Bacterial collective motion near the contact line of an evaporating sessile drop

The near-contact-line dynamics of evaporating sessile drops containing live E. coli cells is studied experimentally. The evaporation of the drop together with its pinned contact-line drives a radially outward fluid flow inside the drop concentrating the suspended cells near the contact-line. Our experiments reveal a collective behavior of the concentrated bacterial population near the contact-line appearing in the form of spatially periodic “bacterial jets” along the circumference of the drop. Based on a physical analysis of the continuum equations of bacterial suspensions, we hypothesize that the patterns result from a concentration instability driven by the active stress of swimming bacteria.

Collection of suspended particles near the pinned contact-line of evaporating drops of particulate suspensions is a ubiquitous observation termed the “coffee ring” effect—a name derived from the appearance of coffee stain which is the archetypal example of this phenomenon.^1–3 The pinning of the contact-line prevents any change in the contact-line radius of the drop. Hence to compensate for the volume of the fluid lost by evaporation, fluid has to be brought toward the contact-line from the bulk of the drop thus setting up a radially outward flow. If there are particles suspended in the fluid, the evaporation driven flow carries them outward and the particles accumulate near the contact-line due to the no-particle-flux boundary condition. In the present work, we experimentally investigate the near-contact-line dynamics of an evaporating drop containing living bacteria instead of passive particles. This investigation is motivated by the fact that recent experiments on bacterial suspensions have shown the existence of organized motion correlated over length scales much larger than the size of individual bacteria if the bacterial concentration is sufficient (see Koch and Subramanian⁴ and references therein). It has been suggested that such large scale motions enhance the mixing process in bacteria suspensions^5–8 resulting in better dispersal of the chemical species vital for bacteria. The evaporation driven flow inside a sessile drop provides a natural mechanism to concentrate the bacterial population. Furthermore, the sessile drop geometry can be important in the context of bacterial swarming and biofilm formation which often involve bacteria confined within thin liquid films.^9,10 In particular, the strain of E. coli employed in the present experiment, RP437, is known to produce biofilms.¹¹ 

Bacteria such as E. coli are composed of a cell-body propelled in fluids in a run-tumble fashion by means of a rotating bundle of helical filaments called flagella.¹² While swimming, a bacterium creates Stokesian hydrodynamic disturbances which behave like force-dipole disturbances in the far-field owing to the absence of any net force or torque on the bacterium.¹³ Numerical studies on hydrodynamically interacting self-propelled particles^14–16 suggest that the collective motion observed in experiments^5,8,17–21 arises from the force-dipoles associated with bacterial swimming. At the continuum level, the effect of swimming induced force-dipoles on bacteria is to provide an “active” stress field in the fluid and many recent theoretical investigations^22–26 have shown the existence of hydrodynamic instabilities driven by the active stress of dipolar swimmers like bacteria. Most of these investigations consider a homogenous base state in which the instability arises from the coupling between the swimmer orientation field and the fluid flow through the shear-induced rotation of swimmers. In the case of suspensions of run-tumble bacteria (see Subramanian and Koch²²), the instability occurs only if the bacterial concentration exceeds a critical value owing to the stabilizing effect of bacterial tumbling. However, such a state with large enough bacterial concentration for the instability is unlikely to persist at macroscopic length scales in bacterial suspensions encountered in practical situations and instead bacteria are more likely to accumulate in a localized environment. Chemotaxis is one of the major reasons behind bacterial accumulation. Many researchers including us^8,18,27–29 have shown recently that inhomogeneous suspensions of chemotactic bacteria are susceptible to a different kind of active stress instability driven by the coupling between the fluid flow and the bacterial concentration field instead of their orientation field as in the case of homogenous suspensions. The fluid flow in this case is driven by the normal stress difference arising in the suspension due to the chemotaxis-induced mean orientation of bacteria parallel to the chemo-attractant gradient direction.^28,29

In contrast to the active accumulation of bacteria as in case of chemotactic bacterial suspensions, the near-contact-line accumulation of bacteria in an evaporating sessile drop of bacterial suspension is passive. In this situation, an inhomogeneous bacteria concentration arises in a radially symmetric, quasi-steady “base state” but any bacterial alignment must result from fluid shearing motion. In this way, the present experiment realizes a physical situation that is distinct from the two aforementioned instabilities leading to collective bacterial motion, namely, the orientation-shear instability^22–26 and the chemotaxis driven instability.^28,29

Our experimental procedure involves placing a 1 μl drop of suspension of fluorescent cells of E. coli on a glass coverslip coated with 0.2% bovine serum albumin solution to reduce adhesion of bacteria onto the glass and the subsequent imaging of the contact-line region using an epi-fluorescence microscope at 10× magnification. The details of bacterial culturing are given in the supplementary material.³⁰ The approximate radius of the drop was 1 mm and the approximate contact angle in the beginning was 40°. The near contact-line dynamics of such a drop containing cells of the wildtype E. coli strain RP437 at a concentration of about 10¹⁰ cells per milliliter is shown in Figure 1 through a series of time-lapsed images.

The images span a period from one and a half minutes after the placement of the drop until the drop has dried out completely (≈10 min) and individual cells cannot be seen in the figure since the bacteria concentration is large. Nevertheless, the fluorescence intensity qualitatively represents the number of bacteria per unit area of the drop. Figure 2 shows the variation of fluorescence intensity in the direction normal to the contact-line (approximately) corresponding to the images at 2, 4, and 6 min in Fig. 1. At early times, when the drop has not evaporated significantly, the bacteria concentration in the drop would be nearly uniform since the suspension is initially well-mixed. This well mixed state, as seen at 2 min in Fig. 1, yields a monotonic increase in the fluorescence intensity with distance from the contact line because the drop thickness and bacteria per unit area increase with distance. As time passes, bacteria accumulate near the contact-line owing to the evaporation-driven radial flow in the drop and the no-flux boundary condition for bacteria concentration as evident in the image at 4 min given in Fig. 1. Soon after this accumulation develops, at around 5–7 min from the placement of the drop, the accumulated bacteria near the contact line organize themselves into spatially periodic clusters along the periphery of the drop as seen in Fig. 1 and the bacterial density is strongly peaked near the contact-line as seen in the intensity profile at 6 min in Fig. 2.

The typical wavelength of the pattern was found to be of the order of 50 μm which is comparable to the length scale over which the bacteria concentration varies in the radial direction near the contact-line and much larger than the length of an individual bacterium (of the order of 10 μm). We took movies of the collective motion and a close examination of them revealed that the periodic clusters of bacteria in images at times 5–7 min in Fig. 1 are in fact dense bacterial jets directed away from the contact-line and the pattern was more or less stationary (see the supplementary movie 1³⁰ (Multimedia view)). Eventually, after 8 min in Fig. 1, the pattern disappeared as the drop evaporated further and the edge of the film may have become too thin to allow suspension motion. The drop is completely dried at a time t_d ≈ 10 min.

A set of control experiments illustrated in Figure 3 demonstrate that the observed collective motion of the bacteria is associated with the evaporation of the drop and the cells’ motility but not their chemotactic response. Experiments using the smooth swimming strain E. coli (RP9535) which is non-chemotactic show (see Fig. 3(a) and also the supplementary movie 2³⁰ (Multimedia view)) a similar pattern to that for wildtype E. coli Fig. 3(d) indicating that aerotaxis is not required for the pattern formation. The much less distinct pattern seen in suspensions of incessantly tumbling cells of E. coli (RP1616) in Fig. 3(c) (also see the supplementary movie 3³⁰ (Multimedia view)) suggests that cell motility plays an important role in the development of angular bacteria concentration fluctuations. Fig. 3(b) demonstrates that the enhancement of cell concentration and the convective patterns near the contact line of the drop can be eliminated in an experiment in which evaporation is suppressed by covering the drop with the lid of a small petri-dish lined with moist paper (also see the supplementary movie 4³⁰ (Multimedia view)). The effect of the initial concentration of cells on the collective motion is shown in Fig. 4. As the initial cell concentration is decreased the thickness of the concentrated layer that develops at the edge of the drop is smaller and the width of the region of collective motion is also reduced.

To characterize the environment within the evaporating drops in our experiments and relate it to predictions from the literature, we measured the contact angle and the velocity of colloidal tracer beads in a drop. The contact angle of a drop with 10¹⁰ bacteria per ml is shown in Fig. 5(a). The contact angle decreases approximately linearly with time as predicted in numerical simulations of drop evaporation by Hu and Larson.³¹ To obtain a crude estimate of the fluid velocity during the time period 0.5t_d–0.7t_d when collective motion is observed in the bacteria-laden drops, we measured the velocity of colloidal tracers in a drop without bacteria and plotted the results as symbols in Fig. 5(b). The observed velocities increase as the edge of the drop is approached and reach a value of approximately 10 μm/s at a distance of 20 μm from the contact line. The lines in Fig. 5(b) show the prediction of the depth-averaged fluid velocity from a lubrication theory for evaporating drops^1,3 

$Vevap=R[4td(1−t/td)r̃]−1[(1−r̃2)−λ(θc)−(1−r2̃)]$ where R is the radius of the drop (≈1 mm), t_d is the drying time (≈6 min),

$\tilde{r} = (R-y)/R$

$r̃=(R−y)/R$ is the non-dimensional radial co-ordinate measured from the center of the drop with y being the distance from the contact-line, and λ(θ_c) = 1/2 − θ_c/180. The contact angle (measured in degrees) observed in the drops during the time period 0.5t_d–0.7t_d is θ_c ≈ 20°. The fluid velocity increases with proximity to the contact-line and at the contact-line (y = 0) the fluid velocity is singular due to the singularity in the diffusive flux of the vapor (see Deegan et al.¹ and Hu and Larson³) and the vanishing depth of the drop. The qualitative nature of the radial velocity variation in the drop is reproduced in our experiments as seen in Fig. 5 and the order of magnitude of the fluid velocity also agrees with the theoretical prediction given above. It is interesting to note that the length scale over which the bacteria concentration varies in Fig. 2 at 6 min is of the order of 100 μm while the balance of the convective flux of bacteria due to evaporation-driven flow with the observed evaporation velocity V_evap = 10 μm/s and a typical bacterial diffusivity D = 100 μm²/s would yield a length scale of D/V_evap = 10 μm. The large thickness of the observed concentrated region may result from the collective motion which drives bacteria away from the contact-line as the bacterial “jets” associated with the collective motion seen in Fig. 1 are directed away from the contact-line.

These experimental observations establish the existence of collective motion of bacteria near the contact-line of an evaporating sessile drop appearing as a periodic variation in the bacteria concentration along the periphery of the drop. The evaporation-driven radial flow inside the drop along with the impenetrability condition at the contact-line concentrate bacteria near the contact-line where the bacterial population organizes into periodic jets directed toward the center of the drop. Experiments show that the bacterial collective motion does not appear when the evaporation of the drop is prevented by enclosing it inside a humid chamber. The observation of collective motion in both wild-type cells and smooth-swimming cells rules out chemotaxis as the reason behind the collective motion. The motility of bacteria nevertheless seems to influence the organized motion since the periodic variation in bacteria concentration observed for cells which tumble incessantly is much more diffuse than that for the wild-type suspension at similar bacteria concentrations. We further observe that decreasing the bacteria concentration results in a decrease in the spatial extent of collective motion. Finally, experiments with fluorescent tracer particles show that the evaporation-driven radial fluid velocity in the region where collective motion is observed is comparable to the swimming speed of bacteria.

These observations indicate that the mechanism leading to collective motion in the present experiments is different from that in previous observations of collective motion in suspensions of aerotactic bacteria near contact-lines of sessile drops as in experiments of Dombrowski et al.¹⁸ and Tuval et al.⁷ and in thin liquid films as in the experiment of Sokolov et al.⁸ In these cases, collective motion was a result of the combination of aerotaxis with either gravity (bioconvection)^7,18 or swimming induced stresses⁸ (also see Kasyap and Koch^28,29). In contrast, aerotaxis is not a prerequisite for the patterns observed here which arise even with smooth swimming, non-chemotactic bacteria and the small thickness of our drops precludes bioconvective instability.⁸ The evaporation of the drop is however needed for pattern formation as revealed by our experiment with non-evaporating drops (see Fig. 3(b)) while the collective motions observed in experiments of Dombrowski et al.¹⁸ and Tuval et al.⁷ were in the absence of any evaporation. We also rule out any surfactant-driven Marangoni effects in our experimental observation since the wildtype E. coli strain RP437 used in our study is not known to secrete any surfactant (see Be'er and Harshey³²).

Since the collective motion observed here is characterized by periodic variation of the bacterial density near the contact-line, it is reasonable to suspect that the observed pattern in the present experiment arises from the concentration instability of an inhomogeneous base state characterized by the balance between evaporation-induced convection and diffusion of bacteria. The following heuristic analysis of the present problem from the point of view of the linear stability of continuum bacteria and momentum conservation equations (see Subramanian and Koch²²) of the bacteria suspension suggests the possibility of an instability driven by the coupling between the bacteria concentration field and the bacteria stress driven fluid flow. First, the inhomogeneous base state resulting from the balance between the diffusion of bacteria and convection by evaporation-driven fluid flow gives rise to a source for bacteria concentration fluctuations appearing as the term u′ · ∇n₀ where u′ is the perturbed fluid velocity field and n₀ is the base-state bacteria concentration field (see Kasyap and Koch²⁸) in the linearized bacteria conservation equation. Second, the bacteria concentration fluctuations can drive the fluid velocity perturbations, u′ in the present problem owing to the shear-induced anisotropy in the bacterial orientation field in the base state. The shear associated with the evaporation driven fluid flow rotates bacteria causing their orientation field to be anisotropic²² and this would lead to a bacteria stress term in the linearized momentum equation of the form

$∇·−CμUsL2n′⟨pp−I/3⟩0$ where n^′ is the perturbed bacteria concentration field and

$\langle {\mathbf {p} \mathbf {p} - \mathbf {I}/3}\rangle _0$

$⟨pp−I/3⟩0$ is the second moment tensor of the base-state orientation field to force the fluid velocity perturbations. Here, C ≈ 0.57 is the non-dimensional bacterial dipole strength, μ is the solvent viscosity, U_s is the bacterial swimming speed, and L is the overall length of the bacterium including its flagella bundle. An isotropic base state on the other hand, would not have such a term since

$\langle {\mathbf {p} \mathbf {p} - \mathbf {I}/3}\rangle _0 = 0$

$⟨pp−I/3⟩0=0$ in that case.

The degree of anisotropy in the orientation field of the bacteria in the present problem is determined by the competition between the rotation of the bacteria by the shear associated with the fluid flow and bacterial tumbling. For small shear rates compared to the bacterial tumbling frequency, bacteria would tend to orient near the extensional axis of the fluid flow and for large shear rates, they would tend to orient along the flow direction. In the inset of Fig. 5(b), we show the variation of the non-dimensional shear rate Γ due to evaporation-driven flow defined as Γ = V_evap/(hτ⁻¹) with h being the drop thickness and τ⁻¹ = 1 s⁻¹ being the typical tumbling frequency of bacteria. The thickness of the drop h as a function of distance from the contact-line is calculated for the typical contact-line radius R = 1 mm and contact angle of θ_c = 20°. Since the contact angle is small, the thickness profile of the drop can be approximated as a parabola so that

$h(y)=h0(1−r̃2)$ where h₀ = R(1/sin (θ_c) − 1/tan (θ_c)) is the maximum height of the drop and

$\tilde{r}(y)$

$r̃(y)$ is the non-dimensional radial coordinate defined earlier.³ The non-dimensional shear rate shown in the figure is of order 1 near the contact-line and it decays quickly as we move farther into the bulk of the drop. Thus, the evaporation-induced flow would tend to orient bacteria lying far away from the contact-line near the extensional axis of the shear flow and their orientation would be progressively brought toward the flow direction as we approach the contact-line. The base state bacterial orientation field will thus be significantly anisotropic and inhomogeneous in the region of collective motion and the resulting active stress facilitates bacteria concentration fluctuation driven fluid flows. The chemotaxis driven instability that we have described previously^28,29 also involved an inhomogeneous base state bacterial orientation. However, it was the normal components of the chemical-gradient induced bacterial stress that drove fluid flows in that case while in the present problem the shear components of the evaporative-flow induced bacterial stress would be the dominant ones.

Thus, in an evaporating drop of bacteria suspension there exists a two-way coupling between the bacteria concentration field and the fluid flow such that bacteria concentration fluctuations can drive a fluid flow which in turn can transport bacteria. This would provide a possible instability mechanism accounting for the temporal and angular variations in the bacteria concentration observed near the contact line of an evaporating drop in our experiments.

We thank Injun Chu for performing some of the preliminary experiments. The work was supported by National Science Foundation (NSF) Grant No. CBET-1066193.