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Shang Yik
Reigh†
*^{ab},
Lailai
Zhu†‡§
*^{c},
François
Gallaire
*^{c} and
Eric
Lauga
*^{a}
^{a}Department of Applied Mathematics and Theoretical Physics, Center for Mathematical Science, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK. E-mail: e.lauga@damtp.cam.ac.uk
^{b}Max-Plank-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany. E-mail: reigh@is.mpg.de
^{c}Laboratory of Fluid Mechanics and Instabilities, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland. E-mail: lailaizhu00@gmail.com; francois.gallaire@epfl.ch

Received
18th July 2016
, Accepted 9th March 2017

First published on 10th March 2017

Inspired by recent experiments using synthetic microswimmers to manipulate droplets, we investigate the low-Reynolds-number locomotion of a model swimmer (a spherical squirmer) encapsulated inside a droplet of a comparable size in another viscous fluid. Meditated solely by hydrodynamic interactions, the encaged swimmer is seen to be able to propel the droplet, and in some situations both remain in a stable co-swimming state. The problem is tackled using both an exact analytical theory and a numerical implementation based on a boundary element method, with a particular focus on the kinematics of the co-moving swimmer and the droplet in a concentric configuration, and we obtain excellent quantitative agreement between the two. The droplet always moves slower than a swimmer which uses purely tangential surface actuation but when it uses a particular combination of tangential and normal actuations, the squirmer and droplet are able to attain the same velocity and stay concentric for all times. We next employ numerical simulations to examine the stability of their concentric co-movement, and highlight several stability scenarios depending on the particular gait adopted by the swimmer. Furthermore, we show that the droplet reverses the nature of the far-field flow induced by the swimmer: a droplet cage turns a pusher swimmer into a puller, and vice versa. Our work sheds light on the potential development of droplets as self-contained carriers of both chemical content and self-propelled devices for controllable and precise drug deliveries.

In related work, droplets have been used to cage motile organisms such as the nematode Caenorhabditis elegans (C. elegans)^{4,5} in order to carry out developmental work. In these studies, the size of an encaged adult C. elegans is comparable to the droplet radius. Despite their mobility, the worms failed to propel their liquid cages, because they were immobilized. In the work of ref. 4, the droplet was tightly squeezed inside a capillary tube, forming a plug thus immobilized hydrodynamically by the lubrication film while in the work of ref. 5, the droplet was anchored mechanically by a microfluidic trap.

Motivated by these droplet-based encapsulations of motile organisms, we raise in this paper a simple question: is it possible for a microswimmer encaged in a droplet to propel its viscous cage and co-swim with it? One could envision setups of this type of interest to the drug delivery community using droplets as small self-contained units propelled and steered by their internal synthetic swimmers.

Recently, microrobots propelled by a magnetically-rotated helical appendage mimicking the flagella of bacteria such as Escherichia coli (E. coli) were fabricated,^{6,7} encapsulated and operated inside a water-in-oil droplet in microfluidic chips.^{8} In this case, the droplets were not mobile, presumably for two reasons: the swimmer was much smaller than the droplet and the droplet was large compared to the height of the micro-fluidic chips so that it was tightly squeezed and thus anchored hydrodynamically.^{4} Excitingly, the same group managed however to use their microrobots to push a droplet of a comparable size from the exterior when the droplet was unbounded or loosely bounded. From the theoretical point of view, the locomotion of a droplet driven by the internal release of a surfactant was previously considered.^{9}

In this paper, we conduct a combined theoretical and numerical study of a three-dimensional (3D) model microswimmer encapsulated in a droplet in free space. The size of the swimmer is of the same order as the radius of the droplet and we attempt to answer the following fundamental questions: will the droplet co-swim with the swimmer? What is the swimming velocity of the droplet compared to that of the swimmer? How are the kinematics and energetics of the microswimmer affected by the confinement due to the presence of the droplet? How stable is the co-movement of the concentric pair of the swimmer and droplet?

We then solve the steady Stokes equations for fluid phase 1 and 2,

∇p^{(i)} = μ^{(i)}∇^{2}v^{(i)}, ∇·v^{(i)} = 0, | (1) |

In the laboratory frame of reference, the fluid velocity components v^{(1)} = (v^{(1)}_{r},v^{(1)}_{θ}) on the swimmer surface, r = a, are given by

(2) |

On the droplet interface r = b, the normal velocities in the droplet frame vanish because the droplet does not deform. In addition, the tangential velocities and tangential stresses are continuous across the interface. These boundary conditions formulated in the laboratory frame are written as

(3) |

Finally, the total hydrodynamic forces exerted on both the swimmer and on the droplet interface are zero, which will be used to determine the values of both swimming velocities, U_{S} and U_{D}. For an unbounded squirmer in a single-phase fluid, the velocity U_{S} ≡ U_{D} is given by^{10,11}

(4) |

(5) |

p_{n}(r,ξ) = _{n}r^{n}P_{n}(ξ), ϕ_{n}(r,ξ) = _{n}r^{n}P_{n}(ξ), |

The radial and tangential velocity components v_{r} and v_{θ} are then obtained as

(6) |

Note that the solution for the flow in region 1 may contain all terms in the brackets of eqn (6) while those in region 2 only contain the last two terms due to the boundary condition at infinity.

Applying this framework to our case, we use eqn (6) for both the inner and outer fluid, solving for the unknown constants ^{(i)}_{n}, ^{(i)}_{n}, ^{(i)}_{−(n+1)} and ^{(i)}_{−(n+1)} (i = 1, 2) using the boundary conditions, eqn (2) and (3), together with the condition at infinity. Taking the n = 0, 1 terms in the series expansion of eqn (6) with the use of eqn (2) and (3) leads to the system for the inner fluid

(7) |

(8) |

Applying the force-free condition for the swimmer, we have

(9) |

(10) |

(11) |

(12) |

In order to complete the calculation and characterize the flow in both fluids, we need to calculate the values of the constants _{n}, _{n}, _{−(n+1)} and _{−(n+1)} for n ≥ 2 in the series expansion from eqn (6). The velocities inside and outside the droplet in the laboratory frame are then obtained to be

(13) |

We can finally calculate the power consumption of the squirmer, , which is equal to the rate of work done by the squirmer on the fluid,

(14) |

(15) |

(16) |

(17) |

The surfaces of the swimmer and droplet are discretized using zero-order flat quadrilateral and second-order curved triangular elements respectively. For the spherical swimmer, a six-patch structured mesh^{23,24} consisting of 600 (before mesh refinement) elements is constructed. The number of elements on the droplet interface is around 2500 (∼5000 discretized points). Gauss–Legendre quadrature is applied on the quadrilateral elements to compute nonsingular integrations; on triangular elements, we compute the integrations using a symmetric Gaussian quadrature rule.^{25} When x_{0} is on the surfaces or Ŝ, the surface integrals become singular and different desingularization strategies are chosen: on the droplet interface , the well-known integral identity for G is exploited and hence the first integral in eqn (16) becomes

(18) |

A crucial numerical difficulty arising from droplet/bubble simulations based on Lagrangian interface representation is to maintain the quality of the mesh of the interface. In order to guarantee the smoothness and orthogonality of the triangle mesh over a long time evolution, we implement a so-called ‘passive mesh stabilization’ scheme.^{28,29} At each time step, the scheme searches the optimal tangential field that is added to the normal velocity to update the Lagrangian points, minimizing a global kinetic-energy-like norm that quantifies the clustering and distortion of the mesh. This scheme significantly slows down mesh degradation. Its effectiveness was proved in the previous study on a squeezed pancake droplet in a microfluidic chip based on an accelerated boundary integral implementation.^{30}

In contrast to the infinite-surface tension assumed in the theory, a large but finite surface tension is adopted in the simulations and hence the numerical droplet is not strictly spherical but slightly deformable. The strength of the typical ratio of viscous stresses to surface tension forces is measured by the capillary number, Ca ≡ μ^{(2)}B_{1}/γ, and Ca = 0 corresponds to the theoretical limit of infinite surface tension. We vary Ca numerically from 10^{−3} to 10^{−2}, without detecting significant changes in the kinematics of the swimmer. We hence use Ca = 10^{−3} throughout our study, and are able to approximate well the Ca = 0 limit from our a posteriori comparison with the theory.

5.1.1 Velocity of the swimmer and droplet: theory.
For a tangential squirmer, the velocities U_{S} and U_{D} are given analytically by

where Δ is defined in eqn (12) and we use the swimming velocity of an unbounded squirmer as the reference scale, U_{0} = 2B_{1}/3. Both velocities are functions solely of the size ratio, χ, and the viscosity ratio, λ.

(19) |

We plot in Fig. 3a the dependence of the swimmer velocity on χ and λ. The velocity decreases monotonically with λ. When the outer and inner phase have matching viscosities (λ = 1), U_{S} is not affected by the presence of the droplet, and is thus equal to the unbounded velocity U_{0} for all values of χ. The squirmer swims faster than the unbounded one when the outer phase is less viscous than the inner (λ < 1), and swims slower in the opposite limit, λ > 1. When λ ≠ 1, the velocity U_{S} varies with the size ratio χ non-monotonically, reaching its maximum value for λ < 1 when the swimmer is tightly confined, χ ≈ 1.1–1.2, namely when the droplet is slightly larger than the swimmer. The result is similar when λ > 1 and the minimum is reached. For any viscosity ratios, U_{S} = U_{0} in the limit of χ = 1 and χ → ∞. The former corresponds to the situation when the droplet exactly encompasses the swimmer and the latter to when the droplet is much larger than the swimmer. In Fig. 3b, we further show that the velocity of the droplet decreases monotonically with χ, as well as with λ. The inset of Fig. 3b presents the ratio U_{D}/U_{S} of the droplet velocity over the swimmer velocity as a function of χ in a log–log form, this ratio decays as χ^{−3} for large χ. It is important to note that for any values of χ or λ the swimmer is always faster than the droplet, U_{S} > U_{D}. The concentric configuration is thus not a steady state if the swimmer only applies tangential forcing.

5.1.2 Comparisons between theory and simulations.
Here we consider the dynamics of a neutral swimmer (β = 0), a pusher with β = −5 and a puller with β = 5 encapsulated inside a same-viscosity droplet (λ = 1). For simplicity we further take B_{n} = 0 for n ≥ 3 and A_{n} = 0 for all n. Since the velocities U_{D} and U_{S} are independent of β, the ratio U_{D}/U_{S} only depends on the value of χ. This functional dependence is plotted in Fig. 4, showing an excellent agreement between the theory (green lines) and numerical data (red squares).

which indicates that the mirror symmetry about the equatorial plane θ = π/2 of the flow field of the pusher with −β is equivalent to the reversed flow field of the puller with β.

and that to the radial velocity of an unbounded pusher/puller is given by ref. 11 as

Their ratio is v^{(2)}_{r}|_{leading}/v_{r}|_{leading} = c_{2}/(Δ_{2}χ^{2}), which is negative for any size ratio χ > 1 hence rationalizing the velocity inversion.

Next in Fig. 5a–c, we plot the flow velocity field, v/B_{1}, in the laboratory frame for the pusher (a), neutral (b) and puller (c) swimmers respectively. The size ratio is χ = 2. Theoretical results are shown on the left panel and numerical data on the right. The numerical predictions show good agreement with the theoretical data in most of the flow domain except very close to the droplet interface where numerical errors arise from the nearly-singular integration.

For the neutral swimmer, note that the velocity field is not affected at all by the presence of the droplet. This is corroborated by the fact that neither the swimming velocity nor the power are impacted by the droplet, as implied by eqn (10) and (15). This results from the vanishing radial velocity in the droplet frame, such that the spherical droplet interface introduces no perturbation and hence does not influence the swimming dynamics.

For the pusher in a drop, similar to a pusher in free space, fluid is locally pushed away from the anterior (θ = 0) and posterior (θ = π) parts of the swimmer and comes to the lateral directions (θ = π/2). Due to the non-penetrating nature of the droplet interface, two counter-rotating toroidal vortices form inside. Outside the droplet the fluid is drawn towards its poles and expelled away on the equatorial plane. Interestingly the flow signature of a local pusher turns therefore into a puller in a far field. More quantitatively one can show that the velocity fields of a puller with β > 0 and a pusher with −β satisfy the relation

v_{r}|_{β}(r,π − θ) + v_{r}|_{−β}(r,θ) = 0, |

v_{θ}|_{β}(r,π − θ) − v_{θ}|_{−β}(r,θ) = 0, | (20) |

We next investigate the spatial variation of v(z)/B_{1} along the z axis in Fig. 5d–f, for the three swimmers. Here again, numerical data (empty red circles) agree very well with the theoretical predictions (solid blue line). The velocity magnitude, |v|/B_{1}, decays in the far-field from the swimmer center as r^{−2} for the pusher/puller and r^{−3} for the neutral swimmer. The velocity distribution v(z) over z for the pusher and that for the puller are symmetric about z = 0, as implied by eqn (20). For both swimmers, two stagnation points appear near the droplet interface r = b, one close to the frontal interface and the other close to the rear. They can be observed in Fig. 5.

It is worth emphasizing the result that the presence of a droplet reverses the direction of the far-field flow with respect to that of a pusher/puller in free space (Fig. 5a and c). This can be made more precise by an analysis of the theoretical predictions in eqn (13). With only B_{1} and B_{2} modes, the leading-order contribution to the radial velocity v^{(2)}_{r} in the outer phase is

(21) |

(22) |

5.1.3 Power consumption.
When the viscosities inside and outside the droplet are equal (λ = 1) and the swimmer uses tangential surface actuations alone, the power consumption based on eqn (15) is simplified to

where

of a similar form to that of an unbounded squirmer^{11}

(23) |

_{n} = 4χ^{2n+3} − (2n + 3)χ^{4} + (2n − 1), |

_{n} = 8χ^{2n+3} − (2n + 1)(2n + 3)χ^{4} + 2(2n − 1)(2n + 3)χ^{2} − (2n − 1)(2n + 1). | (24) |

Restricting then our attention to the simplest squirmer with B_{n} = 0 for n ≥ 3, the power becomes

(25) |

(26) |

Theoretical and numerical values of show excellent agreement, as shown in Fig. 6. The power of an encapsulated squirmer, , always exceeds that of an unbounded one, _{0}. From a practical standpoint, approximately doubles when the radius of the droplet is 50% larger than that of the swimmer. We further observe that is negatively correlated with χ, and the swimmer expends more energy due to a stronger confinement. The inset log–log plot indicates that scaled excessive power /_{0} − 1 decreases with the size ratio as χ^{−3}.

Fig. 6 Similar to Fig. 4, but for the power consumption of the squirmer, , scaled by the unbounded value, _{0}. In contrast to the velocities, depends also on modes |B_{n}| (n ≥ 2). Here |β| = |B_{2}/B_{1}| = 5 and B_{n} = 0 (n ≥ 3). The inset shows the χ^{−3} scaling of the nondimensional excess power /_{0} − 1. |

The results in eqn (10) and (11) imply that the swimming velocities U_{S} and U_{D} only depend on the first modes, A_{1} and B_{1}. We define α ≡ A_{1}/B_{1} to indicate the relative strength of the modes. By comparing eqn (10) and (11), we find that a particular value of α, denoted by α^{co}, allows obtaining equal velocities, namely

(27) |

(28) |

The influence of confinement χ and viscosity ratio λ on the resulting co-swimming speed is depicted in Fig. 7 by plotting the scaled co-moving speed U^{co}_{SD}/U_{0}, where U_{0} = 2B_{1}/3 is the velocity of an unbounded squirmer with pure tangential modes. Even for a small viscosity ratio (λ = 0.1), the co-moving velocity U^{co}_{SD} of the pair remains below 0.7U_{0}. Simulations have been performed to determine the values of α^{co} and U^{co}_{SD} for the λ = 1 case, and here again the numerical results show excellent agreement with the theory (not shown).

The relation between the mode strength α and the size ratio χ required to achieve concentric co-swimming is given by eqn (27) for arbitrary viscosity ratio λ. When λ is fixed, the particular value α^{co} ensuring co-swimming is easily chosen as a function of χ. Conversely, one may determine a particular size ratio χ^{co} as a function of α by solving the quintic equation. In the case of λ = 1, the required size ratio χ^{co} is simply given by

(29) |

Fig. 8 (top row) displays the time evolution of z_{off} for a swimmer which starts initially ahead (blue dot-dashed lines) or behind (red solid lines) using a tangential squirming of β = −5 (pusher, a), β = 0 (neutral, b) and β = 5 (puller, c). The physical characteristic time T = b/B_{1} is used to scale the time t. For the co-moving pusher as shown in Fig. 8a, the offset z_{off}(0) decays to zero regardless of its sign: the concentric co-moving state is recovered and remains stable. The influence of z_{off}(0) for the co-moving neutral swimmer is shown in Fig. 8b. The concentric co-movement is seen to be stable if the swimmer is initially ahead of the droplet, but it is unstable and yields a finite-time collision between the swimmer and the droplet interface, when the swimmer is initially behind. In contrast, for the puller illustrated in Fig. 8c, the swimmer eventually touches the rear interface indicating instability when z_{off}(0) < 0, while when z_{off}(0) > 0, the pair reaches an eccentric co-moving state that is asymptotically stable. In the latter case, the swimmer is close to the front droplet interface but separated by a thin lubrication film which acts to stabilize their co-movement via hydrodynamic interactions. The asymptotically steady thickness of the film is about 0.08a.

The stability properties of the co-moving state seen in Fig. 8 may be interpreted physically by examining the disturbance flow field induced by the swimmer. We plot in Fig. 8 (middle row) the disturbance flow patterns corresponding to the co-moving swimming gaits which consist of normal squirming α^{co} (dashed magenta lines) and tangential squirming β (solid black lines). The disturbance flow of the pusher and puller is characterized by a stresslet oriented in the swimming direction, decaying as 1/r^{2}; that of the neutral swimmer resembles a source dipole along the same direction, decaying faster as 1/r^{3}. The analysis of ref. 11 shows that the flow induced by the A_{1} mode squirming is equivalent to that by a neutral swimmer with B_{1} = A_{1}. The details of this disturbance flow dictate hydrodynamic interactions between the swimmer and its environment. As can be seen in Fig. 8d, a body located in front of or behind a pusher tends to be repelled by it while it will tend to be attracted for a puller. In contrast for a neutral swimmer with A_{1} > 0, ahead of the swimmer will be repulsive while it will tend to be attractive behind it.

We then link in Fig. 8 (bottom row) the disturbance flow of the swimmer and its relative movement with respect to the droplet, where solid/dashed circles denote the swimmer's initial/final location (the dot-dashed circles denote an intermediate position). As seen in Fig. 8g, for a co-moving pusher initially ahead of the droplet center, the repulsive flow in front of the swimmer, consisting of both repulsive flows from tangential squirming of β = −5 and normal squirming of α = 1.4, is stronger than its rear counterpart and brings the swimmer back to the center (stable). For the same swimmer but initially closer to the rear of the droplet as depicted in Fig. 8j, the rear flows dominate. While the flows induced by the two squirming modes are of opposite sign, the repulsive flow arising from tangential squirming is likely to overcome the attractive one of the normal squirming due to the faster-decaying and shorter-ranged disturbance flow of the latter (1/r^{3}vs. 1/r^{2}).

The behavior of the co-moving neutral swimmer can be understood along the same vein, as illustrated in Fig. 8h and k, and similarly for the puller when its is initially located behind that of the droplet (Fig. 8l). The only non-intuitive result is the asymptotically-stable eccentric location of the co-moving puller that is originally closer to the droplet front as illustrated in Fig. 8i. Initially, the gap between the swimmer and interface is relatively large, therefore the longer-ranged attractive flow from the tangential squirming will outweigh the shorter-ranged repulsive one from the normal squirming, and the swimmer will be attracted towards the interface. As the gap width decreases, the repulsive short-range flow becomes stronger, eventually dominating and preventing the swimmer from further approaching the interface. This explains, at least qualitatively, why hydrodynamic interactions lead in this situation to a stable eccentric configuration.

Additional simulations were then performed with 1/χ ranging from 0.3 to 0.7 and β ranging from −5 to 5. These simulations show that the stability properties of the co-moving state are independent of the size ratio χ and depend only on β. As shown in Fig. 9, when β ≤ 0, the concentric co-movement state is stable regardless of the sign of the initial offset z_{off}. When β ≥ 1, the eccentric co-moving state is stable if the swimmer is initially ahead (z_{off} > 0) while no stable co-moving configuration is observed otherwise (z_{off} < 0).

We first plot in Fig. 10a the trajectories of pullers in the laboratory frame with an initial offset (x_{off},z_{off}) = (0.2a,0.2a). Initially the system is not axisymmetric but after a slight rotation the swimmer settles in an axisymmetric configuration. Although the rotational motion is small, it occurs early in the dynamics, in particular before the swimmer closely approaches the droplet. After that, the system becomes equivalent to the axisymmetric situation considered in Fig. 8c and the swimmer reaches a stable state maintaining a thin gap with the droplet.

Next we show in Fig. 10b the trajectories of pushers with an initial offset (x_{off},z_{off}) = (0.2a,−0.2a). The swimmer slightly rotates but in this case does not align with the droplet axisymmetrically. Instead, due to the attractive flows in the lateral directions, the pusher approaches the droplet and eventually collides with it. Other cases with the initial offset (x_{off},z_{off}) = (0.2a,0.2a) or (0.2a,0) exhibit similar behaviors as in Fig. 10b with no stable configurations. Also pullers with the initial offset (x_{off},z_{off}) = (0.2a,−0.2a) or (0.2a,0) and neutral swimmers with (x_{off},z_{off}) = (0.2a,±0.2a) or (0.2a,0) do not settle a stable configuration. Additional simulations by changing the size ratio and stresslet strength lead to similar results.

The analytical solutions provide a useful physical picture of the instantaneous dynamics for the concentric configuration of the squirmer and droplet. For a squirmer using pure tangential surface actuations, although the concentric geometry is only transient, the theoretical results state that the swimmer is always faster than the droplet. When the normal surface velocities are incorporated on top of tangential modes, the squirmer and droplet are able to co-swim with the same velocity and thus to remain concentric.

When the swimmers are slightly displaced from the concentric position, we found that they would either return to the center (stable), deviate further and eventually touch the droplet interface (unstable), or reach an eccentric steady-state position (stable). Such final states depend on swimming gaits or relative locations of swimmers.

The ultimate goal of encaging swimmers is to help transport and deliver small chemical payloads, and thus a lot of future work lies ahead for swimmer–droplet complexes. Questions including swimming near complex boundaries or near walls, or non-axisymmetrically, will have to be tackled. Surfactants, which are commonly used in droplet-based microfluidics to prevent coalescence, could perhaps be used here to prevent collision between the swimmers and the interface, with interesting physical consequences. Finally, if heterogeneous fluid mixtures are to be transported in the droplet, it will be important to quantify their mixing and chemical fate as they move along with the swimmer.

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## Footnotes |

† These authors contributed equally to this work. |

‡ Current address: Linné Flow Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, SE-100 44 Stockholm, Sweden. |

§ Current address: Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ-08544, USA. |

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