Design and actuation of the micro-scallop

Our micro-scallops are constructed from polydimethylsiloxane (PDMS), loaded with phosphorescent pigment, cast into a three-dimensional (3D) printed mold, which permits the use of different materials in a parallel fabrication process. Each micro-scallop consists of two thick (300 μm) shells connected by a thin (60 μm), narrow (200 μm) hinge (Fig. 1b). Rare earth micromagnets (Ø200 μm × 400 μm) are attached to each shell so that when exposed to an external magnetic field the two magnets reorient to align with the field and each other, and thus close the micro-scallop (Fig. 2a, right). When the magnetic field is decreased, the restoring force of the PDMS hinge provides the recovery stroke, opening the micro-scallop (Fig. 2a, left). Alignment with the magnetic field prevents the scallop from pitching and yawing, and ensures that it swims straight. The thick stiff shells and compliant flexible hinge ensure that the deformation is isolated at the hinge. As stated by Purcell: any single-hinge structure can only exhibit reciprocal motion1.

Figure 1: Schematic drawing of the scallop swimmers. (a) Schematic drawing of Purcell’s scallop with reciprocal motion1. (b) 3D model of the submillimetre size ‘micro-scallop’. (c) 3D model of the centimetre size ‘macro-scallop’ for quantitative comparison with theory. Full size image

Figure 2: Actuation mechanism of the micro-scallop. (a) Schematic drawing of the micro-scallop from top view. The green shapes illustrate the opening and closing shape change of the micro-scallop when actuated by an external magnetic field. The shape is a function of the magnetic force aligning the magnetic axes of the two permanent micromagnets and a restoring force because of the induced stress in the PDMS structure. The angle between the two shells α can therefore be controlled by the magnitude of the external field. (b) Top view (microscope image) of the micro-scallop under ultraviolet illumination. The positions of the micromagnets are illustrated by white dashed circles. Scale bar, 200 μm. (c) Time-asymmetric actuation of the micro-scallop. The slow opening and the fast closing cycles are controlled by the external magnetic field, which is generated by an exponentially decaying current (inset). Corresponding images of the micro-scallop during the opening and closing cycles are shown. Full size image

The opening angle α of the micro-scallop is related to the strength of the applied external field. As shown in Fig. 2c, asymmetric actuation of the two shells is achieved by applying a periodic exponentially decaying current to generate the magnetic field. Typically, a 0.5-Hz waveform was used with a slow ~1.9 s exponential decay, followed by a rapid 0.1 s ramp. Since a gradient-free field is used, the micromagnets do not experience any pulling force, which ensures the net displacement of the microswimmer is because of the propulsion caused by its own shape-changing swimming motions.

Propulsion of the micro-scallop in a shear thickening fluid

Forward net displacement of the micro-scallop in a shear thickening fluid (fumed silica in poly(propylene glycol)) was achieved by time-asymmetric actuation of the opening angle as seen in Fig. 3a. Experiments were conducted with the scallop supported at the air–fluid interface (Fig. 3), as well as with micro-scallops that were fully immersed in bulk fluid (Supplementary Fig. 16). Nonreciprocity arising from surface capillary waves in the interface-supported geometry can be ruled out because their phase velocity is too high to be excited by the micro-scallop’s stroke. This is confirmed by the quantitative agreement between the results from the (interface) supported and unsupported (bulk) swimming configurations. Overlaying five frames with the same opening angles taken at 50-s intervals (Fig. 3a) clearly shows net displacement in the x direction parallel to the magnetic field direction (Supplementary Movie 1, upper panel). As a control, the micro-scallop in the same fluid was actuated with a symmetric waveform, and, as expected, no net displacement in the x direction was observed (Fig. 3b and Supplementary Movie 1, lower panel). Furthermore, asymmetric actuation in a Newtonian fluid (glycerol) also yields no net displacement (circles in Fig. 3c).

Figure 3: Displacement of the micro-scallop in a shear thickening and a shear thinning fluid. (a) Forward net displacement of the micro-scallop in a shear thickening fluid and asymmetric actuation (blue curve). The image is a time-lapse composite picture of five frames at an interval of 50 s, with the net displacement indicated along the x direction (see Supplementary Movie 1 upper panel). (b) Corresponding image of the micro-scallop in shear thickening fluid with symmetric actuation (blue curve) and no discernable net displacement (see Supplementary Movie 1 lower panel). Scale bar in a,b, 300 μm. (c) Corresponding displacement curves in the shear thickening, shear thinning and Newtonian fluids, respectively. ω c and ω o are the average angular velocity of closing and opening, respectively. Asymmetric actuations (ω c >ω o , solid squares and ω c <ω o solid triangles) result in net displacement in non-Newtonian fluids, while symmetric actuations (ω c =ω o , hollow squares and triangles) result in no net displacement in the same fluids (see Supplementary Movie 2 for swimming in shear thinning fluid). Asymmetric actuations (ω c >ω o , circles) result in no net displacement in the Newtonian fluid (see Supplementary Movie 3), as stated by the scallop theorem. The error bars correspond to the s.d. of repeated trials. Full size image

Fumed silica particle suspensions were chosen as the shear thickening fluid because of their well-known characteristics28 and relatively high viscosity. The dynamic viscosity of the fluid is in the range of 1–22 Pa s (Fig. 4a), and its density is 1051±2 kg m−3. The micro-scallop swam for more than 100 μm over 10 periods, and the average velocity was 5.2 μm s−1 (3.5% body length per cycle). If we take the characteristic maximum length of the micro-scallop as 1 mm, and the largest forward velocity as 3 mm s−1, then the calculated Re=1.4 × 10−4–3 × 10−3 ≪1. Thus, the microswimmer operates at a very low Re1 using only reciprocal motion.

Figure 4: Comparison between theoretical predictions and experimental measurements of scallop swimming. (a) Experimentally measured apparent dynamic viscosity of the shear thickening fluid (solid squares). The dashed line shows the power law used to fit the transition to the shear thickening regime. (b,c) Dimensionless net displacement over one period plotted against the ratio of angular velocities. D and l are the displacements over one stroke and the characteristic length, respectively. The dashed lines represent the predictions of the scaling theory in Equation (5). When the closing velocities (solid squares) are larger than the opening velocity (hollow triangle), p=ω fast /ω slow =ω c /ω o >1, the swimmer exhibits positive net displacements (b). When the opening velocities (hollow triangles) are larger than the closing velocity (solid square), p=ω fast /ω slow =ω o /ω c >1, the swimmer exhibits negative net displacements (c). In (a–c), the error bars correspond to the standard deviations. Full size image

Propulsion of the micro-scallop in a shear thinning fluid

Propulsion of the micro-scallop operated with a reciprocal but asymmetric actuation sequence is also achieved in hyaluronic acid, a shear thinning solution found in a number of biological systems. In the shear thinning fluid, the micro-scallop only moves forward when the opening–closing cycle is opposite to that of the shear thickening fluid. Now, fast-opening followed by a slow-closing step gives forward propulsion. About 65 μm displacement was covered by the micro-scallop in 10 periods, corresponding to an average velocity of 3.8 μm s−1 (2.5% body length per cycle). As in the shear thickening medium, the micro-scallop showed no significant forward displacement when the opening and closing cycles were symmetric, as is expected (see Supplementary Movie 2).

Analytical theory of propulsion

We consider a simple model of a flapping, Purcell scallop-like swimmer with the typical spatial dimension l composed of two counter-rotating shells that share a common axis. For simplicity, we assume that the shells open and close with different angular speeds, ω slow =ω and ω fast =pω, where ω slow and ω fast stand for the velocity of the slow and fast stroke, respectively, so that p≥1 for all swimming gaits. The maximum rotation angle of one plate is α/2 and the scallop completes one full cycle in period . The momentum transferred to the fluid by the tethered swimmer (or a pump) scales as ~μ app Ul, where U is the fluid velocity and μ app is an apparent (spatially averaged) viscosity of the fluid, which depends on the instantaneous shear rate, and the history of the flow. For the pivoting shells of the scallop, this leads to the scaling relation ~μ app ωl2. Thus, the net force exerted by the pump on the fluid over the stroke period can be estimated (up to an arbitrary function of the opening angle α) as

Note that for a Newtonian fluid, the net force over one cycle in equation (1) is zero, as expected from the ‘scallop theorem’, that is, no net momentum can be transferred to a fluid by a pump using geometrically reciprocal strokes1. However, for a non-Newtonian fluid, the dependence of the apparent viscosity on the shear rate breaks time–reversibility and from equation (1) the net force over a period is

where , are the apparent viscosities time-averaged over the slow and fast strokes respectively, with being their difference.

The propulsion velocity of the force-free swimmer V s can be estimated from the dual drag problem, using F p ≈F d , where F d is the force to be applied to an inactive swimmer in order to drag it with velocity V s 29. We assume that F d ~μ * V s l, where μ * is some apparent viscosity corresponding to the typical shear rate magnitude V s /l. For low values of the shear rate associated with swimming, that is V s /l≪ω, it is reasonable to assume that μ * ≈μ 0 ≈const (the rheological measurements (Fig. 4a) suggest that for shear rates up to 1.7 s−1 viscosity is roughly constant).

Equating F d and F p in equation (2) yields the propulsion velocity up to an unknown function of angle α:

More detailed analysis of a flapping swimmer composed of two infinite plates suggests that the major contribution to in equation (3) is because of an abrupt rise (fall) of apparent viscosity of the shear thickening (shear thinning) fluid sandwiched between two close plates at small openings during the fast phase (either closing or opening, see Supplementary Figs 13 and 14).

To model propulsion through shear thickening fluid, a power law equation30 was applied to fit the transition region of the viscosity in the shear rate range of 1.5–6 s−1 (dotted line in Fig. 4a). The apparent viscosity is , where m, n are constants (m=0.34, n=3.34 as fitted in this case) and is the shear rate defined as , that is the second invariant of the rate-of-strain tensor . Obviously, in this problem the typical shear rate is, respectively, for opening and for closing. Thus, assuming that both (that is, fast and slow) strokes fall within the shear thickening regime, and , equation (3) becomes

Multiplying V s in equation (4) by the stroke period we obtain a simple expression for the scaled displacement-per-stroke, D/l,

where the sign and magnitude of the dimensionless prefactor β depends on the nature of the stroke and fluid properties. The analytical theory therefore clearly confirms that in a shear thickening fluid, the scallop can indeed swim forward using fast closing and slow opening strokes. Alternatively, it can propel in the opposite direction using fast-opening and slow-closing strokes. This is a general result that permits propulsion from symmetric actuation in any of the abundant non-Newtonain fluids, notably those found in biological systems. In order to allow for quantitative comparisons between the predictions of this analytical theory and experiment, we now consider a millimetre-scale low Re scallop-like swimmer actuated by on-board motors. The device, which we call the ‘macro-scallop’ (shown in Fig. 1c and Supplementary Figs 5 and 6), differs from the micro-scallop in the use of rotating hinges, and use of motors, which permit rapid and precisely controlled opening and closing of the swimmer’s shells. It has a characteristic length ~15 mm, an average linear velocity ~25 mm s−1 during the fast closing stroke and 6.25 mm s−1 during the slow opening stroke, so that the Re=0.017 for closing and Re=0.094 for opening, which are still low enough to neglect the inertia of the swimmer.

Tests of the analytical theory were conducted in shear thickening fluid in two regimes: (i) fixed opening speed ω o over a range of closing speeds ω c >ω o (Fig. 4b) and (ii) fixed closing speed ω c at different opening speeds ω o >ω c (Fig. 4c). When the closing speeds (solid squares) are larger than the opening speed (hollow triangle), the net displacement D/l is larger than 0 and the macro-scallop moves in the forward direction (hinge leading). On the other hand, when the opening speeds (hollow triangles) are larger than the closing speed (solid square), the swimmer moves in the opposite direction with a negative net displacement D/l<0 (hinge trailing). Both experimental results are in excellent agreement with the theory of equation (5) based on scaling arguments. It follows that a symmetric scallop can indeed move, provided time-reversal symmetry is broken by the asymmetric actuation speeds of opening and closing in a non-Newtonian fluid. Note that the two actuation gaits (that is, fast opening/slow closing versus fast closing/slow opening) result in a different displacement for the same value of p because of viscosity hysteresis (see Supplementary Note 3 and Supplementary Fig. 10).

Numerical model of propulsion

We also employ numerical simulations to study the propulsion mechanism of reciprocal motion in non-Newtonian fluids using the open-source Computational Fluid Dynamics (CFD) package FeatFlow (detailed method in the Supplementary Note 1). Figure 5a and Supplementary Movie 4 show the fluid velocity and viscosity fields at six frames through the complete 4-s propulsion cycle of the macro-scallop swimming in a shear thickening fluid. The net displacement in one cycle is clear from the difference in swimmer position between the start (0 s) and finish (4 s) of the cycle. The propulsion mechanism is illustrated by comparing the field maps at 0.3 and 2.4 s; these frames both correspond to the same α=115°; however, the higher closing angular velocity of the shells leads to a much higher velocity gradient (shear rate) of the fluid at 0.3 s, which results in larger viscosity between the two shells relative to that during the opening stroke (at 2.4 s). Therefore, the forward displacement during the fast closing half-cycle (upper panel in Fig. 5a) is larger than the backward displacement in the slow opening half-cycle (lower panel in Fig. 5a), which leads to the net displacement over one period. The displacement curves (Fig. 5b) of the simulation show excellent quantitative agreement with the experimental data in both the Newtonian and shear thickening cases. The simulation results demonstrate that the net propulsion is a result of the viscosity differences during the two half-cycles, which is caused by differential apparent fluid viscosity under asymmetric shearing conditions. Importantly, consistent results were achieved using only the simple shear thickening relationship between fluid viscosity and instantaneous shear rate. This suggests that the dominant factor leading to net propulsion in our experiment is differential viscosity rather than fluid energy storage mechanisms such as fluid elasticity or viscosity hysteresis (thixotropy/rheopecty).