# Sondheimer oscillations as a probe of non-ohmic flow in type-II Weyl
semimetal WP2

Maarten R. van Delft  Laboratory of Quantum Materials (QMAT), Institute of
Materials (IMX), École Polytechnique Fédérale de Lausanne (EPFL), 1015
Lausanne, Switzerland  e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Yaxian Wang  Harvard John A.
Paulson School of Engineering and Applied Sciences, Harvard University,
Cambridge, MA 02138, USA.  Carsten Putzke  Laboratory of Quantum Materials
(QMAT), Institute of Materials (IMX), École Polytechnique Fédérale de Lausanne
(EPFL), 1015 Lausanne, Switzerland  Jacopo Oswald  IBM Research - Zurich, 8803
Rüschlikon, Switzerland  Georgios Varnavides  Harvard John A. Paulson School
of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138,
USA.  Christina A. C. Garcia  Harvard John A. Paulson School of Engineering
and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.  Chunyu
Guo  Laboratory of Quantum Materials (QMAT), Institute of Materials (IMX),
École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
Heinz Schmid  IBM Research - Zurich, 8803 Rüschlikon, Switzerland  Vicky Süss
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
01187 Dresden, Germany  Horst Borrmann  Max Planck Institute for Chemical
Physics of Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany  Jonas Diaz
Laboratory of Quantum Materials (QMAT), Institute of Materials (IMX), École
Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland  Yan Sun
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
01187 Dresden, Germany  Claudia Felser  Max Planck Institute for Chemical
Physics of Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany  Bernd
Gotsmann  IBM Research - Zurich, 8803 Rüschlikon, Switzerland  Prineha Narang
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138, USA.  e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Philip J.W. Moll  Laboratory of
Quantum Materials (QMAT), Institute of Materials (IMX), École Polytechnique
Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland  e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

As conductors in electronic applications shrink, microscopic conduction
processes lead to strong deviations from Ohm’s law. Depending on the length
scales of momentum conserving ($l_{MC}$) and relaxing ($l_{MR}$) electron
scattering, and the device size ($d$), current flows may shift from ohmic to
ballistic to hydrodynamic regimes and more exotic mixtures thereof. So far, an
_in situ_ , in-operando methodology to obtain these parameters self-
consistently within a micro/nanodevice, and thereby identify its conduction
regime, is critically lacking. In this context, we exploit Sondheimer
oscillations, semi-classical magnetoresistance oscillations due to helical
electronic motion, as a method to obtain $l_{MR}$ in micro-devices even when
$l_{MR}\gg d$. This gives information on the bulk $l_{MR}$ complementary to
quantum oscillations, which are sensitive to all scattering processes. We
extract $l_{MR}$ from the Sondheimer amplitude in the topological semi-metal
WP2, at elevated temperatures up to $T\sim 50$ K, in a range most relevant for
hydrodynamic transport phenomena. Our data on $\mu m$-sized devices are in
excellent agreement with experimental reports of the large bulk $l_{MR}$ and
thus confirm that WP2 can be microfabricated without degradation. Indeed, the
measured scattering rates match well with those of theoretically predicted
electron-phonon scattering, thus supporting the notion of strong momentum
exchange between electrons and phonons in WP2 at these temperatures. These
results conclusively establish Sondheimer oscillations as a quantitative probe
of $l_{MR}$ in micro-devices in studying non-ohmic electron flow.

## Main text

In macroscopic metallic wires, the flow of electric current is well described
by Ohm’s law, which assigns a metal a spatially-uniform ‘bulk’ conductivity.
The underlying assumption is that the complex and frequent scattering events
of charge carriers occur on the microscopic length scale of a mean-free-path,
which is much smaller than the size of the conductor, $d$, leading to
diffusive behavior. In addition to the scattering processes of bulk systems,
the resistance of microscopic conductors is mostly dominated by boundary
scattering, thereby masking the internal scattering processes of the bulk in
resistance measurements. Here, we present a method to uncover these bulk
processes in micro-scale metals, which are of technological importance for
fabrication of quantum electronic devices, and simultaneously critical to a
fundamental understanding of microscopic current flow patterns. It is
instructive to classify the bulk scattering processes into two categories:
those that relax the electron momentum, such as electron-phonon, Umklapp or
inelastic scattering, occurring at length-scale $l_{MR}$; and those that
conserve the electron momentum, such as direct or phonon-mediated electron-
electron scattering, associated with a length scale $l_{MC}$.

Within a kinetic theory framework, these three length scales, namely $d$,
$l_{MR}$, and $l_{MC}$, can be used to describe the current flow in micro-
scale conductors. When momentum-conserving interactions are negligible, ohmic
flow at the macro-scale ($l_{MC}\gg d\gg l_{MR}$) gives way to ballistic
transport in clean metals where $l_{MR},l_{MC}\gg d$. Conversely, when
momentum-conserving interactions occur over similar or smaller length scales
to momentum-relaxing interactions, a third regime of ‘hydrodynamic’ transport
($l_{MR}\gg d\gg l_{MC}$) is observable [1, 2]. In this regime, the static
transport properties of electron fluids can be described by an effective
viscosity that captures the momentum diffusion of the system [2, 3]. These
electron fluids exhibit classical fluid phenomena such as Poiseuille flow,
whereby the current flow density is greatly decreased at the conductor
boundary. Recently, advances in both experimental probes and theoretical
descriptions have enabled direct observation of these effects using spatially-
resolved current density imaging, and have hinted towards the rich landscape
of electron hydrodynamics in micro-scale crystals [4, 5, 3].

While such local-probe experiments provide means of quantifying electron-
electron interactions, and thus extracting $l_{MC}$, direct measurement of the
intrinsic momentum-relaxing processes ($l_{MR}$) within micron-scale
conductors remains elusive, yet is greatly needed. From a practical
perspective, $l_{MR}$ describes the overall scattering from impurities and the
lattice vibrations within the metallic microstructure, which at low
temperature is an important feedback parameter of quality control in
fabrication. Furthermore, given both the reduction of sample size and the
improved crytal quality, seemingly exotic transport scenarios where $l_{MR}\gg
d\gg l_{MC}$ is satisfied are expected to become more prevalent in technology.
An accurate description of these length scales is necessary to predict the
overall resistance and thus voltage drops and heat dissipation in the
nanoelectronic devices. For example, the resistive processes in a hydrodynamic
conductor occur at the boundaries rather than homogeneously distributed in the
bulk, which alters the spatial distribution of Joule heating and thereby has
significant impacts on thermal design.

Real devices will operate at some intermediate state in the $d$, $l_{MR}$, and
$l_{MC}$ parameter space, departing from the well-understood limiting cases of
ohmic, ballistic and hydrodynamic flow. Rich landscapes of distinct
hydrodynamic transport regimes are predicted depending on the relative sizes
of the relevant length scales [6]. Effective understanding, modeling and
prediction of transport requires an experimental method to estimate these
parameters reliably in every regime. In large, ohmic conductors, the bulk
mean-free-path $l_{MR}$ can be simply estimated from the device resistance
using a Drude model. Yet when $l_{MC},l_{MR}\gtrsim d$, boundary scattering
dominates the resistance, and hence estimates of the bulk scattering
parameters are highly unreliable. This leaves the worrying possibility of
misinterpreting the transport situation in a conductor, in that the
microfabrication itself may introduce defects or changes in the bulk
properties that remain undetected by macroscopic observables such as the
resistance, but have profound impact on the microscopic current distribution.
These effects are already noticeable in state-of-the-art transistors, owing to
the low carrier density of semi-conductors [7], but have similarly been
reported in metallic conductors [5]. With the increased technological interest
in quantum and classical electronics operating at cryogenic temperatures, such
questions about unconventional transport regimes are also of practical
relevance in next generation electronics [8].

In this context, we propose to exploit a magneto-oscillatory phenomenon,
Sondheimer oscillations (SO), as a self-consistent method to obtain the
transport scattering length $l_{MR}$ _in-situ_ , even in constricted channels
when $l_{MR}\gg d$. In general, a magnetic field ($\vec{B}$) applied
perpendicular to a thin metal forces the carriers on the Fermi surface to
undergo cyclotron motion. Those on extremal orbits of the Fermi surface are
localized in space due to the absence of a net velocity component parallel to
the magnetic field. These localized trajectories can become quantum-coherent,
and their interference causes the well-known Shubnikov-de Haas oscillations.
The states away from extremal orbits also undergo cyclotron motion, yet they
move with a net velocity along the magnetic field, analogous to the helical
trajectories of free electrons in a magnetic field (Fig. 1). These states are
responsible for the Sondheimer size effect which manifests itself as a
periodic-in-$B$ oscillation of the transport coefficients, as discovered in
the middle of the past century for clean elemental metals [9].

For any given state, the magnitude of $\vec{B}$ sets the helical radius and
thus determines how many revolutions the electron completes while travelling
from one surface to the other in a microdevice. If an integer number of
revolutions occur, the charge carrier will have performed no net motion along
the channel, and hence is semi-classically localized (Fig. 1a). However, if
the number of revolutions is non-integer, a net motion along the channel
exists, delocalizing the carriers, resulting in oscillatory magnetotransport
behavior. Large-angle bulk scattering events dephase the trajectory, hence the
strong sensitivity of SO to the bulk $l_{MR}$ even in nanostructures. These SO
are an inherent property of mesoscale confined conductors in three dimensions
and have no counterpart in 2D metals like graphene.

The period of the SO is derived by considering a classical charged particle on
a helical trajectory between two surfaces perpendicular to the magnetic field
[10]. One compares the time it takes to travel the distance $d$ between the
surfaces, $t_{d}=d/v_{\parallel}$, to the time to complete a single cyclotron
revolution, $\tau_{c}=2\pi/\omega_{c}=2\pi m^{*}/eB$ ($m^{*}$: effective mass,
$e$: electron charge, $\omega_{c}=eB/m^{*}$: cyclotron frequency). Their ratio
describes the number of revolutions of the trajectory. For certain fields the
helix is commensurate with the finite structure and the number of revolutions
is integer, $n$, such that $t_{d}=n\tau_{c}$. This occurs periodically in
field, with the period given by:

$\Delta B=\frac{2\pi
m^{*}v_{\parallel}}{ed}=\frac{\hslash}{ed}\left(\frac{\partial A}{\partial
k_{\parallel}}\right).$ (1)

The useful identity $v_{\parallel}=\frac{\hslash}{2\pi
m^{*}}\left(\frac{\partial A}{\partial k_{\parallel}}\right)$, derived by
Harrison [11], directly relates the SO period to the Fermi surface geometry,
where $v_{\parallel}$ and $k_{\parallel}$ denote the velocity and momentum
component parallel to the magnetic field and $A$ is the Fermi surface cross-
sectional area encircled by the orbit in k-space. Note the contrast to
conventional quantum oscillations which appear around extremal orbits, where
$\frac{\partial A}{\partial k_{\parallel}}=0$.

All conduction electrons undergo cyclotron motion, yet depending on
$\frac{\partial A}{\partial k_{\parallel}}$, they experience different
commensurability fields with a structure of given size $d$. Hence oscillatory
contributions to the total conductivity are washed out, unless a macroscopic
number of states share the same $v_{\parallel}\propto\left(\frac{\partial
A}{\partial k_{\parallel}}\right)_{E_{f}}$ [10]. In earlier days of Fermiology
[12], geometric approximations for Fermi surfaces, such as elliptical
endpoints, were introduced to identify those generalized geometric features
that lead to extended regions of constant $\frac{\partial A}{\partial
k_{\parallel}}$. The computational methods available nowadays allow a more
modern approach to the problem. Fermi surfaces calculated by ab-initio methods
can be numerically sliced in order to calculate their cross-section
$A(k_{\parallel})$. We propose to extend this routine procedure, used to find
extremal orbits relevant for quantum oscillations $\left(\frac{\partial
A}{\partial k_{\parallel}}=0\right)$, to identify SO-active regions
$\left(\frac{\partial^{2}A}{\partial k^{2}_{\parallel}}\sim 0\right)$, based
on the Fermi-surface slicing code SKEAF [13] (see the supplementary
information for details on implementation).

SO are caused by the real-space motion of charge carriers and hence also pose
some conditions on the shape of the conductor. First, surface scattering needs
to be mostly diffusive. If an electron undergoes specular scattering $N$ times
before scattering diffusively, it contributes towards the SO as if the sample
had an effective thickness $Nd$ [14], leading to overtones. Naturally, SO
vanish in the (unrealistic) limit of perfectly specular boundary conditions,
as such ideal kinetic mirrors remove any interaction of the electron system
with the finite size of the conductor. Secondly, the conductor must feature
two parallel, plane surfaces perpendicular to the magnetic field to select
only one spiral trajectory over the entire structure. The parallelicity
requirement is simply given by a fraction of the pitch of the spiral at a
certain field ($\Delta d<v_{\parallel}\tau_{c}=d\frac{\Delta B}{B}$) [10].
These requirements are naturally fulfilled in planar electronic devices.

It is instructive to briefly compare SO to the more widely known quantum
oscillations of resistance, the Shubnikov-de Haas effect (SdH). Both are
probes of the Fermi surface geometry based on cyclotron orbits, yet the
microscopics are strikingly different. While quantum oscillation frequencies
are exclusively determined by Fermi surface (FS) properties via the Onsager
relation and are thus independent of the sample shape, SO are finite-size
effects. SO emerge from extended regions on the FS, unlike SdH oscillations to
which only states in close vicinity of extremal orbits contribute. While SdH
oscillations are quantum interference phenomena, SO are semi-classical, which
is key to their use as a robust probe of exotic transport regimes. If both can
be observed, powerful statements on the scattering microscopics can be made,
as SdH is sensitive to all dephasing collision events and SO separates out the
large-angle ones [15]. However, the much more stringent conditions of phase
coherence in SdH severely limit their observations at higher temperatures. SO
are observable up to relatively high temperatures at which the rapidly
shrinking $l_{MR}(T)$ leads to a transition into an ohmic state, when
$l_{MR}(T)<d$. As such, they are ideally suited to explore the exotic
transport regimes in which, for example, hydrodynamic effects occur.

Figure 1: Introduction to Sondheimer oscillations. a Illustration of the
Sondheimer effect. Left: the applied magnetic field is $B=3\Delta B$ and the
electron (red) makes an integer number of rotations, with no contribution to
transport. Right: $B\neq n\Delta B$. The electron hits the top surface at a
different position than its origin on the bottom surface, leading to a
contribution to the conductivity. b Resistivity as function of temperature for
a WP2 microdevice. Inset: false-color SEM image of a typical device used in
this study. c Sondheimer oscillations seen in the Hall resistivity of a WP2
microdevice, for different temperatures. The oscillation period of $\Delta
B=$1.6 T is highlighted.

We apply these theoretical considerations to experimentally investigate the
scattering mechanisms in micron-sized crystalline bars of the type-II Weyl
semimetal WP2 [16] exploiting the Sondheimer effect. Bulk single crystals of
WP2 are known for their long $l_{MR}$, in the range of 100-500 $\mu$m [17, 18,
16], comparable to the elemental metals in which SO were initially discovered
[19, 20, 21, 22]. These are an ideal test case for non-ohmic electron flow, as
hydrodynamic transport signatures and nontrivial electron-phonon dynamics have
been observed in various topological semimetals [23, 24, 18, 17, 25]. These
ulta-pure crystals are then reduced in size by nanofabrication techniques into
constricted channels, to study hydrodynamic or ballistic corrections to the
current flow.

Here we employ Focused Ion Beam (FIB) micromachining [26], which allows
precise control over the channel geometry in 3D. In this technique, we
accelerate Xe ions at 30 kV to locally sputter the target crystal grown by
chemical vapor transport (CVT) [27, 16] until a slab of desired dimensions in
the $\mu m$-range remains. This technique leads to an amorphized surface of
around 10 nm thickness, yet has been shown to leave bulk crystal structures
pristine [28]. Naturally, reducing the size of a conductor even without
altering its bulk mean-free-path significantly changes the device resistance
at low temperatures due to finite size corrections [29], both in the ballistic
and hydrodynamic limiting cases. Hence, measurements of the constricted device
resistance alone cannot exclude the possibility of bulk degradation due to the
fabrication step. Thus far, one could only argue based on size-dependent
resistance studies that the values smoothly extrapolate to the bulk
resistivity value in the limit of infinite device size [30, 18]. Measuring SO
directly in the microfabricated devices themselves, however, quantitatively
supports that the ultra-high purity of the parent crystal remains unchanged by
our fabrication. We note that the fundamental question of the bulk parameters
is universal in mesoscopic conducting structures irrespective of the
fabrication technique, and these considerations are thus equally applicable to
structures obtained by mechanically or chemically thinned samples as well as
epitaxially grown crystalline films. SO should provide general insights into
the material quality in the strongly confined regime, allowing to contrast
different fabrication techniques.

At high temperatures, the resistivity measured in $\mu m$-confined devices
agrees well with previous reports on high quality bulk crystals, as expected
given the momentum relaxing limited mean-free-path of charge carriers in this
regime (Fig. 1b). Yet in the low temperature limit, the device resistance
exceeds those of bulk crystals by more than an order of magnitude [17, 31,
32]. Conversely, the residual resistance ratios in our devices
(RRR$\approx$160-300) are also considerably lower than in bulk crystals [32].
The main question we address by SO is whether this excess resistance points to
fabrication induced damage, finite size corrections, or a mixture thereof. At
low temperatures around 3 K, a drop in resistance signals a superconducting
transition. As WP2 in bulk form is not superconducting, this likely arises
from an amorphous W-rich surface layer due to the FIB fabrication similar to
observations made in NbAs [33] and TaP [34]. In Fig. 1c, we show the Hall
resistivity, $\rho_{xy}$, of one of our devices as a function of the magnetic
field, for different temperatures. The Hall signal comprises oscillations with
a period of $\Delta B=$1.6 T, resolved above approximately $B$=2 T.

Figure 2: The staircase device. a, False color SEM image of a staircase
device, used to measure Sondheimer oscillations for different thicknesses. The
crystal is colored in purple, and gold contacts in yellow. b, Main: Schematic
of the staircase device, illustrating all possible measurement configurations
as well as the thickness of each section. Top left: SEM image of the lamella
that will become the device shown in a, prior to extracting it from its parent
crystal. Bottom right: SEM image of the same lamella, glued down onto a
sapphire substrate, ready to define the device geometry. The lamella and glue
are covered in gold (not colored) throughout the full field of view. The
magnetic field is applied perpendicular to the structure, aligned along the
crystallographic [011] direction.

A hallmark signature of SO is their linear frequency dependence on the device
thickness perpendicular to the field. For this reason, we fabricated
crystalline devices with multiple sections of different thickness to study the
$d$-dependence in a consistent manner. This ‘staircase’ device allows the
simultaneous measurement of transport on 5 steps of different thickness $d$,
as illustrated in Fig. 2. SO appear in all transport coefficients,
magnetoresistance and Hall effect alike, yet here we focus on the Hall effect
for two practical reasons. First, the step edges induce non-uniform current
flows, and hence the device would need to be considerably longer to avoid
spurious voltage contributions from currents flowing perpendicular to the
device in a longitudinal resistance measurement. Second, WP2 exhibits a very
large magnetoresistance yet a small Hall coefficient, as typical for
compensated semi-metals. Therefore, the SO are more clearly distinguishable
against the background in a Hall measurement, but they are also present in the
longitudinal channel (see Supplementary Information).

The fabrication process of our WP2 devices follows largely the same procedure
as described in Ref. 26. However, for the staircase device, a few key changes
were made. In the first fabrication step, the FIB is used to cut a lamella
from a bulk WP2 crystal. One side is polished flat, and the other side
polished into five sections, each to a different thickness (Fig. 2b). It is
then transferred, flat side down, into a drop of araldite epoxy on a sapphire
substrate and electrically contacted by Au sputtering (Fig. 2b). In a second
FIB step, the staircase slab is patterned laterally into its final structure
(Fig. 2a).

All segments of the staircase devices show pronounced $B$-periodic
oscillations in the Hall channel, from which the linear background is removed
by taking second derivatives. (Fig. 3). At the lowest fields, a weak,
aperiodic structure is observed. In this regime, the cyclotron diameter does
not fit into the bar, preventing the formation of the Sondheimer spirals. Note
that in all devices of different thickness, this onset field of the SO is the
same. This is a natural consequence of the fact that the lateral size,
perpendicular to the magnetic field, by design, is the same for all steps of
the staircase. Each step, however, differs in thickness $d$ parallel to the
magnetic field, and the period varies accordingly between steps (Fig. 3b). At
even higher fields, the onset of regular SdH oscillations hallmarks a
transition into a different quantized regime. The Sondheimer oscillation
frequency $F=1/\Delta B$ varies linearly with $d$ as expected (Fig. 3c, Eq.
1).

Next we identify the Sondheimer-active region on the Fermi surface from the
ab-initio band structure, which was calculated by density functional theory
(DFT) with the projected augmented wave method as implemented in the code of
the Vienna ab-initio Simulation Package (VASP) [35]. The FS of WP2 consists of
two types of spin-split pockets: dogbone shaped electron pockets and extended
cylindrical hole pockets (see Fig. 4 and supplementary Fig. S2 for a complete
picture of the FS).

Only one area quantitatively agrees with the observed SO periodicity: the four
equivalent endpoints of the dogbone (colored orange in Figs. 3f). Slicing all
Fermi-surfaces using SKEAF [13], their cross-sections $A(k_{\parallel})$ are
obtained. While in quantum oscillation analysis this information is discarded
once the extremal orbits are identified, it forms the basis of the SO
analysis. As the dogbone is sliced from the endpoints, the area continuously
grows until the two endpoint orbits merge and the area abruptly doubles.
Slicing further, the area grows until the maximum orbit along the diagonal is
reached. Inversion symmetry enforces then a symmetric spectrum when slicing
further beyond the maximum. The quasi-linear growth at the endpoints signals
an extended area of Sondheimer-active orbits. Averaging the near-constant
derivative in this region, $\frac{\partial A}{\partial k_{\parallel}}$,
provides via Eq. 1 a tuning-parameter-free prediction of the thickness
dependence of the SO frequency. This ab-initio prediction (red line in Fig.
3c) is in excellent agreement with the observed thickness dependence.

Figure 3: Analysis of Sondheimer oscillations in WP2. a, Second derivative of
the Hall resistivity shown in Fig.1c at $T$=4 K. Inset: Fast Fourier Spectrum
(FFT) corresponding to this data. b, Second derivatives of the Hall
resistivity at three different thicknesses, $d$ = 4.3, 2.7 and 2.0 $\mu m$, at
selected temperatures. c, FFTs corresponding to the data in b. d Dependence of
the Sondheimer frequency on $d$. The red dashed line is calculated from the
Fermi surface as determined from DFT. e Cross-sectional area, $A$, of the
dogbone Fermi surface pocket of WP2 as a function of $k$ parallel to the field
direction of our experiments (top), and its derivative (bottom). f Location of
observed Sondheimer orbits drawn on the dogbone-shaped Fermi surface pocket.
The magnetic field is applied along the [011]-direction, perpendicular to the
current, as indicated by the red line.

Next the temperature-dependence of the SO amplitude is used to gain direct
information about the microscopic scattering processes acting on this region
of the Fermi surface. In Fig. 4a,b, we plot this temperature dependence and
highlight two regimes: that of quantum coherence and that of purely Sondheimer
oscillations. In the first regime, quantum coherence leads to SdH
oscillations; however, for typical effective masses $m^{*}\approx m_{e}$, as
in WP2, they are only observable at very low temperatures ($T<5$ K).
Importantly, their quick demise upon increasing temperature is not driven by
the temperature dependence of the scattering time, but rather by the
broadening of the Fermi-Dirac distribution. This is apparent as their
temperature dependence is well described by the Lifshitz-Kosevich formalism
based on a temperature-independent quantum lifetime, $\tau_{q}$.

This strong temperature-suppression of quantum oscillations severely limits
their use to probe scattering mechanisms at elevated temperatures. SO, on the
other hand, do not rely on quantum coherence and are readily observed to much
higher temperatures, up to 50 K in WP2, while their temperature decay allows a
direct determination of the transport lifetime, $\tau_{MR}=l_{MR}/v_{F}$.
Hence SO make an excellent tool to study materials in the temperature range
pertinent to exotic transport regimes like ballistic or hydrodynamic. They
self-evidence non-diffusive transport as they only vanish when $l_{MR}\sim d$,
and hence are only absent in situations of conventional transport within a
given device.

Key to observable SO is that electrons do not undergo large-angle scattering
events on their path between the surfaces. We therefore have the condition
that $l_{MR}>d$ [36, 37]. As $l_{MR}(T)$ decreases with increasing temperature
and the boundary scattering is assumed to be temperature-independent, the SO
amplitude is suppressed as $e^{-d/l_{MR}(T)}$ which allows us to estimate the
bulk transport mean-free-path within a finite-size sample, even when $d\ll
l_{MR}$. It is extracted as [36]:

$\frac{1}{l_{MR}(T)}=-\frac{1}{d}\ln{\frac{A(T)}{A(0)}},$ (2)

where $A(T)$ is the SO amplitude at temperature $T$. $A(T=0)$ is simply
estimated by extrapolation, which is a robust procedure as the SO amplitude
saturates at low but finite temperatures. This is analogous to the saturation
of the resistivity of bulk metals at low temperatures, once bosonic scattering
channels are frozen out and temperature-independent elastic defect scattering
becomes dominant.

In the following discussion, we focus on the scattering time $\tau_{MR}$ to
facilitate comparison of our results with literature and theory, using the
average Fermi velocity on the dogbone Fermi surface determined from our band
structure calculations self-consistently, $v_{F}=3.6\times 10^{5}$ m/s. The
$\tau_{MR}(T)$ obtained from all devices quantitatively agrees, despite their
strong difference in thickness and hence SO frequency, further supporting the
validity of this simple analysis (see Fig. 4c and Fig. S5). The lifetimes on
the SO devices furthermore agree with measurements on bulk crystals [18],
evidencing that FIB fabrication does not introduce significant changes to the
bulk properties of WP2 that might cause misinterpretations of the scattering
regime.

For our WP2 devices, a standard Dingle analysis of the quantum oscillations
yields a quantum scattering time $\tau_{q}\sim 10^{-13}-10^{-12}$ s (Fig. 4c),
in agreement with published values for bulk crystals WP2 [16]. As $\tau_{q}$
is sensitive to all dephasing scattering events, but $\tau_{MR}$ only to
large-angle momentum relaxing scattering, the microscopics of the scattering
processes in WP2 are brought to light. The four orders of magnitude of
difference between $\tau_{MR}$ and $\tau_{q}$ reflects a common observation in
topological semi-metals such as Cd3As2 [38], PtBi2 [39] or TaAs [40].

Figure 4: Extraction of scattering times from the Sondheimer amplitude. a,
FFTs of the SO at different temperatures for thicknesses of 4.3, 2.7 and 2.0
$\mu$m. b, Temperature dependence of the Sondheimer and SdH oscillation
amplitudes, for different sample thicknesses. The dashed lines are fits used
to extrapolate to the amplitude at zero temperature, $A(0)$ (see the
Supplementary Information for details). The dotted line is a Lifshitz-Kosevich
fit, giving an effective mass of $1.1m_{e}$. Two regimes are highlighted: that
of quantum coherence, where SdH oscillations exist alongside SO, and that of
Sondheimer, where only SO exist. c, Scattering times extracted for WP2 using
Eq. 2 and the Fermi velocity from Ref. [18]. An approximate quantum lifetime
extracted from the SdH oscillations as well as data from Refs. [18, 41] are
included for comparison. d Calculated scattering time for all electron-phonon
scattering ($\tau_{\rm e-ph}$) and e the scattering efficiency determining the
momentum-relaxing scattering lifetimes ($\tau_{MR}$) projected onto the Fermi
surface at $T$=10 K.

Long $\tau_{MR}$, together with a high quality, clean sample, enables the
realization of the hydrodynamic regime where the momentum conserving
scattering dominates. These quantitative measurements of $\tau_{q}$ and
$\tau_{MR}(T)$ can now be directly compared to theoretical models of
scattering. We consider an initial electronic state with energy
$\varepsilon_{n\bf{k}}$ (where $n$ and $\bf{k}$ are the band index and
wavevector respectively) scattering against a phonon with energy
$\omega_{\textbf{q}\nu}$ (where $\nu$ and q are the phonon polarization and
wavevector respectively), into a final electronic state with energy
$\varepsilon_{m\bf{k+q}}$. The electron-phonon scattering time $\tau_{\rm
e-ph}$ describing such an interaction can be obtained from the electron self
energy using Fermi’s golden rule:

$\tau^{-1}_{\rm eph}(n{\bf
k})=\frac{2\pi}{\hslash}\sum_{m\nu}\int_{\mathrm{BZ}}\frac{d\textbf{q}}{\Omega_{\mathrm{BZ}}}|g_{mn,\nu}(\textbf{k},\textbf{q})|^{2}\times\left(n_{\textbf{q}\nu}+\frac{1}{2}\mp\frac{1}{2}\right)\delta\left(\varepsilon_{n\textbf{k}}\mp\omega_{\textbf{q}\nu}-\varepsilon_{m\textbf{k}+\textbf{q}}\right),$
(3)

where $\Omega_{\mathrm{BZ}}$ is the Brillouin zone volume, $f_{n\textbf{k}}$
and $n_{\textbf{q}\nu}$ are the Fermi-Dirac and Bose-Einstein distribution
functions, respectively, and the electron-phonon matrix element for a
scattering vertex is given by

$g_{mn,\nu}(\textbf{k},\textbf{q})=\left(\frac{\hslash}{2m_{0}\omega_{\textbf{q}\nu}}\right)^{1/2}\langle\psi_{m{\bf
k+q}}|\partial_{{\bf q}\nu}V|\psi_{n{\bf k}}\rangle.$ (4)

Here $\langle\psi_{m{\bf k+q}}|$ and $|\psi_{n{\bf k}}\rangle$ are Bloch
eigenstates and $\partial_{{\bf q}\nu}V$ is the perturbation of the self-
consistent potential with respect to ion displacement associated with a phonon
branch with frequency $\omega_{\textbf{q}\nu}$. Plotting these state-resolved
electron-phonon lifetimes at $\sim$10 K on the Fermi surface reveals the
distribution of scattering in the SO-active regions (Fig. 4(d)). Equation 3,
however, accounts, to first order, for all electron-phonon interactions,
irrespective of the momentum transfer or equivalently the scattering angle. To
remedy this, we augment the scattering rate with an ‘efficiency’ factor [42]
given by the relative change of the initial and final state momentum
($1-\frac{v_{n\textbf{k}}\cdot
v_{n\textbf{k}}}{|v_{n\textbf{k}}||v_{n\textbf{k}}|}=1-\cos\theta$), where
$v_{n\textbf{k}}$ is the group velocity and $\theta$ is the scattering angle:

$\left(\tau_{\mathrm{eph}}^{\mathrm{mr}}(n\textbf{k})\right)^{-1}=\frac{2\pi}{\hslash}\sum_{m\nu}\int_{\mathrm{BZ}}\frac{d\textbf{q}}{\Omega_{\mathrm{BZ}}}\left|g_{mn,\nu}(\textbf{k,q})\right|^{2}\times\left(n_{\textbf{q}\nu}+\frac{1}{2}\mp\frac{1}{2}\right)\delta\left(\varepsilon_{n\textbf{k}}\mp\omega_{\textbf{q}\nu}-\varepsilon_{m\textbf{k}+\textbf{q}}\right)\times\left(1-\frac{v_{n\textbf{k}}\cdot
v_{n\textbf{k}}}{|v_{n\textbf{k}}||v_{n\textbf{k}}|}\right).$ (5)

At low temperatures, the thermally activated phonon modes have a tiny q,
therefore the initial and final electronic states only differ from a small
angle. It is thus important to take this momentum-relaxation efficiency factor
into account in addition to $\tau_{\rm e-ph}$, in order to estimate
$\tau_{MR}$ which determines the electron mean free path in the SO-active
regions. In the SO measurements, the electron orbits are located on the
endpoints of the dogbone-shaped electron pockets (Fig. 3e), therefore we
highlight the scattering efficiency distribution on the electron Fermi surface
in Fig. 4e. Indeed, when the orbit is aligned along the diagonal direction,
the Fermi surface cross section features very low scattering efficiency with
an averaged $1-{\rm cos}\theta<0.1$. This supports our observation of
frequently scattering electrons with long transport lifetimes in the SO
measurement.

These results demonstrate the power of the Sondheimer size effect for the
extraction of the momentum relaxing mean-free-path in mesoscopic devices when
$d\ll l_{MR}$ via their temperature dependence. Combined with first-principles
theoretical calculations we were able to locate the states contributing to the
helical motion to the elliptical endpoints of a particular Fermi surface of
WP2. We note however that such analysis as well as the thickness dependence
are only relevant for the academic purpose of robustly identifying these
oscillations as SO. Once this is established, the relevant lifetimes may
straightforwardly be obtained from the resistance oscillations at a single
thickness. Hence, Sondheimer oscillations promise to be a powerful probe to
obtain the bulk mean-free-path in devices with $\mu m$-scale dimensions
without relying on any microscopic model assumptions. This analysis is a clear
pathway to identify scattering processes in clean conductors within operating
devices. It thereby provides important feedback of the materials quality at
and after a micro-/nano-fabrication procedure and disentangles the roles of
bulk and surface scattering that are inseparably intertwined in averaged
transport quantities of strongly confined conductors, such as the resistance.
As their origin is entirely semi-classical, they are not restricted by
stringent criteria such as quantum coherence and thus span materials
parameters of increased scattering rate. In particular, they survive up to
significantly higher temperatures and thereby allow microscopic spectroscopy
in new regimes of matter dominated by strong quasiparticle interactions, such
as hydrodynamic electron transport. With this quantitative probe, it will be
exciting to test recent proposals of exotic transport regimes and create
devices that leverage such unconventional transport in quantum materials.

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Acknowledgements
M.R.v.D. acknowledges funding from the Rubicon research program with project
number 019.191EN.010, which is financed by the Dutch Research Council (NWO).
This project was funded by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (grant no.
715730, MiTopMat). Y.W. is partially supported by the STC Center for
Integrated Quantum Materials, NSF Grant No. DMR-1231319 for development of
computational methods for topological materials. This research used resources
of the National Energy Research Scientific Computing Center, a DOE Office of
Science User Facility supported by the Office of Science of the U.S.
Department of Energy under Contract No. DE-AC02-05CH11231 as well as resources
at the Research Computing Group at Harvard University. P.N. is a Moore
Inventor Fellow and gratefully acknowledges support through Grant No. GBMF8048
from the Gordon and Betty Moore Foundation. C.A.C.G. acknowledges support from
the NSF Graduate Research Fellowship Program under Grant No. DGE-1745303. We
acknowledge financial support from DFG through SFB 1143 (project-id 258499086)
and the Würzburg-Dresden Cluster 274 of Excellence on Complexity and Topology
in Quantum Matter – ct.qmat (EXC 2147, project-id 39085490). B.G. acknowledges
financial support from the Swiss National Science Foundation (grant numner
CRSII5_189924). H.S and B.G thank J. Gooth for discussion and K. Moselund, S.
Reidt, and A. Molinari for support, and received funding from the European
Union’s Horizon 2020 research and innovation program under Grant Agreement ID
829044 "SCHINES".

Author contributions
MRvD, JO, CP, CG, JD performed the transport experiments, as well as the
microfabrication in collaboration with BG and HS. The crystals were grown by
VS and CF, and crystallographically analyzed by HB. YS and CF calculated the
band structures, and YW, CACG, PN performed the electron-phonon scattering
calculations. BG, CF, PN and PJWM conceived the experiment, and all authors
contributed to writing of the manuscript.

Competing interests
The authors declare no competing financial interest.

Supplementary information


Surface patterns in drying films of silica colloidal dispersions

F. Boulognea†∗, F. Giorgiutti-Dauphinéa, L. Paucharda

We report an experimental study on the drying of silica colloidal dispersions.
Here we focus on a surface instability occurring in a drying paste phase
before crack formation which affects the final film quality. Observations at
macroscopic and microscopic scales reveal the occurrence of the instability,
and the morphology of the film surface. Furthermore, we show that the addition
of adsorbing polymers on silica particles can be used to suppress the
instability under particular conditions of molecular weight and concentration.
We relate this suppression to the increase of the paste elastic modulus.

00footnotetext:  a UPMC Univ Paris 06, Univ Paris-Sud, CNRS, F-91405. Lab
FAST, Bat 502, Campus Univ, Orsay, F-91405, France. Fax: +33 1 69 15 80 60;
Tel: +33 1 69 15 80 46; E-mail<EMAIL_ADDRESS>
† Now at: Department of Mechanical and Aerospace Engineering, Princeton
University, Princeton, NJ 08544, USA.

##  1 Introduction

Patterns arising in soft materials such as elastomers, gels and biological
tissues receive a growing attention 1. The understanding and the control of
the underlying instabilities are crucial for technological applications
(microelectronics, microfluidics), for biological systems (wrinkling of human
skin or drying fruit 2) or for medical applications where gels are used as
biological scaffolds for tissues or organs. In addition, various patterns
which affects the surface of films are reported in the literature. One of the
most studied concerns wrinkles observed when a hard skin, sitting on a soft
layer, is compressed. Beyond a critical strain, it results in a periodic
sinusoidal deformation of the interface 3, 4, 5, 6 for which the periodicity
has been derived theoretically 2, 7. When the strain is further increased, a
secondary instability occurs leading to wrinkle-to-fold transition 8, 9; this
results in a localization of the deformation 10. Under a biaxial compressive
stress, it has been shown that a repetitive wrinkle-to-fold transition
produces a hierarchical network of folds 11, 12. These patterns are observed
in various systems such as glassy polymers 13, 3, polyethylene sheets 2 or
foams 14. Another group of patterns arises in soft layers supported by rigid
substrates 1. Under compressive stresses, an instability often called creasing
is manifested by localized and sharp structures at the free surface. These are
commonly observed in elastomers 15, 16, and hydrogels 17, or in the situation
of a rising dough in a bowl 18. Despite similarities in their morphologies,
origins of folds and creases are strongly different. Whereas folds are a
secondary instability of wrinkles, creases are a direct deformation of a flat
film (creases are also known as sulci 19, 20). Moreover drying films can
produce characteristic crack patterns as they are dried21, 22. The drying-
induced cracks can invade the surface and propagate simultaneously into the
volume of the medium with evaporation of solvent, resulting in the division of
the plane into polygonal domains.

In this paper, we experimentally investigate patterns displayed at the surface
of drying films of silica nanoparticles in an aqueous solvent. We discuss the
occurrence of these patterns, and we examine their main features using
different imaging techniques and rheological measurements. Moreover, the
polymer/silica interaction is usually used to tune the mechanical properties
of composite films. Consequently, we propose here a method to suppress these
structures by adding adsorbing polymers to the silica nanoparticles.

##  2 Experimental

Fig. 1: Experimental setup. Dry or moist air is produced by an air flow from
the ambient atmosphere through desiccant or water respectively to the box.
Depending on the humidity measured by the humidity sensor inside the box, a
solenoid valve is actioned to converge the humidity to the desired value. A
honeycomb grid (represented in its actual size) is used to improve the
contrast of the surface corrugations (Schlieren technique).

###  2.1 Controlled drying conditions

Experiments consist in drying a colloidal suspension in a circular glass Petri
dish (inner diameter $2R=5.6$ cm) placed in a chamber which has controlled
temperature ($22\pm 2$ ∘C) and relative humidity ($R_{H}=50\pm 2$%) 23. The
setup is sketched in figure 1. Top and bottom walls of the chamber are
transparent for visualization by transmitted light. The bottom wall is
carefully adjusted horizontally prior each experiment. Since the sample
contrast is very low, the Schlieren technique24 is used: a honeycomb grid is
positioned between the bottom wall and an extended light source. A camera
(Nikon D300), located at the top of the chamber, records a photograph each
minute.

In the following, $m_{i}$ varies in the range $1$ to $9$ g, resulting in a
initial thickness $h_{i}$ varying in the range from $\simeq 0.3$ to $2.5$ mm.

###  2.2 Colloidal dispersions

Table 1: Main properties of silica dispersions used in this paper. The volume fraction is noted $\phi_{0}$. Values of the particle diameters come from reference 25 (p. 324). Silica | diameter $2a$ ($\mathrm{nm}$) | $\phi_{0}$ |  $\rho$ ($\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$) | pH
---|---|---|---|---
SM | 10 | 0.15 | 1180 | 9.9
HS | 16 | 0.19 | 1250 | 9.8
TM | 26 | 0.20 | 1260 | 9.2

We use three aqueous dispersions of silica colloidal particles: Ludox SM-30,
HS-40 and TM-50, commercially available from Sigma-Aldrich. The pH is in the
range of 9-10, and so the particle surface bears a high negative charge
density 26. SM-30 is used without treatments, while HS-40, and TM-50 are
diluted using pure water (milliQ quality, resistivity: $18$ M$\Omega$.cm) at
pH 9.5 by addition of NaOH. A weight ratio of 90/10 (HS-40/water) and 75/25
(TM-50/water) are chosen to obtain similar initial volume fractions. In the
following, SM (unmodified), HS and TM designate these dispersions; their main
properties are reported in Table 1.

The effect of polymer chains on TM particles is investigated using
polyethylene oxyde (PEO) or alternatively polyvinylpyrrolidone (PVP). Indeed,
a high affinity for silica surfaces is known for PEO and PVP resulting in an
adsorption on silica particles. To simplify the notations, polymers are noted
$P_{i}$ as follow. Different molecular weights are studied ($P_{0}$:
$300\text{\,}\mathrm{Da}$, $P_{1}$: $600\text{\,}\mathrm{Da}$, $P_{2}$:
$3350\text{\,}\mathrm{Da}$, $P_{3}$: $6000\text{\,}\mathrm{Da}$, $P_{4}$:
$35\text{\,}\mathrm{kDa}$, $P_{5}$: $600\text{\,}\mathrm{kDa}$ and $P_{6}$:
$8\text{\,}\mathrm{MDa}$) for PEO and $P_{7}$: $40\text{\,}\mathrm{kDa}$ for
PVP. Polymers are purchased from Sigma-Aldrich and are dissolved in pure water
at pH 9.5. This solution, with a weight concentration of polymers noted
$C_{p}$, and is used for the dilution of TM-50. As a result, the final polymer
concentration in TM dispersion is $C_{p}/4$.

The adsorption of polymers on silica particles has been largely studied in the
literature 27, 28, 29, 30. In particular, it has been shown, from adsorption
isotherms, that silica particles are totally covered with polymers for
concentrations above $1$ $\mathrm{mg}\text{\,}{\mathrm{m}}^{-2}$ for PEO 27
and PVP 31. In our experiments, the polymer concentrations do not exceed
$\lesssim 0.01$ $\mathrm{mg}\text{\,}{\mathrm{m}}^{-2}$, which is much lower
than the covering concentration. In this concentration range, polymers adsorb
on particles to form necklaces29; the dispersed state is stable since the
interactions between necklaces are repulsive32.

###  2.3 Visualisation techniques

Observation using an optical microscopy (DM2500 Leica microscope) by
transmitted light is used at a macroscopic scale.

At a microscopic scale, the surface profile is investigated using an Atomic
Force Microscope (AFM, Vicco) in tapping mode on dried TM samples (scanning
area are $80\times 80$ ${\mathrm{\SIUnitSymbolMicro m}}^{2}$) prepared with an
initial weight $m_{i}=6$ g. To complete these measurements at the onset of the
propagation, an optical profiler is used (Taylor Hobson, $10\times$, working
distance $7$ mm). This contact-less method allows us to measure the surface
profile of consolidating materials with a typical surface area $1.6\times 1.6$
mm2.

##  3 Results

###  3.1 Macroscopic observations

####  3.1.1 Temporal evolution

Fig. 2: Time evolution of a flat film of a TM sample (initial weight is
$m_{i}=6$ g). Each image is focused on the film surface (in the center, far
from the petri dish edge) and is taken at the same region of the film. The
arrow in image (4) gives the mean direction of propagation of the structures.
Image (6) shows the final pattern made of sinuous structures superposed to a
network of channeling cracks. Scale bar: $2$ mm.

During water removal, particles concentrate and the film thickness decreases.
At a given time, a network of fine dark lines progressively invades the flat
region of the film from the center to the edges. The time evolution of the
film surface is shown in figure 2. Starting from an homogeneous surface (image
1 in figure 2), surface corrugation can be observed after a few minutes of
evaporation (image 2 in figure 2). The corrugation patterns probably result
from Rayleigh-Bénard or Bénard-Marangoni convective instabilities33. However,
the convective cells disappear after a period of time; the surface recovers
its visual homogeneity (image 3 in figure 2). Then, at time $t_{onset}$,
structures progressively invade the surface of the flat film (image 4 in
figure 2) and results in the pattern shown in image 5 in figure 2. Finally,
the classical crack pattern forms in the film (image 6 in figure 2).

####  3.1.2 Onset time

Starting from an homogeneous surface, the structures appear where the film is
thinner, i.e. in the center, far from the meniscus. For TM dispersions, we
observed that below a critical initial weight $m_{i}^{c}=1.4\pm 0.3$ g, i.e.,
$h_{i}^{c}=0.4\pm 0.1$ mm, no structure forms even if cracks do.

Fig. 3: Onset time $t_{onset}$ of structures as a function of the initial
weight and corresponding initial film thickness, for three samples: TM, TM +
$P_{5}$ (PEO, $600\text{\,}\mathrm{kDa}$) at $C_{p}=0.1$% and TM + $P_{6}$
(PVP, $40\text{\,}\mathrm{kDa}$) at $C_{p}=0.2$%. Lines are guides for the
eye. Error bars reported for TM samples are similar for other samples.

In addition, above this critical initial weight, the onset time $t_{onset}$
needed to observe the structures growth, is measured for dispersions of
different initial weights deposited in the container, ranging from $1$ to $9$
g. Results are reported in figure 3 and show that the onset time increases
linearly with the initial weight.

Table 2: Final state of sample surfaces for the different samples. Sample |  SM |  HS |  TM |  TM + ($P_{0}$, $P_{1}$, $P_{2}$, $P_{3}$) | TM + ($P_{4}$, $P_{5}$, $P_{6}$, $P_{7}$)
---|---|---|---|---|---
|  |  |  |  $\forall C_{p}$ |  $C_{p}<C_{p}^{c}$ |  $C_{p}>C_{p}^{c}$
Observation |  $\forall h$, no pattern |  $\forall h$, no pattern |  patterns if $h>h_{c}$ |  patterns if $h>h_{c}$ |  patterns if $h>h_{c}$ |  $\forall h$, no pattern

Similar experiments have been carried out with the other samples and results
are summarized in table 2. For SM and HS samples, no structure appear at the
film surface, independently of the investigated range of film thicknesses.

Consequently, we studied the addition of adsorbing polymers only on TM
samples. For a sample of initial weight $m_{i}=3$ g deposited in the container
($h_{i}=1$ mm), using PEO with molecular weights larger than $6$ kDa, the
formation of the structures is suppressed above a critical concentration
$C_{p}^{c}=0.13\%$ (within an uncertainty of $0.03\%$ for $P_{3}$ and $0.01\%$
for $P_{4}$, $P_{5}$ and $P_{6}$). Thus, the weight amount of polymers
necessary to suppress the structures is found to be independent of the
molecular weight (for $M_{w}\geq 6\textrm{kDa}$). However, for shorter polymer
chains ($M_{w}<6\textrm{kDa}$), and for concentrations up to $C_{p}=0.6$%, the
structures still form.

Moreover, addition of PVP ($40$ kDa) to TM particles results in suppression of
the structures for $C_{p}^{c}(PVP)=0.33\pm 0.03$%. We notice that the
molecular weight of a polymer unit for PEO $M_{w}^{PEO}=44$ g/mol and for PVP
$M_{w}^{PVP}=111$ g/mol:

$\frac{M_{w}^{PEO}}{M_{w}^{PVP}}\simeq\frac{C_{p}^{c}(PVP)}{C_{p}^{c}(PEO)}$
(1)

As a result, structures are suppressed for a critical number of polymer units
($160\pm 10$ units per colloidal particle, i.e. $7.6\pm 0.6$
$\mathrm{\SIUnitSymbolMicro g}\text{\,}{\mathrm{m}}^{-2}$). We checked that
drying kinetics are not affected by the addition of polymers by weight
measurements 23.

####  3.1.3 State of the material

The drying of colloidal dispersion is a balance between two opposite fluxes:
(a) the solvent flux tends to accumulate the particles to the free surface,
whereas (b) the diffusion process smooths concentration gradients. The
competition between both processes usually states either the formation of a
skin at the surface of a drying film or the consolidation in the bulk phase.
In that way, the relevant dimensionless number is a Péclet number defined as
the ratio of a diffusion timescale $h^{2}/D$ and an advection timescale
$h/V_{e}$: $\textrm{Pe}(h)=\frac{V_{e}h}{D}$ 34, where $h$ is the film
thickness, $V_{e}$ is the evaporation rate and $D$ a diffusion coefficient.
The evaporation rate is found to be equal to $V_{e}\simeq 1\times 10^{-8}$m/s
under the same drying conditions23. The diffusion coefficient is deduced from
the Stokes-Einstein relation $D=k_{B}T/(6\pi\eta_{s}a)$, where $\eta_{s}=$
$1\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$ is the solvent viscosity. From
thicknesses lying in the range $[0.4,2.9]$ mm, the Péclet number is between
$0.2$ and $1.8$. The value close to $1$ does not allow us to distinguish
between skin formation or consolidation in the bulk.

However, to determine the state of the material, we gently collected a
fraction of the material located at the surface of our samples (e.g. TM
sample, $m_{i}=9$ g, typically $1/4$ of the paste thickness); the sample was
operated during the propagation of structures and before the apparition of
cracks. We measured a volume fraction of $\phi=38\pm 2\%$ which is compatible
with a elastic material ($\phi>0.35$) of non-aggregated particles
($\phi<0.61$) 35; this material, deposited in a test tube filled thereafter
with deionized water, can be redispersed after shaking.

###  3.2 Visualisations

####  3.2.1 Optical microscopy

Fig. 4: Dynamics of the structure propagation in the center of a drying TM
film. The time laps between pictures (a) and (b) is $21$ seconds. The
situation indicated by arrows is detailed in the inserts. Picture (c) shows
the final pattern. (d) The series of photographs show a morphogenetic sequence
(time laps between each picture: $1$ s) leading to developed branching
process. Scale bars represent $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The propagation of structures was firstly observed by optical microscopy. The
sequence of images in figure 4(a, b, c) reveals the dynamics of formation of
the structures in the center of the sample. A propagating branch (2)
approaches a prior one (1), then crosses. Moreover, in figure 4(c), the final
pattern in the sequence shows an increase of the structures contrast, and the
formation of a new generation of branches, dividing domains. The series of
photographs in figure 4(d) show a typical detailed process of the splitting of
one branch into two branches.

Fig. 5:  Final pattern (a) in the center of the petri dish and (b) at $1$ cm
from the edge located at the bottom of the picture (The vector $\vec{r}$
designates the radial direction.). Scale bars represent $100$
$\mathrm{\SIUnitSymbolMicro m}$.  Fig. 6: (a) Surface profile (AFM) of a dried
TM sample ($m_{i}=6$ g). The 3D profile on the right corresponds to the region
inside the yellow square in image on the left. (b) Surface profile obtained
using an optical profiler: a drying region with propagating structures from
the background to the foreground are shown in the profiles along the lines 1
and 2.

As shown in previous pictures, the structures exhibit an isotropic pattern in
the center of the sample. However, while structures propagate in a thickness
gradient naturally imposed by the meniscus at the edge of the container, they
become preferentially oriented radially and ortho-radially (figure 5(b)).

A meniscus at the edge of the container extends over few times the capillary
length scale $\kappa^{-1}=\sqrt{\frac{\gamma}{\rho g}}\simeq 2$ mm ($\gamma$
is the surface tension of water). At the final stage, we obtain isotropic
patterns (figure 5(a)) in a region covering $40$% of the total surface area
which corresponds to the flat film area.

####  3.2.2 AFM profilometry

The surface topography of a TM sample is reported in figure 6(a). The typical
structure height ranges between $1$ $\mathrm{\SIUnitSymbolMicro m}$ and $4$
$\mathrm{\SIUnitSymbolMicro m}$. In particular, we notice the asymmetric shape
of the surface profile, as an evidence of two inclined planar surfaces
connected by a transition region of about $10$ $\mathrm{\SIUnitSymbolMicro
m}$. As a result, such measurements during the propagation of structures are
shown in figure 6(b). The red region of the surface corresponds to a thicker
layer; this can be related to the presence of the microscope objective
limiting the evaporation flux 36. In front of the structures, the film is
flat, and an arch-shaped profile is shown at the onset of the instability.
During the drying process, this shape evolves to an asymmetric shape. The
kinks present near the junction of arches (bottom graph) is due to the absence
of interference fringes with the optical profiler at the singularities
resulting in an artifact. Thus, we only consider the parts of the profile
highlighted in red.

Moreover, in the case of experiments carried out with SM and HS samples for
$m_{i}\in[2,12]$ g, no structure can be observed. This statement is also
confirmed using AFM, and optical profiler images.

Finally, measurements using AFM and optical profiler techniques reveal the
absence of structures in TM with polymers above $C_{p}^{c}$ as in the case of
observations using optical microscopy. Note that polymer addition below
$C_{p}^{c}$ leads to a lower structure height. For instance, addition of PEO
($600\text{\,}\mathrm{kDa}$, $C_{p}=0.1$%) decreases the maximum height to
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ ($m_{i}=6$ g).

###  3.3 Rheology

Fig. 7: Rheological measurements of the elastic modulus $G_{0}$ obtained from
small oscillatory shear flow tests (plate-plate geometry) for different
samples of pure particles (SM, HS and TM) and with addition of PEO ($600$ kDa)
in TM. Error bars indicate extreme values.

In the following the rheological properties of pastes are investigated at the
onset of structures formation. In order to compare the elastic moduli of
different silica pastes, measurements are performed at a well-defined
consolidation time: the onset time for TM films and the same consolidation
time for other systems. Samples are prepared using the same protocol: the
initial weights range from $7$ to $9$ g. We collect surface samples
corresponding to the 4/5th of the film thickness in the center of the
container. A part is used for a dried extract in order to deduce the volume
fraction $\phi$. The paste elastic modulus is measured using a rheometer
(Anton Paar MCR501) in the plate-plate geometry (diameter: $24.93$ mm) with a
solvent trap. The gap is adjusted between $0.3$ and $0.7$ mm accordingly to
the amount of paste used for the measurement. Small oscillatory shear flow
tests are performed with an amplitude of $0.2$%, and a frequency sweep from
$100$ Hz to $1$ Hz for $200$ s per point. Under these conditions, the elastic
modulus $G^{\prime}(\omega)$ does not vary significantly with the frequency
(relative variations lower than $10$%) and the deformation amplitude up to few
percent above which the elastic modulus decreases. The average value is
denoted by $G_{0}$ in the following.

In this way, five samples are considered: SM, HS and TM dispersions and binary
mixture of TM and PEO ($600\text{\,}\mathrm{kDa}$) at $C_{p}=0.2$% and
$C_{p}=0.4$%. For TM samples (without polymers), since structures appear at
$\phi=38\pm 2$%, we select other samples (where no structures are developed)
with similar volume fraction. Results are shown in figure 7. Our results on
the HS sample are consistent with values available in the literature37. Note
that, for all studied samples, the loss modulus ($G^{\prime\prime}(\omega)$)
appears to be at least one order of magnitude lower than $G_{0}$. For pure
dispersions, the elastic modulus decreases of nearly one order of magnitude
when the mean particle size varies from $10$ to $26$ nm. On the contrary, the
addition of a small amount of PEO adsorbing on TM particles results in the
increase of the paste elastic modulus.

##  4 Discussion

Our observations concerning the dynamics of formation and the characteristics
of the pattern show that the structures observed here can be scarcely related
to winkles. Indeed, wrinkles are periodic deformations of the interface5
whereas our patterns are clearly localized.

Folds, described as a secondary instability arising from wrinkles8, present
strong similarities such as the localization of the deformation, the
subdivision of domains and the crossing patterns11. However, the observation
of wrinkles is expected but it has never been observed in our systems.
Moreover, wrinkles and folds result from the deformation of a skin in contact
with a softer material whereas at the onset of the structures formation, the
film seems to behave like an homogeneous paste phase.

Observations at the onset of patterning (top graph of figure 6(b)) suggest
some morphological similarities with creases patterns of swelling elastomers
reported in the literature 16 while the mechanism is different. The observed
arches evolve to a“saw-tooth roof shape” as shown by AFM measurements (Fig.
6(a)). This transition is illustrated in the bottom graph of figure 6(b) where
the onset of the lost of symmetry can be seen. Indeed, this transition may be
attributed to the fact that colloidal pastes are able to change their
microstructures. A possible explanation of the symmetry breakage from the
onset of the instability to the final state can be that any further strain
energy is dissipated in a shear of the material (creep flow).

The second question addressed in this paper concerns the absence of structures
for SM and HS particles, as well as the suppression by the addition of
adsorbing polymers.

It results that mechanical properties of pastes depend on the colloidal
dispersion or the concentration of additional polymers. The strain in the
paste phase at the onset time can be deduced its elastic modulus. Indeed,
assuming that the internal stress during the drying process comes from the
flux of water through the porous matrix made by silica particles, the greatest
stress occurs at the drying surface where the pore pressure is the larger38.
There, the stress $\sigma$ scales as follows:

$\sigma\sim\frac{h\eta_{s}}{3k}V_{e}$ (2)

where $k$ is the permeability of the porous material. This expression
justifies that below a critical thickness $h_{i}^{c}$, no structure are
visible because the tensile stress is not large enough to trigger the
mechanical instability. Estimating the permeability from Carman-Kozeny
relation at $\phi=38$ % ($k=6\times 10^{-18}$ ${\mathrm{m}}^{2}$), we deduce a
critical strain $\epsilon_{c}$ at the onset of the structures formation:

$\epsilon_{c}\sim\frac{h_{i}^{c}\eta_{s}}{3kG_{0}}V_{e}\simeq 0.2$ (3)

where $G_{0}=11$ kPa for TM. It is noteworthy that for this deformation, the
elastic modulus of the material is still comparable to the one measured at low
deformation amplitudes. Moreover, the elastic modulus is measured just prior
the propagation of the structures (that occurs over a timescale of 10 min) at
a controlled volume fraction. Thus, this elastic modulus must be considered as
an order of magnitude that could be underestimated, which means that this
critical strain could beoverestimated.

Measurements show that above a critical elastic modulus, structures are
suppressed. This critical value is estimated by $G_{0}^{c}=15\pm 5$ kPa from
figure 7. Indeed, for pastes exhibiting larger elastic moduli, shrinkage
induced by the drying is withstood by the paste network leading to a low
strain. Further shrinkage is frustrated by adhesion of the paste onto the
substrate and results in tensile stress and so cracks formation. However, for
pastes exhibiting lower elastic moduli, the strain is larger leading to
possible surface corrugations. Equation 3 also suggest that $G_{0}^{c}$ should
depend on the film thickness if $\epsilon_{c}$ does not. From our
observations, we cannot conclude on a dependence of $G_{0}^{c}$ with the film
thickness because near the threshold, the presence or absence of structures
are difficult to be stated. Indeed, the structures density decreases with the
film thickness as shown in the inset of figure 3.

In the case of pure colloidal dispersions (SM, HS, TM), the particle size
affects the elastic modulus (Fig. 7), consequently the permeability. Indeed,
fitting the elastic moduli as a function of the particle size with a power
law, $G_{0}\propto a^{p}$, leads to values $p\in[-1,-3]$. The inaccuracy comes
from the large error bars and the little number of available samples. Taking
into account the particle size dependency in the permeability with Carman-
Kozeny model, the deformation varies as $\epsilon\propto a^{-(p+2)}$. A power
coefficient $p$ lower than $-2$ would be consistent with our observations on
the suppression effect. We can also note that our description assumes that
changing Ludox only results in varying particle size. It is worth to note that
other parameters may have effects, such as polydispersity (see reference 25
(p. 324) for measurements) which can modify the permeability 39.

Finally, we also observed that patterns are hierarchical (Fig. 2) which
introduces different lengthscales. The final pattern settles after $250$
minutes in the given example. This suggests that the refinement is related to
the consolidation of the material that can be analyzed by the increase of the
elastic modulus in time which increases the stress in the material.

##  5 Conclusion

In this paper, we report experimental observations on surface instabilities
during the drying of silica colloidal dispersions. These patterns grow if
films are thicker than a critical thickness and that the onset time increases
linearly with the initial film thickness and it is concomitant with a paste
phase sitting on the substrate. Measurements using microscopy techniques
highlight the surface corrugations resulting first in arches that evolves to a
saw tooth roof shape. We also provide a method to prevent these patterns
consisting in the addition of a small amount of polymers which increases the
material elastic modulus. From this technique, we estimated a critical elastic
modulus for structures formation.

As a perspective, it would be interesting to refine the description of the
evolution from arches to the saw-tooth roof shape which is attributed to a
creep flow caused by the nature of the material. Thus, a competition between
the creasing instability and the creep flow might may exist in the selection
of the final characteristic distance separating structures. In particular, in
a future work, the evolution of structures could be related to the
characteristics (onset time, final size) of the different generations in order
to establish a criterion for the distance between structures and to precise
the influence of loss of symmetry in the development of the instability.
Further theoretical developments might focus on similarities and differences
of surface instabilities in visco-elasto-plastic materials compared to visco-
elastic gels.

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##  6 acknowledgments

The authors thank Triangle de la Physique for the rheometer (Anton Paar, MCR
501) and A. Aubertin for designing the experimental setup. We are grateful to
B. Cabane, J.-P. Hulin, G. Gauthier and C. Quilliet for fruitful discussions,
A. Chennevière, C. Poulard and F. Restagno for guidance and sharing their AFM
and optical profiler.