Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

DOI: 10.1039/C7SM01180F
(Paper)
Soft Matter, 2017, Advance Article

Jan-Timm Kuhr*,
Johannes Blaschke,
Felix Rühle and
Holger Stark*

Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany. E-mail: jan-timm.kuhr@tu-berlin.de; holger.stark@tu-berlin.de

Received
14th June 2017
, Accepted 20th August 2017

First published on 2nd October 2017

Active particles, which interact hydrodynamically, display a remarkable variety of emergent collective phenomena. We use squirmers to model spherical microswimmers and explore the collective behavior of thousands of them under the influence of strong gravity using the method of multi-particle collision dynamics for simulating fluid flow. The sedimentation profile depends on the ratio of swimming to sedimentation velocity as well as on the squirmer type. It shows closely packed squirmer layers at the bottom and a highly dynamic region with exponential density dependence towards the top. The mean vertical orientation of the squirmers strongly depends on height. For swimming velocities larger than the sedimentation velocity, squirmers show strong convection in the exponential region. We quantify the strength of convection and the extent of convection cells by the vertical current density and its current dipole, which are large for neutral squirmers as well as for weak pushers and pullers.

For higher densities of microswimmers hydrodynamic interactions become important and collective behavior emerges.^{2,14,23–29} This includes motility-induced phase separation,^{30–34} swarming,^{16,35} and bioconvection,^{36–39} to name but a few phenomena. Furthermore, in real settings interactions with bounding surfaces are important,^{31,34,40–42} especially if the swimmers are not perfectly buoyant.^{43}

In this article we consider systems with thousands of microswimmers under the influence of gravity. We simulate their full hydrodynamic flow fields using the method of multi-particle collision dynamics (MPCD)^{44,45} in order to include hydrodynamic interactions between swimmers as well as between swimmers and bounding walls. As a model microswimmer we use the squirmer,^{46–49} which is versatile enough to model the relevant swimmer types including pushers, pullers, and neutral swimmers.

In the following we concentrate on the case, where passive colloids would just strongly sediment to the bottom. We show how density or sedimentation profiles depend on the ratio of active to sedimentation velocity as well as on the squirmer type. During collective sedimentation squirmers develop densely packed layers in the bottom region of the simulation cell. In contrast, we observe an exponential density profile in the upper region, where squirmers form a more dilute active suspension. The mean vertical orientation of the squirmers depends strongly on their vertical position as well as on their swimmer type. For swimming velocities larger than the sedimentation velocity, we find that hydrodynamic interactions organize squirmers into convection cells. Importantly, both the strength of convection and the extension of the convection cells depend on the squirmer type.

The article is organized as follows. We first introduce the squirmer as our model microswimmer and then shortly address the simulation method of MPCD along with parameter settings and some details of our analysis in Section 2. In Section 3 we present our results of collectively sedimenting squirmers and analyze especially sedimentation and mean vertical orientation (in Section 3.1) and convection (in Section 3.2). Finally, in Section 4 we summarize our findings and conclude.

v_{s}(r_{s}) = B_{1}(1 + βê·_{s})[(ê·_{s})_{s} − ê],
| (1) |

In our MPCD simulations the fluid is modeled by ca. 2 × 10^{7} point particles of mass m_{0}. Their positions r_{i} are updated in a streaming step using their velocities v_{i}: r_{i}(t + Δt) = r_{i}(t) + v_{i}Δt. In the subsequent collision step fluid particles within cubic cells of linear extension a_{0} exchange momentum according to the MPC-AT + a rule.^{60} This conserves linear and angular momentum and also thermalizes velocities to temperature T. Further details of our implementation are described in ref. 31 and 34. We use here the parallelized version of ref. 34, which is suited to simulate large systems with many swimmers.

In the present work, we consider squirmers under gravity. So we have to add an acceleration term aΔt^{2}/2 to the squirmers’ position in the streaming step, where the acceleration a is due to the gravitational force − mge_{z} along the vertical with m the buoyant mass of a squirmer and g the gravitational acceleration. Since gravity does not induce a noticeable density change of the fluid on the micron length scale, we do not apply a gravitational acceleration to the fluid particles. If a fluid particle encounters a bounding wall or a squirmer, the particle's position and velocity are updated according to the “bounce-back rule”,^{62} which implements either the no-slip boundary condition or the surface flow field of eqn (1), respectively. During the streaming step momentum is transferred from the fluid particles to the squirmers, the velocities of which are updated by a molecular dynamics step. This includes steric interactions among squirmers and with bounding walls.

MPCD reliably reproduces the analytical results, including the flow field around passive colloids,^{59} the friction coefficient of a particle approaching a plane wall,^{63} the active velocity of squirmers,^{49} as well as the torque acting on them close to walls, where lubrication theory has to be applied.^{41} It also simulates correctly segregation and velocity oscillations in dense colloidal suspensions under Poiseuille flow.^{64,65} MPCD resolves flow fields on time and length scales large compared to the duration of the streaming step Δt and the mean free path of the fluid particles, respectively. Therefore, using a squirmer radius of R = 4a_{0}, we expect to resolve hydrodynamic flow fields even when squirmers are close to each other.

In the following, an important parameter will be the ratio of active to bulk sedimentation velocity,

(2) |

The Péclet number Pe = v_{0}R/D, where D = k_{B}T/(6πηR) is the translational diffusion coefficient, has the value Pe = 323 in all simulations, thus thermal translational motion is negligible. Furthermore, in all our simulations we have a Reynolds number of Re = v_{0}Rn_{fl}/η = 0.17, where n_{fl} = 10 is the average number of fluid particles per collision cell. This implies Stokesian hydrodynamics where inertia can be neglected. Finally, with the thermal rotational diffusion coefficient D_{r} = k_{B}T/(8πηR^{3}), we introduce the persistence number Pe_{r} = v_{0}/(D_{r}R). It measures the distance in units of particle radius R, where the squirmer moves persistently in one direction, before rotational diffusion changes its orientation. For all our simulations we have Pe_{r} = 430. Thus, without gravity a single isolated squirmer would swim across the vertical extent L_{z} of the simulation cell in an almost straight line.

As explained in the results section, we determine the sedimentation or density profile ρ(z) of the squirmers, from which we identify some layering at the bottom wall of the simulation box, which is followed by a transitional and then an exponential region. In the latter we determine the sedimentation length δ using an exponential fit. The difficulty is to specify a range of heights [z_{b}, z_{t}], in which the exponential fit is performed. We have developed heuristic but robust criteria to identify this range. They ensure that neither layering at the bottom wall nor accumulation of squirmers at the top wall influences the fit values for δ. As a first constraint we demand that z_{t} is at least a distance of 10R away from the top wall. For z_{b} we require twice the height, which the squirmers would assume if they were all perfectly stacked in a hexagonal close packing. Within this first specification for the range [z_{b}, z_{t}] we then determine the final z_{t} as the height, where the density ρ is minimal. For z_{b} we take the smallest height z, where ρ(z) falls below 8% of the hexagonal-close-packed density. The value of 8% is a purely empirical value.

We use the data in the range [z_{b}, z_{t}] to obtain the sedimentation length δ from exponential fitting. In order to also estimate its error, we need to generate several estimates for δ from our data. Therefore, we split up the simulation time after reaching steady state into ten intervals. For each interval we perform exponential fits in four different ranges: (i) [z_{b}, z_{t}], (ii) [z_{b} + 0.1Δz, z_{t}], (iii) [z_{b}, z_{t} − 0.1Δz], and (iv) [z_{b} + 0.1Δz, z_{t} − 0.1Δz], where Δz ≔ z_{t} − z_{b}. We use the modified ranges as an additional measure to ensure that we are in the exponential regime. As an estimate for δ, we then take the mean of all 40 fits, while the corresponding standard deviation specifies the error.

Fig. 2 Semi-logarithmic plot of the sedimentation profile ρ(z) for the system in Fig. 1 (α = 1.0 and β = 0). Different regions are indicated. The green line is an exponential fit to extract the sedimentation length δ. |

In Fig. 3(a) we present sedimentation profiles for a larger swimming speed, α = 1.5, and different swimmer types β to explore the influence of pushers and pullers. At larger α, all profiles show an exponential regime with a larger sedimentation length compared to α = 1. Fig. 4 shows a parametric study of δ in units of squirmer radius R plotted versus β and for three values of α. Clearly, δ decreases with α and is only a fraction of R for α = 0.3, when the activity is too small for particles to swim upwards. However, already the ratio α = 1.5 is sufficient to have sedimentation lengths δ ≈ 10R. To address the robustness of our results, for the case β = 0 and α = 1.5, we reduced the volume density ϕ by decreasing the number of squirmers N to a value where layer formation does not occur, while keeping the height L_{z} of the simulation box fixed. This does not influence the sedimentation length δ significantly, as long as the squirmer density at the bottom of the box is similar to that of the top layers. For β = 0 we also reduced both N and L_{z} by a factor of two, which keeps ϕ constant. We find that the sedimentation length is reduced by about 30% and 40% for α = 1 and α = 1.5, respectively. This is not surprising, since hydrodynamic interactions with the top wall, which were not relevant before, push squirmers downwards and also turn them away from the wall,^{58,69} which makes them swim downwards.

In Fig. 4 we realize that for weak pushers (β = −1) the sedimentation length is the largest and decreases for stronger pushers and also pullers. It has been reported in the literature that the interaction between parallel squirmers grows with |β|.^{48,70} We speculate that the collective interactions of many squirmers will therefore strongly randomize swimming directions and hinder squirmers with large |β| from reaching larger heights, as reflected in the sedimentation length. The larger sedimentation length for weak pushers as compared to neutral squirmers is, however, unexpected. A possible explanation comes from the shape of flow fields for β ≠ 0. Pushers, in their center-of-mass frame, have a stagnation point with vortical flow in front of them, while pullers have it at their back.^{23} Since squirmers are typically pointing up in the exponential regime (see below) and since their density ρ(z) decreases with height z, pullers reorient more nearby squirmers compared to pushers, which decreases δ. Interestingly, the trend of δ for varying β is inverted for small α with the minimum being at β ≈ 1. The reason for the inversion is not clear, but since δ < R, we assume that interactions with the densely packed squirmer layers are relevant.

Fig. 4 Sedimentation length δ in units of R as a function of squirmer parameter β for different ratios of swimming to sedimentation velocities, α. Inset: Semi-logarithmic plot. |

In Fig. 3(a) we also observe how the layer structure in the lower part of the system is influenced by β. For β = 0 and β = 1 the minima between successive layers are less pronounced, which implies less order. In contrast, especially for β = −1 and β = −2 layering is more pronounced. We ascribe this difference to the hydrodynamic interactions between neighboring squirmers, which depend on β.

Finally, in Fig. 5 we show the mean orientation of squirmers as a function of height z for the system illustrated in Fig. 1. The neutral squirmers in the bottom layer at z = 0 have an upright orientation due to hydrodynamic interactions with the bounding wall.^{31,41,42,48} The mean orientation then decreases to zero (see also Fig. 1) and drops to a negative value at the rim of the layering. This is simply because squirmers from above swim into the dense squirmer region and need some time to reorient and swim away. In the transitional region the orientation changes again rapidly to nearly upright and shows only small variations in the exponential regime.

Fig. 5 Mean vertical orientation of squirmers, 〈cosθ〉, as a function of height z for the system in Fig. 1 (α = 1.0 and β = 0). |

The occurrence of polar order in the sedimentation profile of dilute suspensions has been predicted by theory and already occurs without any interactions (i.e. for dilute suspensions of active Brownian particles) just for kinetic reasons.^{2,21} Hydrodynamic interactions between squirmers obviously do not destroy the polar order. In our parametric study of Fig. 3(c), this is also confirmed for other squirmer types β. Differences occur in the layering and in the transitional region. For pullers the upright orientation in the bottom layer decreases for larger β, as expected by hydrodynamic interactions with the bottom wall in lubrication theory.^{41,42,48} In the adjacent layers hardly any polar order is visible in contrast to neutral squirmers. Weak pushers (β = −1 and −2) show a similar but weaker trend compared to neutral squirmers. For strong pushers (β = −5), again, there is hardly any polar order in the layering.

Fig. 6 Distribution of vertical squirmer velocity (blue) and Gaussian fit (black) for α = 1.5 and β = 0 in the exponential regime. |

To quantify convection, we take the vertical squirmer current density and average it along the vertical in the exponential regime:

j_{z}(x) = 〈ρ〉_{∥}(x)〈v_{z}〉_{∥}(x)
| (3) |

Fig. 7 Squirmer current density, , averaged over the exponential regime and time, color-coded in the xy plane for α = 1.5 and β = 0. |

To further characterize the squirmer density current j_{z}(x), we calculate its zeroth and first moment. The zeroth moment is the volume average of the vertical current density:

(4) |

Fig. 8 Volume average of the squirmer current density, 〈j_{z}〉, and its magnitude, 〈|j_{z}|〉, plotted versus time for α = 1.5 and β = 0. |

Fig. 9 Time and volume average of the magnitude of the current density, , as a function of squirmer parameter β for different rescaled swimming speeds α. |

We call the first moment of j_{z}(x) current dipole,

(5) |

Fig. 10 Magnitude j_{D} and orientation angle φ_{D} of the current dipole plotted versus time for α = 1.5 and β = 0. |

Finally, in Fig. 11 we plot the time average versus squirmer type β for different α. Convection is largest for large α and neutral squirmers. The current dipole we define in eqn (5) can be interpreted in analogy with a charge dipole in electrostatics. Its magnitude changes when either the current density (the separated “charges”) changes in magnitude or when the distance between regions of positive vs. negative vertical speed is altered (the distance of “charge separation”). In accordance with Fig. 3(c) we find that for α = 1.5 decreases when the magnitude of the current density j_{z}(x) decreases (weak pusher, β < 0) or the extension of the convection cell becomes smaller (weak puller, β > 0). Strong pushers/pullers with |β| = 5 only show weak convection. For small rescaled swimming speed α ≤ 1 convection is generally small.

Fig. 11 The time average of the magnitude of the current dipole, , as a function of squirmer parameter β for different rescaled swimming speeds α. |

Furthermore, sedimenting squirmers create strong convective currents due to their hydrodynamic interactions. The spatial extension and the strength of convection are again determined by the rescaled swimming speed and by the squirmer type. Neutral squirmers as well as weak pushers and pullers show the strongest convectional flow. In particular, for swimming speeds larger than the sedimentation velocity pronounced convection cells occur, which extend over the whole simulation box. Finally, as another signature of the highly dynamic sedimentation profile in the exponential region, we identified strong temporal fluctuations of the convective currents.

What we could not resolve in our current simulations due to limited computational resources is the question of what determines the lateral extent of the convection cells. Is there an intrinsic length scale, which sets it? For this we would need to increase the simulation cell in the lateral directions. In future work, we plan to include bottom heaviness of the squirmers in our simulations and study in detail the inversion of the sedimentation profile, which was discussed in ref. 22 for very dilute swimmer suspensions. Preliminary results in denser systems show that instabilities occur due to hydrodynamic interactions.^{71} In a harmonic trapping potential a similar instability leads to the formation of fluid pumps by breaking the rotational symmetry of the trap.^{12,14} In the present case we expect convectional patterns to occur. Thereby, we will connect to the phenomenon of bioconvection.^{36–39}

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## Footnote |

† Electronic supplementary information (ESI) available: 2 video files. See DOI: 10.1039/c7sm01180f |

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