# Two step micro-rheological behavior in a viscoelastic fluid

Rohit Jain  Both authors contributed equally to this work.  Institute for
Theoretical Physics, Georg-August Universität Göttingen, 37073 Göttingen,
Germany  Félix Ginot  Both authors contributed equally to this work.
Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany  Johannes
Berner  Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany
Clemens Bechinger  Fachbereich Physik, Universität Konstanz, 78457 Konstanz,
Germany  Matthias Krüger  Institute for Theoretical Physics, Georg-August
Universität Göttingen, 37073 Göttingen, Germany

###### Abstract

We perform micro-rheological experiments with a colloidal bead driven through
a viscoelastic worm-like micellar fluid and observe two distinctive shear
thinning regimes, each of them displaying a Newtonian-like plateau. The shear
thinning behavior at larger velocities is in qualitative agreement with
macroscopic rheological experiments. The second process, observed at
Weissenberg numbers as small as a few percent, appears to have no analog in
macro rheological findings. A simple model introduced earlier captures the
observed behavior, and implies that the two shear thinning processes
correspond to two different length scales in the fluid. This model also
reproduces oscillations which have been observed in this system previously.
While the system under macro-shear seems to be near equilibrium for shear
rates in the regime of the intermediate Newtonian-like plateau, the one under
micro-shear is thus still far from it. The analysis suggests the existence of
a length scale of a few micrometres, the nature of which remains elusive.

††preprint: AIP/123-QED

##  I Introduction

Stochastic processes are of general importance, from a fundamental point of
view but also regarding technical and biological applications. As a result,
they have been the subject of intense research over the past years. A simple
example is Brownian motion which has been intensively studied, both in
equilibrium and in presence of various types of external driving, in
experiments and theoretically Seifert (2012); Sekimoto (1998); Dhont (1996).
In particular in case of entirely viscous, i.e. Newtonian solvents the
treatment of such processes is rather straightforward because it can be
described within the framework of a Markovian theory. The corresponding
Langevin equation is then also linear in the sense that the dissipative parts
(i.e., friction), remain linear in the particle’s velocity. To go beyond these
model systems, a variety of nonlinear properties of complex fluids have been
studied Larson (1999). An important example concerns shearing of complex
fluids Gutsche  _et al._ (2008); Wilson  _et al._ (2011); Leitmann and
Franosch (2013); Harrer  _et al._ (2012); Winter  _et al._ (2012); Gomez-
Solano and Bechinger (2014); Gazuz  _et al._ (2009); Squires and Brady (2005);
Bénichou  _et al._ (2013), which, experimentally, involves macroscopic
rheometers Isa  _et al._ (2007); Besseling  _et al._ (2007); Weiss  _et al._
(1999); Smith  _et al._ (2007). In such investigations, many nonlinear
features have been observed, e.g., shear-thinning or thickening, the
phenomenon of yielding, or the multifaceted nonlinear response to oscillatory
shear. Following these macroscopic studies, the development of microrheology,
where a single microscopic probe particle is driven through a (nonlinear)
medium, allowed to investigate the fluids’ behavior at much smaller length
scales Gutsche  _et al._ (2008); Wilson  _et al._ (2011); Leitmann and
Franosch (2013); Harrer  _et al._ (2012); Winter  _et al._ (2012); Gomez-
Solano and Bechinger (2014); Gazuz  _et al._ (2009); Squires and Brady (2005);
Bénichou  _et al._ (2013). In related experimental and theoretical studies,
several consequences of the nonlinearity of the bath have been reported. For
example, it has been observed that the driven probe experiences an effective
temperature which differs from the true bath temperature Wilson  _et al._
(2011); Démery and Fodor (2019), that it shows superdiffusive behavior
Bénichou  _et al._ (2013); Winter  _et al._ (2012), or shear thinning Wilson
_et al._ (2011); Squires and Brady (2005). More recently, experiments reported
the occurrence of oscillatory modes Berner  _et al._ (2018), which are seen in
a regime of linear, e.g., Newtonian-like behavior. The oscillatory dynamics in
Ref. Berner  _et al._ (2018) was reproduced using a generalized Langevin
equation with negative memory at long times, which can induce persistent
motion Zausch  _et al._ (2008); Mitterwallner  _et al._ (2020) and stress
overshoots Fuchs and Cates (2003). Furthermore, it was noted that the non-
linear properties of a bath can already be detected in equilibrium Müller  _et
al._ (2020). For example, in a nonlinear bath, the effective friction memory
kernel may depend on the external potential Daldrop  _et al._ (2017); Kowalik
_et al._ (2019); Lisỳ and Tóthová (2019); Müller  _et al._ (2020); Müller
(2019); Tóthová and Lisỳ (2021). These findings illustrate the difficulty to
match micro and macro-rheology measurement, and make the study of non-
Markovian baths all the more important.

Here we study the motion of a particle trapped in a viscoelastic fluid, for
different driving velocities, where, compared to previous work Berner  _et
al._ (2018), we extend the experimentally accessible regime towards smaller
velocities. We observe the previously reported linear regime, where the flow
curve shows a plateau. For even smaller velocities, however, the viscosity is
seen to increase in a pronounced manner, so that two distinct plateaus and two
shear thinning processes are found. Theoretically, we reproduce this behavior
by use of a previously introduced stochastic Prandtl Tomlinson Model (SPT)
Müller (2019); Müller  _et al._ (2020). This model also reproduces the
previously observed oscillations Müller (2019); Berner  _et al._ (2018) for
the given shear rates. The model implies that each shear thinning process
corresponds to a de-equilibration of an important set of bath degrees of
freedom, and it allows to estimate the important length scales involved in
these degrees.

##  II Experimental Details

Our experiments were performed in an equimolar solution of the surfactant
cetylpyridinium chloride monohydrate (CPyCl) and sodium salicylate (NaSal) in
deionized water at a concentration of $7\text{\,}\mathrm{m\textsc{m}}$ and at
constant temperatures of $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$, respectively. Under such
conditions, the mixture forms an entangled viscoelastic network of worm-like
micelles Cates and Candau (1990), with a structural relaxation time on the
order $\tau_{s}=$2.5\text{\,}\mathrm{s}$$ as determined by microscopic recoil
experiments Gomez-Solano and Bechinger (2015). Typical length scales of worm-
like micelles are between 100 and $1000\text{\,}\mathrm{nm}$ Walker (2001),
and their typical mesh size is on the order of $30\text{\,}\mathrm{nm}$
Buchanan  _et al._ (2005). A small amount of silica particles with
$2R=$2.73\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ diameter has been added to
the fluid and a single particle has been optically trapped by a focused laser
beam of wavelength of $1064\text{\,}\mathrm{nm}$. This creates a harmonic
potential with trap stiffness $\kappa$. The latter is determined from the
equilibrium probability distribution of the particle in equilibrium, see e.g.
Ref. Berner  _et al._ (2018). To probe the micro-rheological properties, the
sample is translated with constant velocity $-v_{0}$ relative to the static
optical trap by a piezo-driven stage with $v_{0}$ between 4 and
$400\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$. Note that this driving is
equivalent to a trap which is moving at velocity $v_{0}$ with respect to a
fluid at rest. Overall it leads to a local shear rate of $v_{0}/2R$ close to
the particle, and a Weissenberg number of $Wi=v_{0}\tau_{s}/2R$. The so-
obtained Weissenberg numbers range between a few permille to almost unity in
our experiments. We thereby extend the measurements by one decade towards
smaller Weissenberg numbers compared to Ref. Berner  _et al._ (2018).

Figure 1: Microviscosity as a function of the driving speed (flowcurves).
Symbols correspond to experimental data for
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (blue squares) and
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (orange circles). Solid colored
lines correspond to simulations using the stochastic PT model. Both
temperatures exhibit the same two plateau shapes. For
$v_{0}>$200\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, we recover the
shear thinning behavior which is expected from macro-rheological measurements
at $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$, shown as a black dashed line.
However, for $v_{0}<$200\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, the
micro-viscosity further increases towards another plateau reaching values more
than twice as high as the intermediate plateau. Dashed lines and data points
on the $y$-axis give linear response values from simulation and experiment,
respectively.

When driving the particle with constant velocity through the fluid, it
experiences a drag force which leads to a displacement relative to the trap
center. Using Stokes law and the trap stiffness, we can measure the velocity-
dependent microviscosity, given by

$\eta=\frac{\kappa}{6\pi Rv_{0}}\,\left\langle x(t)-v_{0}t\right\rangle.$ (1)

Here, $\left\langle x(t)-v_{0}t\right\rangle$ directly corresponds to the
average position of the particle, relative to the center of the trap. The
resulting flow curves are shown in Fig. 1, for two temperatures
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (blue squares), and
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (orange circles). Both curves
exhibit the same trend. For driving speeds approaching Weissenberg numbers of
order unity, the viscosity shows shear thinning, as expected. This is also
confirmed by the macro-viscosity which is also shown in Fig. 1 for the
temperature of $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for comparison
(black dashed line, obtained by a plane-plane rheometer). Indeed, for that
temperature, the data corresponding to micro and macro viscosities agree
reasonably well for
$v_{0}>$50\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$. We note that this
data range was available in Ref. Berner  _et al._ (2018). From this close
resemblance, together with the fact that the Weissenberg number is of the
order of only $5$% for
$v_{0}=$50\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, at first glance
one might conclude that these velocities are in the regime of linear response.

In contrast to this interpretation, however, the observation of oscillations
in the so called mean conditional displacements (see details below), suggests
that the system is far from equilibrium even at the small driving speed Berner
_et al._ (2018) of $v_{0}=$50\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$.
When extending the range of velocities to smaller values, an astonishing
observation is made: The viscosity shows a pronounced second shear thinning
transition, connected to another plateau in the flow curve, of more than twice
the viscosity value compared to the second plateau. The corresponding shear
thinning process sets in at a Weissenberg number of roughly $2$%, implying
that this dimensionless quantity is not useful to understand that process.
This conjecture is underpinned by the observation that the macro-viscosity
seems to not display the shear thinning at the mentioned small velocities.

Figure 2: a. Sketch of the experimental system, highlighting the probe
suspended in a worm-like micellar fluid. The particle is trapped with an
optical tweezer, using a highly focused laser beam to form a harmonic
potential for the particle. b. Sketch of the Stochastic Prandtl-Tomlinson
model. The tracer particle (bare friction coefficient $\gamma_{0}$) is subject
to a harmonic potential $V_{\textrm{ext}}$ with stiffness $\kappa$ , moving at
constant speed $v_{0}$. The tracer is also linked to several (here two) bath
particles with friction coefficients $\gamma_{1}$ and $\gamma_{2}$,
respectively, through periodic interaction potentials $V_{\textrm{int},i}$.

##  III Theory and model

To rationalize these but also previous experimental observations Müller  _et
al._ (2020), we consider a simple model where the tracer particle is coupled
to a small number of bath particles which mimic the fluid environment (see
sketch of Fig. 2). Each bath particle is coupled to the tracer via a nonlinear
interaction potential $V_{\textrm{int}}$ (specified below). As argued in Ref.
Müller  _et al._ (2020), the potential $V_{\textrm{int}}$ is taken to be
unbounded in the sense that it is finite for any particle separation. This
allows the bath and tracer particles to be arbitrarily far away from each
other, and to travel with different (mean) velocities. Within this model, e.g.
the effect of shear thinning is reproduced, because tracer and bath particles
are able to move at different (average) speeds. For simplicity, in the
following we assume $V_{\textrm{int}}$ to follow a sinusoidal function

$V_{\textrm{int},i}=V_{0,i}\,\cos\left(\frac{2\pi}{d_{i}}(x-q_{i})\right)$ (2)

where $V_{0,i}$ is the amplitude and $d_{i}$ is a length scale corresponding
to bath particle $i$. $x$ and $q_{i}$ are the position coordinates of tracer
and $i$-th bath particle, respectively. The parameter $d_{i}$ in Eq. (2)
mimics an important length scale in the micellar bath, see above for typical
numbers. In addition, the tracer particle is subjected to a harmonic potential
of stiffness which results from the laser tweezer. As the tweezer is moving
with velocity $v_{0}$ relative to the fluid, the trap potential
$V_{\textrm{ext}}$ reads,

$V_{\textrm{ext}}=\frac{1}{2}\kappa\left(x-v_{0}t\right)^{2}.$ (3)

Accordingly, the equations of motion (with $N$ bath particles) are given by

$\displaystyle\gamma_{0}\dot{x}(t)$ $\displaystyle=$
$\displaystyle-\kappa\left(x-v_{0}t\right)-\sum_{i=1}^{N}\frac{2\pi}{d_{i}}\,V_{0,i}\,\sin\left(\frac{2\pi}{d_{i}}(x-q_{i})\right)$
(4) $\displaystyle\,\,+\,\,\xi_{0}(t),$
$\displaystyle\gamma_{i}\dot{q}_{i}(t)$ $\displaystyle=$
$\displaystyle\frac{2\pi}{d_{i}}\,V_{0,i}\,\sin\left(\frac{2\pi}{d_{i}}(x-q_{i})\right)+\xi_{i}(t).$
(5)

Here, $\gamma_{0}$ and $\gamma_{i}$ are the bare friction coefficients of
tracer and bath particle $i$, respectively. These coefficients are linked to
the corresponding random forces $\xi_{0}$ and $\xi_{i}$ via the following
standard properties,

$\left\langle\xi_{i}(t)\right\rangle=0\quad\text{and}\quad\left\langle\xi_{i}(t)\xi_{j}(t^{\prime})\right\rangle=2k_{B}T\gamma_{i}\delta_{ij}\delta(t-t^{\prime}).$
(6)

We assume in the following that the driving started at an infinite time in the
past, so that the system is in a steady state for $t>0$. Practically, this
means that from experimental as well as simulated trajectories, the initial
parts, corresponding to equilibration, are removed.

The interaction potential of Eq. (2) makes this model reminiscent of the so-
called Prandtl-Tomlinson (PT) model, which is used to study dry friction
Prandtl (1928); Tomlinson (1929). Eqs. (4) and (5) extend this model to allow
the background (our bath particles) to be stochastic and dynamic, and to
contain several bath particles. This Stochastic Prandtl Tomlinson (SPT) model
Müller  _et al._ (2020); Müller (2019) thus reduces to the original PT model
when setting $N=1$ and letting $\gamma_{1}$ approach infinity, so that the
bath particle becomes a stationary background potential. This difference is
quite intuitive, as our micellar background is dynamic, while the potentials
considered in dry friction are rather static. Note that the notion ’Stochastic
Prandtl-Tomlinson model’ has been used for other models which are different
from the one used here Jagla (2018); van Spengen  _et al._ (2010).

The solid curves shown in Fig. 1 give the outcomes of the model of Eqs. (4)
and (5) (see Table 1 for parameters used), using Eq. (1). For small velocities
$v_{0}$, we observe a linear response regime, where the micro-viscosity is
independent of $v_{0}$. In this regime, the viscosity can be obtained also via
linear response (see Eq. (35) in Ref. Müller  _et al._ (2020)), shown as the
horizontal dashed lines in the graph. It is insightful to discuss the case of
high potential barriers $V_{0,i}/k_{B}T$, which appear appropriate to fit the
experimental data (see Table 1). Then the regime of small driving speeds
corresponds to the case where all particles move (approximately) with the same
average velocity $v_{0}$. Because of this, the microviscosity in Eq. (1) can
be found to a good approximation from $\eta\approx\frac{1}{6\pi
R}\,\left(\gamma_{0}+\sum_{i}\gamma_{i}\right)$. In the graph, the resulting
value is not shown, as it is indistinguishable from the dashed lines.
Experimentally, this corresponds to the case where the slow colloidal particle
drags its surrounding with it, and thus feels a large friction.

The interaction potential $V_{\textrm{int},i}$ in Eq. (2) supports a maximal
force of $\frac{2\pi}{d_{i}}\,V_{0,i}$. If the force between the tracer and
the bath particle exceeds that value, the bond ruptures, and the velocity of
the bath particle is (on average) smaller than $v_{0}$: As a result, the
system shows a shear thinning behavior. The critical velocity where this
happens can be estimated by balancing the mentioned maximal force with the
drag force $\gamma_{i}v_{0}$ of particle $i$, yielding (in absence of noise)
$v_{i,cr}\approx\frac{2\pi}{\gamma_{i}d_{i}}\,V_{0,i}$ . For the curves in
Fig. 1, we use two bath particles, with distinct critical velocities,
resulting in the two-plateau structure seen in the graph. As mentioned, for
very small velocities, both bath particles move (approximately) with the same
velocity as the tracer. The smaller critical velocity corresponds to the
“larger” bath particle, i.e., the one with a larger value of $d_{i}$. Using
the values of Table 1, we estimate this velocity to be $27$ and
$36\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ for the two temperatures,
respectively, which fits well to the numerical curves. Beyond this velocity,
the viscosity decreases towards the second plateau. On the regime of the
second plateau, the larger particle is thus far from equilibrium, while the
second (“smaller”) bath particle is still close to equilibrium. Once the
second critical velocity (estimated to $321$ and
$655\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$, respectively) is reached,
also the second particle starts shear thinning. The model thus implies the
interpretation that two distinct sets of degrees of freedom of the bath
display very different critical velocities, yielding an intermediate state
with half of them out of equilibrium. Notably, the often employed notion of
fast and slow degrees of freedom Zwanzig (2011) is demonstrated here
explicitly, at least in our model.

The flow curve concerns the average position $\left\langle
x-v_{0}t\right\rangle$ of the particle, as seen from Eq. (1), and we now aim
to address the particle’s fluctuations. Subtracting the particle’s mean
position, i.e. using $X(t)=x(t)-\left\langle x-v_{0}t\right\rangle$, yields a
stochastic variable $X$ with zero mean. Its fluctuations can be quantified
using the so-called Mean Conditional Displacement (MCD) Berner  _et al._
(2018). It is defined by $\left\langle
X\right\rangle_{X_{0}}=\int_{-\infty}^{\infty}dX\,X\,P(X,t|X_{0},0)$, with
$P(X,t|X_{0},0)$ the conditional probability for the particle position $X$. In
previous work, we observed that MCDs show oscillations in this system Berner
_et al._ (2018), which were observed at a temperature of
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$, for Weissenberg numbers in the
range of $0.04$ to $0.34$. Can the SPT model also explain the occurrence of
oscillations? To address this question, we show in Fig. 3 experimental and
simulated MCD curves, this time for a temperature of
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$, to also emphasize the same
phenomenology at the two temperatures. The MCD curves generally are found to a
good approximation linear in $X_{0}$ (both in experimental as well as in
simulated data), so that division by $X_{0}$ yields $X_{0}$-independent curves
as shown. The figure distinguishes between positive and negative values of
$X_{0}$, yielding the positive and negative flanks shown. The observed mirror
symmetry of the data with positive and negative $X_{0}$ further emphasizes the
linearity in $X_{0}$. Fig. 3a) shows a small velocity of
$v_{0}=$16\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, which corresponds
to the regime of linear response in Fig. 1. For this velocity, the MCD curve
decays monotonically to zero, as expected near equilibrium Berner  _et al._
(2018), and also in agreement with the SPT model. Fig. 3b) shows a larger
velocity of $v_{0}=$200\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, a
velocity placed on the intermediate plateau in the flow curve in Fig. 1, thus
in the regime corresponding to the curves shown in Ref. Berner  _et al._
(2018); Müller (2019). This curve shows pronounced oscillations, which are
reproduced by the SPT model. How can these be understood? As described above,
the intermediate plateau is beyond the critical velocity of the larger bath
particle, which is thus far from equilibrium. It thus moves with an average
speed much smaller than $v_{0}$. It can for the sake of argument be assumed to
stand still, so that the tracer is moving in a stationary periodic potential.
It is thus subject to a periodic force, which results in the seen
oscillations. The period of oscillations is in this approximation given by
$d/v_{0}$, which matches well the one observed in Fig. 3b). In Ref. Berner
_et al._ (2018), the frequency of oscillations was indeed found to scale
linearly with $v_{0}$, an observation which can now be understood. This
analysis thus identifies an important length scale in the system, of the order
of $15\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Passing over spatial variations
on that scale (which are almost stationary as seen from the colloidal
particle) seems to cause the observed oscillations.

(a)

(b)

Figure 3: Mean conditional displacements (a) in the linear response regime,
$16\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ and (b) in the regime of
intermediate plateau, $200\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$.
Circles correspond to experiments (at
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$), and lines correspond to the
Prandtl-Tomlinson simulations (solid and dotted lines for $2$ and $3$ bath
particles, respectively). While we recover previously observed oscillations
Berner  _et al._ (2018) for
$200\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$, the linear regime (left,
$16\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$) exhibits none. Adding more
bath particles (solid line for $3$ bath particles), leads to decoherence, and
thus decreases the amplitude of the oscillations.

The amplitude of oscillations is larger in the SPT model as compared to
experiments, which we attribute to a number of idealizations of the model. For
example, the background potential of the SPT model is perfectly periodic with
a sharp length scale $d_{i}$. A real micellar solution, however, will exhibit
a range of length scales, which naturally leads to decoherence. Indeed adding
more bath particles, with slightly different parameters (see Table 2) leads to
a loss of coherence, and the amplitude of the resulting oscillations is
reduced. Notably, adding more bath particles with appropriate parameters keeps
the flow curve unaltered, but changes the MCD. This confirms the expectation
that flow curve and MCD are not one to one related. The flow curve concerns
the mean motion, and the MCD quantifies fluctuations.

In Ref. Müller  _et al._ (2020), a single bath particle was found to be
sufficient to capture experimental observations in the STP model. It is
natural that a system close to equilibrium (as in Ref. Müller  _et al._
(2020)) is easier to model compared to a system far from equilibrium, as
addressed here. Indeed, an open question is whether the larger length scale of
$\sim$15\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ can be detected in
equilibrium.

Finally, we note that, while the MCD curves and the flow curve each can easily
be modelled quantitatively, we had problems fitting both curves with the same
parameters. We attribute this to the different values of trap stiffness used
for flow curve and MCDs of $1.5\text{\,}\mathrm{\SIUnitSymbolMicro
N}\text{\,}{\mathrm{m}}^{-1}$ and $0.3\text{\,}\mathrm{\SIUnitSymbolMicro
N}\text{\,}{\mathrm{m}}^{-1}$, respectively, for experimental reasons. This
makes fitting both curves simultaneously even more challenging. We therefore
allowed the amplitude of flowcurve to be a free parameter in our simulations,
which turned out to be $4.5\text{\,}$.

##  IV Conclusion

In this work we describe micro-rheological experiments where a single colloid
is driven by an optical tweezer within a viscoelastic fluid. The presented
fluid, a micellar solution, shows a flow curve with two distinct shear
thinning regimes, with an intermediate plateau in between. Theoretical
modeling via a stochastic Prandtl Tomlinson model captures the observed
behavior, and implies that, on the intermediate plateau, one set of bath
degrees of freedom is far from equilibrium, while another set is still in
equilibrium. The intermediate plateau corresponds to the linear response
regime of macro-rheology, so that the shear thinning process at even smaller
driving velocities is a purely microscopic effect, and can thus easily be
overlooked. The mean conditional displacements show oscillations on the
intermediate plateau, which are also reproduced in the theoretical model. The
oscillations allow extraction of a length scale, which is as large as
$15\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and whose nature and origin have
to be investigated in future work.

##  V Acknowledgements

The authors thank Boris Müller for discussions at the early stages of this
work. The theoretical parts of this work build on the initial findings
provided in his thesis Müller (2019). Funding: FG acknowledges the support by
the Alexander von Humboldt foundation. This project is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1432 – Project-
ID 425217212. RJ acknowledges the support by the Göttingen Campus QPlus
program. Competing interest: The authors declare no competing interest.

##  VI Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Simulation parameters

In this appendix, we provide the simulation parameters used for generating the
flow-curves in Fig. 1 and MCD curves in Fig. 3 given in the main text.

$T$ |  $\kappa$ |  $(\gamma_{0},\gamma_{1},\gamma_{2})$ |  $(V_{0,1},V_{0,2})$ |  $(d_{1},d_{2})$
---|---|---|---|---
[$\text{\,}\mathrm{\SIUnitSymbolCelsius}$] | [$\text{\,}\mathrm{\SIUnitSymbolMicro N}\text{\,}{\mathrm{m}}^{-1}$] |  [$\text{\,}\mathrm{\SIUnitSymbolMicro N}\text{\,}\mathrm{s}\text{\,}{\mathrm{m}}^{-1}$] |  $[k_{B}T]$ |  [$\text{\,}\mathrm{nm}$]
$25$ | $1.5$ | $(0.06,1.5,1.5)$ | $(7.6,24.0)$ | $(400,15000)$
$30$ | $1.5$ | $(0.04,0.5,1.0)$ | $(4.0,20.6)$ | $(320,15000)$
Table 1: Parameters of the SPT model used to simulate the flow-curves shown in
Fig. 1.

For the MCD curves (dotted lines) in Fig. 3, we use a different trap-
stiffness, $\kappa=$0.35\text{\,}\mathrm{\SIUnitSymbolMicro
N}\text{\,}{\mathrm{m}}^{-1}$$. The additional MCD curves with three bath
particles (solid lines) in Fig. 3 are generated using the parameters in Table
2.

$(\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3})$ |  $(V_{0,1},V_{0,2},V_{0,3})$ |  $(d_{1},d_{2},d_{3})$
---|---|---
[$\text{\,}\mathrm{\SIUnitSymbolMicro N}\text{\,}\mathrm{s}\text{\,}{\mathrm{m}}^{-1}$] |  $[k_{B}T]$ |  [$\text{\,}\mathrm{nm}$]
$(0.04,0.5,0.5,0.5)$ | $(4.0,8.2,10.8)$ | $(320,12000,15000)$
Table 2: Parameters used to simulate the MCD curves with three bath particles
in Fig. 3 at $30\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with
$\kappa=$0.35\text{\,}\mathrm{\SIUnitSymbolMicro
N}\text{\,}{\mathrm{m}}^{-1}$$.

As discussed in the main text, including more bath particles into the model
leads to de-coherence, resulting into the reduction in amplitude of
oscillations in the MCD. We choose the parameters for the third bath particle
(see Table 2) in such a way that the flow curves remain unaffected. In Fig. 4,
we have compared the flowcurve generated with the parameters of Table 1, i.e
with two bath particles with that of Table 2, i.e. with three bath particles.

Figure 4: Comparison of flowcurves simulated at
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with two and three bath particles.

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