Abstract
Spinning objects moving through air or a liquid experience a lift force—a phenomenon known as the Magnus effect. This effect is commonly exploited in ball sports but also is of considerable importance for applications in the aviation industry. Whereas Magnus forces are strong for large objects, they are weak at small scales and eventually vanish for overdamped micrometresized particles in simple liquids. Here we demonstrate a roughly onemillionfold enhanced Magnus force of spinning colloids in viscoelastic fluids. Such fluids are characterized by a timedelayed response to external perturbations, which causes a deformation of the fluidic network around the moving particle. When the particle also spins, the deformation field becomes misaligned relative to the particle’s moving direction, leading to a force perpendicular to the direction of travel and the spinning axis. Our uncovering of strongly enhanced memoryinduced Magnus forces at microscales opens up applications for particle sorting and steering, and the creation and visualization of anomalous flows.
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When a spinning object travels through a fluid, its trajectory is typically curved. Although Isaac Newton was the first to describe it in 1671 (ref. ^{1}), this effect is commonly named after Heinrich Gustav Magnus, who provided a physical explanation on the influence of rotation on the motion of objects^{2}. Nowadays, the Magnus effect is well established and is not only used in ball games but also exploited, for example, as an economic propulsion mechanism for ships^{3,4,5,6} and to provide lifting forces for air vehicles^{7}. In addition to such applications, Magnus effects are also relevant for understanding planet formation in protoplanetary disks^{8} and the behaviour of ions in superfluids^{9,10} and are even discussed in the context of the motion of vortex lines in superconductors^{11,12}. In general, the Magnus force results from an asymmetry of the velocity field in the medium around a translating and simultaneously rotating object. According to the Bernoulli equation, this results in pressure inhomogeneities near the object and a force perpendicular to the direction of travel: that is, the Magnus force F_{M} = f(ω × v), an expression that is also expected to be valid for viscous liquids at small Reynolds numbers^{13}. Here the Magnus coefficient f quantifies the coupling of the particle to its surroundings, and ω and v are the angular and linear velocities of the object relative to the fluid, respectively. Although in most cases f > 0, it can be also negative: for example, at high velocities, when the flow around the object is partially turbulent^{7,14}, or when spinning objects move through rarefied gases or granular media^{15,16,17}. Whereas Magnus forces can be very strong for large objects, they are weak at small scales. In the case of Brownian particles—that is, micrometresized particles in simple fluids—Magnus forces eventually vanish, because viscous forces dominate over inertial effects^{18,19}. Therefore, applications of Magnus forces in such systems are rare.
Here we report the experimental observation of a strong memoryinduced Magnus effect for spinning micrometresized colloidal particles moving through an overdamped viscoelastic fluid. Unlike viscous—that is, Newtonian—liquids, which instantaneously respond to external perturbations, viscoelastic fluids are characterized by stressrelaxation times τ on the order of seconds or more^{20,21}. Similar to the Magnus force F_{M} generated in a viscous liquid, in viscoelastic liquid the memoryinduced Magnus force \({{{\mathbf{F}}}}_{{{{\rm{mM}}}}}=\tilde{f}({{\mathbf{\upomega }}}\times {{\mathbf{v}}})\) is exerted on a translating and spinning object. The coefficient \(\tilde{f} < 0\), and its amplitude is larger than f (that is, the coefficient in a pure viscous fluid) by a factor of more than 10^{6}. Our experimental results are in excellent agreement with a theoretical description in which the timedelayed response of the fluid around a moving object is modelled by a density dipole. Although this dipole points in the direction of v for a pure translational particle motion, it is rotated when the particle also exhibits a spinning motion. As a result, a force component perpendicular to the driving force arises, which eventually leads to F_{mM}. This model also explains our experimental finding that when the particle’s spinning motion is stopped, the force F_{mM} remains and only decays after time τ. Because our findings should apply to many viscoelastic fluids, we expect that this unusual type of Magnus force will lead to new applications—for example, in the field of particle sorting and steering—as well as the creation and visualization of anomalous flows^{22,23,24,25,26}.
In our experiments, we use superparamagnetic colloidal spheres (diameter σ = 4.45 μm) suspended in a viscoelastic fluid (see below) and contained in a thin sample cell. Owing to gravity, the particles sediment towards the bottom surface of the cell, where they remain during our experiments. We note, however, that trajectories similar to those that will be discussed have also been recorded for particles far from any surface (Supplementary Information). The colloid motion is imaged with a video camera mounted to an inverted microscope. After each experiment, the data are analysed using digital video microscopy. The microscope is mounted on a tilting stage, which allows an external (gravitational) drift force \({{\mathbf{F}}}={mg}\sin \alpha \,\hat{{{{\bf{x}}}}}\) to be exerted on each particle. Here mg = 286 fN is the buoyant particle weight (obtained by multiplying the buoyant mass m and the acceleration of gravity g), α the tilting angle and \(\hat{{{{\bf{x}}}}}\) the unit vector along the x direction. The angle α can be tuned between 0^{°} and 35^{°}, which leads to a maximum driving force of about 160 fN for a single particle or 480 fN for a colloidal trimer (see below). Additionally, we use two perpendicular pairs of coils (Fig. 1a,b) that create a rotating magnetic field H(t) in the sample plane, with components \({H}_{x}(t)=H\cos {\omega }_{{{{{H}}}}}t\) and \({H}_{y}(t)=H\sin {\omega }_{{{{{H}}}}}t\). The frequency ω_{H} = 20π s^{−1} is fixed in our experiments. The rotating H induces a rotating magnetization M in the colloidal particles. Owing to a phase lag in M, the rotating magnetic field applies a torque Γ = M × H ∝ H^{2} to the colloid spheres. This results in a spinning motion with angular frequency ω ∝ Γ (see Methods and Supplementary Fig. 1 for details).
Two different types of viscoelastic fluids were used in our experiments: (1) a solution of entangled giant wormlike micelles^{27} composed of about 5 mM equimolar cetylpyridinium chloride monohydrate (CPyCl) and sodium salicylate (NaSal) dissolved in water, and (2) a semidilute aqueous polymer solution of polyacrylamide (PAAM) with molecular weight 18 MDa and mass concentration 0.03%. All our experiments have been performed at a constant sample temperature of 25 ^{°}C, at which both fluids exhibit pronounced viscoelastic behaviours as confirmed by microrheological experiments^{28,29} (see Methods and Supplementary Fig. 2 for details).
To demonstrate that conventional Magnus forces in colloidal systems are vanishingly small when suspended in merely viscous—that is, Newtonian—fluids^{18,19}, we first studied the motion of a colloidal particle in a waterglycerol mixture. Figure 1c shows the corresponding trajectory of a particle subjected to a rotating magnetic field (H = 732 A m^{−1}) and a driving force F = 156 fN. The rotation direction of H is periodically reversed from clockwise to anticlockwise to rule out the possible influence of particle drift in the y direction. In our experimental resolution, no particle deflection from the direction of the driving force is observed. This is consistent with the theoretically predicted conventional Magnus force in viscous liquids in the limit of low Reynolds numbers, F_{M} = πρ(σ/2)^{3}ω × v, where ρ is the fluid mass density^{13,19}. For our liquid, this yields a deflection angle \({\theta }_{{{{\rm{d}}}}}=\arctan ({F}_{{{{\rm{M}}}}}/F)\ll 0.000{2}^{\circ }\) (considering ω ≪ ω_{H}), which is below the experimental resolution. When repeating the experiment in a viscoelastic fluid, however, a pronounced deflection of the trajectory is observed (Fig. 1d). From the measured velocity ratio v_{y}/v_{x}, we determine the deviation angle \({\theta }_{{{{\rm{d}}}}}=\arctan ( {v}_{y}/{v}_{x} )\approx 1{5}^{\circ }\). The deflection changes its direction when we reverse the direction of rotation of H. Although we are particularly interested in the angular motion of the spinning particles, it cannot be resolved for the case of single colloidal spheres. Therefore, in the following, we use colloidal trimers whose angular motion is easily measured and that show almost identical behaviour (Fig. 1e and Supplementary Videos 1 and 2). For details about the formation of such trimers, we refer readers to the Methods section. For trimers with a spinning frequency ω = 0.97 rad s^{−1} and drift velocity v_{x} = 0.16 μm s^{−1}, we find an angular deflection \({\theta_{\rm{d}}\approx 14.4^{\circ}}\), which demonstrates a force pointing in the direction −ω × v.
To quantify our observations, we measured the velocity ratio v_{y}/v_{x} as a function of the trimer spinning velocity ω and for different driving forces F (Fig. 2a). At small ω, we observe for both viscoelastic fluids a linear behaviour that eventually saturates for larger ω (Fig. 2a, inset). In addition, v_{y}/v_{x} is independent of v_{x} when ω is kept constant (Fig. 2b). Both findings indicate that in the linear regime, a memoryinduced Magnus force \({{{{\mathbf{F}}}}}_{{{{\rm{mM}}}}}=\tilde{f}({{\mathbf{\upomega }}}\times {{{\mathbf{v}}}})\) is acting on the colloidal trimer, with prefactor \(\tilde{f}\) depending on the colloid–fluid interaction.
For a theoretical understanding of the above observations, we first consider a nonspinning particle moving at velocity v due to an external force F through a viscoelastic fluid. Owing to the finite stressrelaxation time τ, a frontback inhomogeneity in the fluid builds up around the particle. It can be characterized by a density dipole p_{∥} pointing in the direction of v (Fig. 3a, top, and Supplementary Information)^{20,30,31,32,33}. In linear response theory^{34}, the timedependent magnitude of p_{∥} is given by a history integral of the force
where χ_{T}(t) is a memory kernel that characterizes the dipole relaxation dynamics^{35}. Experimentally, χ_{T}(t) can be determined when the driving force acting on the particle is suddenly removed. This results in a restoring force antiparallel to p_{∥}, leading to a recoil motion opposite to the driving direction. Figure 3b (see also Supplementary Videos 3 and 4) shows such recoil for a colloidal trimer driven by a magnetic field gradient force that was turned off at t = 0. In agreement with previous studies, the recoil is well described by a doubleexponential decay^{33,36}. From these data, we immediately obtain the memory kernel χ_{T}(t), as shown in Fig. 3c (see details in Supplementary Information).
When the particle is also set into a spinning motion by an external torque Γ, the orientation of the density dipole in the xy plane changes by an angle θ_{p} (Fig. 3a, bottom) due to the particlefluid interaction. Accordingly, one obtains a dipole p = p_{∥} + p_{⊥}, with p_{⊥} the component perpendicular to F. Similar to the restoring force caused by p_{∥}, a force perpendicular to F is caused by p_{⊥}. This perpendicular force is identified as the memoryinduced Magnus force F_{mM}, which is proportion to −p_{⊥}. We consider the regime where p_{⊥} is a linear function of F and Γ, for which we find (see details in Supplementary Information)
Here χ_{R}(t) is the memory kernel associated with the relaxation of the dipole component p_{⊥}. Because in the range of our experiments we have Γ ∝ ω and p_{∥} ∝ F ∝ v_{x} (Supplementary Fig. 3), this leads to p_{⊥} ∝ ωv_{x} or \({{{{\mathbf{F}}}}}_{{{{\rm{mM}}}}}=\tilde{f}({{{\mathbf{\upomega }}}}\times {{{\mathbf{v}}}})\), in agreement with the observations in Fig. 2. As a side note, we mention that after the particle starts to move in the y direction under the influence of F_{mM}, this motion creates—similar to p_{∥}—a further density dipole that is opposite to p_{⊥} and reduces the velocity component v_{y}. This effect is already included in equation (2) by considering χ_{R}(t) for a moving particle.
According to equation (2), p_{⊥} (and hence F_{mM}) should not instantaneously vanish when we remove the external torque applied to the particle. This means the particle motion in the y direction decays on a timescale given by χ_{R}(t) that characterizes how fast p_{⊥} decays to zero. To demonstrate this, Fig. 3d shows for t < 0 the motion of a trimer in the y direction under the influence of a force F = 468 fN in the x direction and torque Γ = 66.6 pN μm. When Γ is set to zero at t = 0 s, the cluster’s motion in the y direction decays only on a timescale of several tens of seconds. This decay can be directly compared to the corresponding prediction (solid line) of equation (2) when assuming χ_{R}(t) ∝ χ_{T}(t). Such proportionality is plausible because the susceptibilities describe relaxations of dipole fields in different directions and can thus be expected to be similar due to isotropy of the quiescent fluid.
On the basis of the dipolerotation picture as described in Fig. 3a, we also construct a simple Maxwelllike model for the time dependence of the dipole orientation ϕ, which contains a driving term proportional to the applied torque Γ and a restoring term owing to the relaxation of p_{⊥}:
with γ_{R} the steadystate rotational friction coefficient, so that Γ/γ_{R} = ω is the rotational velocity of the colloid and C ≤ 1 is a constant that describes the coupling between the rotation of the colloid and that of the density dipole. In the absence of Γ, the solution of equation (3) relaxes exponentially with a timescale τ. We remark that recent experiments^{37} as well as Fig. 3b demonstrate that the relaxation process of a colloidal particle in a micellar fluid is governed by two timescales. For the sake of a minimal model, however, here only a single relaxation time is considered. Under steadystate driving conditions, the value of the dipole orientation angle ϕ becomes
Considering that the dipole force, the viscous force and the external force must balance altogether, this leads to a simple relationship between θ_{p} and the velocity ratio v_{y}/v_{x} (see Supplementary Information for details):
Here r is the dimensionless recoil ratio given by the ratio of the particle velocities immediately after and before turning off the driving force. From the typical recoil shown in Fig. 3b for the micellar system, we determine that r ≈ 1.8 ± 0.1 (a similar analysis can be applied to the PAAM solutions to obtain the corresponding r; see Supplementary Fig. 2). Equation (5) allows us to estimate \(\tilde{f}\) defined above by interpreting the motion in y to arise from the force F_{mM}. It yields \(\tilde{f}=rC\tau \gamma\), with γ the translational friction coefficient. By fitting equation (5) to the linear part of our data shown in the Fig. 2a inset, we obtain rCτ = 0.146 ± 0.013 s, or \( \,\tilde{f}\,\) = 0.95 ± 0.08 pN s^{2} μm^{−1}, considering γ = 6.50 ± 0.09 pN s μm^{−1} in 5 mM micellar fluid (Supplementary Fig. 3). This is considerably larger than the Magnus coefficient for a viscous liquid, which is f = πρ(2σ/2)^{3} = 0.28 × 10^{−6} pN s^{2} μm^{−1} (for this estimate, we have approximated the cluster of three colloids, each with diameter σ, with a single spherical particle with diameter 2σ). We have also performed similar experiments in micellar solutions at a lower micellar concentration (4 mM), where the fluid is only slightly viscoelastic. As expected, at such low concentrations, the Magnus coefficient becomes considerably smaller (still much larger than in viscous liquid) compared to that in 5 mM due to weak colloid–fluid interaction (Supplementary Fig. 4).
In the above framework, we also obtain a simple estimate for the upper limit of ω where the linear relation between v_{y}/v_{x} and ω ends. Assuming that the magnitude of p_{⊥} cannot exceed that of the original dipole, this yields v_{y}/v_{x} < r (Supplementary Information). From this, one obtains a rough estimate of the saturation velocity, ω_{sat} ≤ 1/Cτ ≈ 12 s^{−1}, which is consistent with our data in the inset of Fig. 2a. Following a similar procedure, for a colloidal trimer in a polymer fluid, we obtain a saturation velocity of about ω_{sat} ≤ 1/Cτ ≈ 3.4 s^{−1}.
As an alternative approach to Magnus forces, one can introduce a viscosity tensor η_{mM}, which is defined by \({{{{\mathbf{F}}}}}_{{{{\rm{mM}}}}}=\tilde{f}({{{\mathbf{\upomega }}}}\times {{{\mathbf{v}}}})=3\uppi \sigma {{{{{\eta }}}}}_{{{{\rm{mM}}}}}\cdot {{{\mathbf{v}}}}\). Writing the cross product using the LeviCivita symbol, \({F}_{{{{\rm{mM}}}},i}=\tilde{f}\omega {e}_{3ij}{v}_{j}\), shows that η_{mM} is antisymmetric. When also considering the translational friction force −γv to obtain the diagonal components of the viscosity tensor with the friction coefficient γ for ω = 0, introduced below equation (5), this yields, for ω pointing in the z direction,
Such nonsymmetric viscosity tensors (typically referred to as odd viscosity^{22,23,24,25,26,38,39}), which generally result from the violation of Onsager’s reciprocal relations^{40,41}, are caused in our system by the timedelayed dynamics of the density dipole. Accordingly, the offdiagonal elements of the viscosity tensor can easily be tuned by means of the rotation frequency.
In summary, our work demonstrates that Magnus forces, which typically vanish in the realm of small Reynolds numbers, can be surprisingly strong in the case of viscoelastic fluids owing to their timedelayed response to perturbations. In addition to rotating magnetic fields, which have been used in our study to induce a spinning motion, electrical^{42} and optical fields^{43} have also been shown to impose considerable torques on particles down to the nanometre scale. This allows the use of Magnus forces to be extended to the regime of small Reynold numbers, which may lead to new types of microswimmers and strategies for steering and sorting particles as well as visualizing complex flow patterns in liquids. Finally, we remark that our observations bear some phenomenological resemblance to elastic turbulence. Unlike turbulence in Newtonian liquids, which only arises at high Reynolds numbers, turbulent flow patterns in viscoelastic fluids have already been observed for Re ≈ 10^{−3} (ref. ^{44}).
Methods
Sample preparation
We prepare highly diluted colloidal suspensions of superparamagnetic spheres (Dynabeads M450, diameter ~4.5 μm) dispersed in a viscoelastic fluid. The number density (roughly 10^{8} per litre) corresponds to fewer than ten colloid spheres in our field of view, 240 × 300 μm^{2}. For viscoelastic fluid, we use either a polymer solution or a micellar solution. The polymer solution is a semidilute aqueous solution of PAAM with molecular weight 18 MDa and mass concentration 0.03%. The micellar solution is an equimolar, aqueous solution of CPyCl and NaSal. The molar density is 5.5 mM for the data in Fig. 2a and 5.0 mM for the Fig. 2a inset and all other relevant data. To make a colloid sample, we inject the colloid suspension into a glass sample cell about 20 × 10 × 0.2 mm^{3} in size, where 0.2 mm is the sample thickness. After the sample is made, it is transferred to an inverted Nikon microscope where we can observe the motion of the colloid. During the experiment, the temperature of the sample is kept at 25 ± 1 ^{°}C using a flow thermostat.
Formation of colloidal trimers
The rotating magnetic field creates an effective longrange attraction (∝r^{3}) between the magnetic colloids^{45}. In the presence of a rotating magnetic field with H = 732 A m^{−1}, colloidal particles with distances of ~5σ experience a strong attraction and form dense clusters (Supplementary Video 5). To increase the chance of particle encounters during the cluster formation process, the sample stage is tilted by ~10^{°}. This leads to the formation of rigid and stable colloidal clusters with random sizes, including trimers.
Calibration of magnetic torques
Due to the large viscosity in our experiments, the rotating speed of the colloids is much less than the rotating frequency 10 Hz of the magnetic field. This leads to a timeaveraged magnetic torque of Γ = γ_{m}H^{2} (ref. ^{46}) applied by the rotating magnetic field, as shown in Fig. 1b. In steady rotation, the magnetic torque balances the viscous torque of the fluid: that is, Γ = γ_{R}ω, where γ_{R} = 6πησ^{3} for colloid trimers^{47}. This leads to ω = kH^{2}, where k = γ_{m}/γ_{R}. To calibrate the magnetic torque, we first rotate the colloid trimer in water, where the measured rotating speed ω of a colloid trimer as a function of H exactly follows the equation ω = kH^{2}, as shown in Supplementary Fig. 1, with the fitted k = 3.26 × 10^{−5} m^{2} A^{−}^{2} s^{−1}. Considering the measured η = 1.29 × 10^{−3} Pa s for water and the colloid diameter σ = 4.45 μm s^{−1}, we obtain γ_{m} = 6.99 × 10^{−5} pN μm A^{−2} m^{2}.
Colloidal recoils with timedependent magnetic gradients
Because gravitational drift forces on the particles cannot be suddenly changed, the translational recoil curves shown in Fig. 3b are measured using a permanent magnet whose position in the sample plane can be suddenly changed (<0.1 s) with a mechanical springloaded device. To do so, we first put the magnet close to the sample cell, where the magnetic field gradient leads to a constant drift velocity for the trimers. When we activate the springloaded retraction mechanism, the magnet is pulled away and the colloidal recoil sets in.
Data availability
Source data are provided with this paper. All relevant data are also available from the corresponding author on reasonable request.
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Acknowledgements
We acknowledge helpful discussions with M. Fuchs, P. Tierno, G. Junot and A. Ping. This work is funded by the Deutsche Forschungsgemeinschaft (DFG), grant no. SFB 1432 – Project ID 4252172. F.G. acknowledges support from the Humboldt foundation.
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C.B. and X.C. designed the experiments, which were carried out and analysed by X.C. and N.W. The theoretical model was developed by D.D. and M.K. F.G. contributed to overall discussions. All authors contributed to the writing of the paper.
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Supplementary Information
Further information on the data analysis and theory, and Supplementary Figs. 1–5.
Supplementary Video 1
Trajectory deflection of a single colloidal particle in a 5 mM micellar solution.
Supplementary Video 2
Trajectory deflection of a colloidal trimer in a 5 mM micellar solution.
Supplementary Video 3
The observed recoil experiments for a colloid trimer in a 5 mM micellar solution.
Supplementary Video 4
The observed recoil experiments for a colloid trimer in a polymer solution.
Supplementary Video 5
The observed formation of a colloidal trimer.
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Cao, X., Das, D., Windbacher, N. et al. Memoryinduced Magnus effect. Nat. Phys. 19, 1904–1909 (2023). https://doi.org/10.1038/s41567023022131
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DOI: https://doi.org/10.1038/s41567023022131
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