††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Contactless Interfacial Rheology: Probing Shear at Liquid-Liquid Interfaces
without an Interfacial Geometry via Fluorescence Microscopy

Iain Muntz  SUPA School of Physics and Astronomy,
The University of Edinburgh, Edinburgh,
EH9 3FD, Scotland, United Kingdom  Department of Bionanoscience, Kavli
Institute of Nanoscience Delft,
Delft University of Technology, Van der Maasweg 9,
2629 HZ Delft, The Netherlands  James A. Richards  SUPA School of Physics and
Astronomy,
The University of Edinburgh, Edinburgh,
EH9 3FD, Scotland, United Kingdom  Edinburgh Complex Fluids Partnership, The
University of Edinburgh,
Edinburgh EH9 3FD, United Kingdom  Sam Brown  SUPA School of Physics and
Astronomy,
The University of Edinburgh, Edinburgh,
EH9 3FD, Scotland, United Kingdom  Andrew B. Schofield  SUPA School of Physics
and Astronomy,
The University of Edinburgh, Edinburgh,
EH9 3FD, Scotland, United Kingdom  Marcel Rey  SUPA School of Physics and
Astronomy,
The University of Edinburgh, Edinburgh,
EH9 3FD, Scotland, United Kingdom  Department of Physics,
University of Gothenburg, Gothenburg, Sweden
Job H. J. Thijssen<EMAIL_ADDRESS>SUPA School of Physics and
Astronomy,
The University of Edinburgh, Edinburgh,
EH9 3FD, Scotland, United Kingdom

###### Abstract

Interfacial rheology is important for understanding properties such as
Pickering emulsion or foam stability. Currently, the response is measured
using a probe directly attached to the interface. This can both disturb the
interface and is coupled to flow in the bulk phase, limiting its sensitivity.
We have developed a contactless interfacial method to perform interfacial
shear rheology on liquid/liquid interfaces with no tool attached directly to
the interface. This is achieved by shearing one of the liquid phases and
measuring the interfacial response via confocal microscopy. Using this method
we have measured steady shear material parameters such as interfacial elastic
moduli for interfaces with solid-like behaviour and interfacial viscosities
for fluid-like interfaces. The accuracy of this method has been verified
relative to a double-wall ring geometry. Moreover, using our contactless
method we are able to measure lower interfacial viscosities than those that
have previously been reported using a double-wall ring geometry. A further
advantage is the simultaneous combination of macroscopic rheological analysis
with microscopic structural analysis. Our analysis directly visualizes how the
interfacial response is strongly correlated to the particle surface coverage
and their interfacial assembly. Furthermore, we capture the evolution and
irreversible changes in the particle assembly that correspond with the
rheological response to steady shear.

##  I Introduction

Interfacial rheometry is essential when characterising systems with large
interfacial area, such as emulsions or foams [1, 2, 3]. These systems are
ubiquitous in industries such as pharmaceuticals, cosmetics and foodstuffs [4,
5, 6, 7, 8, 9, 10]. In order to probe the rheological properties, one can use
shear rheology [11, 12, 13, 14, 15], dilational rheology [16, 17, 18, 19], or
simultaneously image the interface as shear is applied to connect the
rheological properties to the interfacial microstructure [20, 21].

Previous work on interfacial shear rheology has used probes which directly
attach to an oil-water or air-water interface, such as the magnetic rod
interfacial stress rheometer [22, 23, 24], or the double-wall ring (DWR)
geometry attached to a rotational rheometer [11, 20]. These experimental
setups are both based on the maximisation of the ratio of the surface force to
the sub-phase drag, the Boussinesq number [1, 25]:

$\mathrm{Bo}=\frac{\eta^{s}}{\eta l},$ (1)

where $\eta^{s}$ is the surface viscosity, $\eta$ is the sub-phase viscosity
and $l$ is a characteristic length scale roughly equal to the ratio of contact
area to contact perimeter. In order to accurately measure the surface
properties without unintentionally probing the sub-phase, this ratio must be
maximised for the surface to contribute at least an order of magnitude more
than the bulk. Of the two setups mentioned, the magnetic rod has the larger
Bo, while both have a Bo an order of magnitude larger than that of a rotating
disk, due to a much smaller $l$ [26]. This maximisation of Bo can be
considered as optimising the interface-to-bulk signal to noise ratio. Even
though the magnetic rod set up has higher sensitivity, DWR has the advantages
of using a conventional rotational rheometer combined with a larger dynamic
range [11].

In our work, we take a different approach which makes consideration of Bo less
tangible as we have no contact area or contact perimeter. Rather than affixing
a probe directly to the interface, we shear the upper phase, indirectly
deforming the interface, and measure the response using confocal microscopy: a
fundamentally different approach.

Using this contactless technique, we investigate the efficacy of this method
by studying a jammed core-shell PNIPAM-SiO2–laden interface labelled with
tracer particles, which we compare directly to DWR measurements. We then
demonstrate the advantages of this technique by looking at a weakly
interacting system of interfacially adsorbed colloidal particles. This system
has been studied previously using direct probe techniques [14, 27], and
considering the interparticle interactions [28].

Our technique has two main advantages: (i) the liquid-liquid interface we
probe is not disturbed by a large probe immersed therein, and (ii) this setup
models general applications of these large interfacial area systems, where
interfacial shear is applied indirectly via the continuous phase. A clear
example of this second point is in the application of skin creams, where the
continuous phase is sheared, which indirectly deforms the large area of
interface of the dispersed phase. Notably, the equipment required to perform
these measurements is relatively common. While we use confocal microscopy
coupled to a stress-controlled rheometer, reflection or fluorescence
microscopy and a fixed-rate motor should suffice. We show that our technique
can measure surfaces with lower viscosities than have been measured before
using a DWR geometry, due to the inherent sensitivity of the technique arising
from the absence of direct sub-phase drag. Finally, our setup lends itself to
simultaneous structural analysis, which we show is key to understanding the
rheological properties of a particle-laden interface.

##  II Materials and Methods

###  II.1 Materials

All chemicals were obtained from commercial sources and used as received if
not stated otherwise. N,N’-Methylenebis(acrylamide) (BIS; 99 $\%$, Sigma
Aldrich), ethanol (EtOH, 99.9 $\%$, Sigma Aldrich), ammonium persulfate (APS;
98 $\%$ Sigma Aldrich), tetraethyl orthosilicate (TEOS; 98 $\%$, Sigma
Aldrich), ammonium hydroxide solution (28-30 $\%$ NH3 basis, Sigma Aldrich),
(3-(trimethoxysilyl)propyl methacrylate (MPS; 98 $\%$, Sigma Aldrich) and
isopropyl alcohol (IPA, $>99.8$ $\%$, Sigma Aldrich), were used as received.

N-Isopropylacrylamide (NIPAM; 97 $\%$, Sigma Aldrich) was purified by
recrystallization from hexane (95 $\%$, Sigma Aldrich). Water was distilled
and deionized ($18\text{\,}\mathrm{M\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$)
and n-dodecane (Acros organics, 99% pure) was filtered three times through a
column of alumina (Sigma-Aldrich, activated) to remove polar impurities
following a standard procedure [29]. Red fluorescent carboxyl-functionalized
polystyrene (PS) particles ($2\text{\,}\mathrm{\SIUnitSymbolMicro m}$
diameter, Thermo Fisher) were cleaned twice via centrifugation and
redispersion in water/ethanol (1:1).

###  II.2 Synthesis and Characterisation

####  II.2.1 PNIPAM-SiO2 core-shell particles

Poly(N-isopropylacrylamide)(PNIPAM)-SiO2 core-shell particles were obtained by
growing a PNIPAM shell onto the silica cores via a batch surfactant-free
precipitation polymerization as described in previous work [30]. First,
colloidal silica particles used as cores with a diameter of
$160(10)\text{\,}\mathrm{nm}$ were prepared according to a modified Stöber
process[31]. In a round bottom flask, 250 mL EtOH , 12.5 mL deionised water
and 25 mL NH3 (aq) were stirred together. 18.75 mL of TEOS was stirred in 75
mL EtOH and both solutions were heated to
$50\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and equilibrated for
$30\text{\,}\mathrm{min}$. Next, the TEOS solution was quickly added to the
first mixture under heavy stirring. We let the reaction proceed for
$2\text{\,}\mathrm{d}$ at $50\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The
suspension was functionalised without any further purification by adding
$102.7\text{\,}\mathrm{\SIUnitSymbolMicro l}$ MPS. We allowed the reaction
mixture to stir at room temperature for at least $1\text{\,}\mathrm{d}$ and
then boiled it for $1\text{\,}\mathrm{h}$ to ensure successful
functionalisation. Afterwards, we purified the particles by centrifugation and
redispersed them three times in ethanol and three times in Milli-Q water.

In a 500 mL three-neck round bottom flask, 282.9 mg NIPAM and 19.3 mg BIS
($5\,{\rm mol.\,\%}$) were dissolved in 47 mL Milli-Q water. We added the
2.591 g aqueous SiO2 core dispersion (6.6 ${\rm wt}\,\%$). The solution was
heated to $80\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and purged with nitrogen.
After equilibration for $30\text{\,}\mathrm{min}$, a balloon filled with
nitrogen was used to keep the nitrogen atmosphere. Subsequently, 11 mg APS was
rapidly added to initiate the reaction. We let the reaction proceed for 4 h,
and after it cooled down, we purified the suspension 6$\times$ by
centrifugation and redispersion in deionised water. The hydrodynamic diameter
at $20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ was determined by dynamic light
scattering (Malvern Zetasizer Nano-ZS) to $525(53)\text{\,}\mathrm{nm}$.

####  II.2.2 PMMA particles

Poly(methyl methacrylate) (PMMA) particles, stabilized by poly(lauryl
methacrylate), were used as the hydrophobic system. To synthesize these, the
poly(lauryl methacrylate) stabilizer was fabricated first following the recipe
in Ref. 32, Sec. 3.9.1, and it was kept as a 40% solution in dodecane. To make
the particles, a mixture was created that contained 2.1% w/w poly(lauryl
methacrylate) stabilizer, 41.2% w/w methyl methacrylate, 0.84% methacrylic
acid, 11% butyl acetate, 29.6% hexane, 14.2% dodecane, 0.21% octyl mercaptan
and 0.47% of the dye NBD-MAA (7-nitrobenzo-2-oxa-1,3-diazole-methyl
methacrylate), whose preparation can be found in Ref. 33. This mixture was
placed in a 3-necked round-bottomed flask with a condenser attached, brought
under a nitrogen atmosphere, stirred at 350 rpm and heated to 80°C before 0.4%
w/w of the initiator azo-bis-isobutyronitrile was added to start the
polymerization reaction which was left to proceed for 6 hours. The resultant
particles were filtered through glass wool to remove any coagulum present. The
particles were qualitatively inspected using scanning electron microscopy, and
sized by static light scattering to find a diameter of
$3.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$ with a dispersity of 5%. The
particles were cleaned by repeated centrifugation (5$\times$) in $n$-hexane
followed by repeated centrifugation (5$\times$) in $n$-dodecane. The particles
were kept as a dispersion in $n$-dodecane and sonicated for 30 minutes before
dilution, followed by a further 30 minutes of sonication before use to
minimise the number of aggregates in bulk.

###  II.3 Contactless methods

Figure 1: Interfacial shear geometries. (a) Schematic setup for contactless
interfacial rheology using a parallel-plate geometry rotated at fixed angular
velocity, $\omega$. Interface imaging via confocal microscope. Dimensions:
pinned interface ring radius, $r_{r}=10$ mm; water sub-phase height, $h_{w}=3$
mm; and oil depth, $h_{o}$. $\omega$ and $h_{o}$ vary. $r_{\rm out}$ distance
from imaging region to outer edge varies with experiment from
$4\text{\,}\mathrm{mm}6\text{\,}\mathrm{mm}$. (b) Creep-recovery protocol for
contactless method. Recording (dashed lines) begins before applied $\omega$
(solid line) to $t_{\omega}=$30\text{\,}\mathrm{s}60\text{\,}\mathrm{s}$$,
followed by recovery to recording end. Multiple steps with increasing
$\omega$. (c) Schematic of double-wall ring geometry. Radii: ring, inner and
outer, $R_{r,i}=$35.5\text{\,}\mathrm{mm}$$ and
$R_{r,o}=$34.5\text{\,}\mathrm{mm}$$; trough, $R_{i}=$31\text{\,}\mathrm{mm}$$
and $R_{o}=$39.5\text{\,}\mathrm{mm}$$. Ring width,
$l=R_{r,o}-R_{r,i}=$1\text{\,}\mathrm{mm}$$ for Bo, Eq. (1). Torque, $T(t)$,
and angle, $\theta(t)$, to calculate interfacial stress, $\sigma^{s}$, and
strain, $\gamma^{s}_{0}$. (d) Increasing logarithmic oscillatory strain
amplitude sweep.

Interfaces were prepared in a custom made polytetrafluoroethylene (PTFE) cup
with an aluminium ring insert to pin a flat interface. Fig. 1(a) shows a
schematic representation of the cup. The PTFE cup’s inner radius ($r_{c}$) is
$21\text{\,}\mathrm{m}\mathrm{m}$, the aluminium ring’s inner radius ($r_{r}$)
is $10\text{\,}\mathrm{m}\mathrm{m}$. The aluminium ring has a height of
$3\text{\,}\mathrm{m}\mathrm{m}$ and its edge was roughened using silicon
carbide sandpaper to allow pinning of the interface.

The PTFE cup was filled with water, pinned at the edge of the aluminium ring.
As the first interface, we choose a monolayer of PNIPAM-SiO2 core-shell
particles mixed with fluorescent tracer particles at a fixed surface pressure
of $24\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$. We first created a
suspension by mixing $800\text{\,}\mathrm{\SIUnitSymbolMicro l}$ core-shell
particles (0.1 ${\rm wt}\,\%$) and $100\text{\,}\mathrm{\SIUnitSymbolMicro l}$
fluorescent PS microspheres (1 ${\rm wt}\,\%$) with
$100\text{\,}\mathrm{\SIUnitSymbolMicro l}$ IPA as a spreading agent. The
PNIPAM-SiO2 core-shell particles are smaller and able to spread and extend
once adsorbed to liquid interfaces [34, 35]. Therefore, they occupy more
interfacial area compared to the PS particles. Further, we should point out
that PNIPAM-based microgels are known to adsorb onto PS particles [36] and
accumulate around them when confined at liquid interfaces [37]. Thus, we
expect the PS particles to be integrated within the PNIPAM-SiO2 interface and
only minimally influence the rheological response. We then spread the mixed
suspension on a Langmuir trough and measure the surface pressure using the
Wilhelmy method. We determined that
$3.72\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{l}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$
of prepared suspension is required to obtain a surface pressure of
$24\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$. Thus,
$11.7\text{\,}\mathrm{\SIUnitSymbolMicro l}$ was pipetted onto the air-water
interface pinned by the aluminum ring. Notably, in a control experiment, we
could directly verify the surface pressure of the interface within the cell
using the Wilhelmy method to be
$24\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$. Lastly,
$1.5\text{\,}\mathrm{ml}$ of dodecane was carefully pipetted on top of the
interface to give a depth of the oil phase ($h_{o}$) of
$1.6\text{\,}\mathrm{mm}$.

As a second interface, we choose hydrophobic PMMA colloidal particles. The
aluminum ring was again filled with water and $3\text{\,}\mathrm{ml}$ of a
dilute 0.005 ${\rm vol.}\,\%$ PMMA-in-oil dispersion was pipetted onto the
water sub-phase ($h_{o}=$2.2\text{\,}\mathrm{mm}$$). A 0.005 ${\rm vol.}\,\%$
dispersion generally leads to a low volume fraction interface, however there
is large variability in the final surface fraction for the same initial volume
fraction. To achieve higher surface fractions a higher initial volume fraction
was used. During equilibration for 1 hour, the PMMA particles sedimented to
the oil-water interface and formed a monolayer. The surface fraction, $\phi$,
was adjusted by either adjusting the amount of deposited oil dispersion or by
adjusting the particle concentration.

Surface fractions were measured from microscopy images using a pixel counting
method determining the fraction of foreground (particle) pixels to total
pixels after performing a thresholding procedure. This measurement was made
over multiple frames and the final value for surface fraction was determined
as the mean through one rotation of the interface. We note that this approach
likely overestimates the actual $\phi$, see Appendix A. However, this simple
approach allows systematic comparison between different surface coverages.

Connected pixel clusters can then be identified to assess interface
homogeneity, or aggregation state, via the dispersity [14],

$\mathcal{D}=\frac{s}{\langle A\rangle}\,,$ (2)

with $\langle A\rangle$ the average cluster size and $s$ the standard
deviation. These dispersities were measured from relatively zoomed out images.

A 25 mm diameter parallel-plate geometry was attached to the oil-air interface
in the centre of the PTFE cup. Fixed rotation speeds, $\omega$, were then
applied (MCR 301, Anton Paar), shearing the upper oil phase.

Using a rheoimaging setup, as described by Besseling  _et al._ [38] (although
our rheometer setup lies directly on top of the confocal, as described by
Dutta  _et al._ [39], providing greater stability), rheometry was conducted
while the interface was simultaneously imaged using a Leica SP8 confocal
microscope with a 10$\times$ / 0.3 NA air-immersion objective, at
$1024\\!\times\\!1024$ px2 (PNIPAM-SiO2) or $512\\!\times\\!512$ px2 (PMMA)
($$932\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\\!\times\\!$932\text{\,}\mathrm{\SIUnitSymbolMicro m}$$). The imaging
setup was such that the motion of the interface under shear was horizontally
oriented. Velocimetry of the confocal images was performed using C code
written in house. This splits the images into 10 equally spaced horizontal
bands, i.e. the top 10% of the image, the second 10% of the image, etc. Each
band is offset horizontally by a well defined distance. The Pearson
correlation coefficient of this new offset band with the same band in the
previous frame is calculated [40]. The distance moved between that frame and
the previous is then the horizontal offset which maximises this correlation,
over time this gives the net displacement. Note that, as we use fluorescence
microscopy, and only the particles are fluorescently labelled, the strain we
measure is the strain of the (interfacial) colloidal particles in the field of
view of the microscope.

The interfacial strain is calculated as $x/r_{\rm out}$ for each band and
averaged, where $x$ is the measured displacement of the interface and $r_{\rm
out}$ is the distance from the measurement to the outer, pinned wall; $r_{\rm
out}$ varies for each experiment and is measured in situ, it is always
approximately $6\text{\,}\mathrm{mm}$. To probe the yielding and flow of the
interfaces, a creep-recovery protocol is used [41]. Fixed rotation rates were
set for $t_{\omega}=$60\text{\,}\mathrm{s}$$ (PNIPAM-SiO2) or
$120\text{\,}\mathrm{s}$ (PMMA), applying stress to the interface, before a
further period of fixing the rotation of the rheometer to 0,
$60\text{\,}\mathrm{s}$ (PNIPAM-SiO2) or $30\text{\,}\mathrm{s}$ (PMMA),
allowing the interface to relax, Fig. 1(b). The two steps allow separate
measurement of the elastic response and the plastic, or irrecoverable,
response. The (elastic) recoverable strain ($\gamma^{s}_{\rm rec}$), is given
by the recoil from the peak strain at the end of the applied rotation to the
end of the recovery step. The (plastic) irrecoverable shear rate,
$\dot{\gamma}^{s}$, can be calculated from the total change in strain from the
start of applied rotation to after recovery over the time of the applied
shear. Alternatively, for faster relaxing interfaces (PMMA) $\dot{\gamma}^{s}$
can be calculated from the average shear rate over a $45\text{\,}\mathrm{s}$
window towards the end of the applied rotation. This sequence is repeated at
multiple increasing rotation rates to determine the stress-dependent response
of the interfaces.

###  II.4 Double-wall ring geometry

Conventional interfacial shear rheometry was performed using a double-wall
ring geometry, Fig. 1(c), connected to a stress-controlled rotational
rheometer (TA Instruments, DHR-2). This consists of a Platinum/Iridium ring
(diamond cross-section with inner/outer radius
$R_{r,i/o}=34.5/$35.5\text{\,}\mathrm{mm}$$ and hence width
$l=$1\text{\,}\mathrm{mm}$$) inside a ring-shaped polyoxymethylene trough
(inner/outer radius $R_{i/o}=31/$39.5\text{\,}\mathrm{mm}$$). All surfaces
were cleaned multiple times with ethanol and deionised water. To form an
interface the trough is first filled with the water sub-phase until level and
pinned at the edges of the trough. $70\text{\,}\mathrm{\SIUnitSymbolMicro l}$
of the mixed PNIPAM-SiO2 and PS tracer particle dispersion in a spreading
solvent were then carefully pipetted onto the air-water interface. The ring is
then lowered until pinned and level at the interface, before dodecane is
pipetted on top to cover the ring. The ring is therefore in direct contact
with both the interface and the sub-phase.

Figure 2: Thin ring element of oil-water interface, radius $r$ and width d$r$.
The interfacial stress, $\sigma^{s}$, can be found by considering the torque
balance of bulk oil flow drag and interfacial stress gradient, d$\sigma^{s}$,
with rotation rates (interface, $\omega_{i}$, and top of oil phase, $\omega$),
Sec. II.5.

Oscillatory strain amplitude sweeps were performed using controlled strain at
$f=$0.2\text{\,}\mathrm{Hz}$$ in the low-frequency response region, following
previous protocols for microgel-laden interfaces [16] with one equilibration
cycle and six measurement cycles per point, Fig. 1(d). Strain was increased
logarithmically at 20 points/decade from 0.001 to 1.0 strain amplitude and we
report the strain-dependent elastic ($G^{s\prime}$) and loss
($G^{s\prime\prime}$) moduli from the primary Fourier components. The
sinusoidal oscillation of the ring, $\theta(t)=\theta_{0}\sin(2\pi ft)$, is
converted into an interfacial strain,
$\gamma^{s}_{0}=\theta_{0}[(1-(R_{r,o}/R_{o})^{2})^{-1}+((R_{r,i}/R_{i})^{2}-1)^{-1}]$
using the 2D-equivalent expressions for a Couette cylinder at the position of
the ring [42]. The strain can be approximated as $\gamma^{s}_{0}\approx\theta
R_{r,o}/(R_{o}-R_{r,o})$ 111This can be recovered by factorising the
denominator,
$\gamma^{s}_{0}=\theta_{0}[R_{o}^{2}/[(R_{o}-R_{r,o})(R_{o}+R_{r,o})]+R_{i}^{2}/[(R_{r,i}-R_{i})(R_{r,i}+R_{i})]$,
and approximating to first order by taking $R_{i/o}+R_{r,i/o}\approx
2R_{r,i/o}$, $R_{r,i}\approx R_{r,o}$ and $R_{o}-R_{r,o}\approx
R_{r,i}-R_{i}$., i.e. the displacement of the ring divided by the distance
from the ring to the outer pinned interface, analogous with the expression for
the contactless geometry. As the ring is in direct contact with the interface,
the measured torque can be converted straight to an interfacial stress,
$\sigma^{s}=T/2\pi(R_{r,i}^{2}+R_{r,o}^{2})$. All reported data is at high Bo,
such that sub-phase drag correction is not applied, and low raw-phase angle,
where geometry inertia does not dominate. Due to noise from the strain-control
feedback loop, torque resolution is limited to
$0.07\text{\,}\mathrm{\SIUnitSymbolMicro N}\text{\,}\mathrm{m}$[44] and we
highlight or truncate data below this threshold.

###  II.5 Measuring stress in a contactless geometry

Figure 3: Contactless interfacial rheometry of PNIPAM-SiO2–laden interfaces.
(a) Creep-recovery protocol, applied rotation rate, $\omega=6.31$ rpm, with
time, $t$. Stress,
$\sigma^{s}=$3.2\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$,
for $\omega_{i}\approx 0$, Eq. (9). Highlighted $t$: (i) dashed (blue), start
of applied $\omega$ at $t=$0\text{\,}\mathrm{s}$$; (ii) dot-dashed (grey), end
of applied $\omega$ at $t_{\omega}=$60\text{\,}\mathrm{s}$$; and (iii) dotted
(orange), end of recorded recovery at $t=$120\text{\,}\mathrm{s}$$. (b)
Measured interface response at highlighted $t$. Zoomed-in confocal microscopy
at $t$ in (a). Outline, distinctive particle cluster showing interface motion
along $x$, axis rotated to match (c). (c) Extracted time-dependent strain,
$\gamma^{s}(t)$. Horizontal lines trace strain from: (i) dashed, (ii) dot-
dashed and (iii) dotted. Arrows: recoverable (elastic) strain,
$\gamma^{s}_{\rm
rec}=\gamma^{s}($60\text{\,}\mathrm{s}$)-\gamma^{s}($120\text{\,}\mathrm{s}$)$;
and irrecoverable (plastic) strain, $\gamma^{s}_{\rm
irr}=\gamma^{s}($120\text{\,}\mathrm{s}$)-\gamma^{s}($0\text{\,}\mathrm{s}$)$.

In contrast to the DWR geometry, the interface is not in direct contact with
the geometry in the contactless method. Therefore, the stress cannot be
directly measured by the rheometer. To make meaningful statements on the
rheological properties of the interface, we must first find the interfacial
stress applied at the interface. It is well known that for a parallel-plate
setup, the stress is independent of the height through the sample upon
reaching a steady state [42]. The timescale for momentum diffusion and to
reach steady state is up to $\approx
0.3h_{o}^{2}\rho_{o}/\eta_{o}=\mathcal{O}($1\text{\,}\mathrm{s}$)$ for our
setup [45], where $\rho_{o}$ is the oil density; this is far below the creep
step duration. Considering the upper phase as a Newtonian fluid allows us to,
therefore, find the stress on the interface from the upper fluid using the
applied rotational speed of the geometry.

If the rheometer is rotated at a fixed angular velocity of $\omega$ then the
shear rate of the upper fluid at a radius $r$ (Fig. 2) of the parallel-plate
geometry is given by

$\dot{\gamma}=\frac{\omega r}{h_{o}},$ (3)

where $h_{o}$ is the oil phase depth. This definition relies on the interface
having zero speed. Therefore, this is a first approximation if the speed of
the geometry is much larger than the speed of the interface. If this is not
the case, then we can reduce $\omega$ by the angular speed of the interface at
$r$, $\omega_{i}$, giving,

$\dot{\gamma}=\frac{(\omega-\omega_{i})r}{h_{o}}.$ (4)

The stress induced is given by the product of $\dot{\gamma}$ with the upper
phase bulk viscosity, $\eta_{o}$, i.e.,

$\sigma=\frac{\eta_{o}(\omega-\omega_{i})r}{h_{o}}.$ (5)

To convert this bulk stress into an interfacial stress we consider the torque
applied from the bulk on an area element of a ring in the interface at $r$ of
width ${\rm d}r$, Fig. 2.

We can write the torque element, d$T$, as a product of the force element,
$\sigma{\rm d}A$, and the radius, where d$A=2\pi r{\rm d}r$ is the area of the
infinitesimal ring,

${\rm d}T=\sigma 2\pi r^{2}{\rm
d}r=2\pi\frac{\eta_{o}(\omega-\omega_{i})}{h_{o}}r^{3}{\rm d}r.$ (6)

This torque then gives rise to the interfacial stress, $\sigma^{s}$. With
$T(r)=2\pi r^{2}\sigma^{s}(r)$ the interfacial torque as the product of
$\sigma^{s}$, radius and perimeter, we can write the torque balance

$\begin{split}{\rm d}T=&T(r+{\rm d}r)-T(r)\\\ \approx&2\pi\left[(r+{\rm
d}r)^{2}\sigma^{s}(r+{\rm d}r)-r^{2}\sigma^{s}(r)\right]\\\ \approx&2\pi
r\left[r{\rm d}\sigma^{s}+2\sigma^{s}{\rm d}r\right]\end{split}$ (7)

where second order differential terms, e.g., $({\rm d}r)^{2}$, have been
dropped. Equating Eqs (6) and (7) and rearranging,

$\frac{{\rm d}\sigma^{s}}{{\rm
d}r}+\frac{2\sigma^{s}}{r}=\frac{\eta_{o}(\omega-\omega_{i})}{h_{o}}r.$ (8)

Figure 4: Strain response, $\gamma^{s}(t)$, of PNIPAM-SiO2–laden interface for
contactless rheology protocol, Fig. 3(a), showing transition from elastic
dominated response to plastic flow with increasing rotation rate, $\omega$.
(a) Low
$\omega=$1\text{\,}\mathrm{r}\mathrm{p}\mathrm{m}8\text{\,}\mathrm{r}\mathrm{p}\mathrm{m}$$
(dark to light), see legend, with dominant elastic response, $\gamma^{s}_{\rm
rec}$, after $t_{\omega}=$60\text{\,}\mathrm{s}$$. Shading, error from
standard deviation in image correlation analysis bands. (b) Moderate $\omega$
around yielding with rising plastic response. (c) Strain response at high
$\omega$ dominated by irreversible plastic flow, $\dot{\gamma}^{s}$ (see text
for details).

Equation (8) can be readily solved to give

$\sigma^{s}=\frac{\eta_{o}(\omega-\omega_{i})}{4h_{o}}r^{2}$ (9)

if $\frac{{\rm d}\omega_{i}}{{\rm d}r}\simeq 0$, so there is no interfacial
shear banding, for example; this aligns with our confocal-microscopy
observations, e.g., $\omega_{i}$ is constant in time and across the field of
view (within error). It is evident this expression yields the correct
dimensions for interfacial stress as Pa m, as well as physically reasonable
dependencies on viscosity, applied rotation, oil phase height, and radius. In
order to vary the interfacial stress, we vary $\omega$ while observing at a
fixed $r$, a practically simpler approach.

Finally, we need to ensure that we are measuring surface rather than bulk sub-
phase properties. Typically, this can be checked via Bo [Eq. (1)], but this
assumes a probe in contact with the interface. For our contactless setup, as
we measure interfacial strain via the motion of the interface itself, we focus
instead on the question: is the force on the _interface_ dominated by surface
or bulk sub-phase viscosity? This effect is included in sub-phase drag
corrections [46], but is typically neglected in Bo as drag on the probe
dominates. For our contactless geometry, the stress from the sub-phase can be
crudely approximated as parallel-plate flow with $\omega_{i}$, giving an
equivalent to Bo,

$\mathrm{Bo}^{*}=\frac{\sigma^{s}}{\sigma^{s}_{\rm
drag}}=\frac{\eta_{o}h_{w}}{\eta_{w}h_{o}}\frac{\omega-\omega_{i}}{\omega_{i}}\,\mathrm{for}~{}\sigma^{s}_{\rm
drag}\approx\frac{\eta_{w}\omega_{i}}{4h_{w}}r^{2},$ (10)

where $\eta_{w}$ and $h_{w}$ are the water sub-phase viscosity and depth, Fig.
1(a). The first term in Bo∗ is $\mathcal{O}(1)$ for our bulk phases and
dimensions, and likely most common uses, but the second term can be
arbitrarily large as $\omega_{i}\rightarrow 0$. This makes the technique
suitable for measuring weak interface yielding, a fact that arises from
decoupling stress application and interface motion by not having a direct
probe. For interfaces with significant $\omega_{i}$, Bo∗ can drop and this is
discussed where relevant.

##  III Results and Discussion

###  III.1 Elastic PNIPAM-SiO2–laden interface

To establish the validity of our novel contactless interfacial rheometric
technique we begin by measuring a highly elastic PNIPAM-SiO2–laden interface,
which can also be studied by conventional DWR interfacial rheometry.

####  III.1.1 Contactless rheometry

Using the contactless setup, we perform a creep-recovery test with increasing
rotation rates, $\omega$, from 0.1 to 400 revolutions per minute (rpm) with
logarithmic spacing at 5 pts/decade and higher resolution (20 pts/decade)
where behaviour is observed to be changing from the confocal microscopy
recordings. At each step, while recording, $\omega$ is applied for
$60\text{\,}\mathrm{s}$ before zero rotation rate is set for a further
$60\text{\,}\mathrm{s}$, Fig. 3(a). Imaging tracer particles embedded in the
interface throughout these steps gives the resulting deformation of the
interface. This is illustrated by following a distinctive cluster of particles
in magnified images, Fig. 3(b). From the start of applied rotation (i) to when
rotation is stopped (ii), the interface first moves along the flow direction,
$x$. After cessation and relaxation the interface recoils backwards, along the
previous flow direction, until the recovery step ends, Fig. 3(b)(ii) to (iii).
Over time, this particle motion can be seen as tracing out the strain response
of the interface, $\gamma^{s}(t)=x(t)/r_{\rm out}$, lines from (b)(i)–(iii) to
(c).

At low applied $\omega$, e.g., $6.31$ rpm (Fig. 3), there is an initial jump
in $\gamma^{s}$ as $\omega$ is applied before a further slow increase over
$t_{\omega}$ [the creep step length, Fig. 1(d)], see Fig. 3(c). At the
cessation of applied flow there is a rapid recoil, followed by a slower
further relaxation, approaching a constant value. From the strain profile,
$\gamma^{s}(t)$, we extract two quantities: the recoverable strain as the
decrease from the peak to the final strain [dot-dashed to dotted lines, (ii)
to (iii)], and the irrecoverable strain, $\gamma^{s}_{\rm irr}$, as the
increase from the initial strain, at the start of shear, to the final strain
[dashed to dotted lines, (i) to (iii)]. A strong initial deformation and near
complete elastic recovery can be seen over a range of $\omega\lesssim 8$ rpm,
Fig. 4(a), with both steps growing with $\omega$ (dark to light).

As $\omega$ increases further, up to $20$ rpm [Fig. 4(b) (dark to light)],
$\gamma^{s}_{\rm rec}$ increases proportionally; $\gamma^{s}_{\rm irr}$ also
begins to slowly increase as the interface does not fully recover. At higher
$\omega$ still, $\gtrsim 25$ rpm [Fig. 4(c)] there is a clear change in
behaviour, as $\gamma^{s}_{\rm irr}$ rapidly increases while the elastic
recovery remains unchanged. During applied rotation, the strain is linear in
time, giving a well-defined interfacial shear rate,
$\dot{\gamma}^{s}=\gamma^{s}_{\rm irr}/t_{\omega}$. This behaviour is
indicative of yielding in a creep-recovery test [41].

Using $\gamma^{s}_{\rm rec}$ and $\dot{\gamma}^{s}$ alongside the calculated
interfacial stress ($\sigma^{s}$), Eq. (9), the rheological response can be
quantified. The elastic response, $\sigma^{s}(\gamma^{s}_{\rm rec})$, shows
three regimes, Fig. 5(a) [solid circles]. At the lowest stresses,
$\sigma^{s}<${10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$, no clear
trend is observed. This (shaded) region lies below a minimum strain,
$\gamma_{\rm rec}^{s,\min}\approx 2\times 10^{-3}$, set by noise, e.g.,
vibration or drift. With increasing stress, $\gamma^{s}_{\rm rec}$ increases
linearly until
$\sigma^{s}=$5\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$.
Above this the elastic recovery appears constant, but noisy (i.e. spatially
heterogeneous across the field of view). Within the linear region an elastic
constant, $G^{s\prime}=\sigma^{s}/\gamma^{s}_{\rm
rec}=$4.2(1)\text{\times}{10}^{-4}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$
can be fitted (dashed line). The interface is well described as a linear
elastic solid below
$\sigma^{s}=$5\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$,
this can be emphasised by plotting on linear axes, Fig. 5(b).

However, this is only one side of the measured response. The stress-dependent
plastic flow, $\sigma^{s}(\dot{\gamma}^{s})$, further illuminates the change
around
$\sigma^{s}\approx$5\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$,
Fig. 5(c) (squares). Below this threshold $\dot{\gamma}^{s}$ is limited, but
on further increase the interface begins to flow
$\dot{\gamma}^{s}\sim\mathcal{O}($0.1\text{\,}{\mathrm{s}}^{-1}$)$. The
transition point, or yield stress ($\sigma^{s}_{y}$) can be quantified by
fitting a simple piecewise function to these data. We model the interface as a
Bingham fluid, one of the simplest models capturing suitable non-Newtonian
behaviour, described by:

$\begin{array}[]{lr}\dot{\gamma}^{s}=0&:\sigma^{s}<\sigma^{s}_{y}\\\
\sigma^{s}=\sigma^{s}_{y}+\eta^{s}\dot{\gamma}^{s}&:\sigma^{s}\geq\sigma^{s}_{y}.\end{array}$
(11)

Below $\sigma^{s}_{y}$ there is no flow, above this the excess stress leads to
a shear rate set by the interfacial viscosity, $\eta^{s}$. From this we can
obtain both a yield stress and an interfacial viscosity from the contactless
technique, $\sigma^{s,\rm
Contactless}_{y}=$3.5(1)\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$,
and
$\eta^{s}=$9.4(1)\text{\times}{10}^{-4}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$$.
While the Bingham model is appropriate to capture a clear yield stress
transition, in general more complex interfacial yielding behaviour is often
observed [47, 48].

Figure 5: PNIPAM-SiO2–laden interface rheology. (a) Elastic response, stress
($\sigma^{s}$) vs strain. Points: solid, contactless [$\sigma^{s}$ from Eq.
(9) and strain, $\gamma^{s}_{\rm rec}$, from relaxation, Fig. 4]; open, DWR
elastic stress, [$\sigma^{s}=\gamma^{s}_{0}G^{s\prime}$ for strain amplitude,
$\gamma^{s}_{0}$]. Minimum limits: shading, contactless strain, $\gamma_{\rm
rec}^{s,\min}\\!\approx\\!0.002$; vertical dotted line, DWR torque at
$\gamma_{0}^{s,\min}$. Fit lines: dashed, contactless elastic response,
$G^{s\prime}\\!=\\!\sigma^{s}/\gamma^{s}_{\rm
rec}\\!=\\!$4.2(1)\text{\times}{10}^{-4}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$;
horizontal dotted, yield stress from (c). Fit of linear elastic response for
DWR in Fig. 6. (b) Linear plot of elastic response before yielding, symbols as
in (a). (c) Contactless viscous response. Points, stress vs shear rate,
$\dot{\gamma}^{s}=\gamma^{s}_{\rm irr}/t_{\omega}$ from irrecoverable strain,
Fig. 4(c). Lines: bold dashed, Bingham plastic fit, Eq. 11; fine dashed, yield
stress, $\sigma_{y}^{s,\rm
Contactless}\\!=\\!$3.5\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$;
and dotted, DWR yield stress, $\sigma_{y}^{s,\rm
DWR}\\!=\\!$4.8\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$.

####  III.1.2 Comparison to DWR rheometry

Figure 6: DWR rheology of a PNIPAM-SiO2–laden interface. Elastic
($G^{s\prime}$, dark circles) and loss ($G^{s\prime\prime}$, light squares)
moduli vs strain amplitude, $\gamma^{s}_{0}$. Shading, below instrument
resolution. Lines: dotted,
$\bar{G^{s\prime}}=$3.7(4)\text{\times}{10}^{-4}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$,
mean of $G^{s\prime}(\gamma^{s}_{0}\leq 0.05)$; dashed, yielding at
$G^{s\prime}=G^{s\prime\prime}$, $\sigma^{s,\rm
DWR}_{y}=\gamma^{s}_{0}\sqrt{(G^{s\prime})^{2}+(G^{s\prime\prime})^{2}}=$4.8\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$

As a strong and highly elastic interface we can directly compare the results
of the contactless geometry to DWR interfacial rheology for the same PNIPAM-
SiO2 at equal surface pressures. As the torque resolution for oscillatory
tests is finer than steady shear for rotational rheometers, an oscillatory
strain amplitude sweep is performed. This measures the strain-dependent
elastic and loss moduli, Fig. 6 (symbols). At low $\gamma^{s}_{0}<0.1$, the
elastic modulus is higher than the loss modulus, with $G^{s\prime}$ only
weakly decreasing once above the torque resolution (shaded region), indicative
of a solid elastic material as the stress in phase with the strain. With
increasing $\gamma^{s}_{0}$, $G^{s\prime}$ begins to drop while
$G^{s\prime\prime}$ remains near constant. At $\gamma^{s}_{0}=0.23$ the moduli
become equal, $G^{s\prime}=G^{s\prime\prime}$; above this point $G^{s\prime}$
continues to sharply drop, while $G^{s\prime\prime}$ weakly decreases. In this
region the stress is in phase with the shear rate, i.e. liquid-like.
Therefore, with increasing strain amplitude the interface yields from a solid-
like state to a liquid-like state where $G^{s\prime}=G^{s\prime\prime}$. This
stress, $\sigma^{s,\rm
DWR}_{y}=$4.8\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$,
is an operative yield stress at a finite frequency and shear rate, in contrast
to the ‘static’ measurement of creep recovery [49].

The comparison of $\sigma_{y}^{s,\rm DWR}$ [Fig. 5(c) (dotted line)] with the
contactless rheology value, $\sigma_{y}^{s,\rm Contactless}$ (fine dashed
line), finds similar values, with only a 30% difference, 4.8 vs
$3.5\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$. This is
within the expected variation for different measurement protocols of a non-
linear property 222Oscillatory yielding is ambiguous, as it can be gradual,
with multiple definitions; $G^{s\prime}=G^{s\prime\prime}$ is an upper
(over)estimate [49]. E.g., a tangent analysis appears to give better agreement
with the contactless method, but requires extrapolation from data below
instrument resolution., e.g., in colloidal gels or glasses [51]. The yield
strains, $\gamma^{s}_{0}=0.23$ and $\gamma^{s}_{\rm rec}=0.1$, are also
comparable, Fig. 5(a), but greater than previously observed for microgel-laden
[16] or amorphous jammed interfaces [52]. Above yielding we are not able to
compare DWR and contactless measurements, as the applied shear rates for DWR
($\dot{\gamma}^{s}=2\pi
f\gamma^{s}_{0}\gtrsim$0.25\text{\,}{\mathrm{s}}^{-1}$$) are larger than those
in the contactless geometry. To yield at comparable
$\dot{\gamma}^{s}\approx$0.01\text{\,}{\mathrm{s}}^{-1}$$ would require
$f\approx$0.01\text{\,}\mathrm{Hz}$$, resulting in infeasibly long
experiments.

Figure 7: PMMA-laden interface at low surface coverage. (a) Fluorescent
confocal micrographs of PMMA particles (white) at an oil-water interface with
$\phi=31.0\%$. (b) Corresponding strain vs time, $\gamma^{s}(t)$, at low
imposed rheometer rotation rates, $\omega$, see inset legend. $r_{\rm
out}=$6.6\text{\,}\mathrm{mm}$$. $\omega$ applied from
$t\gtrsim$5\text{\,}\mathrm{s}$$ for $120\text{\,}\mathrm{s}$, followed by
$30\text{\,}\mathrm{s}$ further recording. (c) Corresponding high imposed
$\omega$ response.

In contrast to the yield stress, the linear elastic behaviour is well-defined.
Comparing the DWR elastic modulus below yielding, Fig. 5(a) (open circles),
and the contactless elastic modulus (filled circles) we see excellent
agreement. Fitting over the linear response region ($\gamma^{s}_{0}\leq 0.05$)
the average elastic modulus from DWR is only 12% lower than the contactless
measurement, $\bar{G^{s\prime}}=3.7(4)$ vs
$4.2(1)\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$,
demonstrating that they measure equal quantities within error. As yielding is
approached, the DWR response appears to soften while the contactless
measurement remains linear, Fig. 5(b). However, the oscillatory strain
amplitude will contain both the recoverable elastic strain and plastic flow
[53], in contrast to a creep-recovery measurement that separates the terms.
So, approaching yielding, where plastic flow gradually begins [Fig. 5(c)] they
are no longer directly comparable. The comparable $\sigma^{s}_{y}$ values and
equal elastic moduli demonstrate that the contactless geometry accurately
measures well-defined interfacial rheological properties.

###  III.2 PMMA particle laden interface

####  III.2.1 Low surface coverage

We now turn to an interface laden with solid colloidal hydrophobic PMMA
particles, which are typically challenging to measure using conventional
interfacial rheometric techniques [14]. First, we investigate interfaces with
low surface fraction, e.g. $\phi=31$% [Fig. 7(a)]. Interestingly, even though
the particles are hydrophobic sterically stabilised nearly hard-sphere
particles, they exhibit long-range repulsion when confined at an oil-water
interface and assemble predominantly in a non-close packed arrangement [28].
Some aggregation is present due to attractive capillary and Van der Waals
forces, Fig. 7(a). While capillary forces should be negligible, due to a
vanishingly small Bond number and the use of spherical particles [4], there
will be a certain particle roughness from the variable length of the steric
stabiliser “hairs”. This may cause contact line undulations that lead to
short-range capillary attraction [54, 55].

For this particle system, strain vs time plots show smooth flow with a
constant shear rate over the duration of the creep step, Fig. 7(b). As
expected, at larger imposed rotation rates from the rheometer a larger
interfacial shear rate is measured in response, Fig. 7(c). Note that, when
shear starts the interface appears to immediately (within temporal resolution
of the analysis method) begin flowing at a constant shear rate. Similarly,
when the shear ends the interface immediately stops flowing. This implies that
the interface response in this regime is purely viscous, with no measurable
elasticity.

To quantify the rheology of these particle-laden interfaces, we plot stress vs
shear rate, $\sigma^{s}(\dot{\gamma}^{s})$, with $\dot{\gamma}^{s}$ defined
from the slope of $\gamma^{s}(t)$, Sec. II, due to the immediate response. As
expected, for relatively low $\phi$ we observe a Newtonian response, Fig. 8(a)
[dot-dashed (orange) line]. We can therefore assign a constant
$\eta^{s}=$4.43(9)\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$$
for the interfacial viscosity of this interface. This $\eta^{s}$ is comparable
to that measured using a magnetic rod interfacial rheometer on a similar
system [56].

Figure 8: Newtonian rheological behaviour of low surface coverage interfaces
with varying parameters. (a) Stress–shear-rate behaviour of three interfaces
with varying oil heights ($h_{o}$), surface coverages ($\phi$) and aggregation
states ($\mathcal{D}$). Points, measured data. Lines, fits to constant
viscosity: (i)
$4.1(5)\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$,
solid (green); (ii)
$2.10(10)\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$,
dashed (blue); (iii)
$4.43(9)\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$,
dot-dashed (orange), also in Fig. 7. (b) Interface images: (i) $\phi=38\%$,
$h_{o}=$2.9\text{\,}\mathrm{mm}$$ and $\mathcal{D}=24.7$; (ii) $\phi=29\%$,
$h_{o}=$1.4\text{\,}\mathrm{mm}$$ and $\mathcal{D}=23.4$; and, (iii)
$\phi=31\%$, $h_{o}=$2.2\text{\,}\mathrm{mm}$$ and $\mathcal{D}=2.49$. NB:
surface fractions are determined from a rotational average, so single images
are not wholly representative.

##### Effect of particle assembly.

Next, we have performed repeats of these experiments while varying the oil
phase thickness to test the robustness of our technique, Fig. 8. Confocal
images prior to shearing reveal that the partial presence of aggregates, with
particles in direct contact, varies between samples, Fig. 8(b). The
aggregation state is characterized by the dispersity $\mathcal{D}$, Eq. (2),
where a low $\mathcal{D}$ corresponds to a homogeneous particle distribution
and a high $\mathcal{D}$ to an aggregated assembly. We will discuss the
particle assembly’s influence on the rheology below.

First, we observe a Newtonian response for all three oil thicknesses, Fig.
8(a). This suggests that, as expected, the height of the oil phase does not
seem to have a substantial effect, i.e. the rheology of the interface is not
expected to depend on the depth of the bulk phases. When comparing samples
with different $\phi$ and aggregation states, we observe the following trends.
First, an increase in $\phi$ leads to an increase in interfacial viscosity.
For example, when comparing interfaces with similar aggregation states
($\mathcal{D}=24.7$ and $\mathcal{D}=23.4$), but different $\phi$, the
interfacial viscosity at 38%, (green) solid line, is higher than at 29%,
(blue) dashed, 4.1 vs
$2.1\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$.
This trend is to be expected, as we later demonstrate elastic responses for
high surface fractions of 56.7%, Sec. III.2.2. Second, the aggregation state
seemingly affects the interfacial viscosity. When comparing samples with
similar $\phi$ (29% and 31%), but different aggregation states
($\mathcal{D}=23.4$ and $\mathcal{D}=2.49$ respectively), we measure a
significant difference in $\eta^{s}$ (2.1 vs
$4.4\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$),
cf. (blue) dashed and (orange) dot-dashed lines. This suggests that the
aggregation of PMMA particles at liquid interfaces leads to lower interfacial
viscosities compared to more homogeneously distributed particles.

Our observation that surface coverage and aggregation state make a significant
difference to measured interfacial viscosity demonstrates the utility of the
inherent combination of rheometric measurement with simultaneous imaging for
the contactless geometry. It also aligns with previous reports; for example,
Reynaert, Moldenaers, and Vermant [56] have shown that the complex surface
viscosity magnitude increases with the surface fraction of weakly aggregated
polystyrene particles at a water-oil interface. They also show that, for
surface coverages below 80%, as studied here, the interfacial viscosity of
aggregated particles at a water-oil interface is lower than that for stable
particles. Note that aggregation _can_ lead to higher interfacial viscosities
at high(er) surface coverage, see also, e.g., Ref. 21.

Notably, the results of varying the thickness of the oil phase also imply that
edge effects, i.e. deviations from our assumed parallel-plate flow field, Sec.
II.5, do not play a substantial role in our setup. This is consistent with the
agreement in the measured linear elastic modulus for PNIPAM-SiO2–laden
interfaces between DWR and contactless methods, Sec. III.1.2. A substantial
part of the top surface in our set-up is an oil-plate interface, but the outer
edge of it is an oil-air interface i.e. the sample cup is
$21\text{\,}\mathrm{mm}$ radius at the top, whereas the parallel-plate
geometry attached to the oil-air interface is $12.5\text{\,}\mathrm{mm}$
radius, leaving a radial gap of $9\text{\,}\mathrm{mm}$ between the parallel-
plate geometry and the inner wall of the PTFE cup. To mitigate any edge
effects, we measure interfacial strains at a distance of about 4 mm $\ll$ 12.5
mm from the rotational axis. The results in Fig. 8, i.e. that the interfacial
viscosities do not differ strongly with oil-phase thickness, imply that edge
effects do not substantially affect our results. However, we should note that
even in the contactless method, at low $\phi$, Fig. 7, such low interfacial
viscosities mean that measurements are at a moderate Bo, around
$\mathcal{O}(10)$. This suggests that for precise and absolute determination
of $\eta^{s}$ in this regime a sub-phase drag correction may still be
necessary.

##### Irreversible effect of shear.

Next, we take advantage of the simultaneous confocal imaging to observe the
evolution in particle assembly upon shearing. Before shearing, the particles
are mostly in a non-close packed arrangement with partial aggregation
($\mathcal{D}=2.49$), which we attribute to attractive capillary and Van der
Waals forces, Fig. 9(a). Upon mild shearing ($\omega\leq 6.3$ rpm), the
arrangement is preserved and only minor changes in aggregation state are
observed, Fig. 9(b) and (c). We see from images taken after high shear is
applied, $\omega\gtrsim 10$ rpm corresponding to
$\sigma^{s}=$6.6\text{\times}{10}^{-7}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$
[Fig. 9(d) and (e)], that the interface changes to an inhomogeneous structure
with most particles forming one large aggregate percolating across the region
imaged combined with an increase in dispersity to $\mathcal{D}=13.5$ in the
final state. Importantly, the aggregation appears irreversible and persists
even when higher shear rates are applied, Fig. 9(f).

Figure 9: Structural evolution of PMMA particles at an oil-water interface
with 31% surface coverage via fluorescent confocal micrographs with increasing
applied rotation rate, $\omega$, and, hence, interfacial stress, $\sigma^{s}$.
Interface corresponding to (orange) dot-dashed line in Fig. 8(a). (a) After
low stress, $\omega=0.05$ rpm. Scale bar
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (b) $\omega=4.0$ rpm. (c)
$\omega=6.3$ rpm. (d) Aggregation threshold, $\omega=10$ rpm or
$\sigma^{s}=$6.6\text{\times}{10}^{-7}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$.
(d) Continued aggregation, $\omega=16$ rpm. (e) Highest $\omega=25$ rpm.

Figure 10: PMMA-laden interface at high surface coverage. (a) Fluorescent
confocal micrographs of PMMA particles (white) at an oil-water interface,
$\phi=56.7\%$. (b) Corresponding strain vs time at low imposed rheometer
rotation rates, $\omega$, see inset legend. $r_{\rm
out}=$6.6\text{\,}\mathrm{mm}$$. $\omega$ applied from
$t\gtrsim$5\text{\,}\mathrm{s}$$ for $120\text{\,}\mathrm{s}$, followed by
$30\text{\,}\mathrm{s}$ further recording. (c) Corresponding high imposed
$\omega$ response.

We have seen that applying shear leads to considerable aggregation in this
system. In order for aggregation to occur the shear force must exceed the
maximum repulsive force between these particles. It has been observed that
these particles have an interaction that can be described by a repulsive
screened Coulomb potential [28]. In order to overcome this repulsion we assume
that they must overcome the maximum repulsive force. This maximum force can be
found to be $9.88\text{\times}{10}^{-13}\text{\,}\mathrm{N}$, where we have
rescaled the parameters from Ref. 28, as in this work we use larger particles.
This force can then be converted to an interfacial stress by dividing by the
particle diameter to give a critical aggregation stress of
$3.3\text{\times}{10}^{-7}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$.
Experimentally, we found that aggregation occurs starting at a stress of
$6.6\text{\times}{10}^{-7}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$, which is
in reasonable agreement. This agreement between experiment and prediction
lends further confidence to the use of Eq. (9) when calculating interfacial
stress in our unique geometry. In order to aggregate, the applied stress must
also overcome the steric repulsion, however the steric barrier is considerably
smaller (for a similar particle) than the electrostatic barrier [28, 57]. Once
these particles come into close contact, attractive capillary forces and/or
van der Waals forces are large enough such that this is irreversible.

####  III.2.2 High surface coverage

Now, we investigate the same PMMA particles at higher surface fractions of
56.7%, Fig. 10(a), where the particles assemble into a percolated aggregated
structure. We observe markedly different behaviour, with elastic behaviour
being evident from the strain vs time plots, Fig. 10(b)–(c). Focussing on the
low stress behaviour, Fig. 10(b), an initial jump to a higher strain is
observed, indicative of an elastic material. There is then some erratic motion
in the direction of shear (i.e. the strain is always positive), indicating
that there is some frustrated motion and rearrangements of the interfacial
structure [58]. While the initial elastic response is difficult to measure
precisely due to background noise in the flow, upon cessation of shear the
interface clearly recoils, allowing the elastic strain to be readily measured
[59]. This statement becomes even more apparent when looking at higher applied
stresses, Fig. 10(c), as at these large stresses the flowing behaviour
completely dominates the strain response and the initial elastic jump is
barely visible in the data. However, once the shear has been stopped, the
elastic recoil is clear. At the end of shear at low rotation rate we sometimes
observe motion, e.g., Fig. 10(b) (solid line), perhaps due to thermal
gradients or air flow — note, however, that the shear rates are small.

At higher $\phi$ we observe a more complex rheological response. By plotting
the plastic response, stress vs shear rate, we can infer that the particle-
laden interface behaves as a yield stress fluid, Fig. 11(a), as has been
observed previously using the DWR geometry [14]. By fitting a Bingham plastic
model, Eq. (11), we measure a yield stress of
$1.05(15)\text{\times}{10}^{-7}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$. The
effective interfacial viscosity
($2.16(14)\text{\times}{10}^{-5}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}\text{\,}\mathrm{s}$)
is, as expected, larger than that measured at the lower $\phi$, Fig. 8. We
feel that this is an appropriate model, as the parameter which we are most
interested in comparing to data in the published literature is the yield
stress. The measured $\sigma^{s}_{y}$ is, however, an order of magnitude lower
than the yield stress quoted in Ref. 14. As close agreement between the
techniques is found for a PNIPAM-SiO2–laden interface, Fig. 5(c), this
suggests that the different surface coverages may not be comparable, 56.7%
here vs 74% in Ref. 14. Moreover, there is a difference in PMMA stabilizer,
poly(12-hydroxystearic acid) in Ref. 14 and poly(lauryl methacrylate) here,
though that only makes a small difference in contact angle and a relatively
small difference in interaction potential [28].

When plotting the elastic response at higher $\phi$, stress vs recoverable
strain [Fig. 11(b)], the response is initially linear, with $\sigma^{s}$ and
$\gamma^{s}_{\rm rec}$ proportional up to a strain of 0.03. We fit a linear
dependence of the elastic strain response to the imposed shear stress in the
low-strain regime [inset (orange) points]. This modelled Hookean behaviour
gives us a shear modulus of
$3.16(16)\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$; this
is a factor $10\times$ lower in stiffness compared to the PNIPAM-SiO2
interface, Fig. 5(a), and over a smaller linear region, leading to a far
weaker interface ($\sim 30\times$). Previously reported measurements on a
similar particle-laden interface [14] found interfacial moduli of
$\sim$2\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$, which
is consistent with our measurements here being at a slightly lower surface
coverage, similar to the difference in $\sigma^{s}_{y}$.

Figure 11: Rheological behaviour of an oil-water interface laden with PMMA
particles at $\phi=56.7\%$. (a) Viscous response to applied stress, i.e. after
initial elastic response, but before recoil. Points, data; line, fit to
Bingham fluid model, Eq. (11). (b) Elastic response measured from recoil.
Points, data; line, linear elastic fit to low strain, $\leq 0.03$ (orange and
inset),
$G^{s\prime}=$3.16(16)\text{\times}{10}^{-6}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$.

Finally, we look at the high-stress elastic response, which suggests a complex
structural evolution. At $\gamma^{s}_{\rm rec}\gtrsim 0.03$, Fig. 11(b)
(blue), the recoverable strain initially remains unchanged with $\sigma^{s}$,
no longer increasing linearly. The shift in response at
$\sigma^{s}\\!\approx\\!$1\text{\times}{10}^{-7}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$$,
correlates well with the yield stress, Fig. 11(a). So, just above
$\sigma^{s}_{y}$ a fixed maximum $\gamma^{s}_{\rm rec}$ can be stored, similar
to the PNIPAM-SiO2 interface, Fig. 5(a). However, as $\sigma^{s}$ increases
further, $\gamma^{s}_{\rm rec}$ begins to increase more rapidly, i.e. strain
softening. Qualitatively, this aligns with a variety of literature results.
For example, Reynaert, Moldenaers, and Vermant [56] measured the surface
elastic modulus for polystyrene particle aggregates at a water-oil interface,
which decreased with strain amplitude. Zhang  _et al._ [60] used large
amplitude oscillatory strain rheology and observed strain softening for weakly
attractive silica nanoparticles at an air-aqueous interface. Finally, Orsi
_et al._ [61] used an interfacial shear rheometer on gold nanoparticles at an
air-water interface and also observed strain softening, which they attributed
to breaking of weak bonds in a 2D gel. This suggests that the interface
evolves above yielding, notably, in the stress range for aggregation at low
$\phi$, $3.3\text{\times}{10}^{-7}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$,
which should be $\phi$ independent. It is then possible that the strain
softening is driven by aggregation, consistent with aggregation weakening
interfaces at all but the highest $\phi$ (and lowering $\eta^{s}$, Sec.
III.2.1).

###  III.3 Limits and comparison of techniques

Our results, and the comparison to 1) DWR results using the same PNIPAM-SiO2
system, Fig. 5, and 2) similar colloidal particle laden interfaces using a DWR
[14] or magnetic rod [56], suggest that our setup represents a useful addition
to the field of interfacial rheology. In comparable situations, our results
are highly consistent with results using conventional probes directly attached
to the interface, e.g., the elastic modulus of the PNIPAM-SiO2 interface, or
within expected variation due to differing methods or interface preparation,
i.e. the yield stress for both a PNIPAM-SiO2 or PMMA-particle laden system.
Crucially, our setup allows us to both measure viscosities at lower $\phi$
than have been previously observed using the DWR, and to probe static yielding
at lower stresses. The lower stress limit can be estimated using
Bo${}^{*}\approx 10$, Eq. (10), alongside an estimate of the minimum
interfacial shear rate set by the imaging setup resolution. With a minimum
resolvable strain of $\gamma_{\rm rec}^{s,\min}=0.002$, Fig. 5(a), over a
$120\text{\,}\mathrm{s}$ experiment the minimum shear rate is
$\dot{\gamma}^{s,\min}\sim$2\text{\times}{10}^{-5}\text{\,}{\mathrm{s}}^{-1}$$,
or a rotation rate $\omega_{i}^{\min}=\dot{\gamma}^{s}r_{\rm
out}/(r_{r}-r_{\rm
out})\approx$2.5\text{\times}{10}^{-5}\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$.
For pre-factors in Eq. (10) $\approx 1$, this sets $\omega^{\min}\approx
0.002$ rpm. The minimum rotation rate then sets a lower stress limit, Eq. (9),
$\sigma^{s,\min}=\mathcal{O}(${10}^{-8}\text{\,}\mathrm{Pa}\text{\,}\mathrm{m}$)$,
comparable to more sensitive interfacial techniques, e.g., a micro-needle [24,
44]. To specifically probe low stresses, $\dot{\gamma}^{s,\min}$, and so
$\sigma^{s,\min}$, could be further optimised by using a high magnification or
longer imaging period, together with minimisation of noise (e.g., thermal
gradients and vibrations). Most remarkably, our contactless technique retains
the maximum stresses, Fig. 5(c), attainable for a DWR using a closed feedback
loop [44], and hence has a dynamic range that spans the majority of
interfacial shear rheometry methods. This wide range of measurable stress
combined with in situ determination of interfacial characteristics, either
surface pressure via a Wilhelmy plate or surface fraction via imaging, opens
up the contactless technique to multiple future applications.

##  IV Conclusion

In this work, we have developed a contactless method to perform interfacial
shear rheology on liquid-liquid interfaces without an interfacial geometry.
The shear is applied to the continuous phase using a rotational rheometer and
indirectly deforming the interface and the surface response is measured via
confocal microscopy, either of a fluorescent particle-laden interface or via
tracer particles embedded in the interface, enabling the measurement of a
broad range of interfaces formed from, e.g., proteins, polymers or molecular
surfactants [8]. While we use a confocal microscope and stress-controlled
rheometer, the same results should be achievable using any fixed-rate motor
and reflection or fluorescence microscopy with sufficient resolution and frame
rate. This enhances the applicability of this method as only relatively common
equipment is required.

The method has been verified using a PNIPAM-SiO2–laden interface measured
using both our novel contactless geometry and a conventional DWR method, with
equal elastic moduli found and comparable yield stress values. Our contactless
setup allows us to both measure interfacial viscosities at lower surface
fractions than have been previously observed using the DWR, owing to the high
sensitivity achieved by having no probe attached directly to the interface,
while maintaining the ability to apply large interfacial stresses.
Additionally, we have linked the rheological behaviour to the structural
behaviour of PMMA particle interfaces with different initial conditions. At
low surface coverage, the interface behaves as a two-dimensional Newtonian
fluid and is subject to aggregation above a certain shear threshold. At higher
surface coverage the interface begins to behave as an elastic sheet with a
measurable shear modulus, up to a yield stress where the interface begins to
flow. In addition, our results suggest that both surface coverage and
interfacial particle aggregation state affect the rheology of the interface,
in line with results in the literature.

This work has focussed on the motion of the particles in the plane of the
interface under steady shear. As the setup presented here does not have a
probe attached to the liquid interface, the effect of the interface on how
shear is propagated from the oil to the water phase can now be studied. This
would facilitate observation of how the inside of an emulsion droplet is
influenced by shear of the continuous phase, thereby greatly increasing the
understanding and predictability of the flow behaviour of these systems, which
are encountered ubiquitously in many formulation applications.

## Acknowledgments

IM acknowledges studentship funding from the EPSRC Centre for Doctoral
Training in Condensed Matter Physics (CM-DTC, EP/L015110/1). SB acknowledges
studentship funding from the EPSRC Centre for Doctoral Training in Soft Matter
and Functional Interfaces (SOFI-CDT, EP/L015536/1). MR acknowledges funding
from the Marie Sklodowska-Curie Individual Fellowship (Grant No. 101064381).
The authors acknowledge R. O’Neill, R. Van Hooghten, Damian Renggli and Jan
Vermant for useful discussions, A. Garrie for the PTFE cup and aluminium ring,
J. Royer for assistance with the rheoimaging setup, and M. Hermes for the
velocimetry C code. For the purpose of open access, the authors have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising from this submission.

## Author declarations

### Conflicts of Interest

The authors declare that they have no conflicts of interest.

### Data Availability

The data that support the findings of this study are openly available in
Edinburgh DataShare at https://doi.org/10.7488/ds/3759. *

##  Appendix A Surface Coverage Measurements

As small differences in surface fraction can cause significant differences in
rheological response, and surface fractions for non-close packed PMMA
monolayers formed by sedimentation are challenging to reproduce, the surface
coverage $\phi$ is determined for each interface prepared. First, microscopy
images are analysed, the magnification level being a balance between
individual particle resolution, which improves upon magnification, and
statistics of particle counting, which decreases upon magnification. Examples
are shown in Fig. 8(b). Surface fractions are quoted in terms of a pixel
counting method, as described in the main text. To determine how accurately
surface coverage can be defined, we compare the pixel fraction method with a
particle counting method. Here, we can count the number of particles and use
knowledge of particle size and image size to calculate surface fraction. For
the sample in Fig. 7(a), measurements of surface fraction via the pixel
fraction and the particle counting methods give respectively 31.0% and 23.0%
while for the sample in Fig. 10(a) these give respectively 56.7% and 46.1%.

Note the significant difference in these measurements, highlighting the
challenge in defining the surface coverage. With perfect particle resolution,
the particle counting method should yield the correct answer for that
particular region of the interface, however, perfect particle resolution is
rarely achieved (especially at high $\phi$), for instance because an aggregate
could be mistaken for one particle. The pixel fraction method is also flawed
in that it assumes a direct match between area of emitted light with area of
the particle. This however is not true due to the point spread function of the
imaging setup, a question over whether the particles are exactly in the focal
plane, and the brightness of the fluorophore itself, among other
considerations. Comparing these images with other, similar images taken using
the same imaging setup allows us to have an estimate for the surface fraction
and certainly allows us to observe trends in flow behaviour as surface
fraction changes.

It is also worth noting in Fig. 10(a) there appears to be two types of
particle with different intensity levels. There are a few possible reasons:
firstly, dispersity in particle size, fluorophore intensity, or contact angle
[62] may lead to this effect. This aligns with some particles in Fig. 7(a)
appearing smaller but with less appreciable change in intensity, presumably as
the excitation signal is at saturation in these imaging conditions. Secondly,
the particles, while in close contact, may be pushed out of the surface
leading to variations in their vertical $z$-position. This, however, is
unlikely as the $z$ resolution is $\gg$ particle diameter [63],

$\delta z=\frac{0.88\lambda_{\rm exc}}{1-\sqrt{(n^{2}-{\rm
NA}^{2})}}=$9.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$,$ (12)

for excitation wavelength $\lambda_{\rm exc}=$488\text{\,}\mathrm{nm}$$, dry
objective refractive index $n=1$, and numerical aperture ${\rm NA}=0.3$.
Finally, aggregates might cause a (local) curvature in the liquid-liquid
interface, which could lead to some particles remaining interfacial but moving
relative to the imaging plane, which would result in particles appearing
smaller and/or less bright.

These possible artefacts highlight the challenge in precisely determining the
surface fraction from images. However, the pixel counting method gives a
consistent result that allows a systematic comparison with increasing surface
coverage of particles, Sec. III.2, and a reasonable comparison with the
literature results of e.g., Ref. 14.

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