# The rheology of confined colloidal hard discs

Ian Williams  Department of Physics, University of Surrey, Guildford GU2 7XH,
UK  H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK
Centre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol, BS8
1FD, UK  School of Chemistry, University of Bristol, Cantock’s Close, Bristol,
BS8 1TS, UK  Erdal C. Oğuz  School of Mechanical Engineering, Tel Aviv
University, Tel Aviv 6997801, Israel  The Sackler Center for Computational
Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel  Key
Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  Hartmut Löwen  Institut für Theoretische
Physik II, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany  Wilson C.
K. Poon  SUPA and School of Physics & Astronomy, The University of Edinburgh,
Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK  C. Patrick Royall
<EMAIL_ADDRESS>Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL,
75005 Paris, France  H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol,
BS8 1TL, UK  Centre for Nanoscience and Quantum Information, Tyndall Avenue,
Bristol, BS8 1FD, UK  School of Chemistry, University of Bristol, Cantock’s
Close, Bristol, BS8 1TS, UK

###### Abstract

Colloids may be treated as “big atoms” so that they are good models for atomic
and molecular systems. Colloidal hard disks are therefore good models for 2d
materials and although their phase behavior is well characterized, rheology
has received relatively little attention. Here we exploit a novel, particle-
resolved, experimental set-up and complementary computer simulations to
measure the shear rheology of quasi-hard-disc colloids in extreme confinement.
In particular, we confine quasi–2d hard discs in a circular “corral” comprised
of 27 particles held in optical traps. Confinement and shear suppress
hexagonal ordering that would occur in the bulk and create a layered fluid. We
measure the rheology of our system by balancing drag and driving forces on
each layer. Given the extreme confinement, it is remarkable that our system
exhibits rheological behavior very similar to _unconfined_ 2d and 3d hard
particle systems, characterized by a dynamic yield stress and shear-thinning
of comparable magnitude. By quantifying particle motion perpendicular to
shear, we show that particles become more tightly confined to their layers
with no concomitant increase in density upon increasing shear rate. Shear
thinning is therefore a consequence of a reduction in dissipation due to a
weakening in interactions between layers as the shear rate increases. We
reproduce our experiments with Brownian Dynamics simulations with Hydrodynamic
Interactions (HI) included at the level of the Rotne–Prager tensor. That the
inclusion of HI is necessary to reproduce our experiments is evidence of their
importance in transmission of momentum through the system.

††preprint: AIP/123-QED

##  I Introduction

Confined fluids often exhibit modified flow behavior compared to the bulk due
to coupling to the boundary. This may be static (_i.e._ structural alterations
due to energetic or entropic interactions) or dynamic (_e.g._ the hydrodynamic
influence of the wall). The shear viscosity of simple liquids increases by
orders of magnitude in films less than $\sim 7$ molecules thick Hu, Carson,
and Granick (1991); Demirel and Granick (1996); Klein and Kumacheva (1998).
This is a consequence of confinement-induced solidification near the boundary,
driven by van der Waals interactions Horn and Israelachvili (1981);
Christenson  _et al._ (1987); Klein and Kumacheva (1995). By contrast, the
viscosity of water increases more modestly under similar conditions Raviv,
Laurat, and Klein (2001); Antognozzi, Humphris, and Miles (2001) as
confinement-induced solidification is suppressed by the hydrogen bond network.

On the mesoscopic scale, rheological measurements of two-dimensional soft
materials typically focus on interfacially adsorbed components including
nanoparticles Maestro  _et al._ (2015); Feng, Hoagland, and Russell (2016),
proteins Cicuta, Stancik, and Fuller (2003); Ariola, Krishnan, and Vogler
(2006); Allan  _et al._ (2014); Williams and Squires (2018), lipid Kim  _et
al._ (2011); Sachan  _et al._ (2017); Williams, Zasadzinski, and Squires
(2019); Kim  _et al._ (2018), surfactant Zell  _et al._ (2014) and polymer
Anseth  _et al._ (2003); Cappelli  _et al._ (2016); Feng, Hoagland, and
Russell (2016); Pepicelli  _et al._ (2017) monolayers or asphaltene films Fan,
Simon, and Sjöblom (2010); Chang  _et al._ (2018, 2019), 2d foams Katgert,
Möbius, and van Hecke (2008); Raufaste, Foulon, and Dollet (2009), Hele-Shaw
emulsions Desmond and Weeks (2015) lipid bilayers Hormel  _et al._ (2014)
dusty plasmas Feng, Goree, and Liu (2010) and colloids Cicuta, Stancik, and
Fuller (2003); Masschaele, Fransaer, and Vermant (2011); Buttinoni  _et al._
(2015); Deshmukh  _et al._ (2015); Buttinoni  _et al._ (2017). The latter can
be tailored to exhibit interactions similar to simple models of atomic and
molecular systems, and since they explore phase space in much the same way
colloids provide a means to probe the same phenomena, for example the effect
of confinement, and other external fields such as shear, upon phase behavior
Rice (2009); Ivlev  _et al._ (2012); Löwen (2009). Due to their mesoscopic
size, colloids may be confined by walls that are smooth on the particle scale
or by walls that are rough (like atomic and molecular systems).

Another class of materials at a somewhat larger lengthscale is (athermal)
granular matter and here too individual particles may be studied, and even the
force networks between them Majmudar and Behringer (2005). Under vibration,
granular matter can mimic the phase behavior of thermal systems Reis, Ingale,
and Shattuck (2006). The rheology of granular matter has received considerable
attention Midi (2004); Forterre and Pouliquen (2008); Daniels and Behringer
(2005); Dijksman  _et al._ (2011); Boyer, Guazzelli, and Pouliquen (2011);
Watanabe, Kawasaki, and Tanaka (2011); Mandal and Khakhar (2017). Like
suspensions of colloids, granular materials become very much more viscous at
high packing fraction, however the limiting case of divergence viscosity is
jamming in the case of grains, rather than the thermal glass transition (which
occurs at a lower packing fraction) in the case of colloids Ikeda, Sollich,
and Berthier (2012). At the microscopic level, the shear response of colloidal
and granular systems is distinguished in that colloidal particles usually do
not come into physical contact with each other or with the walls of the sample
cell. Therefore, dissipation occurs through hydrodynamic coupling and Brownian
motion rather than friction due to direct contacts as in granular matter. Very
occasionally, under extreme shear rates for example, contact can occur,
leading to very sudden shear thickening Mari  _et al._ (2015); Morris (2019).

In colloidal systems at volume or area fractions where glassy dynamics are
encountered, researchers report both increases and decreases in viscosity with
respect to the bulk depending on the degree of confinement and the boundary
details. Relaxation may be accelerated (decelerated) near smooth (rough) walls
Scheidler, Kob, and Binder (2003); Nugent  _et al._ (2007); Sarangapani and
Zhu (2008). These effects are attributed to boundary-induced structural
modification dependent on the shape, roughness, wetting characteristics or
interaction potential of the wall (to name but a few possibilities) Oğuz  _et
al._ (2012); Isa, Besseling, and Poon (2007); Huber (2015). As a general rule,
the formation of well-defined particle layers is associated with faster
relaxation or reduced viscosity.

The bulk rheology of a hard-sphere-like (or hard-disc-like) colloidal
suspension depends on its volume (or area) fraction, $\phi$ Meeker, Poon, and
Pusey (1997), and the shear rate, $\dot{\gamma}$. At low shear rates,
viscosity decreases with $\dot{\gamma}$ Brader (2010); Cheng  _et al._ (2011);
Pham  _et al._ (2008); Zackrisson  _et al._ (2006). For volume (area)
fractions approaching the hard sphere (disc) glass transition, shear thinning
occurs after the applied stress exceeds a yield stress on the scale of
$k_{\mathrm{B}}T/a^{3}$, where $k_{\mathrm{B}}T$ is the thermal energy and $a$
is the particle radius Bonn  _et al._ (2017); Zackrisson  _et al._ (2006);
Pham  _et al._ (2008); Henrich  _et al._ (2009); van der Vaart  _et al._
(2013); Le Grand and Petekidis (2008). Upon increasing $\dot{\gamma}$, hard
sphere suspensions exhibit a Newtonian plateau followed by shear thickening at
very high shear rates Cheng  _et al._ (2011); Bender and Wagner (1996);
D’Haene, Mewis, and Fuller (1993); Brader (2010). Shear thinning in these
systems is attributed to stratification of particles decreasing resistance to
flow, while shear thickening at high $\dot{\gamma}$ is a consequence of
frictional particle surfaces Guy, Hermes, and Poon (2015).

Figure 1:  (a-b) Annotated micrographs and (c) side-view schematic. (a) Shear-
melted, layered system for $\operatorname{\mathrm{P\kern-0.80002pte}}\geq
5.25$. (b) Hexagonal structure at
$\operatorname{\mathrm{P\kern-0.80002pte}}=1.75$. Red crosses label optically
trapped particles which are translated along the circular path indicated by
the red arrow. Polar co-ordinates, $(r,\theta)$ are defined from the centre.
$R$ is the boundary radius. Orange dashed lines in (a) and (c) demarcate
particle layers, numbered 1 to 5 from the centre. Pink lines in (b) indicate
hexagonal structure. (d) Histograms of radial particle location in experiments
at $\operatorname{\mathrm{P\kern-0.80002pte}}$ indicated in legend. $y$-axis
scale is arbitrary and data are offset in $y$ for clarity. (e) Angular
velocity profiles corresponding to experiments in (d). Vertical dashed lines
demarcate layers and zeros represent the inter-layer boundaries. (f) Area
fraction as a function of layer number measured from average Voronoi cell
areas in experiment. Lines are colored according to driving
$\operatorname{\mathrm{P\kern-0.80002pte}}$. Black data represent the
unsheared system where differing local structure leads to a change the
distribution of local area fraction across the layers [see (d)]. (g) Area
fraction from Voronoi cell area as a function of driving
$\operatorname{\mathrm{P\kern-0.80002pte}}$ for each layer, measured in
experiment. (h-i) Illustrated interpretation of shear thinning. Coupling
between layers depends on the range of radial motion within adjacent layers.
In a slowly driven system, (h), particles explore a larger radial extent than
in a quickly driven system, (i). For larger local shear rate, particles in
adjacent layers are further apart, interlayer interactions are weaker and the
effective viscosity is lower. This manifests as shear thinning.

While the rheology of (3d) colloidal hard spheres has been extensively studied
Besseling  _et al._ (2007); Schall, Weitz, and Spaepen (2007); Koumakis  _et
al._ (2012) attention to their 2d analogue, hard discs has focussed on
quiescent (unsheared) systems Brunner  _et al._ (2002); Collins  _et al._
(2015); Stoop and Tierno (2018); Brunner  _et al._ (2003); Stratton  _et al._
(2009); Thorneywork  _et al._ (2014); Tamborini, Royall, and Cicuta (2015);
Williams  _et al._ (2015); Gray  _et al._ (2015); Williams  _et al._ (2018).
Here, we focus on the rheology of confined colloidal hard discs. We present
experiments on a quasi-hard-disc system of spherical colloids adjacent to a
solid substrate confined by 27 identical particles optically trapped along a
circular boundary and subjected to shear by boundary rotation. These are
accompanied by computer simulation of a 2d system using Brownian Dynamics with
hydrodynamic interactions (HI) included at the level of the Rotne–Prager
tensor. Under quiescent conditions, our system has a bistable state between a
hexagonal configuration that distorts the flexible walls [Fig. 1 (b)] and a
layered fluid where hexagonal ordering is suppressed [Fig. 1 (a)]. Hexagonal
configurations exhibit voids adjacent to the wall, enabling relaxation
mechanisms that are absent in the bulk Williams  _et al._ (2015). Under shear,
hexagonal configurations rotate as a rigid body, slipping only at the wall
interface, while layered fluid configurations slip between each layer. At
higher shear rates, the system is shear melted and only the layered fluid is
observed Williams  _et al._ (2016). It has subsequently been shown that
simulations of a similar system exhibit distinct regions of shear thinning and
thickening as a function of shear rate Ortiz-Ambriz  _et al._ (2018); Gerloff
_et al._ (2020). Here, we develop a layer-by-layer approach to determine local
rheological properties directly from experiments and simulations via drag
forces, and we characterise motion perpendicular to shear via mean squared
displacements. Our approach opens a route to particle-level analysis of
rheological properties in experimental systems.

The key finding of this work is that, in contrast with many molecular liquids
Hu, Carson, and Granick (1991); Demirel and Granick (1996); Klein and
Kumacheva (1998), strong confinement has little effect on the shear rheology
of colloidal quasi-hard-discs. Our measurements are qualitatively and
quantitatively aligned with simulations of hard discs under steady shear
Henrich  _et al._ (2009). We measure a flow curve indicative of a yield
stress, and shear thinning towards a Newtonian plateau. Motion perpendicular
to shear is suppressed with no change in local density as shear rate is
increased. We hypothesize that this suppression of perpendicular motion
reduces the coupling between adjacent particle layers, reducing drag and
dissipation, and consequently we measure a reduction in viscosity. This
interpretation is illustrated in Fig. 1 (h-i). Our simulations reveal that the
mechanism of momentum transfer in this system is dominated by hydrodynamic
interactions, and the excluded volume interactions between the particles play
a rather minor role.

This paper is organised as follows. Section II describes the experimental and
simulation procedures, and details of our stress calculation. Full details of
the drag coefficient determination, simulations and interlayer hopping are
provided in the Appendices. Section III.1 presents key structural and dynamic
measurements required for the measurements of local shear rate, stress and
viscosity in Section III.2. Section III.3 quantifies the motion perpendicular
to shear. Finally, Section IV combines these measurements to develop our
explanation of shear thinning. We conclude this work in section V.

##  II Methods

###  II.1 Experiment

Figures 1 (a) and (b) show micrographs and (c) shows a side-view schematic of
the system. Polystyrene spheres of radius $a=2.5\;\mathrm{\upmu m}$ and
polydispersity 2% are suspended in a $3:1$ mass ratio mixture of deionised
water and ethanol, loaded into a cell constructed using three glass coverslips
and a microscope slide and sealed with epoxy. Owing to their density mismatch
with the solvent, the particles quickly sediment, forming a quasi-two-
dimensional layer adjacent to the lower glass coverslip, which is treated with
Gelest Glassclad 18 to prevent particle adhesion. The gravitational length is
$l_{g}/a=0.030\pm 0.002$, resulting in negligible out-of-plane particle motion
in the $z$ direction, while still being far enough from the substrate that any
direct interactions with the substrate through contacts can be safely
neglected. Interparticle interactions are of Yukawa form with a Debye length
$\lambda_{\mathrm{D}}\approx 25\;\mathrm{nm}$, which is sufficiently short
that they may be considered quasi-hard-discs Williams  _et al._ (2013). In the
dilute limit, the time for a particle to diffuse its radius (the Brownian
time) in the substrate-adjacent, quasi-two-dimensional layer is measured to be
$\tau_{\mathrm{B}}\approx 70\;\mathrm{s}$.

Computer-controlled holographic optical tweezers in an inverted microscope are
used to manually gather $N=75$ particles, $27$ of which are optically trapped
to form a circular boundary of radius $R$ which confines the remaining
$n_{\mathrm{conf}}=48$. The spring constant of the optical traps maintaining
this boundary is extracted from the trapped particle trajectories and is
determined to be $k=105(1)\;k_{\mathrm{B}}Ta^{-2}$. The colloids sediment to
the bottom of a sample cell, and thus there is a (relatively) large amount of
solvent above. We presume that any local heating effect is swiftly dissipated
to the surrounding solvent. When we tested the system, for example at
different strengths of the laser tweezers, we saw no signs of any such
heating.

Using the optical tweezers, the boundary particles are translated along a
circular path through the periodic displacement of the optical tweezers array
in discrete steps of length $a/4$. The frequency with which the array of traps
is updated defines the boundary rotation speed, which we characterise using
the boundary Péclet number,
$\operatorname{\mathrm{P\kern-0.80002pte}}=\tau_{\mathrm{B}}/\tau_{D}$, where
$\tau_{D}$ is the time taken to drive a boundary particle a distance $a$. We
consider experiments with $\operatorname{\mathrm{P\kern-0.80002pte}}$ in the
range $1.75\leq\operatorname{\mathrm{P\kern-0.80002pte}}\leq 19.25$.

Following the initiation of boundary rotation, the system is left to complete
$5$ full rotations before micrographs are acquired at a rate of $2$ frames per
second for up to $3$ hours. No indication of anything other than a steady-
state behavior was found. Particle trajectories are extracted from micrographs
using standard algorithms Crocker and Grier (1996). Circular polar co-
ordinates, $r$ and $\theta$, are defined from the system centre.

###  II.2 Simulation

We perform 2d Brownian Dynamics simulations of strongly screened charged
colloids interacting via a Yukawa pair potential under highly ionic
conditions.

$V(r)=V_{0}\dfrac{\mathrm{e}^{-\kappa r}}{\kappa r},$ (1)

with $r$ denoting the inter-particle separation. The inverse screening length
$\kappa$ is chosen as $\kappa a=14.85$ and the contact potential
$V(r=2a)\approx V_{0}k_{B}T$ where $V_{0}=0.85$, informed by the experimental
parameters, ensuring quasi-hard-disc behavior Williams  _et al._ (2016). In
particular, we found empirically that $\kappa a=14.85$ was sufficiently hard
to closely reproduce key dynamical properties of the experimental system such
as the way in which layers of particles slip past one another. Further details
are available in Appendix A and Ref. Williams  _et al._ (2016). We neglect
polydispersity. Boundary particles experience harmonic potentials mimicking
optical traps and are translated along a circular path at a prescribed rate.
Hydrodynamic effects are accounted for at the Rotne-Prager level Gauger,
Downton, and Stark (2009); von Hansen, Hinczewski, and Netz (2011) in the
presence of a planar substrate (Blake’s solution Blake (1971)), giving the
overdamped equation of motion for a particle trajectory $\textbf{r}_{i}$ in a
time step $\delta t$

$\mathbf{r}_{i}(t+\delta
t)=\mathbf{r}_{i}(t)+\left(\sum_{j=1}^{N}\tensor{\mathbf{\mu}}_{ij}\mathbf{F}_{j}\right)\delta
t+\delta\textbf{W}_{i}.$ (2)

The mobility tensor $\tensor{\mathbf{\mu}}_{ij}$ comprises both the self-
mobility and the entrainment of particle $i$ by the hydrodynamic flow field
created by conservative forces $\mathbf{F}_{j}$ on particle $j$. This force
stems from pair interactions and, for the boundary particles, the harmonic
potentials. The random displacement, $\delta\textbf{W}_{i}$, is sampled from a
Gaussian distribution with zero mean and variance $2D_{0}\delta t$ (for each
Cartesian component) fixed by the fluctuation-dissipation relation, where
$D_{0}$ is the diffusion coefficient. Further details are provided in Appendix
A. We have previously shown that these simulations faithfully reproduce
experiments both qualitatively and quantitatively Williams  _et al._ (2016).

###  II.3 Stress Calculation

Figure 1(a) shows that the particles organize into concentrated layers. This
motivates our determination of the stress, by considering each layer, $i$. We
presume that the angular velocity and radial position are constant within each
layer, which, as we shall show in Section III is reasonable for our purposes.
We thus perform a series of force balances to obtain the stress across each
layer, $\sigma_{i}$. Within each layer there are three forces which must sum
to zero at steady state. There is some driving force (either optical forces or
the transmitted force due to the motion of an external layer),
$F_{\mathrm{ext}}$. This is balanced by some self-drag force representing
dissipation within the layer, $F_{\mathrm{self}}$, and some additional force
that drives the next internal layer, $F_{\mathrm{int}}$. In the case of layer
$1$ (the central layer), there is only self-drag.

The dilute limit single particle drag coefficient near the substrate,
$\zeta_{\mathrm{emp}}=1.9\times 10^{-7}\;\mathrm{kg\,s^{-1}}$, is extracted
from measurements of Brownian motion in the dilute limit, without any optical
traps. However, multiple particles moving along a circular path of radius $r$
near a substrate each experience reduced hydrodynamic drag due to the presence
of the other particles Ladavac and Grier (2005). This is the _drafting
effect_. This drag reduction depends on the number of particles and $r$, which
suggests that particles in different layers experience different drag
coefficients. Based on the discussion in Appendix B we assume a constant drag
coefficient throughout the system, $\zeta=0.34\zeta_{\mathrm{emp}}$, which is
reduced compared to the dilute limit drag coefficient due to many-body
hydrodynamic effects, but is independent of radius. This effect is discussed
in greater detail and this approximation is justified in the Appendix B. We
therefore assume that the self-drag force has the same form for all particles
and is proportional to velocity, $F_{\mathrm{self}}=\zeta r\omega$.

Consider layer $5$, the boundary, which consists of $n_{5}=27$ particles, each
of which is located at radial position $r_{5}$ and subjected to an optical
force $F_{\mathrm{opt}}$. All $n_{5}$ particles move with angular velocity
$\omega_{5}$ and each experiences a self-drag proportional to its velocity.
Balancing the forces on layer $5$ yields:

$n_{5}F_{\mathrm{opt}}-n_{5}\zeta r_{5}\omega_{5}-F_{4}=0,$ (3)

where $F_{4}$ is the as of yet unknown force transmitted inwards to drive the
motion of all $n_{4}=21$ particles forming layer $4$. The force balance for
layer 4 is then:

$F_{4}-n_{4}\zeta r_{4}\omega_{4}-F_{3}=0,$ (4)

where, once again, $F_{3}$ is the unknown force required to drive layer 3.
Propagating the layer-by-layer force balance inwards to layer $1$ reveals the
retrospectively trivial result that the optical driving forces are exactly
balanced by the self-drags experienced by all of the particles:

$n_{5}F_{\mathrm{opt}}-\zeta\sum_{i=1}^{5}n_{i}r_{i}\omega_{i}=0.$ (5)

Replacing the optical forces by the sum of the drag forces gives the unknown
force required to drive layer $i$, $F_{i}$, as the sum of the drag forces on
layer $i$ and all layers that are internal to $i$:

$F_{i}=\zeta\sum_{j=1}^{i}n_{j}r_{j}\omega_{j}.$ (6)

Assuming that the inter-layer forces act at circular contact lines between
layers at radial locations intermediate between the two layer centres, we
calculate the two-dimensional tangential stress on layer $i$:

$\sigma_{i}=\frac{\zeta}{\pi(r_{i}+r_{i+1})}\sum_{j=1}^{i}n_{j}r_{j}\omega_{j}.$
(7)

##  III Results

We begin by recapitulating the structural and dynamic features necessary for
our rheological analysis. We subsequently implement the analysis described in
section II.3 to measure the shear rheology combining particle-level structural
and dynamic information to extract stresses from drag forces. Finally, we
quantify particle motion in the radial direction, perpendicular to shear.

###  III.1 Angular Velocity Profiles and Structure

Without shear, the system can adopt either layered fluid or locally hexagonal
structures, shown in Fig. 1 (a) and (b) respectively Williams  _et al._
(2015). When sheared at $\operatorname{\mathrm{P\kern-0.80002pte}}\lesssim 3$,
hexagonal structures can persist and rotate as a rigid body Williams  _et al._
(2016). The hexagonal structure is characterised by multiple peaks in the
radial density profile [black line in Fig. 1 (d)] and a flat angular velocity
profile [black line in Fig. 1 (e)]. For
$\operatorname{\mathrm{P\kern-0.80002pte}}\gtrsim 3$, the system is shear
melted and only layered structures are observed [coloured data in Fig. 1 (d)].
The location of the $i$th peak in the layered density profile is labelled
$r_{i}$. Layer populations are fixed at $n_{1}=3$, $n_{2}=9$, $n_{3}=15$,
$n_{4}=21$ and $n_{5}=27$. In these shear melted experiments, angular velocity
decreases in a step-like manner from the boundary to the centre coloured data
in Fig. 1 (e)], representing slipping between adjacent layers. The average
angular velocity within layer $i$ is denoted $\omega_{i}$.

The area fraction is estimated for layer $i$ as $\phi_{i}=\pi a^{2}/\langle
A_{\mathrm{V}}\rangle_{i}$ where $\langle A_{\mathrm{V}}\rangle_{i}$ is the
average Voronoi cell area for particles in layer $i$ Gray  _et al._ (2015).
Figure 1 (e) shows that $\phi$ varies from $\phi_{1}\approx 0.8$ at the
centre, to $\phi_{4}\approx 0.76$ adjacent to the boundary. At these
densities, bulk hard discs are crystalline Bernard and Krauth (2011). However,
in shear-melted experiments, hexagonal ordering is inhibited by the curved
boundary and the application of shear and our system is liquid-like Williams
_et al._ (2015, 2016).

Shearing the system modifies the area fraction profile compared to the
unsheared system [Fig. 1 (e)], enhancing $\phi_{4}$ and suppressing $\phi_{3}$
and $\phi_{2}$. However, once the system is shear melted, the area fraction
profile is insensitive to $\operatorname{\mathrm{P\kern-0.80002pte}}$ in the
range of $\operatorname{\mathrm{P\kern-0.80002pte}}$ investigated, as shown in
Fig. 1 (f).

Figure 2:  (a) Dimensionless stress and (b) viscosity as a function of
dimensionless strain rate from experiments (squares) and simulations
(crosses). Solid (dashed) line shows Herschel-Bulkley fit to experimental
(simulated) data. Points are colored according to the driving
$\operatorname{\mathrm{P\kern-0.80002pte}}$. Error bars are determined from
angular velocity fluctuations in each layer. Insets show same data colored
according to particle layer as indicated in the legend.

###  III.2 Shear Rheology

To characterize the shear rheology, we follow the analysis in section II.3 and
treat the shear-melted system as a series of coupled layers, following
reference Katgert, Möbius, and van Hecke (2008). Layers have fixed
populations, $n_{i}$. We assume all particles are located at radial position
$r_{i}$ corresponding to the peaks in the density profile [Fig. 1 (d)] and
move with angular velocity $\omega_{i}$ [Fig. 1 (e)]. Viscosity is
$\eta=\sigma/\dot{\gamma}$ where $\sigma$ is the stress and $\dot{\gamma}$ is
the shear rate. The shear rate experienced by layer $i$ is obtained from the
angular velocity profile as

$\dot{\gamma}_{i}=r_{i}\frac{\Delta\omega_{i+1,i}}{\Delta r_{i+1,i}},$ (8)

where $\Delta\omega_{i+1,i}=\omega_{i+1}-\omega_{i}$ is the step down in
$\omega$ between layer $i+1$ and layer $i$, and the radial separation between
the layers $\Delta r_{i+1,i}=r_{i+1}-r_{i}$.

We determine the stress throughout the system as discussed in section II.3. In
particular, the corresponding stresses, $\sigma_{i}$, are obtained via a force
balance on each layer, the key result is given in Eq. 7, that the force
required to drive layer $i$ is the sum of the drag forces on layer $i$ and all
layers internal to $i$. This force acts along a circular interlayer contact
line, giving the stress on layer $i$, $\sigma_{i}$, from which we define an
effective viscosity, $\eta_{i}=\sigma_{i}/\dot{\gamma}_{i}$, for each layer.

Figure 2 shows the results of this analysis for experiments (squares) and
simulations (crosses). Stress is scaled by $\zeta^{-1}\,\tau_{\mathrm{B}}$,
shear rate by $\tau_{\mathrm{B}}$ and viscosity by $\zeta^{-1}$ to facilitate
the comparison. Points in the main panels are coloured according to the
driving $\operatorname{\mathrm{P\kern-0.80002pte}}$, while the insets present
the same data coloured by the layer number. Error bars are determined from
angular velocity fluctuations in each layer determined from particle tracking.
We see in Fig. 2 that the simulations do not reach such low shear rates low
rates as the experiments. Now the only parameter that is directly set in both
the experiments and the simulations is the rotation speed of the outer
particle layer. Shear rates experienced by internal layers are determined by
the physics of the system, ie the coupling between adjacent layers. So, the
shear rates plotted are measurements, not directly controlled quantities.

The viscosity of hard particle suspensions depends on shear rate and
volume/area fraction Meeker, Poon, and Pusey (1997); Brader (2010); Cheng  _et
al._ (2011); Pham  _et al._ (2008); Zackrisson  _et al._ (2006); van der Vaart
_et al._ (2013); Le Grand and Petekidis (2008); Henrich  _et al._ (2009). The
shear rate is largest in layer 4 and smallest in layer 1 and lower viscosity
is measured for greater shear rate. However, Fig. 1 (f) and (g) show that
local area fraction is largest in layer 1 and decreases towards the boundary.
Therefore, measuring larger viscosity nearer the centre of the system could be
a consequence of increased density, and unrelated to the local shear rate.
However, Fig. 1 (g) shows that the area fraction of each layer is independent
of $\operatorname{\mathrm{P\kern-0.80002pte}}$, and the inset to Fig. 2 (b)
shows that the viscosity of a given layer _does_ decrease as the shear rate
increases. We expect the spatial variation in area fraction to make some
contribution to the viscosity, but the shear rate dependence is unambiguous.

The data in Fig. 2 are remarkably consistent with theoretical predictions and
simulations of unconfined binary hard discs under steady shear Henrich  _et
al._ (2009). The flow curve suggests a dynamic yield stress Fuchs and Ballauff
(2005); Varnik and Henrich (2006); van der Vaart  _et al._ (2013), beyond
which, shear thinning is found. At high $\dot{\gamma}$, the response
approaches a Newtonian region. The solid (dashed) black lines show Herschel-
Bulkley fits to the experimental (simulated) data of the form
$\sigma=\sigma_{\mathrm{y}}+k\dot{\gamma}^{\nu}$, where $\sigma_{\mathrm{y}}$
is the dynamic yield stress and $\nu$ is the high shear rate exponent, which
is unity for Newtonian flow. The fits yield $\nu=0.82\pm 0.03$ in experiment
and $\nu=0.97\pm 0.06$ in simulation, consistent with weakly shear thinning
(experiment) and Newtonian (simulation) flow at the largest shear rates
studied. We relate the yield stress to the change in structure upon the
application of shear, from a hexagonal to layered fluid configuration. It is
possible to fit the data in Fig. 2 with a power–law
$\sigma=k^{\prime}\dot{\gamma}^{\nu^{\prime}}$ (rather than the
Herschel–Bulkeley fit). While the error bars in the case of the data points
corresponding to low shear rates are large enough that the this regime can be
fitted with a power law, in fact the fit at high shear rate is much worse than
the Herschel-Bulkeley fit shown. Further dependencies are of course possible,
such as a power–law with a shear–rate dependent exponent. While this would be
most interesting to explore in the future, the quality of our existing data
means that it may be challenging to accurately discriminate between such more
complex dependencies.

The dynamic yield stress is $\sim 0.1k_{\mathrm{B}}T/a^{2}$. A dynamic yield
stress is the stress required to maintain flow, and is smaller than the static
yield stress, which must be exceeded during flow start-up. Although we focus
on yielded systems, at very low $\operatorname{\mathrm{P\kern-0.80002pte}}$
the system is unyielded and rotates as a rigid body Williams  _et al._ (2016).
Rigid-body rotation requires the local stress to be less than the static yield
stress. Therefore, the stress measured in unyielded systems is a lower limit
estimate of the static yield stress. In experiment, the largest stress for
which the system does not yield, is $\sim 1.9\,k_{\mathrm{B}}T/a^{2}$, which
is comparable to yield stresses at the scale $k_{\mathrm{B}}T/a^{3}$ in hard
spheres Zackrisson  _et al._ (2006); Pham  _et al._ (2008); van der Vaart  _et
al._ (2013); Le Grand and Petekidis (2008).

Figure 3:  Radial mean squared displacements. All experimental data in (a)
layer 4, (b) layer 3, (c) layer 2, and (d) layer 1. Black points and grey
shaded regions represent average and standard deviation of behaviour in 5
unsheared experiments. All simulation data in (e) layer 4, (f) layer 3, (g)
layer 2, and (h) layer 1. (i) All layers in a single experiment driven at
$\operatorname{\mathrm{P\kern-0.80002pte}}=14$. (j) All layers in a single
free centre simulation driven at
$\operatorname{\mathrm{P\kern-0.80002pte}}=14$. Points in (a-h) coloured
according to driving $\operatorname{\mathrm{P\kern-0.80002pte}}$ and in (i)
and (j) according to layer number as indicated in the legends. Lines in (i)
and (j) show fits to Eq. 9 as described in the text.

The crossover from shear thinning to Newtonian flow occurs in the range
$\dot{\gamma}\tau_{B}\sim 10^{-1}$ to $10^{0}$, which is coincident with this
crossover in simulations and theory of hard discs Henrich  _et al._ (2009),
hard spheres Cheng  _et al._ (2011), and charged colloids Laun (1984);
Kawasaki, Ikeda, and Berthier (2014). That the rheology under such strong
confinement bears even a qualitative resemblance to bulk behaviour is a
remarkable finding. Many molecular liquids and complex fluids exhibit
increases in viscosity of many orders of magnitude when confined to a few
particle layers Hu, Carson, and Granick (1991); Demirel and Granick (1996);
Klein and Kumacheva (1998); Scheidler, Kob, and Binder (2003); Nugent  _et
al._ (2007); Sarangapani and Zhu (2008), but this is not the case for
colloidal hard discs, which reproduce their predicted bulk rheology when
confined to a system only $8$ particles across. We emphasize that the yield
stress is likely related to the shear–induced melting of hexagonal order in
this system. This ordering we have investigated previously Williams  _et al._
(2013, 2016).

###  III.3 Motion Perpendicular to Shear

At the particle population of interest, and on the timescale of the
experiment, particles are confined to layers (Appendix C). However, within the
layers, they exhibit positional fluctuations in the radial direction as
indicated by the width of the peaks in Fig. 1 (d). The mean squared
displacement (MSD) in the radial co-ordinate is shown in Fig. 3 for all
experiments (a-d) and simulations (e-h). Black points in the experiment panels
show the radial MSD for $\operatorname{\mathrm{P\kern-0.80002pte}}=0$,
averaged over 5 experiments. These MSDs grow with time up to a plateau which
represents the confinement within layers. In the case of the experiments, the
increase to a (somewhat noisy) plateau is continuous, in some simulation data,
notably for layer 2, there is some evidence of two timescales in the radial
MSD [Fig. 3(g)]. Given that this two timescale behavior is only evidenced in
layer 2 of the simulations, and not at all in the experiments, we focus on a
single timescale, $\tau_{\mathrm{rad}}$ as determined below in Eq. 9.

Within each layer, the radial MSD approaches its plateau more quickly as
$\operatorname{\mathrm{P\kern-0.80002pte}}$ is increased, indicating a
coupling between radial and tangential motion. The shear rate is largest in
layer 4 and decreases towards the system centre. The long-time plateau is
reached most quickly in layer 4, and progressively more slowly in layers 3 and
2. Therefore, larger $\dot{\gamma}$ causes faster radial motion. Additionally,
in all but the central layer, shearing increases the plateau above that
measured in the unsheared system (black data) indicating that the amplitude of
radial motion is increased under shear.

Figure 4:  Parameters extracted from fits to radial MSDs as a function of
shear rate. (a) Plateau height $A$ (Eq. 9). (b) Timescale
$\tau_{\mathrm{rad}}$ (Eq. 9). Solid (dashed) lines guide the eye to trends in
experimental (simulated) data. Points are colored according to layer number
indicated in the legend.

Radial MSDs are fit with a function of the form

$\langle\Delta r^{2}(t)\rangle=A\left(1-e^{-t/\tau_{\mathrm{rad}}}\right),$
(9)

which captures their growth with $A$ the plateau height. Example fits for all
four layers in an experiment and a simulation at
$\operatorname{\mathrm{P\kern-0.80002pte}}=14$ are shown in Fig. 3 (i) and
(j). We note that not all of the fits in (i) and (j) are of high quality.
While we did find an improved fitting with the use of two timescales (i.e. two
independent contributions to the right hand side of Eq. 9), in fact the small
timescale proved to be very scattered, and its physical significance was
unclear. A more sophisticated treatment than we have performed here with Eq. 9
would be interesting to explore in the future.

Figure 4 shows the radial MSD plateau height $A$ and timescale
$\tau_{\mathrm{rad}}$ as a function of the local shear rate for all
experiments and simulations. The fit to the plateau height [Fig. 4 (a)]
reveals a downward trend with increasing $\dot{\gamma}$ in both experiment and
simulation for all layers except layer 1. Figures 3 (a-d) show that shear
enhances radial motion compared to the unsheared case (with the exception of
layer 1). But, fitting reveals that this enhancement is greatest for lower
shear rates. Particles are maximally radially mobile at the onset of shear
melting to a layered fluid, and become increasingly confined in their layers
at higher $\dot{\gamma}$.

The timescale $\tau$ shows a similar dependence on $\dot{\gamma}$ [Fig. 4
(b)], decreasing as $\dot{\gamma}$ increases, indicating a faster approach to
the long-time plateau. When subjected to faster shear, particles more quickly
explore their full range of motion perpendicular to shear. This is very
clearly evident for layers 2 to 4 in experiments. Once again, layer 1 does not
follow the trend evident in layers 2 to 4.

##  IV Discussion

The rheology of our system is similar to that of bulk colloidal hard discs,
with evidence for a yield stress and shear thinning Henrich  _et al._ (2009).
Our particle-resolved approach allows us to address the origins of this yield
stress and shear thinning. The former we believe is related to the breakdown
of hexagonal configurations found in the absence of shear. The latter seems to
be similar to the 3d case of shear-induced stratification Cheng  _et al._
(2011). We have shown that motion parallel and perpendicular to shear are
coupled. At larger local shear rate, particles more quickly explore their
layer radially. Simultaneously, the extent of radial motion within the layer
is reduced and they are more tightly confined. This is concurrent with shear
thinning.

In colloidal hard sphere (and disc) systems, shear thinning is attributed to
increased stratification parallel to shear as shear rate is increased Wagner
and Brady (2009); Kawasaki, Ikeda, and Berthier (2014). When particles are
organised into layers, interactions between them become weaker and
consequently the resistance to flow is reduced. Concentric particle layers are
enforced by the boundary of our system, and therefore the flow resistance is
inherently lower than in a bulk suspension at comparable density. However,
Fig. 2 shows that shear thinning _is_ observed. We explain this fact using the
data of Section III.3. Radial motion is suppressed as shear rate is increased,
but Fig. 1 (f) shows that local area fraction is independent of
$\operatorname{\mathrm{P\kern-0.80002pte}}$. Faster shearing does not increase
the packing density of particles, yet they exhibit more tightly confined
dynamics. Radial motion brings particles to interact with particles in
adjacent layers, which dissipates energy and resists flow. Suppressed radial
motion with no concomitant increase in area fraction means that particles in
adjacent layers are, on average, further apart and their interactions with one
another are weakened. This results in a reduction in drag between layers, and
therefore flow resistance, or viscosity. This interpretation is illustrated in
Fig. 1 (h) and (i). Thus, the radial dynamics suggest the rheology — at larger
shear rates, particles are more tightly confined to their layers at constant
density. This reduces interlayer drag and leads to shear thinning.

What is the nature of the interactions that dominate the rheology of our
system? Our simulations include hydrodynamic interactions at the Rotne–Prager
level for the particle–particle interactions and also the Blake tensor for the
coupling to the substrate. We found it necessary to include HI at this level
to reproduce the behavior of the experiments. Pure Brownian Dynamics (without
HI between the particles) leads to weak momentum transfer through the system,
due to a far higher level of slip between the layers than is encountered in
the experiments. Therefore we conclude that momentum transfer is dominated by
HI and that steric effects due to the excluded volume of the particles play a
secondary role. Thus, compared to molecular systems, where van der Waals
interactions can lead to solidification near the boundary, here instead there
are two important differences: (_i_) there is no equivalent of the long–ranged
van der Waals interactions in our hard discs, with no mechanism for
solidification. (_ii_) The momentum transfer is dominated by solvent–mediated
hydrodynamic coupling, which is absent in molecular systems. The importance of
hydrodynamic coupling between the particles suggests that similar behavior
might be encountered in _wet_ granular matter Mari  _et al._ (2015); Morris
(2019). However in our system, the lack of direct particle–particle contacts
(and increased ordering) enables a purely shear–thinning regime.

##  V Conclusion

We have investigated the rheology of colloidal hard discs confined in a
_layered fluid_ configuration under shear and hexagonal configuration with
very weak or no shear in experiment and simulation. Using a particle-level
analysis, we infer the inter-layer forces from drag forces and the driving
force exerted by optical tweezers. Since this system is dissipative and in
steady state, balancing the forces on each layer allows the measurement of the
local viscosity. This is the first experimental measurement of the viscosity
of a hard-disc-like colloidal system under steady shear.

The flow curve is indicative of a dynamic yield stress and shows that the
confined hard disc system exhibits shear thinning at low shear rates and
approximately Newtonian behavior for $\dot{\gamma}\tau_{\mathrm{B}}\gtrsim
0.1$. The dynamic yield stress is $\sim 0.1\,k_{\mathrm{B}}T/a^{2}$. We find
evidence for a static yield stress with a lower bound of
$1.9\,k_{\mathrm{B}}T/a^{2}$. This is remarkably similar to sheared,
unconfined, bidisperse hard discs and bulk hard spheres. This is by no means
an anticipated result as strongly confined systems regularly exhibit very
different rheological responses to their bulk counterparts Hu, Carson, and
Granick (1991); Klein and Kumacheva (1998); Nugent  _et al._ (2007);
Sarangapani and Zhu (2008). In our system, there is a change in structure, in
that the system undergoes shear melting, which we relate to the yield stress.
In the future, it would be intriguing to explore the response of this system
to oscillatory shear and to investigate any yield stress in more detail.

Shear thinning in colloidal hard particle systems is due to shear-induced
layering progressively reducing off-axis interparticle collisions as shear
rate is increased, reducing the viscosity. By measuring the particle motion
perpendicular to the direction of shear, we show that this is also the case in
our system. At higher shear rate, radial motion is suppressed without a change
in local density and therefore interlayer particle collisions are reduced,
reducing the coupling and dissipation between layers, and the therefore the
viscosity.

Colloidal hard discs and spheres under extreme confinement behave remarkably
similarly to their bulk counterparts in both 2 and 3 dimensions and are
qualitatively different to systems dominated by van der Waals interactions,
for which viscosity increases massively on increasing confinement Hu, Carson,
and Granick (1991); Demirel and Granick (1996); Klein and Kumacheva (1998). We
attribute this to an absence of long–ranged vdW interactions in our system.
Furthermore, we infer from our computer simulations that hydrodynamic coupling
between the particles is the dominant mechanism of momentum transfer with
excluded volume interactions playing a secondary role. This latter observation
suggests that similar behavior might be observed in _wet_ granular matter, in
the case that interactions due to contacts between particles are not dominant
Mari  _et al._ (2015); Morris (2019).

We have considered a particular geometry here, where a population of quasi-
hard discs are effectively “corralled” by 27 tweezer particles arranged in a
circle. Of course other geometries are possible. In the case of planar shear.
we expect that behavior we observe would also be found, as one would expect
particles to form layers parallel to the confinement, as is the case here.
Depending on the particle spacing one might expect coupling between the
packing of free particles and of wall particles.

In the future, it would be attractive to carry out a more complete inclusion
of the HI than we have done here, for example with Lattice-Boltzmann dynamics
or Stochastic Rotation dynamics. In particular, it would be useful to enquire
whether such a description would exhibit the shear–thinning behavior seen in
the experiments, and furthermore to explicitly probe lubrication phenomena
neglected in the simulations we have performed here. It would also be
interesting to develop a better description than the fit we have used to
describe the radial MSDs in Eq. 9.

## Acknowledgements

The authors are grateful for enlightening discussions with Paul Bartlett, Wuge
Briscoe, Olivier Dauchot, Jens Eggers, Yael Roichman, Thomas Speck and Todd
Squires. HL was supported by the German Research Foundation (DFG) within the
project LO 418/20-2. IW and CPR acknowledge support from the European Research
Council (ERC Consolidator Grant NANOPRS, project number 617266).

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##  Appendix A Simulation Details

We perform Brownian dynamics simulations of $N$ particles interacting via a
Yukawa pair potential (Eq. 1). Additionally, each of the $27$ particles in the
outermost boundary layer is exposed to a harmonic potential mimicking the
optical traps employed in experiment, given by

$V_{t}(|\mathbf{r}_{i}-\mathbf{r}_{i,0}|)=\dfrac{k}{2}|\mathbf{r}_{i}-\mathbf{r}_{i,0}|^{2},$
(10)

where $\mathbf{r}_{i}$ is the position of $i$th particle and
$\textbf{r}_{i,0}$ the center of its potential well, with $k$ denoting the
trap strength. At each time step $\delta t$, the locations of the $27$
harmonic potential minima, $\mathbf{r}_{i,0}$, are translated a predetermined
arc length, $l$, along the boundary, resulting in a rotation velocity
$l/\delta t$. The velocity, and thus the Péclet number, is controlled by
altering this arc length.

The hydrodynamic interactions between colloids of radius $a$ are modeled on
the the Rotne-Prager level Gauger, Downton, and Stark (2009); von Hansen,
Hinczewski, and Netz (2011). To account for the hydrodynamic effect of the
planar substrate that is present in experiments we first consider Blake’s
solution $\tensor{\mathbf{G}}^{B}$ Blake (1971), which uses the method of
images to obtain the Green’s function of the Stokes equation satisfying the
no-slip boundary condition at $z=0$. Furthermore, the hydrodynamic entrainment
effect of the motion of particle $j$ at $\mathbf{r}_{j}$ on another particle
$i$ at $\mathbf{r}_{i}$ is approximated by a multipole expansion Kim and Netz
(2006); von Hansen, Hinczewski, and Netz (2011) to the second order in $a$
leading to the Rotne-Prager level of the Blake tensor

$\displaystyle\tensor{\mathbf{G}}^{\mathrm{RPB}}(\mathbf{r}_{i},\mathbf{r}_{j})$
$\displaystyle\equiv$
$\displaystyle\left(1+\dfrac{a^{2}}{6}\nabla^{2}_{\mathbf{r}_{i}}\right.+\left.\dfrac{a^{2}}{6}\nabla^{2}_{\mathbf{r}_{j}}\right)\tensor{\mathbf{G}}^{\mathrm{B}}(\mathbf{r}_{i},\mathbf{r}_{j})$
$\displaystyle=$
$\displaystyle\tensor{\mathbf{G}}^{\mathrm{RP}}(\mathbf{r}_{ij})-\tensor{\mathbf{G}}^{\mathrm{RP}}(\mathbf{R}_{ij})+\tensor{\Delta\mathbf{G}}(\mathbf{R}_{ij}),$

where $\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$ is the vector between
particles $i$ and $j$, and
$\mathbf{R}_{ij}=\mathbf{r}_{i}-\overline{\mathbf{r}}_{j}$ is the vector
between particle $i$ and the image of particle $j$ at
$\overline{\mathbf{r}}_{j}=(x_{j},y_{j},-z_{j})$. The Rotne-Prager tensor
$\tensor{\mathbf{G}}^{\mathrm{RP}}$ is given as Rotne and Prager (1969)

$\tensor{\mathbf{G}}^{\mathrm{RP}}(\mathbf{r})=\dfrac{1}{8\pi\eta|\mathbf{r}|}\left(\tensor{\mathbf{I}}+\dfrac{\mathbf{r}\otimes\mathbf{r}}{|\mathbf{r}|^{2}}\right)+\dfrac{a^{2}}{4\pi\eta|\mathbf{r}|^{3}}\left(\dfrac{\tensor{\mathbf{I}}}{3}-\dfrac{\mathbf{r}\otimes\mathbf{r}}{|\mathbf{r}|^{2}}\right).$

with fluid viscosity $\eta$. The last term in Eq. A involves the Rotne-Prager
correction terms to the Stokes and source doublets, with its (off-)diagonal
elements reading as von Hansen, Hinczewski, and Netz (2011)

$\displaystyle\Delta G_{\alpha\alpha}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4\pi\eta}\left[\dfrac{-z_{i}z_{j}}{R^{3}}\left(1-3\dfrac{R^{2}_{\alpha}}{R^{2}}\right)\right.$
(11) $\displaystyle+$
$\displaystyle\left.\dfrac{a^{2}R_{z}^{2}}{R^{5}}\left(1-5\dfrac{R^{2}_{\alpha}}{R^{2}}\right)\right],$
(12) $\displaystyle\Delta G_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4\pi\eta}\left(\dfrac{3z_{i}z_{j}R_{\alpha}R_{\beta}}{R^{5}}-5a^{2}\dfrac{R_{\alpha}R_{\beta}R^{2}_{z}}{R^{7}}\right),$
(13)

where $\alpha,\beta\in\\{x,y\\}$, and $R_{\alpha}$, $R_{\beta}$ corresponding
to $\alpha$\- and $\beta$-component of $\mathbf{R}_{ij}$, and $z_{i}$
specifying the $z$-coordinate of particle $i$. Note that in simulations all
particles have the same vertical distance $z$ from the substrate as adjusted
according to the experimental gravitational length.

The no-slip boundary at the wall alters the particles’ self-mobilities. We
therefore employ a Rotne-Prager level self-mobility tensor
$\tensor{\mathbf{G}}^{\mathrm{RPB}}_{self}(z)\equiv\mu^{\mathrm{RPB}}_{\|}(z)\tensor{\mathbf{I}}$
to obtain an expression for the dependance of the self-mobility of a colloid
separated from a wall by distance $z$ with the diagonal element being von
Hansen, Hinczewski, and Netz (2011)

$\mu^{\mathrm{RPB}}_{\|}(z)=\mu_{0}\left(1-\dfrac{9a}{16z}+\dfrac{1}{8}\left(\dfrac{a}{z}\right)^{3}\right)+O(a^{4}),$
(14)

and $\mu_{0}=1/(6\pi\eta a)$ describing the Stokes self-mobility.

Finally, the equation for the trajectory $\textbf{r}_{i}$ of a colloidal
particle $i$ obeying Brownian motion after a time step $\delta t$ reads

$\mathbf{r}_{i}(t+\delta
t)=\mathbf{r}_{i}(t)+\left(\sum_{j=1}^{N}\tensor{\mathbf{\mu}}_{ij}\mathbf{F}_{j}\right)\delta
t+\delta\textbf{W}_{i},$ (15)

where $\tensor{\mathbf{\mu}}_{ij}$ is given as

$\tensor{\mathbf{\mu}}_{ij}=\tensor{\mathbf{G}}^{\mathrm{RPB}}_{self}(z_{i})\delta_{ij}+(1-\delta_{ij})\tensor{\mathbf{G}}^{\mathrm{RPB}}(\mathbf{r}_{i},\mathbf{r}_{j}),$
(16)

which comprises both the self-mobility part and the entrainment of particle
$i$ by the hydrodynamic flow-field created by conservative forces
$\mathbf{F}_{j}$ acting on particle $j$. This force stems from the pair
interactions, $V$, and for the $27$ driven wall particles also from the
harmonic trap potential $V_{t}$. The random displacement
$\delta\textbf{W}_{i}$ is sampled from a Gaussian distribution with zero mean
and variance $2D_{0}\delta t$ (for each Cartesian component) fixed by the
fluctuation-dissipation relation, where $D_{0}$ is the diffusion coefficient.

In the simulations, the length scale is set by $\kappa$, the energy scale by
$k_{B}T$, and the time scale by $\tau=1/(\kappa^{2}D_{0})$. The inverse
screening length $\kappa$ has been chosen as $\kappa a=14.85$, where the
experimental value of the radius served as a reference. Consequently, the
corral radius has been set to $\kappa R_{0}=128$ yielding the experimental
ratio of $R_{0}/(2a)\approx 4.31$. The high screening at $\kappa a=14.85$
together with the contact potential chosen as $V(r=2a)\approx 0.85k_{B}T$
ensures the quasi hard-disc-behaviour. Another crucial parameter in our system
is the trap strength which has been set to $\lambda=0.42\kappa^{2}k_{B}T$ in
order to mimic the laser trap strength in the experiments. The time step is
chosen as $\delta t=10^{-4}\tau$. Our simulations run for up to $8\times
10^{3}\tau$, corresponding to approximately $35.5\tau_{B}$ with $\tau_{B}$
being the experimental Brownian time, ie., the time one colloid needs to
diffuse a length equal to its radius.

##  Appendix B Determining the Drag Coefficient

Figure 5:  (a) Mean squared displacement in the dilute limit in the absence of
optical traps. Line is linear fit used to extract the empirical drag
coefficient. (b) Drag coefficient correction term of the form in Ladavac and
Grier (2005) as a function of radial location, calculated using experimentally
measured layered structure. Horizontal dashed line shows the average over
layers 2 to 5.

The analysis described in the next section relies on knowledge of the drag
forces acting on each each particle to extract dimensional forces, and
therefore stresses and viscosities. Thus, it is necessary to know the single
particle drag coefficient for our $a=2.5\;\mathrm{\upmu m}$ radius polystyrene
spheres undergoing quasi-two-dimensional motion near the substrate. To this
end, we track the diffusive motion of these particles in the dilute limit,
without any optical traps, and show the resulting mean squared displacement in
Fig. 5 (a). The Stokes-Einstein relation says that this is linear in time,
with gradient $4k_{\mathrm{B}}T/\zeta$. Thus, we perform a linear fit to the
experimental data in Fig. 5 (a) and obtain an empirical measurement of the
single particle drag coefficient to be $\zeta_{\mathrm{emp}}=1.9\times
10^{-7}\;\mathrm{kg\,s^{-1}}$. This represents an approximate four-fold
increase over the Stokes drag coefficient $\zeta_{0}=6\pi\eta a=4.7\times
10^{-8}\;\mathrm{kg\,s^{-1}}$ for spheres of $2.5\;\mathrm{\upmu m}$ radius in
water.

However, it is known that multiple particles moving along a circular path of
radius $r$ each experience reduced hydrodynamic drag due to the presence of
the other particles Ladavac and Grier (2005). This is the drafting effect.
Ladavac & Grier give an approximate result at the level of the Stokeslet
approximation for the single-particle drag coefficient when $n$ particles of
radius $a$ are equally spaced on a ring of radius $r$, where $r\gg a$, and
near a planar substrate Ladavac and Grier (2005).

$\displaystyle\frac{\zeta_{N}^{w}}{\zeta_{0}^{w}}$ $\displaystyle=$
$\displaystyle\left\\{1+\frac{3a\zeta_{0}^{w}}{8r\zeta_{0}}\sum_{j=2}^{n}\left[\frac{(1+\cos\theta_{1j})(1-3\cos\theta_{1j})}{\sqrt{2-2\cos\theta_{1j}}}\frac{h}{r}+\frac{8\cos\theta_{1j}}{(2-2\cos\theta_{1j})^{3/2}}\frac{h^{2}}{r^{2}}+\frac{6(1+\cos\theta_{1j})(5\cos\theta_{1j}-3)}{(2-2\cos\theta_{1j})^{3/2}}\frac{h^{3}}{r^{3}}\right]\right\\}^{-1}$
(17)

This result is truncated at order $(h/r)^{3}$. Here
$\theta_{ij}=(2\pi/n)(j-i)$ is the angular separation between particles $i$
and $j$ and $h$ is the distance between the substrate and the particle
centres.

Using the radial location of each layer’s peak, $r_{i}$, and the layer
populations $n_{i}$, we calculate this modification to the drag coefficient
for our system, assuming $h=a+l_{g}$, that is that the particles are located
one gravitational length above the substrate. The dependance on the radius of
the circular path and the number of particles suggests that particles in
different layers of our system will experience different drag coefficients.
The results of this calculation are shown in Fig. 5 (b). The drag correction
calculated using equation 17 for layer 1 is negative, an unphysical result
that is a consequence of the fact that equation 17 is valid only for $r\gg a$,
which is not true for layer 1 as $r_{1}\approx a$. Therefore, ignoring the
invalid layer 1 result, it is evident that the drafting effect results in a
reduced drag coefficient compared to the dilute limit single-particle wall-
corrected drag coefficient $\zeta_{0}^{w}$. Furthermore, the anticipated
dependence of $\zeta_{N}^{w}$ on radial position is evident, though weak. If
we identify our empirically measured wall-corrected drag coefficient,
$\zeta_{\mathrm{emp}}$, with $\zeta_{0}^{w}$, then we anticipate that
particles in our system will experience a drag coefficient of approximately
$0.34\,\zeta_{\mathrm{emp}}$ due to the drafting effect, where $0.34$ is the
average of $\zeta_{N}^{w}/\zeta_{0}^{w}$ over layers 2 to 5, indicated by the
dashed line in Fig. 5 (b).

Estimating the true drag correction is further complicated by the fact that
our system consists of a series of concentric and closely interacting layers.
Equation 17 considers a single circular particle layer in isolation, and
therefore cannot be a true description of our system. The effective drag
coefficient experienced by a particle in layer $i$ likely depends on the
motion of particles in layers $i+1$ and $i-1$ in addition to the other
particles in layer $i$. Therefore, the calculated reduction in drag is really
only an order-of-magnitude estimate of the drafting effect. The sum in
equation 17 is dominated by contributions from particles $j=2$ and $j=n$,
which are the neighbouring particles of particle $1$. Since the separation
between neighbouring particles is approximately the same in all layers, and
since the radial dependence of of $\zeta_{N}^{w}$ is predicted to be weak, we
treat the drag coefficient as being independent of radial position and equal
to $\zeta=0.34\,\zeta_{\mathrm{emp}}$. This assumption is necessary to
estimate the drag coefficient in layer 1, for which equation 17 is invalid, as
indicated its prediction of a negative drag coefficient in this region.

##  Appendix C Interlayer Hopping at Lower Population

Figure 6:  Time evolution of layer populations $n_{i}$ measured in experiments
driven at $\operatorname{\mathrm{P\kern-0.80002pte}}=17.5$ with confined
populations (a) $n_{\mathrm{conf}}=44$ and (b) $n_{\mathrm{conf}}=48$.

Figure 6 shows the time evolution of layer populations in experiments driven
at $\operatorname{\mathrm{P\kern-0.80002pte}}=17.5$ for confined populations
(a) $n_{\mathrm{conf}}=44$ and (b) $n_{\mathrm{conf}}=48$. In the more densely
packed system at $n_{\mathrm{conf}}=48$ the populations of all layers are
constant in time, as required for the rheological analysis described in the
main manuscript. When the population is reduced to $n_{\mathrm{conf}}=44$,
however, particles occasionally move between layers. When layer populations
are not fixed, the rheological analysis as described in the main manuscript
cannot be applied, and so we have focused our attention on the
$n_{\mathrm{conf}}=48$ system.