2. Theory

For active colloidal particles there are three characteristic lengths: (i) the macroscopic length scale $L$ , (ii) the run length $\ell =U_{0}{\it\tau}_{R}$ and (iii) a microscopic length ${\it\delta}=\sqrt{D_{T}{\it\tau}_{R}}$ , where $D_{T}$ is the translational diffusivity of the active particles. Although in a typical application we expect $L>\ell \gg {\it\delta}$ , the theory we present is valid for any ratio of length scales.

Active Brownian particles (ABPs) are governed by the Smoluchowski equation for the probability density for finding a swimmer at $\boldsymbol{x}$ with orientation $\boldsymbol{q}$ :

The translational and rotational fluxes are: $\boldsymbol{j}^{T}=(U_{0}\boldsymbol{q}+\boldsymbol{F}^{P}/{\it\zeta}-D_{T}\boldsymbol{{\rm\nabla}}\ln P)P$ and $\boldsymbol{j}^{R}=-D_{R}\boldsymbol{{\rm\nabla}}_{R}P$ , where $\boldsymbol{{\rm\nabla}}_{R}=\boldsymbol{q}\times \boldsymbol{{\rm\nabla}}_{\boldsymbol{q}}$ is the orientational gradient operator. For a spherical swimmer of radius $a$ in a Newtonian solvent of viscosity ${\it\eta}$ , ${\it\zeta}=6{\rm\pi}{\it\eta}a$ , $D_{T}=k_{B}T/{\it\zeta}=k_{B}T/6{\rm\pi}{\it\eta}a$ , $D_{R}\;(=1/{\it\tau}_{R})=k_{B}T/8{\rm\pi}{\it\eta}a^{3}$ and ${\it\delta}=\sqrt{D_{T}/D_{R}}=\sqrt{4/3}a$ .

At a boundary surface the normal component of the translational flux must vanish, $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{j}^{T}=0$ . If there were no translational Brownian motion or boundary force ( $\boldsymbol{F}^{P}=0$ ), then $U_{0}(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{q})P=0$ , which means that either (i) $U_{0}=0$ or (ii) $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{q}=0$ or (iii) $P=0$ at the surface, none of which is true in general. It is essential to have a strong enough boundary force or translational Brownian diffusion (or both, or hydrodynamics) to prevent particle crossing. As is well known in colloidal dynamics, a hard-particle repulsive force is infinite and non-zero only at the boundary surface and the no-flux condition is satisfied via the Brownian flux.

Rather than having a boundary force of finite range and amplitude or hydrodynamic lubrication interactions to prevent particle flux, we choose to use $D_{T}$ as this is the simplest to implement theoretically and most easily reveals the underlying physics. It is important to note that whatever means is used to prevent active particles from crossing a boundary it will introduce a microscopic length scale ${\it\delta}$ . As we shall see, for pressures and forces, ${\it\delta}$ will not appear in the final results. Any of the mechanisms would produce the same behaviour.

Indeed, Ezhilan, Alonso-Matilla & Saintillan (Reference Ezhilan, Alonso-Matilla and Saintillan2015) recently examined active particles in 2D confined between two walls without translational Brownian motion ( $D_{T}\equiv 0$ ) and showed that the problem could be modelled with two regions: freely swimming bulk behaviour connected to a singular surface layer of particles in contact with the walls. The action of translational Brownian motion is to spread this singular surface layer over the microscopic thickness ${\it\delta}$ adjacent to the walls, as is standard in boundary-layer theory. Our planar 2D results are in agreement with their findings.

Although we speak in terms of translational Brownian motion and forces proportional to $k_{B}T$ , this is not necessary. One can simply replace $k_{B}T$ with ${\it\zeta}D_{T}$ and the results are unchanged; the translational diffusion, like the rotary diffusion $D_{R}$ , need not be thermal in origin. The Smoluchowski equation only requires that the random ‘Brownian’ displacements be small compared to any other length scale (e.g. the swimmer’s size).

The Smoluchowski equation applies for all length and time scales but its solution in any but the simplest situations is challenging. We need a simplified description that captures the essential physics and, more importantly, provides insight into the general behaviour and can explain phenomena without detailed calculations.

Consider a body immersed in a dilute suspension of ABPs. With $\boldsymbol{F}^{P}=0$ , the force the active colloidal particles exert on the body is given exactly by (Brady Reference Brady1993; Squires & Brady Reference Squires and Brady2005) $\boldsymbol{F}=-k_{B}T\int _{S_{B}}P(\boldsymbol{x},\boldsymbol{q},t)\boldsymbol{n}\,\text{d}S$ , where $\boldsymbol{n}$ is the outer normal to the body surface as shown in figure 3. The force averaged over the orientation of the active particles is

where $n(\boldsymbol{x},t)\equiv \int P(\boldsymbol{x},\boldsymbol{q},t)\,\text{d}\boldsymbol{q}$ is the number density of swimmers.

The conservation equations for the zeroth and first moments of the Smoluchowski equation are (Saintillan & Shelley Reference Saintillan, Shelley and Spagnolie2015)

where $\boldsymbol{m}(\boldsymbol{x},t)=\int \boldsymbol{q}P(\boldsymbol{x},\boldsymbol{q},t)\,\text{d}\boldsymbol{q}$ is the polar-order field and $\unicode[STIX]{x1D64C}(\boldsymbol{x},t)=\int (\boldsymbol{q}\boldsymbol{q}-\unicode[STIX]{x1D644}/3)P(\boldsymbol{x},\boldsymbol{q},t)\,\text{d}\boldsymbol{q}$ is the nematic order field. Since the force on a body only involves the number density at the surface, we can use the simplest closure of the hierarchy $\unicode[STIX]{x1D64C}=0$ . We show below (and discuss in appendix B) that this closure is sufficient to achieve good accuracy and reveals the essential physics.

Two remarks will help in understanding the structure of the moment equations. First, when departures from uniformity vary slowly, the $\boldsymbol{m}$ -field equation has a balance between the ‘sink’ term and the gradient in the concentration, $2D_{R}\boldsymbol{m}\approx -(U_{0}\boldsymbol{{\rm\nabla}}n)/3$ , which gives a diffusive flux in the concentration field that incorporates the swim diffusivity: $\boldsymbol{j}_{n}\approx -(D_{T}+(U_{0}^{2}{\it\tau}_{R})/6)\boldsymbol{{\rm\nabla}}n$ . Second, at the other extreme when variations are rapid, the $\boldsymbol{m}$ -field has a natural screening length where diffusion balances the sink: $D_{T}{\rm\nabla}^{2}\boldsymbol{m}\approx 2D_{R}\boldsymbol{m}$ . This screening length is proportional to the microscopic length ${\it\delta}=\sqrt{D_{T}/D_{R}}$ . The screening length plays a fundamental, but unusual, role in active matter – it is essential in order to have a well-posed problem and there will be rapid variations in properties on the scale of ${\it\delta}$ , but in the limit where ${\it\delta}\ll \ell ,L$ , the microscopic length does not appear in the active pressure or in the forces and torques on boundaries. The athermal limit ( $D_{T}\rightarrow 0$ ) is singular and $D_{T}$ (or $k_{B}T$ ) can only be set to zero after the limit is taken; the force from (2.2) will then be independent of  $k_{B}T$ .

3. Examples

First, we consider an infinite flat plate with normal along the $z$ -direction; there is no macroscopic length scale. The $n$ - and $\boldsymbol{m}$ -fields are subject to no flux at $z=0$ : $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{j}_{n,m}=0$ and a uniform concentration and no polar order as $z\rightarrow \infty$ : $n\sim n^{\infty }$ and $\boldsymbol{m}\sim 0$ . The concentration and polar-order fields are simple exponentials decaying on the screening length

where ${\it\lambda}=\sqrt{2(1+(\ell /{\it\delta})^{2}/6)}/{\it\delta}$ is the inverse screening length.

The concentration at the wall, $n(0)=n^{\infty }(1+(\ell /{\it\delta})^{2}/6)$ , is independent of the closure (which follows directly from (2.1) in 1D), always exceeds that far away and can become very large as $\ell /{\it\delta}\rightarrow \infty$ . This ‘infinite’ concentration applies for a dilute suspension. It is not a build-up associated with a finite concentration of active particles. Rather, it is the singularity alluded to earlier that results if translational Brownian motion (or a microscopic length) is not considered. (The active particle size $a$ must be taken into account in defining the no-flux surface $z=0$ .)

Even though the concentration can become arbitrarily large, the force per unit area or pressure on the wall from the microscopic force definition (2.2) is finite and independent of ${\it\delta}$ : ${\it\Pi}^{W}=n(0)k_{B}T=n^{\infty }(k_{B}T+k_{s}T_{s})$ , where we have defined the swimmer’s ‘activity’ $k_{s}T_{s}={\it\zeta}U_{0}^{2}{\it\tau}_{R}/6=k_{B}T(\ell /{\it\delta})^{2}/6$ . We recognize the pressure on the wall as the active pressure – the sum of the osmotic pressure of Brownian particles plus the swim pressure. And note that this is true regardless of the relative magnitudes of $k_{B}T$ and $k_{s}T_{s}$ (including the singular athermal limit $k_{B}T=0$ ). This same independence of $k_{B}T$ occurs in the analogous hard-sphere rheology problem at high Péclet numbers (Brady & Morris Reference Brady and Morris1997; Squires & Brady Reference Squires and Brady2005). Also, the ratio $(\ell /{\it\delta})^{2}=6D^{swim}/D_{T}=U_{0}\ell /D_{T}=Pe_{\ell }$ is a Péclet number based on the run length that measures the relative importance of swimming to Brownian diffusion.

The second problem is active Brownian particles confined between two parallel plates separated by a distance $H$ . The concentration distribution is

where the constant $n_{0}$ is related to the average number density of ABPs in the channel $\langle n\rangle =\int _{0}^{H}n(z)\,\text{d}z/H$ . The concentration is identical at both walls and is the same as for a single wall with $n_{0}$ replacing $n^{\infty }$ . In the limit of large ${\it\lambda}H$ , corresponding to ${\it\delta}\ll H$ , and when ${\it\delta}\ll \ell$ , $n_{0}\sim \langle n\rangle [1+(\ell /H)/\sqrt{3}]^{-1}$ and the pressure at the walls becomes

As for a single wall, the pressure is independent of the microscopic length scale ${\it\delta}$ but now depends on the ratio of the run length to the macroscopic scale $\ell /H$ . We shall see that this behaviour is generic – the influence of the run length enters as $\ell /L$ . In a simulation study Ray, Reichhardt & Reichhardt (Reference Ray, Reichhardt and Reichhardt2014) observed that the pressure in a channel depends on the gap spacing as predicted by (3.3) (in 2D the coefficient is $1/\sqrt{2}$ ).

Figure 1. ${\it\Pi}^{W}$ of ABPs confined between parallel walls in 2D. The inset shows the area fraction distribution ${\it\phi}(z)$ . Here, $\ell =U_{0}{\it\tau}_{R}$ is the run length and ${\it\delta}=\sqrt{D_{T}{\it\tau}_{R}}$ is the microscopic length. The swimmer’s radius is $a$ .

Figure 1 compares the concentration profile and pressure for a channel from the theory with results from ABP dynamic simulations (appendix A). Also shown are the theoretical predictions from closing the hierarchy at the next level including the nematic order field $\unicode[STIX]{x1D64C}$ as described in appendix B. The $\boldsymbol{m}$ -field closure is sufficient, both qualitatively and quantitatively.

The next problems are the concentration and pressure distribution in 3D outside and inside a sphere, and in 2D outside and inside a circle, of radius $R$ . Symmetry dictates that $n(\boldsymbol{x})=n^{\infty }f(r)$ and $\boldsymbol{m}(\boldsymbol{x})=n^{\infty }\boldsymbol{x}g(r)$ , where $f(r)$ and $g(r)$ are scalar functions of $r$ . The exterior solution in 3D has the form of an exponentially screened concentration reminiscent of Debye screening:

and similarly for the $\boldsymbol{m}$ -field. In 2D Bessel functions replace the exponential:

where the 2D inverse screening length ${\it\lambda}^{\prime }=\sqrt{1+(\ell /{\it\delta})^{2}/2}/{\it\delta}$ , $\text{K}_{0,2}$ are the modified Bessel functions and $n_{A}^{\infty }$ is the area number density at infinity. For large ${\it\lambda}^{\prime }r$ the concentration disturbance decays as ${\sim}\text{e}^{-{\it\lambda}^{\prime }r}/\sqrt{r}$ .

The pressure at the sphere surface in the dual limits ${\it\delta}\ll \ell$ and ${\it\delta}\ll R$ , but for arbitrary $\ell /R$ , is

while for the circle

where $k_{s}T_{s}^{\prime }={\it\zeta}U_{0}^{2}{\it\tau}_{R}/2$ is the activity in 2D. We again see the effect of the finite run length entering as $\ell /R$ .

For the spherical interior problem the concentration field is given by

while for the interior problem in 2D

with $\text{I}_{0,2}$ modified Bessel functions. In the dual limits ${\it\delta}\ll \ell$ , ${\it\delta}\ll R$ , the interior pressure in 2D is identical to (3.7) with $\langle n_{A}\rangle$ replacing $n_{A}^{\infty }$ .

Figure 2. ${\it\Pi}^{W}$ of ABPs outside and inside a circle. Legends are as in figure 1.

Figure 3. Theoretical predictions of the force on an asymmetric body in 2D with curvatures $R$ and $2R$ compared to ABP simulations. The symbols are simulation results and the solid lines were obtained by numerically solving for the $n,\boldsymbol{m}$ fields. The force is calculated from (2.2) with body normal as $\boldsymbol{n}$ and is scaled by the bulk active pressure ${\it\Pi}^{\infty }=n(k_{B}T+k_{s}T_{s})$ .

Figure 2 compares the predicted results in 2D for the exterior and interior problems with ABP simulations and the next level $\unicode[STIX]{x1D64C}$ theory (by symmetry $\unicode[STIX]{x1D64C}=h(r)\unicode[STIX]{x1D644}+s(r)\boldsymbol{x}\boldsymbol{x}$ ). Again, the $\boldsymbol{m}$ -level theory is quantitatively accurate unless $R/{\it\delta}<5$ , which is not unexpected.

By symmetry there is no net force on a sphere or a circle in an active suspension. The Brownian osmotic pressure is independent of both ${\it\delta}$ and $\ell$ (as it must be) and thus the integral of the constant Brownian osmotic pressure over the surface of any body will be zero.

From the examples the swim pressure has a correction due to the finite run length, ${\it\Pi}^{swim}\sim k_{s}T_{s}/[1+{\it\alpha}(\ell /R)]$ , where ${\it\alpha}$ is a constant and $R$ is the curvature of the body. Thus, in the limit $\ell /R\ll 1$ the swim pressure is a constant at each point on the body surface and there will again be no force no matter what its shape. This is as one would expect from the pressure for a macroscopic object. Only when the run length is comparable to the local radius of curvature of a body is it possible to have a net force from the swimmers’ activity.

Equation (2.2) for the force applies to any body shape and for any body size. Figure 3 compares the force on an axisymmetric body in 2D determined by solving the $n,\boldsymbol{m}$ fields numerically with ABP simulations. The unsteady equations (2.3)–(2.4) were solved numerically with a standard Galerkin P2-FEM method with adaptive mesh refinement. Implicit time-stepping was used to ensure solution stability, and the solution is tracked long enough ( ${\sim}150{\it\tau}_{R}$ ) to reach a steady state. The agreement is excellent. If the body were free to move, its speed would result from the balance of its Stokes drag, $-\unicode[STIX]{x1D64D}_{FU}\boldsymbol{\cdot }\boldsymbol{U}$ , with the active force, where $\unicode[STIX]{x1D64D}_{FU}$ is the hydrodynamic resistance tensor for the body. A body may also rotate if the active pressure exerts a net torque on the body, which is given by $\langle L\rangle _{\boldsymbol{q}}=-k_{B}T\oint _{S_{B}}(\boldsymbol{x}-\boldsymbol{x}_{0})\times n\boldsymbol{n}\,\text{d}S$ , where $\boldsymbol{x}_{0}$ is the point about which the torque is taken. The angular velocity then follows from the hydrodynamic resistance tensor coupling torque and rotation, $-\unicode[STIX]{x1D64D}_{L{\it\Omega}}\boldsymbol{\cdot }{\it\bf\Omega}$ .