# Buckling instability causes inertial thrust for spherical swimmers at all scales

###### Abstract

Microswimmers, and among them aspirant microrobots, generally have to cope with flows where viscous forces are dominant, characterized by a low Reynolds number (). This implies constraints on the possible sequences of body motion, which have to be nonreciprocal. Furthermore, the presence of a strong drag limits the range of resulting velocities. Here, we propose a swimming mechanism, which uses the buckling instability triggered by pressure waves to propel a spherical, hollow shell. With a macroscopic experimental model, we show that a net displacement is produced at all regimes. An optimal displacement caused by non-trivial history effects is reached at intermediate . We show that, due to the fast activation induced by the instability, this regime is reachable by microscopic shells. The rapid dynamics would also allow high frequency excitation with standard traveling ultrasonic waves. Scale considerations predict a swimming velocity of order 1 cm/s for a remote-controlled microrobot, a suitable value for biological applications such as drug delivery.

###### pacs:

Besides their playful aspect, artificial microswimmers present undeniable fundamental and practical interests, mostly driven by a constant race toward increasing miniaturization with potential applications such as targeted drug delivery. Comprehensive studies aim to identify the efficient strategies for small scale displacement in liquids Stone and Samuel (1996); Alouges et al. (2008); Avron et al. (2004); Farutin et al. (2013); Ishimoto and Gaffney (2014); Walker et al. (2015); Chisholm et al. (2016), which can possibly be exploited for the conception of synthetic microswimmers. Sticking to the strict definition of swimming as performing a displacement induced by body deformation, quite a few realizations of synthetic microswimmers can be found in literature Najafi and Golestanian (2004); Dreyfus et al. (2005); Peyer et al. (2012, 2013); Ahmed et al. (2016). A growing attention toward the simplicity of their fabrication Qiu et al. (2015); Bertin et al. (2015); Kaynak et al. (2017) opens possibilities for transfer in the industrial arena. The two main external sources of power are magnetic Peyer et al. (2012, 2013); Qiu et al. (2015) and acoustic Bertin et al. (2015); Ahmed et al. (2016); Kaynak et al. (2017) fields, which are probably more suitable for medical applications and less expensive. The major conceptual difficulty lies in the low Reynolds flows usually associated with microscopic scales; the scallop theorem Purcell (1977) then imposes that a non zero displacement may only occur via a nonreciprocal succession of shapes. Except in chiral systems Peyer et al. (2012, 2013), this necessary condition requires at least two degrees of freedom, which commonly implies two control parameters. Such heavy double steering could indeed be bypassed if flow rates can be rendered high enough so that inertia cannot be neglected anymore, or if any hysteresis in the deformation ”naturally” prevents reciprocity.

We suggest fulfilling these two conditions together with simple spherical colloidal shells full of air that are microscopic objects quite easy to manufacture Zoldesi and Imhof (2005); Pisani et al. (2009). Deflation from a spherical geometry occurs via buckling, which is a subcritical instability, likely to provide both swiftness and hysteresis during a deflation-re-inflation cycle driven by a single scalar control parameter: pressure. We investigate the swimming that results from these deformations thanks to macroscopic shells placed in a set of fluids with varying viscosity so that relevant dimensionless numbers could be kept unchanged from the microscopic scale.

## Design and actuation of the swimmer

— The swimmer was a hollow sphere of thickness and external radius mm, made in an elastomer of Young modulus MPa. The pressure inside the shell was controlled by a pressure controller while the shell was attached to a frictionless rail. A weak spot for buckling was oriented in the rail direction ( axis, inset of Fig. 1). Deformation without displacement was also studied by controlling the external pressure so as to discuss the anticipated microscopic situation (activation by pressure waves). In that case, the shell was immersed in a pressurized tank (see sup () for more details on the method).

After a pressure cycle of sufficient amplitude so that buckling occurs, the shell and its support always move in the same direction whatever the shell thickness and the fluid viscosity (Fig. 1). Deeper insight into this swimming motion requires first to focus on the deformation dynamics.

## Shell deformation cycle

— A stress-free elastic spherical shell of radius and thickness submitted to an outside-inside pressure difference first shrinks while keeping its spherical symmetry, which corresponds to a quasi-linear relationship between and the volume variation (path A-B in Fig. 2-a)Knoche and Kierfeld (2011); Marmottant et al. (2011). Then, over a threshold pressure difference Hutchinson (1967); Landau and Lifschitz (1986); Knoche and Kierfeld (2011); Quilliet (2012), an instability occurs toward a highly deflated conformation with a depression of extent Knoche and Kierfeld (2011); Quilliet (2012). In practice, the final state depends on the possibility to compress the inner medium (see e.g. C and C in Fig. 2-a). The depression often appears repeatedly on a weak spot, at a pressure difference possibly lower than Lee et al. (2016); Zhang and Ru (2016). If is then decreased back to 0, another stable branch is followed, along which the radius of the depression decreases progressively much below (D in Fig. 2-a). Then, a small amplitude unbuckling instability brings the shell shape back to the isotropic branch.

## Shape hysteresis and shape dynamics

— The evolution in the height and width of the shell during a pressure cycle exhibits the hysteresis that is a necessary condition for swimming at low Reynolds number (Fig. 2-a).

After buckling, shape oscillations of frequency can be observed in most liquids (Fig. 2-b). We define the buckling velocity as , which should be close to the maximum velocity of the buckling spot. Fig. 2-c shows that this velocity is almost constant for fluid viscosities up to 1 Pa.s after which the influence of fluid damping on the shell dynamics cannot be neglected.

## Displacements

— Displacements as a function of fluid viscosity are shown in Fig. 3, for both phases of the cycle (increase or decrease of ), and three different relative thicknesses .

At low fluid Reynolds number (in Ucon oil of viscosity 37 Pa.s, for ), displacements are important in both phases (deflation and re-inflation), but they almost compensate within one cycle, with a final displacement of around 1% of the radius due to shape hysteresis (Fig. 1). In this Stokes regime, the displacement is quite similar for all shell thicknesses, as is the sequence of shapes.

On the opposite end of the viscosity range ( in water), the inertial thrust should scale like , where is the mass of the accelerated fluid in the vicinity of the (un)buckling area and its typical acceleration.

In our macroscopic model, this thrust serves first to accelerate the whole system (ball+gliding support) of mass kg. For the shell, where is of order 20% of the total volume sup (), this results in a typical swimming velocity m/s, in very good agreement with the velocity measured during buckling (Fig. 4-b).

The accelerated volume is an increasing function of sup (), which makes the inertial regime eventually more efficient than the Stokes regimes for thick enough shells (Fig. 1). Interestingly, this efficiency is also a consequence of the shape hysteresis: the pre-unbuckling shape (D in Fig. 2-a), which is obtained after a slow decrease of the depression, is much closer to the spherical shape than the post-buckling shape (C in Fig. 2-a). This leads to a much smaller amount of accelerated fluid at unbuckling, hence a negligible contribution of the re-inflation phase to the motion (Fig. 3). For instance, for the shell, the unbuckling volume change and velocity are smaller by a factor 8 and 2, respectively sup ().

Last, the displacement after one full cycle appears to be even larger at an intermediate Reynolds number, for which the displacement due to buckling is enhanced. This points to the need for better knowledge of the surrounding fluid hydrodynamics.

## Flows

— Flows around a shell attached to a fixed support were studied during buckling by time-resolved particle imaging velocimetry. They are qualitatively different according to the liquid viscosity (Fig. 4-a), which echoes to distinct evolutions of the displacement during shape oscillations (Fig. 4-b).

In water, the flow reverses during inward and outward shell oscillations, in phase with the boundary conditions. The buckling induces a displacement of the moving support that oscillates transiently in a synchronous way with the shape.

In glycerol, the not-fully-negligible viscous effect () induces a qualitatively different scenario. The collapse of the shell during phase (ii) (backward motion of both back and front ends) creates a shear flow near the translating flank. This tangential flow is still present during the outward oscillation (iii). As a result, the outward motion of the expelled fluid is not directed towards the axis. This fluid does not contribute to the inertial thrust, which lowers the backward displacement (see phase (iii) in Fig. 4-b). Thus, displacements due to oscillations do not counter each other as in water, but contribute, thanks to this delay effect, to a displacement lasting more than 100 ms (Fig. 4-b).

The delay effect is characterized by the Womersley number , defined as , that compares the oscillation period with the viscous damping time. In the classical Stokes problem of a plate oscillating with frequency in a viscous liquid, shear waves propagate in the normal direction with a wavenumber and a damping , where Batchelor (1967). At a given distance , the shear waves are damped at high and they are in phase with the plate at low . For intermediate values of , the waves are nor damped nor synchronized. In our case, the phase shift gives rise to the complex pattern observed in glycerol where , while in water . Since , in our configuration ; thus, the interplay between inertial, non-stationary dynamics and viscous damping of shear waves will always take place at an intermediate Reynolds number regime, and will imply an enhancement of the thrust during deflation.

## Discussion and miniaturization

— We discuss the implications of our results for the motion of a colloidal armored bubble in a water-like fluid, for which the control of the pressure difference would be attained by an external acoustic field.

## Toy model

— To that purpose, we developed a simplified model to describe the post-buckling dynamics of the shell sup (). If we consider a viscoelastic Voigt material of loss and storage modulus and , with (which, in practice, is almost always the case Lakes (2009)), and if the surrounding fluid does not influence the shell dynamics, the post-buckling frequency obeys

(1) |

with . Here, holds for the shell material volume mass. The buckling speed obeys

(2) |

with . Note that by dimensionality arguments scales necessarily like .

For our shell with kg/m, if we assume that , we find Hz, which, firstly is comparable the measured pulsation Hz and secondly validates the above assumption since, for elastomeric materials, the stiffness is almost constant up to the kHz Lakes (2009). We find that the buckling speed is comprised between 1.9 and 4.4 m/s, which perfectly surrounds the experimental value of m/s in water.

## Flow regime

— From the preceding calculation, we expect the Reynolds number to be , with a prefactor . Displacement enhancement is controlled by the Womersley number and in this problem.

For a shell of radius µm, and considering for the maximum value of 1 bar to avoid cavitation by the bar pressure wave, . Eq. 1 shows that, as scales as , miniaturization down to µm propels to the MHz. Usual values of compliance for elastomers Lakes (2009) indicate that may reach 400. Microscopic shells could then swim in the 1-600 intermediate regime where inertial thrust is enhanced by the coupling between flow and shape oscillations. Besides, this happens at frequencies compatible with sonographic devices that are already known to induce repeated buckling on armored bubbles Marmottant et al. (2011).

## Expected displacements

— A microscopic shell would be controlled by variations of the external pressure, while our macroscopic model was activated by varying the internal pressure.

At low Reynolds number, the sequence of shapes is quasi similar for both ways of controlling the pressure difference (Fig. 2-a) so we anticipate our result of a displacement per cycle of 1% of (which is a slight underestimation due to friction on the arm holding the swimmer) to also hold for a microscopic shell.

At higher Reynolds number, the mass of accelerated fluid is given by the loss of shell volume during buckling, which is limited by the resistance of the inner gas to compression. However, it can be shown that to the first two orders in , the lost volumes are identical whatever the way the pressure is controlled sup (). The estimate done for the final displacement in the intermediate (and ) regime is then valid, and is even a minor bound for a microswimmer that would not be attached to a heavy support.

Finally, a microswimmer subjected to an ultrasonic wave of amplitude and driving frequency will swim under the condition that , where is the post buckling spontaneous frequency of the shape oscillations, so as to allow time for the material to react to pressure variations. While the displacement per cycle is rather low, the high frequency that is allowed, thanks to the fast activation due to the instability, may lead to high velocities. We find that swimming velocity is at least equal to (the Stokes case). With MHz, this potentially leads to a net velocity per second, that is mm/s for a µm shell, much faster than that of Janus particles (10 µm/s Paxton et al. (2004)), helicoidal microrobots (10 µm/s Peyer et al. (2012)), microrobots with acoustically activated flagella (50 µm/s Ahmed et al. (2016) to 1 mm/s but for much larger swimmers Kaynak et al. (2017)), microrobots propelled by metachronal waves (3 µm/s Palagi et al. (2013)).

A microscopic shell designed in such a way that the buckling pressure is of the order 1 bar would be in the intermediate Reynolds regime and would even swim faster. In addition, we anticipate that the swimming amplification observed in the moderate Womersley number regime opens a path for active amplification by a fine tuning of the pressure cycle period, so as to make it comparable to the viscous decay time.

Finally, we extrapolate that multidirectional remote control in a compound of several spheres of different characteristics, which may be built using smart self-assembly properties of colloidal particles Yang et al. (2008); Cademartiri et al. (2012); Yi et al. (2013), can be reached by playing on the wave amplitude (with strong non-linear on/off response depending on whether the buckling pressure has been reached or not) and/or on the wave frequency.

## Conclusion

— We have proposed and experimentally tested a swimming mechanism active at all numbers, that relies on the intrinsic property of shape deformation hysteresis of a spherical shell upon a deflation/re-inflation cycle. Thanks to the fast deformations associated with shape instabilities, an inertial regime is reachable even at small scales. Hysteresis in the deformation velocity sequence implies fast propulsion in this regime, which can be amplified by the coupling between shape oscillations and flow patterns.

###### Acknowledgements.

This research has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 614655 ÒBubbleboostÓ. We thank C. Grégoire for his contribution as an internship student, L. Vignale and N. Mordant for their help and advice for the PIV, T. Combriat, P. Peyla, S. Rafaï and T. Podgorski for the numerous discussions, O. Stephan and S. Lecuyer for their help in the shell molding process, and all the workshop employees for their commitment to build up the different experimental setups.## References

- Stone and Samuel (1996) H. A. Stone and A. D. T. Samuel, Phys. Rev. Lett. 77, 4102 (1996).
- Alouges et al. (2008) F. Alouges, A. DeSimone, and A. Lefebvre, J. Nonlinear Sci. 18, 277 (2008).
- Avron et al. (2004) J. E. Avron, O. Gat, and O. Kenneth, Phys. Rev. Lett. 93, 186001 (2004).
- Farutin et al. (2013) A. Farutin, S. Rafaï, D. K. Dysthe, A. Duperray, P. Peyla, and C. Misbah, Phys. Rev. Lett. 111, 228102 (2013).
- Ishimoto and Gaffney (2014) K. Ishimoto and E. A. Gaffney, Phys. Rev. E 90, 012704 (2014).
- Walker et al. (2015) D. Walker, M. Kübler, K. I. Morozov, P. Fischer, and A. M. Leshansky, Nano Lett. 15, 4412 (2015).
- Chisholm et al. (2016) N. G. Chisholm, D. Legendre, E. Lauga, and A. S. Khair, J. Fluid Mech. 796, 233 (2016).
- Najafi and Golestanian (2004) A. Najafi and R. Golestanian, Phys. Rev. E 69, 062901 (2004).
- Dreyfus et al. (2005) R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, and J. Bibette, Nature 437, 862 (2005).
- Peyer et al. (2012) K. Peyer, A. Mahoney, L. Zhang, J. Abbott, and B. Nelson, Bacteria-Inspired Microrobots (Elsevier Inc., 2012), pp. 165–199.
- Peyer et al. (2013) K. E. Peyer, S. Tottori, F. Qiu, L. Zhang, and B. J. Nelson, Chem. ï¿½ Eur. J. 19, 28 (2013).
- Ahmed et al. (2016) D. Ahmed, T. Baasch, B. Jang, S. Pane, J. Dual, and B. J. Nelson, Nano Lett. 16, 4968 (2016).
- Qiu et al. (2015) T. Qiu, S. Palagi, and P. Fischer, in 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC) (2015), pp. 4922–4925.
- Bertin et al. (2015) N. Bertin, T. A. Spelman, O. Stephan, L. Gredy, M. Bouriau, E. Lauga, and P. Marmottant, Phys. Rev. Applied 4, 064012 (2015).
- Kaynak et al. (2017) M. Kaynak, A. Ozcelik, A. Nourhani, P. E. Lammert, V. H. Crespi, and T. J. Huang, Lab Chip 17, 395 (2017).
- Purcell (1977) E. M. Purcell, Am. J. Phys. 45, 3 (1977).
- Zoldesi and Imhof (2005) C. Zoldesi and A. Imhof, Adv. Mat. 17, 924 (2005).
- Pisani et al. (2009) E. Pisani, C. Ringard, V. Nicolas, E. Raphaël, V. Rosilio, L. Moine, E. Fattal, and N. Tsapis, Soft Matt. 5, 3054 (2009).
- (19) See Supplemental Material [url]
- Knoche and Kierfeld (2011) S. Knoche and J. Kierfeld, Phys. Rev. E 84, 046608 (2011).
- Marmottant et al. (2011) P. Marmottant, A. Bouakaz, N. de Jong, and C. Quilliet, J. Ac. Soc. Am. 129, 1231 (2011).
- Hutchinson (1967) J. W. Hutchinson, ASME. J. Appl. Mech. 34, 49 (1967).
- Landau and Lifschitz (1986) L. Landau and E. Lifschitz, Theory of Elasticity (Elsevier Butterworth-Heinemann, Oxford, 1986), 3rd ed.
- Quilliet (2012) C. Quilliet, Eur. Phys. J. E 35, 48 (2012).
- Lee et al. (2016) A. Lee, F. López Jiménez, J. Marthelot, J. Hutchinson, and P. M. Reis, ASME. J. Appl. Mech. 83, 111005 (2016).
- Zhang and Ru (2016) L. Zhang and C. Q. Ru, Phys. Rev. E 93, 062403 (2016).
- Batchelor (1967) G. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, 1967).
- (28) See Supplemental Material [url] for a description of the model, based on the theory of viscoelastic thin rods Baker et al. (1967)
- Baker et al. (1967) \BibitemOpen\bibfieldauthor W. E. Baker, W. E. Woolam, and D. Young, \bibfieldjournal Int. J. Mech. Sci. 9, 743 (1967).
- Lakes (2009) R. S. Lakes, Viscoelastic Materials (Cambridge University Press, 2009).
- Gazzola et al. (2014) M. Gazzola, M. Argentina, and L. Mahadevan, Nat. Phys. 10, 758ï¿½761 (2014).
- Paxton et al. (2004) W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. St. Angelo, Y. Cao, T. E. Mallouk, P. E. Lammert, and V. H. Crespi, J. Am. Chem. Soc. 126, 13424 (2004).
- Palagi et al. (2013) S. Palagi, E. W. Jager, B. Mazzolai, and L. Beccai, Bioinspir. Biomim. 8, 046004 (2013).
- Yang et al. (2008) S.-M. Yang, S.-H. Kim, J.-M. Lim, and G.-R. Yi, J. Mater. Chem. 18, 2177 (2008).
- Cademartiri et al. (2012) L. Cademartiri, K. J. M. Bishop, P. W. Snyder, and G. A. Ozin, Philos. T. Roy. Soc. A 370, 2824 (2012).
- Yi et al. (2013) G.-R. Yi, D. J. Pine, and S. Sacanna, J. Phys.: Cond. Matt. 25, 193101 (2013).