a slip cap after absorbing hydrophobic proteins. This might affect its spinning behavior and hence the subsequent molecular transport on its surface. In either case, it is necessary to know how much torque is needed to rotate an SSJP or how fast an SSJP can spin when it is subject to torsion. In the unsteady situation, since the applied torsion is not balanced by the viscous torque, the resulting rotational velocity may reveal more information about the particle. For rotary oscillations, since there exists a phase difference between the rotational velocity and the torque, the behavior of such phase difference for an SSJP is expected to vary with the stick-slip partition. This might provide a distinct fingerprint for an SSJP to differentiate from no-slip and uniform slip particles hydrodynamically.
As both no-slip and slip effects will join to influence the hydrodynamic responses of an SSJP, it is instructive to review basic features for no-slip and slip spheres under rotary oscillations. For a no-slip sphere, it is well known that it can experience a Basset history force 11 varying as a/δ due to a thin boundary layer of thickness δ = (2ν/ω) 1/2 , especially at the oscillation frequency ω much higher than the viscous damping frequency ν/a 2 , where a is the sphere radius and ν is the kinematic viscosity. More precisely, this a/δ Basset force arises from a much larger shear stress μΩ 0 a/δ across the boundary layer with the peak angular velocity Ω 0 (with μ being the fluid viscosity). The resulting torque thus also varies as a/δ in the high ω regime. 12

ARTICLE scitation.org/journal/adv
For a slip sphere, on the contrary, the boundary layer can become much thinner than the slip length λ at ω > ν/λ 2 to make slip effects much stronger. This leads to a constant shear stress μΩ 0 a/λ on the sphere surface. 13 As a result, the torque also becomes constant and persists until the slip-stick transition (SST) point at δ ∼ λ when ω ∼ ν/λ 2 below which the usual Basset decay reappears. 14 Such plateau and slip-stick transition are not noticed in the previous investigation on the unsteady rotation of a slippery sphere. 15 As such, purely no-slip and slip particles have completely different characteristics in their torque responses. This raises a question, what if a particle is comprised of both no-slip and slip faces like an SSJP? In our recent study on an oscillatory translating SSJP, 16 we found that the force response can be mixed with both no-slip and uniform slip contributions. While a similar torque response might be expected to occur to an oscillatory spinning SSJP, the two problems have quite distinct physics, reflected by the following aspects. First, for the translation problem the total viscous force on the particle is nonzero, whereas the spinning problem is force free. It follows that the flow field in the latter will decay at a much faster rate than that in the former. Second, for the translation problem, the added mass of O((a/δ) 2 ), which is purely of the potential flow origin, always exists in the force response. 12 However, there is no counterpart in the torque for the spinning problem since there is no net fluid entrainment by a force couple with a constant pressure everywhere.
Because of the above distinctions, the flow characteristics of the spinning problem are generally quite different from those of the translation problem. Such differences, in particular, manifest when there is a flow past a rotating sphere where a secondary flow emerges 17 or in the nonvanishing Reynolds number situation where steady streaming often occurs. 18 Prior to extending the uniform sphere situation to SSJP, it is necessary to analyze the leading order flow characteristics of SSJP under the Stokes flow condition. This is another reason why we would like to pursue this yet-explored spinning SSJP problem in this work.

II. PROBLEM FORMULATION
Motivated by the above, we consider the oscillatory spinning motion of a spherical SSJP of radius a at angular velocity Ω(t) = Ω 0 e −iωt in an incompressible viscous fluid of density ρ and viscosity μ, where Ω 0 is the peak angular velocity and ω is the oscillation frequency. As illustrated in Fig. 1, the particle is partially covered with a slip surface of polar angle θ 0 , with the remaining stick portion satisfying the no-slip boundary condition. Assume that the spinning is around the axis of the symmetry. Described by the spherical coordinates (r, θ, ϕ) with the origin at the center of the particle, the fluid velocity only occurs in the azimuthal ϕ direction. Having length, time, and velocity scaled by a, ω −1 , and Ω 0 a, respectively, the azimuthal fluid velocity w ′ is governed by the unsteady Stokes equation in the dimensionless form, with η = cos θ. Note that for this axisymmetric spinning, because the total viscous force on the particle is zero, the pressure is constant everywhere. Hence, there is no pressure term in (1).

FIG. 1.
Geometry of a spherical stick-slip Janus particle and coordinate system.

III. MIXED STICK AND SLIP TORQUE RESPONSES
First of all, it can be verified analytically that for constant β, (9) is reduced to the uniform slip result, 14 Here,δ = (2ν/ωa 2 ) 1/2 measures the extent of the boundary layer relative to the particle radius a. As revealed by (10), when the particle is no-slip withλ = 0, the torque varies as 1/δ asδ → 0 due to the Basset shearing with phase π/4 ahead of the particle rotational movement.
In the case of uniform slip, however, the torque amplitude becomes a constant plateau of value 1/3λ in theδ → 0 limit due to the constant shearing resulted from strong slip effects. 14 Figure 2(a) plots the torque amplitude againstδ for a half cap SSJP (θ 0 = 90 ○ ). First of all, purely no-slip and uniform slip cases give torque amplitudes of 1/δ and 1/3λ, respectively, as given by (10). As for a half cap SSJP, we find that all the curves with different values ofλ tend to approach the same Basset-like 1/δ decay asδ → 0, but in a reduced amplitude compared to the no-slip case due to drag reduction imparted by the slip face. Since such a Basset torque disappears when no-slip changes to uniform slip but reappears for an SSJP, it can be thought of as a reentry Basset torque to distinct from the usual no-slip Basset torque. In terms of the phase χ = tan −1 [Im(Tz(ω))/Re(Tz(ω))], Fig. 2(b) shows that it basically varies from the no-slip result χ = −π/4 to the uniform slip result χ = 0 though changes can be nonmonotonic. Figure 3(a) plots how the torque amplitude varies as gradually decreasing the stick portion by increasing θ 0 . When increasing θ 0 to 150 ○ where the stick portion becomes small, we observe a reentrant history torque transition (RHTT) in which the torque first follows a Basset-like 1/δ decay in the smallδ regime and then turns into a slip plateau at larger values ofδ prior to the SST pointδ ∼λ. How the phase χ varies withδ in this case appears even more nonmonotonic, as shown in Fig. 3(b).

IV. MATCHED ASYMPTOTIC BOUNDARY LAYER THEORY
To explain the observed re-entry Basset torque and RHTT, we further develop a matched asymptotic theory to resolve how the flow behaves within a thin boundary layer when ωa 2 /ν is large. Let ε ≡δ/ √ 2 = (ωa 2 /ν) −1/2 be the small parameter. We stretch the radial coordinate with r = 1 + εy and expand the azimuthal velocity as Substituting (11) into (2) and (3), we obtain the leading order governing equation and boundary condition as In (13), because the effective slip coefficient β/ε can either vanish for the stick face or be large for the slip face, it is necessary to retain the driving slip velocity sin θ on the left hand side so that a transition from slip to no slip can be captured when ε is varied. The solution to (12) satisfying (13) is where k = e −iπ/4 and g(θ) = (1 + kβ(θ)/ε) −1 . The shear stress is thus Using (8) and (15), the leading order torque amplitude can be readily determined as where g 0 = (1/2)∫ 1 −1 g(η)P 0 (η)dη and g 2 = (5/2)∫ 1 −1 g(η)P 2 (η)dη are the monopole and quadrupole contributions, respectively, and the functions Pn represent the Legendre functions. Writing β(η) =λH (η − η 0 ) in terms of the Heaviside step function H, (16) can be evaluated as where C = (2 − η 0 )/4. In the ε → 0 limit, (17) is reduced to This is exactly the reduced Basset torque shown in Figs. 2 and 3. The amplitude of this torque is found to depend only on the stick-slip partition. More importantly, the torque appears more sensitive to the coverage of the stick face, η 0 + 1. In the purely no-slip case, i.e., η 0 = 1, (18) is reduced to the usual Basset torque value e −iπ/4 /3ε. However, if the particle is completely slippery, i.e., η 0 = −1, then (17) yields a constant torque plateau 1/3λ for |k|λ/ε ≫ 1 or for ε below the SST point, Hence, if this slip particle is covered with a tiny stick patch (i.e., η 0 is close to −1), the slip torque plateau 1/3λ will start to rise toward the reduced Basset torque (18) when ε is decreased to the RHTT point, (20) Similar to (18), ε RHTT is also sensitive to the converge of the stick face. When the stick face is small, ε RHTT will be much smaller than ε SST . This will in turn make the torque exhibit a reduced Basset torque followed by a slip torque plateau, which explains the θ 0 = 150 ○ curve shown in Fig. 3(a). However, if the stick face is not small, ε RHTT becomes comparable to ε SST . The torque in the small ε regime will be dominated by the reduced Basset torque (18) without seeing a slip plateau, which explains Fig. 2(a).
For an arbitrary time-dependent spinning motion, we can express the angular velocity Ω(t) as a Fourier integral followed by its conversion to Laplace transform with −iω → s. Further with the aid of the convolution theorem, we transform (17) into with the memory kernel Here, tν = a 2 /ν is the viscous diffusion time.λ 2 tν = λ 2 /ν ≡ t SST is the SST time corresponding to (19) when the boundary layer thickness δ ∼ (νt) 1/2 grows to the size of the slip length λ. t SST is typically shorter than tν since λ < a.

V. CONCLUDING REMARKS
We have demonstrated that the viscous torque responses of an oscillatory spinning SSJP are in fact of neither no-slip nor slip type but mixed with both. Because part of the particle surface is noslip, the response in the high frequency regime is always dominated by the reduced Basset torque that varies inversely with the Stokes boundary layer thickness δ. However, if the stick face becomes sufficiently small, the reduced Basset torque can turn into a plateau due to strong slip effects at larger values of δ prior to changing the usual no-slip Basset torque that prevails at δ greater than the slip-stick transition point δ ∼ λ. This reduced Basset to slip plateau transition and the slip-stick transition seems to be generic features for SSJPs. These features may provide more robust means for better characterizing or manipulating these heterogeneous particles.