Binary pusher-puller mixtures of active microswimmers and their collective behavior

TL;DR

This study investigates how mixtures of pusher and puller microswimmers exhibit collective behavior, revealing that orientational order can be tuned by adjusting the composition of these active particles.

Contribution

It provides the first detailed analysis of binary pusher-puller microswimmer mixtures, showing how their collective orientation depends on composition and hydrodynamic interactions.

Findings

Puller suspensions spontaneously develop collective order.

Ordered motion persists up to pusher-puller ratios of about 1:2.

Pushers surrounded by pullers show enhanced alignment.

Abstract

Microswimmers are active particles of microscopic size that self-propel by setting the surrounding fluid into motion. According to the kind of far-field fluid flow that they induce, they are classified into pushers and pullers. Many studies have explored similarities and differences between suspensions of either pushers or pullers, but the behavior of mixtures of the two is still to be investigated. Here, we rely on a minimal discrete microswimmer model, particle-resolved, including hydrodynamic interactions, to examine the orientational ordering in such binary pusher-puller mixtures. In agreement with existing literature, we find that our monodisperse suspensions of pushers do not show alignment, whereas those of solely pullers spontaneously develop ordered collective motion. By continuously varying the composition of the binary mixtures, starting from pure puller systems, we find that ordered collective motion is largely maintained up to pusher-puller composition ratios of about 1:2. Surprisingly, pushers when surrounded by a majority of pullers are more tightly aligned than indicated by the average overall orientational order in the system. Our study outlines how orientational order can be tuned in active microswimmer suspensions to a requested degree by doping with other species.

Equations

$$\displaystyle\left(\begin{array}[]{c}\textrm{d}\bm{R}\\ \textrm{d}\bm{\theta}\end{array}\right)$$

$$\displaystyle\left(\begin{array}[]{c}\bm{v}_{det}\\ \bm{\omega}_{det}\end{array}\right)=$$

$$\displaystyle\mathds{M}=\left(\begin{array}[]{c}\mathds{M}^{tt}\ \mathds{M}^{rt}\\ \mathds{M}^{tr}\ \mathds{M}^{rr}\end{array}\right)\ \mbox{and}\ \mathds{A}=\left(\begin{array}[]{c}\mathds{A}^{tt}\ \mathds{A}^{rt}\\ \mathds{A}^{tr}\ \mathds{A}^{rr}\end{array}\right).$$

$$\displaystyle\left\langle\left(\begin{array}[]{c}\textrm{d}\bm{R}\\ \textrm{d}\bm{\theta}\end{array}\right)\right\rangle$$

$$⟨\text{d}\mathbf{R}\text{d}\mathbf{R}⟩$$

$$⟨\text{d}\mathbf{R}\text{d}\mathbf{\theta }⟩$$

$$⟨\text{d}\mathbf{\theta }\text{d}\mathbf{\theta }⟩$$

$${\text{M}}_{ii,\alpha \beta }^{tt}={\mu }^t{\delta }_{\alpha \beta },{\text{M}}_{ii,\alpha \beta }^{rr}={\mu }^r{\delta }_{\alpha \beta },$$

$${\text{M}}_{ii,\alpha \beta }^{tr}=0,{\text{M}}_{ii,\alpha \beta }^{rt}=0,$$

$${\mu }^t=\frac{1}{6\pi \eta a},{\mu }^r=\frac{1}{8\pi \eta a^3},$$

$${\text{M}}_{ij,\alpha \beta }^{tt}(\mathbf{r})$$

$$\displaystyle\qquad+\frac{1}{2}{\left(\frac{a}{r}\right)}^{3}\left(\delta_{\alpha\beta}-3\frac{r_{\alpha}r_{\beta}}{r^{2}}\right)\Bigr{]},$$

$${\text{M}}_{ij,\alpha \beta }^{rt}(\mathbf{r})$$

$${\text{M}}_{ij,\alpha \beta }^{rr}(\mathbf{r})$$

$${\mathbf{f}}_i^{+}=|f|{\mathbf{u}}_i,{\mathbf{f}}_i^{-}=-|f|{\mathbf{u}}_i,$$

$${\mathbf{R}}_i^{+}={\mathbf{R}}_i+\frac{f}{|f|}\alpha L{\mathbf{u}}_i,{\mathbf{R}}_i^{-}={\mathbf{R}}_i-\frac{f}{|f|}(1-\alpha )L{\mathbf{u}}_i,$$

$${\mu }_{\alpha \beta }^{tt}(\mathbf{r})$$

$$+\frac{a^2}{24\pi \eta r^3}\left({\delta }_{\alpha \beta }-3\frac{r_{\alpha }{r}_{\beta }}{r^2}\right),$$

$${\mu }_{\alpha \beta }^{rt}(\mathbf{r})$$

$${\text{A}}_{ij,\alpha \beta }^{tt}=f\left[{\mu }_{\alpha \beta }^{tt}({\mathbf{r}}_{ij}^{+})-{\mu }_{\alpha \beta }^{tt}({\mathbf{r}}_{ij}^{-})\right],$$

$${\text{A}}_{ij,\alpha \beta }^{rt}=f\left[{\mu }_{\alpha \beta }^{rt}({\mathbf{r}}_{ij}^{+})-{\mu }_{\alpha \beta }^{rt}({\mathbf{r}}_{ij}^{-})\right],$$

$${\text{A}}_{ij,\alpha \beta }^{tr}={\text{A}}_{ij,\alpha \beta }^{rr}=0.$$

$$v_0=\frac{f{\mu }^{t}a}{2L}\left[3\frac{1-2\alpha }{\alpha (1-\alpha )}-\frac{a^2}{{\alpha }^3{L}^2}+\frac{a^2}{(1-\alpha {)}^3{L}^2}\right].$$

$$V^{st}(\mathbf{r})={ϵ}_0\exp \left(-\frac{{|\mathbf{r}|}^4}{{\sigma }^4}\right),$$

$$\text{Pe}=\frac{v_0\sigma }{{\mu }^t{k}_{B}T}$$

$$\varphi =\frac{N\pi {\sigma }^2}{4A},$$

$$P(t)=\left|\frac{1}{N}\sum _{i=1}^N{\mathbf{u}}_i(t)\right|,$$

$$P_{\infty }=\frac{1}{M}\sum _{n=1}^{M}P(t_n)$$

Full text

microswimmers; active suspensions; binary mixture; pushers and pullers; orientational order

I.  Introduction

The field of self-propelled particles and active matter has developed into a prime area to study the properties of non-equilibrium systems. Examples that have been addressed in detail are the statistics of the migrational behavior of individual self-propelled agents involving stochastic fluctuations Howse et al. (2007); ten Hagen et al. (2011); Romanczuk et al. (2012); Cates (2012); Bechinger et al. (2016) or of their collective motion, including their dynamical phase behavior Vicsek et al. (1995); Chaté et al. (2008); Fily and Marchetti (2012); Stenhammar et al. (2013); Mognetti et al. (2013); Romensky et al. (2014); Speck et al. (2014); Menzel (2015, 2016); Bechinger et al. (2016); Siebert et al. (2017).

The vast majority of studies on the collective behavior in this field concentrates on monodisperse systems. To some extent, mixtures of active and passive particles have been investigated. This concerns the collective behavior of mixtures of self-propelled and passive rods, in which, for instance, laning of the active rods in the passive background is observed McCandlish et al. (2012). Moreover, the separation into dense and more dilute regions in mixtures of active and passive spherical particles was addressed Takatori and Brady (2015); Kümmel et al. (2015); Stenhammar et al. (2015); Wysocki et al. (2016), as was the coarsening of crystal domains when systems of passive particles were doped by active agents van der Meer et al. (2016). A related topic is the study of mixtures of particles of different temperatures Grosberg and Joanny (2015); Weber et al. (2016); Smrek and Kremer (2017).

Investigations on mixtures of different types of active particles are exceptions. For instance, multi-species swarms of microorganisms were addressed Ben-Jacob et al. (2016), predator-prey scenarios were analyzed Mecholsky et al. (2010); Sengupta et al. (2011), mixtures of active rotors of opposite sense were considered Nguyen et al. (2014) including doping by passive particles Yeo et al. (2016), a stochastic description of mixtures of particles of different activity was outlined Wittmann et al. (2018), the alteration of the transition to polarly ordered collective motion with increasing polydispersity of the aligning self-propelled agents was investigated Guisandez et al. (2017), and the mutual support between different species in their orientational ordering and collective motion was studied in the context of imposed alignment interactions Menzel (2012). Mostly, these works concentrate on “dry” systems of self-propelled particles, not taking into account the role of a surrounding medium between the individual agents.

Active microswimmers represent one special type of such self-propelled particles Elgeti et al. (2015); Menzel (2015). These objects are suspended in a surrounding fluid. Examples are given by artificial colloidal Janus particles that propel by localized asymmetric concentration or temperature gradients induced in their environment Howse et al. (2007); Jiang et al. (2010); Buttinoni et al. (2012). Biological microswimmers are found in nature in the form of mechanically propelled bacteria or algae Polin et al. (2009); Min et al. (2009). Their mechanism of self-propulsion sets the surrounding fluid into motion. As a first coarse classification, one may distinguish between two different types of active microswimmers. If, to leading order, the induced flow field describes fluid pushed out along the propulsion axis and is dragged in from the sides, the swimmer is called a pusher Lauga and Powers (2009). In the opposite case of fluid being pulled inwards towards the swimmer along the propulsion axis and ejected to the sides, it is classified as a puller Lauga and Powers (2009). Via these induced fluid flows, hydrodynamic interactions Batchelor and Green (1972); Batchelor (1976); Dhont (1996); Reichert and Stark (2004); Lauga and Powers (2009); Malgaretti et al. (2012); Bechinger et al. (2016) arise between the individual swimmers that can affect the overall collective behavior Nash et al. (2010); Ezhilan et al. (2013); Zöttl and Stark (2014); Menzel et al. (2014); Matas-Navarro et al. (2014); Hennes et al. (2014); Blaschke et al. (2016); Menzel et al. (2016); Hoell et al. (2017). Due to the small dimensions of microswimmers, the relevant fluid flows are typically characterized by low Reynolds numbers Purcell (1977).

In the present work, we combine the two aspects described above. That is, we study mixtures of simplified active model microswimmers that hydrodynamically interact with each other through self-induced fluid flows in suspension. More precisely, we investigate binary mixtures of pusher- and puller-type swimmers. We concentrate on the microscopic swimmer-scale level, explicitly taking into account the hydrodynamic interactions on this scale. The swimmers are resolved individually in a discretized description using a minimal swimmer model Menzel et al. (2016); Hoell et al. (2017). On this basis, we evaluate the global orientational behavior.

The arising orientational ordering in crowds of microswimmers due to hydrodynamic interactions has been analyzed before for suspensions of either pushers or pullers separately Evans et al. (2011); Saintillan and Shelley (2011); Ezhilan et al. (2013); Alarcón and Pagonabarraga (2013). Here we study this effect in mixtures of the two types. In our computer simulations Frenkel and Smit (2001) we find, for instance, that pushers surrounded by a majority of pullers exhibit tighter orientational ordering than the surrounding pullers. Underlying details like possible intermittent or spatially localized orientational ordering of the swimmers can be analyzed accordingly in more detail in the future.

Below, we proceed in the following way. First, we describe the equations of motion for our suspended pusher and puller microswimmers. Afterwards, we analyze the collective behavior of binary mixtures of the two swimmer species for varying amounts of mixing ratio. In this context, also the impact of temperature and area fraction is addressed. Some conclusions are added in the end.

II.  Model

We consider a total of $N$ self-propelled microswimmers, $N_{A}$ of which are pushers and $N_{B}=N-N_{A}$ are pullers, with positions $\bm{R}_{i}=(R_{i}^{x},R_{i}^{y})$ and normalized orientational vector $\bm{u}_{i}=(u_{i}^{x},u_{i}^{y})$ ($i=1\dots N$). For an undisturbed swimmer, $\bm{u}_{i}$ coincides with its propulsion direction. All positions and orientations are confined to a two-dimensional plane generated by the directions $\widehat{x}$ and $\widehat{y}$. Still, three-dimensional hydrodynamic interactions apply. For brevity, we introduce the multi-dimensional vectors $\bm{R}$ and $\bm{u}$, the components of which are given by the positional and orientational coordinates, respectively, of all swimmers. Moreover, in the two-dimensional plane, the orientational vector $\bm{u}_{i}$ of each swimmer can be represented by its angle $\theta_{i}$ with the $x$-axis such that $\bm{u}_{i}=(\cos\theta_{i},\sin\theta_{i})$. In a similar fashion, we denote by $\bm{\theta}$, $\bm{v}$, and $\bm{\omega}$ the multi-dimensional vectors containing the angles, the linear, and the angular velocities of all swimmers. The microswimmers are confined to a two-dimensional periodic square box of area $A$.

In the low-Reynolds-number regime of active microswimmers, dissipation dominates, and the motion is governed by an overdamped, stochastic Langevin equation. It is to be integrated forward in time according to Stratonovich calculus Van Kampen (1992). Here we employ a simple Euler integration scheme at the cost of introducing a “spurious drift” term Ermak and McCammon (1978); Hennes et al. (2014).

By integrating the Langevin equation over a small time interval $\textrm{d}t$, we obtain the following expressions for the discrete increments $\textrm{d}\bm{R}$ and $\textrm{d}\bm{\theta}$ Makino and Doi (2004); De Corato et al. (2015)

[TABLE]

with the deterministic linear and angular velocities

[TABLE]

as well as the mobility and active mobility matrices

[TABLE]

The first term on the right-hand side of Eq. (14) determines the contributions of the conservative forces $\bm{F}$ and torques $\bm{T}$ to the deterministic velocities $\bm{v}_{det}$ and angular velocities $\bm{\omega}_{det}$. The second term includes the contribution of self-propulsion along each particle axis $\bm{u}_{i}$. The last term is the spurious drift Ermak and McCammon (1978), i.e., the divergence of the diffusion matrix $\mathds{D}=k_{B}T\mathds{M}$. In the case of our hydrodynamic interactions, see below, the drift term vanishes Ermak and McCammon (1978). Finally, the matrix $\mathds{H}$ is obtained by Cholesky decomposition Press et al. (2007) to satisfy $\mathds{H}\cdot\mathds{H}^{T}=2\mathds{D}$. The components of the vector $\bm{\xi}$ are uncorrelated Gaussian random numbers of zero mean and of variance unity.

Thus, we obtain the correct deterministic mean displacements

[TABLE]

and, in the absence of deterministic driving forces and torques, the correct mean squared displacements

[TABLE]

($i,j=1\dots N$) that reproduce the correct time evolution of the corresponding Smoluchowski equation Van Kampen (1992); Ermak and McCammon (1978).

III.  Details of the hydrodynamic and steric swimmer interactions

Hydrodynamic couplings between the swimmers are considered on the Rotne-Prager level Menzel et al. (2016); Dhont (1996); Hoell et al. (2017). Each swimmer consists of a spherical body of no-slip surface conditions for the surrounding fluid. Non-hydrodynamic forces and torques acting on such a swimmer body are transmitted to the surrounding fluid, set it into motion, and in this way affect the motion of all other swimmer bodies, see Eq. (14). Examples are conservative forces originating from steric repulsion or forces and torques resulting from external fields. For the $i$-th swimmer, the corresponding components of $\mathds{M}$ are

[TABLE]

where $i=1\dots N$, $\delta_{\alpha\beta}$ denotes the Kronecker delta, and $\alpha,\beta=x,y$ label the different Cartesian coordinates. Here, we introduced the translational and rotational mobility coefficients

[TABLE]

with $a$ denoting the hydrodynamic radius of the swimmer body and $\eta$ the viscosity of the surrounding fluid. The remaining components of $\mathds{M}$ are given by Dhont (1996)

[TABLE]

for $i\neq j$ ($i,j=1\dots N$), $\bm{r}=\bm{R}_{j}-\bm{R}_{i}$, $r=|\bm{r}|$, and $\epsilon_{\alpha\gamma\beta}$ the Levi-Civita tensor.

So far, we have described passive particles interacting hydrodynamically with each other. Now we include self-propulsion. For this purpose, two point-like force centers are rigidly connected to each swimmer body, see Fig. 1.

The two force centers are separated by a distance $L$ and exert on the fluid two oppositely oriented forces of equal magnitude along the symmetry axis of each swimmer. Since the force centers are located at different distances from the sphere, the resulting flow field leads to a transport of the swimmer body in the self-induced fluid flow Baskaran and Marchetti (2009); Menzel et al. (2016); Hoell et al. (2017). In principle, the rigid swimmer bodies affect the self-induced flow field Adhyapak and Jabbari-Farouji (2017). Here, we do not include this effect. That is, we only address the situation to lowest order in the length scale $a/L$ Hoell et al. (2017).

The two active forces of the $i$-th swimmer are parametrized as

[TABLE]

with their centers located at the positions

[TABLE]

respectively. Here, $\alpha\in\ ]a/L,\ 1-a/L[$ quantifies the asymmetry in the propulsion mechanism. The case of $\alpha=0.5$ recovers the symmetric “shaker” configuration Baskaran and Marchetti (2009). Moreover, following Eq. (34), the sign of $f$ determines whether the swimmer is a pusher or a puller, i.e., whether it pushes the fluid outward or pulls the fluid inward along the symmetry axis. To calculate the effect of the active forces of swimmer $j$ on the motion of swimmer $i$, we use the mobility matrices of components Menzel et al. (2016); Hoell et al. (2017)

[TABLE]

Using these expressions, we obtain the components of the active mobility matrix $\mathds{A}$ in Eq. (14) as Menzel et al. (2016); Hoell et al. (2017)

[TABLE]

$\bm{r}_{ij}^{\pm}=\bm{R}_{j}^{\pm}-\bm{R}_{i}$ is the vector connecting the $j$-th $\pm$ active force site to the center of particle $i$. The elements of $\mathds{A}^{rt}$ and $\mathds{A}^{rr}$ vanish because the propulsion forces are aligned with and are located on the symmetry axis of the swimmer and, thus, exert no active torque Hoell et al. (2017).

In the case of extremely diluted (i.e., non-interacting) swimmers, their self-propulsion speed $v_{0}$ follows as

[TABLE]

To position the force centers outside of the swimmer body, we require $\alpha\in\ ]a/L,\ 1-a/L[$.

Finally, our swimmers sterically interact with each other via the pair potential of the generalized exponential model of index 4 (GEM-4) Archer et al. (2014)

[TABLE]

where $\epsilon_{0}$ and $\sigma$ measure, respectively, strength and range of the steric repulsion. Although the steric interaction is soft, we indicate by $\sigma$ the size of the swimmers. In the following, for convenience, we use $v_{0}$ as the unit of measure of velocities. We set $a=\sigma/4\sqrt{3}$, $L=\sigma/2$, $\alpha=0.3$, and swimming forces $|f|\simeq 2.41f_{0}$. Distances, times, and forces are measured in multiples of $\sigma$, $t_{0}=\sigma/v_{0}$, and $f_{0}=v_{0}/\mu^{t}$, respectively.

Moreover, to compare our study with other theoretical investigations as well as with experimental results, we introduce the following dimensionless numbers. First, the Péclet number

[TABLE]

quantifies the strength of self-propulsion with respect to Brownian diffusion. Furthermore the area fraction is given by

[TABLE]

and the fraction of overall pushers by $\chi_{A}=N_{A}/N$. Thus, we indicate by $\chi_{A}=0$ and $\chi_{A}=1$ pure monodisperse systems of pullers and pushers, respectively. Finally, unless specified otherwise, all of the following results are obtained for simulations with a total of $N=1024$ particles. This, together with Eq. (43) and for a given area fraction $\phi$, sets the area of our periodic square box ${A}$.

IV.  Results

In our simulations, a suspension of active microswimmers can spontaneously develop collective motion into a common direction, see the example snapshot in Fig. 2. To describe the degree of such collective orientational ordering quantitatively, we define the global polar order parameter

[TABLE]

which is equal to $1$ in the case of complete polar alignment of all swimmers and [math] if the orientations do not show a net global polar order. In all our simulations, we start from an initial configuration of isotropically distributed orientations, implying $P(t=0)=0$. In agreement with the results obtained via a Lattice-Boltzmann scheme in Ref. Alarcón and Pagonabarraga (2013), and as shown in Fig. 3, suspensions of only pullers spontaneously develop a steady polar order, which here seems to saturate around $P\sim 0.8$.

In the case of only pushers, instead, we in our system do not observe the polar order parameter to spontaneously increase; moreover, if initialized by an aligned state $P(t=0)=1$, $P(t)$ quickly decays to almost zero.

The overall area fraction $\phi$ affects the dynamics of developing ordered collective motion. At low area fractions, e.g., $\phi=0.01$ in Fig. 3, the swimmers eventually reach an equally high amount of alignment as for $\phi\sim 0.03$, but reaching this value takes a noticeably longer time. The ordering process involves the induced flow fields acting on the other swimmers. Lower area fractions imply larger interparticle distances, weaker hydrodynamic interactions, and longer time needed for the swimmers to develop the collective behavior. At higher area fractions, instead, the time necessary to reach the steady state further decreases, see $\phi=0.1$ in Fig. 3. The attained orientational order, however, is lower, presumably, because for denser systems collisions between the swimmers become more relevant and affect the overall order

Mostly, the results that we report here were obtained at vanishing temperature $k_{B}T=0$, i.e., for infinite Péclet number $\textrm{Pe}=\infty$. In all considered cases, in which we examined the influence of finite temperature, we found it to lower the limiting value of $P(t)$ and increase its fluctuations, see Fig. 3.

We now move on to the central concern of our study, i.e., the collective behavior of pusher-puller mixtures. For this purpose, we vary the fraction of pushers $\chi_{A}=N_{A}/N$ from [math] to $1$. We sample the average polar order parameter in the stationary state, i.e.,

[TABLE]

with $M>3000$. Sampling is performed over a time interval $t\in[t_{1},t_{M}]$ in the long-time regime, for which a stationary state has been reached. The effect of increasing mixing ratio $\chi_{A}$ for different area fractions is shown in Fig. 4.

As mentioned above, high area fractions (see $\phi=0.2$ in Fig. 4) hinder the orientational ordering of the swimmers regardless of the swimmer species. On the contrary, at low to intermediate area fractions, collective motion spontaneously emerges for $\chi_{A}=0$ and is more or less preserved even upon introduction of relatively large amounts of pushers. Even up to a total of $\sim 30\%$ of pushers, see the curves for $\phi=0.03162$ and $\phi=0.05623$ in Fig. 4, $P_{\infty}$ remains as high as $0.4$, indicating still a significant degree of alignment. As $\chi_{A}$ further increases beyond this point, $P_{\infty}$ quickly decays to zero and the absence of polarly ordered collective motion in our pure pusher suspensions ($\chi_{A}=1$) is recovered.

Remarkably, when a large set of pullers is doped with a small amount of pushers, the latter are observed to align themselves along the collective direction of motion more tightly than the surrounding pullers. To illustrate this behavior, we show in Fig. 5 the polar order parameters for the two species separately, $P_{A}(t)$ for pushers and $P_{B}(t)$ for pullers.

In the stationary regime, we find $P_{A}(t)>P_{B}(t)$. As a consequence of this higher degree of alignment, the distributions of the pusher and puller swimming orientations (see the inset of Fig. 5) are centered on the same direction, but the pusher distribution is narrower. Even for $\chi_{A}$ as high as $0.3$, we found the polar order parameter $P_{A}(t)$ to be systematically higher than $P_{B}(t)$.

We remark that an increased orientational ordering and mutual support in collective motion by interactions between different species in a binary mixture of self-propelled particles has been previously reported in a “dry” system Menzel (2012), analyzing a variant of the Vicsek model Vicsek et al. (1995). In our case, such an effect of mutual support in orientational ordering would need to result from the presence of the hydrodynamic interactions due to the self-induced flow fields. In the inset of Fig. 4, we enlarge the curves for elevated polar order at low fractions of pushers $\chi_{A}$. Whether also the overall polar orientational order increases by the initial addition of pushers at low values of $\chi_{A}$ cannot be statistically resolved by our present means. This question needs further clarification in the future.

Partial answer to this question can be obtained by evaluating the different pair distribution functions Härtel et al. (2018) $g_{{XY}}(r,\varphi)$, with X and Y either A (pushers) or B (pullers). $g_{{XY}}(r,\varphi)$ represents the probability to find a swimmer of species Y around a swimmer of species X at distance $r$ and in direction $\varphi$ with respect to the swimmer orientation of X, see Fig. 6.

The amount of doping by pushers in Fig. 6 was $\chi_{A}=0.1$ and the overall total pair distribution function is virtually identical to $g_{{BB}}(r,\varphi)$.

The functions $g_{XY}(r,\varphi)$ feature a ring around the center, most likely due to the soft steric interaction introduced in Eq. (41), which was cut at $r\sim 2\sigma$. $g_{BB}(r,\varphi)$ shows a central maximum at the front, which is presumably related to collisions between the self-propelled swimmers. Interestingly, $g_{{AB}}(r,\varphi)$ features three distinct maxima: one at the front and two lateral ones at $\varphi\approx\pm 3\pi/4$. This may indicate a preferred arrangement for pushers when surrounded by a majority of pullers, namely, one or more pullers in front of each pusher and two behind at $\varphi\approx\pm 3\pi/4$. A similar triangular-like configuration is found for pullers when considering the probability to find a nearby pusher, see the function $g_{{BA}}(r,\varphi)$. Remarkably, there is no pronounced maximum at the front for $g_{{AA}}(r,\varphi)$ for the pusher-pusher spatial correlation. This may reflect the propulsion mechanism associated with the ejection of fluid along this axis. Such flows will counteract the mutual approach of two pushers along this axis.

V.  Conclusions

To summarize, we employed particle-resolved simulations to address the behavior of binary mixtures of self-propelled particles of different propulsion mechanisms (pushers and pullers). The effect of mutual support between the two species concerning the polar orientational ordering of their propulsion directions was analyzed. So far, this question of mutual inter-species coupling has been investigated within a variant of the famous Vicsek model for “dry” self-propelled particles Menzel (2012). Here, we have explicitly included the contribution of hydrodynamic interactions to the collective orientational behavior.

Via our minimal hydrodynamic microswimmer model, we can readily realize both pusher- and puller-like propulsion mechanisms. In agreement with previous studies Alarcón and Pagonabarraga (2013), we observe the spontaneous polar orientational ordering of pure monodisperse suspensions of pullers, while no polar ordering could be found in our monodisperse suspensions of pushers. Furthermore, we point out that increased area fraction or temperature counteract the polarly ordered collective motion. We remark that at very low area fractions the swimmers weakly interact and a common orientation could not be reached within observable times.

By doping a system of pullers even with significant amounts of pushers (up to $30\%$) the overall polar collective motion is largely preserved. Surprisingly, we find that the polar ordering of pushers in this case is higher than the overall polar orientational order in the rest of the system. Such an effect is possibly connected to some preferred spatial arrangement of the pushers relatively to the surrounding pullers. One hint to support the existence of such preferred arrangements can be inferred from the inter-species pair distribution functions. Further work is necessary in the future to determine the mechanism that drives the pushers into a more ordered state than the enclosing pullers. A way to shed further light onto the internal structure of such mixtures could result from more analytical investigations, based, for instance, on dynamical density functional theory Menzel et al. (2016); Hoell et al. (2017).

Several developments may follow on the basis of the present study. First, polydispersity in size of both pushers and pullers could be considered, as well as different continuous combinations of the parameters $\alpha$ and $f$ related to propulsion efficiency and activity. Moreover, an additional doping by passive particles should be addressed. The effect of using other microswimmer models Babel et al. (2016); Najafi and Golestanian (2004) could likewise be assessed in subsequent investigations. In this way, a large set of parameters is to be explored to devise mixtures of different active and passive particles to adjust at will the structural and dynamic properties of the system. Achieving tunable degrees of alignment for specific subsets of active particles could, for instance, allow to modify the transport properties or selectively separate the different species. Via improved particle-resolved simulations, we hope to gain a better understanding and to develop elaborated predictions on the dynamic and structural behavior of real active systems.

Acknowledgements.

This paper is dedicated to Daan Frenkel on the occasion of his 70th birthday. The authors thank M. Puljiz and C. Hoell for helpful discussions and the Deutsche Forschungsgemeinschaft for support of this work through the SPP 1726, grant nos. LO 418/17 and ME 3571/2.