# Sedimentation dynamics of passive particles in dilute bacterial suspensions:
emergence of bioconvection

Bryan O. Torres Maldonado1  Shravan Pradeep2  Ranjiangshang Ran1  Douglas
Jerolmack 1,2  Paulo E. Arratia1<EMAIL_ADDRESS>1Department of
Mechanical Engineering & Applied Mechanics, University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA 2Department of Earth and Environmental
Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

###### Abstract

Microorganisms are ubiquitous in nature and technology. They inhabit diverse
environments ranging from small river tributaries and lakes to oceans, as well
as wastewater treatment plants and food manufacturing. In many of these
environments, microorganisms coexist with settling particles. Here, we
investigate the effects of microbial activity (swimming E. coli) on the
settling dynamics of passive colloidal particles using particle tracking
methods. Our results reveal the existence of two distinct regimes in the
correlation length scale ($L_{u}$) and the effective diffusivity of the
colloidal particles ($D_{e\\!f\\!f}$), with increasing bacterial concentration
($\phi_{b}$). At low $\phi_{b}$, the parameters $L_{u}$ and $D_{e\\!f\\!f}$
monotonically increases with increasing $\phi_{b}$. Beyond a critical
$\phi_{b}$, second regime is found in which both $D_{e\\!f\\!f}$ and $L_{u}$
are independent of $\phi_{b}$. We demonstrate that the transition between
these regimes is characterized by the emergence of bioconvection. We use
experimentally-measured particle-scale quantities ($L_{u}$, $D_{e\\!f\\!f}$)
to predict the critical bacterial concentration for the diffusion-
bioconvection transition.

###### keywords:

active fluids, sedimentation dynamics, bacteria, bioconvection

##  1 Introduction

Microorganisms are ubiquitous in both natural environments and technological
applications. They are highly adaptable and inhabit diverse ecosystems such as
swamps, oceans, and rivers [Schallenberg & Kalff (1993); Zhang et al. (2020);
Roberto et al. (2018); Nealson (1997); Herndl & Reinthaler (2013)]. They also
play a significant role in numerous technological applications [Falkowski
(2012); Savage (2011); Falkowski et al. (1998)] including fermentation
processes for vaccine and food production and wastewater treatment processes
[Lorenzo et al. (2018); Swartz (2001); van Loosdrecht & Brdjanovic (2014);
Honda et al. (2023)]. Controlling the sedimentation process in biofuel
production, for instance, can lead to improvements in algae separation from
the fluid medium and an accumulation of algal biomass at the bottom of the
production pond [Savage (2011)].

Suspensions of swimming microorganisms (e.g., E. coli) or active particles
(e.g., phoretic colloids) are typical examples of so-called active fluids
[Ramaswamy (2010); Marchetti et al. (2013); Patteson et al. (2016a); Moran &
Posner (2017)]. In these fluids, the constituents inject energy, generate
mechanical stresses, and create flows within the fluid medium. Thus, active
fluids are systems that are inherently out of equilibrium even in the absence
of external forces [Dabelow et al. (2019)]. Bacterial activity can lead to
many complex phenomena such as reduced shear viscosity [Gachelin et al.
(2013); López et al. (2015); Rafaï et al. (2010)], hindrance in mixing and
transport of passive scalars [Ran et al. (2021)], formation of biofilms
[Pradeep & Arratia (2022)], and diffusion enhancement. [Leptos et al. (2009);
Miño et al. (2011); Jepson et al. (2013); Kurtuldu et al. (2011); Thiffeault &
Childress (2010); Kim & Breuer (2004)]. Particle (or tracer) diffusivity, in
particular, has been shown to increase linearly with bacteria concentration
(in the dilute regime) Wu & Libchaber (2000) but to depend on microorganism
swimming behavior Chen et al. (2007) and particle size Patteson et al.
(2016b). Less understood are the effects of bacterial activity on the settling
dynamics of tracer particles.

Initial studies have largely focused on steady-state analysis Palacci et al.
(2010); Ginot et al. (2015); Maggi et al. (2013). These include sedimentation
studies with active Janus particles that show (steady-state) density profiles
that decay exponentially with system height [Palacci et al. (2010); Ginot et
al. (2018, 2015)]. The resulting length scale is larger than expected for
thermal equilibrium systems, which is explained through the introduction of an
effective temperature (and diffusivity) with values that far exceeds that of
passive systems [Palacci et al. (2010)]. These findings are corroborated by
theory and simulations with no (or limited) hydrodynamic interactions [Ginot
et al. (2018); Nash et al. (2010); Tailleur & Cates (2008); Wang et al.
(2014); Vachier & Mazza (2019)]. Experiments with swimming E. coli show that
bacteria aggregation (due to extra-cellular polymer) can significantly
enhanced sedimentation rates [Maggi et al. (2013)], while particles settling
in suspensions of C. reinhardtii show the familiar exponential concentration
profile but with an effective gravitational length (or diffusivity) that is
proportional to algae concentration Jeanneret et al. (2016). These studies
show that the concept of effective diffusivity (and temperature) can be useful
in characterizing the steady sedimentation profiles of active suspensions.

Only recently the time-dependent sedimentation (i.e.,concentration) profiles
of bacterial suspensions have been investigated [Singh et al. (2021); Torres
Maldonado et al. (2022)]. These studies show that, in the dilute regime,
bacteria activity can significantly hinder the sedimentation process [Singh et
al. (2021)]. At vanishingly low bacteria concentration ($\phi_{b}$), the
(time-dependent) concentration profiles can be accurately described by an
advection-diffusion equation; however, as $\phi_{b}$ is increased, sink-source
terms must be introduced in the formulation to account for bacteria population
dynamics, including a dispersivity parameter that increases with $\phi_{b}$
[Singh et al. (2021)]. In this regime, the bacterial suspension undergoes
phase separation and bacteria setting speed is found to be strongly correlated
to a timescale associated with oxygen depletion within the system [Torres
Maldonado et al. (2022)]. These studies highlight the role of oxygen in the
sedimentation of living suspensions; microorganisms are known to exhibit
aerotaxis, a phenomenon that prompts aerobic bacteria to seek an oxygen source
[Wager (1911); Adler et al. (2012); Bouvard et al. (2022); Baracchini &
Sherris (1959)]. When suspended in a relatively deep layer of fluid,
microorganism can deplete the oxygen concentration in the water while oxygen
continually diffuses in from the air-water interface above. In response,
microorganisms swim towards the oxygen concentration gradient and generate
fluid motion, a phenomenon referred to as bioconvection driven by aerotaxis
[Jánosi et al. (1998); Hillesdon et al. (1995)]. During bioconvection, oxygen-
enriched water descends from the interface, giving rise to a complex
convection-dependent gradient that guides the swimming bacteria. Analysis
predict that the emergence of bioconvection occurs beyond a critical Rayleigh
number [Hillesdon & Pedley (1996)], a dimensionless number that quantifies the
ratio of the buoyancy forces to viscous forces [Hill et al. (1989)]. While
extensive research has focused on elucidating the mechanisms behind
bioconvection formation [Bees (2020); Hill & Pedley (2005); Pedley & Kessler
(1992)], limited experimental work has been conducted to corroborate these
theoretical predictions.

In this contribution, we experimentally investigate the sedimentation dynamics
of colloidal particle and the formation of bioconvection patterns in dilute
bacterial suspensions using particle tracking methods. We find that bacterial
activity has a significant impact on particle diffusivity, leading to the
identification of two dynamical regimes. Initially, as bacterial concentration
is increased, we observed a corresponding linear increase in diffusivity.
However, as bacterial concentration is further increased (but still dilute),
particle diffusivity becomes independent of bacterial concentration. This
transition coincides with the inception of bioconvection patterns above a
critical Rayleigh number ($\Gamma_{cr}$) [Hillesdon & Pedley (1996)].

##  2 Materials and Methods

We experimentally investigate the sedimentation dynamics of spherical colloids
in the presence of bacterial activity. Active fluids are mixtures of bacterial
suspensions and passive particles suspended in deionized (DI) water to reach
the desired volume fractions of bacteria and particles, denoted as $\phi_{b}$
and $\phi_{p}$. Bacterial suspensions are prepared using wild-type Escherichia
coli (K12 MG1655), which are cultured to saturation ($10^{9}$ cells/mL) in LB
broth (Sigma-Aldrich). Passive particles are polystyrene spheres (density
$\rho_{p}=1.05$ g/cm3, Thermo Scientific) with a diameter of $d=3.2$ m, which
are gently cleaned by centrifugation (Centrifuge 5430, Eppendorf). All active
suspensions are contained within a custom-made quasi two-dimensional
sedimentation cell constructed from optically transparent polymethyl
methacrylate (PMMA) material, offering optimal clarity [see Fig. 1(a)]. The
container measures $100~{}$mm in height, $28~{}$mm in width, and $400~{}\umu$m
in thickness and is assembled using precision laser cutting and chemical
bonding techniques [Sun et al. (2007)].

Figure 1: Experimental setup and sample particle trajectories: (a) A schematic
of the setup and bacteria/particle suspensions (active fluid). Sedimentation
experiments are conducted in an optically clear square container. The
particles are $3.2~{}\umu$m polystyrene spheres, and the bacteria are
$2~{}\umu$m rod-shaped E. coli, both are subjected to gravity. Particles
located at the center and at a height $\approx 20~{}$mm from the bottom of the
container are observed through a microscope. These spherical colloidal
particles are tracked through particle tracking velocimetry method. Particle
trajectories ($\phi_{p}=0.04\%$) are shown in (b) without bacteria and in (c)
with bacteria ($\phi_{b}=0.45\%$). When particles and E. coli are combined,
the passive particles trajectories in the lateral direction undergo
significant modifications compared to their trajectories in the absence of
bacteria.

Active fluids are manually homogenized using a pipette before being introduced
into the custom-made PMMA container. A volume of $\approx 1.1$ mL of the
prepared fluid is then transferred into the container, as schematically
depicted in Fig. 1(a). Subsequently, the container is immersed in a water tank
to maintain the fluid at a room temperature of $24$ °C. Images are captured at
a frame rate of $30~{}$fps with a high-resolution camera (IO Industries, Flare
4M180) and a microscope (Infinity, K2) equipped with a $10\times$ objective.
The light source used is a single-color cold visible LED (M490L4, Thor Labs).
The initial volume fraction $\phi_{b}$ of motile bacteria in the container
varies from $0\%$ to $0.75\%$, while the volume fraction of spherical colloids
is consistently maintained at $\phi_{p}=0.04\%$. It is worth noting that both
$\phi_{b}$ and $\phi_{p}$ concentrations are within the dilute regime
($\phi<1\%$), and no significant macroscopic collective behavior is observed
in these bacteria-particle suspensions under the microscope Singh et al.
(2021); Patteson et al. (2016b). The selected range of $\phi_{b}$ in this
study remains below the threshold at which collective motion is typically
observed ($\approx 10^{10}$ cells/mL) [Kasyap et al. (2014)]. All captured
images for the suspensions are acquired within a 30-minute time frame.
Importantly, within this designated time frame, no noteworthy reduction in
bacteria motility is observed, a finding that has been corroborated by recent
investigations [Ran et al. (2021); Singh et al. (2021)].

##  3 Results and Discussion

The main goal is understand the interplay between bacterial activity and the
settling dynamics of passive spherical particles. To this end, we use particle
tracking techniques to visualize colloidal particle’s Lagrangian trajectories
[Crocker & Grier (1996)]. Sample particle trajectories from passive
($\phi_{p}=0.04\%$, no bacteria) and active ($\phi_{p}=0.04\%$,
$\phi_{b}=0.45\%$) fluids are shown in Fig. 1(b) and (c), respectively. The
passive fluid case show particle trajectories predominately in the downward
direction ($-y$) with apparent noise along the $x$-axis. For the active fluid,
on the other hand, particle paths show amplified displacement along the
$x$-axis, indicating that bacterial activity induces modifications to the
colloidal particle downward trajectories. This heightened trajectory
enhancement (in the $x$-axis) consistently manifests itself across all
experiments, providing an explanation for the previously observed hindrance in
the speed of the particle front [Singh et al. (2021); Torres Maldonado et al.
(2022)]. These qualitative findings show that bacteria modify the settling
trajectories of colloidal particles through hydrodynamic interactions, even in
the dilute regime ($\phi_{b}<1\%$).

###  3.1 Characterizing Length Scales of Particle Correlations

We quantify the sedimentation dynamics of our fluid by computing their time-
dependent velocity fields using an in-house particle imaging velocimetry (PIV)
method [Ran et al. (2022); Li et al. (2021); Brosseau et al. (2022)]. The goal
is to understand how bacterial activity ($\phi_{b}$) alters the spatial
correlation functions of particle fluctuations [Segrè et al. (1997); Tee et
al. (2002)]. Here, spatial correlation functions of the velocity fluctuations,
$C_{u}(r_{x})$, is defined as:

$C_{u}(r_{x})=\langle{u}(x)\,{u}({x}+{r_{x}})\rangle/\langle u^{2}\rangle,$
(1)

where $x$ represents the direction perpendicular to gravity,
$\langle...\rangle$ denotes an ensemble average across all spherical
particles, and ${u}({x})$ denotes the local velocity of a particle at position
${x}$. Notably, ${u}({x})$ incorporates contributions from both particle self-
diffusivity and hydrodynamic interactions. In the absence of bacteria
($\phi_{b}=0\%$), we observe a decay in the spatial correlation length scales
at small distances (Fig. 2a). With an increase in $\phi_{b}$, we identify a
corresponding increase in spatial correlation functions over larger distances.
However, as $\phi_{b}\geq 0.45\%$, the spatial correlation functions becomes
independent of $\phi_{b}$. To quantitatively analyze this trend, we calculate
the integral length scale of velocity $L_{u}$ as shown in Fig. 2(b). This
integral length scale is defined as
$L_{u}=\int_{0}^{\infty}C_{u}(r_{x})\,dr_{x}$. Results shows that $L_{u}$
increases linearly as $\phi_{b}$ is increased [Fig. 2(b)], but this
relationship weakens and $L_{u}$ becomes independent of $\phi_{b}$ for
$\phi_{b}\geq 0.45\%$. These findings show a positive correlation between
$L_{u}$ and $\phi_{b}$ up to $\phi_{b}\geq 0.45\%$. This outcome aligns with
previous research, highlighting the presence of two distinct regimes within
the sedimentation process [Torres Maldonado et al. (2022)].

Figure 2: (a) The spatial correlation functions of particle velocities ($u$)
in the lateral direction across the $x$-axis. (b) The integral length scale of
the lateral velocities at different bacteria volume fraction $\phi_{b}$.
Presence of bacteria leads to an increase in correlation functions, and
subsequently, to an increase in length scales.

###  3.2 Mean Square Displacements and Diffusivity

Next, we use experimentally measured particle tracks to compute the mean
square displacements (MSD) of the passive particles as a function of
$\phi_{b}$. The mean square displacement is defined as MSD$(\Delta t)$ =
$\langle|\textbf{r}(t_{R}+\Delta t)-\textbf{r}(t_{R})|^{2}\rangle$, where
$t_{R}$ is denoted as the reference time. First, we extract the distance
vector r in the direction perpendicular to gravity, $r_{x}$, yielding the mean
square displacement along the $x$-axis (MSDx). As shown in Fig. 3(a), MSDx
exhibits diffusive behavior (at sufficiently long time intervals) that
increases with $\phi_{b}$. We can estimate an effective diffusivity $D_{eff}$
by fitting MSD${}_{x}=2D_{e\\!f\\!f}\Delta t$ to the experimental data and
find that, similar to $L_{u}$, $D_{eff}$ grows linearly with $\phi_{b}$
followed by an asymptote for $\phi_{b}\geq 0.45\%$ (Fig. 3c). Similarly, we
compute the mean square displacements of the tracked particles relative to the
$y$-axis (MSD${}_{y^{\prime}}$) as a function of $\phi_{b}$. Notably, we
subtract the mean sedimentation rates along the $y$-axis since our focus is on
understanding how bacteria activity affect particle fluctuations, i.e.,
$y^{\prime}=y-\langle y\rangle$. Figure 3(b) shows that MSD${}_{y^{\prime}}$
exhibits diffusive behavior that increases with increasing $\phi_{b}$. We
apply a similar fitting approach, relating MSD${}_{y^{\prime}}$ to
$D_{e\\!f\\!f}$ as shown in Fig. 3(c) (filled symbols). Our results reveals
that for the control case ($\phi_{b}=0\%$), $D_{e\\!f\\!f}=0.150\pm
0.001~{}\umu$m2/s for MSDx and $0.145\pm 0.001~{}\umu$m2/s for
MSD${}_{y^{\prime}}$. These measurements align with the theoretically
predicted value from the Stokes-Einstein relation, $D_{0}=k_{B}T/(3\pi\mu
d)=0.150~{}\umu$m2/s [Einstein (1905)], where $k_{B}$ represents the Boltzmann
constant, $\mu$ is the fluid viscosity, and $T$ denotes the temperature. These
results indicate that hydrodynamic interactions among passive particles in our
study are relatively weak, which is not surprising given the dilute particle
volume fraction ($\phi_{p}=0.04\%$).

Figure 3: (a) Mean square displacement in the $x$-axis (MSDx) of spherical
particles at different bacteria volume fractions $\phi_{b}$. (b) Mean square
displacement of particle fluctuations in the $y$-axis (MSD${}_{y^{\prime}}$)
at different $\phi_{b}$. (c) Effective particle diffusivities $D_{e\\!f\\!f}$
from MSDx (Open symbols) and MSDy (closed symbols) as a function of
$\phi_{b}$. The presence of swimming E. coli increases particle fluctuations
in both the $x$ and $y$ directions, resulting in higher $D_{e\\!f\\!f}$. (d)
Péclet number (Pe) as a function of $\phi_{b}$. The results demonstrate that
the presence of bacteria enhances diffusion transport in the settling process,
eventually reaching a plateau regime where both advection and diffusion
transport are in close balance (Pe$\,\approx 1$).

Overall, the mean square displacement (MSD) measurements of particles in the
presence of motile bacteria exhibit an increasing trend as $\phi_{b}$ rises,
as shown in Fig. 3(a) and (b). This enhancement on particle displacement leads
to a corresponding increase in $D_{e\\!f\\!f}$, illustrated in Fig. 3(c). By
comparison, a similar experiment was conducted using non-motile bacteria
($\phi_{d,b}=0.35\%$), as shown in Fig. 3(a) and (b). Here, we observe a
negligible increase in MSD compared to the absence of bacteria
($\phi_{b}=0\%$), resulting in $D_{e\\!f\\!f}=0.159\pm 0.001~{}\umu$m2/s along
the $x$-axis and $D_{e\\!f\\!f}=0.163\pm 0.002~{}\umu$m2/s along the $y$-axis
for $\phi_{d,b}=0.35\%$ [Fig. 3(c)]. This results indicates that the presence
of non-motile bacteria exerts negligible impact on particle diffusivity and
hydrodynamic interactions. In other words, the observed enhancement in
particle diffusivity across the experiments with bacteria is primarily
attributed to bacterial activity within the suspension; as mentioned before,
$D_{e\\!f\\!f}$ exhibits an nearly linear relationship with bacteria
concentration up to $\phi_{b}\leq 0.45\%$. Surprisingly, however,
$D_{e\\!f\\!f}$ (and $L_{u}$) become independent of $\phi_{b}$ for
$\phi_{b}\geq 0.45\%$ (Fig. 2b and Fig. 3c). The macroscopic signature of this
behavior can be capture by computing the Péclet number (Pe), defined as
Pe$\,=V_{p}d/(2\langle D_{e\\!f\\!f}\rangle)$. Here, $V_{p}$ is the average
settling speed of particles, which has been previously measured for the same
fluids and conditions [Torres Maldonado et al. (2022)]. A small Pe signifies
the dominance of diffusion in particle transport, while a large Pe indicates
the prevalence of advection. Figure 3(d), shows that Pe decreases as
$\phi_{b}$ increases but it reaches a plateau at Pe$\,\approx 1$, indicating a
point in which advection and diffusion transport are in close balance.

This asymptotic behavior has also been observed in macroscopic quantities such
as the hindering settling function, $H(\phi)$, for the same type of active
fluids and in similar range of $\phi_{b}$ [Torres Maldonado et al. (2022)].
These results are puzzling since prior studies of (passive) particles in
active suspensions demonstrated that $D_{e\\!f\\!f}$ increases linearly with
$\phi_{b}$ up to $\phi_{b}\approx 1\%$ [Leptos et al. (2009); Patteson et al.
(2016b); Wu & Libchaber (2000)]. A possible explanation is that motile
bacteria likely undergo a transition in their swimming motion, shifting from a
random pattern to “organized” trajectories, within the range of $\phi_{b}\geq
0.45\%$ (still dilute). One possibility is the development of bioconvection
patterns [Hillesdon & Pedley (1996); Hill et al. (1989)]. We will explore this
possibility below.

###  3.3 Unveiling the Phenomenon of Bioconvection

Our study involves a suspension containing E. coli, which is known for its
oxygen-seeking behavior termed aerotaxis. Following previous work [Hillesdon &
Pedley (1996)], we define a dimensionless parameter $\Gamma$ that is analogous
to the Raleigh number in thermal convection problems:

$\mathrm{\Gamma}=\frac{(\rho_{b}-\rho_{w})\phi_{b}gL_{u}^{3}}{\rho_{w}\nu
D_{e\\!f\\!f}},$ (2)

where $\rho_{b}=1.105~{}$g/cm3 [Martínez-Salas et al. (1981)] and
$\rho_{w}=1.0$ g/cm3 represents the densities of bacteria and water
respectively; $g$ is the constant of gravity, and $\nu$ denotes the kinematic
viscosity of water. The only difference between our formulation and prior work
is the length scale we employ [Hillesdon & Pedley (1996)]. Our present
approach incorporates a length scale associated with the particle dynamics
(i.e., $L_{u}$), which is directly measured from the spatial correlation
functions. Figure 4(a) show that $\Gamma$ increases linearly with $\phi_{b}$,
as expected.

Next, following a linear stability analysis, one can define a critical
Rayleigh-like number ($\Gamma_{cr}$) that describes relative importance of
bacterial diffusion to bioconvection (aerotatic behavior plus density
difference) behavior [Hillesdon & Pedley (1996)]. Initially, it is necessary
to establish two dimensionless constants stemming from the equations governing
cell and oxygen conservation. The first of these constants, denoted as
$\beta$, characterizes the relationship between the rate of oxygen consumption
and the rate of oxygen diffusion. It is defined as:

$\beta=\frac{K_{0}n_{0}L_{u}^{2}}{D_{c}C_{0}},$ (3)

where $K_{0}$ is the oxygen consumption rate, $n_{0}$ is the initial cell
concentration, $D_{c}$ represents oxygen diffusivity, and $C_{0}$ denotes the
initial oxygen concentration. Here, we assume $K_{0}\approx 2\times
10^{-18}~{}$mol/min/cell [Schwarz-Linek et al. (2016)], $D_{c}\approx
200~{}\umu$m2/s [Han & Bartels (1996)], and $C_{0}\approx
0.0058~{}$mLO${}_{2}/$mLH2O [Carpenter (1966)].

The second dimensionless constant, $\gamma$, quantifies the strength of
oxytactic swimming relative to random diffusive swimming. It is expressed as:

$\gamma=\frac{cV_{sb}}{D_{b}},$ (4)

where $c$ is the chemotaxis constant, $V_{sb}$ represents bacteria swimming
speed, and $D_{b}$ is the bacteria diffusivity. Here, we assume
$c=V_{sb}\tau_{r}$, where $\tau_{r}=0.95~{}$s stands for the mean run time of
E. coli [Patteson et al. (2015)]. Additionally, we take $D_{b}\approx
10~{}\umu$m2/s and $V_{sb}\approx 10~{}\umu$m/s [Ran et al. (2021)].

Figure 4: (a) Experimental Rayleigh number ($\Gamma$), and theoretical
critical Rayleigh number ($\Gamma_{cr}$) as a function of bacteria volume
fraction ($\phi_{b}$). Results show a crossover of Ra across $\phi_{b}\approx
0.45\%$. When $\Gamma\,<\,$$\Gamma_{cr}$, swimming bacteria are not
significantly affected by the oxygen at the top of the container. However,
when $\Gamma\,>\,$$\Gamma_{cr}$, oxygen plays a significant role in the
swimming behavior of E. coli, leading to bioconvection. This explains the
emergence of the plateau regime observed in settling experiments when
$\phi_{b}\geq 0.45\%$. (b) Péclet number (Pe) as a function of experimental
Rayleigh number ($\Gamma$), where it shows that convective transport of
colloidal particles is reduced with increased bacterial-driven convection. The
inset shows the validity of the Eq. 5 by using the inequality
$\gamma\beta\leq\xi\,$tan${}^{-1}\,\xi$.

Our experiment can be approached as a shallow chamber, since it satisfy the
inequality $\gamma\beta\leq\xi\,$tan${}^{-1}\,\xi$, wherein $\xi$ is defined
as $\xi^{2}=e^{\gamma}-1$ [Fig. 4(b), inset]. Therefore, the analytical
approximation, following [Hillesdon & Pedley (1996)], $\Gamma_{cr}$ can be
expressed as:

${\Gamma_{cr}}=\frac{576}{\gamma\beta}.$ (5)

We proceed to compute the experimental values of $\Gamma$ and $\Gamma_{cr}$ as
a function of $\phi_{b}$, as shown in Fig. 5(a). The data illustrates an
increase in $\Gamma$ with rising $\phi_{b}$, while $\Gamma_{cr}$ decreases
with higher $\phi_{b}$. This aligns with the expectation that higher
$\phi_{b}$ leads to accelerated oxygen consumption by E. coli within the
fluid. Remarkably, an intersection between $\Gamma$ and $\Gamma_{cr}$ occurs
around $\phi_{b}\approx 0.45\%$, signifying the onset of bioconvection when
$\phi_{b}\geq 0.45\%$. This observation could elucidate the plateau observed
in $D_{e\\!f\\!f}$ and $L_{u}$, and in the macroscopic hindering settling
function ($H(\phi)$) seen in previous experiments [Torres Maldonado et al.
(2022)]. Concurrently, with the increase in $\Gamma$, the Péclet number (Pe)
reaches a plateau at unity for $\phi_{b}\geq 0.45\%$, as shown in Fig. 5(b).
This strongly suggests that the reduction in convective transport of colloidal
particles (Pe) results from the increased convective motion driven by swimming
bacteria.

Figure 5: Representative snapshots of the time evolution of the settling
suspensions with dye at the top of the settling container. Left column: No
bacteria ($\phi_{b}=0\%$); middle column: active bacteria at
$\phi_{b}=0.35\%$; right column: active bacteria at $\phi_{b}=0.75\%$. Dye in
passive suspensions shows that it settles straight downwards as a function of
time. When bacteria are added, as shown in the middle column
($\phi_{b}<0.45\%$), the results show that dye settling is slightly hindered
due to the random motion of bacteria. However, no significant bioconvection
patterns were observed. When $\phi_{b}\geq 0.45\%$, as shown in the right
column, dye gets convected in a counterclockwise motion. Dye return to the
field of view from below at long times, showing that bioconvection is observed
in this case.

To further illustrate the existence of bioconvection in settling active
fluids, we perform dye mixing experiments for a range of $\phi_{b}$. These
experiments involved introducing $100~{}\umu$L of dye ($2.5\times 10^{-3}M$
fluorescein aqueous solution) into bacterial solutions at the upper portion of
the square chamber within the experimental setup. Image capture occurred every
2 minutes over a 14-hour period using a Nikon D7100 camera equipped with a 105
mm Sigma lens. Illumination was achieved using black light (USHIO, F8T5/BLB),
peaking at $368~{}$nm in the ultraviolet (UV) range. Importantly, $90\%$ of
the UV energy falls within the long-wave UVA-I range (340 to 400 nm), and it
minimally impacted suspension activity [Ran et al. (2021); Vermeulen et al.
(2008)]. As expected, experiments with no bacteria ($\phi_{b}=0\%$)
demonstrated dye settling dominated by diffusion and moving directly downward,
evident in the left column of Fig. 5. Similarly, $\phi_{b}=0.35\%$
experiments, with $\Gamma~{}<~{}\Gamma_{cr}$, displayed no significant
convection patterns (middle column). However, experiments at
$\phi_{b}=0.75\%$, where $\Gamma~{}>~{}\Gamma_{cr}$, exhibited pronounced
convection patterns (right column). In the early stages, the dye exhibited
counterclockwise motion due to convection, transitioning to complex
bioconvection patterns in the upward direction at later times.

We quantify the dynamics observed in these images by computing the image
correlation with respect to the initial image ($t=0$ min). Figure 6
illustrates that in the absence of bacteria ($\phi_{b}=0\%$), the correlation
is reminiscent of pure diffusive behavior throughout most of the observation
period (the time window is selected so that most, if not all, of the initial
dye is still in the field of view). For $\phi_{b}=0.35\%$, decorrelation rates
are initially similar to the no-bacteria case but start to deviate and slow
down at about $\Delta t\approx$ 30 minutes. That is, bacterial activity
hinders the settling behavior of dye particles as the suspension transitions
from a diffusive-dominated behavior to one dominated by bacterial-activity
induced convection. Macroscopically, this behavior manifests itself by
decreasing the hindered settling function $H(\phi)$ (where
$H(\phi)=V_{p}(\phi)/V_{0}$, $V_{p}(\phi)$ is the mean sedimentation speed of
the particles in the presence of bacteria, and $V_{0}$ is the Stokes’ settling
speed) as reported in our earlier work [Torres Maldonado et al. (2022)]. As
$\phi_{b}$ is further increase ($\phi_{b}=0.75\%$), however, the correlation
data shows two distinct parts: at short times the correlation shows a a
significant delay due to bioconvection-induced motion of the dye particles
perpendicular to the gravity. At longer time scales image correlations mirror
the samples with diffusion-like behavior emerges due to the dye particles
following the downward draft in bioconvection rolls. These results shows
delicate interplay between bacterial activity, oxygen diffusion (i.e.,
aerotactic behavior), and density differences. Our results clarifies the
existence of two regimes in settling dynamics parameters observed in our
present study ($L_{u}$, $D_{p}$, and $Pe$) and the previous work ($H(\phi$))
[Torres Maldonado et al. (2022)], indicating that active suspensions are
purely diffusive at $\phi_{b}<0.45\%$ and are characterized with complex
bioconvection patterns for $\phi_{b}\geq 0.45\%$.

Figure 6: Temporal evolution of image correlation in dye experiments as a
function of time, for no bacteria ($\phi_{b}=0\%$), $\phi_{b}=0.35\%$, and
$\phi_{b}=0.75\%$. For concentrations $\phi_{b}<0.45\%$, the correlation is
reminiscent of pure diffusive behavior throughout most of the observation
period. However, for $\phi_{b}>0.45\%$, the image correlation showed a
significant delay due to bacteria, which convects the dye perpendicular to
gravity’s direction and subsequently exhibits diffusive behavior.

Our experimental findings demonstrate that the presence of bacteria leads to
an increase in both particle correlation length-scales (Fig. 2) and their
effective diffusivity values, $D_{e\\!f\\!f}$ (Fig. 3). As mentioned before,
these measurements exhibit a linear increase up to approximately
$\phi_{b}\approx 0.45\%$. Beyond this threshold ($\phi_{b}\geq 0.45\%$), both
$L_{u}$ and $D_{e\\!f\\!f}$ remain constant, establishing their independence
from $\phi_{b}$. These experimental results align with our prior findings,
wherein the hindered settling function $H(\phi)$ of the particle front showed
a linear decrease with bacterial concentration for $\phi_{b}<0.45\%$. The
hindered settling function $H(\phi)$ of the particle front ultimately reaches
a plateau at $\phi_{b}\geq 0.45\%$ [Torres Maldonado et al. (2022)]. We
assessed the impacts of advection and diffusion by quantifying the Péclet (Pe)
number. Within the linear regime ($\phi_{b}<0.45\%$), we noticed a decrease in
advection effects due to particle hindrance, accompanied by an increase in
diffusion effects owing to increased particle fluctuations. In the vicinity of
the plateau regime, both settling (advection) and random motion (diffusion)
exhibit near-equivalence ($Pe\approx 1$). The emergence of this plateau regime
can be explained by Eq. 2, quantifying $\Gamma$, and Eq. 5, determining
$\Gamma_{cr}$. Interestingly, our results manifest a crossover around
$\phi_{b}\approx 0.45\%$ between the experimental Rayleigh number and the
estimated critical Rayleigh number (Fig. 4). This suggests that under
$\phi_{b}<0.45\%$, bacteria engage in random swimming, with minimal oxygen-
related effects at the upper portion of the container. However, at
$\phi_{b}\geq 0.45\%$, oxygen plays a prominent role, giving rise to intricate
bioconvection patterns as depicted in Fig. 5. In both these regimes, the input
of energy via bacterial swimming motions significantly influences the
sedimentation process, thereby enhancing particle diffusivities.

##  4 Conclusion

We experimentally investigate the sedimentation dynamics of colloidal
particles within dilute suspensions of swimming bacteria. Our overall
observation reveals that bacterial activity enhances the effective
diffusivities ($D_{e\\!f\\!f}$) and the correlation length $L_{u}$ of
colloidal particles. Moreover, our experimental analysis of spherical particle
behavior exhibits two distinct regimes. The first regime showcases a linear
increase in both $D_{e\\!f\\!f}$ and $L_{u}$ in correlation with bacterial
concentration ($\phi_{b}$) up to $\phi_{b}<0.45\%$. In contrast, once
$\phi_{b}\geq 0.45\%$, both $D_{e\\!f\\!f}$ and $L_{u}$ remain unaltered
despite varying bacterial concentration. To uncover the foundation of these
distinct regimes, we computed the experimental Rayleigh number ($\Gamma$) and
the critical Rayleigh number ($\Gamma_{cr}$), respectively, using
experimentally measured quantities. Solving for these non-dimensional
quantities reveals a transition point in bacterial concentration that divides
the two observed regimes within our experiments. This transition suggests that
the system becomes unstable beyond $\phi_{b}\geq 0.45\%$, consequently giving
rise to complex bioconvection patterns. This inference gains further
validation through a separate set of experiments utilizing dye as a tracer
within the fluid. For $\phi_{b}<0.45\%$, the variance demonstrates diffusive
behavior. Intriguingly, when $\phi_{b}\geq 0.45\%$, the variance initially
exhibits diffusive-like behavior for short times, followed by a transition to
convection-like behavior. In summary, our experimental findings shows that the
reduction in convective transport of colloidal particles is attributed to
increased bacterial-driven convective motions. Future work involving diverse
colloidal sizes and swimmer actuation modes (such as pusher vs puller) holds
the potential to elucidate whether the insights obtained from our particle-
bacteria interaction study can be generalized to others.

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