Emergent behavior in active colloids

To date a variety of different active colloidal systems have been realized experimentally. Paxton et al [6] and Fournier-Bidoz et al [7] were the first to create micrometer-sized active bimetallic rods. They propel themselves by consuming fuel such as ${{\text{H}}_{2}}{{\text{O}}_{2}}$ at one end of the rod due to the different surface chemistry³. Here, a local ion gradient creates an electric field, which initiates fluid flow close to the rod and hence electrophoretic self-propulsion⁴. Spherical active colloids [48, 71] and sphere dimers [80] are able to swim by self-diffusiophoresis, where a gradient in chemical species close to the surface generates a pressure gradient along the surface and thereby fluid flow [51].

When heated with laser light, metal-coated Janus colloids are able to swim via self-thermophoresis [49, 84–86]. They establish a non-uniform temperature field around themselves, which again induces fluid flow near the surface of the colloid and thus leads to self-propulsion.

Water droplets can become active and move in oil by consuming bromine fuel supplied inside the droplet. The bromination of mono-olein surfactants at the water–oil interface induces gradients in surface tension and thus Marangoni stresses, which initiate surface flow and thereby self-propulsion [8, 55, 61, 82]. Even pure water droplets are able to move by spontaneous symmetry breaking of the surfactant concentration at the interface [81], as do liquid-crystal droplets [61]. In both cases, surfactant micelles are crucial to initiate the spontaneous symmetry breaking [57].

When swimmers depend on fuel, they stop moving after it has been consumed. Instead of supplying fuel in the solution, permanent illumination with laser light as a power source can sustain self-propulsion for an arbitrarily long time. For example, in [50, 87] the authors suspend Janus particles with a golden cap in a water–lutidine mixture. Heating the cap with laser light demixes the binary liquid and initiates self-diffusiophoretic motion. Recent theoretical calculations have been performed in [88, 89].

Finally, the release of bubbles behind an active colloid [90, 91] or from cone-shaped microtubes [92, 93] can also lead to self-propulsion. Moreover, self-propelled colloids that are powered by ultrasound have been designed [94].

Currently the number of experimentally realized colloidal swimmers is increasing fast. We refer to recent reviews, which also discuss more details of the different mechanisms for generating self-locomotion [4, 26–33].

We discuss in more detail the concept of self-phoresis, by which solid active colloids move forward, and also comment on the Marangoni propulsion of active emulsion droplets.

The gradient of an external field $X\left(\mathbf{r}\right)$ induces phoretic transport of a passive colloid of radius R, since the field determines the interaction between the colloidal surface and the surrounding fluid [22]. Within a thin layer of thickness $\delta \ll R$ the field gradient $\nabla X\left(\mathbf{r}\right)$ induces a tangential near-surface flow, which increases from zero in the radial direction and saturates at an effective radius ${{R}_{\text{e}}}=R+\delta$ to a constant value (see figure 2(b)). Typically, the external field $X\left(\mathbf{r}\right)$ is an electric potential $\phi \left(\mathbf{r}\right)$, a chemical concentration field $c\left(\mathbf{r}\right)$, or a temperature field $T\left(\mathbf{r}\right)$, and the respective colloidal transport mechanisms are called electrophoresis, diffusiophoresis, and thermophoresis.

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In contrast, active colloids create the local field gradient $\nabla X\left(\mathbf{r}\right)$ and the resulting near-surface flow by themselves. In the simplest case, half-coated Janus spheres are used, as sketched in figure 2(a). The catalytic cap (red) catalyzes a chemical reaction of the fuel towards a chemical product (shown as green dots), which diffuses around. In steady state a non-uniform concentration profile around the particle is established [51–53, 95–102]. Since the Janus colloid produces the diffusing chemical by itself, the propulsion mechanism is called self-diffusiophoresis. Recent works showed how details of the surface coating and ionic effects determine the swimming speed and efficiency of catalytically driven active colloids [70, 100, 103–107].

Janus colloids coated with metals such as gold are able to move by heating the metal cap [49, 69, 84–86]. The resulting local temperature gradient induces an effective slip velocity, which is known as the Soret effect, and the active colloid moves by self-thermophoresis. Furthermore, bimetallic nano- and microrods move by self-electrophoresis [108–110]. Here an ionic current near the surface drags fluid with it and thereby induces an effective slip velocity at the surface.

All the systems mentioned above create a tangential slip velocity ${{\mathbf{v}}_{\text{s}}}$ close to the particle surface and proportional to the field gradient ${{\nabla}_{\text{s}}}X\left(\mathbf{r}\right)$ along the surface [22],

$$\begin{eqnarray}&&{{\mathbf{v}}_{\text{s}}}\left({{\mathbf{r}}_{\text{s}}}\right)=-\kappa \left({{\mathbf{r}}_{\text{s}}}\right){{\nabla}_{\text{s}}}X\left(\mathbf{r}\right){{|}_{|\mathbf{r}|={{R}_{\text{e}}}}}\,.\end{eqnarray} \tag{ 14 }$$

The slip-velocity coefficient κ depends on the specific phoretic mechanism and material properties of the particle–fluid interface, which vary with the location ${{\mathbf{r}}_{\text{s}}}$. For spherical active colloids with axisymmetric surface coating, ${{\mathbf{v}}_{\text{s}}}(\theta )$ depends on the polar angle θ of the position vector ${{\mathbf{r}}_{\text{s}}}$ relative to the orientation vector $\mathbf{e}$ (see figure 2(a)).

To determine the swimming speed and the flow field around the active colloid, one takes the slip velocity at the effective radius ${{R}_{\text{e}}}=R+\delta$. Concretely, one solves the homogeneous Stokes equations for $r>{{R}_{\text{e}}}$ with the boundary condition $\mathbf{v}\left({{\mathbf{r}}_{\text{s}}}\right)={{\mathbf{v}}_{\text{s}}}+{{v}_{0}}\mathbf{e}+\mathbf{\Omega }\times {{\mathbf{r}}_{\text{s}}}$, where v₀ is the constant swimming speed and $\mathbf{\Omega }$ is the angular velocity of the active colloid. Since phoretic transport is force and torque free [22], both the hydrodynamic force and torque of equations (3) and (4) should be zero when evaluated along the surface with radius R_e. This gives the translational and angular velocity of a spherical (self-)phoretic colloid [22],

$$\begin{eqnarray}\begin{array}{*{35}{l}} \mathbf{V} & ={{v}_{0}}\mathbf{e}=-\langle {{\mathbf{v}}_{\text{s}}}\rangle \end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \mathbf{\Omega } & =-\frac{3}{2R}\langle {{\mathbf{v}}_{\text{s}}}\times \mathbf{n}\rangle \,, \end{array}\end{eqnarray} \tag{ 16 }$$

where $\langle \cdots \rangle$ denotes the average taken over the surface with normal $\mathbf{n}$. Note that, for axisymmetric surface velocity profiles, $\mathbf{\Omega }=\mathbf{0}$ and the active colloid moves on a straight line. This is, for example, true for half-coated Janus spheres. Nevertheless, rotational Brownian motion reorients the swimmer and it performs a persistent random walk, as we will discuss in section 3.1.

To gain a better insight into self-phoretic motion on the microscopic level, explicit hydrodynamic mesoscale simulation techniques have been used to study self-diffusiophoretic and self-thermophoretic motion. In particular, reactive multi-particle collision dynamics (R-MPCD) is a method to explicitly simulate the full hydrodynamic flow fields and chemical fields around a swimmer. It also includes chemical reactions between solutes and thermal noise [111]. The method has been used to simulate the self-phoretic motion of spherical Janus colloids [112, 113] and self-propelled sphere dimers [79, 80, 113–121]. The simulated flow field of a sphere-dimer swimmer agrees with that of a force dipole [113, 122], which has recently been confirmed by an analytic calculation [122]. Dissipative particle dynamics was used to determine the effect of particle shape on active motion [123], and to calculate the flow fields initiated by self-propelled Janus colloids for different fluid–colloid interactions [124].

In active emulsion droplets spontaneous symmetry breaking generates gradients in the density of surfactant molecules at the droplet–fluid interface and thus gradients in surface tension σ. They drive Marangoni flow at the interface [8, 55–57, 81]. The slip velocity field does not simply follow from an equivalent of equation (14), which links slip velocity to ${{\nabla}_{\text{s}}}\sigma \left({{\mathbf{r}}_{\text{s}}}\right)$. Instead, one has to solve the Stokes equations inside and outside the droplets and match the difference in tangential viscous stresses by ${{\nabla}_{\text{s}}}\sigma \left({{\mathbf{r}}_{\text{s}}}\right)$ [57]. So, in contrast to the case of colloidal particles, fluid flow also evolves inside the droplet [8, 58, 60, 125].

We have seen that the active motion of self-phoretic colloids is mainly determined by the surface velocity field ${{\mathbf{v}}_{\text{s}}}$, independent of the underlying physical or chemical mechanisms to realize ${{\mathbf{v}}_{\text{s}}}$. In the simplest case one assumes a prescribed surface velocity field without taking the underlying mechanism into account. Such model swimmers are useful to study how hydrodynamic flow fields influence the (collective) dynamics of microswimmers.

One prominant example is the so-called squirmer [126, 127]. It was originally proposed to model ciliated microorganisms such as Paramecium and Opalina. Nowadays, the squirmer is frequently used as a simple model microswimmer. In its most common form, the squirmer is a solid spherical particle of radius R with a prescribed tangential surface velocity field [128], which is expanded into an appropriate set of basis functions involving spherical harmonics [57, 129]. In the important case of an axisymmetric velocity field with only polar velocity component ${{v}_{\theta}}$, the expansion uses derivatives of Legendre polynomials ${{P}_{n}}(\cos \theta )$ [126–128],

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{v}_{\theta}} & =\underset{n=1}{\sum}\,{{B}_{n}}\frac{2}{n(n+1)}\sin \theta \frac{\text{d}{{P}_{n}}(\cos \theta )}{\text{d}\cos \theta} \end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} ={{B}_{1}}\sin \theta (1+\beta \cos \theta )+\ldots \end{array}\end{eqnarray} \tag{ 18 }$$

where the coefficients B_n characterize the surface velocity modes and ${{\mathbf{v}}_{\text{s}}}={{v}_{\theta}}\widehat{\boldsymbol{\theta }}$ is directed along the longitudinals (see figure 3(a)). We also introduced the dimensionless squirmer parameter $\beta ={{B}_{2}}/{{B}_{1}}$, which defines the strength of the second squirming mode relative to the first one. Without loss of generality we set B₁  >  0 and the squirmer swims in the positive z direction, as we will demonstrate below. Typically, only the first two modes are considered and B_n  =  0 for $n\geqslant 3$ [128, 130]. The reason is that B₁ and B₂ already capture the basic features of microswimmers. If B₁ is the only non-zero coefficient, the resulting flow field in the bulk fluid is exactly that of a source dipole and is illustrated in figure 1(f). The coefficient B₁ is linearly related to the source dipole strength q in equation (11), $q=8\pi \eta {{B}_{1}}{{R}^{3}}/3$. If we add the second mode, ${{B}_{2}}\ne 0$, the far field of the swimmer is that of a force dipole shown in figure 1(b). It decays more slowly with distance than the source dipole field. To match the boundary condition at the swimmer surface, an additional term proportional to r⁻⁴ is needed [128]. Figure 1(e) illustrates the complete squirmer flow field for a pusher with $\beta =-3$. The second mode coefficient B₂ is related to the force dipole strength p introduced in equation (10) by $p=-4\pi {{R}^{3}}\eta {{B}_{2}}$. So, ${{B}_{2}}\propto \beta <0$ describes a pusher (p  >  0), while for ${{B}_{2}}\propto \beta >0$ the squirmer realizes a puller (p  <  0).

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We note that a coordinate-free expression of the surface velocity field reads [128]

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mathbf{v}}_{\text{s}}} & =\underset{n=1}{\sum}\,{{B}_{n}}\frac{2}{n(n+1)}\left[\left(\mathbf{e}\centerdot {{{\mathbf{\hat{r}}}}_{s}}\right){{{\mathbf{\hat{r}}}}_{s}}-\mathbf{e}\right]\frac{\text{d}{{P}_{n}}\left(\mathbf{e}\centerdot {{{\mathbf{\hat{r}}}}_{s}}\right)}{\text{d}\left(\mathbf{e}\centerdot {{{\mathbf{\hat{r}}}}_{s}}\right)} \end{array}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} ={{B}_{1}}\left(1+\beta \mathbf{e}\centerdot {{{\mathbf{\hat{r}}}}_{s}}\right)\left[\left(\mathbf{e}\centerdot {{{\mathbf{\hat{r}}}}_{s}}\right){{{\mathbf{\hat{r}}}}_{s}}-\mathbf{e}\right]+\ldots \,, \end{array}\end{eqnarray} \tag{ 20 }$$

where we introduced the swimming direction $\mathbf{e}$ and the unit vector ${{\mathbf{\hat{r}}}_{s}}={{\mathbf{r}}_{s}}/r$. We calculate the velocity of the squirmer by inserting equation (19) into equation (15) and obtain ${{v}_{0}}=2{{B}_{1}}/3$. Hence, the swimming velocity only depends on the first mode B₁.

A simple model of an active Janus particle also uses a prescribed surface velocity field (see figure 3(b)). While the inert part (I) of the particle surface fulfills the boundary condition of a passive colloid (${{\mathbf{v}}_{\text{s}}}=0$), the chemically active cap (A) generates nearby slip flow and ${{\mathbf{v}}_{\text{s}}}\ne 0$ [67].

Notably, numerical simulations were also performed with ellipsoidal and rodlike swimmers driven by a surface flow [67, 131–138].

Self-phoretic active colloids are surrounded by a cloud of self-generated field gradients but also respond to external gradients $\nabla X\left(\mathbf{r}\right)$. Again, this can be realized by diffusio-, electro-, or thermophoresis. Then, spherical colloids at position $\mathbf{r}$, either passive or active, translate with velocity ${{\mathbf{V}}_{\text{C}}}$ and rotate with angular velocity ${{\mathbf{\Omega }}_{\text{C}}}$ [22],

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mathbf{V}}_{\text{C}}} & =\left[\langle \kappa \left({{\mathbf{r}}_{\text{s}}}\right)\rangle \mathbf{1}-\frac{1}{2}\langle \left(3\mathbf{n}\otimes \mathbf{n}-\mathbf{1}\right)\kappa \left({{\mathbf{r}}_{\text{s}}}\right)\rangle \right]\nabla X\left(\mathbf{r}\right)\,, \end{array}\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mathbf{\Omega }}_{\text{C}}} & =\frac{9}{4R}\langle \mathbf{n}\kappa \left({{\mathbf{r}}_{\text{s}}}\right)\rangle \times \nabla X\left(\mathbf{r}\right)\,, \end{array}\end{eqnarray} \tag{ 41 }$$

where $\kappa \left({{\mathbf{r}}_{\text{s}}}\right)$ is the slip-velocity coefficient introduced in equation (14) and $\langle \ldots \rangle$ is again the average taken over the particle surface with normal $\mathbf{n}$. Finally, the total colloidal velocity reads $\mathbf{V}={{v}_{0}}\mathbf{e}+{{\mathbf{V}}_{\text{C}}}$, and the angular velocity is simply $\mathbf{\Omega }={{\mathbf{\Omega }}_{\text{C}}}$. This is sketched in figure 5. Note, however, that the velocity v₀ will in general depend on the local concentration $X\left(\mathbf{r}\right)$ [188].

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Experimentally, the chemotactic response of active Janus colloids in a concentration gradient $\nabla c\left(\mathbf{r}\right)$ of ${{\text{H}}_{2}}{{\text{O}}_{2}}$ was observed [32, 189–195]. A self-propelled nano-dimer motor moving in a chemical gradient was simulated in [196]. Recent studies on interacting self-phoretic active colloids (see section 4) considered phoretic attraction and repulsion but also reorientation of a particle due to field gradients produced by others [9, 10, 188, 197–200].

The chemical field $c\left(\mathbf{r},t\right)$, which a self-diffusiophoretic particle creates, diffuses on the characteristic time scale ${{t}_{\text{c}}}\propto D_{\text{c}}^{-1}{{R}^{2}}$, where D_c is the diffusion constant of the chemical species and R is the particle radius. Since the chemical diffuses fast, one typically has ${{t}_{\text{c}}}\ll {{\tau}_{\text{r}}}$, where ${{\tau}_{\text{r}}}$ is the orientational persistence time introduced in section 3.1. The particle interacts with its own chemical trail, which is known as autochemotaxis. Thus, the chemical field couples back to the motion of the microswimmer, which can drastically modify its ballistic and diffusive movement. The implications were mainly studied in the context of biological autochemotactic systems [178, 201–204] and recently discussed for active colloids [53].

In [53] the interaction of an active colloid with its self-generated chemical cloud was investigated. Since the active colloid performs rotational diffusion and the released product molecules at the chemically active surface diffuse on a finite time t_c, a non-stationary concentration profile around the particle evolves. As a result, the mean square displacement of equation (30) is modified and shows anomalous diffusion at intermediate time scales and a modified effective diffusion in the long-time limit [53].

Active particles in a (semi-)dilute suspension collide much more frequently compared to passive particles, with a rate that increases linearly with colloidal density and, in particular, with the Péclet number [155, 182, 287, 288]. Therefore, self-propulsion, also in combination with swimmer shape and interactions, can generate self-assembled bound states of active colloids, which autonomously translate and rotate depending on the specific shape of the assembled clusters [193, 289–293]. Interestingly, recent theoretical works indeed suggest the importance of swimmer shape, surface chemistry, and hydrodynamic interactions for the structures formed by self-assembled active colloids [294–299].

Passive hard-sphere colloidal systems phase-separate at relatively large densities and in a narrow density region into a fluid and a crystalline phase favored by an increase in entropy [300, 301]. However, if one turns on activity in active particles, phase-separation can occur at lower densities [301]. This so-called motility-induced phase separation was first discussed in the context of run-and-tumble bacteria [182] but seems to be generic for active particles, which slow down in the presence of other particles [40, 154, 155, 302]. Typically, they phase-separate into a gaslike and a fluid-/solidlike state at sufficiently high density and swimming velocity. This happens even in the absence of any aligning mechanism [154, 155].

Simple model systems to experimentally study the influence of activity on the phase behavior of self-propelled particles are active colloidal suspensions confined to a monolayer, which is either sandwiched between two bounding plates [8, 12] or sedimented on a substrate [9–11]. These quasi-two-dimensional systems allow a relatively easy tracking of particle positions and thereby the monitoring of particle dynamics.

Probably the simplest experimental system showing motility-induced phase separation was introduced by Buttinoni et al using light-activated spherical Janus particles that swim in a binary fluid close to the critical demixing transition (see also section 2.2.1). Tuning the laser intensity, the swimming speed—and therefore the Péclet number—can be adjusted. Since phoretic and electrostatic interactions between colloids seem to be negligible in these experiments, the Janus particles serve as a simple experimental model for active hard spheres swimming in a low-Reynolds-number fluid. At sufficiently low Péclet number and areal density, the particles do not form clusters but rather behave as an active gas. However, for higher particle density and swimming speed, densely packed clusters emerge, which coexist with a gas of active particles (see figure 9(a)). Increasing the Péclet number, the size of these clusters grows, as predicted earlier by simulations and theory (see sections 4.2 and 4.3).

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Very dynamic particle clusters form in the experiments of Theurkauff et al with self-phoretic spherical colloids half-coated with platinum, which become active when adding ${{\text{H}}_{2}}{{\text{O}}_{2}}$ [9] (see figure 9(b)). Effective particle interactions exist, both attractive and repulsive, which in combination can produce pronounced dynamic clustering already at very low area fractions of a few percent [9, 197, 199]. These effective forces arise from diffusiophoresis, which the colloids experience in non-uniform chemical fields produced by their neighbors while consuming ${{\text{H}}_{2}}{{\text{O}}_{2}}$ (see also section 3.3). Therefore, such artificial systems can mimic chemotactic processes found in biological systems. Aggregation of chemotactic active colloids was also reported in further experiments [10, 189, 191, 194, 303].

Most recently, the sedimentation profile of interacting self-phoretic colloids under gravity was studied in detail [184]. Here, far from the bottom surface the density of swimmers is very small and shows an exponential decay as described in section 3.2. Closer to the bottom clusters are formed due to phoretic interactions at semi-dilute particle suspensions. The cluster formation could be mapped to an adhesion process of a corresponding equilibrium system.

Palacci et al investigated the collective motion of colloidal surfers (see section 2.3) on a substrate. Here again, the combination of self-propulsion and steric and chemical interactions triggers the formation of clusters [10], which show a well defined crystalline structure (see figure 9(c)). These living crystals are highly dynamic; they form, rearrange, and break up quickly.

All of these systems show the emergence of positional order through the formation of clusters, but no significant orientational order in the swimming direction has been reported. Nevertheless, even for spherical active particles polar order can emerge in the presence of hydrodynamic, electrostatic, or other interactions between nearby particles. They lead to a local alignment of swimmer orientations, as we report in the following section.

Local mechanisms for aligning active particles give rise to new collective phenomena. We summarize them here.

Quite surprisingly, even for spherical active colloids, where an aligning mechanism is not obvious, polar order was reported in a dense suspension of swarming active emulsion droplets [8] (see figure 9(d)). The observed large-scale structures and swirls show some behavior reminiscent of biological swarms [1]. Hydrodynamic interactions between active droplets, due to the flow fields they create, are expected to play an important role. However, their details are not fully clear. Marangoni flow at the droplet surfaces causes hydrodynamic flow fields in the bulk fluid (see section 2.2.2), but this Marangoni flow might be modified by the presence of other active droplets. It remains to be investigated how this affects the collective dynamics. Notably, the system reported in [8] is strongly confined in the plane by a curved petri dish, which seems to be important for observing the swarming dynamics. Similar behavior was noted for vibrated granular matter [304–306]. Finally, swimming liquid crystal droplets form crystalline rafts that float above the substrate [61].

Bricard et al studied the collective motion of Quincke rollers (see section 2.3) in a racetrack geometry [11]. Here, both electrostatic and hydrodynamic interactions between the rollers determine their collective motion. For constant strength of the applied electric field, the collective motion can be tuned by modifying the area fraction of the particles. While at sufficiently low densities the particles form an apolar active gas, at a critical density propagating polar bands emerge (see figure 9(e)). At even higher densities a polar liquid is stabilized, where all particles move in the same direction. Interestingly, when the confining geometry is changed to a square or to a circular disc, a single macroscopic vortex forms [11, 307]. Similar vortices occur in circularly confined bacterial suspensions [137, 308].

In a very recent work, Nishiguchi and Sano observed active turbulence in a monolayer of swimming spherical colloids [13]. Here, the Janus colloids sediment on a substrate and start to swim when an AC electric field is applied. While both hydrodynamic and electrostatic interactions seem to play a role for generating turbulence, the detailed mechanisms are not well understood yet. Notably, up to now active turbulent states have only been observed for non-spherical active particles.

The most basic realization for studying interacting active colloids in theory and simulations is active Brownian particles (ABPs) (see also section 2.4). The equations of motion for a single, free ABP are given in equations (21) and (22), and the solution is a persistent random walk (see section 3.1).

The dynamics of interacting ABPs can be implemented in different ways using Brownian dynamics or even kinetic Monte Carlo simulations [309]. To account for the fact that particles cannot interpenetrate each other due to steric repulsion, hard or soft potentials between the ABPs are typically used. For example, the Weeks–Chandler–Anderson (WCA) potential $U(r)$ is employed frequently to model soft-core or (almost) hard-core repulsion between two particles. It is a Lennard-Jones potential acting between spherical particles of radius R and cut off at distance ${{d}^{\ast}}$, where the Lennard-Jones potential has its minimum [310],

$$\begin{eqnarray}U(r)=\left\{\begin{array}{*{35}{l}} 4\epsilon \left[{{\left(\frac{\sigma}{r}\right)}^{12}}-{{\left(\frac{\sigma}{r}\right)}^{6}}\right]+\epsilon, & \text{for}~r<{{d}^{\ast}} \\ 0, & \text{for}~r\geqslant {{d}^{\ast}}. \end{array}\right. \end{eqnarray} \tag{ 48 }$$

Here, ${{d}^{\ast}}={{2}^{1/6}}\sigma$, ε is the potential strength, and $\sigma =2R/{{2}^{1/6}}$ is chosen such that the interaction force $\mathbf{F}=-\nabla U$ becomes non-zero at ${{d}^{\ast}}<2R$, i.e. when two ABPs overlap. A simple alternative to implement hard-core interaction is the following [10, 197]: whenever particles overlap during a simulation, one separates them along the line connecting their centers.

Then, in a system consisting of N active particles the equation of motion for the ith particle reads

$$\begin{eqnarray}&&{{\overset{\centerdot}{{\mathbf{r}}}\,}^{(i)}}={{v}_{0}}{{\mathbf{e}}^{(i)}}+\mu \underset{j\ne i}{\sum}\,{{\mathbf{F}}^{(ij)}}+\sqrt{2D}\boldsymbol{\xi }\,,\end{eqnarray} \tag{ 49 }$$

where the index j runs over all the other N  −  1 particles, ${{\mathbf{F}}^{(ij)}}$ is the interaction force from particle j on i, and $\mu ={{(6\pi \eta R)}^{-1}}$ is the mobility coefficient of the ABP. Together with the equation for the stochastic reorientation of the particle orientations ${{\mathbf{e}}^{(i)}}$ (see equation (22)) equation (49) is solved, e.g. by Brownian dynamics simulations. Additional attractive and/or repulsive interactions may be included [157, 311].

While elongated ABPs tend to align with neighbors after steric collisions [163, 164, 166], active Brownian spheres and disks lack any intrinsic aligning mechanism. Nevertheless, different mechanisms such as aligning, phoretic, or hydrodynamic interactions alter the particle orientations ${{\mathbf{e}}^{(i)}}$ and can be included in equation (22) for the orientation vector (see, for example, [10, 197, 312–314]).

In theory and simulations, the simplest system for studying collective motion of active Brownian particles consists of active Brownian disks, which interact only via hard-core repulsion in two dimensions (2D). This minimal model has frequently been used to investigate the collective dynamics of active particles [12, 152, 154, 155, 159, 161, 162, 287, 288, 311, 315–319]. Bialké et al [152], Fily et al [154], and Redner et al [155] were the first to explore dynamic structure formation within this model. The two relevant parameters in the system are the areal density ϕ and the Péclet number $\text{Pe}$. The latter is linear in the persistence number $\text{P}{{\text{e}}_{\text{r}}}$ for pure thermal noise (see equation (29)). While at low densities the ABPs form an active gas, they start to phase-separate into a gaslike and a crystalline phase at $\phi \gtrsim 0.3$ and sufficiently large $\text{Pe}$ [12, 41, 154, 155, 161, 162, 287, 288, 311]. Figure 10 shows a typical snapshot of the phase-separated state for $\phi =0.64$ and $\text{Pe}=360$. The crystalline structure is not perfect but also contains defects. For sufficiently small Péclet numbers the dense cluster phase is rather liquidlike [152, 155].

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In three dimensions (3D) the system also phase-separates. However, the cluster phase does not have crystalline order but is rather fluidlike and the local density can reach the random-close-packing limit [160]. Furthermore, the coarsening dynamics of the clusters clearly differ in 2D and 3D. While in 3D the mean domain size grows as  ∼  t^1/3 in time [162], similar to equilibrium coarsening dynamics, a cluster coarsens more slowly in 2D according to $\sim$t^0.28 [162, 287, 311].

We note that the minimal model can be extended in several ways leading to new emergent collective behavior. For example, introducing polydispersity in the ABPs results in active glassy behavior [156, 158, 309, 320–322]. Additional attractive interparticle forces lead to gel-like structures [311] or very dynamic crystalline clusters at low densities [10, 157, 197]. The collective motion of self-propelled Brownian rods has been studied extensively [163–166, 168, 305, 323–327]. Due to the local alignment of the rods, one observes the formation of dynamic swarming clusters [163], moving bands [324], and even turbulent states [168]. Finally, the self-assembly of active particles with more complex shapes was investigated [173, 328].

To summarize, the simple model of ABPs is able to qualitatively reproduce important aspects of the observed emergent behavior in active colloids such as motility-induced phase separation (see section 4.1.2) by only accounting for self-propulsion and steric hindrance. However, it neglects the effect of flow fields generated by active colloids and does not include phoretic interactions, which we will discuss in sections 4.2.2 and 4.2.5.

As we have discussed in section 2, active colloids moving in a Newtonian fluid create a flow field around themselves. In the following, we discuss how these flow fields determine the collective dynamics of microswimmers.

Hydrodynamic interactions between active colloids.

We briefly introduce here the basic principles of hydrodynamic interactions between microswimmers (see also [5, 16, 17, 20, 128, 313, 329]). Active colloids at positions ${{\mathbf{r}}_{i}}$ and swimming in bulk with velocities ${{\mathbf{V}}_{i}}={{v}_{0}}{{\mathbf{e}}_{i}}$ create undisturbed flow fields ${{\mathbf{v}}_{i}}\left(\mathbf{r};{{\mathbf{r}}_{i}},{{\mathbf{e}}_{i}}\right)$ around themselves when they are far apart from each other. The flow fields depend on the swimmer type, as discussed in section 2. To leading order of hydrodynamic interactions, each microswimmer is advected by the undisturbed flow fields from its neighbors and the colloidal velocity becomes $\mathbf{V}_{i}^{\prime}={{\mathbf{V}}_{i}}+\mathbf{V}_{i}^{\text{HI}}\left(\left\{{{\mathbf{r}}_{j}},{{\mathbf{e}}_{j}}\right\}\right)$ with $\mathbf{V}_{i}^{\text{HI}}={\sum}_{j\ne i}{{\mathbf{v}}_{j}}\left({{\mathbf{r}}_{i}};{{\mathbf{r}}_{j}},{{\mathbf{e}}_{j}}\right)$. In addition, the vorticities of the undisturbed flow fields add up to determine the angular velocities $\mathbf{\Omega }_{i}^{\prime}=\mathbf{\Omega }_{i}^{\text{HI}}\left(\left\{{{\mathbf{r}}_{j}},{{\mathbf{e}}_{j}}\right\}\right)$ to leading order. When particles come closer together, the undisturbed flow fields do not satisfy the no-slip boundary condition at the surfaces of neighboring particles. They have to be modified and thereby the hydrodynamic contributions, $\mathbf{V}_{i}^{\text{HI}}\left(\left\{{{\mathbf{r}}_{j}},{{\mathbf{e}}_{j}}\right\}\right)$ and $\mathbf{\Omega }_{i}^{\text{HI}}\left(\left\{{{\mathbf{r}}_{j}},{{\mathbf{e}}_{j}}\right\}\right)$, to the colloidal velocities change.

Still in the dilute limit, where the distances between the swimmers are much larger than their radii, one can improve on the far-field hydrodynamic interactions using Faxén's law [64]. For example, for spherical particles of radius R the hydrodynamic contributions $\mathbf{V}_{i}^{\text{HI}}$ and $\mathbf{\Omega }_{i}^{\text{HI}}$ to the colloidal velocities read

$$\begin{eqnarray}\begin{array}{*{35}{l}} \mathbf{V}_{i}^{\text{HI}} & =\underset{j\ne i}{\sum}\,\left(1+\frac{{{R}^{2}}}{6}\nabla _{i}^{2}\right){{\mathbf{v}}_{j}}\left({{\mathbf{r}}_{i}};{{\mathbf{r}}_{j}},{{\mathbf{e}}_{j}}\right), \end{array}\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \mathbf{\Omega }_{i}^{\text{HI}} & =\frac{1}{2}{{\nabla}_{i}}\times \underset{j\ne i}{\sum}\,{{\mathbf{v}}_{j}}\left({{\mathbf{r}}_{i}};{{\mathbf{r}}_{j}},{{\mathbf{e}}_{j}}\right)\, \end{array}\end{eqnarray} \tag{ 51 }$$

where ${{\mathbf{v}}_{j}}\left({{\mathbf{r}}_{i}};{{\mathbf{r}}_{j}},{{\mathbf{e}}_{j}}\right)$ is again the undisturbed flow field created by swimmer j and evaluated at the position ${{\mathbf{r}}_{i}}$ of swimmer i. Similar expressions hold for ellipsoidal particles [63, 66].

In more dense suspensions of active colloids the approximation of far-field hydrodynamics (equations (50) and (51)) is not valid any more. In contrast, $\mathbf{V}_{i}^{\text{HI}}$ and $\mathbf{\Omega }_{i}^{\text{HI}}$ are mainly determined by near-field hydrodynamic interactions. Their calculation is much more complicated and strongly depends on the specific swimmer model. Hence, they usually have to be determined via hydrodynamic simulations, which capture the hydrodynamic near fields correctly. For squirmers lubrication theory can be applied to calculate $\mathbf{V}_{i}^{\text{HI}}$ and $\mathbf{\Omega }_{i}^{\text{HI}}$, but it only holds for interparticle distances $d\ll R$ [128].

Collective motion of squirmers.

The squirmer model introduced in section 2.2.3 is a simple model to study how hydrodynamic interactions influence the collective motion of active colloids. Ishikawa and Pedley were the first to use the boundary element method and Stokesian dynamics simulations for investigating the collective motion of squirmers in bulk fluids [330–337]. Typically, in a suspension of squirmers the reorientation rates $\mathbf{\Omega }_{i}^{\text{HI}}$ evolve chaotically in time, and hydrodynamic interactions thus modify rotational diffusion and an increased effective diffusion constant $D_{\text{r}}^{\text{HI}}$ results [330, 338]. For squirmer pullers polar order can emerge [333], which was also quantified further in [138, 339–341]. This is in contrast to hydrodynamic simulations of self-propelled rods, which show local polar order for generic pushers but not for pullers (see the last paragraph of this section). The different behaviors of squirmer and active rod suspensions might be due to different types of near-field hydrodynamic interaction as discussed in section 3.5.

Notably, in large suspensions of squirmer pullers temporal density variations emerge, where a large cluster periodically forms and breaks apart [340]. The time correlation function of these density fluctuations shows oscillatory behavior with a well defined frequency. Recently, the dynamics of many squirmers confined between two hard walls has also been studied [161, 254, 342–344]. For a separation distance much larger than the swimmer size, a huge dynamically evolving cluster again emerges. It travels between the walls and has been interpreted as a propagating sound wave [344].

Recent investigations studied the collective motion of squirmers moving either in 2D [345–347] or in quasi-2D [161] in order to reveal the influence of hydrodynamic interactions on the dynamics of active colloidal suspensions. The quasi-2D geometry constrained the squirmers to a monolayer similar to experiments in [12] (see section 4.1). 2D simulations showed that long-range hydrodynamic interactions result in strong reorientation rates $\mathbf{\Omega }_{i}^{\text{HI}}$ that are sufficient to entirely suppress motility-induced phase separation of squirmers [346]. Simpler non-squirming swimmers simulated with the lattice Boltzmann method in 2D [348, 349] and with Stokesian dynamics in a monolayer in 3D [350] showed some clustering.

We performed three-dimensional MPCD simulations of squirmers and strongly confined them between two parallel plates, such that they could only move in a monolayer (quasi-2D geometry) [161]. The simulations were based on [130, 177, 210], where single squirmers were implemented with MPCD. Our results showed that the phase behavior of squirmers strongly depends on the swimmer type, characterized by the squirmer parameter β, and on areal density ϕ (see figure 11). While neutral squirmers ($\beta =0$) and weak pushers ($\beta <0$) phase-separate at a sufficiently high density, pullers ($\beta >0$) only form small and short-lived clusters. Strong pushers do not cluster at all and only develop one crystalline region at high areal densities. They tend to point with their swimming directions perpendicular to the bounding walls, which significantly reduces their in-plane persistent motion so clustering does not occur. Pullers are oriented more parallel to the walls, but their rotational diffusivity is strongly enhanced so the persistent motion is again too small to exhibit phase separation. In contrast, neutral squirmers and weak pushers also swim parallel to the walls and their hydrodynamic rotational diffusion $D_{\text{r}}^{\text{HI}}$ is sufficiently small to allow stable clusters to form and hence they phase-separate. We do not observe polar order in any of the studied systems. This could be due to the presence of the walls or thermal noise, which was absent from all the other simulations on the collective dynamics of squirmers. We currently perform large-scale simulations, which confirm all these findings and clearly demonstrate the first-order phase transition associated with phase separation.

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Collective motion of elongated microswimmers and actively spinning particles.

As discussed in section 3.2, a suspension of non-interacting active colloids under gravity shows an exponential sedimentation profile [71] and develops polar order against gravity [180]. The presence of hydrodynamic interactions in a dilute suspension of sedimenting run-and-tumble swimmers does not significantly modify such a sedimentation profile [258]. However, when active Brownian particles are strongly bottom heavy, they collect in a layer while swimming against the upper boundary of a confining cell. This layer is unstable, when hydrodynamic interactions are included, and the particles move downwards in plumes similar to observations made in bioconvection [207, 359].

A similar instability occurs for run-and-tumble swimmers [258] and active Brownian particles of radius R [313], when they move in a harmonic trap potential while interacting hydrodynamically. The microswimmers form a macroscopic pump state that breaks the rotational symmetry of the trap. The radial force $\mathbf{F}=-k\mathbf{r}$ (k  >  0) confines the particles to a spherical shell of radius ${{r}_{\text{hor}}}=R\text{Pe}/\alpha$, where $\alpha =k{{R}^{2}}/{{k}_{\text{B}}}T$ is the trapping Péclet number [313]. When started with random positions and orientations, the active particles first accumulate at the horizon at radial distance ${{r}_{\text{hor}}}$, where they point radially outwards [258, 313]. Due to the externally applied trapping force, each particle initiates the flow field of a stokeslet, to leading order. However, a uniform distribution of stokeslets on the horizon sphere, all pointing radially outward, is not stable against small perturbations. In particular, particles' swimming directions are rotated by nearby stokeslets. The particles move towards each other, creating denser regions, which are advected towards the center. At sufficiently large swimmer density the active particles align in their own flow field and thereby generate a macroscopically ordered state (see figure 12(a)), quantified by the global polar order parameter $\mathcal{P}$ (see also section 4.3.4). The coarse-grained flow field, created by the swimmers, has the form of a regularized stokeslet and pumps fluid along the polar axis to the center of the trap. The pump formation does not occur if the Péclet number, the trapping strength, or the density is too small, as shown in figure 12(b).

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Interestingly, the form of the distribution $\Phi(\theta )$ of orientation angles θ measured against the pump axis can be calculated analytically from the corresponding Smoluchowski equation. One obtains $\Phi(\theta )\sim \exp (A\cos \theta )$, reminiscent of dipoles aligning in a field A, which is produced by the swimmers themselves. In simulations we find $A\sim {{\left(\text{Pe}\mathcal{P}\right)}^{\gamma}}$, with $\gamma \approx 1$ for low densities and decreasing for higher densities. Thus, the mean flow field created by all the swimmers and the associated polarization $\mathcal{P}$ play the same respective role as the mean magnetic field and the magnetization in Weiss's theory of ferromagnetism [313].

In [360] the collective behavior of purely repulsive active particles in two-dimensional traps was mapped on a system of passive particles with modified trapping potential and then formulated as a dynamical density functional theory [360]. In very good agreement with the numerical solution of the corresponding Langevin equations, one could show that the radial distribution in the trap including packing effects strongly depends on the Péclet number.

We have discussed the response of a single microswimmer to an externally applied flow field $\mathbf{v}$ in section 3.4. In turn, a suspension of microswimmers is able to modify $\mathbf{v}$ and the rheological properties of the fluid.

In pure Newtonian fluids the shear stress tensor $\boldsymbol{\tau }$ is linear to the strain rate tensor, $\boldsymbol{\tau }={{\eta}_{0}}\left[\nabla \otimes \mathbf{v}+{{\left(\nabla \otimes \mathbf{v}\right)}^{t}}\right]$, where ${{\eta}_{0}}$ is the shear viscosity. Adding a dilute suspension of passive colloids with volume fraction $\phi \ll 1$ is known to increase the effective viscosity η. To leading order in ϕ the increase is linear, $\eta ={{\eta}_{0}}\left(1+\frac{5}{2}\phi \right)$, as calculated by Einstein [361, 362]. In contrast, adding a suspension of swimming bacteria [363–365] or algae [366, 367] to water can lead to both an increase and a decrease of the viscosity, an effect which was first predicted in [368]. Strikingly, at high shear rates η approaches zero for swimming E. coli suspensions [365], which thus become a superfluid. This was also predicted before for sheared active gels [369].

While to date no experimental studies exist that measure the effective viscosity η of active colloidal suspensions, numerical simulations have been performed using the squirmer model. The rheological properties of squirmer suspensions with hydrodynamic interactions were simulated using Stokesian dynamics [330] and lattice Boltzmann simulations [343]. In [330] a semi-dilute suspension of squirmers in unbounded shear flow show almost the same, but slightly reduced, apparent viscosity as for a semi-dilute passive colloidal suspension. In wall bounded shear flow, as studied in [343], the viscosity of squirmer suspensions depends strongly on the applied shear rate $\overset{\centerdot}{{\gamma}}\,$ and on the swimmer type. While for strong $\overset{\centerdot}{{\gamma}}\,$ the viscosity of all squirmer suspensions is comparable to the effective viscosity of passive suspensions, it can be strongly decreased or increased at lower $\overset{\centerdot}{{\gamma}}\,$. Namely, for all squirmer suspensions with sufficiently small $|\beta |\lesssim 20$ the effective viscosity is increased ($\eta /{{\eta}_{0}}>1$), but for apolar pushers ($\beta \to -\infty$) and apolar pullers ($\beta \to \infty$) the effective viscosity is strongly decreased ($\eta /{{\eta}_{0}}<0$) and almost vanishes for very small $\overset{\centerdot}{{\gamma}}\,$.

As discussed in section 3.3.1, an active colloid that experiences a non-uniform background concentration field $X\left(\mathbf{r}\right)$ moves with an additional phoretic velocity ${{\mathbf{V}}_{\text{C}}}$ and angular velocity ${{\mathbf{\Omega }}_{\text{C}}}$, which both are linear in the local field gradient $\nabla X\left(\mathbf{r}\right)$. Self-phoretic colloids act as sinks and sources in a chemical field and respond to local field gradients created by neighboring particles. This introduces an effective interaction, which can be either attractive or repulsive. In particular, active colloids can orient parallel to or against a field gradient and thereby either swim towards or away from other particles.

The collective behavior of chemotactic active particles has been studied numerically with particle-based models using Langevin dynamics simulations [10, 197, 199, 370], and theoretically by continuum models [188, 197, 200]. Depending on whether the active colloids are chemoattractive or chemorepulsive, or whether they tend to rotate towards or against a chemical gradient, qualitatively different collective behavior emerges [10, 188, 197, 199, 200, 370].

To capture the basic effects, we follow [197] and include diffusiophoretic attraction/repulsion and reorientation into equations (21) and (22). Then the equations of motion for N interacting particles ($i=1,\ldots,N$) read [197, 199]

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\overset{\centerdot}{{\mathbf{r}}}\,}_{i}} & ={{v}_{0}}{{\mathbf{e}}_{i}}-{{\zeta}_{\text{tr}}}\nabla c\left({{\mathbf{r}}_{i}}\right)+\sqrt{2D}\boldsymbol{\xi } \end{array}\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\overset{\centerdot}{{\mathbf{e}}}\,}_{i}} & =-{{\zeta}_{\text{rot}}}\left(\mathbf{1}-{{\mathbf{e}}_{i}}\otimes {{\mathbf{e}}_{i}}\right)\nabla c\left({{\mathbf{r}}_{i}}\right)+\sqrt{2{{D}_{\text{r}}}}\boldsymbol{\xi }_{{\text{r}}}\times {{\mathbf{e}}_{i}}\,, \end{array}\end{eqnarray} \tag{ 53 }$$

where ${{\zeta}_{\text{tr}}}$ is the translational diffusiophoretic parameter, which quantifies chemoattraction (${{\zeta}_{\text{tr}}}>0$) and chemorepulsion (${{\zeta}_{\text{tr}}}<0$), and ${{\zeta}_{\text{rot}}}$ is the rotational diffusiophoretic parameter, which describes reorientation along (${{\zeta}_{\text{rot}}}>0$) and against (${{\zeta}_{\text{rot}}}<0$) chemical gradients. Here, the chemical field $c\left(\mathbf{r}\right)$ created by the swimmers is assumed to be a stationary sink, which moves around with the particles. To leading order it decays linearly with the distance from the swimmer [52] but is screened due to the presence of other particles [10, 197]. Dipolar contributions and higher multipoles are neglected. We discuss here the two-dimensional motion of relatively dilute systems (area fraction $5\%$) similar to experiments [9, 10].

First, for sufficiently small ${{\zeta}_{\text{tr}}}$ and ${{\zeta}_{\text{rot}}}$ the particles are in a gaslike state, where they hardly cluster (see figure 13(a)), similar to active Brownian disks which interact purely due to steric hindrance [154, 155]. Second, for sufficiently strong attraction all the particles end up in a collapsed state and form a single stable cluster [188, 197, 370] reminiscent of the chemotactic collapse obtained from the Keller–Segel model [371] (see figure 13(b)). Third, for ${{\zeta}_{\text{rot}}}<0$ and sufficiently large ${{\zeta}_{\text{tr}}}>0$ dynamic clustering is observed (see figure 13(c)), in good agreement with experiments [9, 10]. Dynamic clusters form since chemoattraction is balanced by the reorientation of the swimmers such that they try to swim away from each other. The cluster-size distribution (see also section 4.3.2) follows a power-law decay. Fourth, in the opposite case, for chemorepulsion (${{\zeta}_{\text{tr}}}<0$) but alignment against the chemical gradient (${{\zeta}_{\text{tr}}}>0$) an oscillating state can emerge, where all particles periodically collapse and dissolve [199]. Further decreasing ${{\zeta}_{\text{tr}}}$, the particles ultimately arrange in a stable core–corona state as shown in figure 13(d), which also occurs when screening is turned off. Oscillating states reminiscent of plasma oscillations and aster formation were also observed in a continuum model [188]. Different scenarios for pattern formation in chemorepulsive active colloids have recently been explored in [200].

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The collective behavior of a dilute suspension of light-heated active colloids interacting by thermoattraction and thermorepulsion was analyzed in [372]. While thermorepulsive colloids show a depletion zone in the center of a confining region, thermoattractive colloids aggregate in the center and collapse at sufficiently high density [372]. Using Brownian dynamics simulations, the formation of a dense cometlike swarm, which consists of interacting thermally active colloids moving towards an external light source, was reported [373]. Finally, [198] discusses the orientation of thermally active Janus particles in a temperature gradient, as well as the effect of Péclet number on aggregation and dispersion of the particles.

The collective motion or pattern formation of active particles on the liquid–air interface of a thin liquid film was studied in [374, 375], while passive but chemically active particles at the interface were treated in [264]. A 2D continuum approach to couple chemical signaling and flow fields created by pushers and pullers was used in [376].

A first work by Thakur and Kapral includes full hydrodynamic and phoretic interactions, which is computationally very expensive [118]. They studied the collective dynamics of up to ten dimer motors in a quasi-2D geometry in the presence of thermal noise and observed dynamic cluster formation, which is mainly triggered by phoretic interactions but altered by hydrodynamic interactions [118].

Stenhammar et al investigated the structure and dynamics of a monodisperse mixture of active and passive Brownian disks for a broad range of Péclet numbers Pe, total area fractions ${{\phi}_{0}}$, and fraction of active particles $0\leqslant {{x}_{\text{A}}}\leqslant 1$ [377]. They observed phase separation induced by the active particles. However, in contrast to motility-induced phase separation of purely active Brownian disks (${{x}_{\text{A}}}=1$), cluster formation, cluster dynamics, and cluster break-up are much more dynamic for active–passive particle mixtures (${{x}_{\text{A}}}<1$). At sufficiently high Pe phase separation is already possible for a relatively low number of active particles (${{x}_{\text{A}}}>0.15$). In the phase-separated state the active particles are mainly concentrated at the borders of the clusters. This has recently been confirmed in experiments with a very small amount (${{x}_{\text{A}}}<3\%$) of active colloids [269]. In contrast, above the threshold density for the crystallization of passive colloids, active particles accumulate at grain boundaries. When passing through crystalline domains, active particles are able to melt them locally and create defects along their trajectory [269].

Very recently, travelling fronts in polydisperse phase-separating mixtures of active and passive disks were observed [378]. Finally, by performing event-driven Brownian dynamics simulations, Ni et al [379] realized that a small number of active particles helps to crystallize passive polydisperse glassy hard spheres in 3D.

Various types of segregation in a mixture of passive and self-propelled hard rods moving in 2D were observed by McCandlish et al [326], i.e. swarming, coherent clustering, and transient lane formation. In dissipative particle dynamics simulations a mixture of passive and active particles showed the emergence of turbulence, polar order, and vortical flows [380].

Motility-induced phase separation of active particles can mainly be understood as a self-trapping mechanism: when active particles collide, they slow down due to steric hindrance and a cluster can form for sufficiently high persistence number [12, 40, 302, 381].

In theory, the separation of purely repulsive active Brownian particles into a liquidlike and a gaslike phase has recently been investigated by mapping the system to an equilibrium system, for which an effective free energy functional was constructed [182, 287, 288, 302, 317, 382, 383]. Here, the velocities of the particles in the presence of neighbors are typically approximated by a function $v(\rho )$ which decays almost linearly with density ρ [154, 155, 162, 287, 288],

$$\begin{eqnarray}&&v(\rho )={{v}_{0}}\left(1-\frac{\rho}{\rho {{\ast}^{}}}\right)\,,\end{eqnarray} \tag{ 54 }$$

where ${{\rho}^{\ast}}\approx {{\rho}_{\text{rcp}}}$ is the density close to random close packing [40]. In the limit of large Péclet numbers, phase coexistence occurs when the following relation holds [40, 182]:

$$\begin{eqnarray}&&\frac{\text{d}v(\rho )}{\text{d}\rho}<-\frac{v(\rho )}{\rho}.\end{eqnarray} \tag{ 55 }$$

The coexistence of a gaslike and a liquidlike phase emerges via spinodal decomposition; similarly to the case in equilibrium thermodynamics, two-phase coexistence with the binodal densities ${{\rho}_{1}}$ and ${{\rho}_{2}}$ arises from the tangent construction for an effective free energy functional [182, 287, 288, 302]. Interestingly, by linearizing the underlying hydrodynamic equations [288] an effective Cahn–Hilliard equation for active phase separation can be constructed [317]. Wittkowski et al formulated a scalar ${{\phi}^{4}}$ field theory for the order parameter field $\phi \left(\mathbf{r},t\right)$ to study phase separation and its coarsening dynamics by an active model B [382], which violates detailed balance. This procedure has recently been extended to include hydrodynamics by coupling $\phi \left(\mathbf{r},t\right)$ to the Navier–Stokes equations (active model H [384]).

A kinetic approach for determining a condition for motility-induced phase separation was formulated by Redner et al [155, 311]. The incoming flux ${{j}_{\text{in}}}$ of active particles from the gas phase of density ${{\rho}_{\text{g}}}$ onto an existing cluster is compared with the outgoing flux ${{j}_{\text{out}}}$ from the border of a cluster into the gas phase. While the incoming flux, ${{j}_{\text{in}}}\propto {{\rho}_{\text{g}}}\text{Pe}$, is proportional to particle speed and density, the outgoing flux, ${{j}_{\text{out}}}\propto \text{Pe}_{\text{r}}^{-1}$, depends on the inverse persistence number and thus on rotational diffusion. Note that rotational diffusion can be strongly enhanced in the presence of hydrodynamic interactions [333] and thereby reduce or suppress phase separation [161, 346]. The condition for phase separation and the fraction of particles in the cluster are determined from the steady state condition ${{j}_{\text{in}}}={{j}_{\text{out}}}$ [155, 311].

Finally, we comment on how to quantify a phase-separated state obtained from experimental or simulation data of active particles. A phase-separated state consists of a high-density (liquid- or solidlike) and a low-density (gaslike) region and has, by definition, a bimodal distribution $p\left({{\phi}_{\text{l}}}\right)$ of local densities ${{\phi}_{\text{l}}}$. The local density $\phi _{\text{l}}^{(i)}$ of particle i at position ${{\mathbf{r}}_{i}}$ is typically defined via the size A_i of its corresponding Voronoi cell [385], $\phi _{\text{l}}^{(i)}={{A}_{0}}/{{A}_{i}}$, where ${{A}_{0}}=\pi {{R}^{2}}$ in 2D or ${{A}_{0}}=4\pi {{R}^{3}}/3$ in 3D and R is the particle radius.

To characterize if active particles are in a gas-, liquid- or crystal-like phase, further structural properties are determined from the particle positions ${{\mathbf{r}}_{i}}(t)$.

Pair correlation function.

The pair correlation function (or radial distribution function) g(r) measures the correlations of particles at distance r and has been used to characterize the structure of collectively moving active colloids [12, 118, 155, 157, 269, 333, 340]. It is defined as the local density $\rho (r)$ of active particles in a spherical shell with radius r around a central particle, normalized by the mean particle density ${{\rho}_{0}}$, thus $g(r)=\rho (r)/{{\rho}_{0}}$. For example, in 2D it can be calculated from

$$\begin{eqnarray}&&g(r)=\frac{1}{4\pi {{R}^{2}}(N-1){{\phi}_{0}}}\langle \underset{j}{\sum}\,\underset{i\ne j}{\sum}\,\delta \left(|{{\mathbf{r}}_{i}}-{{\mathbf{r}}_{j}}|-r\right)\rangle \,,\end{eqnarray} \tag{ 56 }$$

where ${{\phi}_{0}}$ and N are the total area fraction and number of particles, respectively. Peaks in g(r) indicate favored interparticle distances. While for a dilute gas no distances are favored and $g(r)\approx 1$, g(r) shows a finite number of peaks with decreasing intensity for a liquid (see figure 14(a)), and pronounced peaks for solids, where the actual peak positions indicate the structure of the solid (e.g. hexagonal).

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Static structure factor.

The Fourier transform of the pair correlation factor is the static structure factor $S\left(\mathbf{k}\right)$, which identifies periodic modulations of a material with wave vector $\mathbf{k}$. It is defined as

$$\begin{eqnarray}&&S\left(\mathbf{k}\right)=\frac{1}{N}\langle \underset{j}{\sum}\,\underset{i}{\sum}\,{{\text{e}}^{i\mathbf{k}\centerdot \left({{\mathbf{r}}_{i}}-{{\mathbf{r}}_{j}}\right)}}\rangle \,,\end{eqnarray} \tag{ 57 }$$

and has been used to characterize the structure of active colloidal clusters [9, 11, 154–156, 162, 288].

Bond orientational order.

Clusters of active colloids moving in a monolayer often show hexagonal crystal-like ordering; see sections 4.1.2 and 4.2 and figures 10, 11 and 13. The local sixfold bond-orientational order around particle i is measured using [152, 161, 311, 386]

$$\begin{eqnarray}&&|q_{6}^{(i)}{{|}^{2}}\quad \text{with}\quad q_{6}^{(i)}=\frac{1}{6}\underset{j\in N_{6}^{(i)}}{\sum}\,{{\text{e}}^{i6{{\alpha}_{ij}}}}\,,\end{eqnarray} \tag{ 58 }$$

where $|q_{6}^{(i)}{{|}^{2}}\in (0,1)$, the sum goes over the six nearest neighbors⁶ of particle i, and the angle ${{\alpha}_{ij}}$ is measured between the distance vectors ${{\mathbf{r}}_{i}}-{{\mathbf{r}}_{j}}$ and an arbitrarily chosen axis (see figure 14(b)). For example, for particles in a cluster with perfect hexagonal order, $|q_{6}^{(i)}{{|}^{2}}\,=1$, as indicated in figures 10 and 11. The mean value, $\langle |{{q}_{6}}{{|}^{2}}\rangle$, can be used as an order parameter to measure how many particles are in a dense crystal-like cluster phase [161].

Spatial correlations in the sixfold bond-orientational order measured using $\langle q_{6}^{\ast}\left(\mathbf{r}\right){{q}_{6}}\left({{\mathbf{r}}^{\prime}}\right)\rangle$ [155] and correlations of particle i with its six nearest neighbors, $\text{Re}\frac{1}{6}{\sum}_{j\in N_{6}^{(i)}}q_{6}^{(i)}q_{6}^{\ast (\,j)}$ [152, 387] were used to distinguish between liquidlike, hexaticlike, and solidlike structures [152, 155]. The global bond-orientational order parameter $|\langle {{q}_{6}}\rangle {{|}^{2}}$ indicates an overall structural order in the system. Note, however, that in general $|\langle {{q}_{6}}\rangle {{|}^{2}}$ depends on the system size. Only for a perfect crystal is $|\langle {{q}_{6}}\rangle {{|}^{2}}=1$, independent of system size.

Cluster size distributions.

In thermal equilibrium the number of particles N in a volume V in a grand canonical ensemble fluctuates with standard deviation $\Delta N=\sqrt{\langle {{N}^{2}}\rangle -{{\langle N\rangle}^{2}}}\sim \sqrt{N}$ [388]. In contrast, self-propelled particle systems that are out of equilibrium often show giant number fluctuations indicated by $\Delta N\sim {{N}^{\alpha}}$ with $0.5<\alpha \leqslant 1$. This was predicted [46, 351, 389, 390] and first demonstrated in experiments with vibrated granular rods [304] (see also the discussion in [391]). Giant number fluctuations were also reported in dynamic clustering of active colloids [10] and in simulations of active Brownian particles [153, 154, 168]. However, electrostatic and hydrodynamic interactions suppress giant number fluctuations in dense suspensions of polarly ordered Quincke rollers [11].

So far, in section 4.3, we have described quantitative measures using particle positions ${{\mathbf{r}}_{i}}$. In the following, we consider particle orientations ${{\mathbf{e}}_{i}}$ and velocities ${{\mathbf{v}}_{i}}$ to characterize orientational order in the system.

Polar order.

To quantify global polar order in a suspension of active colloids, the order parameter

$$\begin{eqnarray}&&P(t)=\,|\langle \mathbf{e}(t)\rangle |\,=\frac{1}{N}|\underset{i}{\sum}\,{{\mathbf{e}}_{i}}(t)|\end{eqnarray} \tag{ 59 }$$

was used in squirmer suspensions [333, 339–341] or for active particles in a harmonic trap [313]. In all these cases the particles are spherical and the aligning mechanisms have a pure hydrodynamic origin. In the long-time limit P(t) typically reaches a steady state with $\mathcal{P}=\text{li}{{\text{m}}_{t\to \infty}}P(t)$. For fully developed polar order $\mathcal{P}=1$. However, even an ensemble of randomly oriented particles shows a residual value for the order parameter: $\mathcal{P}\approx {{N}^{-1/2}}$ [333]. Interestingly, as reviewed in section 4.2.3 and demonstrated for collectively moving Quincke rollers, the formation of polar order at a critical density or a critical Péclet number can be mapped to the onset of ferromagnetic order treated within mean-field theory [11, 313]. In the continuum limit the polarization becomes $\mathcal{P}=\,|{{{\int}^{}}_{{{\mathbb{S}}_{2}}}}\Phi\left(\mathbf{e}\right)\mathbf{e}\text{d}S|$ [11, 313], where $\text{d}S$ is the surface element on the unit sphere ${{\mathbb{S}}_{2}}$, and $\Phi\left(\mathbf{e}\right)$ the orientational distribution function in steady state.

To measure the temporal correlations of the orientation vector $\mathbf{e}$, the autocorrelation function ${{C}_{\text{p}}}=\langle \mathbf{e}(t)\centerdot \mathbf{e}(0)\rangle$ from equation (25) for a single particle is calculated and then averaged over all particles, ${{\bar{C}}_{\text{p}}}=\overline{\langle \mathbf{e}(t)\centerdot \mathbf{e}(0)\rangle}$. If the decay of ${{\bar{C}}_{\text{p}}}$ is exponential, ${{\bar{C}}_{\text{p}}}={{\text{e}}^{-2D_{\text{r}}^{\text{eff}}t}}$, $D_{\text{r}}^{\text{eff}}$ defines an effective rotational diffusion constant. For example, due to hydrodynamic or phoretic interactions it can have a larger value than the coefficient D_r of an isolated swimmer [118, 161, 333]. Hence, $D_{\text{r}}^{\text{eff}}$ quantifies the orientational persistence of interacting microswimmers.

Spatial correlations of particle orientations at distance r and at time t are described with the equal-time polar pair correlation function [132]

$$\begin{eqnarray}&&{C_e}(r,t) = \frac{{\langle \sum\nolimits_{i \ne j} {{{({{\bf{e}}_i}(t) \cdot {{\bf{e}}_j}(t)\rangle }^2})\delta (|{{\bf{r}}_i} - {{\bf{r}}_j}| - r)} \rangle }}{{\langle \sum\nolimits_{i \ne j} {\delta (|{{\bf{r}}_i} - {{\bf{r}}_j}| - r)} \rangle }}.\end{eqnarray} \tag{ 60 }$$

The average is taken over all pairs of swimmers i, j. Note that the maximum distance r in equation (60) is set by the systems size of the experiment or simulation. In experiments it is often more convenient to measure particle velocities ${{\mathbf{V}}_{i}}$ instead of their orientations ${{\mathbf{e}}_{i}}$. Then, the spatial correlation function C_V(r, t) is defined as C_e(r, t) in equation (60) but with orientation vectors replaced by velocities.

From the decay of the time-averaged correlation functions ${{C}_{e}}(r)=\frac{1}{T}{\int}_{0}^{T}{{C}_{e}}(r,t)\text{d}t$ and ${{C}_{V}}(r)=\frac{1}{T}{\int}_{0}^{T}{{C}_{V}}(r,t)\text{d}t$, one can determine correlation lengths l_c for orientations and velocities in the system [8, 132, 276, 333, 334, 353, 392, 393]. Anticorrelations (C_e  <  0 or C_V  <  0) indicate the formation of vortices, for example, in turbulent bacterial suspensions [276, 394].

To characterize the mobility of collectively moving active colloids, the dynamic order parameters

$$\begin{eqnarray}&&{{V}_{1}}(t)=\,|\langle \mathbf{V}(t)\rangle |\,=\frac{1}{N}|\underset{i}{\sum}\,{{\mathbf{V}}_{i}}(t)|\end{eqnarray} \tag{ 61 }$$

and

$$\begin{eqnarray}&&{{V}_{2}}(t)=\langle \mathbf{V}(t)\centerdot \mathbf{e}(t)\rangle =\frac{1}{N}\underset{i}{\sum}\,{{\mathbf{V}}_{i}}(t)\centerdot {{\mathbf{e}}_{i}}(t)\end{eqnarray} \tag{ 62 }$$

are used. While V₁ simply measures the mean speed of the particles (see, e.g. [332, 346]), V₂ determines how fast they move along their intrinsic directions ${{\mathbf{e}}_{i}}$ (see, e.g. [118, 161, 346]). Possible deviations of V₂ from the swimming velocity v₀ always indicate colloidal interactions.

We note that equal-time correlation functions C_v(r, t) and ${{C}_{\omega}}(r,t)$ are formulated for the hydrodynamic flow field $\mathbf{v}\left(\mathbf{r},t\right)$ and vorticity field $\boldsymbol{\omega }\left(\mathbf{r},t\right)=\nabla \times \mathbf{v}\left(\mathbf{r},t\right)$ created by the motion of microswimmers (see, e.g. [395]). Particularly interesting are the Fourier transforms of the time-averaged correlation functions, ${\int}^{}{{C}_{v}}(r,t)\text{d}t$ and ${\int}^{}{{C}_{V}}(r,t)\text{d}t$, which are used to define the energy spectrum E(k) in an active fluid. It was applied to characterize active turbulence in bacterial [169] and active colloidal [13] suspensions. An order parameter for the strength of vorticity in the fluid is the so-called enstrophy, which is the spatial average of $\boldsymbol{\omega }^{{2}}$.

In order to quantify the temporal evolution of N interacting active colloids by means of their trajectories ${{\mathbf{r}}_{i}}(t)$, the particle-averaged mean square displacement

$$\begin{eqnarray}&&\langle \Delta {{r}^{2}}(t)\rangle =\frac{1}{N}\underset{i=1}{\overset{N}{\sum}}\,|{{\mathbf{r}}_{i}}(t)-{{\mathbf{r}}_{i}}(0){{|}^{2}}\rangle \end{eqnarray} \tag{ 63 }$$

is used [118, 154, 155]. Recall that for a single particle (equation (30)) the motion is ballistic (∼t²) at small times and diffusive ($\sim$t) at large times. Collective motion can introduce an intermediate superdiffusive regime  ∼  t^3/2 [118, 155] and interactions between active particles typically reduce the long-time diffusion constant extracted from equation (63) compared to free swimmers.

In equilibrium thermodynamics macroscopic state variables such as temperature T and pressure p are well defined quantities. The values calculated from their microscopic definitions coincide with the values obtained by introducing them as first derivates of the thermodynamic potentials. Temperature and pressure are connected via an equation of state; for example, for an ideal gas one has pV  =  nRT.

In non-equilibrium thermodynamics it is in general not possible to define state variables such as T and p in a meaningful way. Nevertheless, very recently quite a lot of effort has been made in formulating a thermodynamic description of active matter systems (see, e.g. the discussions in [40, 42, 184, 396–398]). For example, as already discussed in section 4.3.1, motility-induced phase separation can sometimes be addressed with an effective free energy [40] and the activity-induced collisions of the particles mapped to an effective attraction potential [319]. Another example is the controversial discussion about the concept of effective temperature in active particle systems [71, 184, 399–402]. Indeed, sometimes one can assign an effective temperature to the stochastic motion of active colloids. Examples are the persistent random walk in bulk (see section 3.1) or the sedimentation profile of non-interacting active particles in a gravitational field (see section 3.2). In general, however, it is not possible to treat the stochastic dynamics of microswimmers with an effective temperature.

Active pressure.

Very recently some research groups have started to work on defining pressure in active particle suspensions [42, 159, 270, 271, 315, 316, 396, 398, 403–411] by using various definitions for pressure. First, one views pressure as the mechanical force which an active particle system in a container exerts per unit area on bounding surfaces. Second, in bulk it should be the trace of the hydrodynamic stress tensor. Third, pressure can be calculated as the derivative of a free energy with respect to volume. Since, in general, a free energy does not exist for active particle systems, only the first two, microscopic definitions of pressure are usually accessible.

However, although all pressure definitions coincide in thermal equilibrium, they do not for active particle systems out of equilibrium. Based on work from [113, 315, 316], Solon et al showed [404] for interacting active Brownian disks that the mechanical pressure acting on the walls of a container contains the sum of three terms including the active pressure (also called swim pressure) [113, 316, 404, 408]

$$\begin{eqnarray}&&{{p}_{\text{A}}}=\frac{\gamma {{v}_{0}}}{2V}\underset{i}{\sum}\,\langle {{\mathbf{r}}_{i}}\centerdot {{\mathbf{e}}_{i}}\rangle \,,\end{eqnarray} \tag{ 64 }$$

where v₀ and γ are particle speed and friction constant, respectively. V is the area (2D) or volume (3D) of the simulation box and the average $\langle \ldots \rangle$ is taken over time. This expression is independent of the swimmer–wall interaction potential and hence a state function [404]. However, in general, pressure is not a state function, as demonstrated, for example, for active particles, which transfer torques to bounding walls during collision [398].