# Extending depolarized DLS measurements to turbid samples

Antara Pal  Division of Physical Chemistry, Department of Chemistry, Lund
University, Lund, Sweden.  Peter Holmqvist  Division of Physical Chemistry,
Department of Chemistry, Lund University, Lund, Sweden.  Andrea Vaccaro  LS
Instruments, Fribourg, Switzerland.  Peter Schurtenberger
<EMAIL_ADDRESS>Division of Physical Chemistry, Department of
Chemistry, Lund University, Lund, Sweden.  Lund Institute of advanced Neutron
and X-ray Science LINXS, Lund University, Lund, Sweden

###### Abstract

The application of dynamic light scattering to soft matter systems has
strongly profited from advanced approaches such as the so-called modulated 3D
cross correlation technique (mod3D-DLS) that suppress contributions from
multiple scattering, and can therefore be used for the characterization of
turbid samples. Here we now extend the possibilities of this technique to
allow for depolarized light scattering (Mod3D-DDLS) and thus obtain
information on both translational and rotational diffusion, which is important
for the characterization of anisotropic particles. We describe the required
optical design and test the performance of the approach for increasingly
turbid samples using well defined anisotropic colloidal models systems. Our
measurements demonstrate that 3D-DDLS experiments can be performed
successfully for samples with a reduced transmission due to multiple
scattering as low as 1%. We compare the results from this approach with those
obtained by standard DDLS experiments, and point out the importance of using
an appropriate optical design when performing depolarized dynamic light
scattering experiments with turbid systems.

††preprint: APS/123-QED

## Introduction

Anisotropic building blocks with different shapes such as rods, cylinders and
plates are frequently found in biological macro-molecules, micelles and other
colloidal systems [1, 2, 3]. The shape of the building blocks affects not only
the spatial arrangement for self-assembled structures but also the
translational and rotational dynamics for these systems. For spherical
colloids, their dynamic behavior is well understood, and a very good agreement
has been established between theory, simulations, and experiments [4, 5, 6].
For anisotropic particles, well established theory and predictions can be
found for dilute systems [7, 8, 9, 10], which also have been successfully
compared with experiments for both translational and rotational motions [11,
12, 13, 14, 15, 16] mainly using dynamic light scattering (DLS) and
depolarized dynamic light scattering (DDLS) [17]. DDLS is the most useful tool
for characterizing nonspherical particles, and thus has been applied to
optically anisotropic particles such as the tobacco mosaic virus [18], gold
nanorods [19], spheres with internal optical anisotropy [20, 21, 22, 23],
carbon nanotubes [24], claylike particles [12, 25] and cellulose nanocrystals
[26]. However, all these measurements are limited to rather low
concentrations, beyond which multiple scattering and or absorption by the
samples play a crucial role to obtain reliable data.

Today there are several dynamic light scattering-based measurement techniques
available to measure translational diffusion in turbid samples [27, 28, 29,
30, 31]. However, the rotational diffusion for opaque samples or samples
suffering from multiple scattering is still somewhat inaccessible by the
available measurement techniques. We have now implemented the depolarized
measurement option on the modulated 3D cross correlation technique (mod3D)
[31] in order to perform DDLS experiments with turbid samples to characterize
the rotational motions of anisotropic colloids even in the multiple scattering
regime. With this development one can now investigate the depolarized
scattering on samples with a substantial amount of turbidity. This will not
only greatly increase the concentration range at which these investigations
can be performed, but also the different type of systems that can be measured,
e.g. systems which are made up of inorganic materials.

In our case study we have investigated aqueous dispersions of colloidal
hematite/silica core/shell ellipsoids with two different aspect rations at
several different concentrations as a strongly scattering model system. We
demonstrate the feasibility of 3D-DDLS measurements down to a transmission as
small as 1%. We highlight the importance of an approach capable of suppressing
multiple scattering in DDLS by comparing our data with standard DDLS
measurements and theoretical predictions. We also discuss the experimental
set-up and the technical and experimental challenges that are posed by these
measurements.

## Experimental section

### Materials

### Synthesis and Transmission Electron Microscopy (TEM)

Silica/hematite core/shell ellipsoidal particles of two different aspect
ratios were synthesized in a two step process. First the hematite ellipsoids
were synthesized in water following the approach described by Ocana et al.
[32]. They were then coated with silica layer(s) in ethanol using the method
described by Graf et al. [33]. The aspect ratio of the particles was
controlled by tuning the thickness of the silica shell. Particles were
purified by repeated centrifugation/re-dispersion cycles in water and were
kept in water as a stock dispersion.

Figure 1: (a), (b) Representative TEM images for core-shell hematite-silica
ellipsoidal particles of two different aspect ratios $\rho_{1}$ =2.9, and
$\rho_{2}$=3.7, respectively. (c), (d) represent the statistical distribution
of length scales as has been obtained from (a) and (b).

A transmission electron microscope (TEM) (JEOL 3000F microscope operating at
300 kV) was used to characterize both the size and shape of the particles.
Particle size distributions were calculated by measuring at least 100
particles from TEM micrographs using the software ImageJ. For the batch of
particles that we have named $\rho_{1}$, we found the long and short axes to
be L1 = 316 $\pm$ 26.3 nm and D1 = 108 $\pm$ 7 nm respectively, leading to an
aspect ratio of $\rho_{1}$ = 2.9, while for another batch, named $\rho_{2}$,
L2 = 266 $\pm$ 19 nm and D2 = 72 $\pm$ 6 nm corresponding to $\rho_{2}$ = 3.7.
Fig. 1 (a) and (b) show representative TEM images for the ellipsoids and (c)
and (d) shows the corresponding size distributions.

### 3D-DLS and 3D-DDLS

Figure 2: a) Schematic representation of the 3D-DLS set up with the addition
of polarizers before and after the sample to enable 3D-DDLS measurements. b)
The scattering geometry of the 3D set-up where Ia and Ib represent the vectors
of the two incident beams and Sa and Sb scattering vectors at scattering angle
$\theta$. The transparent triangle represents the mirroring plane between the
two beams and is the plane at which a 2D measurement is performed. $\alpha$ is
the angle between the normal of the mirroring plane and the incident beam and
determine the angle of the scattering cone in 3D c) The scattering geometry of
one of the beams in the scattering volume for the 3D-DLS set-up. $\alpha$ is
the cone angle and $\omega$ is the turning of the polarization. d) Photograph
of the actual motorized set-up at the beam splitter side of the 3DDLS set up.

The DLS measurements were performed on a 3D-DLS Spectrometer (LS Instruments,
Switzerland) that implements the modulated 3D cross correlation technology
[31], equipped with 660 nm Cobolt laser with a maximum power of 100 mW. In the
standard set up the measurements are performed in a VU polarization geometry
where the single beam from the laser is vertically polarized. This means that
the split beams are vertically (V) polarized for the incident beams in 3D
geometry. There are no polarizers at the detector side giving an unpolarized
(U) detector condition. In our modified set-up we have added two Glen-Thomson
polarizers, with an extinction ratio larger than 105, and vertically(V)
orientated them at the detector side, thus giving us a VV polarization
geometry. For the 3D depolarized dynamic light scattering, 3D-DDLS, two
additional identical Glen-Thomson polarizers, were mounted after the beam
splitter in the 3D geometry (figure 2(a)). A VH geometry was then achieved by
minimizing the detected scattering intensity of an isotropic sample containing
spherical particles, prior to experiments at each angle. 5 mm cylindrical
glass cells were used and placed in the temperature-controlled index matching
bath containing decalin. The scattered light was detected within an angular
range of 30 – 135∘ by avalanche photodiodes and processed by an LS instrument
correlator.

### Polarization in 3D-DLS set-up

The 3D cross-correlation technology uses two laser beams that cross each other
within the sample as shown schematically in Fig. 2(a). This then allows for
performing two independent DLS experiments in the same scattering volume using
exactly the same scattering vector $\overrightarrow{q}$ both in magnitude and
in direction (Fig. 2(b)). The intensities measured by the detectors of these
two experiments are then cross-correlated, resulting in an effective
suppression of all multiple scattering contributions. As a result, the
measured cross-correlation function corresponds to the correct intermediate
scattering function without multiple scattering contributions, while the
multiply scattered light only results in a decrease in the intercept of the
correlation function. In order to perform both standard 3D-DLS as well as
3D-DDLS experiments, polarizers need to be added accordingly as shown in Fig.
2(a).

However, due to the non-normal incidence of the incident and scattered beams,
the combination of 3D cross correlation technology and DDLS requires a
different instrument design. In order to perform accurate DDLS measurements
the polarizers at the detectors have to be aligned such that they are
perpendicular to the polarization of the incident beam. In standard 2D
scattering geometry there is a single scattering plane defined by the incident
beam and the scattered light, and the polarizer at the (single) detector needs
to be turned by 90∘ away from the initial vertical polarization direction for
all scattering angles in order to achieve the correct VH polarization
geometry. In contrast, in 3D geometry, the incident beam points towards the
sample at an angle (90∘ \- $\alpha$), which makes the wave vector of the
scattered light follow the surface of a cone (Fig. 2(c)) with an angle
$\alpha$. In order to achieve a correct VH geometry, this then requires an
adjustment of the direction of the polarizers in front of the detectors with a
tilt angle that depends on the scattering angle $\theta$. This is illustrated
in Fig. 2(c), where the direction of polarization for the scattering angle
$\theta=0^{\circ}$ is shown by green lines, whereas pink lines indicate the
same for another scattering angle ($0^{\circ}<\theta<180^{\circ}$). This
change in the direction of polarization given by the tilt angle $\omega$ (as
shown in Fig. 2(c) inset) can be derived analytically as a function of
$\theta$. The corresponding expression for $\omega(\theta)$ is given by
(detailed calculations are included in Appendix):

$\omega=\cos^{-1}\left(\frac{\cos^{2}\alpha\;\cos\theta\;+\;\sin^{2}\alpha}{\sqrt{1-\sin^{2}\alpha\>\cos^{2}\alpha\>(1-\cos\theta)^{2}}}\right)$
(1)

Here $\alpha$ is the angle of the scattering cone due to focusing lens (Fig.
2(a)). This renders DDLS experiments more demanding as the direction of the
polarizers need to be adjusted at each scattering angle, and we have therefore
designed a motorized set of holders that allow for high precision alignment of
the polarizers according to the relation given by eqn. 1 as illustrated in
Fig. 2(d).

Figure 3: Shift of the polarizer, $\omega$, from classical cross polarization
as a function of scattering angle, $\theta$.

### Transmission measurements

The transmission measurements were performed in 5 mm glass tubes using a 660
nm Cobolt laser of 100 mW. The transmission is defined as the ratio between
the intensity measured after a sample with only solvent, $I_{0}$, and after a
sample with particles, $I_{s}$, The transmission, T, is then defined as
$T=I_{s}/I_{0}$.

## Results

To investigate the validity of eqn. 1, an isotropic sample containing
spherical particles was mounted and the incident beams were adjusted to be
vertically polarized. The polarizers at the detectors were turned horizontally
(90∘ in relation to the incident polarization) and then carefully readjusted
from that position until the detected scattering intensity was minimized. In
figure 3 the experimentally observed tilt angle or rotation away from
classical cross polarization, i.e. 90∘ degree turned in relation to the
incident beam, is shown (red symbols) as a function of scattering angle
$\theta$. The line in figure 3 represents the theoretical calculation using
eqn. 1 for an incident angle of the laser beam of $\alpha$ = 2.4∘. It can be
noted that the polarization is not affected at 0∘ and 180∘ and the tilt angle
$\omega$ is symmetric around 90∘ where it has its largest value of about 2.4∘.

This readjustment of the polarizer orientation has the same magnitude for the
two different beams, but they should be turned in opposite directions: one
turns clockwise while the other counterclockwise.

Figure 4: Photograph of aqueous suspensions of silica-coated hematite
ellipsoids (axial ratio $\rho_{2}=3.7$) at 4 different concentrations: from
left to right $c$ = 0.005, 0.05. 0.1, 0.2 wt$\%$.

The performance of the 3D-DDLS implementation described in Fig. 2 was
investigated from a series of test measurements with well-chosen model systems
made by inorganic ellipsoidal colloids with very large scattering cross
section that exhibit multiple scattering already at quite low concentrations,
where interparticle interactions are still negligible. These particles are
made up by a hematite core and an added silica shell which provides both
improved colloidal stability and low polydispersity as well as allows for a
convenient selection of the axial ratio. As described in the materials and
methods section, two batches with different axial ratios were synthesized and
characterized, and the details of their size distributions are given in Fig.
1. A number of samples with different concentrations were then prepared, and
3D-DLS and 3D-DDLS measurements were performed to effectively suppress
multiple scattering. Polarized ($g_{vv}(t)$) and depolarized ($g_{vh}(t))$
intensity crosscorrelation functions were obtained at different angles, and
the corresponding translational and rotational diffusion coefficients were
extracted from the angular dependence of $g_{vv}(t)$ and $g_{vh}(t)$,
respectively.

Due to the high refractive index of the particles, the system gets turbid
already at low colloidal concentrations. This is demonstrated in Fig. 4 with
pictures from different samples in the 5 mm cylindrical glass cells at
concentrations between 0.005 - 0.2 wt$\%$. The black line behind the cells is
only visible for the lowest concentration, whereas the turbidity of the
samples at larger concentrations makes them completely intransparent. From an
optical inspection the particles were however well dispersed and did not
sediment visibly during the measurements. All the samples were measured at
scattering angles between 30∘ and 135∘ scattering angles in steps of 15∘.

### 3D-DLS measurements

Figure 5: Normalized correlation functions, $g_{2}$ \- 1, for VV (black
symbols) and VH (red symbols) measurements for two different aspect ratios,
$\rho_{2}$ = 3.7, concentration 0.05wt$\%$, transmission = 0.3540, (a-c) and
$\rho_{1}$ = 2.9, concentration 0.05wt$\%$, transmission = 0.4145 (d-f). Three
different scattering angles, $\theta$, are shown: 60∘ (a,d), 90∘ (b,e) and
120∘ (c,f).

We first perform 3D-DLS measurements where the measured intensity cross-
correlation function $g_{2}(q,t)$ corresponds to a VV geometry, i.e. we
determine $g_{vv}(t)$. Measuring in this geometry at different angles $\theta$
would require in principle a change in the orientation of the polarizer in
front of the detectors according to eqn. 1 and Fig. 3. While the polarizers
can be adjusted automatically to the correct position according to eq. 1 when
using motorized polarizers, this renders correct 3D-DLS more tedious if the
polarizers need to be changed manually. However, performing these experiments
with a constant orientation that corresponds to the one of the direction of
polarization of the incident beam, i.e. a tilt angle $\omega=0^{\circ}$, will
have a very small effect on the detected scattered light. With a maximum
change of the polarization of 2.4∘ the maximum loss of intensity if the
polarizers are kept in the same orientation as the incident light is less than
0.1%.

Typical examples of the measured normalised intensity cross-correlation
functions, $g_{2}(q,t)-1$, in VV geometry are shown in Fig. 5 for both aspect
ratios $\rho_{1}$ and $\rho_{2}$ and different scattering angles. The
correlation functions are well described by a single exponential:

$g_{2}(q,t)-1=fe^{-2\Gamma t}$ (2)

where $f$ is the amplitude of the correlation function and $\Gamma$ is the
relaxation rate of the density fluctuation (see black lines in Fig. 5). It is
important to point out that in general $g_{vv}(t)$ is not characterized by a
single exponential decay for optically anisotropic particles, since it
contains also contributions from $g_{vh}(t)$ [34, 22]. For the present system
the contribution from the depolarized scattering is more than 100 times
smaller than the polarized part and cannot be seen in $g_{vv}(t)$. However,
for particles with a much larger depolarization ratio such as gold
nanoparticles the depolarized contribution can be seen also in the VV
measurements, resulting in a double exponential decay of $g_{vv}(t)$ [15, 22].

Eqn. 2 was used to analyse the data obtained from the 3D-DLS measurements, and
relaxation rates $\Gamma$ were extracted accordingly. The $q$-dependence of
$\Gamma$ is given by

$\Gamma=D_{T}\;q^{2}$ (3)

Therefore the translational diffusion coefficient $D_{T}$ of the particles can
be obtained from the angular dependence of $\Gamma$ for each sample, as it is
directly related to the slope of the data plotted as $\Gamma$ vs q2. This is
shown in Fig. 6 for the two different aspect ratios at one concentration
(0.05wt$\%$).

Figure 6: The extracted relaxation rate, $\Gamma$, as a function of q2 for VV
(black) and VH (red) measurements for the two different aspect ratios,
$\rho_{2}$ = 3.7 (a) and $\rho_{1}$ = 2.9 (b). Lines are fits to Eqn. 3 for VV
(black) and Eqn. 4 for VH (red). Concentrations used were 0.05wt$\%$ for both
samples.

### 3D-DDLS measurements

In a next step we then also measured $g_{vh}(t)$ for the same samples in
3D-DDLS mode (3D-DLS in VH geometry). For every detection angle $\theta$ we
first determined the polarizer orientation manually using the same approach as
already described for the experimental verification of eqn. 1. Note that no
attempt was made here to also adjust the polarizer orientation for the
incident beam for these experiments, but we will subsequently demonstrate
below the negligible error that arises from this approach for these
experiments. With the motorized polarizers, the variation of the polarizer
tilt angle can of course be controlled remotely, rendering 3D-DDLS
measurements of the angular dependence of $g_{vh}(t)$ much less time
consuming. Examples for the thus obtained correlation functions $g_{vh}(t)$
are shown in Fig. 5 in red, and can be compared with the corresponding
$g_{vv}(t)$ measured for the same systems and scattering angles. If the cross
polarization of the instrument is set up properly, only a single exponential
form of correlation function $g_{vh}(t)$ is expected for dilute samples. Only
for a system close to a phase transition has a non-single exponential shape of
the correlation function in VH been reported [12]. The measured correlation
functions obtained in VH geometry are indeed well-described by a single
exponential given by Eqn. 2, as demonstrated by the red lines in Fig. 5. From
theses fits the relaxation rates $\Gamma$ in VH geometry are obtained and can
be expressed as,

$\Gamma=6\>D{{}_{R}}+D{{}_{T}}\>q^{2}$ (4)

where DR is the rotational diffusion coefficient. In Fig. 6, the
experimentally determined relaxation rates for VH are also plotted as $\Gamma$
vs q2 (red circles), and fitted with eq. 4. Both the VV and VH data show a
clear q2 dependence. Further, as expected from Eqns. 3 and 4, the VV and VH
data are parallel, i.e. they have the same slope.

Figure 7: Transmission (circles) and amplitude of the correlation function for
the VV (squares) and VH (triangles) geometry as a function of ellipsoid
concentration. Figure 8: The extracted relaxation rate, $\Gamma$, as a
function of q2 for VV (black) and VH (red) measurements for $\rho_{2}$ = 3.7
at a concentration of 0.35 wt%, resulting in a transmission of 0.75 %. Lines
are fits to Eqn. 3 for VV (black) and Eqn. 4 for VH (red).

While the strong multiple scattering and thus high turbidity and low
transmission of our samples renders standard DLS and DDLS measurements
impossible, with the Mod3D technology such measurements can still be
performed. The 3D-DLS cross-correlation technique isolates the single
scattering contribution, while all contributions from multiple scattering
enter the base line only and result in a decrease of the signal-to-base line
ratio and thus the intercept of the correlation function. However, for samples
with a transmission $T\gtrsim 10\%$ the quality of the correlation function
data is still excellent and allows for accurate determination of the
rotational and translational diffusion coefficients of the particles as
demonstrated by the data shown in Fig. 6.

There is a direct relation between the transmission and the intercept of the
correlation function, which is demonstrated in figure 7, where the measured
transmission and the amplitude of the correlation functions at
$\theta=90^{\circ}$ are shown for the VV and VH mode, respectively. The
decrease in intercept and thus in signal-to-noise increases the uncertainty in
the values of $\Gamma$ in particular for the VH geometry. This is the result
of a combination of a reduced intensity and thus a reduced signal to noise
ratio for the depolarized scattering contributions as well as an enhanced
proportion of multiple scattering due to the fact that multiple scattering
also randomizes the polarization. At about a transmission of 1%, the signal to
noise ratio in the VH mode is so small that the resulting statistical error of
the extracted relaxation rate $\Gamma$ becomes to large, while in VV mode
somewhat lower transmission values can still be measured. This is further
illustrated with data for particles with $\rho_{2}=3.7$ and a concentration of
0.35 wt%. The sample has a transmission of 0.75 %, and the corresponding
$\Gamma$ vs $q^{2}$ plots for VV and VH are shown in Fig. 8. Fig. 8 clearly
shows that while the data quality for the VV measurements is still very high
and allows for a quantitative determination of $D_{T}$ even at such a low
transmission value, the individual values of $\Gamma$ obtained in VH carry a
larger individual error, making a corresponding determination of $D_{R}$ (and
$D_{T}$) also much more uncertain.

Figure 9: Extracted diffusion coefficients DR (a) and DT (b) (symbols) as a
function of concentration for particles with aspect ratio $\rho_{2}$ = 3.7.
The green line represents the theoretical prediction found in ref [7].

While almost all samples measured were highly turbid, the corresponding
concentrations are very low, almost two orders of magnitude below the disorder
to order transition or kinetic arrest [35]. Therefore interaction or crowding
effects should not affect the diffusion coefficients measurably at these
concentrations, and $D_{T}$ and $D_{R}$ should be independent of
concentration, provided that multiple scattering contributions are properly
suppressed. The experimentally obtained values of DT and DR are shown in Fig.
9 for all concentrations for the aspect ratio $\rho_{2}$ (= 3.7). Both DT and
DR are indeed independent of the concentration, which confirms that 3D-DLS
using the mod3D technique is capable of reliably suppressing multiple
scattering effects and allows us to correctly determine translational and
rotational diffusion coefficients for highly turbid suspensions with
transmission values as low as around 1%. Using the TEM data as input, DT and
DR were calculated using the theoretical approach of Ortega et al. for
elongated particles, and the results are also shown as the green line in Fig
9. Given the known shape imperfections of these particles [36], the agreement
between the theoretical calculation and the measured values is surprisingly
good. Note that two alternative expressions for different geometrical models,
ellipsoids and spherocylinders, as described in Martchenko et al. [13],
resulted in significantly larger discrepancies between theoretical prediction
and experimental results, with DT overestimated by 5-15%, and even larger
discrepancies for DR.

As demonstrated by these measurements, 3D-DLS allows for precise measurements
of translational and rotational diffusion coefficients for anisotropic
particles even with samples that exhibit strong multiple scattering and thus
high turbidity or low transmission. This is especially the case when combining
this technique with the so-called modulated 3D cross-correlation technique
(mod3D-DLS and mod3D-DDLS) that guarantees high amplitudes for the correlation
function in the absence of multiple scattering. However, due to the 3D cross-
correlation geometry measurements in VV and VH geometry require a relatively
tedious measurement protocol, where the orientation of the polarizers and
analyzers need to be adjusted for each scattering angle. We therefore also
investigated the influence of contributions from multiple scattering as well
as incorrect orientation of the polarizers on the outcome of the experiments.
The results are summarized in Table 1. For this purpose a 0.1 wt% samples of
ellipsoids with an average aspect ratio of $\rho_{2}$ = 3.7 was used.

| VV-3D | VH-3D | VH-3D (90∘) | VH-3D (0∘) | VV-2D | VH-2D | VH-3D∗
---|---|---|---|---|---|---|---
$D_{T}/D_{T,0}$ | $1.0$ | $1.0$ | $0.99$ | $0.96$ | $1.05$ | $0.68$ | $1.0$
$D_{R}/D_{R,0}$ | $-$ | $1.0$ | $0.99$ | $1.12$ | $-$ | $1.16$ | $1.0$
Table 1: A comparison of the calculated normalized translational
(D${}_{T}/D_{T,0}$) and rotational (D${}_{R}/D_{R,0}$) diffusion coefficients
obtained for different geometries with a sample of particles with $\rho_{2}$ =
3.7 at a weight fraction of 0.1% and a transmission of 19%, where DT,0 and
DR,0 are the correct values in the absence of effects from multiple scattering
or interaction effects. VV-3D: measurement using mod3D-DLS with the correct
adjustment of the polarizer orientation at each scattering angle. VH-3D:
measurement using mod3D-DDLS with the correct adjustment of the polarizer
orientation at each scattering angle. VH-3D(90∘): measurement using mod3D-DDLS
with a polarizer tilt angle that was correctly determined at 90∘ scattering
angle and then used for all the other scattering angles. VH-3D(0∘):
measurement using mod3D-DDLS with a polarizer tilt angle that was determined
with the direct beam (i.e., at 0∘ scattering angle) and then used for all
scattering angles. VV-2D: standard DLS experiment in 2D with VV geometry.
VH-2D: standard DDLS experiment in 2D with VH geometry. VH-3D∗: measurement
using mod3D-DDLS with the correct adjustment of the polarizer orientation for
the incident beam as well as at each scattering angle

We see clear and characteristic deviations for the different measurement
protocols. The largest errors are obtained when performing DDLS experiments in
standard 2D geometry, i.e. without suppressing contributions from multiple
scattering (VV-2D and VH-2D). This leads to significant errors of up to 30% in
the translational and 16% in the rotational diffusion coefficient. The use of
the mod3D-DLS and mod3D-DDLS techniques lead to a significant reduction of the
error, even without correctly adjusting for the correct tilt angle of the
polarizers at each scattering angle independently. Due to our choice of the
range of angles measured (30${}^{\circ}\leq\theta\leq$ 135∘) and the
symmetrical shape of the function describing the tilt angle $\omega$ around
90∘ scattering angle (Fig. 3), the effect is larger when initially adjusting
the positions of the analyzers with the direct beam, i.e. at 0∘ scattering
angle (VH-3D(0∘)), compared to a protocol where the position of the analyzer
is determined with an isotropic scattering sample at a scattering angle of 90∘
(VH-3D(90∘)) and then used at all angles without further adjustment. However,
it is important to point out that these results are specific for the chosen
sample (optical anisotropy, range of scattering angles measured, degree of
multiple scattering), and the only way to safely exclude artifacts in 3D-DDLS
measurements originating from an incorrect alignment of the polarizers is to
use the correction factor for the tilt angle given by eq. 1 for each angle
measured.

The tilt angle of the polarizers is much less important for the VV geometry.
Here, if the polarizers are not changed with angle $\theta$ according to eq.
1, a part of the polarized light will be blocked differently at each angle and
consequently the intensity vs. scattering angle $\theta$ (or scattering vector
$q$) will not be correct. This effect is however very small for the samples
investigated, $<$0.1%, over the whole angle range and can therefore be
neglected. Secondly, by only turning the polarizers at the detectors, the
incident vertical polarization and the horizontal polarization at the
detectors are not in the same scattering plane. As a result, the scattering
matrix is not nonzero for the non-eigenvectors. We experimentally turned both
the incident polarizers and detector polarizers such that the polarization for
both the incident and the detected scattered beam were in the same scattering
plane (VH-3D*). We could see no difference between those measurements and the
measurements where only the detector polarizers were turned (VH-3D).

## Conclusions

We have extended the established mod3D-DLS technique with its ability to
suppress contributions from multiple scattering in DLS and SLS experiments to
also perform depolarized dynamic light scattering measurements, and therefore
to measure translational and rotational diffusion even for turbid samples. Due
to the optical set-up of the 3D-DLS experiment, the direction of vertical and
horizontal polarization of the scattered light changes with the scattering
angle. While this has minor consequences for measurements using a VV geometry,
it requires a readjustment of the orientation of the polarizers in front of
the detectors at every investigated scattering angle in order to get the pure
depolarized signal. While this renders the corresponding 3D-DDLS measurements
tedious when performing them using manual adjustment of the polarizers, we
have also built a motorized set-up that fully implements the required
measurement protocols needed to perform such experiments. Using well-
characterized colloidal model systems we have then tested this approach, and
in particular investigated the performance as well as the magnitude of
possible artefacts in case the measurement protocol would not be implemented
completely. As a result, we now have a method that allows the conduction of
DDLS experiments also with turbid samples, where multiple scattering would
result in significant artefacts if standard DDLS experiments in 2D would be
performed instead.

Using the 3D-DDLS set-up we have shown that it is possible to perform
depolarized dynamic light scattering on samples with a transmission as low as
1% with good accuracy and reproducibility. The current limitation comes from
the increasing contribution from the multiple scattering, which decreases the
intercept and thus the signal to noise ratio to a level, where in the end no
reliable relaxation times can be extracted. The actual limit does of course
depend on the system used, and turbid samples with a larger depolarization
ratio would likely allow us to push the measuring limit to lower transmission
values. This approach clearly opens up new possibilities to study dynamics of
colloidal systems under non-ideal conditions. In the past, the
characterization of the influence of interparticle interactions on rotational
motion using DDLS was limited to systems that remained transparent up to very
high packing fractions. With the extension of the mod3D crosscorrelation
technique to DDLS experiments this requirement can now be relaxed, and we can
also investigate systems that are not optically matched.

###### Acknowledgements.

We gratefully acknowledge financial support from the European Union’s Horizon
2020 research and innovation program through the European Soft Matter
Infrastructure grant (731019-EUSMI). Crispin Hetherington is acknowledged for
his help during TEM measurement.

## Appendixes

The detailed derivation of the change in the polarization angle for different
scattering angles for the 3D-DLS set-up as presented in eq. 1 is given below.
The 3D set-up has two beams that are focused on the scattering volume of the
sample, see fig. 2 main text. The two beams are mirror images of each other at
the plane where the scattering volume is. Due to this the change in
polarization angle for the two beams are the same but in opposite directions.
Therefore, this derivation is the same for both the beams. The incident beams
and the detectors lay in the same plane. One lens before the sample focuses
the two beams at the scattering volume. A second lens after the sample makes
the two beams parallel again before they are detected. The beams now come in
an angle in the scattering volume due to the scattering geometry and the
scattering geometry of one of the beams can be described as a cone. The other
beam creates the same scattering geometry mirrored in the center of the
scattering volume. The direction of the “beam” along the cone can be described
as a vector, $\vec{S_{c}}$:

$\vec{S_{c}}=\begin{pmatrix}-\tan\alpha\cos\theta\\\ -\tan\alpha\sin\theta\\\
1\end{pmatrix}$ (5)

Where $\alpha$ is the cone angle (the angle of the laser is (90 - $\alpha$))
and $\theta$ is the rotational angle of the cone, i.e. scattering angle.
Normalizing the $\vec{S_{c}}$;

$\left\|S_{c}\right\|=\frac{1}{\cos\alpha}$ (6)

Giving;

$\hat{S_{c}}=\begin{pmatrix}-\sin\alpha\cos\theta\\\ -\sin\alpha\sin\theta\\\
\cos\alpha\end{pmatrix}$ (7)

Vector for the incident beam where $\theta$ = 0:

$\hat{I_{0}}=\begin{pmatrix}-\sin\alpha\\\ 0\\\ \cos\alpha\end{pmatrix}$ (8)

and scattering “beam”:

$\hat{I}_{S_{c}}=\begin{pmatrix}-\sin\alpha\cos\theta\\\
-\sin\alpha\sin\theta\\\ \cos\alpha\end{pmatrix}$ (9)

The vertical polarization of the scattering“beam”, ($\hat{I}^{\bot}_{S_{c}}$)
is normal to the ($\hat{I}_{S_{c}}$) at any $\theta$. To specify
($\hat{I}^{\bot}_{S_{c}}$), the plane at which ($\hat{I}^{\bot}_{S_{c}}$) is
normal to has to be determined. This plane is determined by
($\hat{I}_{S_{c}}$) and ($\vec{I}_{tan}$), where ($\vec{I}_{tan}$) is the
tangent vector of the cone circle. The line that is the tangent of a circle:

$x\cdot x_{0}+y\cdot y_{0}=r^{2}$ (10)

where $x_{0}$ and $y_{0}$ is the point of the circle and r is the radius of
the cone at a distance z from the top of the cone:

$r^{2}=(-z\>tan\alpha)^{2}$ (11)

The vector along the tangent, ($\vec{I}_{tan}$), is:

$\displaystyle\vec{I}_{tan}$ $\displaystyle=\begin{pmatrix}x_{0}\\\ y_{0}\\\
z\end{pmatrix}-\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}$ $\displaystyle x_{0}$
$\displaystyle=-z\tan\alpha\cos\theta$ $\displaystyle y_{0}$
$\displaystyle=-z\tan\alpha\sin\theta$ $\displaystyle y$
$\displaystyle=\frac{(-z\tan\alpha)^{2}+z\tan\alpha\cos\theta\>x}{-z\tan\alpha\sin\theta}$
$\displaystyle=-\frac{z\tan\alpha+\cos\theta\>x}{\sin\theta}$

Thus,

$\displaystyle\vec{I}_{tan}$
$\displaystyle=\begin{pmatrix}-z\tan\alpha\cos\theta\\\
-z\tan\alpha\sin\theta\\\ z\end{pmatrix}-\begin{pmatrix}x\\\
-\frac{z\tan\alpha+\cos\theta\>x}{\sin\theta}\\\ z\end{pmatrix}$ (12)
$\displaystyle=\begin{pmatrix}-(z\tan\alpha\cos\theta+x)\\\
\frac{\cos\theta}{\sin\theta}(z\tan\alpha\cos\theta+x)\\\ 0\end{pmatrix}$ (13)

Normalizing ($\vec{I}_{tan}$) gives:

$\hat{I}_{tan}=\begin{pmatrix}-\sin\theta\\\ \cos\theta\\\ 0\end{pmatrix}$
(14)

The vector normal to the plane, $\hat{I}^{\bot}$, defined by $\hat{I}_{S_{c}}$
and $\vec{I}_{tan}$ is:

$\hat{I}^{\bot}=\hat{I}_{S_{c}}\times\hat{I}_{tan}=\begin{pmatrix}-\sin\alpha\cos\theta\\\
-\sin\alpha\sin\theta\\\
\cos\alpha\end{pmatrix}\times\begin{pmatrix}-\sin\theta\\\ \cos\theta\\\
0\end{pmatrix}$ (15)

The vector that would correspond to the vertical component of the of the cone
at the incident beam ($\theta$=0), ($\hat{I}^{\bot}_{0}$) and the “beam” at
the detector ($\theta=\theta$), ($\hat{I}^{\bot}_{S_{c}}$) would have the
vector representation in the cone geometry as:

$\displaystyle\hat{I}^{\bot}_{S_{c}}$
$\displaystyle=\begin{pmatrix}\cos\alpha\cos\theta\\\ \cos\alpha\sin\theta\\\
\sin\alpha\end{pmatrix}$ $\displaystyle\hat{I}^{\bot}_{0}$
$\displaystyle=\begin{pmatrix}\cos\alpha\\\ 0\\\ \sin\alpha\end{pmatrix}$

To find out how much the scattering polarization deviates from vertical
polarization at the detection due to the cone geometry when the incident beam
is vertically polarized, we project $\hat{I}^{\bot}_{0}$ on $\hat{I}_{S_{c}}$:

$\vec{I}^{s}_{proj}=\frac{\left(\hat{I}^{\bot}_{0}\cdot\hat{I}_{S_{c}}\right)}{\left(\left\|{I}_{S_{c}}\right\|\right)^{2}}\hat{I}_{S_{c}}$
(16)

Since $\left\|{I}_{S_{c}}\right\|^{2}$ = 1,

$\displaystyle\vec{I}^{s}_{proj}=$ $\displaystyle\begin{pmatrix}\cos\alpha\\\
0\\\ \sin\alpha\end{pmatrix}\cdot\begin{pmatrix}-\sin\alpha\cos\theta\\\
-\sin\alpha\sin\theta\\\ \cos\alpha\end{pmatrix}\hat{I}_{S_{c}}$
$\displaystyle\phantom{\vec{I}^{s}_{proj}}=$
$\displaystyle\sin\alpha\cos\alpha(1-\cos\theta)\hat{I}_{S_{c}}$ (17)

The normal to $\hat{I}_{S_{c}}$, $\vec{I}^{\bot}_{proj}$ of this projection of
the vertical component of the incident beam:

$\displaystyle\vec{I}^{\bot}_{proj}=$
$\displaystyle\hat{I}^{\bot}_{0}-\hat{I}^{s}_{proj}$
$\displaystyle\phantom{\vec{I}^{\bot}_{proj}}=$
$\displaystyle\begin{pmatrix}\cos\alpha\\\ 0\\\
\sin\alpha\end{pmatrix}-\sin\alpha\cos\alpha(1-\cos\theta)\begin{pmatrix}-\sin\alpha\cos\theta\\\
-\sin\alpha\sin\theta\\\ \cos\alpha\end{pmatrix}$
$\vec{I}^{\bot}_{proj}=\begin{pmatrix}\cos\alpha\>(1+\sin^{2}\alpha\cos\theta\>(1-\cos\theta))\\\
\sin^{2}\alpha\cos\alpha\sin\theta\>(1-\cos\theta)\\\
\sin\alpha-\sin\alpha\cos^{2}\alpha\>(1-\cos\theta)\end{pmatrix}$ (18)

The angle between $\vec{I}^{\bot}_{proj}$ and $\hat{I}^{\bot}_{S_{c}}$ gives
the turning of the polarization due to the cone geometry:

$\cos\omega=\frac{\vec{I}^{\bot}_{proj}\cdot\hat{I}^{\bot}_{S_{c}}}{\left\|{I}^{\bot}_{proj}\right\|\left\|{I}^{\bot}_{S_{c}}\right\|}$
(19)

where $\omega$ is the turning of the polarization. With
$\left\|{I}^{\bot}_{S_{c}}\right\|$ = 1,

$\left\|{I}^{\bot}_{proj}\right\|=\sqrt{1-\sin^{2}\alpha\cos^{2}\alpha\>(1-\cos\theta)^{2}}$
(20)
$\displaystyle\vec{I}^{\bot}_{proj}\cdot\hat{I}^{\bot}_{S_{c}}$
$\displaystyle=\begin{pmatrix}\cos\alpha\>(1+\sin^{2}\alpha\cos\theta\>(1-\cos\theta))\\\
\sin^{2}\alpha\cos\alpha\sin\theta\>(1-\cos\theta)\\\
\sin\alpha-\sin\alpha\cos^{2}\alpha\>(1-\cos\theta)\end{pmatrix}\cdot\begin{pmatrix}\cos\alpha\cos\theta\\\
\cos\alpha\sin\theta\\\
\sin\alpha\end{pmatrix}\phantom{\vec{I}^{\bot}_{proj}\cdot\hat{I}^{\bot}_{S_{c}}}=\cos^{2}\alpha\cos\theta+\sin^{2}\alpha$
(21)

This gives the turning of the polarization, $\omega$, as a function of
scattering angle, $\theta$, and the incident beam due to the focusing,
$\alpha$:

$\omega=\cos^{-1}\left(\frac{\cos^{2}\alpha\;\cos\theta\;+\;\sin^{2}\alpha}{\sqrt{1-\sin^{2}\alpha\>\cos^{2}\alpha\>(1-\cos\theta)^{2}}}\right)$
(22)

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critical point($\delta=0$) by transforming it from an integral equation(Eq.
(S116)) to an algebraic equation(Eq. (S119)). Even though this is right to do
at $\delta=0$, the price one has to pay is that the solution gives only the
modulus of the gap function $\Delta_{h}(\mathbf{k})$ leading to a functional
degeneracy to the obtained solution, as any function of the form
$\Delta_{h}(k)\,e^{i\,\Phi(\mathbf{k})}$ is also another possible solution.
Away from the critical point ($\delta\neq 0$), one has to solve the integral
equation. In this section, we present our numerical result for the solution of
the non-linear integral gap equation for non-zero values of $\delta$. We first
focus on the linearized gap equation

$\displaystyle\dfrac{1}{\lambda}\Delta_{h}(\theta_{\mathbf{k}})=\bigintsss\dfrac{d\theta_{\mathbf{p}}}{2\pi}\dfrac{\cos^{2}\Bigr{(}\theta_{\mathbf{k}}+\theta_{\mathbf{p}}\Bigr{)}}{\delta^{2}+4\sin^{2}\Bigr{(}\theta_{\mathbf{k}}-\theta_{\mathbf{p}}/2\Bigr{)}}\Delta_{h}(\theta_{\mathbf{p}})=\bigintsss\dfrac{d\theta_{\mathbf{p}}}{2\pi}\,M(\theta_{\mathbf{k}},\theta_{\mathbf{p}})\,\Delta_{h}(\theta_{\mathbf{p}}),$
(S158)

cast into a matrix equation in angular space with eigenvalue labeled as
$\lambda$ and kernel matrix

$\displaystyle
M(\theta_{\mathbf{k}},\theta_{\mathbf{p}})=\dfrac{\cos^{2}\Bigr{(}\theta_{\mathbf{k}}+\theta_{\mathbf{p}}\Bigr{)}}{\delta^{2}+4\sin^{2}\Bigr{(}\theta_{\mathbf{k}}-\theta_{\mathbf{p}}/2\Bigr{)}}.$
(S159)

Here, we suppress the $\delta$ dependence of the matrix $M$ for the simplicity
of the representation. We break Eq. (S158) into different angular momentum
channels and solve for the largest eigenvalue in each channel to find the
leading instability. The relevant channels consistent with the pairing
interaction, Eq. (S115) are $A_{1g}$ (eigen functions are like: $1,\cos
4\theta,\cos 8\theta$ etc), $B_{1g}$ (eigen functions are like: $\cos
2\theta,\cos 6\theta,\cos 10\theta$ etc), and $E_{g}$ (eigen functions are
like: $\cos\theta,\cos 3\theta,\cos 5\theta$ etc). We will refer them as $s,d$
and $p-$ waves respectively. In Fig.S12a, we plot how the largest eigenvalue
$\lambda_{s},\lambda_{d}$ and $\lambda_{p}$ in $s,d,$ and $p-$ wave channel
respectively vary with the nematic mass parameter $\delta$. We find the
leading instability happens in the $s-$ channel followed by $d-$ and $p-$ wave
respectively from the linearized gap equation. In the inset, we plot the ratio
of $\lambda_{d}/\lambda_{s}$ and $\lambda_{p}/\lambda_{s}$ with varying
$\delta$. As $\delta$ goes to zero, these eigen values get closer to each
other and become degenerate at $\delta=0$. Below the transition point, there
could be $2$ situations possible: (a) after the gap in $s-$ wave channel
develops, it acts against the subleading $d$ and $p$ waves and never let them
win. As a result, $s-$ wave remains down to zero temperature, or (b) at some
low temperature there is a second phase transition from $s-$ to either
$s+i\,d$ or $p$ wave. We solve the full non-linear gap equation Eq. (S116)
numerically for $\delta=0.01$ at different temperatures and show our result in
Fig.S12b. Our results support the first case of $s-$ wave gap symmetry being
intact at all the temperature below the critical temperature $T_{c}$.

Even though $s-$ wave wins, we argue there is a possibility of having only
$p-$ wave gap symmetry if one includes the repulsive component of the pairing
interaction presented in Eq. (S110). The repulsive regular part of the intra-
pocket pairing interaction does not scale with the nematic susceptibility and
has only $s-$ and $d-$ wave components. As a result, the pairing strength for
the $p-$ wave is larger than for $s-$ and $d-$ wave when one includes both the
attractive ($V^{h}_{\text{singular}}$) and repulsive
($V^{h}_{\text{regular}}$) components of the effective pairing interaction in
the solution of the non-linear gap equation. The $p-$ wave has to be of the
form $p_{x}\pm i\,p_{y}$, as $|\Delta_{h}(\mathbf{k})|$ is same on the $k_{x}$
and $k_{y}$ axis found from the analysis of section.A5.

We define the gap anisotropy
$\alpha=\Delta_{h}(\theta=\pi/4)/\Delta_{h}(\theta=0)$ as the ratio of the gap
along the $k_{x}=\pm k_{y}$ axis($\theta=\pi/4$) to $k_{x}$ axis($\theta=0$)
show its variation with the nematic mass parameter $\delta$ in Fig.S12c. We
fit our result upto second order in $\delta$. We find a good fit with:
$\alpha(\delta)=2.12\,\delta^{2}+0.44\,\delta$. We numerically obtain the
specific heat coefficient as a function of the reduced temperature $T/T_{c}$
for two different values of $\delta=0.01$ and $0.1$ and show the result in
Fig.S12d. For small but non-zero $\delta$, there will be jump at the
transition point as mentioned before. On the other hand, at the low
temperature the behavior of the specific heat coefficient will be
qualitatively similar to the $\delta=0$ case because of the presence of the
exponentially small gap near the cold spot for small values of $\delta$.

Figure S12: In (a) we plot the largest eigenvalue $\lambda$ of the matrix $M$
defined in Eq. (S159) in different angular momentum channels labeled as
$\lambda_{s},\lambda_{d}$ and $\lambda_{p}$ for the $s,d$ and $p-$ wave
respectively. We represent the ratio of these eigenvalues in the inset and
find $\lambda_{p}/\lambda_{s}<\lambda_{d}/\lambda_{s}<1$ which indicates that
$s-$ wave is the leading instability. With nematic mass parameter decreasing,
the ratio of the eigenvalues becoming closer to each other and approaching
unity which indicates degeneracy among different pairing channels. In (b) we
plot the angular variation of the gap $\Delta_{h}(\theta)$ on the hole pocket
for a set of temperatures. In (c) we plot(brown dots) the ratio of the gap
along the $k_{x}=\pm k_{y}$ axis($\theta=\pi/4$) to $k_{x}$ axis($\theta=0$)
defined as $\alpha$ in the main text as a function of $\delta$ and fit (blue
line) our result upto second order in $\delta$ with the fitting parameters
$\Delta(\pi/4)/\Delta(0)=2.12\,\delta^{2}+0.44\,\delta$. In (d) we plot the
variation of the specific heat coefficient divided by the density of state,
$N_{0}$ with the reduced temperature $T/T_{c}$ for a set of values of
$\delta=0.01$(blue) and $0.1$(orange). The black dashed line represents the
normal state result for the specific heat coefficient and is equal to
$2\pi^{2}/3$.


section [ beforeskip=-1afterskip=.1]section subsection [
beforeskip=-1afterskip=.1]section subsubsection [
beforeskip=-1afterskip=.1]section

Experimental study of humidity influence on triboelectric charging of
particle-laden duct flows
Wenchao Xu, Holger Grosshans

Physikalisch-Technische Bundesanstalt (PTB), Braunschweig, Germany

E-mail<EMAIL_ADDRESS>

## Abstract

Electrostatic charge on powders arises during pneumatic transport due to
particle-particle and particle-surface interactions via triboelectrification.
This is a potential threat to the safety of industrial productions and the
source of numerous fires and dust explosions in the past. Triboelectric
charges are affected by environmental conditions, such as ambient temperature
and relative humidity. In this work, we experimentally investigated the
influence of ambient humidity on the particle charge of gas-solid flows in a
square-shaped duct. Monodisperse PMMA particles are fed into a fully developed
airflow in a PMMA duct and then pass through a metallic duct section. The
charge of particles is measured at the outlet of the metallic duct via a
Faraday cup. By measuring the electrostatic charge under various environmental
conditions, we observed that the electrostatic charge first increases with the
humidity and then decreases when the humidity becomes higher.

Keywords: electrostatics, particle charge, triboelectricity, particle-laden
flow

##  1 Introduction

Dust explosion is one of the most serious and widespread explosion hazards in
the processing industry. In an industrial production involving pneumatic
conveyance of powders, electrostatics require to be paid particular attention
(Abbasi & Abbasi, 2007). In the past, electrostatics has caused numerous fires
and dust explosions (Eckhoff, 2003).

During pneumatic transport, powders undergo undesired charging via
triboelectrification due to particle-particle and particle-surface
interactions. The charge can accumulate on non-conductive materials. When such
an insulator with a high concentration of charge moves close to a blunt
grounded conductor, a "brush discharge" can happen and ignite dust clouds
(Larsen et al., 2001). The charged particles can also agglomerate and deposit
at the inner surface of pipelines or ducts due to particle-wall adhesion.
These dust deposits can result in the plugging of conveyors and lead to a
frequent cleaning of the system (Sippola et al., 2018). Moreover, the dust
deposits can accumulate heat, develop high temperatures in a spot and even
cause internal smouldering fires, which serve as another ignition source for
dust explosions (Eckhoff, 2003).

Although the debate on the mechanism of triboelectrification is still ongoing
(Lacks & Shinbrot, 2019), it is widely acknowledged that the triboelectric
effect is an extremely sensitive phenomenon and differs from numerous
conditions, including the material properties, methods by which the materials
come into contact, temperature and humidity of the surrounding environment,
and other properties.

The influence of environmental conditions on triboelectric charging of
polymers, i.e. ambient temperature and humidity, has been reported in many
publications. Kolehmainen et al. (2017) investigated the triboelectric charge
of polyethylene (PE) particles in a glass container subjected to vertical
vibration at different humidity levels. Applying the effective work function
theory, they established a model predicting the effective work function
difference regarding the humidity and showed the charge in the vibrating
vessel, as well as fluidized bed, decreases non-linearly when the relative
humidity increases. Jantač et al. (2019) reported that saturation charge of PE
particles reduces with increasing air humidity in a shaking apparatus for a
relative humidity ranging from 46% to 67% at 22 ${}^{\circ}\mathrm{C}$.
Although electron pair interaction is widely acknowledged as the fundamental
mechanism of triboelectric charging of polymers, Németh et al. (2003) proposed
that the water adsorption of polymer particles influences the charging
mechanism by introducing an additional ion conductivity into the process.
Recent research reveals that the ambient humidity can even change the polarity
of the tribocharge of non-polar polymers (PC and PVC) after rubbing with
Aluminum samples (Tilmatine et al., 2022).

With the increase of humidity, the saturation charge of a particle can rapidly
drop off after reaching a certain level of humidity. The experimental study by
Cruise et al. (2022) shows that the cut-off humidity level is dependent on
particle size. Smaller particles are more sensitive to humidity and begin to
discharge at a lower humidity level than larger particles.

Despite the plentiful studies related to the influence of ambient humidity on
triboelectric charging, most of them focus on the saturation charge under
various humidity. The particles in this case are charged to saturation via
sufficient frictions or collisions. However, the conclusions from these
studies can not be directly applied to the scenario, in which particles are
charged via a limited number of impacts instead of adequate collisions.

In this paper, we present an experimental study of the influence of humidity
on triboelectric charging of particles in gas-solid flows in a short square-
shaped duct. In this pneumatic conveying setup, the particle samples are
charged by a limited number of collisions.

##  2 Experimental setup

###  2.1 Pneumatic conveying test bench

Figure 1(a) illustrates a schematic sketch of the experiment facility. The
present configuration consists of an air blower, a powder feeder, a square
test duct, and a Faraday cage. The air blower (Moro MHR 452) is placed at the
inlet of the test duct to generate the conveying airflow. The blower is
equipped with a frequency converter (Danfoss FC 51) to control the rotation
speed of the blower, as well as the airflow velocity. A Pitot tube anemometer
is installed along the channel center upstream of the particle inlet to
measure the stream-wise air velocity.

Air from blowerTest ductPMMAObstacleSteelPowder feedingFaradayChargeamplifier&
electrometer300 mm1500 mm500 mm

(a)

Airflow

(b)

Fig. 1: (a) Schematic sketch of the pneumatic conveying system (Grosshans et
al., 2022). The Faraday measures the charge of the powder collected by a
filter at the outlet of the metallic duct. (b) The green cylinder in the
enlarged sketch shows the position of the obstacle in the metallic section.

The test duct includes two sections with the same inner side length of 45
mm$\times$45 mm but different materials. The upstream section is made of
polymethyl methacrylate (PMMA) and has a length of 1.8 m, whereas the
downstream section is made of S195T steel section with a length of 0.5 m.
Particles are fed into the duct in the PMMA section at the feeding position
noted in fig. 1(a) with a distance of 0.3 m from the airflow inlet. In
addition, a steel cylindrical obstacle with a diameter of 17 mm is placed at
the middle of the entrance of the metallic section (see fig. 1(b)). The
obstacle functions as a generic representation of components that are
frequently built in pneumatic conveying systems, such as sensors, screws, or
other pipes. The obstacle together with the metallic duct is grounded so that
the electric potential of the metal surface remains constant.

###  2.2 Measurement equipment

To measure the charge of the particles, a 50 $\mathrm{\upmu m}$ pore size
filter bag is attached at the end of the duct and meanwhile covered by a
Faraday cup (Monroe Electronics 284/22A). The Faraday cup is connected to a
charge amplifier (PCB 44302). Particles are separated from the airflow by the
filter and their charge is measured by the charge amplifier with a systematic
error of 2%. It is worth mentioning that the charge of the electrical circuit
leaks consistently. During the experiment, the time-scale of the charging
measurement was set to be significantly smaller than the charge leakage time-
scale of the electrical circuit, so that the error due to circuit leakage was
negligible.

To reduce the error introduced by the leakage, the time-scale of the charging
measurement is required to be significantly smaller than the charge leakage
time-scale of the electrical circuit.

The temperature and relative humidity are measured by a high accuracy
temperature and humidity sensor (Lascar EL-WiFi-TH+) with a tolerance of
$\mathrm{\pm 0.2\leavevmode\nobreak\ ^{\circ}C}$ for temperature and
$\mathrm{0.1\%}$ for relative humidity.

###  2.3 Experimental procedure

In the experiment, the airflow velocity is regulated to
$13\leavevmode\nobreak\ \mathrm{m/s}$. Due to the character of the blower, the
velocity fluctuates persistently with $\pm 0.15\leavevmode\nobreak\
\mathrm{m/s}$ during the experiment. This velocity is measured by the Pitot
tube anemometer at the centre-line, which corresponds to the maximum velocity
in the cross-section of the airflow. The mean flow velocity, i.e. the mean
conveying velocity, can be estimated by multiplying a coefficient of 0.8
(Susanti & Grosshans, 2020).

After the airflow is stabilized, particle samples are fed into the duct. In
this study, we use monodisperse spherical PMMA particles from EPRUI Biotech
with a diameter of 100 $\mathrm{\upmu m}$. The particles have a material
density of $\mathrm{1.15\leavevmode\nobreak\ g/cm^{3}}$ and a bulk density of
$0.8\leavevmode\nobreak\ \mathrm{g/cm^{3}}$. To avoid measurement error due to
charge leakage and meanwhile not exceed the measurement range of the charge
amplifier, we inject $\mathrm{0.5\leavevmode\nobreak\
g-1.5\leavevmode\nobreak\ g}$ particles into the PMMA duct within less than 10
s in each measurement.

Considering the short duration of each measurement, the feeding of particles
is accomplished manually with a syringe instead of a vibrating feeder. Before
and after each injection, the weight of the syringe is measured by a precise
scale (Kern PCB 3500-2) with $0.01\leavevmode\nobreak\ \mathrm{g}$ accuracy
and the differential of the two measurements returns the weight of the fed
powder. After a set of measurements, the particles collected by the filter are
weighted to verify the measurement during each injection. Because the
conveying system uses a positive pressure configuration, part of the particles
might be blown away during the feeding process. The fine particles may also
agglomerate on the inner wall of the duct. In our experiment, the ratio
between the collected mass and the fed mass is higher than $95\%$, i.e., the
mass loss during the feeding and the pneumatic conveying is less than $5\%$.
Because the majority of the mass loss occurs during the feeding process due to
the positive pressure in the conveying system, the influence of the mass loss
on the charge measurement should be small.

##  3 Experimental repeatability

Being an extremely sensitive process, triboelectric charging is influenced by
various conditions and parameters. To ensure the repeatability of the
experiment, the whole process needs to be precisely controlled. In this
section, we discuss the parameters we suspect to affect the repeatability of
the charge measurement the most, namely the conveying velocity, powder mass
flow rate, and the state of the duct’s surface.

###  3.1 Conveying velocity

The impact velocity of particles on the conveying system, which increases with
the conveying velocity, affects the electrostatic charging. This parameter is
strongly correlated to the practical production process, since a high
conveying speed during pneumatically transporting powdery materials is usually
preferred to avoid deposition and stagnation of powders.

Figure 2 depicts the effect of conveying velocity on particle charge under the
same environmental condition. For an air velocity of 15 m/s, the particles
obviously receive more charge than the lower velocity of 13 m/s. This result
is consistent with the experiments reported by Matsusaka et al. (2000) and
Watanabe et al. (2006). Higher conveying velocity contributes to an increase
of the normal component of the impact velocity, which, according to Hertz
theory, proportionally increases the maximal contact area of an elastically
deformed particle impacting against a plane (Watanabe et al., 2006).
Consequently, it increases the contact area between the particle and the and
promotes triboelectrification.

Therefore, to receive repeatable results, the conveying velocity should remain
constant during all the measurements.

Fig. 2: Particle charge with different airflow velocities at the same ambient
condition ($RH=54\%$, $T=16.3\leavevmode\nobreak\ ^{\circ}C$). The dashed line
represents the average charge at each velocity.

###  3.2 Powder mass flow rate

To ensure a small feeding quantity and a short feeding duration in each
measurement, we injected the powders into the duct manually using an syringe.
In this case,the mass flow rate can not be precisely controlled and inevitably
varies upon each attempt. Our previous study reported that the mass flow rate
affects the powder charging in the particle-laden airflow (Grosshans et al.,
2022). According to the numerical simulations presented by Grosshans et al.
(2022), the wall-normal velocity of particles is faster for a higher mass flow
rate in the dilute particle-laden flow. As a consequence, triboelectric
charging is also promoted, which is analogous to the procedure of increasing
conveying velocity. Moreover, when the mass flow rate further increases, a
high concentration of charged particles can give rise to the surrounding
electric field, and, according to the charge relaxation theory (Matsuyama &
Yamamoto, 1995), reduce the charge a particle can hold.

In the experiment, to reduce the influence of varying mass flow rate, we
repeated multiple times measurements for each experiment and meanwhile tried
to cover a wide range of mass flow rates.

###  3.3 State of the duct’s surface

We suspect the state of the conveying duct’s surface, cleanliness and charge
spots, to have a major impact on the experimental repeatability.

Fine particles can adhere or deposit on the inner wall of the pneumatic
conveying system. With the presence of electrostatic force, particles adhere
stronger (Hays, 1995). The adherence on the surface not only increases the
energy loss during pneumatic transport by increasing surface friction, but
also influences the triboelectric charge.

In our experiment, we occasionally observed significant drops of particle
charge during successive measurements, see blue dots in fig. 3(a). After
properly cleaning the duct with ethanol and drying the test rig, the measured
specific charge is recovered temporally and then continues decreasing (orange
triangles in fig. 3(a)). Nevertheless, such a dramatic decrease of the charge
seldom occurred in the experiment compared to the "normal" experiment (see
fig. 3(b)), in which the measured charging is not correlated with the
cumulative number of measurements.

To avoid the influence of surface contamination and deposition, the duct was
cleaned with ethanol regularly. The experimental data, when such a phenomenon
occurred, were regarded as invalid data and excluded from subsequent
discussions.

(a)

(b)

Fig. 3: Particle charge as a function of the measurement sequence. The charge
of each measurement $\left<Q\right>_{i}$ is normalized by the first
measurement $\left<Q\right>_{1}$. The gray area in (b) marks the measurements
within one standard deviation of the mean charge.

##  4 Charging distribution in the conveying duct

In the reported configuration, particles are released upstream in the PMMA
duct section, so that the particle-laden airflow is fully developed before
entering the metallic section. The volumetric flow rate of the particles
($10^{-1}\leavevmode\nobreak\ \mathrm{cm^{3}/s}$) is much smaller than of the
air ($10^{5}\leavevmode\nobreak\ \mathrm{cm^{3}/s}$). In such a dilute phase
gas–solid duct flow, the effect of particle–particle interactions on particle
charging is negligible. Each particle can freely collide with the inner wall
without hindering from surrounding particles. Although the study focuses on
triboelectrics in the metallic section, charging in the PMMA duct is worth to
be noticed.

(a)

(b)

Fig. 4: (a) Particle charging with different setups. Squares: normal setup
with PMMA duct (PD), metallic duct (MD), and metallic obstacle (MO); circles:
test duct with PMMA duct and metallic duct (no obstacle); triangles: test duct
only with the PMMA section. (b) Charge generation per meter as a fraction of
the total powder charge along the streamwise direction with mass flow rate
0.091 kg/h and centre-line velocity 14.7 m/s from numerical simulation.

Figure 4(a) shows the charge-to-mass ratio of particles,
$\left<\mathrm{Q}\right>$, measured under different configurations with the
same parameter. The blue squares represent the standard configuration, the
orange dots correspond to the setup without the metallic obstacle at the
entrance of the metallic duct, and the green triangles are the setup in which
the metallic components, including the obstacle and the metallic duct, are
completely removed. it is noticed that the charging differences between the
orange dots and green triangles are very small, which indicates the
triboelectrical charging between particles and the metallic duct is very weak.
The blue squares, on the other hand, denote a much higher level of
triboelectrical charging for the setup with the metallic obstacle compared to
the setups without the obstacle. Obviously, the metallic obstacle plays an
essential role in the triboelectrical process.

This result is consistent with our numerical simulations. Figure 4(b) shows
the percentage of particle charges in the powder flow along the streamwise
direction from simulations using our open-source CFD tool pafiX (Grosshans,
2022). Pafix uses an Eulerian–Lagrangian approach, in which the fluid and the
particles are solved respectively in Eulerian and Lagrangian framework. The
triboelectrification is calculated with an empirical charging model, which
approximates the impact charge of a particle in the simulation directly from
similar impacts measured in a single-particle experiment (Grosshans et al.,
2022). A detailed description of the mathematical model and numerical methods
implemented in pafiX was given by Grosshans et al. (2021). According to fig.
4(b), more than half of the triboelectric charging occurs at contacts between
the particles and the cylindrical obstacle.

This can be attributed to the following two reasons. First, due to the short
length of the metallic duct (0.5 m), the particle-wall contacts in the
metallic section occur less frequently on the inner wall of the duct than on
the cylindrical obstacle. Second, the wall-normal velocity of particles is
much higher when colliding with the cylinder compared with contacting the
walls parallel to the stream direction.

Moreover, we observed that the PMMA particles are charged negatively in
measurements without the cylindrical obstacle, see fig. 4(a), which suggests
the PMMA particles acquire electrons when colliding with a wall made by the
same material. This phenomenon is analog to the scenario where particles of
different sizes and the same insulating material contact with each other.
Large particles usually tend to charge positively whereas small particles
charge negatively (Waitukaitis et al., 2014). The PMMA particles can be
regarded as "small particles" when colliding on the PMMA duct, therefore they
get a negative charge after contact.

##  5 Influence of humidity

The laboratory for the experiment is not climate-controlled so that the room
temperature is affected by the day-to-day weather variation. To study the
influence of humidity on particle charge, we carefully recorded the
temperature ($T$) and relative humidity ($RH$) variations in the lab and
performed experiments under different weather conditions. Due to the slow
variation of the room temperature and the short duration of each experiment
(around 30 min), change of the ambient condition during an experiment can be
neglected and the temperatures of the test rig and the room are considered
identical. Thus, we were able to measure the influence of the air properties
on the particle charge for a range of $RH$ = $50\%\leavevmode\nobreak\
-\leavevmode\nobreak\ 74.4\%$ and $T$ = 10.3
${}^{\circ}\mathrm{C}\leavevmode\nobreak\ -\leavevmode\nobreak\
20\leavevmode\nobreak\ ^{\circ}\mathrm{C}$. The temperature and relative
humidity during the experiments are plotted in fig. 5(a).

(a)

(b)

Fig. 5: (a) Distribution of temperature and relative humidity of the
experiment attempts; (b) distribution of temperature and absolute humidity of
the experiment attempts.

To exclude the influence of the fluctuating room temperature on the moisture
content in the ambiance, we introduce the absolute humidity (AH), which is
obtained from the following empirical equation (Ogino et al., 2019):

$AH=\frac{217}{T+273.15}\times
6.1078\times\exp\left(\frac{17.2694T}{T+237.3}\right)\times\frac{RH}{100}$ (1)

where $T$ is the room temperature in ${}^{\circ}\mathrm{C}$ and $RH$ is in
$\%$. The obtained $AH$ is in the unit of $\mathrm{g/m^{3}}$. As shown in fig.
5(b), the experiments are performed in an absolute humidity range of
4.97$\leavevmode\nobreak\ \mathrm{g/m^{3}}$ \- 8.98$\leavevmode\nobreak\
\mathrm{g/m^{3}}$.

(a)

(b)

Fig. 6: Charge-to-mass ratio as a function of (a) relative humidity and (b)
absolute humidity. Each dot represents one measurement attempt. The red dashed
curve represents a second order least squares polynomial fit.

The experimental results are presented in fig. 6, which depicts charge-to-mass
ratio against relative humidity and absolute humidity. In the diagram, each
point represents one measurement by feeding a certain portion of particles in
the duct flow. The feeding mass is not constant but varies with each attempt.
The dots at the same humidity correspond to repeated measurements for each
environmental condition.

In fig. 6, the charge-to-mass ratio is solely described by relative humidity
and shows to be increasing monotonically with the rising relative humidity.
However, given the varying room temperature, the same relative humidity by
different room temperatures does not represent the same moisture content in
the ambient circumstance. Therefore, applying the relative humidity
exclusively is not sufficient to compare the influence of the ambient
moisture.

Figure 6(b) shows the charge-to-mass ratio as a function of absolute humidity,
which is a measure of the actual amount of moisture in the air regardless of
the air’s temperature. As indicated with the polynomial fit curve, the charge-
to-mass ratio first increases and then decreases with increasing absolute
humidity. At first glance, the result seems to exhibit a trend inconsistent
with the prevalent theory, that increased humidity reduces the saturation
charge of polymer particles and increases the decay of electrostatic charge.
However, the scenario in our experiment is different from the typical
experiments with shakers or fluidized beds. Particles in the test duct acquire
only a limited number of contacts or collides with other particles/surfaces.
Therefore, the particles passing through the test rig might not be
sufficiently charged and the measured charge must not be equal to the
saturation charge.

Németh et al. (2003) proposed a theory of the function of water molecules on
tribocharges, that the charge transfer is dominated by electrons at low
humidity, whereas adhered water and ions in water promote triboelectric
charging with increased humidity. This theory well explains the relationship
between the specific charge and the humidity. Electron transfer is the
fundamental process of polymer charging. At lower humidity, the tribocharging
is marginally influenced by water molecules in the ambient. With increased
humidity, the water can swell the surface of the polymer or form absorption
layers onto polymer surfaces. Due to the auto-dissociation of water and the
solvation of impurities on the particle surface during production, this formed
water-containing layer introduces ionic species into the charging process. The
ions can decrease the surface conductivity and intensify the triboelectric
charging.

The thickness of the water-absorption layer is correlated to the amount of
available water in the surrounding air. When the humidity further increases,
the layers get thicker, thus, decrease the upper limit of the surface charge
reduces and impedes the particles from obtaining more charge. Therefore the
specific charge drops at $AH=9\leavevmode\nobreak\ \mathrm{g/m^{3}}$.

Moreover, PMMA is sensitive to the moisture in the surrounding atmosphere due
to its good ability in absorbing water (Németh et al., 2003). Therefore the
PMMA particles are able to quickly form water-containing swollen layers by
absorbing water molecules from the air.

According to our previous single-particle experiment (Grosshans et al., 2022),
in which a 200 $\mathrm{\upmu m}$ PMMA particle was shot onto two parallel
metallic plates, the charging of the particles heavily relies on the first few
contacts when the particle fast collides with a surface. Figure 7(a) displays
the charge and velocity of one particle in a succession of collisions. The
charge transfer during the first two collisions is much stronger than the
following collisions. This behavior suggests that the particle is saturated
after the second impact. Such an efficient charge transfer might be also
related the water-absorption layer on the particle surface, which increases
the efficiency of the triboelectrical charge during an impact.

This effect can be also elucidated using the classical condenser model. In the
condenser model, the charge transferred in a sequence of impacts is
$\mathrm{d}q_{c}/\mathrm{\leavevmode\nobreak\ d}n=kCV$, where $n$ is the
number of impacts, $k$ is the electrification efficiency, $C$ is the
capacitance between the contact bodies and $V$ is the potential difference
between the contact bodies (Matsusaka et al., 2010). The existence of the
water layer promotes the surface charge via increasing the electrification
efficiency $k$, therefore increases the charge during a single contact.

To further validate the influence of the moisture on tribocharging, we
prepared two particle samples for a comparison experiment. Both samples
applied identical 100 $\mathrm{\upmu m}$ spherical PMMA particles. One sample
was stored in a shallow glass tray and exposed in the air at a room
temperature of $12\leavevmode\nobreak\ ^{\circ}\mathrm{C}\leavevmode\nobreak\
-\leavevmode\nobreak\ 16\leavevmode\nobreak\ ^{\circ}\mathrm{C}$ and relative
humidity around $50\%$ for 24 hours, while the other sample was stored in a
dry sealed container.

(a)

(b)

Fig. 7: (a)Charge and velocity of a 200 $\mu m$ PMMA particle in a single
particle experiment (adapted from Grosshans et al. (2022)). (b)Charge-to-mass
ratio of particles stored in different conditions. Blue triangles: particles
stored in a hermetical container; yellow dots: particles exposed to air for 24
hours. ($RH=50.9\%$, $T=12.5\leavevmode\nobreak\ ^{\circ}C$).

Both of the samples were fed into the test duct with the same operating
parameter and the specific charge after conveying was measured, see fig. 7(b).
As expected, the water-contained layer due to the long-time exposure in the
air results in a higher particle charge compared to the particles stored in
the closed container. Moreover, the charge of the previous sample is seemly
more stable, which implies the particles stored in a moister environment are
closer to saturation at the outlet of the duct.

##  6 Conclusion and outlook

In this work, we reported a series of experiments on triboelectric charging of
PMMA particles in a pneumatic conveying duct under various temperatures and
humidities. To investigate the influence of humidity at different
temperatures, instead of relative we applied absolute humidity, which provides
a quantitative indication of the moisture content in the atmosphere.

According to the experimental results, the particle charge first increases
with the rise of the absolute humidity and then decreases when the humidity
exceeds a threshold. This phenomenon can be attributed to the surface layer of
PMMA particles formed by absorbed water molecules from the surrounding air.
This water-containing layer increases surface conductivity, promotes
triboelectric charging, and meanwhile decreases the saturation of surface
charge. In the conveying system with a limited number of collisions, the
particles are not sufficiently charged, therefore the existence of water-
containing layers increases the triboelectric charge. However, when the
humidity becomes higher, the saturation charge decreases dramatically, thus
resulting in a lower charge.

Moreover, we discussed several effects that can hinder reproducing the
experiments. Changing conveying velocity and mass flow rate can influence
particle charging. Adhesion on the inner surfaces of the conveying duct may
reduce the charge of particles. For a repeatable experiment, the conveying
velocity and the mass flow rate should remain consistent through all
measurements, meanwhile, the duct should be cleaned regularly for a consistent
surface state. Our results also reveal that increasing conveying velocity can
raise risks of dust explosion due to stronger triboelectrification,
particularly when there are surfaces or obstacles on the flow path, on which
the particles can collide with a high wall-normal velocity. On the other hand,
from the perspective of preventing static electricity, direct wall-normal
collisions should be avoided by designing a pneumatic system.

The experiment shows that in a system where the particles are not sufficiently
charged, increasing the humidity can intensify triboelectric charging.
However, this study has only applied 100 $\upmu\mathrm{m}$ PMMA particles for
the measurements. It remains an open question if particles with different
sizes or a material with different abilities to take up water can change the
behavior of triboelectric charging in response to varying humidity. This will
be part of a future investigation.

##  7 Acknowledgement

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 947606 PowFEct).

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rewritten, and so a larger solution will not change the satisfiability of
(49). Now, if the $⊆$-constraint is rewritten as in (48) we know that:

  1. 1.

All the $y_p^i$ are of sort $Int$.

  2. 2.

All the $y_p^i$ are such that $k^i ≤y_p^i ≤m^i$.

  3. 3.

There are at least $v_i$ integer numbers in $[k^i,m^i]$—because all the
$y_p^i$ are different from each other.

Therefore, if (49) is unsatisfiable when (50) is satisfiable it must be
because $Φ_1 ∧Φ_2$ implies at least one of the following:

  1. 1.

$x ∈˙A_i ∧ninteger(x)$

  2. 2.

$x ∈˙A_i ∧(x < k^i ∨m^i < x)$

  3. 3.

$size(˙A_i,q) ∧m^i-k^i+1 < q$

$m^i-k^i+1 < q$, $size(˙A_i,q) ∧˙A_i ⊆[k^i,m^i]$ is trivially unsatisfiable.

  4. 4.

$0 ≤p ∧(⋀_j=1^p+1 x_p ∉˙A_i ∧k^i ≤x_p ≤m^i) ∧size(˙A_i,m^i-k^i+1 - p)$

    1. (a)

$⋀_j=1^p+1 k^i ≤x_p ≤m^i$ implies that there are $p+1$ elements in
$[k^i,m^i]$.

    2. (b)

$size(˙A_i,m^i-k^i+1-p) ∧˙A_i ⊆[k^i,m^i]$ implies that there are $m^i-k^i+1-p$
elements in $[k^i,m^i]$.

    3. (c)

$⋀_j=1^p+1 x_p ∉˙A_i$ implies that the $p+1$ elements of 4a are disjoint from
the $m^i-k^i+1-p$ elements of 4b.

    4. (d)

Then, there are $p+1+m^i-k^i+1-p = m^i-k^i+2$ elements in $[k^i,m^i]$ when its
cardinality is $m^i-k^i+1$.

If any of the above makes (49) unsatisfiable, any solution larger than the
minimal will not change that because 1-4 are implied by $Φ_1 ∧Φ_2$ and so any
of its solutions will imply the same. ∎

The next is the proof of Theorem 4.

###### Proof.

Termination of $SAT_[ ]$ is a consequence of: _a)_ termination of $SAT_|⋅ |$
[7, Theorem 3]; _b)_ the individual termination of each new rewrite rule added
to $STEP_S[ ]$; and _c)_ the collective termination of all the rewrite rules
of $STEP_S[ ]$.

Assuming _b)_ and _c)_ , the same arguments used in [7, Theorem 3] can be
applied to Algorithm 1. That is, Algorithm 1 uses $STEP_S[ ]$ instead of the
$STEP_S$ procedure used by $SAT_|⋅ |$ and adds the then branch after the main
loop. $STEP_S[ ]$ differs from $STEP_S$ in the new rewrite rules introduced in
Section 4. Therefore, it is enough to prove that $STEP_S[ ]$ terminates as
$STEP_S$ does. In turn, this entails to prove _b)_ and _c)_ —as done when the
termination of CLP($\mathit{SET}$) and $SAT_|⋅ |$ were proved.

Concerning _b)_ (i.e., individual termination), observe that in Figures 2-5
and A the only self recursive rule is (24). However, termination of (24) is
guaranteed as, at some point, it arrives at the set part of the extensional
set. At this point (24) becomes disabled and (22) or (23) are executed. Note
that (24) is not mutually recursive.

Concerning _c)_ (i.e., collective termination), we can observe the following:

  1. 1.

Rewrite rules of $SAT_|⋅ |$ do not call the new rewrite rules as $L_|⋅ |$ does
not admit integer intervals. Then, if one of the new rewrite rules generates a
constraint dealt by the old rewrite rules, the latter will not call the
former. Then, termination is proved in what concerns the interaction between
new and old rewrite rules.

  2. 2.

Rules (10), (12), (13), (15), (16), (17), (18), (20), (22), (27), (28) and
(29), produce only integer constraints or $true$. All these constraints are
processed by CLP(Q) which only generates integer constraints. Then,
termination is proved.

  3. 3.

Rules (16) and (17) trivially terminate. This is important because other rules
in Figures 2-5 produce constraints of the form $⋅∈[⋅,⋅]$ or $⋅∉[⋅,⋅]$. So when
that happens, termination is proved.

  4. 4.

Rules (22)-(24) are either non-recursive or self recursive, but they never
make a recursion on another rule. This is important because these are the
rules called when the identity (3) is applied. Then, when (22)-(24) are
called, termination is proved.

  5. 5.

Rule (11) generates a $size$-constraint in which case 1 applies; and a
constraint of the form $⋅⊆[⋅,⋅]$ in which case 4 applies. Then, termination is
proved.

  6. 6.

Rule (14) calls rules (16) and (17), in which case 3 applies. Then,
termination is proved.

  7. 7.

Rule (19) generates a $size$ and a $∥$ constraints in which case 1 applies;
and a constraint of the form $⋅⊆[⋅,⋅]$ in which case 4 applies. Then,
termination is proved.

  8. 8.

Rule (21) generates a constraint of the form $⋅∈[⋅,⋅]$ in which case 3
applies. Then, termination is proved.

  9. 9.

Rules (25) and (26) generate equality, $size$ and $un$ constraints in which
case 1 applies. They also generate constraints of the form $[⋅,⋅]= ⋅$ in which
case 5 applies; and constraints of the form $⋅⊆[⋅,⋅]$ in which case 4 applies.
Then, termination is proved.

∎

##  Appendix C Two Operators Definable as $L_[ ]$ Formulas

In this section we show two more operators definable as $L_[ ]$ formulas.

$lb ub(S,Lb,Ub)$ holds if the set $S$ is partitioned into two equal halves one
containing the lower elements ($Lb$) and the other the higher elements ($Ub$).
In particular, if the cardinality of $S$ is odd, the predicate does not hold.

$\begin{split}lb\leavevmode\vbox{\hrule
width=5.0pt}ub&(S,Lb,Ub)\mathrel{\widehat{=}}\\\
&un(Lb,Ub,S)\mathrel{\wedge}Lb\parallel Ub\\\
&\mathrel{\wedge}Lb\subseteq[\leavevmode\vbox{\hrule
width=5.0pt},m]\mathrel{\wedge}size(Lb,k)\\\ &\mathrel{\wedge}m<n\\\
&\mathrel{\wedge}Ub\subseteq[n,\leavevmode\vbox{\hrule
width=5.0pt}]\mathrel{\wedge}size(Ub,k)\end{split}$ (51)

$ssucc(A,x,y)$ holds if $y ∈A$ is the successor of $x ∈A$.

$\begin{split}ssucc&(A,x,y)\mathrel{\widehat{=}}\\\ &x<y\\\
&\mathrel{\wedge}A=\\{x,y\mathbin{\scriptstyle\sqcup}B\\}\mathrel{\wedge}x\notin
B\mathrel{\wedge}y\notin B\\\
&\mathrel{\wedge}un(Inf,Sup,B)\mathrel{\wedge}Inf\parallel Sup\\\
&\mathrel{\wedge}Inf\subseteq[\leavevmode\vbox{\hrule width=5.0pt},x-1]\\\
&\mathrel{\wedge}Sup\subseteq[y+1,\leavevmode\vbox{\hrule
width=5.0pt}]\end{split}$ (52)