To describe the microbe’s motion, we model microswimmers as active Brownian particles (ABP) in two dimensions [37]. Here, each swimmer moves at constant speed u ₀ with a persistent orientation e ̂ ( t ) that undergoes rotational diffusion. That is, the orientation changes smoothly on time as compared to run-and-tumble particles (RTP), for which the director changes abruptly at discrete events. Although different, it has been shown that at large times, both models present similar dynamics and there is a direct mapping of the tumbling rate to rotational diffusion coefficient D _R [38]. The microswimmers are circular particles with diameter 2a that interact between them by excluded volume only, and no mutual alignment takes place. The ABP model, despite its simplicity, is known to reproduce many of the observed properties of microswimmers, in particular the accumulation near walls, regime where it has also been shown that the key features are equivalent to other models of active particles [39]. On the other hand, the simplicity of the ABP model, characterized by a few parameters, allows for systematic analysis and to unveil the key features of relevant phenomena for a wide range of microswimmers, without needing to model specific details of each microswimmer under study. Finally, we restrict to spherical swimmers as it has been shown that considering the ellipticity only changes quantitatively the results, with the same phenomenology as for spherical particles for the study of accumulation in surfaces [23]. The use of this model here shows how different accumulation regimes appear as a function of the self-propulsion speed compared to the imposed flow.

There is a single circular obstacle of radius R, which is impenetrable by the swimmers. At short distances, due to hydrodynamic interactions, pusher swimmers, like bacteria, are attracted to solid surfaces and they are aligned to swim parallel to them [2]. To correctly describe this interaction, for example, to get finite-induced velocities, near field hydrodynamics should be considered [40, 41], which are specific for each microbe. Instead, to mimic this effect in a more general way, without introducing hydrodynamic interactions, which are also computationally expensive, we introduce a short-range attractive force that exerts the obstacle on the swimmers and a torque that aligns them. The whole system is subject to an imposed external flow. We assume that the swimmer concentration is low enough such that the induced flow generated by them can be neglected. Hence, the form of this velocity profile is simply the one that results from the interaction of the external flow with the obstacle. Finally, the modeling is done in two spatial dimensions; the extension to three dimensions is direct.

The swimmers’ motion is completely described by the low Reynolds dynamics, i.e., inertia can be completely neglected. Hence, instead of forces and torques, it is more convenient to describe interactions by the induced linear and angular velocities they generate. Thus, the equations of motion for the position r ⃗ i of swimmer i are d r ⃗ i d t = u 0 e ̂ i + u r ⃗ i + u ⃗ O r ⃗ i + ∑ i ≠ j u ⃗ S | r ⃗ i − r ⃗ j | , (1) where the first term is the self-propulsion along the director, the second term is the drift produced by the external flow, and the last two terms are the induced velocities produced by the interaction with the obstacle and other swimmers, respectively. Similarly, for the director e ̂ i , d e ̂ i d t = 1 2 ∇ × u ⃗ r ⃗ i + Ω ⃗ O r ⃗ i , e ̂ i + 2 D R ξ ⃗ × e ̂ i , (2) where the first term describes the rotation induced by the flow vorticity, that for a spherical swimmer adopts this simple form [42, 43], the second one accounts for the reorientation of the swimmer by the interaction with the obstacle, and the last one is a stochastic term, with ξ a white noise, that produces rotational diffusion with a diffusion coefficient D _R . The cross product with the director guarantees that remains unitary. No swimmer-swimmer torques are considered.

For an ambient fluid with density ρ, viscosity η, and a small imposed flow U ∞ x ̂ far from the obstacle, the presence of the obstacle centered at the origin of the coordinate system results in the velocity profile u ⃗ ( r ⃗ ) = U ∞ 1 + R 3 2 r 3 − 3 R 2 r cos ⁡ θ r ̂ − 1 − R 3 4 r 3 − 3 R 4 r sin ⁡ θ θ ̂ . Here, r = | r ⃗ | is the distance to the origin, θ is measured with respect to the x ̂ axis and ( r ̂ , θ ̂ ) is the pair of polar unit vectors.

The obstacle has radius R and the interaction is modeled with a potential U _O , such that u ⃗ O ( r ⃗ ) = − ∇ U O ( r ⃗ ) . The potential is attractive in the adhesion region, R < r _i < ɛ ₀ + R, and is repulsive for r < R, describing the rigidity of the surface (see Figure 1A). This interaction mimics, in a simple way, electrostatic, hydrodynamic, or substrate effects on surfaces [16, 32, 33, 35], and allows for reversible adhesion to the surface. For simplicity, we use a Morse potential to have a simple expression valid for all distances: U O r ⃗ i = U O e − 2 β M r i − r M − 2 e − β M r i − r M . (3)

When the microswimmer is close to the surface of the obstacle, there is also a torque that aligns the swimmer with the surface. For simplicity, we consider that the induced angular velocity is Ω ⃗ O r ⃗ i , e ̂ i = Ω O r ̂ i × e ̂ i , if  | r ⃗ i | < R + ε 0 0 , otherwise , (4) with the same range as the interaction potential.

For the swimmer-swimmer interaction, we use a simple repulsive Yukawa potential U ⃗ S r ⃗ = U S e − r / 2 a / r , (5) which gives the induced velocity u ⃗ S ( r ⃗ ) = − ∇ U S ( r ⃗ ) .

The simulation is performed in a stripe of size L _y = 4R in the vertical direction and unbounded in the x direction. Periodic boundary conditions are used in the y direction. To generate a continuous injection of microswimmers that approach the obstacle, particles are released periodically, every τ _wave, at a distance d ₀ = 3R, randomly distributed along L _y . Each wave is composed of N = 100 microswimmers, uniformly distributed in the chamber all pointing initially in the positive x ̂ direction (see Figure 1A). The distance to the obstacle is sufficient for the swimmers to randomize and in the different observables that quantify the accumulation of swimmers in the obstacle, and there is no signature of the periodicity τ _wave.

The model has several parameters, characterizing the motion of the swimmers, their mutual interaction, and the interaction with the wall, as well as the properties of the imposed flow and obstacle size. In this study, we focus on varying the swimmer’s speed u ₀, the imposed flow U _∞ , and the obstacle radius R. The rest of the parameters are fixed to represent typical experimental and natural conditions.

We set the microswimmer’s diameter 2a = 1 µm in the Yukawa potential, as the length scale of the problem. The time scale of the problem is set by the rate of particle injections τ _wave = 1 s. Hence, in what follows, all lengths and times, and the derived units, are expressed as dimensionless quantities.

We define the obstacle adhesion region in ɛ ₀ = 7, which accounts for the typical hydrodynamic effects in the vicinity of the obstacle [19, 20, 28]. The intensity of the interactions is rather arbitrary as it is only needed that excluded volume is accurately achieved. We use U _S = 2, U _O = 3.2, β _M = 1.44, and Ω _O = 0.28, which are sufficient to enforce the excluded volume with the integration time step Δt/τ _wave = 1 × 10^–3 and a rapid alignment with the obstacle. We consider two microswimmer types, with very different persistences, characterized by their rotational diffusion coefficient: more-persistent microswimmers with D _R = 0.16 [44, 45], and less-persistent ones with D _R = 0.6 [46]. This classification is related with different bacterial strains modified and used for medical or experimental tasks [17].

We solve the equations of motion (1) and (2) using molecular dynamics simulations with the Euler-Maruyama integration method, for a total time of 200 s. To improve the computational efficiency, we implemented cell lists for the particle-particle interactions and an effective cut-off for the particle-obstacle interaction in order to avoid unnecessary interactions when their distance is large [47].

To study how the activity u ₀, obstacle radius R, and external flow U _∞ affect the first adhesion of microbes, we performed three different studies varying different parameters, for both microswimmer’s types.

i. Microswimmer activity: in this case, we will fix the obstacle radius R = 100 and the external flow U _∞ = 40, unless otherwise indicated. We study bacterial activity in the range u ₀ = 14, … , 65.

With these set of parameters, the concentration of swimmers in the bulk of the system is dilute. Yet, still accounts a considerable accumulation of microswimmers on the obstacle surface. We performed 24 different simulations for each studied parameter combination and, for all cases, we show the average results.

For all considered cases of velocities and obstacle radii, the temporal dynamics is rather similar. First, it takes a time t ₀ ∼ 3R/(u ₀ + U _∞ ) for the first swimmers that were injected into the system to reach the obstacle. After this time, there is a continuous income of swimmers to the obstacle. Some of them will reach the adhesion region and remain there while swimming and being advected by the flow. Interactions between swimmers create crowded environments that enhance the residence time in this zone but, also, it is possible to scatter bacteria from the surface after an encounter, helping their erosion by the external flow. As a whole, the total number of particles in the adhesion zone N(t) starts to increase steadily after t ₀ until it saturates to the steady value N _trap (see Figure 1B and the Supplementary Video S1). In all cases, the average growth curves can be well fitted to the model ⟨ N t ⟩ = N trap 1 − e − t − t 0 / τ trap , (6) where τ _trap gives the relaxation time to reach the steady state, similar to the probability of successful interaction presented in Refs. [28, 48]. Considering that the incoming rate is constant, having Eq. 6 as solution of the balance equation implies that the desorption rate is proportional to the actual number of particles in the adhesion layer. In the steady state, the obstacle is saturated and ready for microbes to form the first adhesion [6, 8]. From the simulations, we will obtain τ _trap and N _trap, which are important parameters to characterize and optimize the microbe’s adhesion in convex surfaces.

Figure 2 presents snapshots of the system in the three regimes that are described in the text for the transient at t = τ _trap and in the steady state. In the transient, the distribution is not uniform with particles still being transported along the perimeter, except for the direct interception regime, where the distribution is uniform, although with less particles than in the steady state. In all cases, it is seen that the steady-state distribution in rather uniform in the circle, contrary to other studies where there is a larger accumulation in the back [20, 23]. The three regimes differ notably on the number of accumulated particles.

Another relevant observable is the contact time of microswimmers with obstacle’s surface, τ _contact. This parameter gives the average residency time of microbes on the surface and therefore the time available to realize an irreversible adhesion to prompt a biofilm. It is measured, for each set of parameters, as the mode considering 24 realizations of the time that particles spend inside the adhesion region.

The number of trapped particles can be compared to the maximum occupation in a monolayer, N _max ≡ 2πR/(2a), which allows us to define the dimensionless average number of deposited layers λ _trap = N _trap/N _max. Similarly, the relaxation time and the contact time can be compared to the time it takes a swimmer to travel the obstacle by its own, τ _s ≡ πR/u ₀.

Using dimensional analysis, we expect that the microbial behavior depends on the Péclet number which compares advective transport with diffusion Pe = u ₀/(RD _R ). Then, in the limit of Pe → 0, we expect that Brownian diffusivity dominates microswimmer’s exploration of the medium, the phenomenon is known as “Brownian deposition.” While in the other limit Pe → ∞, the advection dominates and particles encounter the obstacle surface by “direct interception” [13, 14]. When varying the microswimmer’s activity, in a biological range of velocities [8], we are changing the Péclet number in a narrow window for each microswimmer’s type, and the two limiting cases are not always achieved. Furthermore, the external velocity allows to define new dimensionless parameters. Therefore, for simplicity, we present the results in terms of the control parameters, where the transition between both regimes can also be appreciated.

Here we keep the obstacle radius constant to R = 100 and vary the swimmer’s speed u ₀. For the imposed flow, we consider three different values: U _∞ = 20, 40, and 60. We found that depending on the microswimmer’s activity and external flow there are, basically, two different regimes. In one of them, the microswimmer’s velocity is smaller compared with external flow, yet particles diffuse across the streamlines and settle around the obstacle covering the whole perimeter and forming multiple layers, this regime is known as Brownian deposition [13, 14]. In the second regime, when particle’s activity is larger than the external flow, particles scatter faster forming approximately one layer on the obstacle surface. The particle’s capture now depends only on the direct interception with the obstacle. In Figure 3, we show λ _trap, τ _trap, and the contact time τ _contact for the case U _∞ = 40, for both values of persistence. These three observables decrease with the parameter u ₀/U _∞ . Naturally, as the swim speed increases, the relaxation and contact times decrease accordingly. Also, the thickness of the deposited layer decreases as particles can escape more easily due to excluded volume interactions with other microswimmers.

Now, we vary the obstacle radius R, while keeping fixed the microswimmer’s activity to u ₀ = 20 and the external flow to U _∞ = 40, unless otherwise indicated. The number of layers λ _trap decreases with R (Figure 5A). Small pillars are capable to adhere more than two layers of microbes and for large radii, the number of layers saturates to a value slightly larger than one. The results are well fitted to the expression λ _trap(R) = A[1 + exp(−R/R ₀)], with A = 1.24 and R ₀ = 65.2, independent of microswimmer’s type and external flow. With this, the total number of accumulated particles increases monotonically with R. Microbe’s capture is in agreement with some experimental results in cylindrical pillars, either for biological microswimmers such as bacteria or algae [33, 34] or artificial microswimmers [35], where there is a critical radius for constant particle’s capture located at R* ≈ 140.

Regarding the relaxation time, the more-persistent microswimmers reach the steady state faster than the less-persistent ones, and in both cases τ _trap grows with the radius. The results show that for more-persistent microswimmers follows τ _trap/τ _s ∼ 0.5 for all R (see Figure 5B), while for less-persistent microswimmers the steady state increases linearly with the obstacle radius. The time that particles remain in contact with the surface τ _contact increases also with obstacle’s radius, similar to Refs. [32, 35], but it does not follow the simple law τ _contact(R) ≈ τ _s (see Figure 5C). Instead, the contact time increases with R for both microswimmer’s types, yet there is not linear dependence on its growth. Thus, the microbe’s adhesion is enhanced with small pillar radius and less-persistent microswimmers.

Here, we vary the external flow U _∞ , while we keep fixed the bacterial activity to u ₀ = 20 and the obstacle radius to R = 100. The number of captured layers presents a non-monotonic behavior, with a pronounced maximum for the less-persistent swimmers at U* ≈ 1, where the microswimmer and the flow velocities are similar. For the more-persistent swimmers, the maximum is less pronounced and it is located at U* ≈ 2 (see Figure 6A). For larger external velocities, microbe’s adhesion decreases due to erosion by the flow [19, 20, 49]. For the less-persistent swimmers, the erosion is well fitted to the expression λ _trap(U _∞ /u ₀) = 1 + 0.66 exp(−0.15U _∞ /u ₀), according with microbe’s erosion of the surface [20, 49]. For the more-persistent microswimmers, the decrease of λ _trap is slower and well fitted to λ _trap(U _∞ /u ₀) = 1.62–0.037(U _∞ /u ₀), similar to the experimental limit for erosion observed by Miño et al. [19].

In the case of relaxation time, we observe a very different behavior for the two analyzed persistences. For the less-persistent microswimmers, τ _trap is non-monotonic, with a maximum at U* and larger values than for the more-persistent swimmers (see Figure 6B). In the erosion phase, the time that takes to reach the steady state, for less-persistent microswimmers, decays as τ trap ( U ∞ / u 0 ) / τ s = 0.9 ( u 0 / U ∞ ) 1 / 6 . For more-persistent microswimmers, the τ _trap time increases following a law τ _trap(U _∞ /u ₀)/τ _s = 0.13 log(25.7U _∞ /u ₀) in the first phase for U _∞ /u ₀ < 3 and then, in the erosion zone, it is constant with τ _trap/τ _s ≈ 0.56.

Surprisingly, the contact time is constant (see Figure 6C), even in the erosion zone, with small variations around τ _contact/τ _s ≈ 0.63 for the more-persistent swimmers and τ _contact/τ _s ≈ 0.71 for the less-persistent ones. We state that the existence of this almost constant value in the contact time is related with the accumulation of microswimmers in the front and in the back of the obstacle, where there are stagnation points. The particles are expelled from these regions then by their own activity but not on by the flow [20]. Consistent with this hypothesis, the less-persistent swimmers present larger contact times. Also, in the transport of the swimmers along the adhesion zone, the imposed flow almost vanishes there, resulting in that the contact time is dominated by the travel time τ _s.