# Fluidisation of yield stress fluids under vibration

Ashish Garg1,2 Nico Bergemann2,3 Beccy Smith4 Matthias Heil2,3 Anne Juel∗1,2
1Department of Physics & Astronomy, The University of Manchester, Oxford Road,
Manchester M13 9PL, UK;  2Manchester Centre for Nonlinear Dynamics, The
University of Manchester, Oxford Road, Manchester M13 9PL, UK;  3Department of
Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
4Mondelez International, Bournville Place, Birmingham B30 2LU, UK.

###### Abstract

Motivated by the industrial processing of chocolate, we study experimentally
the fluidisation of a sessile drop of yield-stress fluid on a pre-existing
layer of the same fluid under vertical sinusoidal oscillations. We compare the
behaviours of molten chocolate and Carbopol which are both shear-thinning with
a similar yield stress but exhibit very different elastic properties. We find
that these materials spread when the forcing acceleration exceeds a threshold
which is determined by the initial deposition process. However, they exhibit
very different spreading behaviours: whereas chocolate exhibits slow long-term
spreading, the Carbopol drop rapidly relaxes its stress by spreading to a new
equilibrium shape with an enlarged footprint. This spreading is insensitive to
the history of the forcing. In addition, the Carbopol drop performs large-
amplitude oscillations with the forcing frequency, both above and below the
threshold. We investigate these viscoelastic oscillations and provide evidence
of complex nonlinear viscoelastic behaviour in the vicinity of the spreading
threshold. In fact, for forcing accelerations greater than the spreading
threshold, our drop automatically adjusts its shape to remain at the yield
stress. We discuss how our vibrated-drop experiment offers a new and powerful
approach to probing the yield transition in elastoviscoplastic fluids.

##  I Introduction

Yield-stress fluids are amongst the most common and versatile materials in
everyday life, ranging from food and healthcare products to concrete and crude
oil. Their relevance extends from the geophysical scale, e.g., mudslides and
lava flow Balmforth et al. (2014) to the microscale where, e.g., self-
assembled peptide gels are used as a vehicle for targeted drug delivery Gao et
al. (2017). These materials all exhibit a transition from solid to fluid-like
behaviour when subjected to stresses exceeding the material’s yield stress but
they may have widely different mesostructures, e.g. microgels such as
Carbopol, suspensions, emulsions Bonn et al. (2017). Industrial processing of
yield-stress materials often involves their temporary fluidisation under
mechanical stress in order to facilitate moulding, e.g., of chocolate or
concrete, and bottling and packaging, e.g. cosmetics and food products Coussot
(2005). In chocolate manufacture, vibration is routinely imposed at key stages
along the production line in order to shape or ensure uniform distribution of
the material Chevalley (1975); Bergemann (2015). However, vibrational
parameters such as amplitude, frequency and direction of forcing are typically
selected empirically. Thus, there is scope to optimise this process by gaining
a fundamental understanding of the mechanics of fluidisation under vibration.
In this paper, we study a configuration relevant to chocolate production
lines, namely the fluidisation under vertical oscillations of a drop of yield-
stress fluid initially at rest on a thin layer of the same material.

Thus far, the study of yield-stress fluid flows has focussed mostly on uniform
flows in rheometric and conduits geometries, flows in non-uniform vessels of
varying cross-section, transient flows such as gravity currents and spin-
coating processes and some elongational flows involving droplet formation
Coussot (2014). Thus, literature on the fluidisation of yield-stress fluids
under vibration is scarce. Flow enhancement and liquefaction of pastes and
complex fluids in vibrated pipes have been explored Deysarkar and Turner
(1981); Deshpande and Barigou (2001) and the effect of vertical oscillations
on a layer of complex fluid has generated growing interest in recent years.
For example, a signature of elasticity was identified in the viscoelastic
Faraday problem using a wormlike micelle solution Ballesta and Manneville
(2005). Convection-driven heap formation and cracking was observed in dense
slurries of bronze microspheres Schleier-Smith and Stone (2001) while
persistent holes were found to form in vibrated layers of cornstarch or glass
sphere suspensions Merkt et al. (2004). In yield-stress materials, Shiba et
al. (2007) observed convective motion in drops of shear-thinning polymer gels
when driven by vertical oscillations above a critical acceleration, which
resulted in organised patterns of periodic deformation of the drop. The onset
of this convective motion is governed by a constant ratio, marginally greater
than unity, of the vibrational stress due to inertia to the yield stress of
the material. Wolf et al. (2015) generated regular patterns of holes in a more
extended layer of Carbopol microgel subject to vertical vibration, which
persisted when the forcing was discontinued, returning the layer to a solid
state. These spatial features suggest that different modes of deformation of
the free-surface may be accessed depending on the vibrational parameters. The
mechanics of yielding under vibration, however, remain poorly explored.

In the last decade or so, intense rheological research coupled to flow
visualisation has provided new insight into yielding from a microscopic point
of view Bonn et al. (2017), and shown that the increase of stresses in the
material towards the yield threshold leads to its restructure, indicated by a
plethora of intriguing phenomena such as unsteady flows, creep, transient
shear-banding and slip at solid boundaries Balmforth et al. (2014); Divoux et
al. (2011); Coussot (2014). Understanding the detailed mechanics of yielding
is key to the development of accurate constitutive models, which can in turn
be applied to predict flow in a variety of practically-relevant
configurations. Such models may be derived from the microscopic dynamics
Fielding (2020) or the relationship between stress, strain and rate of strain,
formulated at a macroscopic scale to capture the mechanics observed
experimentally. Arguably the most successful constitutive model to date is the
Saramito model Saramito (2009), which treats the material as a linear
viscoelastic solid (with a characteristic relaxation time) for stresses below
yield and an elastoviscoplastic fluid with a Herschel-Bulkley viscosity above
yield. Direct experimental evidence of the role of viscoelasticity around the
yield point has only emerged recently Balmforth et al. (2014); Dinkgreve et
al. (2017); Ewoldt et al. (2008); Luu and Forterre (2009); Piau (2007) and
views diverge on whether a linear-to-nonlinear viscoelastic transition occurs
prior to reaching the yield threshold Fernandes et al. (2017).

In this paper, we investigate fluidisation of yield-stress materials under
vibration by comparing the behaviours of molten chocolate and Carbopol
microgel, which are both shear-thinning yield-stress fluids with the same
yield stress but exhibit widely different elastic properties. We perform
experiments on drops initially at equilibrium under gravity on a layer of the
same fluid, by oscillating the substrate vertically. Below the onset of
spreading, the chocolate drop remains at rest in the frame of reference of the
oscillating substrate whereas the Carbopol drop exhibits periodic viscoelastic
deformations at the frequency of the forcing. Although both chocolate and
Carbopol drops fluidise for similar values of the forcing acceleration, we
observe qualitatively different spreading behaviours. Whereas in the chocolate
drop an initial phase of rapid spreading is followed by slow, long-term
dynamics, the Carbopol drop swiftly spreads to a new equilibrium shape about
which it continues to oscillate viscoelastically at the forcing frequency,
featuring large-amplitude stretching and compression of the drop. This results
in much reduced spread of the Carbopol drop compared to the chocolate drop.

We also find that the threshold for the onset of spreading is dependent on the
process by which the initial drop shape is obtained. However, for each
subsequent increase in forcing acceleration, the vertically oscillated drop
automatically adjusts its shape by spreading in order to relieve its stress so
that it remains at the yield threshold. This implies that compared to the
standard procedures used for oscillatory rheometry Hyun et al. (2011) – the
most widely used methodology for the characterisation of non-Newtonian fluids
– our experimental setup provides unique access to the effects of nonlinear
viscoelasticity near the yield threshold. This is difficult in oscillatory
shear rheometry because for yield-stress fluids that are capable of undergoing
large viscoelastic deformations, the periodic shear strain imposed on the
sample can be absorbed in viscoelastic or plastic deformations. Hence the
oscillatory nature of the forcing ensures that even plastic deformations are
effectively reversed over each cycle of the forcing. Our measurements suggest
a reduction in the storage modulus of Carbopol at the yield point with
increasing forcing, which concurs with rheometric measurements Fernandes et
al. (2017); Varges et al. (2019); Giuseppe et al. (2015). We demonstrate that
our vibrated-drop setup offers a new and powerful experimental approach to
probing the yield transition in elastoviscoplastic fluids.

The outline of the paper is as follows. The experimental methods used to
create drops, oscillate them and image them are described in §II. Results are
presented in §III where we compare the behaviours of chocolate and Carbopol
drops under vibration in §III.1. We show that both materials fluidise at a
critical forcing acceleration in §III.2. In §III.3 we demonstrate that the
viscoplastic spreading of Carbopol is insensitive to its forcing history and
in §III.4, we investigate the viscoelastic oscillations of the Carbopol drop.
We summarise our findings in §IV and discuss the potential for our vibrated-
drop setup to act as a rheometer providing elastoviscoplastic material
properties near the yield point.

##  II Experimental methods

A schematic diagram of the experimental system is shown in figure 1. A drop of
yield-stress fluid is at rest on a thin layer of the same material contained
within a circular trough on a rigid, horizontal substrate. We oscillate the
substrate sinusoidally so that the vertical displacement of the platform is
$z(t)=A\sin(2\pi ft)$, where $A$ and $f$ are the amplitude and frequency of
oscillation, respectively, and we monitor the evolution of the drop with side
and top-view cameras.

Figure 1: Schematic diagram of the experimental setup. The vertical
displacement of the oscillating platform from its equilibrium position is
given by $z(t)=A\sin(2\pi ft)$.

###  II.1 Materials, drop deposition and substrate preparation

We studied two yield-stress fluids: (i) a commercially available, tempered
milk chocolate with a fat content of 30% (in molten form), and (ii) Carbopol
Ultrez-21 microgels (Lubrizol) with different concentrations.

Materials |  $\rho$ (kg $m^{-3}$) |  $\tau_{y}$ (Pa) |  $K$ (Pa sn) | $n$ |  $G$ Pa |
---|---|---|---|---|---|---
Chocolate | 1,247 | $36.1\pm 0.6$ |  $8.7\pm 0.02$ | $1$ | $\sim 44,000$ |
Carbopol (2.2 g/L) | 1,020 | $35\pm 5$ |  $8.2\pm 2.0$ | $0.49\pm 0.02$ | 350 |
Carbopol (3 g/L) | 1,030 | $60\pm 6$ |  $12\pm 3$ | $0.50\pm 0.02$ | 450 |
Carbopol (6 g/L) | 1,050 | $120\pm 10$ |  $17\pm 4$ | $0.51\pm 0.02$ | 700 |
Table 1: Density and rheological properties of the commercially available,
tempered milk chocolate with 30% fat content, and Carbopol microgels of
different concentrations. The yield stress $\tau_{y}$, the power index $n$ and
the constant $K$ were obtained by fitting a one-dimensional Herschel-Bulkley
model of the form $\tau=\tau_{y}+K|\dot{\epsilon}|^{n}$ ($\tau\geq\tau_{y}$)
Herschel and Bulkley (1926) to flow curves measured experimentally. In the
case of chocolate, the coefficients were obtained from a two-parameter fit to
the Bingham model ($n=1$) Bergemann et al. (2018). $G$ is the elastic shear
modulus of the material obtained with constant-shear rheometry for
$\tau<\tau_{y}$ in chocolate by Bergemann et al. (2018). For Carbopol we list
the values measured by Giuseppe et al. (2015); Garg et al. (2020) with
oscillatory shear rheometry in the limit of an approximately elastic response
($\tau\ll\tau_{y}$).

The chocolate was tempered using a Model Prima tempering machine (FBM, Italy)
which delivered chocolate at its outlet at a temperature of $(28.2\pm
0.3)^{\circ}$C. The temper level was determined prior to deposition of each
drop onto the substrate, using an EXOTHERM 7400 temper meter (Systech
Analytics SA, 170 Switzerland). The tempered chocolate has been extensively
characterised by Bergemann et al. (2018), who performed rheological
measurements using a Brookfield R/S-Plus (SST) rheometer with a four-bladed
vane spindle rotating in a large-diameter cylindrical glass beaker. In that
study, shear was applied cyclically by successively increasing and decreasing
the angular velocity or the torque with constant rate. Values of angular
velocity and torque were converted into shear rate $\dot{\epsilon}$ and stress
$\tau$, respectively. The dynamic yield stress $\tau_{y}$ was determined as
the value of decreasing stress at which the fluid came to rest. For
$\tau\geq\tau_{y}$, the resulting flow curve of stress as a function of strain
rate was found to be best fitted by a two-parameter Bingham model (see table 1
for parameter values). For $\tau<\tau_{y}$, the material deformed elastically,
and the measured elastic modulus was found to be moderate ($G\sim 700$ Pa)
prior to initial fluidisation, but increased to $G\sim 44,000$ Pa in
subsequent cycles, indicating the mesoscopic reorganisation of the material
following fluidisation.

We prepared three Carbopol solutions with different concentrations by adding
Carbopol Ultrez-21 powder in concentrations of 2.2 g/L, 3 g/L and 6 g/L to a
0.021 Molar NaOH solution ($pH=12.3$) at the laboratory temperature of 20 oC.
The powder was initially dissolved by vigorous stirring with an electric whisk
for a few minutes, followed by intermittent manual stirring over a period of
$2-3$ hours in order to achieve homogeneous solutions. The resulting Carbopol
solutions had $pH\simeq 8.0\pm 0.5$. They were each placed in an airtight
container and rested for at least 3 days before they were used in the
experiments. Prior to experiments, we performed rheometric tests analogous to
those reported by Bergemann et al. (2018) for tempered chocolate, which we
present in Appendix A. They were carried out repeatedly over a period of
several months and did not reveal measurable aging of the material, consistent
with previous studies Varges et al. (2019); Giuseppe et al. (2015). The flow
curve was accurately captured by a Herschel-Bulkley constitutive relation and
the values of the fitting parameters are listed in Table 1.

The tempered chocolate and the 2.2 g/L Carbopol solution have a similar yield
stress of $\tau_{y}\simeq 35$ Pa. Table 1 also gives the values of the elastic
shear modulus below yield. For Carbopol, we list values from Giuseppe et al.
(2015); Garg et al. (2020) obtained with oscillatory shear rheometry in the
limit of an approximately elastic response for different Carbopol grades,
which range from $G\sim 350$ Pa to 700 Pa. This means that the elastic shear
modulus of chocolate is approximately two orders of magnitude larger than that
of the 2.2 g/L Carbopol solution.

Figure 2: Schematic diagram of the substrate assembly and illustration of the
procedure used to deposit the drop.

A schematic diagram of the experimental apparatus used to prepare the drop is
shown in figure 2(a). A square Perspex substrate plate of width 105 mm was
secured to a Perspex base plate with three finger-tight nylon screws. It
featured a circular trough with a diameter of $60.5$ mm and a depth of $1$ mm
where a uniform film of yield-stress material of depth $0.9\pm 0.1$ mm was
deposited prior to experimentation by initially overfilling the trough and
then slowly scraping off the excess fluid with a square-edged ruler at an
angle of around $30^{\circ}$, taking care to minimise the washboard
instability at the free surface Hewitt et al. (2012). The drop of yield-stress
fluid was deposited onto this thin layer by extrusion from a standard 10 mL
plastic syringe with a nozzle enlarged to an inner diameter of $8\pm 0.05$ mm.
To ensure reproducible deposition, we placed the syringe inside a vertical,
tightly-fitting, removable holder which was rigidly mounted on three vertical,
threaded poles above the substrate plate. We ensured the axisymmetry of the
deposited drop by levelling the entire assembly to less than $0.5^{\circ}$
from the horizontal using a digital inclinometer.

The syringe was partially filled with $4.5\pm 0.3$ mL of yield-stress fluid,
taking care to avoid trapping any air bubbles, and placed in the holder
(figure 2(a)) so that the end of the nozzle was 12 mm above the substrate. The
plunger of the syringe was then slowly pushed down manually for approximately
3 seconds to empty the syringe. The material spread onto the substrate
yielding a drop at equilibrium under gravity following deposition (figure
2(b)); see §II.4. Finally, the deposition assembly holding the syringe was
removed from the apparatus, which was then ready for the vibration experiments
(figure 2(c)).

###  II.2 Oscillatory forcing

We used two experimental rigs to vibrate our sessile drop, as shown
schematically in figure 3. In Rig 1 the rotation of a DC brushless servo motor
(McLennan M644CI500L) is converted to sinusoidal translation motion via the
Scotch yoke mechanism (figure 3(a)), while in Rig 2 oscillations are applied
with an electromagnetic shaker (LDS V455) (figure 3(b)). Both pieces of
apparatus were bolted onto heavy steel tables with adjustable feet positioned
onto sand contained in four steel cylinders, in order to minimise the transfer
of vibration through the floor. Rig 1 was limited to moderate frequencies and
accelerations ($f\leq 10$ Hz, $A(2\pi f)^{2}\leq 5g$, where $g$ is the
acceleration of gravity), which were sufficient to perform the experiments on
tempered chocolate. In contrast, frequencies and accelerations of up to $f=50$
Hz and $A(2\pi f)^{2}=12.5g$ were applied in experiments with Carbopol using
Rig 2.

Figure 3: (a, b) Schematic diagrams of the two experimental rigs used to
impose the vertical, oscillatory motion of the substrate. (a) Rig 1: brushless
DC motor attached to a Scotch yoke mechanism. (b) Rig 2: Electromagnetic
shaker. (c, d) Time-series of the vertical position of the platform started
from rest showing the different ramp-up procedures available in the two
systems, which are applied over a time-interval $t_{\rm ramp}$ for (c) Rig 1,
(d) Rig 2.

####  II.2.1 Rig 1: brushless DC motor with Scotch yoke mechanism

Rig 1(figure 3(a)) was originally designed for the tempered chocolate
experiments. Thus, the apparatus was enclosed in a temperature-controlled
cabinet which was held at $29\pm 0.5^{o}$C during experiments with tempered
chocolate and at the laboratory temperature of $20.2\pm 0.5^{o}$C during
experiments with Carbopol.

The substrate assembly described in §II.1 was adjustably mounted on three
threaded poles 20 mm above a square aluminium plate (thickness 8 mm and width
250 mm) which was coupled to the vertical shaft of a Scotch yoke mechanism.
The Scotch yoke was driven by a computer-controlled brushless DC motor via a
5:1 planetary gearbox (HPE70-S-5:1). For a constant rotation rate $2\pi f$ of
the motor shaft, the Scotch yoke produced a sinusoidal displacement of the
platform in the vertical direction. The frequency $f$ could be incremented
during an experimental run by varying the rotation rate of the motor but the
amplitude was set manually in the range $6\;\mathrm{mm}\leq A\leq
12.5\;\mathrm{mm}$ (to within of $\pm 0.1$ mm) prior to switching the motor
on. The oscillations of the platform were calibrated to the input signal by
measuring the displacement of the platform with a Linear Voltage Differential
Transformer (LVDT, Solartron, Mach 1) mounted onto the platform, see figure
3(a). Accelerometer data (PCB Piezotronics) recorded at the centre of the
substrate plate indicated negligible harmonic content over the entire range of
frequency investigated ($f\leq 10$ Hz). A camera (Pulnix TM-6740CL) was
mounted on the laboratory wall to capture side view images of the drop which
was backlit by a halogen lamp, see §II.3.

####  II.2.2 Rig 2: electromagnetic shaker

To reach the larger platform accelerations required for the Carbopol
experiments, we used a permanent magnet shaker (LDS, V455) as shown in figure
3(b). The substrate assembly described in §II.1 was adjustably mounted on
three threaded poles 20 mm above a circular aluminium plate (thickness 8 mm
and diameter 250 mm) which was coupled to the shaker. The drop was lit
uniformly by two LED panels, a horizontal panel below the substrate assembly
and a vertical panel mounted on the laboratory wall. It was imaged using
cameras in side-view (Pulnix TM-6740CL) and top-view (DALSA HM1400).

The shaker was driven by a sinusoidal waveform generated in LabVIEW which was
converted to an analog output with a data acquisition board (NI PCI-6221) and
amplified to the power required by the shaker with a linear amplifier (LDS, PA
1000L). As in Rig 1, the oscillations of the platform were calibrated to the
input signal with an LVDT (Solartron, Mach 1) mounted onto the platform, see
figure 3(b). The maximum displacement amplitude imposed by the shaker in our
experiments was $A=9.5$ mm and the maximum frequency was $f=50$ Hz.

The waveform of the platform oscillation was sampled with an accelerometer
(PCB Piezotronics) mounted at the centre of the platform. Fast Fourier
Transform (FFT) of the accelerometer signal gave a percentage power of the
first harmonic relative to the fundamental mode of less than 0.2 % for all
amplitudes if $f\geq 30$ Hz, increasing to approximately 1.5 % for $A=2$ mm
and 3.2 % for $A=8$ mm at $f=10$ Hz. Thus for $f\leq 10$ Hz, we only used this
rig to perform experiments at small amplitudes $0.15\;\mathrm{mm}\leq A\leq
5.5\;\mathrm{mm}$ which were not accessible in Rig 1 and for which the
harmonic power ratio was less than 2%.

####  II.2.3 Start-up and shut-down of the forcing

In each rig, start-up and shut-down protocols were required to allow the
system to reach its set oscillation parameters from rest or when the peak
acceleration was increased or decreased from a previous setting. Typical time
series of the vertical displacement of the platform starting from rest are
shown in figures 3(c, d) for the brushless DC motor (Rig 1) and the shaker
(Rig 2), respectively. The shut-down procedure is not shown but was the same
as the start-up procedure in reverse. In each graph, the duration of the
start-up phase is indicated by $t_{\rm ramp}$. In Rig 1, the amplitude is
constant and the rotation rate increases with an imposed acceleration of
either $20\pi$ rad s-2 (chocolate experiments) or $40\pi$ rad s-2 (Carbopol
experiments) towards its set frequency $f$ so that the ramping time is either
$t_{\text{ramp}}=0.1f$ s or $t_{\text{ramp}}=0.05f$ s. In figure 3(c), the
imposed acceleration is $40\pi$ rad s-2 and $f=10$ Hz so that
$t_{\text{ramp}}=0.5$ s. In Rig 2, a fixed frequency was imposed by the input
signal but it was necessary to ramp up the platform oscillation amplitude in
order to avoid damaging the shaker. We followed the manufacturer’s
specifications and linearly increased the voltage to the amplifier until the
value associated with the chosen parameter values ($f$, $A$) was reached,
which led to ramp-up times $t_{\text{ramp}}\leq 0.8$ s. In figure 3(d), the
oscillating platform reaches $A=9$ mm in $t_{\text{ramp}}=0.3$ s.

###  II.3 Image analysis

We used the top-view camera to check the axisymmetry of the drops with a
resolution of 13.4 pixels/mm. The mean ratio of minimum to maximum diameter of
the drops presented in this paper was
$D_{\mathrm{min}}/D_{\mathrm{max}}=0.986\pm 0.004$. In some experiments, we
inclined the camera by $30^{\circ}$ to capture the free-surface deformation of
the drop at rates between 20 and 40 frames per second (fps); see figure 9 in
§III.2. The side-view camera was levelled and aligned with the substrate in
its reference position, which was the maximum and mean height in Rigs 1 and 2,
respectively. Vertical cross-sections of the drop were imaged with a
resolution of 11.2 pixels/mm. The maximum difference in viewing angle when the
platform was at its maximum and minimum heights was less than $1^{\circ}$,
which led to differences in the measured height of a sessile drop of $<1$ %.
However, we measured the height of the drop relative to the height of the
substrate in the same vertical plane of view to mitigate the effects of
perspective arising due the variation in the viewing angle. The side-view
camera recorded images at between 100 and 140 fps and we synchronised it to
the drive to record images stroboscopically at a specific phase of the
platform oscillation cycle.

A typical side-view image of a Carbopol drop is shown in figure 4(a). A MATLAB
routine based on the Canny algorithm was used to determine the contour of the
drop and the position of the horizontal top surface of the platform, see
figure 4(b). We measured the diameter $D$ of the drop $5$ pixels above the top
surface of the platform and defined the mean drop height $\bar{H}$ as the
local drop height $h(x)$ integrated over the central half of the drop,

$\bar{H}=\frac{2}{D}\int_{\frac{-D}{4}}^{\frac{D}{4}}h(x)dx.$

This metric was chosen in order to reduce the influence on the drop height of
the central material thread resulting from the deposition process, and allow
comparisons between chocolate and Carbopol, see figure 5. By analysing series
of successive images, we obtained time series of the vertical substrate
displacement $z(t)$, the drop height $\bar{H}(t)$ and the drop diameter
$D(t)$. The error from image analysis was on the order of a pixel, and thus in
the range from $1.0\%$ to $2.5\%$ for the drop height and from $0.5\%$ to
$1.5\%$ for the substrate displacement and drop diameter.

Figure 4: Processing of side-view images: (a) raw image; (b) contour extracted
using MATLAB. $h(x)$ is the local height of the drop, $D$ its diameter and the
mean height $\bar{H}$ is the integral of the drop height for $-D/4\leq x\leq
D/4$.

###  II.4 Initial drop shape

Figure 5: Initial shape of the droplet (a) chocolate, (b) Carbopol 2.2 g/L,
(c) Carbopol 3.0 g/L, (d) Carbopol 6.0 g/L. The processed images show the drop
interface and the platform edge (red line). The pink line is the mean height
of the drop averaged over the central half of the drop. The mean height and
diameter of each drop are: (a) $H_{0}=8.1\pm 0.2$ mm, $D_{0}=30\pm 0.6$ mm;
(b) $H_{0}=9.8\pm 0.3$ mm, $D_{0}=23.8\pm 0.6$ mm; (c) $H_{0}=10.3\pm 0.3$ mm,
$D_{0}=22.4\pm 0.6$ mm; (d) $H_{0}=11.1\pm 0.3$ mm, $D_{0}=19.6\pm 0.6$ mm.

Figure 5 shows the drop shapes associated with each material listed in table
1. We denote the mean height and diameter of the drops measured prior to
imposing the oscillatory forcing by $\bar{H}_{0}$ and $D_{0}$, respectively.
The chocolate drop (a) exhibits a more pronounced slump than the 2.2g/L
Carbopol drop (b), which has a similar yield stress but an approximately 20%
smaller density (see table 1). The reduced spread of Carbopol drops at higher
concentrations is due primarily to the associated increase in yield stress.
The shape of the chocolate drop profile differs significantly from the
Carbopol drops (b–d) in that the thread of material that is pinched off upon
removal of the syringe collapses and merges into the drop, whereas for
Carbopol it remains erect.

##  III Results

###  III.1 Drop spreading under vibration

Figure 6: Evolution of the drop height of (a) chocolate, and (b) Carbopol (2.2
g/L) for different values of acceleration, where the forcing was imposed at
$t=0$. Data for chocolate is shown for $a=0.87g$ ($A=6$ mm and $f=6$ Hz),
$a=1.16g$ ($A=8$ mm and $f=6$ Hz), $a=1.18g$ ($A=6$ mm and $f=7$ Hz),
$a=1.58g$ ($A=8$ mm and $f=7$ Hz), $a=2.06g$ ($A=8$ mm and $f=8$ Hz),
$a=2.58g$ ($A=8$ mm and $f=9$ Hz), $a=2.61g$ ($A=10$ mm and $f=8$ Hz) and
$a=3.26g$ ($A=10$ mm and $f=9$ Hz). Data for Carbopol is for $a=0.35g$ (blue:
$A=5.5$ mm and $f=4$ Hz) and $a=6.40g$ (red: $A=5.5$ mm and $f=17$ Hz). In
(b), the maximum extension and compression are traced with solid lines of the
same colour (envelopes). The mean of the envelopes is shown with a solid black
line. The equilibrium heights $\bar{H}_{\mathrm{eq}}/\bar{H}_{0}$ measured
after interruption of the forcing in both experiments are denoted by dashed
black horizontal lines.

Figure 6 shows the time evolution of the drop height for (a) chocolate and (b)
Carbopol (2.2 g/L) from the instant substrate oscillations were imposed
($t=0$) for different values of the forcing acceleration. For low forcing
acceleration ($a\leq 1.16g$), the chocolate drop remains essentially
undeformed, with $\bar{H}/\bar{H}_{0}$ varying by less than $\pm 0.5\%$. This
is in contrast with the Carbopol drop which deforms periodically with the
substrate oscillations, with amplitudes on the order of 10% of the initial
drop height for $a=0.35g$. Once the forcing was interrupted, the Carbopol drop
returned to its initial height (dashed horizontal line) to within the
experimental resolution, thus indicating the absence of permanent deformation
(spreading). The lack of spreading of the drops under weak forcing suggests
that in this regime both chocolate and Carbopol behave as viscoelastic solids.
The absence of measurable oscillations in the chocolate drop is consistent
with its larger elastic modulus; see table 1.

In chocolate, the decrease of $\bar{H}/\bar{H}_{0}$ with time for $a\geq
1.18g$ indicates the spreading of the chocolate drop. For $a=1.18g$ this
reduction becomes apparent only for $t>6$ s which is much larger than the
start-up time $t_{\mathrm{ramp}}=0.7$ s. Conversely for the largest
acceleration ($a=3.26g$), spreading occurs much sooner (from $t\simeq 1$ s,
which is on the order of the start-up time, $t_{\mathrm{ramp}}=0.9~{}s$). For
all values of $a$, the spreading is sufficiently slow that the drop does not
reach a constant height by the end of the experimental recording of up to $12$
s. However, for $a\geq 2.06g$ this slow long-term spreading is preceded by a
rapid decrease in drop height during which the drop oscillates with the
frequency of the drive. The amplitude of these viscoelastic oscillations
diminishes as the drop spreads and falls below experimental resolution between
$t=4$ and $5$ s. Their transient nature is likely associated with the
mesoscopic reorganisation of the chocolate following fluidisation. This was
already observed by Bergemann et al. (2018) who found a considerable increase
(by a factor of 6) of the pre-yield elastic modulus of the material following
the first cycle of rheometric yield measurements. In the vibrated drop
experiment a larger number of the faster substrate oscillation cycles during
which the drop is momentarily fluidised may be required to achieve a similar
effect. This would be consistent with the observed progressive reduction in
oscillation amplitude. For the largest value of $a=3.26g$, an apparent initial
increase in mean height is visible because the extension of the drop is larger
than its compression during each cycle of the oscillation. Stroboscopic side-
view images of this experiment are shown in figure 7(a) to highlight the
spreading of the fluidised drop. The drop height is reduced by $50\%$ at $t=3$
s, and more than $70\%$ at $t=8$ s. Moreover, the shape of the initial drop
rapidly evolves to a smooth profile of uniformly positive curvature.

Figure 7: Snapshots of chocolate (a) and Carbopol (2.2 g/L) (b) drops
spreading under vertical oscillations, taken with the substrate is in its mean
position. The processed images are similar to those shown in figure 5. (a)
Chocolate: $a=3.26g$, $A=10$ mm, and $f=9$ Hz. (b) Carbopol 2.2 g/L:
$a=3.25g$, $A=5.6$ mm, $f=12$ Hz.

A typical example of the spreading of a Carbopol drop is shown in figure 6(b)
for $a=6.40g$. The drop develops large-amplitude shape oscillations during the
ramp-up of the forcing, before rapidly spreading, while continuing to
oscillate, to reach a time-periodic state of constant mean value (indicated by
the solid black line) for $t\gtrapprox 3$ s. This time-periodic state is
similar to that observed at low forcing ($a=0.35g$), albeit with significantly
larger oscillation amplitude. It indicates that the stress in the drop has
decreased below $\tau_{y}$ so that the drop behaves as a viscoelastic solid.
Upon stopping the forcing, the drop relaxes to a new equilibrium height
indicated by the horizontal dashed line. Figure 6(b) shows that, as for the
chocolate drop, the mean height of the drop is larger than its equilibrium
height because the amplitude of the extensional deformation is larger than its
compressional deformation. This feature is also visible in the ramp-up
oscillations. We performed experiments for 13 values of acceleration up to
$a=8g$, corresponding to frequencies in the range $8\;\mathrm{Hz}\leq f\leq
23$ Hz, and found that although the extent of the spreading increased with
increasing forcing acceleration, the duration of the spreading remained
approximately constant with $t_{\rm spreading}=3.2\pm 0.8$ s.

Figure 7 shows that spreading is considerably reduced in Carbopol compared
with chocolate. For $a=3.25g$, the final height reduction of the Carbopol drop
(after interruption of the forcing) is $12\%$, while in chocolate it exceeds
$70\%$ by $t=8$ s. A further increase of the substrate acceleration to
$a=6.40g$ (not shown) results only in a final height reduction of the Carbopol
drop of $36\%$. Moreover, the Carbopol drop retains its characteristic pointed
shape which suggests only partial fluidisation in contrast with chocolate.

Figure 8: Height of the chocolate drop relative to its initial height
(measured at $t=9.5$ s after imposition of the forcing) as a function of
frequency (a) and acceleration (b). Each data point corresponds to the mean of
at least 3 experiments and the error bar is the standard deviation.
Experiments with different amplitudes of forcing are denoted by different
coloured symbols: $A=6.0$ mm (red square), $A=8.0$ mm (black circle), $A=10.0$
mm (blue triangle). In (b), the vertical dashed black line indicates the
threshold acceleration beyond which axisymmetric spreading occurs. This
threshold was estimated by linear extrapolation of the data for $1.4\leq
a/g\leq 2.5$ to a unit non-dimensional height.

###  III.2 Yield threshold

Figure 9: Equilibrium height of Carbopol (2.2 g/L) drops scaled by the initial
equilibrium drop height as a function of: (a) forcing frequency, (b) amplitude
and (c) acceleration. Measurements were taken after interruption of the
forcing imposed for $t>t_{\rm spreading}$. Each data point corresponds to the
mean of at least 3 experiments and the error bars are the standard deviations.
In (a), the forcing frequency is varied for fixed amplitudes (blue data):
$A=0.15$ mm (hexagon), $A=1.0$ mm (left-pointing triangle), $A=1.5$ mm
(circle), $A=2.5$ mm (right-pointing triangle), $A=4$ mm (diamond) , $A=5.5$
mm (square), $A=12.5$ mm (downward-pointing triangle). In (b), the forcing
amplitude is varied for fixed frequencies (red data): $f=1-4$ Hz (downward-
pointing triangle), $f=10$ Hz (square), $f=13$ Hz (left-pointing triangle),
$f=17$ Hz (circle), $f=27$ Hz (upward-pointing triangle), $f=45$ Hz (diamond).
In (c), different regimes are illustrated by side-view snapshots of the
Carbopol drop. The onset of spreading $a_{c}(Carb)/g=1.59\pm 0.2$ is
calculated by extrapolating the data in the axisymmetric spreading region to a
scaled drop height of unity. A transition to higher wavenumber oscillations
occur for $a/g=8.0\pm 0.2$, and non-axisymmetric spreading and rupture of the
drop sets in for $a/g=10.9\pm 0.3$.

In order to determine the forcing threshold beyond which chocolate and
Carbopol (2.2 g/L) drops spread, we performed experiments for a range of
frequencies and amplitudes of forcing. For chocolate we measured the height of
the drop at $t=9.5$ s after imposition of the forcing, by which time
oscillations had decayed and most of the spreading had occurred. For Carbopol
we measured the final height of the drop by stopping the forcing for $t>t_{\rm
spreading}$, i.e., once the drop had spread to reach a new oscillatory state
of constant mean height; see §III.1.

Results for chocolate are shown as a function of frequency in figure 8(a).
Different coloured symbols are used to indicate the three datasets with
different amplitudes of forcing, which are displaced from each other along the
frequency axis. Figure 8(b) shows that the data collapses onto a master curve
when plotted as a function of the forcing acceleration. For sufficiently small
accelerations, the drop does not exhibit measurable spreading. Once the
forcing acceleration exceeds a threshold value, $\bar{H}(t=9.5\;{\rm
s})/\bar{H}_{0}$ decreases steeply with the acceleration. The threshold value
for the onset of spreading, indicated with a vertical dashed line, was
determined by linearly extrapolating the data for $1.4\leq a/g\leq 2.5$ to
yield a value of $a_{\rm c(choc)}/g=1.28\pm 0.1$.

Similar data is reported for Carbopol in figure 9. The equilibrium height
$\bar{H}_{\mathrm{eq}}/\bar{H}_{0}$ is shown as a function of frequency (a)
and amplitude (b) of forcing, corresponding to series of experiments performed
at constant amplitude by varying the frequency, and at constant frequency by
varying the amplitude, respectively. As for the chocolate experiments,
individual data sets are displaced with respect to each other along the
horizontal axis. Figure 9(c) shows that the data approximately collapses again
onto a master curve when plotted as a function of the forcing acceleration. As
before, the threshold forcing acceleration for the onset of spreading was
determined by linearly extrapolating the data for accelerations in the range
$1.6\leq a/g\leq 8$, yielding $a_{\rm c(Carb)}/g=1.59\pm 0.2$.

Beyond this acceleration, the drop rapidly spread to a new equilibrium shape
about which it continued to oscillate. In fact the experiments were performed
by incrementally increasing either frequency or amplitude of forcing and
recording the new equilibrium at each step by temporarily interrupting the
forcing. For accelerations in the range $1.6\leq a/g\leq 8.0$, the data points
suggest a continuum of decreasing equilibrium heights corresponding to
axisymmetric drops without spatial features. For $8.0\pm 0.2\leq a/g\leq
10.9\pm 0.3$, the drop oscillations remained axisymmetric but gained
additional spatial features by transitioning to higher spatial modes. Finally,
for $a/g\geq 10.9\pm 0.3$, the oscillating drop lost its axisymmetry and
exhibited localised rupture; see images in figure 9(c). We did not investigate
these regimes in further detail.

A summary of all chocolate and Carbopol experiments is provided in figure 10
where the critical forcing acceleration is plotted as a function of yield
stress (see table 1). We note that Carbopol (2.2 g/L) and chocolate, which
have very similar values of the yield stress, also exhibit similar values of
the critical acceleration. We find that the critical acceleration increases
approximately proportionally to the yield stress as shown by the solid blue
line, which is a one-parameter linear fit to the data. This confirms that the
yield stress uniquely determines the onset of spreading in our experiments.
However, the existence of a critical acceleration for the onset of spreading
is in itself significant. If prior to the imposition of oscillatory forcing,
the drop had reached its equilibrium by slumping due to gravity forces alone,
we would expect the maximum stress in the drop to have relaxed to $\tau_{y}$
at equilibrium. Hence, any additional acceleration imposed during the
oscillatory forcing cycle would be sufficient to fluidise the drop.
Observation of a critical acceleration suggests that the initial stress
distribution within the drop must be significantly below yield because an
additional acceleration greater than that of gravity has to be imposed in
order for the drop to spread. This implies that the constant of
proportionality between critical acceleration and yield stress shown in figure
10 is set by the process by which the drop is generated. The vibrated-drop
experiment can be therefore be used to determine the yield stress of other
materials based on measured critical acceleration values provided that the
same drop deposition process is retained and that the initial drop shape are
similar.

Figure 10: Critical forcing acceleration required to spread drops of different
yield-stress fluids (see table 1) as a function of their yield stress. The
solid line indicates a least-square linear fit to the data of the form
$a_{\mathrm{c}}/g\propto\sigma_{\mathrm{y}}$.

###  III.3 Dependence of Carbopol spreading on the history of forcing

Figure 11: Influence of the forcing history on the spreading of a Carbopol
drop (2.2 g/L), starting from the sessile configuration shown in figure 5. The
final forcing is $a=8.00g$ in each panel. This forcing is applied directly in
(a), and following 1, 3 and 6 increments in (b–d). After each period of
spreading, the forcing was interrupted to measure the new equilibrium shape of
the drop (inset images). The envelopes of the drop oscillations are shown with
solid lines of the same colour as the data, while the mean of the envelopes is
shown with a yellow solid line. The successive equilibrium heights of the drop
are indicated with dashed lines. The red arrows highlight that all forcing
protocols result in the same final equilibrium drop heights (and profiles).

We showed in §III.1 that Carbopol drops spread when the value of the forcing
acceleration exceeds a critical threshold. We now turn to investigating the
effect of forcing history upon the final equilibrium shape adopted by the drop
post-spreading.

Figure 11 shows four spreading experiments on a Carbopol drop (2.2 g/L)
starting from the sessile shape of height $\bar{H}_{0}$ shown in §II.4. In (a)
the drop was subjected to a forcing acceleration $a=8.0g$ for the entire
duration of the experiment. In (b–d), this acceleration was reached via 1, 3
and 6 acceleration increments, respectively. The forcing was temporarily
interrupted after each increment in order to record the new equilibrium height
$\bar{H}_{\mathrm{eq}}$ of the drop. Despite these different experimental
procedures, the value of $\bar{H}_{\mathrm{eq}}/\bar{H_{0}}$ following
interruption of the forcing at $a=8.0g$ is the same in all experiments, as
indicated by the dashed horizontal lines on the pink experimental data
(highlighted by red arrows). This indicates that spreading of the drop is not
sensitive to the forcing history. It suggests that upon exceeding the critical
acceleration, the Carbopol drop rapidly reaches a new equilibrium shape about
which it oscillates periodically while the stress remains below the yield
stress. Because this new drop equilibrium is at its yield threshold, any
subsequent increase in forcing acceleration causes the drop to spread further
to another flatter equilibrium shape. Further experiments with forcing
accelerations in the range $1.70g\leq a\leq 9.68g$ and between 1 and 14
acceleration increments (not shown) confirmed that the final equilibrium
height of the drop is indeed independent of the time-history of forcing to
within experimental resolution provided that all experiments start from the
same initial sessile drop shape.

Figure 12: Spreading of a Carbopol drop (2.2 g/L) for $a=4.83g$ ($f=10$ Hz and
$A=12$ mm) when the forcing was applied continuously (a) and intermittently
(b). In (b), the duration of the intermittent bursts of oscillatory forcing
are less than $t_{\mathrm{spreading}}\simeq 3.2$ s required for the droplet to
reach an oscillatory state about a new equilibrium. Each time the forcing is
interrupted, the drop settles into a new equilibrium state. The total duration
of the intermittent oscillations during which spreading occurs in (b) is
approximately equal to the continuous spreading time in (a), with small
effects of start-up and shut-down of the forcing visible in (b).

We also examined the effect of applying forcing in short, successive bursts of
5 periods of oscillation of the substrate. This meant that the forcing was
interrupted before the drop had spread to a new equilibrium shape. The
comparison in figure 12 between continuous forcing (a) and short bursts (b) at
$a=4.83g$ shows that spreading occurs either continuously or intermittently,
but that the drop reaches the same long-term equilibrium (highlighted by the
black arrows) about which the drop can oscillate. Remarkably, the cumulative
time required to spread to the new equilibrium while the forcing is applied is
approximately the same ($\simeq 3$ s) in both cases, although in figure 12(b)
the ramp-up and ramp-down of the forcing result in small alterations of the
forcing signal.

In a final set of experiments, we investigated the influence of the position
at which the substrate came to rest on the equilibrium shape of the drop
reached in figure 12. We imposed approximately two periods of oscillation of
the substrate (with additional ramp-up and down) at $a=4.83g$, starting with
the substrate either in its neutral position, its bottom position or its top
position. We then ensured that the substrate came to rest in each of these
three positions. Figure 13 shows that the ultimate equilibrium shape of the
drop reached after interruption of the forcing was the same irrespective of
the forcing protocol.

Figure 13: Comparison between the drop shape upon interruption of the forcing
and its final equilibrium shape after relaxation. The height of the drop
normalised by its equilibrium height is plotted as a function of the
normalised radius. The blue and red lines show two instantaneous drop shapes
at the end of the substrate motion. They both relax to the same final
equilibrium shape (black).

###  III.4 Viscoelastic oscillations of the Carbopol drop

Figure 14: Time-periodic deformation of a Carbopol drop (2.2 g/L) under
vertical vibration during one period of oscillation $1/f$ for (a)
$a=0.86g<a_{c}$ ($f=5$ Hz and $A=8.5$ mm) and (b) $a=7.11g>a_{c}$ ($f=14$ Hz,
and $A=9$ mm), after the drop has reached a state of constant mean height
following spreading. The processed images show the drop interface and the
platform edge (red line). The dashed blue lines, which indicate the height and
diameter of the initial drop at $t=0$, highlight the compression and extension
of the drop over the period of oscillation.

We have established that upon exceeding a critical forcing acceleration,
Carbopol drops fluidise and rapidly spread until they reach a new equilibrium
drop shape about which they continue to oscillate. These observations suggest
that the change in equilibrium drop shape due to spreading results in a
reduction of the stress within the drop to below the yield stress, so that
Carbopol recovers the behaviour of a viscoelastic solid. In this section, we
investigate the viscoelastic oscillations of the drop about the different
equilibrium states that are reached as the oscillatory forcing is increased.
Figure 14 shows typical Carbopol (2.2 g/L) drop shapes over one period of the
oscillation for $a<a_{c}$ (a) and $a>a_{c}$ following the initial spreading
phase (b). The maximum variation in the height of the drop over the period of
the oscillation increases significantly with the imposed acceleration. Whereas
the height oscillation is barely visible for $a=0.86g<a_{c}$ (figure 14(a)),
when $a=7.11g>a_{c}$ (figure 14(b)) the drop extends significantly above its
equilibrium height (dashed blue horizontal line) and reduces its diameter
below its equilibrium value (e.g., at $t=0.38f^{-1}$). Conversely, during
compression the height change is smaller but an increase in the diameter is
still clearly visible ($t=0.90f^{-1}$). In addition, the phase of the drop’s
response differs for the two values of oscillatory forcing in that the drop
extends vertically with the rising platform for $a>a_{c}$ while it compresses
with the rising platform for $a<a_{c}$.

Figure 15: Time series of the height of a Carbopol drop (2.2 g/L) relative to
its equilibrium height (blue) for increasing values of forcing acceleration.
Each time series represents multiple cycles of oscillation over one period
$f^{-1}$ of the substrate oscillation which is shown in red in the top panel
(in terms of the normalised substrate displacement as a function of
dimensionless time $tf$). The forcing amplitude is $A=5.5$ mm. The frequencies
are (a) $f=3$ Hz, (b) $f=4$ Hz, (c) $f=5$ Hz, (d) $f=6$ Hz, (e) $f=7$ Hz, (f)
$f=8$ Hz, (g) $f=9$ Hz, (h) $f=11$ Hz, (i) $f=12$ Hz, (j) $f=13$ Hz, (k)
$f=14$ Hz, (l) $f=15$ Hz, (m) $f=17$ Hz, (n) $f=19$ Hz and (o) $f=22$ Hz. Note
the increase in vertical axis range in successive rows. Figure 16: Comparison
between the normalised Fast Fourier Transform of the time series of drop
height shown in figure 15 (blue) and the vertical displacement of the platform
(red). Forcing parameters are the same as in figure 15. Figure 17: Lissajous
curves of the data presented in figure 15 obtained by plotting the height of
the drop relative to its equilibrium height as a function of the vertical
displacement of the platform. Forcing parameters are the same as in figure 15.

Figure 15 shows a sequence of time series of the drop height (blue) for
increasing values of the forcing acceleration. Multiple cycles are plotted
over a single period of the sinusoidal forcing, which is shown in red in the
top panel. For the lowest accelerations (a, b), the oscillations of the
viscoelastic solid drop are approximately sinusoidal and lag approximately
$180^{\circ}$ behind the position of the substrate. For $a<a_{c}$, the peak-
to-peak amplitude of the drop oscillations appears to vary only weakly with
forcing acceleration. However, figures 15(c–f) indicate a weakening of the
fundamental mode of frequency $f$, and the appearance of higher frequency
modes in the signal as the forcing acceleration is increased towards the yield
threshold. This change in the drop response is quantified in figure 16: the
normalised Fast Fourier Transforms (FFT) of the time series shown in figure 15
reveal harmonics up to $5f$. Figure 16 indicates that higher harmonics first
appear in (c) and their contribution increases considerably in (d–f) so that
the amplitude of the first harmonic of frequency $2f$ increases and approaches
that of the fundamental mode of frequency $f$ as the forcing acceleration
reaches $a=1.42g<a_{c}$. Although the harmonic content of the signal
progressively decreases as $a$ increases above $a_{c}$, the first harmonic
persists up to large forcing accelerations (g–n) and becomes indistinguishable
from the signal noise only for $a=10.7g$ (o). The rich harmonic content of the
drop response near the yield threshold suggests significant nonlinear
viscoelastic effects in this region.

Returning to the time-series of figure 15, the gradual reduction in the
amplitude of the fundamental mode up to $a=1.42g$ (f) is reversed in (g) (note
the change in the vertical axis range) where it begins to strengthen again but
with a phase-shift of approximately $100^{\circ}$ compared with (a–f). A
pictorial illustration of this evolution is provided by the Lissajous curves
in figure 17, where the drop height relative to its equilibrium value is
plotted as a function of the vertical displacement of the substrate. In (a–c),
the maximum and minimum drop heights occur for the minimum and maximum
vertical substrate positions, respectively. The limit cycle (which appears as
a line in (a) due to the small amplitude of the drop oscillations) then
rotates anti-clockwise with increasing forcing acceleration towards an
approximately horizontal structure (d–g). This rotation continues in (h–l) so
that the maximum drop extension and compression occur near the maximum and
minimum substrate heights, respectively. The strong second harmonic component
of the signal means that the cycle is a distorted figure of eight (g–j), which
evolves into a simple cycle as the second and higher harmonics subside. In
(m,n), the maximum compression of the drop still occurs at the minimum
vertical substrate position but the maximum extension occurs closer to the
neutral substrate position. In (o) the cycle has reshaped so that the maximum
compression of the drop occurs increasingly close to the neutral position of
the substrate and the response of the drop to the forcing is now close to
sinusoidal again.

Figure 18: Dependence on forcing acceleration of (a) the phase difference
between drop oscillations and vertical substrate acceleration, (b) the maximum
extension of the drop. Each symbol corresponds to a fixed amplitude of
oscillation, so that increasing values of forcing acceleration are achieved by
increasing the frequency of forcing. The error bars in (b) indicate the
standard deviation of at least three experiments.

We summarise the dependence of the viscoelastic oscillations of Carbopol drops
(2.2 g/L) on the forcing acceleration in figure 18. Figure 18(a) shows the
phase lag of the fundamental harmonic of the oscillatory drop response
relative to the substrate acceleration, which was determined from the FFT
data. The phase angle increases steeply from values near $0^{\circ}$ to nearly
$180^{\circ}$ on the approach to the yield threshold as previously observed in
figures 15 and 17.

Figure 18(b) shows that the maximum extension of the drop increases
monotonically with forcing acceleration. The maximum compression of the drop
follows a similar relation (not shown). The maximum extension (and the smaller
compression) provide measures of the maximum strain in the drop, by accounting
for the decrease in equilibrium height of the drop which occurs through
spreading as the forcing acceleration is increased.

##  IV Discussion and conclusion

We have studied experimentally the response to sinusoidal vertical
displacement of drops of molten chocolate and Carbopol initially at rest on a
thin layer of the same fluid. These shear-thinning yield-stress fluids have
very different pre-yield elastic moduli, up to a factor 100 larger for
chocolate than for Carbopol, because of their different mesostructures.
Carbopol is also more strongly shear-thinning with a steeper decrease of its
viscosity to lower values than chocolate. We adjusted the Carbopol
concentration so that its yield stress matched that of chocolate and found
that drops of both materials first spread for approximately the same value of
forcing acceleration $a_{c}$. For $a<a_{c}$, the chocolate drop is at rest in
the frame of reference of the oscillating substrate while the Carbopol drop
undergoes viscoelastic stretching and compression periodically with the
forcing. For $a\geq a_{c}$, rapid initial spreading of the chocolate drop
gives way to long-term slower motion, whereas in Carbopol, the drop rapidly
relaxes its stress by spreading to a new equilibrium shape of larger
footprint, which continues to undergo large-amplitude viscoelastic stretching
and compression. Similar viscoelastic oscillations are also observed
transiently in the chocolate drop for sufficiently large forcing but they
become weaker as the drop spreads. This reduction in overall strain is
consistent with an increase of the elastic modulus, which is known to occur in
chocolate over successive fluidisation cycles through mesoscopic
reorganisation of the material Bergemann et al. (2018). The strong
viscoelastic effects observed in Carbopol result in a striking reduction in
spreading: for $a=3.25g$, the total height reduction of the Carbopol drop is
12% compared with 70% for the chocolate drop after 8 s, which then only
spreads weakly at larger times; see figure 6.

The existence of a forcing threshold for the onset of spreading in our
experiment indicates that the initial drop shape at equilibrium under gravity
(reached after deposition from the syringe) has a stress distribution below
the yield stress. We found that the threshold for the onset of spreading is
uniquely determined by a critical forcing acceleration $A(2\pi f)^{2}$ and is
proportional to the yield stress. In practice, this means that our
experimental setup can be used to measure the yield stress of materials
provided that prior calibration of the rate of increase of forcing
acceleration with yield stress is performed with a fluid of known yield
stress. Our threshold criterion differs from that of experiments at much
larger forcing Shiba et al. (2007), where yield phenomena occur above a
critical velocity of forcing, i.e. when the stress $\rho(2\pi fA)^{2}$ exerted
by the substrate due to inertia of the material exceeds the yield stress.

Figure 19: Estimate of the variation of the absolute value of the effective
complex modulus $|G_{\mathrm{eff}}|$ of Carbopol as a function of forcing
acceleration. Figure 20: Estimate of the variation as a function of forcing
acceleration of the effective storage (a) and loss moduli (b), which are the
real and imaginary parts of the effective complex modulus of Carbopol and are
calculated using the data in figure 18.

Remarkably, we found that the spreading of Carbopol drops did not depend on
the history of forcing. We were able to interrupt and resume forcing as well
as increase the forcing acceleration with multiple increments without
affecting the final equilibrium drop shape. If the forcing was discontinued
before a new equilibrium drop shape had been reached, spreading towards this
equilibrium state continued when reapplying the forcing. This confirms that
spreading continues until the stress within the drop has reduced to the yield
stress, which is indicated by a new equilibrium drop shape undergoing large
amplitude viscoelastic oscillations.

Hence, the Carbopol drop (and presumably also the chocolate drop)
automatically adjusts its shape to remain at the yield stress for increasing
values of the forcing acceleration $a\geq a_{c}$. Our experiment is therefore
uniquely taylored to investigate the rheology of the material at its yield
stress. Although our experimental evidence suggests that after each increase
in forcing acceleration the material returns to a viscoelastic solid state
following short transients, we cannot exclude the existence of recirculating
viscoplastic flows within the drop. Shiba et al. (2007) observed free-standing
convection rolls in vibrated drops of a different elastoviscoplastic gel in
addition to viscoelastic oscillations, but the convective flow was associated
with time scales much longer than the period of the forcing. We also find a
complex nonlinear viscoelastic response of the drop to the oscillatory forcing
near $a_{c}$: a steep increase in the phase angle of the fundamental
oscillation mode and oscillations at higher harmonic frequencies.

Increasing the forcing acceleration increases the amplitude of the drop
oscillations about the equilibrium state and thus the strain in the drop,
without exceeding the yield stress. We use the data of figure 18(b) to
estimate the absolute value of the effective complex modulus of the Carbopol
microgel as

$|G_{\mathrm{eff}}|=\frac{\tau_{y}}{(\bar{H}_{\mathrm{max}}-\bar{H}_{\mathrm{eq}})/\bar{H}_{\mathrm{eq}}},$

which is plotted as a function of forcing acceleration in figure 19. We also
decompose the effective complex modulus into the storage and loss moduli
$G^{\prime}_{\mathrm{eff}}$ and $G^{\prime\prime}_{\mathrm{eff}}$ in figure 20
using the phase angle shown in figure 18(a). The most striking feature of
these plots is the monotonic decrease within our parameter range of
$|G_{\mathrm{eff}}|$ and $G^{\prime}_{\mathrm{eff}}$. Remarkably, the values
of $G^{\prime}_{\mathrm{eff}}$ and $G^{\prime\prime}_{\mathrm{eff}}$ follow
similar qualitative trends to the storage and loss moduli of Carbopol
microgels as functions of either imposed strain or stress, measured by
oscillatory-shear rheometry Fernandes et al. (2017); Varges et al. (2019);
Giuseppe et al. (2015). Although the values obtained for $a<a_{c}$
overestimate $|G_{\mathrm{eff}}|$ because the stress within the drop is less
than the yield stress, $|G_{\mathrm{eff}}|$ is of the correct order of
magnitude of hundreds of Pa (see table 1). Hence, our vibrated drop offers a
route to simple measurements of the nonlinear viscoelastic properties of the
material at the yield stress. We also find that the strain in the drop
eventually reaches a maximum with increasing forcing acceleration for $a\geq
10.9g$ where the drop ruptures. We therefore conclude that our vibrated-drop
setup offers unique features to help deepen insight into the rheology of
elastoviscoplastic fluids at their yield threshold and could act as a simple
rheometer for fluids undergoing large viscoelastic deformations. In order to
deepen understanding of the drop behaviour, it would be particularly
interesting to explore whether the experimental dynamics reported in this
paper can be reproduced numerically using an appropriate constitutive model
such as the Saramito Saramito (2009) model extended to include nonlinear
viscoelasticity.

##  Appendix A Quasi-steady shear rheometry

We performed quasi-steady shear-rheometry measurements of the yield stress and
viscosity curves of the Carbopol microgels used in our experiments by applying
the methodology of Bergemann et al. (2018). We used a Brookfield R/S-Plus
(SST) rheometer with a four-bladed vane spindle (blade height of $20\pm 0.01$
mm) rotating in a stationary cylindrical glass beaker of inner radius $42.5\pm
0.25$ mm, filled with Carbopol microgel to a height of $50\pm 2$ mm. We refer
the interested reader to Bergemann et al. (2018) for details of the
experimental protocol and associated parameter values.

In order to measure yield stress, we imposed cycles of linearly increasing and
decreasing torque and measured the angular velocity. The shear stress $\tau$
was readily calculated from the imposed torque and the geometry of the
spindle, with a correction for spindle end-effects applied by adding a virtual
length to actual spindle length Bergemann et al. (2018). Figure 21(a) shows
the measured angular velocity of the spindle as a function of the applied
shear stress for the 2.2 g/L Carbopol microgel. The period of the forcing
cycle was $240$ s and the maximum stress applied in each cycle was 160 Pa. The
error bars correspond to standard deviations over 5 consecutive cycles of 4
repetitions of the experiments.

Figure 21: Measured angular velocity as a function of shear stress in Carbopol
microgel (2.2 g/L). Shear stress was imposed periodically through linear ramp-
up and down phases indicated in (a) by red and blue symbols and in (b) by
circles and squares, respectively. (a) Cycle period of $\tilde{T}=240$ s and
ramping rate $\alpha=1.33$ Pa/s. (b) Ramping rates and cycle periods of
$\alpha=0.83$ Pa/s, $\tilde{T}=240$ s (blue), $\alpha=1.00$ Pa/s,
$\tilde{T}=240$ s (green), $\alpha=1.16$ Pa/s, $\tilde{T}=240$ s (magenta),
$\alpha=2.67$ Pa/s, $\tilde{T}=120$ s (cyan).The data shown corresponds to
mean values indicate of 4 repetitions of individual experiments of 5 cycles,
and the error bars are the standard deviations from the mean. The vertical
black line in (b) indicates the value of the dynamic yield-stress $\tau_{y}$
of the material.

During the ramp-up phase (red symbols), a modest increase of the angular
velocity $\omega$ for $\tau\lesssim 30$ Pa is followed by a steep and smooth
increase beyond this threshold. Large fluctuations in $\omega$ occur for
$\tau\lesssim 30$ Pa, which were absent in chocolate Bergemann et al. (2018).
For $\tau\lesssim 30$ Pa, the Carbopol microgel deforms as a solid, and the
large variability in the rotation rate measurements can be attributed to
viscoelasticity and stick-slip of the Carbopol microgel on the spindle walls
Ovarlez et al. (2013); Balmforth et al. (2014); Dinkgreve et al. (2017);
Birren and Reber (2019). The ramp-down phase (blue symbols) shows a smooth
decrease of $\omega$ which retraces the red curve to within 5%. However, the
blue dataset continually steepens with decreasing stress. Thus, it is
associated with a minimum stress value required for maintaining flow known as
the dynamic yield-stress, $\tau_{y}=35\pm 3$ Pa, which we obtained by
extrapolation of the blue dataset.

Figure 21(b) shows that our results are insensitive to the details of the
ramping cycle. Experiments performed with different rates of change of the
applied stress, $0.83\leq\alpha\leq 2.67$ Pa/s, and cycle periods of either
120 and $240$ s all collapse approximately onto a master curve above the yield
stress. The hysteresis between the ramp-up and ramp-down parts of the forcing
cycle is on the order of the measurement error, which suggests that thixotropy
is negligible within our parameter range. The dynamic yield stress estimated
from all the experiments performed with the 2.2 g/L Carbopol microgel is
$\tau_{y}=35\pm 5$ Pa and the values of yield stress obtained with the same
method for the other Carbopol microgels used in our experiments are listed in
table 1 in the main text.

In order to measure the viscosity curve of the Carbopol microgel post-yield,
we performed cyclic rheometric measurements by imposing either torque or
angular velocity. We converted angular velocity into shear rate by
approximating our rheometric set up to a circular Couette flow geometry. We
solved the associated inverse problem for the unknown shear rate function
$\dot{\epsilon}(\tau)$ numerically, where the yield stress $\tau_{y}$ was
unknown and had to be determined as part of the solution Yeow et al. (2000);
Bergemann et al. (2018). The resulting values of $\tau_{y}$ were within $5\%$
of the values measured directly from figure 21. The benefit of this method is
that it does not require any prior assumption about a specific rheological
model.

Figure 22: (a) Constitutive behaviour of Carbopol microgel (2.2 g/L) during
constant rate ramp-down of either torque or angular velocity. The data is
averaged over 4 experiments of 5 cycles and the error bars indicate the
standard deviation from the mean. Ramping rates were applied in the range
$0.83\leq\alpha\leq 2.67$ Pa/s. Variation with shear rate of (a) the shear
stress, and (b) the viscosity. The blue lines indicate a two-parameter fit to
the data of the Herschel-Bulkley model $\tau=\tau_{y}+K|\dot{\epsilon}|^{n}$
fit, using the value of $\tau_{y}$ previously determined. Inset: Log–log plot
of the same graph.

Figure 22(a) shows that shear stress increases monotonically as a function of
shear rate. Each symbol corresponds to the mean of 4 repetitions of ramp-down
experiments (including 5 consecutive cycles) and the error bars indicate the
standard deviation from the mean. As in figure 21(b), the data is insensitive
to different ramping rates applied in the range $0.83\leq\alpha\leq 2.67$
Pa/s. The Carbopol microgel is fluid during ramp-down and its flow curve of
shear stress as a function of shear rate is accurately captured by a Herschel-
Bulkley model, consistent with the literature (Putz and Burghelea, 2009;
Varges et al., 2019). Carbopol is shear-thinning as indicated by the monotonic
decrease of its viscosity $\displaystyle\mu=\tau/\dot{\epsilon}$ with
increasing shear rate shown in figure 22(b). The straight line on the log-log
plot in the inset indicates the power-law behaviour of the Herschel-Bulkley
model. A two-parameter fit to the Herschel-Bulkley model gave consistency and
shear incidices of $K=8.2$ Pa sn and $n=0.49\pm 0.02$ for 2.2 g/L Carbopol
microgel. Results for different concentrations of the Carbopol microgels are
listed in table 1 in the main text.

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