1

YAKOVLEV et al. Cavity Formation in silica-Filled Rubber Compounds Observed
During Deformation by Ultra Small-Angle X-Ray Scattering

Corresponding author Ilya Yakovlev, European Synchrotron Radiation Facility,
Avenue des Martyrs 71, Grenoble, 38043, France , ORCID: 0000-0003-1730-6113

H2020 Marie Skłodowska-Curie Actions grant number 847439 468 to the InnovaXN
project.

# Cavity Formation in Silica-Filled Rubber Compounds Observed During
Deformation by Ultra Small-Angle X-Ray Scattering

Ilya Yakovlev  Michael Sztucki  Frank Fleck  Hossein Ali Karimi-Varzaneh
Jorge Lacayo-Pineda  Christoph Vatterott  Ulrich Giese<EMAIL_ADDRESS>European Synchrotron Radiation Facility, Grenoble, France,
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Continental Reifen Deutschland GmbH, Hanover, Germany,  jorge.lacayo-
<EMAIL_ADDRESS>Faculty of Natural Sciences, Gottfried Wilhelm Leibniz
University Hannover, Hanover, Germany<EMAIL_ADDRESS>Deutsches
Institut für Kautschuktechnologie e. V., Hanover, Germany,
<EMAIL_ADDRESS>
(Date Month Year; Date Month Year; Date Month Year)

###### Abstract

[Abstract]When silica-filled rubber compounds are deformed, structural
modifications in the material’s bulk lead to irreversible damage, the most
significant of which is cavitation appearing within the interfaces of
interconnected polymer and filler networks. This work introduces a new method
to analyze cavitation in industrial-grade rubbers based on Ultra Small-Angle
X-ray Scattering. This method employs a specially designed multi-sample
stretching device for high-throughput measurements with statistical relevance.
The proposed data reduction approach allows for early detection and
quantification of cavitation while providing at the same time information on
the hierarchical filler structures at length scales ranging from the primary
particle size to large silica agglomerates over four orders of magnitude. To
validate the method, the scattering of SSBR rubber compounds filled with
highly dispersible silica at different ratios was measured under quasi-static
strain. The strain was applied in incremental steps up to a maximum achievable
elongation or breakage of the sample. From the measurements performed in
multiple repetitions, it was found that the minimum strain necessary for
cavity formation and the size evolution of the cavities with increasing strain
are comparable between these samples. The sample with the highest polymer
content showed the lowest rate of cavity formation and higher durability of
silica structures. The structural stability of the compounds was determined by
the evolution of the filler hierarchical structures, obtained by fitting data
across the available strain range.

###### keywords:

Ultra Small-Angle X-Ray Scattering (USAXS), Filled Rubber Compounds,
Cavitation, In Situ Straining

††articletype: Research Article††footnotetext: Abbreviations: SAXS, Small
Angle X-ray Scattering; USAXS, Ultra Small Angle X-ray Scattering; $\mu$CT,
Micro-Computed X-ray tomography; TLM, Transmitted Light Microscopy; TEM,
Transmission Electron Microscopy; SEM, Scanning Electron Microscopy; AFM,
Atomic Force Microscopy; ESRF, European Synchrotron Radiation Facility; SBR,
Styrene Butadiene Rubber; HD, Highly Dispersible; TDAE, Treated Distillate
Aromatic Extract; DIN, German Institute for Standardization; ISO,
International Organization for Standardization; M300, stress at 300 % strain;
USF, Unified Scattering Function

##  1 Introduction

Figure 1: Scheme of the set-up used for the experiments: a specially designed
multi-sample stretching device was positioned on the 3-axis sample table,
allowing consecutive measurement of up to 15 samples; a monochrome camera was
installed in front of the device to register the position of reflecting dots
for strain calibration.

Figure 2: Relationship between stretcher opening and visually measured strain
(distance between reflecting dots): the almost linear relationship between
stretcher opening and strain confirms the reliability of the anti-slip
patterns milled to the clamps that hold the samples.

Rubber and its mixtures are a vital component in industrial use. The
versatility of this material allows it to be utilized in almost all known
methods of transportation (such as cars, planes, boats, trains, etc.), in the
construction of buildings, medical and sports devices, and general consumer
products. Within such a wide range of applications, its main advantage is the
viscoelastic behavior which allows the creation of products with a high degree
of flexibility and high resistance to deformations.

Nevertheless, even the most resistant rubber composites are not perfect: such
factors as aging, temperature variation,1 wear and fatigue under stress,2, 3,
4 or inadequate reinforcement/mixing inevitably introduce damage, that starts
from the cavitation on molecular and sub-molecular levels. The presence of
cavitation phenomena poses a significant challenge across the entire industry.
It often shortens the life span of the final product and negatively affects
desired characteristics. Tires are especially affected by cavitation due to
the technologically complex production process, harsh usage environment, and
extremely high loads during the performance on the road. Therefore, there has
been an ongoing industrial pursuit for the method of detection, evaluation,
and quantification of cavitation, to achieve the final product with optimized
qualities.

A multitude of methods are usually used for the evaluation of cavitation. One
of the oldest and most popular methods for evaluating cavitation, which also
provided the theoretical basis for the phenomena, is tensile testing.4, 2, 5,
6, 7 Although this method allows the collection of statistically reliable
descriptions of the physical properties of the rubber compounds very fast,
direct observation of what is happening in the bulk of the material on a
nanometer scale is beyond its scope. Dilatometry was reported to be used to
validate established models.8 Microscopy methods, including TLM, TEM, SEM,
AFM,9, 10, 11, 12, 13 and dynamic image correlation are often used together
with tensile tests.14, 15 While dynamic image correlation provides insight
into the multi-axial behavior of the sample, microscopy methods offer a
micron-range resolution to observe the cavitation. However, sample preparation
is necessary and these methods enable only the observation of the surface of
the rubber rather than its internal structure. Micro-computed X-ray tomography
($\mu$CT) is often used to access the bulk of rubber composites.16, 17, 2, 15,
13 It allows for analyzing the internal arrangement of rubber composites on a
volumetric scale and observing the structural origin of volume changes in 3D.
However, it requires reasonably fast scanning of the sample and can only
deliver this information within the limits of the available spatial
resolution.

Small-angle X-ray Scattering (SAXS), on the other hand, is a proven method
that can describe hierarchical structures over several orders of magnitude
within a single scattering image. Therefore, it is often used to characterize
the bulk hierarchical structure of the filler within the rubber and follow
structural changes in the sample during in situ experiments on multiple length
scales at the same time.18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 The
technique has few requirements for sample preparation and almost all shapes
standardized by ISO for rubber evaluation can be used for the measurements.
SAXS has already been explored as a means to characterize cavitation within
rubber compounds.1, 30, 31, 32, 33, 34 However, the previous research was
limited by instrumental restrictions on available length scales, and generally
low throughput of the setup, used for measurements.

In this work, we use for the first time the unique capabilities of Ultra
Small-Angle X-Ray Scattering (USAXS) as provided by the beamline ID02 at the
European Synchrotron Radiation Facility (ESRF) in Grenoble, France to study
during in situ stretching both cavitation effects and structural changes in
rubber on the length scale from nanometers to tens of microns,
simultaneously.35 For high throughput, a new sample stretcher has been
developed. By utilizing the state-of-the-art set-up, we were able to obtain
new information related to cavitation phenomena and discuss a comprehensive
approach to the analysis of the recorded data.

##  2 Materials and Methods

A custom-made device has been designed for in situ stretching of rubber, which
enables high sample throughput. Ultra Small-Angle X-ray Scattering is used to
follow the cavitation phenomena and structural changes in the bulk of the
rubber samples during the application of strain. Samples with systematically
varying compositions have been produced to validate the performance of the
proposed experimental set-up and analysis.

###  2.1 Custom-made stretching device

To ensure robust material characterization with a statistical approach that
meets industrial standards, a special (U)SAXS stretching device was developed
to observe structural changes, occurring cavitation, and the breaking of
samples during the application of strain (Figure 1). The in-house developed
sample stretcher enables accurate clamping between two metal plates with
engraved anti-slip patterns that increase the effective surface area and
reduce sample slipping during measurements. Our approach involves the repeated
measurement of multiple individual samples of the same material to ensure
statistical robustness.

The device offers the capability to simultaneously clamp and measure up to 15
samples with a maximum opening of 120 mm. For the experiment described in this
article, the initial sample length was fixed at 15 mm in order to apply strain
theoretically up to 700%. The strain was applied in 50 steps (of 2.1 mm each)
from 15 mm to the maximum opening of 120 mm. At each strain level, the
experimental stage was moved horizontally and SAXS patterns were registered at
the central part of each installed sample. Measurement of one strain state of
the fully loaded stretching device required approximately 1 minute 40 seconds.
Therefore, we consider our measurements quasi-static. All 50 strain states
were measured in about 1.5 hours.

In order to observe the possibility of the sample slipping in the clamping
mechanism during the experiments and to correct it if necessary, we have
introduced a method using light-reflecting markers. These markers were applied
to the center of the sample surface at an initial distance of 10 mm from each
other and tracked optically by a small monochrome camera. In this way, we
could track the actual deformation (strain) of the samples during the
stretching experiments and guarantee precise and reliable strain measurements
throughout the stretching process. Figure 2 shows the evolution of the
measured strain by analyzing the movement of the reflecting dots as a function
of the nominal opening of the stretching device for 3 different samples and 3
measured repetitions for each sample. The good linearity of the curves proves
that sample slipping can be almost neglected. In any case, for maximum
accuracy, all strain values referred to in this article are based on the
optically measured evolution of the reflecting markers.

Figure 3: Schematic layout of a (U)SAXS set-up depicting the incident
($\boldsymbol{k}_{i}$) and scattered ($\boldsymbol{k}_{s}$) wave vectors,
transmitted beam, scattering angle $\theta$, the 2D detector, and the
scattering vector (q) with
$\textbf{\emph{q}}=\boldsymbol{k}_{s}-\boldsymbol{k}_{i}$, where
$|\boldsymbol{k}_{s}|=|\boldsymbol{k}_{i}|=2\pi/\lambda$

###  2.2 (Ultra) Small-Angle X-ray Scattering measurements

The USAXS/SAXS measurements were carried out at the European Synchrotron
Radiation Facility (ESRF) located in Grenoble, France. The Time-Resolved Ultra
Small-Angle X-Ray Scattering (TRUSAXS) beamline ID02 was utilized to obtain 2D
scattering patterns of the specimens using a state-of-the-art 2D photon-
counting pixel array detector Eiger2 X 4M from Dectris, Switzerland.35 The
X-ray wavelength for the measurements was $\lambda$ = 0.101 nm. The set-up was
collimated in the pinhole configuration. The incoming beam was attenuated
using a 50 $\mu$m Molybdenum filter due to the high scattering of samples. The
experiments were performed at room temperature with a variable sample-to-
detector distance of 1 m, 8 m, and 31 m to cover the entire available range of
the scattering vector q:

q
$\displaystyle=|\textbf{\emph{q}}|=\frac{4\pi}{\lambda}\cdot\sin\left(\frac{\theta}{2}\right)$
(1)

from 0.0016 nm-1 to 7 nm-1. The set-up used a circular beam stop of 1 mm in
diameter for 31 m distance and a circular beam stop of 3 mm in diameter for 8
m and 1 m (Figure 3). The sample transmission was accurately recorded using a
diode situated behind the entrance window of the evacuated detector tube.

The measured 2D scattering patterns were normalized to an absolute intensity
scale, cleared from detector gaps, azimuthally regrouped, and averaged to one-
dimensional scattering curves I(q) or $I(\psi)$, with $\psi$ being the
azimuthal angle, using the ID02 standard data reduction process and the
$SAXSutilities\leavevmode\nobreak\ 2$ software package.36, 37 A partial
averaging was often applied as discussed in more detail in section 3. All
measured (U)SAXS data was normalized to mm-1, taking into account the
reduction in sample thickness due to the stretching of the sample. The
correction factor for the sample thickness is proportional to the sample
transmission according to the Lambert-Beer law:

$\displaystyle I_{1}=I_{0}\leavevmode\nobreak\ exp(-\mu t)\ $ (2)

with $I_{0}$ and $I_{1}$ being the incident and transmitted intensity, $\mu$
being the absorption coefficient of the sample, and t the sample thickness.38
Assuming a constant $\mu$ within the sample, the transmission coefficient $T$:

$\displaystyle T=\frac{T_{1}}{T_{0}}$ (3)

has a power law dependence on the change of the sample thickness during the
stretching, and therefore allows calculating the sample thickness from the
measured change in the transmission:

$\displaystyle t_{n\%}=\frac{\ln(T_{n\%})}{\ln(T_{0\%})}\cdot t_{0\%}$ (4)

with $n$ indicating the strain applied to the sample.

###  2.3 Samples

Table 1: Recipes of samples (in Vol%). Components |  Sample A |  Sample B |  Sample C
---|---|---|---
SSBR |  37.68 |  44.42 |  54.50
Silica |  26.21 |  25.05 |  22.80
TDAE |  27.96 |  21.97 |  13.48
6PPD |  0.84 |  0.99 |  1.22
Wax |  0.96 |  1.14 |  1.39
ZnO |  0.19 |  0.22 |  0.27
Steric acid |  0.83 |  0.98 |  1.21
TESPD |  3.83 |  3.61 |  3.32
DPG |  0.60 |  0.56 |  0.52
TBBS |  0.55 |  0.65 |  0.79
Soluble sulfur |  0.35 |  0.41 |  0.50
Table 2: Results of additional standard measurements performed at Continental AG. Measurement type |  Sample A |  Sample B |  Sample C
---|---|---|---
Stress at 300 % strain (M300) |  6.2 MPa |  6.9 MPa |  12.0 MPa
Hardness |  69.5 Shore A |  71 Shore A |  71 Shore A

Three types of rubber compounds were provided by the European automotive parts
manufacturing company Continental AG, located in Hanover, Germany. The basic
idea behind the sample composition was to have controlled variations of the
three main components of the rubber: polymer matrix (styrene butadiene rubber
(SBR)), silica (HD silica SA 160), and oil (treated distillate aromatic
extract (TDAE)). The laboratory compounds were mixed using an internal mixer
applying a standard mixing protocol. In the first mixing step, rubber, filler,
silane, and chemicals are added – the mixing time is sufficiently long to
allow a chemical reaction between silane and silica. Accelerator and sulfur
are added in the final mixing step. Rubber samples have been cured at 160°C
for 20 minutes. Recipes of samples and their respective names can be found in
Table 1. Values for additional standard measurements, performed at the
laboratories of Continental AG for the sample systems can be found in Table 2.
The hardness was measured according to DIN 53 505 and the tensile test was
performed on a universal testing machine according to DIN 53 504. The former
represents a standard measure for the static stiffness of a rubber compound
and is within 1.5 Shore A points very similar between all three samples. From
the tensile test, the stress at 300 % strain (often referred to as M300) is
reported in Table 2 and it is apparent that with increasing amounts of oil and
filler, the M300 is significantly decreasing and hence the overall material
stiffness is reduced.

Sample A has a significantly reduced amount of the polymer in relation to the
filler components: 37.6 vol % of polymer versus 26.2 and 27.9 vol % of silica
and oil; the relation of the polymer to silica and oil is 0.695. Sample B has
a larger SSBR volume with 44.4 vol % of polymer versus 25.0 and 21.9 vol % of
silica and oil; the relation of the polymer to silica and oil is 0.945. Sample
C represents the system where the polymer amount dominates over the filler
amounts within the composite: it has 54.5 vol % of polymer versus 22.7 and
13.4 vol % of silica and oil; the relation of the polymer to silica and oil is
1.5.

All dumbbell-shaped specimens were provided for the X-ray study in a state
without any fatigue history and measured using the stretching device described
in section 2.1. The geometrical parameters of the samples correspond to
American Society for Testing and Materials (ASTM) standard D4482-11(2021).39

##  3 Quantitative analysis

USAXS/SAXS patterns have been recorded over the available q-range between
0.0016 nm-1 and 7 nm-1 (Figure 4) at 3 sample-to-detector distances (31 m, 8
m, 1 m). The scattering at high q (above 1 nm-1) is dominated by scattering
from the polymer matrix and is not further analyzed in this article. The
signal originating from primary silica particles is visible in the q-range
from 0.1 nm-1 to 1 nm-1. In the following analysis, we focus on the analysis
of the USAXS regime between 0.0016 nm-1 and 0.4 nm-1 where we can observe the
hierarchical structure of silica filler particles, their clusters, and fractal
branched structures. Moreover, in a given range the onset of cavitation can be
observed, and the evolution of the number density, and size of cavities can be
determined. This q-range is mostly covered by measurements taken at 31 m
sample-to-detector distance with some complementary information from data
recorded at 8 m (Figure 4, black lines).

Figure 4: I(q) of Sample B at 0% strain: comparison of the scattering
intensity partially averaged in the vertical direction obtained at 3 different
sample-to-detector distances of 31 m, 8 m, and 1 m at the ID02 beamline. The
q-range, where most of the structural changes (including cavitation) are
happening is limited by vertical black lines and mostly covered by
measurements at a 31 m sample-to-detector distance.

Already in the absence of strain (Figure 5, top left), the scattering of
rubber compounds displays an oriented alignment often referred to as a
"butterfly" pattern.18, 19, 30, 32, 34, 33 The pre-orientation is explained by
the rearrangement of silica clusters and interpenetrating mass-fractal
agglomerates during standard mixing procedures and vulcanization.18

Upon stretching, the pre-oriented USAXS "butterfly" pattern reorients in the
direction of the applied strain and the scattering intensifies as the strain
enforces the alignment of the filler network within the rubber to follow the
vector of applied tension (Figure 5, top row 0%, 70%, and 140%). The
reorientation effect has been previously observed in various
investigations.18, 19, 30, 32

When the applied strain goes beyond typically 140% of the original length we
start to see noticeable changes happening in the patterns. A streak,
perpendicular to the strain direction appears and rapidly grows in intensity
(Figure 5, bottom row), superimposing over the "butterfly" scattering of the
filler. Prior studies have reported similar effects, associated with
refraction phenomena occurring at elongated oriented interfaces within the
material’s bulk.40, 41, 42, 43, 44 Based on these previous findings, it can be
concluded that the origin of the observed streak is the interaction of X-rays
with interfaces within the bulk of the material, like the walls of cavities
opened by the applied strain - whereas the "butterfly" patterns originate from
elastic scattering of X-rays from the silica filler network.

Figure 5: Magnification of the evolution of scattering patterns at ultra-small
angles with the applied stain as indicated in the scattering images for the
Sample B. The insets show the full scattering pattern recorded at a sample-to-
detector distance of 31 m in a logarithmic intensity scale. The initial
orientation of the scattering patterns (observed at 0% strain, top left) gets
oriented with increasing strain. The strain was applied in the vertical
direction. An intensity streak in the horizontal direction starts to appear
from 140% strain, indicating the appearance of cavities.

For the quantitative analysis of the observed phenomena, we propose two
methods. The first one analyzes the azimuthal dependence of the scattering
intensity under gradually rising strain. The second method focuses on
analyzing the q dependence of the scattering signal at different strain levels
in the direction parallel and perpendicular to the applied strain. By
employing both methods, we gain complementary information about cavitation as
a function of strain and related structural changes within the rubber matrix.

###  3.1 Analysing the azimuthal dependence of the scattering intensity in the
USAXS regime

By looking into the scattering patterns as a function of the azimuthal angle
we aim to determine the earliest strain state where a streak originating from
the cavitation can be observed. The main difficulty at this step is that the
onset of streak formation has to be detected while being superimposed on an
evolving scattering pattern originating from the alignment of the silica
filler particles.

As illustrated in Figure 6, we average the measured intensity as a function of
the azimuthal angle (indicated by the arrow in Figure 6 (2)) in a q-range in
the USAXS regime between 0.0018 and 0.0035 nm-1. This averaging area is shown
in Figure 6 (2) and (3) as a hatched area, where Figure 6 (3) is an
azimuthally regrouped representation of (1). The resulting averaged azimuthal
profile is shown in Figure 6 (4), identifying the streak at 0°and 180°. The
observed streaks are oriented exactly perpendicular to the applied strain, due
to the fact that most of the cavities are oriented in the direction of the
pull. However, the streaks are superimposed on a strong scattering
contribution originating from the filler particles. The two contributions are
separated by searching for the minimum background value across the integrated
scattering intensity and subtracting it from the resulting azimuthal profile
(indicated by the red line in Figure 6 (4)). With increasing strain, we
observe the streak progressively rising over the signal of the filler and we
can determine precisely the moment when cavities appear. To better visualize
the evolution of the streak formation, we stack the averaged azimuthal
intensity profiles(Figure 6 (4)) as a function of the strain applied to our
samples and create a 3D representation of our averaged intensity (Figure 7 \-
top row).

Figure 6: Procedure for the evaluation of the streak intensity for Sample B at
320% strain (sample strained vertically): (1) \- full scattering pattern
recorded at a single sample-to-detector distance of 31 m; (2) \- zoom in the
central area: the arrow indicates the azimuthal angle, the hatched area
indicates the q-range between 0.0018 nm-1 and 0.0035 nm-1 which was used for
averaging along the scattering vector q; (3) \- azimuthally regrouped
representation of the scattering image (1) with the q-range used for averaging
(hatched); (4) \- averaged intensity as a function of the azimuthal angle. The
streak indicating the presence of cavities is found at 0°and 180°in the
horizontal direction (hatched). The red line indicates the level of scattering
background, originating mostly from the filler particles, that is later
subtracted from the obtained intensity profile.

Figure 7: The top row shows 3D plots of the averaged intensity (determined by
the procedure presented in Figure 6) as a function of the azimuthal angle and
strain for samples A, B, and C. The area under the streaks as a function of
the strain for repeated sample measurements is presented in the bottom row.
The rise of this integrated intensity during the strain application (indicated
by the blue arrow) can be used as a precise measure for the onset of the
cavitation in the samples. Different repetitions of stretching the same
material give very reproducible results with an onset of 140% for all samples
reported in this article. On the other hand, the integrated intensity of the
highest strain states decreases from Sample A to Sample C, indicating a lower
number density of cavities created during the straining of the material. Due
to sample breakage, not all strain values could be measured for some samples.

By looking at the area under the streak (Figure 7, bottom row) as a function
of the applied strain, we learn about the evolution of the number density of
the cavities present in the probed rubber volume. The onset of cavitation is
the point where the intensity of the streak area starts to rise. By looking at
the resulting strain-area graphs, we discovered that the onset of cavitation
in the rubber systems studied in this article stays the same at around 140%
for all three samples. The same onset value can be connected to the fact that
all samples are comparable in the static stiffness usually measured at low
deformation (i.e. hardness (Table 2)). Hence, the formation of cavities at
140% for samples A, B, and C points towards a balance between stiffness
contribution from fillers on the one side and the polymer network on the
other. This is not too unexpected because the samples have been designed to be
balanced regarding these two contributors: an increasing filler amount is
counterbalanced by an increased oil amount (decreasing the network
contribution).

Beyond the indicated onset of cavitation, we also obtain the rate at which the
number density of cavities evolves within the sample as the slope of the
strain-area graphs. Here, the network contribution dominates the stiffness and
therefore the potential resistance towards the evolution of damage. Thus the
observed rate of cavity formation is significantly different between the three
samples, which is clearly observable in Figure 7 (bottom row). This is due to
the fact that the internal stress distribution within the samples is
significantly different due to the varying share of polymer within the volume
of the sample. Since the polymer network has to carry most of the stress after
the filler structures have been broken down, a sample with a reduced amount of
polymer (e.g. sample A) will experience more stress per each supporting
polymer bond. Its vulnerability to a higher number of damages is well shown in
Figure 7 by an increased growth slope.

Figure 8: Calculated average size of cavities as a function of applied strain.
Calculations were performed for multiple measurements of the same material and
showed a very comparable trend.

To determine beyond the onset and cavitation rate also their size we apply the
Ruland Streak Method. This method allows a description of the average
longitudinal extension of cavities, rods, or micro-fibrils from streak-like
scattering signals in case refraction is observed.41, 32 Refraction in our
case comes from interfaces of cavities that are strongly oriented in the
direction of the pull and creates an intensity streak that is well-oriented in
the direction perpendicular to the strain. According to the method, the size
$L$ of the cavities can be calculated from the measured integral breadth of
the streak $B_{obs}$ as a function of the azimuthal angle:

$\displaystyle L$ $\displaystyle=\frac{1}{q_{\text{av}}\cdot B_{\text{obs}}}$
(5)

where $q_{\text{av}}$ is the scattering vector where the angular cut was
calculated - an average value in the USAXS regime between 0.0018 nm-1 and
0.0035 nm-1 for this study. The integral breadth was determined as the area of
a peak divided by the peak height after subtraction of the scattering signal
of the silica filler. Using this calculation we can obtain the average sizes
$L$ of the cavities for each strain value where the streak is visible as
presented in Figure 8.

The evolution of the cavity size between Samples A, B, and C during stretching
is very comparable and does not show differences as strong as in the
previously discussed cavitation rate. Our samples exhibit a substantial
difference in polymer concentration, however, the cavity size experiences only
a slight increase in samples with a higher polymer fraction (Sample C). This
observation leads us to the conclusion that the cavity size manifests as a
more universally applicable effect of a polymer (SSBR), showing less
dependence on the overall composition of the rubber compound. Consequently,
the evolution of the number density of cavities visible through the cavitation
rate emerges as a more crucial parameter for the comprehensive evaluation and
comparison of samples.

By paying attention to the q-dependent analysis presented in the following
section, we will try to clarify the influence of cavity formation on the
interconnected filler structures of the rubber compound.

###  3.2 Analyzing I(q) as a function of the scattering vector in the USAXS
regime.

In general, the scattered intensity as a function of the scattering vector
I(q) delivers valuable information about the hierarchical structure of the
silica filler network within rubber compounds over many length scales. In the
case of stretched rubber, two contributions are observed within the (U)SAXS
scattering patterns. The first one is the butterfly-shaped scattering
originating from the alignment of primary particles, clusters, and larger
mass-fractal silica agglomerates. The second contribution is the refraction
induced by cavities and observed perpendicular to the strain direction. It is
superimposed on an evolving scattering of the filler particles as soon as the
onset of cavitation is reached. Due to its preferential direction
perpendicular to the applied strain, it can be (partly) separated from the
scattering intensity by partially averaging the acquired SAXS patterns.
Therefore, we focus here on two angular ranges for the azimuthal averaging of
the scattered intensity as indicated in Figure 9: The first one is centered
$\pm$10∘ around the strain axis (purple range) and gives information of the
strain-oriented hierarchical arrangement of the silica filler particles in
that direction. It will be referred to as scattering in parallel direction
($\parallel$). The second angular range is centered $\pm$10∘ around the
expected direction of the streak originating from the cavities (red range) and
will be referred to as scattering in the perpendicular direction ($\perp$).
Before the onset of cavitation, structural information about the arrangement
of the silica filler network in that direction can be obtained. Beyond onset,
the signal at higher strain levels is dominated by the refraction originating
from the cavities.

Figure 9: Partial integration scheme: 20 degrees integration ranges in
directions parallel and perpendicular to the strain were calculated. In the
vertical direction, only structural information from filled rubber is
detected. In the horizontal direction refraction coming from the streak
additionally contributes to the signal

To quantitatively describe the structures within the filled rubber compounds,
we employ the Unified Scattering Function (USF) as a well-established
method.45 The USF enables us to assess the dimensions, spatial distribution,
and organization of the different-sized filler structures within the polymer
matrix.18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 46 In rubbers, silica
fillers are usually arranged hierarchically, creating a system composed of $n$
structural levels. In our analysis, we will follow the evolution of a system
composed of 4 structural levels: the first level is located at high q values
and describes the primary silica particles. The second level, between 0.06
nm-1 and 0.15 nm-1, corresponds to the clusters of silica primary particles.
The third level describes structures of higher order observed in the q-range
between 0.004 nm-1 and 0.03 nm-1. The fourth level was used to describe
ramified filler agglomerates with tens of $\mu$m in size.

In USF, each described level is represented as a combination of Guinier and
Power laws. For our compounds, we use the extended version of the function
described by Beaucage (Equations 6 and 7) with additional corrections by
Hammouda (Equation 8) that help us to follow an assumption of silica
fractality within the rubber-embedded filler networks.46, 47, 48

Figure 10: Evolution of the scattering curves for Samples A, B, and C in the
direction parallel ($\parallel$) to the applied strain for different strain
states as indicated in the legend. The continuous lines are unified fits using
4 structural levels.

Figure 11: Evolution of the scattering curves for Samples A, B, and C in the
direction perpendicular ($\perp$) to the applied strain for different strain
states as indicated in the legend. The continuous lines are unified fits using
4 structural levels.

$\displaystyle I(\textit{q})$ $\displaystyle=\sum_{i=1}^{n}\left(G_{i}\cdot
e^{-\frac{q^{2}R_{g,i}^{2}}{3}}+B_{i}\cdot
e^{-\frac{q^{2}R_{g,i+1}^{2}}{3}}\cdot(q_{i}^{*})^{-P_{i}}\right)$ (6)

$\displaystyle q_{i}^{*}$
$\displaystyle=\frac{q}{\left\\{\text{erf}\left(\frac{kqR_{g,i}}{\sqrt{6}}\right)\right\\}^{3}}$
(7)

$\displaystyle B_{i}$ $\displaystyle=G_{i}\cdot
e^{-\frac{1}{2}P_{i}}\cdot\left(\frac{3P_{i}}{2}\right)^{\frac{P_{i}}{2}}\cdot\frac{1}{R_{g}^{P_{i}}}$
(8)

In those equations, I(q) is the scattering intensity, which represents the sum
of contributions from $n$ structural levels within the rubber compound. These
structural levels are characterized by their respective contrast values ($G$
and $B$), radius of gyration ($R_{g}$), and power law exponent ($P$). The
radius of gyration ($R_{g}$) describes the characteristic size of the
structural units, reflecting their spatial distribution within the material.
q* in the equation involves the error function (erf), which enables a smooth
transition between power law and Guinier regions. The parameter $k$ is
empirical and has specific values depending on the nature of the structure,
with approximately 1.06 for $P=2$ and 1 for $P>3$.

The size of the primary silica particles has been fitted from independent SAXS
measurements of the raw silica material (not shown). Their radius of gyration
$R_{g}$ was determined to be 11.5 nm. This size is used consistently in the
USF fits of samples A, B, and C. For the fitting, we used scattering data
recorded at 31 m and 8 m which covers well, after merging them in unique
scattering curves, the relevant part of the full available q-range (Figure 4,
q-range of interest). Fit parameters were determined independently for the
partially averaged data in parallel $\parallel$ and perpendicular $\perp$
direction using the IRENA package in IgorPro.49 Figures 10 and 11 show merged
I(q) curves in logarithmic scale with resulting fits in the corresponding
partial integration ranges. It should be noted again that in the $\perp$
direction, the USF allows us to characterize scattering features of
hierarchical structures in the scattering signal only until the refraction
streak appears. Afterward, the contribution of the streaks dominates and does
not allow any more to extract structural information.

Having a closer look at the evolution of the scattering curves with increasing
strain in $\parallel$ direction (Figure 10) one observes that the scattering
curves of all three samples do not show any evolution at high q values. This
is due to the fact that the scattering curves have been normalized for the
decreasing sample thickness as explained before. Moreover, the size of the
primary particles that contribute in this range is not expected to change.

More significant is the evolution of the scattering curves in the center of
the plots. In the corresponding q-range, we observe the signal of the silica
clusters (second structural level in the USF). A clear shift of the Guinier
level to the left is observed with increasing strain. This is explained by the
fact that the silica clusters get elongated in the same way as the rubber is
stretched in this direction. Further to the left another bump is observed in
the scattering curves. This bump corresponds to the third structural level in
the USF and is later discussed in the text. The rise of the scattering curves
at the lowest q values is due to large silica agglomerates which are, with a
size of more than 10 $\mu$m, beyond the detectable size range of the
instrument.

Looking at the scattering in the $\perp$ direction in Figure 11 one observes
that again no change is observed at the level of primary particles. However,
the Guinier level of the silica clusters moves only very little. A third
structural level is detected further to the left as is the case in the
$\parallel$ direction. The strong intensity increase from lowest q values up
to 0.1 nm-1 at strains of more than 140% is explained by the appearance of the
refraction streak. Therefore, data is not fitted anymore.

Figure 12: Radius of gyration of the silica clusters (second structural level)
of Samples A, B, and C in the parallel and perpendicular direction to the
applied strain. The solid black line indicates the theoretical elongation of
the silica clusters assuming a pure affine deformation without any relaxation
effects. The hashed line shows the moment when the streak is detected in the
scattering patterns. Its presence does not allow the determination of a radius
of gyration in the perpendicular direction beyond this point.

Figure 13: Radius of gyration of the structures observed in the third
structural level of Samples A, B, and C in parallel and perpendicular
direction to the applied strain. The hashed line shows the moment when the
streak is detected in the scattering patterns. Its presence does not allow the
determination of a radius of gyration in the perpendicular direction beyond
this point.

The detailed fit results of the radius of Gyration of the clusters are
reported in Figure 12 for samples A, B, and C. During deformation, stresses
from the polymer matrix are transmitted via the silica-polymer bonds to the
silica clusters. Thus, there will be a balance between these intra-cluster
forces and the forces at the cluster-polymer interface. At low strain, one
observes for all samples a steep rise of the curves following the solid black
line which was added as a guide to the eye. The line indicates the theoretical
elongation of the silica clusters assuming a pure affine deformation without
any relaxation effects, i.e. linear with strain. The increase of $R_{g}$ for
the clusters in the $\parallel$ direction affinely follows the elongation of
the rubber up to about roughly 50%. Until this strain level, the clusters have
a certain degree of freedom to be deformed affinely due to rearrangements of
the silica particles that follow the applied strain direction within the
cluster.

Beyond 50 % strain the size changes are less pronounced with strain and a sub-
affine deformation regime of the clusters can be observed. Most likely, the
observed change in $R_{g}$ growth is caused either by the relaxation of local
stresses due to rearrangements and breakage of silica structures or due to the
formation of early-stage cavities that are still too small to generate a
refraction streak.

With further increasing strain, the $R_{g}$ of the clusters continues to
increase linearly again, however, with different rates according to the
composition of the samples. With decreasing filler levels from sample A to C,
the slope formed by the radius of Gyration is continuously increasing since
there is the most potential for rearrangements and breakage in highly filled
compounds, and hence, more stress relaxation.

Beyond a strain of 200%, the radius of the clusters does not increase anymore
for all three samples thus indicating that the overall forces acting on the
clusters exceed the yield strength of the interfacial bonds, which leads to a
pronounced local breakage within the inter-phase (cavitation) rather than
breakage of the intra-silica bonds of the cluster.

Consequently, beyond this critical macroscopic strain, there is no resulting
deformation of silica clusters possible, which is in good agreement with the
observed evolution of the $R_{g}$. Furthermore, $R_{g}$ is smallest for the
highly filled sample A and largest for the lowest filled sample C, which is
best explained with a local strain amplification due to the presence of filler
and hence a rather early critical strain in a highly filled sample and a more
stable interphase in a low filled compound.

The large achievable radius for Sample C also confirms the high intrinsic
toughness of the silica clusters which is confirmed by the measurements
performed at Continental AG (Table 2) indicating that Sample C is the
stiffest.

We can assume that there exists an optimal polymer-to-filler ratio that is
beneficial for stronger intra- and inter-cluster interactions resulting in a
slower fracture rate of clusters. For Samples A and B, a disproportional
amount of filler volume fraction as compared to polymer explains the lack of
polymer connectivity between silica and polymer. This leads to all stress
being transferred to the filler instead of balancing between filler and
polymer, resulting in filler jamming and a higher cavitation rate. Therefore,
the radius of Gyration for Samples A and B does not reach the same level as
for Sample C.

In perpendicular $\perp$ direction, an affine deformation leads to a decrease
of sizes according to $R_{g}\propto E^{-1/2}$ (with $E=1+\epsilon$ being the
extension ratio and $E$ being the strain), which is in good agreement with the
observed decrease for all samples. Beyond strains close to 100%, there is no
change in observable size most likely due to local stress relaxation effects
that lead to sub-affine deformations of the clusters in $\parallel$ direction.

Extending the analysis to lower q values between 0.004 nm-1 and 0.03 nm-1, we
observe a broad bump already well visible in the unstrained samples. It can be
fitted with a third structural level in the USF. The fit parameters for the
characteristic size in $\parallel$ and $\perp$ directions are shown in Figure
13.

For samples A and B, we observe a continuous decrease of this fit parameter in
the $\parallel$ direction from about 150 nm to 70 nm, whereas in Sample C it
remains constant around 150 nm and rises at strains above 250 % to $R_{g}$
more than 200 nm.

We cannot unambiguously determine the origin of the structural parameter
represented by the observed broad bump. However, we have reason to think that
it represents another hierarchical silica structure, i.e. aggregated silica
clusters, rather than a distance between ordered silica clusters. In the
latter case, the characteristic distance between more or less densely filled
regions in the polymer matrix would be very close to the cluster size and the
fact that it decreases with increasing strain would be difficult to explain.

Interpreting this third structural level as a second order of silica cluster
aggregates, formed from mass-fractal silica clusters, can give us further
insight into the structural stability of the investigated rubber compound
where the polymer-to-filler ratio seems to play an important role.

For samples A and B, the radius of gyration of these cluster aggregates is
rather constant up to 100 % strain. Afterwards, it decreases until an applied
strain of 300 % which is interpreted as a breakdown into smaller sub-units.
For sample C, it remains constant until about 250 % of strain before the
aggregates even expand. As seen before in Fig. 12, the size of the silica
clusters increases at the same time continuously until 200 % of strain.

One has to keep in mind that for samples A and B the share of polymer is less
than 50 vol.%. Hence the majority of the sample is filled with silica or oil
which strongly decreases the overall connectivity within the polymer phase and
most stresses have to be carried by the various structures forming the filler
network. The large-scale filler structures are however not very stable under
higher stresses and break down. 46, 50

In contrast, sample C consists of more than 50 vol.% of polymer and therefore
shows an overall higher connectivity of the polymer network. Better
connectivity helps support filler at higher deformations within the polymer
matrix resulting in an expansion of the cluster aggregates when the expansion
of the silica clusters has reached its maximum size (Fig. 12). Eventually, at
very high strains these stable filler structures of aggregated clusters also
break down due to high local stresses, and hence one observes a decrease in
size.

In the $\perp$ direction, the $R_{g}$ for all three sample types is
independent of the applied strain within the error bars of the fit. This
aligns well with the assumption of the incompressibility of rubber. 51, 52

##  4 Conclusions

For the first time, we report a systematic study of cavity formation in rubber
during in situ stretching in the USAXS regime. The increased sample-to-
detector distance of 31 m allows us to surpass the limitations observed in
previous SAXS studies by probing lower q values, thus facilitating the early
detection of very weak refraction streaks only visible very close to the
direct beam. We describe a methodology to detect the earliest stages of
cavitation by identifying the characteristic intensity streak occurring in the
direction perpendicular to the applied strain and analyzing the angular
dependence of the scattering intensity as a function of the azimuthal angle.
Simultaneously from the same datasets, we extract and describe structural
information about the hierarchical structure of the silica filler particles
within the rubber compound.

To gain statistically robust data, we developed and utilized a sample
environment capable of performing high-throughput measurements under quasi-
static deformation up to the breaking of the sample.

The newly introduced methodology allowed us to characterize in multiple
repetitions the first set of industrial-grade rubber samples with
systematically varying production parameters. Samples, where the filler and
oil content predominates over the polymer content (Samples A and B), showed
higher cavitation rates and instability within filler structures, indicating
that the silica filler network is the main recipient of the incoming stresses.
On the other hand, the sample, where polymer content is dominant over filler
and oil content (Sample C), showed better stability of the filler network due
to the overall higher connectivity of interconnected polymer-filler networks
resulting in better inter-phase interactions between hierarchical silica
structures and the polymer.

The described approach will allow us to perform research on more systematic
compound variation, leading to a more profound understanding of material
physics related to structural changes caused by adjustments in compound
ingredients and applied strain. The introduced methodology simplifies the
analysis of damage evolution and aids in enhancing industrial rubber
compounds.

To improve the level of confidence regarding the discussed silica-structure
mechanics during deformation, more systematic studies using e.g. a wider range
of silica concentration has to be performed. This will help identify more
precisely the critical concentrations of polymer and filler in a filled rubber
nano-composite.

*Author contributions

Ilya Yakovlev: Conceptualization; Investigation; Visualization; Data Curation;
Formal Analysis; Methodology; Writing – Original Draft Preparation. Michael
Sztucki: Conceptualization; Investigation; Supervision; Formal Analysis;
Methodology; Writing – Original Draft Preparation. Hossein Ali Karimi-
Varzaneh: Conceptualization; Methodology; Resources; Supervision; Validation;
Writing – Review & Editing. Jorge Lacayo-Pineda: Conceptualization;
Validation; Resources; Supervision; Writing – Review & Editing. Frank Fleck:
Conceptualization; methodology; Validation; Writing – Original Draft
Preparation. Christoph Vatterott: Conceptualization; Resources; Validation.
Ulrich Giese: Supervision; Validation; Writing – Review & Editing.

*Acknowledgments We want to express our gratitude to Continental AG for providing in-kind support, training, data, and samples; Theyencheri Narayanan, Manfred Burghammer, and all beamline staff for support during development of the sample environment, experiments, and data analysis; ESRF for provision of beamtime on beamlines ID02 and ID13; Deutsches Institut fur Kautschuktechnologie for providing theoretical and practical training on rubber material science; the InnovaXN for the support of the PhD project.

*Financial disclosure

None reported.

*Conflict of interest

The authors declare no potential conflict of interest.

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