Hydrodynamic attraction and repulsion between asymmetric rotors

At low Reynolds numbers, the hydrodynamic interaction between dumbbells driven by an external rotating field can be attractive or repulsive. Dumbbells of dissimilar asymmetric shape or different coupling to the external field undergo conformational rearrangements that break the time reversal symmetry. The parameter ranges leading to attraction or repulsion are explored numerically. The results of our simulations suggest that rotating fields may be a useful avenue for the assembly, disassembly, and sorting of particles of different shape as well as for the study of collective micro-swimmers.


Introduction
Hydrodynamic interactions in suspensions of active or externally driven nanoparticles or microorganisms are involved in collective, biological, chemical and technical motion [1,2,3,4,5,6,7,8,9] of fascinating complexity. The propulsion of microparticles and microorganisms differs substantially from the motion of macroscopic objects like fishes or birds due to the small scale motion at low Reynolds numbers [2]. As a consequence swimming macro-and microorganisms transform chemical into translational or rotational energy in different ways. Considerable progress has been made in understanding the propulsion of microorganisms by studying artificial active swimmers [9,10,11] or active rotors [12]. Self-organization phenomena of assemblies of micro-swimmers can teach us how hydrodynamic interactions synchronize the motion of swimmers or cilia The importance of hydrodynamic interactions (HI) at small Reynolds numbers has been emphasized by showing that they may cause a lift force during the motion of vesicles close to a wall [13,14,15]. They also lead to cross-streamline migration of droplets, vesicles and deformable bead-spring models in Poiseuille flow [16,17,18]. Additionally, the interplay between Brownian motion of small particles and hydrodynamic interactions can cause surprising anti-correlations and cross-correlations [19,20]. Brownian dynamics of hydrodynamically interacting polymers may cause elastic turbulence that is used for mixing in microfluidic devices [21,22].
Macroscopic rotors experience the attractive Magnus effect that can easily be understood using the concept of dynamic pressure. The attraction or repulsion between rotors in the limit of Stokes flow is less obvious. Magnetically driven rotors can attract each other due to their magnetic interaction [6,23,24]. They form dynamically induced self-organized structures depending on the particles' contours [7]. Magnetic dipolehexapol interactions may be used for separating rotors with respect to their shape [25]. Rotors also play an important role for bacterial motion such as the propulsion of Escherichia coli [3] with its rotating flagella. The dynamics of single flagella has recently been modeled in numerical calculations [26]. There hydrodynamic interactions are the source of the propulsion the flagella create. One fundamental problem for active and driven low Reynolds number motion is the understanding of the consequences of hydrodynamic interactions on the collective motion of an assembly of active components.
The aim of this work is to explore the effect of hydrodynamic interactions on two rotating dumbbells. Symmetry considerations forbid a net conformational change over one rotational period for similarly shaped dumbbells with similar couplings to the external field. One major question explored here is the influence of differing shapes on the hydrodynamic interactions between two particles. Therefore we investigate two interacting dumbbells of dissimilar shape and coupling to an external rotating field.
We demonstrate by numerical calculations that for dissimilar rotating dumbbells the time reversal symmetry is broken. As a consequence both dumbbells can attract or repel each other. Transitions between dumbbell attraction and repulsion are induced by changing the torques or the dumbbell shapes. Phase diagrams separating regimes of attraction from regions of repulsion between the dumbbells are constructed for generic sets of parameters.
We suggest to test our theoretical exploration with experiments on anisotropic birefringent particles. Such particles may be rotated by circularly polarized light [27,28]. Another possibility to apply a torque is to place paramagnetic or ferromagnetic particles in a rotating magnetic field. Like in nuclear magnetic resonance, magic angle spinning may be used to eliminate time averaged dipole-dipole interactions that might overshadow the hydrodynamic attraction or repulsion.

Model
We investigate the motion of a single dumbbell and of two hydrodynamically interacting dumbbells in a fluid of viscosity η in the low Reynolds number limit. Due to a torque created by a rotating external field f the dummbbells perform a planar rotational motion. Each dumbbell is composed of two beads which are connected by a bar of constant length as indicated in figure 1.
The beads with the effective hydrodynamic radii a i (i = 1, 2, 3, 4) at the positions r i (i = 1, 2, 3, 4) are described as point particles having drag coefficients ζ i = 6πηa i . The vectors r 21 = r 2 − r 1 and r 43 = r 4 − r 3 describe the two dumbbell axes, which we assume to be of constant length b = |r 21 | = |r 43 |. The hydrodynamic centers of the dumbbells, c 21 and c 43 , are given by c 21 = r 1 + a 2 a 1 + a 2 r 21 and c 43 = r 3 + a 4 a 3 + a 4 r 43 .
(1) Figure 1. Sketch of two dumbbells with unequal asymmetries, a 1 = a 3 and a 2 = a 4 . The dumbbells are rotated by a driving field f (t), cf. Eq. (5). ∆φ is the angle between the axesr 21 andr 43 of the dumbbells. The crosses indicate the hydrodynamic centers c 21 and c 43 of the two dumbbells. The distance between them is denoted by c h .
The equation of the over-damped motion is given bẏ with the Rotne-Prager mobility matrices for unequal spheres [29,30] the connection unit vectorsr ij = r ij /|r ij | = r ij /r ij and the external forces F j (j = 1, 2, 3, 4) acting on the beads. In our model the rotation of the dumbbells is caused by the potential with the vector rotating at a frequency ω in the xy plane. The forces (2) can be expressed in terms of the rotating field f as The potential in Eq. (4) yields the following asymmetric relations between the forces According to this property together with Eq. (6) the dumbbells are force free. Additionally the forces acting on the individual beads are perpendicular to the dumbbell axes, which results in a rotation of the dumbbells. The form of the potential V for the driving field is similar to that of magnetic dipoles in an external magnetic field where the interaction between the dipoles is neglected.
But other means to propel the dumbbells are possible. For example, non-spherical birefringent particles can be rotated by circularly polarized light. The magnitude of the torques imposed on the dumbbells can easily be tuned by the applied power of the light [27,28].

Dumbbell dynamics
The balance between the driving and the viscous torques leads to phase angles φ 21 and φ 43 between the dumbbell orientations r ij and the driving field f (t) with cos(φ ij ) = r ij ·f /|r ij ·f |. The dynamics of the dumbbells is analyzed in this section. The conditions under which both dumbbells attract or repel are discussed.

Single dumbbell
The two beads of a single dumbbell (V 43 = 0 and V 21 = V 0 ) move on circular trajectories around their hydrodynamic center c 21 . The radii R 1,2 of those trajectories depend on the effective hydrodynamic radii a 1,2 of the two beads and the length b of the dumbbell axis. In the Oseen approximation they are given by For a sufficiently strong external field the dumbbell axis r 21 rotates synchronously with the driving field f (t) at the same frequency Ω = ω after a transient period. The stationary phase angle φ 21 is given by the expression

Two dumbbells
In the case of two rotated dumbbells each one experiences the perturbed liquid flow created by the respective other one. Due to this interaction the hydrodynamic centers c ij are set in motion. In the mean this results in a circular or a spiral like motion as indicated in figure 2. In the following considerations we fix the effective radii of one bead of each dumbbell at the value a 2 = a 4 = 1 and the length of the axes of both dumbbells at b = 3 for simplicity. The (asymmetric) shapes of the dumbbells and the couplings to the external field can be adjusted independently by varying the bead radii a 1 and a 3 as well as the coupling parameters V 21 and V 43 . The viscosity η, the angular frequency ω, the coupling V ij and the length b form a dimensionless group ηωb 3 /V ij . It is therefore sufficient to vary the coupling and keep the viscosity η = 10 and the angular frequency ω = 5 × 10 −4 fixed. The complete conformation of the system (up to a rotation of the whole system) can be described by the distance D(t) between the hydrodynamic centers of the dumbbells, as well as the conformation angles ψ 21 and ψ 43 , which are given by The hydrodynamic interactions between the dumbbells induce a slow rotation of c h and oscillations of D(t) around some mean distance as indicated by figure 2. The frequency Ω ′ of these oscillations is the rotation frequency of the dumbbell axes r 21 and r 43 with respect to c h . Therefore the frequency of the slow rotation of c h is equal to the difference Ω − Ω ′ . 3.2.1. Two dumbbells having identical shapes and couplings. The dumbbell dynamics is considerably simplified for equally asymmetric shapes of the two dumbbells, a 1 = a 3 , and equal couplings, V 21 = V 43 . If the two dumbbells are also equally oriented, as sketched in figure 2(a), the phase angles are equal, φ 21 = φ 43 = φ, and the conformation angles are identical for all times, ψ 21 = ψ 43 = ψ. For two oppositely oriented dumbbells the phase difference has the constant value ∆φ = π whereas the conformation angles add up to ψ 21 + ψ 43 = π.
According to the symmetry of the system, the distance D(t) between the hydrodynamic centers of two equally oriented dumbbells oscillates around a mean distance with the frequency 2Ω ′ as indicated by the sketch in figure 2(a). The corresponding oscillation period is π/Ω ′ , which is shown for a specific parameter set in figure 3(a). In contrast to this, the distance D(t) for two dumbbells with opposite shape orientation oscillates with a period of τ Ω ′ = 2π/Ω ′ being twice as long as in the former case. For comparison we show the x component of the driving field f x (t) in figure 3(c), which illustrates the inequality of the frequencies, Ω ′ < Ω, and thus the inequality of the oscillation periods τ Ω ′ > τ Ω = 2π/Ω. In figure 3(b) the oscillatory contributionψ to the phase ψ = Ω ′ t +ψ is shown. Since the coupling strengths of the dumbbells are constant, the oscillation of ψ about a linearly growing part confirms that the rotational hydrodynamic drag changes as a function of the relative orientation of the dumbbell axes with respect to c h . In the case of equally oriented dumbbells the viscous drag is maximal in the ranges about the conformation angles ψ = (2n−1)π/4 (n = 1, 2, 3, 4) where fluid between the dumbbells is squeezed out or sucked into the region between the dumbbells. These are also the phases of the rotation during which the repulsion (resp. attraction) between the dumbbells is strongest.
According to the symmetry of the system the attractive forces acting between the two dumbbells in one quarter of a period (0 < t < τ Ω ′ /4) are compensated during the consecutive quarter (τ Ω ′ /4 < t < τ Ω ′ /2) by repulsive forces of the same magnitude. This causes an oscillation of D(t) with the frequency 2Ω ′ around some mean value, but without a net attraction or repulsion. In the case of oppositely oriented dumbbells two consecutive maxima of the viscous drag have different amplitudes and therefore the attractive and repulsive hydrodynamic forces do not cancel each other completely during one half of an oscillation period but during a whole rotation with the period τ Ω ′ .
During the rotation the viscous drag acting on the inner beads, which are closer to the respective other dumbbell, is higher than the drag on the outer beads of the dumbbells. These differing drag forces induce an oscillation of the phase angles φ ij during the rotation.
In the ranges of increasing (decreasing) viscous drag the rotation frequency Ω ′ of the dumbbell axes decreases (increases). When Ω ′ is larger (smaller) than ω the phase delays φ ij of the dumbbells decrease (increase).
The oscillations of D(t) and φ(t) are shown as a function of time for two different values of the coupling constants in figure 4. In Part (a) the coupling to the driving field is weak, V 21 = V 43 = 0.7, and in part (b) the coupling is strong, V 21 = V 43 = 100. The amplitudes of the oscillations of the distance are the same in both cases. Is contrast to that the phase lag φ of the dumbbell axes with respect to the orientation of the external field is smaller in the case of large values of the coupling. Moreover the diagrams in figure 4 show that the oscillations of φ are in phase with the oscillations of the distance in the case of strong couplings to the external field. Contrary to that φ(t) is delayed with respect to D(t) for weak couplings.
The reason for this delay is the following: While the distance between the dumbbells increases the drag on the dumbbells also grows. So the rotation frequency Ω ′ of the dumbbells must decrease. As soon as it is below ω the phase angle φ grows and so does the torque on the dumbbells (∼ sin φ). But for weak couplings the torque builds up much slower than for strong couplings. So for small coupling constants even in the beginning of the domain where the distance (and thus the drag) decreases φ is still growing because the torque is too weak. At some point the torque is large enough to overcome the drag so that φ decreases again afterwards. An analogous argument holds in the ranges where D(t) is increasing. The shift between the curves of D(t) and φ(t) has to be smaller than a quarter of an oscillation period in our considerations because otherwise the field f (t) would be too weak to enforce a synchronous rotation of the dumbbells at all.
For strong couplings there are small dips at the maxima of φ(t). These correspond to local minima of the drag, which occur when the dumbbells are aligned with each other so that the four beads lie on a single line. In this conformation the distance between the dumbbells is maximal. An equivalent phenomenon can be seen in the plot of φ(t) for weak coupling. Of course there is also a local minimum of the drag when the dumbbells are aligned with each other and the distance is maximal. But due to the weak torque acting at that stage there is no dip but only a slight flattening of the curve in this case.
In the coordinate system rotating with the connection vector c h the conformation after one period τ Ω ′ maps on top of the original conformation. The reciprocity of the motion is illustrated in figure 5, where the distance D is plotted as a function of the conformation angle ψ for weak couplings, V 12 = V 34 = 0.7, in part a) and strong couplings, V 12 = V 34 = 100, in part b). Both curves are invariant under the  transformation ψ → −ψ. This underlines the reciprocity of the motion as it can already be seen in the plot of D(t) which is invariant under time reversal t → −t. In addition to that the curves for weak and strong couplings are identical in the conformation space D − ψ as shown in figure 5. In fact the trajectories shown are universal for all values of the coupling constants as long as they are large enough so that the dumbbells can follow the field f (t) synchronously.
However, the rate of change of the conformation angleψ along the curves depends very much on the coupling parameters (cf. figure 4). This is indicated by the color code along the two curves, where red and blue mark the ratesψ below and above the angular frequency Ω ′ . This shows that irrespective of the strength of the driving field the dumbbells pass through the same set of conformations during one cycle, although each conformation is passed at a different rate, as indicated by the distribution of the different colors along both lines. The reciprocity of the motion ensures that for a given rotational frequency the distance of the dumbbells described by the Stokesian dynamics is locked to the individual orientation of the dumbbells with respect to c h and returns to the same value after one period.  In the previous section D(t) was either oscillating with the frequency 2Ω ′ or Ω ′ , but in both cases D(t) was symmetric with respect to a reflection of time, cf. figure 4. Here both, D(t) and ∆φ(t), are not symmetric anymore with respect to time reflection.
As in the case of equally asymmetric dumbbells the viscous drag is a function of the angles which the dumbbell axes enclose with the vector c h . There are also ranges of the phase angles, in which liquid is squeezed out between the dumbbells and others where fluid is sucked in. As a consequence the length of the vector c h is also oscillating here, i. e. there are phase ranges of dumbbell attraction, which are followed by phases of dumbbell repulsion and so on.
For finite values of ∆φ = 0, which are a consequence of the broken symmetry, the conformation angles ψ 21 and ψ 43 differ from each other. Combined with the hydrodynamic interactions, which depend nonlinearly on the distances between the  Figure 8. The Fourier modes of the function ∂D ∂ψ21 are shown. There is a nonvanishing zeroth order coefficient that leads to an attraction or a repulsion between the dumbbells. However, there are eight higher order coefficients that have larger values which indicate a quite complex motion.
beads, the magnitude and the directions of the forces between the beads during one rotation cycle are quite complex. Therefore the trajectories in the configurational space are not universal anymore.
For one parameter set the distance D(t) is plotted as a function of the conformation angle ψ 21 in figure 7(a). It is clear from first sight that the behavior is much more complex here than in the case of equally asymmetric dumbbells with equal couplings, which was shown in figure 5. Especially the magnitudes of the attraction and repulsion between the dumbbells during the different stages of the motion are different here. So a priori it is not clear anymore whether the dumbbells can attract or repel each other during a whole rotational period. In order to illustrate the complex behavior of D as a function of ψ 21 in more detail, figure 7(b) shows the derivative ∂D ∂ψ 21 (ψ 21 ). In the ranges of the conformation angle ψ 21 where this function is positive the dumbbells repel each other and when the function is negative they attract each other. The Fourier modes of this function are given in figure 8. In this spectrum one can clearly see that there is a non-vanishing coefficient of zeroth order which means that the distance between the dumbbells changes during a complete dumbbell rotation. But it is also obvious that this net repulsive or attractive effect is superimposed by much stronger oscillations. So it is not possible to determine in which phase of the motion the decisive effect takes place that causes the overall repulsion or attraction. In fact all coefficients from n = 1 (∼ e iψ 21 ) up to n = 8 (∼ e 8iψ 21 ) are larger than the zero order coefficient.
As shown above the mean distance between two unequally shaped dumbbells may increase, as sketched in figure 2(b), or decreases as a function of time. In figure 9 we have plotted D(t) for the same parameter sets as in figure 6, but for a longer time window. One can easily see that in the mean the dumbbells repel each other in part a) and attract each other in part b). How the hydrodynamically induced repulsive or attractive dumbbell motion depends on the parameters, describing the system's asymmetry, is illustrated in terms of diagrams for representative sets of parameters in figures 10-12. In figure 10 areas of dumbbell attraction and repulsion are shown as a function of the two coupling parameters V 21 and V 43 for the fixed parameters, a 1 = 1.2 and a 3 = 1.1, corresponding to unequal dumbbell shapes. For combinations of the two couplings along the solid lines in figure 10, the bead asymmetries of the dumbbells are compensated by the difference between V 21 and V 43 , so that the distance D(t) is constant in the mean. If one coupling parameter is much larger than the other one (V 21 ≫ V 43 or V 43 ≫ V 21 ) the dumbbells attract each other. Between those two regimes there is a range of parameters for which the dumbbells repel each other. For increasing coupling constants the repulsive area widens.
In figure 11 the regions of dumbbell attraction and repulsion are shown as functions of the two bead radii a 1 and a 3 for fixed coupling constants, V 21 = 1.2 and V 43 = 1.0. For these parameters there is only a narrow domain, in which the dumbbells attract each other. In this area the dumbbell asymmetries differ only about 20 − 30%. The phase diagram in figure 11 also shows that for identical asymmetries of the two dumbbells, In figure 12 the coupling V 43 = 1.0 and the asymmetry a 3 /a 4 = 1.1 of one of the dumbbells were fixed, whereas the asymmetry a 1 /a 2 as well as the coupling parameter V 21 is varied. For a symmetric or slightly asymmetric dumbbell, i.e. a 1 /a 2 ≈ 1, there is always attraction, but for stronger asymmetries there are also regions, where the two dumbbells repel each other. Even for equal couplings (V 21 = V 43 ) one can find areas of dumbbell attraction as well as regions in which the dumbbells repel each other.
Away from the solid lines of vanishing attraction and repulsion in figure 10, 11 or 12 the modulus of the relative velocity between the dumbbells increases with the distance from these lines. This is demonstrated in figure 13, where the mean relative velocity between the dumbbells is shown as a function of the coupling V 43 . In this figure the same parameter set as in figure 10 was used and V 21 = 1.2 was fixed. In this case the transitions from attraction to repulsion and back occur at the values V 43 ≈ 1.17 and V 43 ≈ 1.59 and there is only a small range of values of V 43 for which the dumbbells repel each other. Is there a simple mechanism by which the attractive or repulsive behavior can be explained? As shown before, the broken time reversal symmetry of D(t) is necessary for the non-reciprocal motion of the dumbbells to occur. Furthermore for some parameter sets the phase shift ∆φ between the dumbbells as well as its modulation amplitude has a minimum in the repulsive regime. However, this depends very much on the parameters being used. For other parameter sets ∆φ takes its maximum value in the range in which the dumbbells repel each other. For a third class of parameter sets there is no obvious correlation between the phase shift and the fact that the dumbbells behave in an attractive or a repulsive way. All in all the mechanism leading to dumbbell attraction or repulsion is a complex interplay between the applied torques resulting from the coupling parameters and the drag forces, which depend on the bead radii and thus on the asymmetries of the dumbbells.

Conclusions and discussion
Hydrodynamic interactions between two rotating asymmetric dumbbells in a fluid at a low Reynolds number were investigated. We found either a temporally averaged attraction or repulsion between the dumbbells if they have different shapes and/or their couplings to the driving field differ. Differences in shape and/or coupling break the time reversal symmetry, and are hence a pre-condition for attraction or repulsion. No generic rule could be identified of whether a specific form of symmetry breaking leads to dumbbell attraction or repulsion. We presented phase diagrams separating parameter regions of attraction from regions of repulsion.
We suggest experiments where anisotropic birefringent particles are rotated by circularly polarized light, similar to recent experiments [27,28], in order to explore the hydrodynamic attraction and repulsion between rotating small particles. In such experiments the torque on an anisotropic particle may easily be tuned by varying the laser power. Different torques may also be applied to the particles by using different shapes or materials for the particles. A spatially varying power of the circularly polarized light can also cause different torques on neighboring particles. Dumbbells or anisotropic particles can also be constructed from super-paramagnetic particles which can be rotated using a magnetic field. In this case the magnetic dipoledipole interactions have to be taken into account. Magic angle spinning may suppress the magnetic dipole or multipole interactions [25] such that the hydrodynamic interactions dominate.
If the hydrodynamic interactions of an assembly of asymmetric objects are investigated, collective dynamics such as chaotic motion can be expected. Our findings might find applications in an efficient way to micromix a fluid or to separate particles on the small scale. It might also be useful for studies on the collective assembly of micro-swimmers.