# Alignment of colloidal rods in crowded environments

Vincenzo Calabrese<EMAIL_ADDRESS>Stylianos Varchanis  Simon J.
Haward  Amy Q. Shen<EMAIL_ADDRESS>[

###### Abstract

Understanding the hydrodynamic alignment of colloidal rods in polymer
solutions is pivotal for manufacturing structurally ordered materials. How
polymer crowding influences the flow-induced alignment of suspended colloidal
rods remains unclear when rods and polymers share similar length-scales. We
tackle this problem by analyzing the alignment of colloidal rods suspended in
crowded polymer solutions, and comparing against the case where crowding is
provided by additional colloidal rods in a pure solvent. We find that the
polymer dynamics govern the onset of shear-induced alignment of colloidal rods
suspended in polymer solutions, and the control parameter for the alignment of
rods is the Weissenberg number, quantifying the elastic response of the
polymer to an imposed flow. Moreover, we show that the increasing colloidal
alignment with the shear rate follows a universal trend that is independent of
the surrounding crowding environment. Our results indicate that colloidal rod
alignment in polymer solutions can be predicted based on the critical shear
rate at which polymer coils are deformed by the flow, aiding the synthesis and
design of anisotropic materials.

Unknown University] Okinawa Institute of Science and Technology, Onna-son,
Okinawa 904-0495, Japan

##  1 Keywords

Cellulose nanocrystals, polymer rheology, flow-induced birefringence,
rotational diffusion, colloidal rod alignment, Weissenberg number, Péclet
number.

##  2 Introduction

The ability to control the hydrodynamic alignment of colloidal rods is
critical to produce structurally ordered soft materials that possess desirable
mechanical, thermal, optical, and electrical properties.1, 2, 3, 4, 5, 6, 7, 8
These anisotropic materials are promising in applications ranging from
electronic sensors and soft robotics, to tissue engineering and biomedical
devices. 5, 9, 10, 11 In material science and engineering, colloidal rods are
used in combination with other polymers that impart specific functionality to
the final composite material (e.g., increasing ductility and mitigating
embrittlement).12, 13, 1 As such, understanding and controlling the
hydrodynamic alignment of colloidal rods in polymer matrices becomes of
pivotal importance in large-scale processing operations.

Existing literature has shown that the most important control parameter for
the onset of hydrodynamic alignment of rigid colloidal rods is the Péclet
number $Pe=\dot{\gamma}/Dr$, a dimensionless number quantifying the relative
strength between the imposed deformation rate (e.g., the shear rate,
$\dot{\gamma}$) and the rotational diffusion coefficient of the rods
($Dr$).14, 15, 16, 17, 18 In dilute suspensions, the Péclet number can be
defined as $Pe_{0}=\dot{\gamma}/Dr_{0}$, with the rotational diffusion
coefficient of the rods given as

$Dr_{0}=\frac{3k_{b}T\mathrm{ln}(l/d)}{\pi\eta_{s}l^{3}}~{},$ (1)

where $d$ and $l$ are the hydrodynamic diameter and length of the colloidal
rod, respectively, $k_{b}=1.38\times 10^{-23}$ J/K is the Boltzmann constant,
$T$ is the absolute temperature, and $\eta_{s}$ is the solvent viscosity. For
$Pe_{0}<1$ Brownian flocculation dominates, whilst at $Pe_{0}\geq 1$
convective forces are strong enough to induce alignment of the colloidal rods
in the flow direction. However, this criterion is only valid under the
assumption that the colloidal rods perceive the surrounding fluid as a
continuum medium, i.e., the characteristic length scale of the colloidal rods,
such as the radius of gyration
$Rg_{r}=\sqrt{\frac{d^{2}}{2}+\frac{l^{2}}{12}}$, must be much larger than
that associated with the suspending medium.19, 20, 21 Colloidal rods suspended
in low $Mw$ solvents such as water generally satisfy this assumption. However,
in many industrial and biological processes, colloidal rods flow in crowded
environments of polymers in solution where the characteristic length scale of
suspended rods is similar to those of the surrounding macromolecules, e.g.,
the polymer radius of gyration, $Rg_{p}$, or the polymer mesh size, $\xi_{p}$
(also referred to as the correlation length).20, 22, 23, 24, 25, 26, 27 In
this scenario, the continuum assumption breaks down and the rods experience a
local viscosity ($\eta_{local}$) that lies between the solvent viscosity and
the bulk viscosity of the polymeric solution.23, 27, 22, 24 In principle, by
knowing $\eta_{local}$ it is possible to predict the shear rate for the onset
of colloidal alignment based on the criterion of $Pe_{0}=1$ using
$\eta_{local}$ in place of $\eta_{s}$ in eqn.1. However, the main hurdle in
predicting the alignment of colloidal rods suspended in macromolecular
solutions based on the criterion $Pe_{0}=1$ stems from the fact that the value
of $\eta_{local}$ is not known a priori. Consequently, to date, it is
challenging to predict the onset of flow-induced alignment of colloidal rods
with comparable length scales to the suspending polymeric fluids.

The flow-induced alignment of rigid elongated particles suspended in
viscoelastic polymeric solutions has been studied experimentally and
numerically for particles with relevant length scales larger than that
associated with the suspending fluid; thus considering the suspending
polymeric fluid as a continuum medium for the colloidal rods.28, 26, 29, 30,
31, 32 In this length-scale context, theories predict a critical deformation
rate above which the elastic forces of the suspending fluid cause the particle
alignment to drift from the flow direction to the vorticity direction.33, 34,
28, 26 However, contrasting experimental results have been reported for
relatively small particles suspended in shear-thinning polymeric fluids.26,
29, 30 For instance, elongated hematite particles (with $l=600$ nm) in
entangled polyethylene oxide solutions30 displayed the particle alignment in
the vorticity direction as expected by theory. However, shorter hematite
particles (with $l=360$ nm) suspended in entangled polystyrene solutions26 did
not, casting doubts on the validity of continuum theories in conditions where
colloids and polymers have similar characteristic length-scales.

In this work, we elucidate the mechanism driving the onset of colloidal rod
alignment in semi-dilute polymer solutions where polymers act as the crowding
agents to tracer colloidal rods. We use cellulose nanocrystals (CNC) as rigid
colloidal rods as they are widely used in synthesis of composite polymer
materials with high performance and functionalities.35, 36, 13 To understand
the effect of crowding on the CNC alignment, we adopt two approaches: (i)
increasing the CNC mass fraction, $\phi$, in an aqueous Newtonian solvent
spanning from the dilute regime where interparticle interactions are
negligible up to the semidilute regime where interparticle interactions are at
play, providing _self-crowding_ of the CNC by other analogous particles; (ii)
Using shear-thinning (non-Newtonian) polymer solutions as the suspending
fluid, while keeping $\phi=1.0\times 10^{-3}$ so that the CNC is in the dilute
regime with negligible interparticle interactions, and the confinement acting
on the CNC is provided only by the surrounding polymer chains, which we refer
to as _polymer-crowding_. Specifically, we use high $Mw$ neutral polymers
(polyethylene oxide, PEO, and polyacrylamide), with a radius of gyration
($Rg_{p}$) comparable to the CNC length scale (i.e., $Rg_{r}$).37, 38, 39, 40,
23 For the polymer crowding case, we show that the onset of CNC alignment is
linked with the relaxation time of the polymer solution ($\tau_{p}$).
Specifically, we show that the Weissenberg number $Wi=\tau_{p}\dot{\gamma}$,
quantifying the strength of the elastic response of the fluid to an imposed
deformation rate, controls the onset of CNC alignment in polymeric media.

Figure 1: (a) Snapshots of the microfluidic platform from bottom and side
view. (b) Schematic drawing with coordinate system and relevant dimensions.
(c, d) Time averaged flow-induced birefringence (FIB) across the channel width
for a representative test fluid composed of CNC at $\phi=1\times 10^{-3}$
suspended in water at an average velocity $U=1.6\times 10^{-3}$ m s-1. In (c)
the normalized birefringence intensity ($\Delta n/\phi$) and in (d) the
orientation of the slow optical axis ($\theta$), given by the contour plot (in
full resolution) and depicted by the solid segments to guide the eye. The
horizontal dashed lines in (c) and (d) indicate the location where the
spatially averaged birefringence, $\langle\Delta n\rangle$, and the spatially
averaged angle of slow axis $\langle\theta\rangle$ are obtained for
quantitative analysis.

##  3 Experimental section

###  3.1 Test fluids

The test fluids were prepared using an aqueous CNC stock suspension
(CelluForce, Montreal, Canada, pH 6.3 at 5.6 wt%). The CNC has an average
length $\langle l\rangle=260\pm 180$ nm, a maximum length $l_{max}=700$ nm, an
average diameter $\langle d\rangle=4.8\pm 1.8$ nm as detected from atomic
force microscopy.18 The effective hydrodynamic diameter is computed as
$d=\delta+\langle d\rangle=27.4$ nm, considering the estimated contribution of
the electric double layer $\delta=22.6$ nm in deionised water.41 Assuming a
cylindrical shape, the number density of the CNC was calculated as
$\nu=(4~{}\phi_{volume})/(\langle d\rangle^{2}~{}l~{}\pi)$ where $l$ is the
hydrodynamic length of the CNC obtained experimentally through $Dr_{0}$
(specified in the main text), and $\phi_{volume}$ is the volume fraction of
the suspended CNC (calculated using a CNC density of 1560 Kg/m3).42 CNC
suspensions at different mass fraction, $\phi$, were prepared by dilution of
the mother CNC stock with deionised water and mixed on a laboratory roller for
at least 24 h at $\sim$ 22 ∘C. Where specified, the $\phi=1\times 10^{-3}$ CNC
suspension was prepared in a Newtonian solvent composed of a glycerol:water
mixture containing 17.2 vol% glycerol (Sigma-Aldrich 99% with $\eta_{s}=$ 1.7
mPa s as measured via shear rheometry).

Polyethylene oxide with $Mw\approx 4$ MDa, polyethylene oxide with $Mw\approx
8$ MDa, and polyacrylamide with $Mw\approx 5~{}$MDa, referred to as PEO4, PEO8
and PA5, respectively, were purchased from Sigma-Aldrich in powder form and
solubilized in deionised water on a laboratory roller for at least 48 h at
$\sim 22$ ∘C (stock solution). Polymer solutions at different concentrations
($c$ in mg/mL) containing a constant amount of CNC were prepared by diluting
the polymer stock solution with deionised water followed by dilution of the
CNC stock suspension to a final $\phi=1\times 10^{-3}$, followed by mixing on
a laboratory roller for 24 h at $\sim 22$ ∘C. Polymer solutions without the
CNC were prepared by following the same procedure described above. The polymer
concentration where polymers begin to overlap was estimated as
$c^{*}=Mw/((4\pi/3)Rg_{p}^{3}N_{A})$, where $N_{A}$ is the Avogadro’s number.
The radius of gyration for the PEO4 and PEO8 was 135 and 202 nm, respectively,
estimated as $Rg_{p}=0.02Mw^{0.58}$.37, 38 For PA5 $Rg_{p}=7.5\times
10^{-3}Mw^{0.64}=154$ nm.39 For the PEO8, the polymer concentrations tested
spanned between the dilute ($c/c^{*}<1$) and semidilute unentangled regime,
where $c/c^{*}>1$ and the tube diameter $t_{d}=b_{k}\phi_{p}^{-0.75}>Rg_{p}$,
where $\phi_{p}$ is the polymer volume fraction and $b_{k}\approx 3.7$ nm is
the experimentally determined tube diameter in a PEO melt.40, 23, 43 For PEO8
with $Rg_{p}=202$ nm, the highest polymer concentration tested is
$c/c^{*}=10.5$ which yields $t_{d}\approx 250$ nm, thus in the semidilute
unentangled regime ($c/c^{*}>1$ and $t_{d}>Rg_{p}$).

###  3.2 Methods

To assess the CNC alignment in crowded environments, we measure the flow-
induced birefringence (FIB) as a function of the shear rate in a straight
microfluidic channel etched in fused silica (Fig. 1(a,b)). The microfluidic
channel has a rectangular cross section with height $H=2$ mm along the
$z$-axis, corresponding to the optical path, and width $W=0.4$ mm, along the
$y$-axis, thus providing an approximation to a two-dimensional (2D) Poiseuille
flow.44, 18 The flow in the microfluidic channel is driven along the channel
length (x-axis) by a syringe pump (Cetoni Nemesys) and Hamilton Gastight
syringes infusing and withdrawing at an equal and opposite volumetric flow
rate ($Q$) from the inlet and outlet, respectively. The average flow velocity
in the channel is $U=Q(WH)^{-1}$.

####  3.2.1 Flow-induced birefringence (FIB)

Time-averaged FIB measurements were performed using an Exicor MicroImager
(Hinds Instruments, Inc., OR) using a 5$\times$ objective at room temperature
($\sim 22$ ∘C). The channel is illuminated through the $z$-axis (vorticity
direction) using a monochromatic beam of wavelength $\lambda=450$ nm or
$\lambda=630$ nm (see Fig. 1(a,b)). The retardance, $R$ in nm, and the
orientation of the slow optical axis (extraordinary ray), $\theta$, were
obtained from 7 images acquired at 1 s interval and time-averaged. The
spatially-resolved (spatial resolution of $\Delta n$ and $\theta$ is $\approx
2$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$/pixel), and time-averaged,
birefringence ($\Delta n=R/H$) and the orientation of the slow optical axis,
$\theta$, are obtained in the flow-velocity gradient plane ($x$-$y$ plane) as
shown in Fig. 1(c) and (d), respectively, for a representative test fluid of
dilute CNC in water. For each flow rate, the birefringence $\Delta n$ and the
absolute value of the orientation of the slow optical axis, $|\theta|$, are
spatially averaged along 1 mm of the $x$-axis at $y=\pm 0.1~{}$mm (see dashed
lines in Fig. 1(c, d)) and referred to as $\langle\Delta n\rangle$ and
$\langle\theta\rangle$, respectively. The background value of $\Delta n$
acquired at rest was subtracted for all the analysis presented and
quantitative analysis of $\theta$ is restricted to $\Delta n>3\times 10^{-6}$.
The orientation angle $\langle\theta\rangle$ probes the CNC orientation with
respect to the flow direction whilst the birefringence intensity,
$\langle\Delta n\rangle$, probes the extent of anisotropy in the system. For
fully isotropically oriented particles $\langle\Delta n\rangle=0$ with
colloidal alignment occurring for $\langle\theta\rangle<45^{\circ}$ and
$\langle\Delta n\rangle>0$. 45, 46 The error related to the spatially averaged
$\langle\Delta n\rangle$ and $\langle\theta\rangle$ is the standard deviation
from the averaging process.

####  3.2.2 Rheology and flow simulations

Shear rheometry of the test fluids was performed using a strain-controlled
ARES-G2 rotational rheometer (TA Instrument Inc.) equipped with a stainless
steel cone and plate geometry (50 mm diameter and 1∘cone angle). The test
fluids were covered with a solvent trap and measured at 25$\pm$0.1 ∘C
(controlled by an advanced Peltier system, TA Instruments). The shear
viscosity data were fitted to the Carreau-Yasuda (CY) generalized Newtonian
model:

$\eta=\eta_{\infty}+\frac{\eta_{0}-\eta_{\infty}}{[1+(\dot{\gamma}/\dot{\gamma}^{*})^{a}]^{(1-n)/a}}~{},$
(2)

where $\eta_{0}$ is the zero-shear-rate viscosity, $\eta_{\infty}$ is the
infinite-shear-rate viscosity, $\dot{\gamma}^{*}$ is the characteristic shear
rate for the onset of shear thinning, $n$ is the power law exponent and $a$ is
a dimensionless fitting parameter that controls the transition to the shear-
thinning region. The velocity field along the channel width ($W$, $y$-axis)
and the value of shear rate at $y=|0.1|$ mm, (the channel location where
$\langle\Delta n\rangle$ and $\langle\theta\rangle$ are obtained) were
computed using numerical simulation. The simulations were performed assuming
steady, 1-dimensional, fully developed flow in a planar channel with width
$W$, under the imposition of an average velocity $U$. The $x$-component of
momentum equation and the CY constitutive equation were discretized and solved
using an in-house Finite Element solver 47. In all simulations, 200 linear
elements were used across the width of the channel. For each average velocity
used during the FIB experiment, $\langle\Delta n\rangle$ and
$\langle\theta\rangle$ are compared with the effective value of
$|\dot{\gamma}|$ obtained from the simulation at $y=|0.1|$ mm. The channel
location of $y=\pm 0.1$ mm is chosen to be the midpoint between the side walls
($y=\pm 0.2$ mm) and the centerline ($y=0$ mm) so to provide relatively high
values of shear rate while avoiding undesired wall effects in the FIB
experiment. For very weakly shear-thinning fluids, for which the CY model
could not be fitted to the rheological data, the shear rate was computed as
for a Newtonian fluid.

##  4 Results and discussion

Figure 2: Spatially averaged orientation angle $\langle\theta\rangle$ (a, b,
e, f) and normalized birefringence $\langle\Delta n\rangle/\phi$ (c, d, g, h)
for CNC suspensions in the crowding-free (a-d) and self-crowding (e-h) regime.
Open and green filled stars represent CNC in water and in water:glycerol
mixture (17.2 vol% glycerol), respectively (crowding-free). For the samples
prepared in water the CNC rotational diffusion coefficient in the crowding-
free regime is $Dr_{0}=59$ s-1 whilst $Dr_{0}=31$ s-1 for the sample prepared
in the water:glycerol mixture labeled by the green star symbol and used as a
reference sample (Ref.). For the crowding-free regime, $\langle\theta\rangle$
(a) and $\langle\Delta n\rangle/\phi$ (c) as a function of $|\dot{\gamma}|$
and scaled as $Pe_{0}=|\dot{\gamma}|/Dr_{0}$ in (b) and (d), respectively. For
the self-crowding regime, $\langle\theta\rangle$ (e) and $\langle\Delta
n\rangle/\phi$ (g) as a function of $Pe_{0}$ and scaled as
$Pe=|\dot{\gamma}|/Dr$ in (f) and (h), respectively. The dashed lines in (a,
b, e, f) are the fittings to eqn. 3 with $\alpha=0.39$. In (b) eqn. 3 is also
plotted using $\alpha=1$ (solid line). Vertical lines at $Pe_{0}=1$ and $Pe=1$
are drawn as a reference. Error bars indicate the standard deviation of the
measurement.

####  4.0.1 Crowding-free

We begin with the evaluation of the FIB of dilute CNC at $\phi=1.0\times
10^{-3}$ suspended in an aqueous Newtonian media, either in water
($\eta_{s}=0.9$ mPa s) or water:glycerol mixture (17.2 vol% glycerol,
$\eta_{s}=1.7$ mPa s), shown in Fig. 2(a-d). At this relatively low CNC
concentration, interparticle interactions are negligible so that each CNC can
be considered as isolated and crowding-free.48 Fig. 2(a) and (c) present the
$\langle\theta\rangle$ and $\langle\Delta n\rangle/\phi$ as a function of
$|\dot{\gamma}|$ for the CNC in the crowding-free regime respectively. For
both media, the orientation angle, $\langle\theta\rangle$, displays a gradual
decrease with $|\dot{\gamma}|$ (Fig. 2(a)) and, for a given value of
$|\dot{\gamma}|$, the greater solvent viscosity of the water:glycerol mixture
favours the CNC to align with a smaller value of $\langle\theta\rangle$
compared to water as the solvent. Moreover, the greater viscosity of the
water:glycerol media triggers the onset of CNC alignment at a lower shear rate
than in water, as indicated by the onset of birefringence occurring at lower
values of $|\dot{\gamma}|$ (Fig. 2(c)).

The effective rotational diffusion coefficient $Dr$ of the CNC is obtained
experimentally based on

$\langle\theta\rangle=\frac{\pi}{4}-\frac{1}{2}\mathrm{arctan}\bigg{[}\bigg{(}\frac{|\dot{\gamma}|}{6Dr}\bigg{)}^{\alpha}\bigg{]}~{},$
(3)

where $0<\alpha\leq 1$ is a stretching exponent that accounts for particle
polydispersity.46, 49, 50, 51 For monodisperse particles $\alpha=1$ whilst for
polydisperse particles $\alpha<1$ (see solid and dashed line in Fig. 2(b)).
Fitting the data in Fig. 2(a) with eqn. 3 yields $\alpha=0.39$ for both
Newtonian solvents, while $Dr_{0}=59$ s-1 and $Dr_{0}=31$ s-1 for the water
and water:glycerol solvents respectively. The subscript “0” to $Dr$ indicates
the crowding-free regime (i.e., dilute CNC in a continuum medium) and is used
to distinguish it from the case where $Dr$ is affected by the surrounding
crowding agent. By solving eqn.1 for $l$, using an effective CNC diameter
$d=27.4$ nm accounting for the contribution from the electric double layer,41
and $T=22~{}^{\circ}$C, we obtain $l=610$ nm, in accordance with the longest
CNC population detected via microscopy in our previous work.18 Our
experimental data collapse onto master curves when we plot
$\langle\theta\rangle$ and $\langle\Delta n\rangle/\phi$ as a function of the
respective Péclet number in the crowding-free regime,
$Pe_{0}=|\dot{\gamma}|/Dr_{0}$, and, the birefringence signal increases
sharply at $Pe_{0}=1$, indicating that the $Pe_{0}$ scaling is correct (Fig.
2(b, d)). It is expected that the $Pe_{0}$ scaling yields master plots of
$\langle\theta\rangle$ and $\langle\Delta n\rangle/\phi$ only for the systems
where the CNC is not experiencing any confinement. It is important to note
that, $\langle\Delta n\rangle$ and $\langle\theta\rangle$ are complementary
parameters to quantify particle alignment during flow. The magnitude of
birefringence $\langle\Delta n\rangle$ is linearly related with the number of
aligned particles in a given illuminated volume, thus presented in a
normalized form as $\langle\Delta n\rangle/\phi$. Contrarily, for a uniform
particle alignment, $\langle\theta\rangle$ captures a geometrical property of
the system that is independent from the number of aligned particles in a given
illuminated volume.45

####  4.0.2 Self-crowding

Following the same approach as for the crowding-free case, we investigate the
CNC alignment upon increasing CNC mass fraction $\phi$, so that analogous
particles restrain the motion of each other; a regime that we refer to as
self-crowding (Fig. 2(e, h)). As a reference sample we use the crowding-free
suspension in water:glycerol mixture (see green star symbol). Increasing the
CNC concentration, the values of $\langle\theta\rangle$ decrease with
increasing $\phi$ (Fig. 2(e)). Analogously, the birefringence onset occurs at
smaller values of $Pe_{0}$, corresponding to lower values of $|\dot{\gamma}|$
(Fig. 2(g)). This behaviour can be explained by the restrained rotational
motion of the CNC above the overlap concentration due to particle confinement,
leading to a decreasing $Dr$ with increasing $\phi$. For each sample, $Dr$ can
be obtained by fitting the curves in Fig. 2(e) with eqn. 3 using a fixed value
of $\alpha=0.39$ as established in the crowding-free regime. By plotting
$\langle\theta\rangle$ and $\langle\Delta n\rangle/\phi$ as a function of the
Péclet number $Pe=|\dot{\gamma}|/Dr$, the curves collapse onto single master
curves (Fig. 2(f, h)) and eqn. 3 leads to the dashed line in Fig. 2(f). We
note that for $\langle\theta\rangle$, the master curve is set by only
considering $Pe$ as scaling factor. Contrarily, for the birefringence
$\langle\Delta n\rangle$, the CNC mass fraction $\phi$ needs to be considered,
indeed only scaling the birefringence as $\langle\Delta n\rangle/\phi$ enables
to collapse the curves onto a master curve.

Figure 3: Spatially averaged birefringence $\langle\Delta n\rangle$ and
orientation angle $\langle\theta\rangle$ for PEO8 solutions, at different
$c/c^{*}$ , seeded with CNC at $\phi=1.0\times 10^{-3}$. The PEO8 solutions
are prepared in water, thus the rotational diffusion coefficient in the
crowding-free regime is $Dr_{0}=59$ s-1, whilst $Dr_{0}=31$ s-1 for the
reference water:glycerol mixture, labeled by the green star symbol (Ref.).
Orientation angle $\langle\theta\rangle$ (a) and, normalized birefringence
$\langle\Delta n\rangle/\phi$ (c) as a function of
$Pe_{0}=|\dot{\gamma}|/Dr_{0}$ and scaled as $Pe=|\dot{\gamma}|/Dr$ in (b) and
(d), respectively. In (e) the birefringence $\langle\Delta n\rangle$ is scaled
as $\langle\Delta n\rangle/(\phi-\phi_{cage})$ where
$\phi-\phi_{cage}=\phi_{eff}$ and plotted as a function of $Pe$. (f) The
$\phi_{cage}/\phi$ (circles) and $Rg_{r}/(\xi_{p}+Rg_{r})$ (dashed blue line)
as a function of polymer concentration ($c/c^{*}$). $Rg_{r}$ is the radius of
gyration of the CNC and $\xi_{p}$ is polymer mesh size (eqn. 4). The dashed
line in (a) is an instance of the fittings to eqn. 3 with $\alpha=0.39$. In
(b) the curve described by eqn. 3 collapse on to a master curve. Vertical
lines at $Pe_{0}=1$ and $Pe=1$ are drawn as a reference. Error bars from panel
(a-e) indicate the standard deviation of the measurement, whilst in (f) error
bars indicate uncertainty associated with $\phi_{cage}$ calculation.

####  4.0.3 Polymer-crowding

For the case of polymer-crowding we follow $\langle\theta\rangle$ and
$\langle\Delta n\rangle$ arising from the alignment of diluted CNC
($\phi=1.0\times 10^{-3}$) in polymer solutions exposed to a shearing flow. We
consider polymers with different $Mw$ and concentrations to achieve
viscoelastic polymer solutions with viscosity up to $400\times$ the viscosity
of water, and relaxation times $0.04<\tau_{p}<0.7$ s, to provide a large span
of crowding environments to the CNC. We use aqueous solutions of polyethylene
oxide with $Mw\approx 4$ MDa and $Mw\approx 8$ MDa, referred to as PEO4 and
PEO8, respectively, and polyacrylamide with $Mw\approx 5.5$ MDa, referred to
as PA5. In the absence of CNC, these polymer solutions do not display any
significant birefringence, ensuring that $\langle\Delta n\rangle$ and
$\langle\theta\rangle$ signals measured for the CNC dispersions arise
exclusively due to the CNC alignment. Here we focus on PEO8 solutions;
analysis for the PEO4 and PA5 solutions are given in the ESI Section 2. The
confinement imposed on the CNC is tuned by the polymer concentration ($c$),
expressed in normalized form as $c/c^{*}$, where $c^{*}$ is the polymer
overlap concentration (Fig. 3). For the PEO8, $c^{*}=0.39$ mg/mL. The PEO8
concentration in the polymer crowded CNC dispersions is varied between the
dilute regime, $c/c^{*}<1$, and the semidilute unentangled regime, where
$c/c^{*}>1$ and the tube diameter ($t_{d}$) is greater than the PEO8 radius of
gyration, $Rg_{p}=202$ nm.43 Upon increasing the PEO8 concentration,
$c/c^{*}$, the $\langle\theta\rangle$ decreases for a given shear rate, thus
$Pe_{0}=|\dot{\gamma}|/Dr_{0}$ (see Fig. 3(a)), and the onset of birefringence
shifts towards lower values of $Pe_{0}$ (Fig. 3(c)). Interestingly, by
plotting $\langle\theta\rangle$ as a function of $Pe=|\dot{\gamma}|/Dr$, using
the values of $Dr$ obtained from the fitting of eqn. 3, the curves collapse
onto a master curve (Fig. 3(b)), revealing that the alignment of the CNC
occurs in a similar manner for different PEO8 concentrations. Elliptical
hematite particles (with $l=600$ nm and cross-section of $130$ nm) in
entangled PEO solutions have been reported to first orient along the flow
direction and then evolve to orientations in the vorticity direction.30
However, within our experimental window, we could only observe $\langle\Delta
n\rangle/\phi$ increasing as a function of $Pe$, without the birefringence
drop associated with the particle alignment drifting from the flow direction
to the vorticity direction.30, 29, 26 Indeed, perfect alignment along the
vorticity direction ($z$-axis) for particles with circular cross-sections
would lead to isotropic projections in the flow-velocity gradient plane
($x$-$y$ plane) with $\langle\Delta n\rangle=0$. Similarly to our results,
Johnson et al.26 only observed the orientation in the flow direction for
elliptical hematite particles (with $l=360$ nm and cross section of $100$ nm)
in entangled polystyrene solutions.

Different from the self-crowding cases, scaling $\langle\Delta n\rangle/\phi$
by $Pe$ is insufficient to collapse the data onto a single master curve (Fig.
3(d)). We re-analyze the data based on the assumption that the failure to
collapse might be caused by a small amount of caged CNC ($\phi_{cage}$) within
the PEO8 matrix that remain in an isotropic state, which do not contribute to
$\langle\Delta n\rangle$ intensity. Indeed, scaling the $\langle\Delta
n\rangle$ by an effective CNC mass fraction $\phi_{eff}=\phi-\phi_{cage}$
enables the data to collapse onto a master curve (Fig. 3(e)). For each
$c/c^{*}$, $\phi_{eff}$ is obtained by minimizing the sum of squared residuals
between the reference water:glycerol curve and the polymer-containing samples
(detailed procedure is given in the ESI Section 3). The $\phi_{cage}/\phi$
increases with $c/c^{*}$ and at $c/c^{*}\approx 5$, it seems to approach a
plateau value $\phi_{cage}/\phi\approx 0.65$ (Fig. 3(f)). To corroborate the
hypothesis that a portion of CNC is entrapped within the PEO8 network without
contributing to the birefringence intensity, it is instructive to understand
the evolution of $\phi_{cage}$ from a topological perspective. As such, we
compare the radius of gyration of the CNC,
$Rg_{r}=\sqrt{\frac{d^{2}}{2}+\frac{l^{2}}{12}}=177$ nm, using $l=610$ nm as
previously determined from $Dr_{0}$, and $d=27.4$ nm, with the mesh size of
the PEO8 38

$\xi_{p}=Rg_{p}(c/c^{*})^{-0.75}~{},$ (4)

as a function of $c/c^{*}$, where $Rg_{p}=202$ nm is the radius of gyration of
PEO8. Specifically, we use the ratio $Rg_{r}/(\xi_{p}+Rg_{r})$ to yield values
between 0 ($\xi_{p}\rightarrow\infty$) and 1 ($\xi_{p}\rightarrow 0$). Both
$\phi_{cage}/\phi$ and the theoretical curve $Rg_{r}/(\xi_{p}+Rg_{r})$, based
on eqn. 4, display a similar trend as a function of $c/c^{*}$, suggesting that
$\phi_{cage}$ is linked to the topological confinement exerted by the polymer
mesh size $\xi_{p}$. Thus, at $c/c^{*}<5$, $\xi_{p}$ is a strong function of
the polymer concentration, and $\phi_{cage}/\phi$ increases likewise.
Contrarily, at $c/c^{*}>5$, $\xi_{p}$ becomes a weaker function of $c/c^{*}$,
accordingly, $\phi_{cage}$ starts to plateau.

Overall, our observations imply that the PEO8 provides a two-way topological
hindrance to the CNC rotation, where a fraction of the CNC, $\phi_{eff}$,
perceives the confinement but is able to align at $Pe>1$, whilst the other
remaining fraction, $\phi_{cage}$, is unable to align in the range of
investigated $Pe$. Contrarily, for the case of self-crowding, all the CNC
contribute to the birefringence signals, i.e., $\phi=\phi_{eff}$. This
significant difference between self- and polymer-crowding is associated with
the network formed from flexible PEO8 polymer chaining versus the network
composed of rigid rod-like CNC. In the following sections we examine the
dependence of $\phi_{eff}$ on relevant length- and time-scales.

Figure 4: (a, b) Rotational diffusion coefficient of CNC, $Dr$, obtained from
FIB (eqn.3), normalized by the respective rotational diffusion coefficient in
the crowding-free regime, $Dr_{0}$, as a function of the mesh size of the
crowding agent $\xi$. (a) The case of self-crowding, given by an increasing
CNC concentration with the corresponding mesh size $\xi_{r}$ (eqn. 5). (b) The
case of polymer-crowding, given by an increasing PEO8 concentration with the
corresponding mesh size $\xi_{p}$ (eqn. 4). Relevant regimes for the (a) self-
and (b) polymer-crowding are annexed above the respective panels. Line in (a)
is the prediction from eqn. 7. In (b), the triangle indicates the scaling
prediction from Cai et al.20 where $Dr\propto\xi_{r}^{2}$ in the regime
$\xi_{p}<Rg_{r}<t_{d}$. (c) Comparison between the normalized zero shear
viscosity $\eta_{0}/\eta_{H_{2}O}$ measured by bulk shear rheology (empty
symbols), and the viscosity experienced by the CNC,
$\eta_{local}/\eta_{H_{2}O}$ (filled symbols), obtained by eqn. 8. Error bars
in (a, b) indicate uncertainty associated to $Dr$ obtained from the fitting
procedure of eqn. 3.

####  4.0.4 Length-scale dependence

The rotational diffusion coefficient $Dr$ obtained from eqn. 3 captures the
impact of the crowding environment on the CNC. To compare the $Dr$ for the
case of self- and polymer-crowding, we plot $Dr$ against the statistical mesh
size $\xi$ of each crowding agent. The mesh size of the polymer matrix is
given by $\xi_{p}$ (eqn. 4), whilst for the CNC suspension it is estimated as

$\xi_{r}=(l\nu)^{-0.5}~{},$ (5)

where $\nu$ is the number density of the CNC (number of CNC per unit
volume).52, 17 For both cases $Dr$ decreases progressively with the decreasing
$\xi$, indicating that the rods progressively sense the local confinement with
the decreasing $\xi_{r}$ or $\xi_{p}$. However, the CNC follows a much sharper
decrease in $Dr$ as a function of $\xi$ for the case of self-crowding than
that of the polymer-crowding, indicating that the mesh size $\xi$ alone is not
able to fully capture the dependence of $Dr$ from the crowding agent. The $Dr$
trend for the self-crowding case shown in Fig. 4(a) meets the expectation from
rigid-rod theory where in the dilute regime $Dr$ is concentration independent
and $Dr/Dr_{0}=1$. Whilst in the semidilute (self-crowding) regime, the CNC
motion is constrained by the surrounding rods and $Dr$ becomes concentration
dependent.53 The dependence of $Dr$ with the rod concentration has been
described by the tube model in the framework of the Doi-Edwards theory,
assuming that particles are rigid and monodisperse rods in the semidilute
regime, as

$Dr/Dr_{0}=\beta(\nu l^{3})^{-2}~{},$ (6)

where $\beta$ is a length-independent prefactor.53, 54, 55 Recently, Lang et
al.17 experimentally validated for monodisperse colloidal rods that
$\beta=1.3\times 10^{3}$ as previously found from computer simulation.56
Rearranging eqn. 6 with eqn. 5, we obtain

$Dr/Dr_{0}=\beta\xi_{r}^{4}l^{-4}.$ (7)

Although polydisperisity is not accounted for in the Doi-Edwards theory, it is
interesting to note that the $\xi_{r}^{4}$ scaling captures the $Dr/Dr_{0}$
trend for the self-crowding case (Fig. 4(a)). Moreover, using $\beta=1.3\times
10^{3}$ and $l=610$ nm in eqn. 7, it is possible to quantitatively capture the
increasing $Dr/Dr_{0}$ with $\xi_{r}$ in the semidilute (self-crowding) regime
(see line in Fig. 4(a)).

For the polymer-crowding case, we take inspiration from prior work on the
translational and rotational diffusions of nanorods in PEO solutions.20, 22
Specifically, we adopt the scaling law developed by Cai et al.20, 22, by
considering $Rg_{r}$ as the characteristic dimension of CNC. The majority of
the data fall in the “intermediate regime” where $\xi_{p}<Rg_{r}<t_{d}$,
annotated above Fig. 4(b). In this regime, the scaling theory predicts
$Dr\propto\xi_{p}^{2}$, which captures our trend reasonably well.20 We have
also used other polymeric solutions (PEO4 and PA5) to gain further
understanding of the polymer-crowding cases. We confirm that the mesh size
alone is unable to fully describe the dependence of $Dr$ from the crowding
agent, thus, the curves of $Dr/Dr_{0}$ vs. $\xi_{p}$ do not collapse onto a
master curve (see ESI, Fig. S3). Since CNC in the PEO8 solutions are in the
dilute regime ($\phi=1.0\times 10^{-3}$), it is instructive to evaluate the
local viscosity experienced by the particles, $\eta_{local}$, as 57, 24, 58,
23

$Dr/Dr_{0}=\eta_{H_{2}O}/\eta_{local}~{}.$ (8)

We use $Dr_{0}=59$ s-1 determined in water with $\eta_{H_{2}O}$ being the
water viscosity and $Dr$ obtained experimentally from eqn. 3 (plotted in Fig.
4(a)), to retrieve $\eta_{local}$. In Fig. 4(c), the zero-shear-rate viscosity
$\eta_{0}$ obtained from bulk rheology (details in Fig. 5(b)) is compared to
$\eta_{local}$, displaying $\eta_{local}<\eta_{0}$ in the investigated range
of $\xi_{p}$. The mismatch between $\eta_{0}$ and $\eta_{local}$ indicates
that the CNC do not perceive the surrounding medium as a continuum, thus
experiencing a viscosity that lies between the water viscosity
($\eta_{H_{2}O}$) and the macroscopic bulk viscosity $\eta_{0}$ of the polymer
solutions. Consequently, predicting the minimum shear rate required to induce
CNC alignment in polymer crowds, based on the criterion $\dot{\gamma}=Dr$
(i.e., $Pe=1$), using the bulk viscosity $\eta_{0}$ as the solvent viscosity
in eqn. 1, would fail by underestimating $Dr$.

Figure 5: Steady shear viscosity measurements for (a) CNC suspended in water
at different values of $\phi$, and (b) PEO8 solutions at different values of
$c/c^{*}$ (without CNC) presented as $\eta/\eta_{H_{2}O}$ vs. $\dot{\gamma}$.
Solid lines in (a, b) describe the CY model (eqn. 2). The onset of shear
thinning, $\dot{\gamma}^{*}$, obtained from the CY, and the value of $Dr$
obtained from FIB measurements are plotted on the respective viscosity values.
(c, d) Comparison between the $Pe$ (filled symbols) and $Wi$ (empty symbols)
scaling for the normalized birefringence, $\langle\Delta
n\rangle/(\phi-\phi_{cage})$ , and orientation angle, $\langle\theta\rangle$,
for the CNC in PEO8 solutions at different values of $c/c^{*}$. (e) Schematic
of relevant time-scales involved in the alignment of colloidal rods in
polymer-crowding cases. The schematic is valid for polymers with a relaxation
time $\tau_{p}>Dr_{0}^{-1}$.  Figure 6: (a) Orientation angle,
$\langle\theta\rangle$, and normalized birefringence, $\langle\Delta
n\rangle/\phi_{eff}$, as a function of $Pe$ for the self-crowding (from Fig.
2(f, h)) and polymer-crowding (from Fig. 3(b, e) of the main manuscript) for a
total of 16 data sets including the reference sample. The dashed line in (a)
is the plot of eqn. 3 using $\alpha=0.39$.

####  4.0.5 Time-scale dependence

Flow curves obtained from bulk shear rheometry were used to probe the two
crowding agents, CNC suspensions in water and PEO8 solutions in the absence of
CNC, under different shear rates and related flow time scales (i.e.,
$t=\dot{\gamma}^{-1}$), see Fig. 5(a, b). It is important to note that the
presence of CNC at $\phi=1.0\times 10^{-3}$ in the PEO8 solutions did not
alter the bulk rheology (see ESI, Fig. S1). With increasing concentrations of
the crowding agent ($\phi$ and $c/c^{*}$ for the CNC and PEO8, respectively),
the shear viscosity increases and the onset of shear-thinning, captured by
$\dot{\gamma}^{*}$ from the CY model (eqn. 2), shifts to lower values of shear
rate (depicted by the yellow squares in Fig. 5(a, b)). The $Dr$ obtained via
birefringence measurements (as shown in Fig. 4(a, b)) are marked as triangular
red symbols on the respective plots of the CNC suspensions and PEO8 solutions.

For the CNC suspensions (Fig. 5(a)), $\dot{\gamma}^{*}$ obtained from shear
rheometry and $Dr$ obtained via birefringence measurements have similar values
because the onset of shear-thinning correlates with the onset of CNC
alignment. Therefore, the physical interpretation for the decreasing
$\dot{\gamma}^{*}$ as a function of $\phi$ mirrors the one given for $Dr$.
Specifically, in the semi-dilute (self-crowding) regime, the rods perceive the
surrounding rods, enabling the onset of alignment at values of shear rate that
decrease progressively with increasing $\phi$. For the PEO8 solutions (Fig.
5(b)), the onset of shear-thinning, $\dot{\gamma}^{*}$, corresponds to the
longest relaxation time of the polymer solution as
$\tau_{p}=\dot{\gamma}^{*-1}$. In good approximation, the CNC alignment in
PEO8 occurs at a shear rate $\dot{\gamma}=\dot{\gamma}^{*}=Dr_{0}$ (see yellow
squares and red triangles in Fig. 5(b)), suggesting that the onset of CNC
alignment is coupled with the polymer relaxation time as
$\tau_{p}=Dr_{0}^{-1}$. As such, a suitable control parameter for the CNC
alignment is the Weissenberg number, $Wi=\tau_{p}\dot{\gamma}$, quantifying
the strength of the elastic response of the fluid to an imposed deformation
rate, where for $Wi<1$ the polymers are in their equilibrium conformation
whilst for $Wi\geq 1$ the relatively high flow rate drives the polymers out of
their equilibrium conformation.59, 60, 61 Based on the relationship
$\tau_{p}=Dr_{0}^{-1}$, the Weissenberg and Péclet numbers become identical
because the rotational diffusion time-scale of the CNC rods is the same as the
longest relaxation time of the polymer for the regime $\xi_{p}<Rg_{r}<t_{d}$
investigated here (analogous results are also obtained for the PEO4 and PA5
solutions presented in ESI, Fig. S4). Consistently, the trend of CNC alignment
in PEO8 solutions captured by $\langle\Delta n\rangle/\phi$ and
$\langle\theta\rangle$ as a function of $Pe$ is also well described by $Wi$,
with the onset of CNC alignment occurring at $Pe=Wi=1$ (Fig. 5(c, d)). This is
remarkably different from previous reports of elliptical hematite particles
suspended in PEO solutions in the regime $Rg_{r}>t_{d}$, where the onset of
alignment occurs at $Wi\ll 1$ 30. Similarly, the onset of alignment of carbon
nanotubes in sheared polymer melt was observed at $Wi\ll 1$, ruling out the
coupling of $\tau_{p}$ with $Dr$ for relatively large colloids
($Rg_{r}>t_{d}$).29 From a topological perspective, the coupling between
tracer particles and the polymer dynamics is predicted by Cai et al.20 in the
regime $\xi_{p}<Rg_{r}<t_{d}$ with the scaling $Dr\propto\xi_{p}^{2}$ (see
Fig. 4(b)). It is possible to conceptualize the coupling of $Dr$ with
$\tau_{p}$ by analysing three distinct time-scales at play during flow as
conceptualized in Fig. 5(e). For $t>\tau_{p}$
($\dot{\gamma}<\dot{\gamma}^{*}$) the polymer is relaxed and in its
equilibrium configuration as the probed time scales are long enough to enable
polymer relaxation, during which the CNC is able to escape from the transient
confinement provided by the polymer mesh, thus Brownian diffusion dominates.
Increasing the shear rate we reach
$t=\tau_{p}$($\dot{\gamma}=\dot{\gamma}^{*}$) where the polymer is driven out
of its equilibrium conformation and deformed by the flow. At this time scale
the CNC perceives the surrounding polymer as a static mesh that provides
confinement and aids CNC alignment at values of $\dot{\gamma}^{*}=Dr$
($\tau_{p}=Dr^{-1}$). At higher values of shear rate, $t<\tau_{p}$
($\dot{\gamma}>\dot{\gamma}^{*}$), the CNC continues to align with the flow,
following a universal curve of $\langle\theta\rangle$ and $\langle\Delta
n\rangle/\phi_{eff}$ with respect to $Pe$ for a wide range of polymer
concentrations in the semidilute unentangled regime as shown by the master
curves in Fig. 3(b, e). We note that the CNC alignment for the polymer- and
self-crowding case display analogies. In Fig. 6 we plot together
$\langle\theta\rangle$ and $\langle\Delta n\rangle/\phi_{eff}$ as obtained for
all the polymer- and self-crowding cases presented above. The CNC alignment
can be described by the same master curve of $\langle\theta\rangle$ and
$\langle\Delta n\rangle/\phi_{eff}$ vs. $Pe$. This observation implies that
both self- and polymer-crowding of the CNC only alters the critical shear rate
for the onset of alignment, but the subsequent trend in alignment for $Pe>1$
remains the same for both cases. This universal trend with respect to the
crowding agent is likely caused by the inability of the CNC to explore the
surrounding confinement once the alignment is triggered by a sufficiently high
shear rate (i.e., $Pe=1$).17 Note that in our present study the characteristic
polymer time-scale is greater than the rotational diffusion time-scale of the
CNC in the crowding-free regime, $\tau_{p}>Dr_{0}^{-1}$
($\dot{\gamma}^{*}<Dr_{0}$), see vertical line in Fig. 5(b). However,
colloidal rods with slower rotational dynamics in the crowding-free regime
compared to the characteristic polymer time-scale, will be in the regime
$\tau_{p}<Dr_{0}^{-1}$ ($\dot{\gamma}^{*}>Dr_{0}$). Practically, this regime
can be achieved by increasing the length of the colloidal rods ($Dr_{0}\propto
l^{-3}$), and/or decreasing the polymer molecular weight. As the regime
$\tau_{p}<Dr_{0}^{-1}$ is approached, by for instance increasing $l$, the
colloidal rods will progressively experience the surrounding environment as a
continuum rather than a discrete medium and $\eta_{local}\rightarrow\eta_{0}$.
Therefore, for $\tau_{p}\ll Dr_{0}^{-1}$ ($\dot{\gamma}^{*}\gg Dr_{0}$) the
onset of colloidal alignment is expected to be dominated by the bulk viscosity
of the surrounding polymer solution.19

###  4.1 Conclusions

We tackle an industrially relevant problem from a fundamental perspective: the
control over the alignment of rod-like colloids in polymeric matrices.
Specifically, we compare the flow-induced alignment of rigid colloidal rods,
namely CNC, in two contrasting crowded environments referred to as self- and
polymer-crowding. By analysis of the length- and time-scale, we find that
rotational diffusion coefficient, $Dr$, of CNC in high molecular weight
polymeric crowds is coupled with the longest relaxation time of the
surrounding polymer, $\tau_{p}$. On this ground, we propose the Weissenberg
number $Wi$ as the control parameter for the alignment of colloidal rods that
possess similar length-scales as the suspending polymer fluid,
$\xi_{p}<Rg_{r}<t_{d}$; i.e., in conditions where the continuum approach
breaks down. Specifically, we show that by knowing $\tau_{p}$ from rheological
tests, it is possible to predict the critical shear rate for the onset of
colloidal alignment in polymeric fluids as $\dot{\gamma}=\tau_{p}^{-1}$,
equivalently $Wi=1$, without relying on the knowledge of the local viscosity
experienced by the colloidal rods, $\eta_{local}$. In conclusion, our results
provide crucial insights on the dynamics of colloidal rods under shearing
flows that will aid the production of composite materials with desired
structural organization. Additionally, the ability of tracer colloidal rods to
probe the relaxation times of the surrounding polymers opens the opportunity
to perform in-situ and spatially-resolved characterisation of the dynamics of
polymeric fluids under flow using tracer colloidal rods. With further
optimisation (e.g., size and composition of the colloidal rods), this
technique is promising for analysing polymer dynamics in complex flows
encountered in real-life conditions where the investigation is a significant
challenge. As a natural next step to our work, we envisage future studies
where self- and polymer-crowding are at play jointly, mirroring actual
industrial conditions. We use CNC as industrially relevant colloidal rods but
the basic principles will also apply to other anisotropic, rod-like, colloidal
particles.

##  5 Acknowledgements

The authors gratefully acknowledge the support of Okinawa Institute of Science
and Technology Graduate University with subsidy funding from the Cabinet
Office, Government of Japan. V.C, SV, S.J.H. and A.Q.S. also acknowledge
financial support from the Japanese Society for the Promotion of Science
(JSPS, Grant Nos. 22K14738, 22K14184, 18K03958, 18H01135, and 21K03884) and
the Joint Research Projects (JRPs) supported by the JSPS and the Swiss
National Science Foundation (SNSF).

##  6 Competing interests

The authors declare no competing interests.

##  7 Additional information

Supplementary information is available for this paper.

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