# Phase-separation during sedimentation of dilute bacterial suspensions†

Phase-separation during sedimentation of dilute bacterial suspensions

  * Bryan O. Torres Maldonado1, Ranjiangshang Ran 1, K. Lawrence Galloway1, Quentin Brosseau1, Shravan Pradeep2, and Paulo E. Arratia1†

  * 1)Department of Mechanical Engineering & Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104.
2)Department of Earth and Environmental Science, University of Pennsylvania,
Philadelphia, PA 19104.
<EMAIL_ADDRESS>
  *

Numerous natural systems depend on the sedimentation of passive particles in
presence of swimming microorganisms. Here, we investigate the dynamics of the
sedimentation of spherical colloids at various E. coli concentration within
the dilute regime. Results show the appearance of two sedimentation fronts, a
spherical particle front and the bacteria front. We find that the bacteria
front behave diffusive at short times, whereas at long times decays linearly.
The sedimentation speed of passive particles decays at a constant speed and
decreases as bacteria concentration ($\phi_{b}$) is increased. As $\phi_{b}$
is increased further, the sedimentation speed becomes independent of
$\phi_{b}$. The timescales of the bacteria front is associated with the
particle settling speeds. Remarkably, all experiments collapse onto a single
master line by using the bacteria front timescale. A phenomenological model is
proposed that captures the sedimentation of passive particles in active
fluids.

##  1 Introduction

The settling of particles in the presence of microorganisms is a ubiquitous
natural process from small river tributaries to lakes to oceans 1, 2, 3, 4, 5,
6. Sedimentation of biological matter, for instance, plays an important role
on the distribution of plankton in oceans, which is key part of the carbon
cycle (i.e. ocean’s biological pump) that transports carbon from the ocean’s
surface to depth 7, 8, 9. From a technological stand point, microbial activity
can affect many process such as the production of food, biofuels, and even
vaccines 10, 11, 12, 13, 14. While the settling of passive particles, as a
function of suspension volume fraction ($\phi$) and interparticle
interactions, have been extensively studied 15, 16, 17, 18, 19, 20, 21, 22,
23, much less is known about particle sedimentation in the presence of
swimming microorganisms.

At infinite dilution, the particle sedimentation speed is set by the balance
between the gravitational and viscous drag forces, and the associated velocity
called the Stokes’ settling speed is given as, $V_{0}=2a^{2}\Delta\rho
g/9\eta$, where $a$ is the particle radius, $\Delta\rho$ is the density
difference between the particle and the solvent, $g$ is the acceleration due
to gravity, and $\eta$ is the solvent viscosity 17. As $\phi$ increases, the
competition between the interparticle forces and thermal motion determines the
suspension microstructure, affecting the hydrodynamic interactions between the
particles, and in turn the suspension sedimentation speed. These effects are
on suspension sedimentation speed are usually described by the hindered
settling function, given as $H(\phi)=V_{p}(\phi)/V_{0}$, where $V_{p}(\phi)$
is the mean sedimentation speed due to the upward fluid flow between the
settling particles 21.

Obtaining the exact form of $H(\phi)$ for quiescent particle suspensions is
quite challenging and has been the subject of much investigation 24, 18. One
of the first empirical functional form that captures the $\phi$-dependence on
the hindered settling function was proposed by Richardson and Zaki nearly 70
years ago 16, as $H(\phi)=(1-\phi)^{n}$ where $n\approx 4.65$ for particles
Reynolds numbers ($Re$) below 0.2 when wall effects can be neglected. This
relationship was found to be valid for both sedimentation and fluidization
processes. It is worth emphasizing that there is still a debate on the exact
value of the exponent $n$. Recently, a comprehensive study has shown that the
exponent has in fact two values: $n\approx 5.5$ for Brownian particles and
$n\approx 4.5$ for non-Brownian particles. At very dilute particle
concentration, the exponents converge to $n\approx 6.5$, a value that is close
to Batchelor’s theoretical prediction,
$H(\phi)=1-6.55\phi+\mathcal{O}(\phi^{2})$, obtained by including many-body
hydrodynamic interactions between monodisperse constituent particles 15.

While there is a rich history (and much still to understand) on the subject of
sedimentation of passive suspensions 23, 19, 20, 25, 21, 18, there has been a
growing interest in systems that emulate the sedimentation of natural
processes, particularly those that include living matter such as swimming
microorganisms 26, 27, 28, 29, 30, 31. Presence of active particles (living or
synthetic) inject energy to a suspending solution. Activity can drive the
fluid out of equilibrium even in the absence of external forcing and lead to
many poorly understood phenomena including low values of reduced viscosity 32,
33, enhancement in particle diffusivity 34, 35, 36, 37, 38, and collective
motion (at high densities) 39, 40, 41. Thus, one expects activity to also
affect the sedimentation of passive particles and the associated hindered
settling function $H(\phi)$.

Experiments with synthetic active Janus particles has shown some intriguing
results 26, 31, 29. These investigations show that the particle steady-state
density profiles are significantly affected by activity and can be described
by an exponential decay 26. Increasing particle activity leads to a slower
exponential decay, which is described by using an effective temperature (and
diffusivity) that is much larger than for passive systems 26. Varying
sedimentation speed while maintaining a constant system activity also led to
exponential decay in particle diffusivity 31. These steady-state results agree
well with theory and simulations 42, 43, 44, 30, 31.

Sedimentation experiments with live organisms have been far less common but
also show unusual phenomena. For example, bacteria has been shown to enhance
sedimentation rates in the presence of polymers due to aggregation 45. In
mixtures of swimming algae and particles, the (exponential) steady-state
sedimentation profile of passive particle is found to be described by an
effective diffusivity (or temperature) that increases linearly with the
concentration of swimming microbes 46, similar to results in artificial active
particles. Recently it was found that the sedimentation speed of passive
particles can be significantly hindered by the presence of swimming
microorganisms (E. coli) in the dilute regime 28; the hindered settling
function for these settling active suspensions is captured by a Richardson-
Zaki-type equation 16 with an exponent that is much larger than for passive
systems and a function of bacteria activity. The time-dependent concentration
profiles can be adequately described by an advection-diffusion equation
coupled with sink-source terms to describe bacteria population dynamics, and
with a dispersivity parameter that increases linearly with bacteria
concentration 28. Nevertheless, the effects of bacterial activity on
suspension sedimentation dynamics is still not fully understood.

Here, we experimentally investigate the sedimentation dynamics of dilute
active suspensions using mixtures of passive colloidal spheres and swimming E.
coli. These initially well-mixed suspensions settle for a period up to three
days (depending on bacteria concentration $\phi_{b}$) and image analysis
techniques are used to track the height of the particle sedimentation front.
We find that the suspension phase separate into a particle- and bacteria-rich
fronts. A simple model is presented that can capture the phase separation time
based on species dispersivities and settling speeds.

##  2 Experimental Methods

In this contribution we experimentally study sedimentation of particles in the
presence of bacteria. Active suspensions are prepared by mixing Escherichia
coli (wild-type K12 MG1655) and polystyrene spheres (density $\rho=1.05$
g/cm3, Thermo Scientific) of diameter $d=2$ m in deionized (DI) water.
Bacteria are grown to saturation ($10^{9}$ cells/mL) in culture media (LB
broth, Sigma-Aldrich). Both spheres and the saturated culture are centrifuged
(Centrifuge 5430, eppendorf) and mixed with DI water to reach the desired
bacteria volume fraction.

Figure 1: Experimental setup and sample images are shown from left to right:
(a) A schematic of the setup and bacteria/particle suspensions. Passive
particles are 2 m polystyrene spheres, subject to gravity; volume fraction
$\phi_{p}=0.04\%$ kept constant for all experiments. Swimming activity is
provided by E. coli, a 2 $\umu m$ rod shaped bacterium. Sample snapshots of
sedimentation experiments with (b) a mixture of passive particles
($\phi_{p}=0.04\%$) with non-motile bacteria (E. coli) at $\phi_{d,b}=0.20\%$,
(c) mixture of passive particles with motile bacteria (E. coli) at
$\phi_{b}=0.20\%$ and (d) a mixture of passive particles with motile bacteria
(E. coli) at $\phi_{b}=0.85\%$. All vials samples are shown at time $t=40$ hr.
Note the appearance of two sedimentation fronts, bacteria and particle.

We transfer 1.5 mL of the prepared suspensions into glass vials (9.7 mm in
diameter, 35 mm in height), as shown schematically in Fig. 1(a); the liquid
column is approximately 20 mm tall. The suspensions are homogenized by hand
with a pipette so that the bacteria and colloids are uniformly distributed at
the start of the experiment. These vials are placed in a water tank to
maintain the vials at room temperature $T_{0}=22$ °C and to remove optical
aberrations during imaging (cylindrical vials are preferred to avoid effects
from sharp edges). Images are taken every 10 min for a period of 2-3 days with
a Nikon D7100 camera that is equipped with a 105 mm Sigma lens. The light
source is a Edge-lit Collimated Backlights (CX,Advanced Illumination).

The initial motile bacteria volume fraction $\phi_{b}$ in the vials ranges
from $0.15\%$ to $1.0\%$, while the spherical colloids volume fraction is kept
constant at $\phi_{p}=0.04\%$. Both $\phi_{b}$ and $\phi_{p}$ concentrations
are considered dilute, and no large scale collective behavior are observed for
these bacteria-particle suspensions. The range of $\phi_{b}$ used here are
below which collective motion is generally observed ($\approx 10^{10}$
cells/mL) 47. Under these dilute conditions, E. coli swims at speeds ranging
from $10$ m/s to $20$ m/s by the actuation of a helical flagellar bundle. This
mode of swimming, under steady conditions, can be described in the far field
by an extensile dipole flow (i.e., "pusher") 40.

##  3 Results and Discussion

Sample snapshots of the sedimentation experiments are shown in Fig. 1(b) – (d)
at $t=40$ hr. The control passive case, (mixture of non-motile bacteria at
$\phi_{d,b}=0.2\%$ with spherical colloidal particles at $\phi_{p}=0.04\%$) is
shown in Fig. 1(b), while motile bacteria at $\phi_{b}=0.20\%$ and
$\phi_{b}=0.85\%$ cases are shown in Fig. 1(c) and Fig. 1(d), respectively.
The images show that bacterial activity hinders the settling speed of passive
particles, as previously reported 28. We find that the passive particle
sedimentation front sits higher in the $\phi_{b}=0.20\%$ case than in the non-
motile bacteria case ($\phi_{d,b}=0.20\%$). This trend continues for the
$\phi_{b}=0.85\%$ case, which indicates that particle settling speed decreases
with increasing $\phi_{b}$ (see SM for more information). We also observe the
formation of two sedimentation fronts, one that is rich in passive particle
and one that is rich in bacteria, as shown in Fig. 1(b) – (d). These two
fronts settle at different speeds. These qualitative results show that (i)
bacteria significantly hinders particle sedimentation speed and (ii) the two
species phase separate during the sedimentation process. While the appearance
of the bacteria front has been previously reported 28, here we will focus on
quantifying and understanding the appearance of the two sedimentation fronts
and its relationship to particle sedimentation front speed.

Figure 2:  Kymographs for sedimentation at fixed particle concentration
($\phi_{p}=0.04\%$) and different bacterial concentrations. (a) With non-
motile bacteria $\phi_{d,b}=0.20\%$, (b) motile bacteria at $\phi_{b}=0.20\%$,
and (c) motile bacteria at $\phi_{b}=0.85\%$. Two fronts are seen when
$\phi_{b}\geq 0.15\%$. Note that activity hinders the settling speed of
spherical particles.

###  3.1 Front Sedimentation Heights & Speeds

We begin by measuring the sedimentation height of the passive spheres front as
a function of time at different $\phi_{b}$. The particle front sedimentation
height, $h_{p}(t)$, is located in the vial by identifying the peak in the
vertical gradient of the image intensity. By plotting $h_{p}(t)$ as a function
of time, we can extract the sedimentation speed of the particle front $V_{p}$
and define a hindered settling function $H(\phi_{b})$ as a function of
bacteria concentration. The hindered settling function is defined as the ratio
of the mean sedimentation speed $V_{p}$ of the particle front over the
sedimentation speed of a single particle such that
$H(\phi_{b})=V_{p}(\phi_{b})/V_{0}$. We note that the Stokes’ sedimentation
speed $V_{0}$ is $0.12\leavevmode\nobreak\ \umu$m/s for the 2 m polystyrene
particles in water.

Figure 3: Height of particle sedimentation front, $h_{p}(t)/h_{0}$, where
$h_{0}$ is the initial height tracked on the sedimentation process as a
function of time for a range of initial motile bacterial volume fractions
$\phi_{b}$. Data shows the decrease in front speed with an increase of
bacteria volume fraction $\phi_{b}$. The slope of $h_{p}(t)/h_{0}$ does not
vary significantly when $\phi_{b}>0.40\%$. Solid lines are obtained by solving
Equation 5. Solution to this expression is shown in Equation 7, and it
captures relatively well the height of the front of passive particles for a
wide range of bacterial concentrations $\phi_{b}$. Inset shows particle front
speed ($V_{p}$) as a function of $\phi_{b}$ for short times ($t<t_{c}$).

Figure 2(a) – (c) shows sedimentation kymographs ($h$ vs $t$) for the mixture
of particles with non-motile bacteria ($\phi_{d,b}=0.20\%$) case and two with
active suspensions, $\phi_{b}=0.20\%$ and $\phi_{b}=0.85\%$, respectively;
these kymographs correspond to the data shown in Fig. 1(b) – (d). These
experimental kymographs are obtained by the juxtaposition of a single pixel
line at the center of the vial for each image of the time series. As noted
before, all samples have a fixed particle volume fraction ($\phi_{p}=0.04\%$).
In each case a sedimentation front is formed and moves downward from the top
of the container at a speed $V_{p}$. A first look at the corresponding
kymographs reveal that the velocity $V_{p}$, here visible as the slope of the
colloid front (black dashed line), decreases as the concentration of bacteria
increases. We again observe the formation of a bacteria front that separates
from spherical particles after a time $t_{c}$ (red dashed line). The timescale
$t_{c}$ is defined as the time that the bacteria-rich front diverge from the
particle-rich front. Experiments with non-motile bacteria and spherical
particles shows that the two phases separate at relatively short time-scales
($t<1$ hr), as shown in Fig. 2(a). Suspensions of motile bacteria (with
particles) on the other hand phase separate at much longer time-scales ($t>10$
hr). That is, activity seems to delay phase separation. Moreover, the phase
separation time, $t_{c}$, seems to decrease with $\phi_{b}$, that is,
$t_{c}=t_{c}(\phi_{b})$.

The particle front sedimentation speed $V_{p}$ is found by linear fits to the
kymograph plots such that

$h_{p}(t)=h_{0}-V_{p}t,$ (1)

where $h_{0}$ is the initial front position; in all cases studied here, this
linear relationship works relatively well even at long times ($t>24$ hr). In
absence of bacteria ($\phi_{b}=0\%$), the front sedimentation speed
$V_{p}=0.11\leavevmode\nobreak\ \umu$m/s, which is relatively close to the
sedimentation speed of a single sphere, $V_{0}=0.12\leavevmode\nobreak\
\umu$m/s. This result indicates that hydrodynamic interaction among passive
particles are relatively weak in our study. This not surprising since particle
volume fraction is quite dilute, $\phi_{p}=0.04\%$.

Figure 3 shows the normalized height of the particle sedimentation front,
$h(t)/h_{0}$, for different bacteria volume fractions $\phi_{b}$, where
$h_{0}$ is the front initial height. We find that the settling height of the
particle front is linear through all experiments, so it is described by a
average speed, $V_{p}$. A constant sedimentation speed is not a trivial result
because the decrease in bacteria activity is commensurable with the time of
the experiment. At 48 hr of experiment, approximately 70$\%$ of the population
has lost motility and behave as a passive colloid 28. This has been previously
characterized by the bacteria motility loss rate $k$, which is approximately
$6\times 10^{-6}\leavevmode\nobreak\ \mathrm{s}^{-1}$ (see Ref. 28). We
believe that the decrease in activity promotes a faster sedimentation;
concurrently, the increase in passive particles (non-motile bacteria) should
decrease the overall sedimentation speed. These two processes are not
necessarily of the same magnitude.

The data also show that $V_{p}$ decreases as $\phi_{b}$ increases (Fig. 3).
Surprisingly, this trend weakens and $V_{p}$ becomes independent of $\phi_{b}$
for $\phi_{b}>0.40\%$ at long times ($t>t_{c}$). This transition, however, is
not observed at short times ($t<t_{c}$). To better illustrate this trend, we
plot $V_{p}$ as a function of $\phi_{b}$ for all cases (Fig. 3, inset); we
find that, for $t<t_{c}$, $V_{p}$ decreases linearly with $\phi_{b}$. Overall,
we find two different functionalities regarding the effects of bacteria
activity on $V_{p}$: (i) for $t<t_{c}$, $V_{p}$ decreases linearly with
$\phi_{b}$ for all cases; (ii) for $t>t_{c}$, $V_{p}$ decreases linearly with
$\phi_{b}$ for $\phi_{b}\leq 0.4\%$ followed by an asymptote for $\phi_{b}\geq
0.45\%$.

###  3.2 Hindered Settling Function for Active Suspensions

We now compute the suspension hindered settling function $H$ as a function of
bacteria concentration, $\phi_{b}$; note that $\phi_{p}$ is kept constant at
0.04% for all cases. This variable is explored as a touchstone to previous
literature and as a tool to develop our phenomenological model for the
sedimentation of active suspensions. Similar to the $V_{p}$ data (and as
expected), we find two regimes for $H(\phi_{b})$. (i) For $t<t_{c}$, $H$
decreases linearly with $\phi_{b}$ for all cases [Fig. 4(a), black triangles];
(ii) for $t>t_{c}$, $H$ decreases linearly with $\phi_{b}$ for $\phi_{b}\leq
0.4\%$ followed by an asymptote ($H\approx 0.45$) for $\phi_{b}\geq 0.45\%$
[Fig. 4(a), red diamonds]. These data show that bacterial activity can reduce
the particle sedimentation speed by a factor of 2, which illustrates the
dramatic effect of bacterial activity on sedimentation dynamics. The first
regime ($t<t_{c}$) can be described by a linearized version of the Richardson-
zaki equation 16 such that $H(\phi_{b})=(1-n\phi_{b})$, with a best fit of
$n=95$ [Fig. 4(a), black line]. This exponent is significantly larger than the
expected fitting parameter for Brownian particles ($n\approx 5.5$) 21 and
Batchelor’s result ($n=6.5$) 15, but in line with our own previous results
($n\approx 120$) 28. While the mechanisms leading to the linear dependence of
$H$ on $\phi_{b}$ was previously investigated in our previous work 28, the
transition of $H$ from a linear dependence to a $\phi_{b}$-independent regime
at $t>t_{c}$ has yet to be reported and/or understood.

The observed $\phi_{b}$-independent regime in $H$ may be due to the increase
in the amount (or volume fraction) of non-motile bacteria within the
suspension during the sedimentation process. If we assume a a constant
motility loss rate $k$ 28, it follows that an increase in $\phi_{b}$ would
result in the production of more non-motile bacteria for the same experimental
time $\Delta t$. This increase in non-motile concentration within the system
may overwhelm the effect of activity ($\phi_{b}$), although it is not enough
to significantly affect particle sedimentation speed. That is, the dependence
of $H$ on passive (non-motile) particle volume fraction is much weaker than on
motile bacterial volume fraction.

###  3.3 Phase Separation During Sedimentation

Figure 4: (a) Hindered settling function $H(\phi_{b})$ as a function of
bacterial volume fraction $\phi_{b}$. $H(\phi_{b})$ was obtained at short
times (black triangles) and at long times (red diamonds). At short times
($t<t_{c}$) we observed that sedimentation is hindered linearly in the
experiments. However, at long times ($t>t_{c}$), $H$ decreases linearly with
$\phi_{b}$ for $\phi_{b}\leq 0.4$ followed by an asymptote ($H\approx 0.45$)
for $\phi_{b}>0.45$ (b) Phase separation times-scale $t_{c}$ (blue circles) as
a function of bacteria volume fraction $\phi_{b}$. Solid line is the times-
scale prediction obtained by Equation 3 with a time dependent diffusivity that
decays with time. Experimental results shows that the timescale $t_{c}$ has
the same functionality as $H(\phi_{b})$.

The images and kymographs shown in Fig. 1 and Fig. 2, respectively, show that
after a certain period of sedimentation time, the active suspension separates
into particle-rich and bacteria-rich fronts. Figure 4(b) [blue circles] shows
the experimentally measured phase separation time $t_{c}$ as a function of
$\phi_{b}$ for all cases. Similar to $H(\phi_{b})$, the quantity $t_{c}$ shows
a decreasing trend with $\phi_{b}$, suggesting a deeper underlying relation at
play. We now provide a simple argument to understand the origins of $t_{c}$ by
noting that the behavior of the bacteria front is diffusive and can be
captured by an equation of the form:

$h_{b}(t)=h_{0}-V_{b}t-2\sqrt{D_{b}t}.$ (2)

Here, $D_{b}$ is a dispersivity related to bacterial activity and $V_{b}$ is
the sedimentation speed of the bacteria front. A dispersion length scale can
thus be defined as $L_{D}(t)=2\sqrt{D_{b}t}$. At earlier times, the bacteria
front [or Eq. 2] settles faster than the particle front [or Eq. 1], but slower
at later times (see SM Fig. 3). The phase separation time, $t_{c}$, is the
time at which the particle front crosses over the bacteria front, or
$h_{b}(t_{c})=h_{p}(t_{c})$. By equating Eq. 1 and Eq. 2, we obtain the
following equation for the phase separation time:

$t_{c}=\frac{4D_{b}}{(V_{p}-V_{b})^{2}}.$ (3)

All variables in the above equation can be measured or estimated. For example,
$V_{p}$ is experimentally measured as a function of $\phi_{b}$ (Fig. 3) and
$V_{b}=0.02\pm 0.002\leavevmode\nobreak\ \umu$m/s is approximately independent
of $\phi_{b}$ in the dilute regime investigated here (see SM for more
information). The dispersivity $D_{b}$ can be obtained by fitting Eq. 2 to the
bacteria front shown in the kymograph. The data shows that $D_{b}$ decays
exponentially with time as such that $D_{b}(t)=D_{0}\exp(-\beta t)$, where
$D_{0}$ is the initial dispersivity, and $\beta$ is a dispersivity decay rate;
the values of $D_{0}$ and $\beta$ can be found in the SM. The decay in
dispersivity over time may be due to nutrient and oxygen depletion, or even
aerotactic response near the air-liquid interface 48, 49 (see SM for more
information). For simplicity, we will proceed with an average dispersivity
$\overline{D_{b}}$ defined as:

$\overline{D_{b}}=\frac{1}{t_{0}}\int_{0}^{t_{0}}D_{b}(t)\,dt,$ (4)

We plot the calculated phase separation time using $\overline{D_{b}}$ in Fig.
4(b) [black diamonds]. Results show that this simple argument is able to
capture the nonlinear experimental data relatively well, including the
decreasing trend of $t_{c}$ with $\phi_{b}$. Here, $t_{0}$ is a long enough
time to provide a statistically meaningful average, longer than the
experimentally measured phase separation time.

####  3.3.1 Scaling for Hindering Settling Function

The data show above makes it clear that there is a relationship between the
timescale associated with phase separations $t_{c}$ and the hindering settling
function $H(\phi_{b})$. The other time scale present in the experiments is the
bacteria motility loss rate, $k$. We can use these two time-scales, $t_{c}$
and $k$, to non-dimensionalize both $h_{p}(t)$ and $t$. The quantity $t_{c}$
is used to obtain a "separation" length-scale, $l_{0}=t_{c}V_{0}$, to
normalize $h_{p}(t)$ such that $h_{p}(t)/l_{0}$. This quantity tells us how
far or close the front is from separating. Next, we non-dimensionalize time
$t$ with the motility loss rate, $k$. The quantity $tk$ provides a measure of
activity left in the sample. Figure 5(a) shows the normalized particle
sedimentation height, $h_{p}(t)/l_{0}$, as a function of scaled time, $tk$,
for a range of $\phi_{b}$. The data shows that $h_{p}(t)/l_{0}$ decreases
linearly with $tk$, suggesting that activity prevents phase separation likely
by mixing the two phases during the sedimentation process. Also, we find an
average slope of $-1.97\pm 0.050$ through all experiments, indicating that
these sample share similar dynamics.

Figure 5: (a) Normalized height $h_{p}(t)/l_{0}$ as a function of normalized
time $tk$. Length-scale $l_{0}$ is defined as $l_{0}=V_{0}t_{c}$. Normalized
plot shows a constant slope on all experiments, the slope as a value of
$1.97\pm 0.050$. (b) Normalized height as a function of the normalized time.
The height was initially normalized by the length-scale $l_{0}$, and the time
by the motility loss rate $k$. This normalization gave a constant slope of
$1.97\pm 0.050$ as shown in (a). Now, the normalized height is shifted by a
constant $C$ divided by the hindered settling function $H(\phi_{b})$. The
shift on the normalized height of the experimental data gives a collapse into
a single line at different $\phi_{b}$. The constant has a value of $C=1.955\pm
0.045$.

Next, we account for the intercept in Fig. 5(a) by noticing that it changes
similarly to the inverse of the hindered settling function, $H(\phi_{b}$) (see
SM for more information). Therefore, we subtract the re-scaled height
($h_{p}(t)/l_{0}$) by a constant divided by the hindered settling function
such that $C/H(\phi_{b})$. The constant $C$ is obtained by equating the
intercept of $h_{p}(t)/l_{0}$ with $C/H(\phi_{b})$, giving a value of
$1.955\pm 0.045$. Remarkably, all experiments collapse onto a single master
curve [Fig. 5(b)]. Putting together these results, the sedimentation front of
passive particles can be described by the following differential equation:

$\frac{h_{p}(t)}{t_{c}V_{0}}-C\frac{V_{0}}{dh/dt}=-2kt,$ (5)

which can be rearranged to be

$\frac{dh_{p}(t)}{dt}\left(\frac{h_{p}(t)}{t_{c}V_{0}}+2kt\right)=CV_{0}.$ (6)

We take the linear solution to this equation to obtain

$h_{p}(t)=-2kt_{c}V_{0}t+\frac{CV_{0}}{2k}.$ (7)

Equation 5 shows that the sedimentation front of the passive particles are
described by three main terms. The first term is (sedimentation) height
normalized by a diffusion-based length-scale associated with the bacteria-
particle settling process. The second term quantifies passive settling under
gravity via the hindered settling function. The third term is time normalized
by the motility decay rate, and quantifies the level of activity left in the
sample. We note that a solution to Equation 5 is a line and is shown in
Equation 7. If we solve the equation and revisit front height over time [Fig.
3], we observe that the solution captures all cases of active suspensions.

Our experimental observations show that sedimentation and separation of
passive particles and bacteria suspensions are described by using
sedimentation and diffusive length scales. The quantity $t_{c}$ of active
suspensions occurs over very long time-scales compared to mixtures of passive
particles that have two different masses and is reminiscent of the effects of
centrifuging. This timescale plateau above a critical bacteria concentration
($\phi_{b}=0.40\%$). We find that when $t_{c}$ plateau when $\phi_{b}>0.40\%$,
and $H(\phi_{b})$ has the same functionality. A possible mechanism that
explains our experimental observations is due to an increased presence of non-
motile bacteria which overwhelm the effects of the increased activity. The
injection of energy by the bacteria swimming motion plays an important role in
the sedimentation process; its decay is quantified by a motility loss rate
$k$, which is used to describe the settling of spherical particles.

Preparation of new experiments that track spherical particles while they
sediment would help to understand these observations. This could be achieved
by using a confocal microscope with a smaller vial. The downside of this
proposed approach is that controlling the temperature at long times is a
challenge. With this approach we would be able to obtain 3-D imaging as a
function of time, which would allow to track the spherical particles and
explore particle fluctuations.

##  4 Conclusions

The sedimentation of passive particles in the presence of swimming bacteria is
experimentally investigated. We find that bacterial activity (even in the
dilute regime) hinders the sedimentation of passive particles and phase
separates at large time-scales, whereas the bacteria front with non-motile
bacteria shows at smaller time-scales ($t<1$ hr). Particle hindered settling
function, $H(\phi_{b}$), as well as $t_{c}$, show two distinct regimes: (i) a
monotonic decay by increasing $\phi_{b}$ and (ii) a regime in which both
$H(\phi_{b}$) and $t_{c}$ are no longer dependent of $\phi_{b}$. A simple
model based on particle and bacteria length scales is proposed which describes
$t_{c}$ for a range of $\phi_{b}$. Using non-dimensional analysis, we collapse
all sedimentation front data onto a single master line that captures both
bacteria concentration regimes. A phenomenological model is proposed to
describe the height of the particle sedimentation front. The solution of this
model captures relatively well the height of the sedimentation front of the
passive particles for all $\phi_{b}$ within the dilute regime. In summary, an
addition of a diffusive length-scale can captures the short time behavior of
the bacteria front, and a sedimentation speed length scales captures the long
time behavior of the bacteria front and the particle front. Further study on
this field with different colloid concentrations and swimmer actuation modes
(pusher vs puller) would presumably shed light on how the hydrodynamic
interactions between the passive particles and active bacteria underlay the
dynamics of our system and be generalized to others.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We thank D. Jerolmack, C. Kammer, and A. Patteson for fruitful discussions.
S.P. and P.E.A. acknowledge support from Army Research Office (ARO, Grant
W911NF2010113). B.O.T.M., R.R., and P.E.A. acknowledge support by the National
Science Foundation grant DMR-1709763.

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