11institutetext: Department of Earth and Planetary Science, Weizmann Institute
of Science, Rehovot, Israel
11email<EMAIL_ADDRESS>22institutetext: Hamburger
Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany
33institutetext: Department of Astronomy, Faculty of Physics, Sofia University
“St. Kliment Ohridski”, 5 James Bourchier Blvd., BG-1164 Sofia, Bulgaria
44institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, 69117
Heidelberg, Germany 55institutetext: Department of Physics, Ariel University,
Ariel 40700, Israel 66institutetext: Zentrum für Astronomie der Universtät
Heidelberg, Landessternwarte, Königstuhl 12, 69117 Heidelberg, Germany
77institutetext: Astrophysics Geophysics And Space Science Research Center,
Ariel University, Ariel 40700, Israel

# Analysis of the public HARPS/ESO spectroscopic archive

Ca ii H&K time series for the HARPS radial velocity database††thanks: Tables 1
and 4 are only available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/.

V. Perdelwitz 112255 T. Trifonov 334466 J. T. Teklu 55 K. R. Sreenivas 55 L.
Tal-Or 5577

(Received 13/10/2023; accepted 16/1/2024)

###### Abstract

Context. Magnetic activity is currently the primary limiting factor in radial
velocity (RV) exoplanet searches. Even inactive stars, such as the Sun,
exhibit RV jitter of the order of a few m s-1 due to active regions on their
surfaces. Time series of chromospheric activity indicators, such as the Ca ii
H&K lines, can be utilized to reduce the impact of such activity phenomena on
exoplanet search programmes. In addition, the identification and correction of
instrumental effects can improve the precision of RV exoplanet surveys.

Aims. We aim to update the HARPS-RVBank RV database and include an additional
$3.5$ years of time series and Ca ii H&K lines ($R_{\mathrm{HK}}^{\prime}$)
chromospheric activity indicators. This additional data will aid in the
analysis of the impact of stellar magnetic activity on the RV time series
obtained with the HARPS instrument. Our updated database aims to provide a
valuable resource for the exoplanet community in understanding and mitigating
the effects of such stellar magnetic activity on RV measurements.

Methods. The new HARPS-RVBank database includes all stellar spectra obtained
with the HARPS instrument prior to January 2022. The RVs corrected for small
but significant nightly zero-point variations were calculated using an
established method. The $R_{\mathrm{HK}}^{\prime}$ estimates were determined
from both individual spectra and co-added template spectra with the use of
model atmospheres. As input for our derivation of $R_{\mathrm{HK}}^{\prime}$,
we derived stellar parameters from co-added, high signal-to-noise ratio (S/N)
templates for a total of 3 230 stars using the stellar parameter code SPECIES.

Results. The new version of the HARPS RV database has a total of 252 615 RVs
of 5 239 stars. Of these, 195 387 have $R_{\mathrm{HK}}^{\prime}$ values,
which corresponds to 77% of all publicly available HARPS spectra. Currently,
this is the largest public database of high-precision (down to $\sim 1$ m s-1)
RVs, and the largest compilation of $R_{\mathrm{HK}}^{\prime}$ measurements.
We also derived lower limits for the RV jitter of F-, G-, and K-type stars as
a function of $R_{\mathrm{HK}}^{\prime}$.

###### Key Words.:

stars: activity – planetary systems – techniques: radial velocities – stars:
late-type

##  1 Introduction

While the transit method has led to the detection of a greater number of
exoplanets compared to the radial velocity (RV) method, the RV technique
remains a valuable tool in the detection and characterization of exoplanetary
companions. This is because it does not require continuous observations over
multiple planetary periods, enabling the discovery of planets with orbital
periods of the order of the time span over which spectra have been collected.
This time span can be several decades for stars observed by multiple
facilities. As such, the RV method enables the creation of a representative
sample of planetary companions without the need for continuous coverage.

The yield of a search for substellar companions via the radial velocity (RV)
method using high-resolution spectrographs such as the High Accuracy Radial
velocity Planet Searcher (HARPS) (Mayor et al., 2003) is increased by
optimizing the RV extraction method (e.g. Zechmeister et al., 2018) and by
correcting the RVs for nightly instrumental effects (e.g. Courcol et al.,
2015; Tal-Or et al., 2019). Based on the publicly available HARPS spectra,
Trifonov et al. (2020) published the HARPS-RVBank, a catalogue of $>212\,000$
corrected RVs, which have since been used by the community to confirm or
discover planet candidates (e.g. Palatnick et al., 2021; Sreenivas et al.,
2021) and estimate planet occurrence rates (Bashi et al., 2020).
Aside from the detrimental influence of instrumental effects, stellar magnetic
activity has long been known to be the primary cause of RV jitter for a long
time (Saar et al., 1998). In combination with stellar rotation, the presence
of active regions on the stellar surface can result in quasi-periodic RV
signals mimicking those caused by a substellar companion (Queloz et al., 2001;
Dumusque et al., 2011; Reiners et al., 2013). While this RV noise is louder in
younger and more active stars (Tal-Or et al., 2018; Brems et al., 2019), it
has a more complex dependence on the spectral type (Dumusque et al., 2011).
Even slow rotators with low levels of magnetic activity, such as our Sun,
exhibit RV jitter of a few m s-1 (Marchwinski et al., 2015; Haywood et al.,
2016; Collier Cameron et al., 2019), thus exceeding the nominal precision of
state-of-the-art planet-hunting facilities such as the Echelle Sectrograph for
Rocky Exoplanets and Stable Spectroscopic Observations (ESPRESSO) (Pepe et
al., 2014), and upcoming facilities.

Though efforts have been made to mitigate the effect of magnetic activity with
various methods (Collier Cameron et al., 2021; Binnenfeld et al., 2021), the
most reliable way to exclude false positive detections to date is to measure
the stellar rotation period and demonstrate it to be different from the
planetary candidate signal. This can be achieved by analysing photospheric
activity indicators (i.e. light curves, McQuillan et al., 2014), or
chromospheric lines such as H$\alpha$ or Ca ii H&K (e.g. Amado et al., 2021).
In this publication, we present an update to the HARPS-RVBank, which
incorporates newly available observational data, including 252 615 RVs of 5
239 stars and measurements of the relative emission in the Ca ii H&K lines,
$R_{\mathrm{HK}}^{\prime}$, for 77% of all spectra. In Sect. 2, we provide
details of our re-processing scheme with the Spectrum radial velocity analyser
(SERVAL) pipeline, the nightly zero point (NZP) correction process, and the
derivation of stellar parameters and R’HK from the public spectra. In Sect. 3,
we present our results and provide a discussion on our findings. Finally, in
Sect. 4, we present a summary and our conclusions.

##  2 Updating the HARPS-RVBank

We obtained all available HARPS stellar spectra from the ESO archive
(http://archive.eso.org/wdb/wdb/adp/phase3_main/form) that were observed prior
to January 2022. These spectra were processed in the same manner as described
in Trifonov et al. (2020). In addition to the data reduction software (DRS)
products provided by HARPS, we calculated SERVAL RVs and various activity
indices using the methodology described in Zechmeister et al. (2018, 2020). We
also determined the data-driven NZP corrections for both the DRS and SERVAL
RVs. Figure 1 shows the zero-point variations of post-SERVAL RVs. The $774$
good NZPs (out of $1395$ nights with useful data) have a weighted rms scatter
of $1.8$ m s-1 and a median error of $1.1$ m s-1. Despite the $2020$–$2021$
COVID-related observing gaps, the instrument’s zero-point behaviour does not
differ much from what was reported in Trifonov et al. (2020).

In the first release of the HARPS-RVBank, we only included targets with at
least three usable spectra, as this was the minimum required for constructing
a spectral template and obtaining precise RVs with SERVAL. In this second
release of the HARPS-RVBank, we extended the inclusion criteria to include
targets with a single usable spectrum, significantly increasing the number of
targets. While we could not compute meaningful SERVAL RVs and activity indices
for these targets, we provide the original HARPS-DRS estimates and used their
spectra to estimate Ca ii H&K values, which is the primary focus of this work.
From 2003 to the present, we retrieved 309,425 unique spectra from the ESO
archive. Of those spectra, 12,136 were excluded due to their classification as
solar spectra, Solar System objects, quasars (QSOs), and various transients.
Additionally, 36,765 spectra were not included in the HARPS-RVBank due to
failing the quality control check, which includes criteria – for example –
being taken with an I2 cell, having an extremely low or high signal-to-noise
ratio (S/N), or exhibiting other issues that prevent the extraction of
meaningful results. The new HARPS-RVBank consists of 252 615 unique spectra of
5 239 targets111Instant access to the HARPS-RVBank products is made available
through the Exo-Striker exoplanet toolbox (Trifonov, 2019), which can be
freely accessed at https://github.com/3fon3fonov/exostriker. that have passed
the semi-automated quality control process (see, Trifonov et al., 2020, for
details).

Figure 1: Temporal evolution of NZPs. Black errorbars: HARPS-post SERVAL NZPs.
Cyan dots: Stellar zero-point subtracted RVs of all RV-quiet stars (weighted
RV rms scatter $<10$ m s-1). Red boxes: NZPs that were calculated with too few
($<3$) RVs. Red circles: NZPs with too large uncertainties ($>1.2$ m s-1).
Magenta circles: Significantly deviating NZPs. The RVs in the red-marked
nights were corrected by using a smoothed version of the NZP curve, which was
calculated with a moving weighted average (21-d window). For the ten
significantly deviating NZPs, we adopted their individual (unsmoothed) NZP
values, regardless of their uncertainties. Green dots: NZP values that were
eventually used for the correction.

###  2.1 Derivation of stellar parameters

Table 1: Stellar parameters. The full table is available on the CDS. Name | ra | dec |  Teff |  $\Delta$Teff | log g |  $\Delta$log g | [Fe/H] |  $\Delta$[Fe/H] | v sini |  $\Delta$v sini | $R_{\mathrm{HK}}^{\prime}$ | $\Delta R_{\mathrm{HK}}^{\prime}$ | vflag
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| [deg] | [deg] | [K] | [K] |  |  | [dex] | [dex] |  [km s-1] |  [km s-1] |  |  |
BD+012494 | 163.0321195 | 0.49329 | 5524 | 50 | 4.1 | 0.1 | 0.29 | 0.042 | 4.0 | 0.3 |  1.07$\cdot 10^{-5}$ |  5.63$\cdot 10^{-7}$ | 0
BD+01316 | 26.63371 | 2.69887 | 6370 | 50 | 4.2 | 0.1 | 0.26 | 0.047 | 5.5 | 0.4 |  6.98$\cdot 10^{-6}$ |  3.59$\cdot 10^{-7}$ | 0
BD+034525 | 318.47394 | 4.45293 | 5754 | 61 | 4.1 | 0.1 | 0.05 | 0.05 | 3.9 | 0.2 |  8.80$\cdot 10^{-5}$ |  1.61$\cdot 10^{-5}$ | 0
BD+053805 | 278.947839 | 5.33738 | 5163 | 50 | 3.0 | 0.1 | -0.09 | 0.043 | 4.2 | 0.4 |  6.49$\cdot 10^{-6}$ |  6.87$\cdot 10^{-7}$ | 0

The approach developed by Perdelwitz et al. (2021) requires precise stellar
parameters, namely effective temperature ($T_{\mathrm{eff}}$), surface gravity
($\log{g}$), metallicity ($\mathrm{[Fe/H]}$), and rotational velocity
($v\sin{i}$). In order to ensure that the parameters are derived in as
homogeneously a manner as possible, we used the publicly available code
Spectroscopic Parameters and atmosphEric ChemIstriEs of Stars
SPECIES222https://github.com/msotov/SPECIES (Soto & Jenkins, 2018; Soto et
al., 2021). The single-exposure HARPS spectra for each target were shifted to
the rest frame and co-added. Only the resulting templates with an S/N $\geq
20$ were then processed with SPECIES, using the default line lists implemented
in the code. Fig. 2 shows the distribution of the S/N of all template spectra.

Figure 2: Distribution of the S/N of the spectral templates used to compute
stellar parameters.

Since SPECIES is known to converge towards false effective temperatures for
stars with $T_{eff}\leq 4300$ (Soto & Jenkins, 2018; Soto et al., 2021), we
adopted stellar parameters for all stars with color $B-V>1.24$ from the latest
version of the Transiting Exoplanet Survey Satellite (TESS) input catalogue
(TIC v8.2 Stassun et al., 2019; Paegert et al., 2021). Out of a total of 5 239
stars in the sample, we were able to derive stellar parameters for 3 230 of
them. The discrepancy is caused in most cases by a low S/N of the co-added
spectrum, or stellar contamination by a close companion in others. Figures 3,
4, and 5 show the comparison of the resulting effective temperature, surface
gravity, and metallicity with those published in the latest version of the
TESS input catalogue. While there is generally good agreement between the sets
of parameters, we do advise that systematic errors in any derivation of
stellar parameters may be larger than measured uncertainties (Torres et al.,
2012).
The sample of targets for which all stellar parameters have been derived
solely based on the HARPS spectra corresponds to a total of 172,948 single
spectra, or 68.6% of the total. In order to increase this fraction, the
remaining stars were cross-matched with the TESS input catalogue, which
contains the effective temperature, surface gravity, and metallicity, but no
rotational velocity. We, therefore, crossmatched the remaining sources with
the Gaia DR3 catalogue (Gaia Collaboration et al., 2021a, b), yielding an
additional 54 rotational velocity values. In the stellar parameter catalogue
published along with this paper (see Table 1), which is available at the
CDS333https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/vvv/App, these targets
are marked with $vflag=2$ and those without any rotational velocity with
$vflag=1$.

Figure 3: Comparison of the effective temperature with literature values. The
y-axis represents the values published in Stassun et al. (2019); Paegert et
al. (2021), and the black dashed line represents equality. We note that the
apparent tail of deviations towards lower temperatures is comprised of stars
for which the RV was not known, and for which the stellar parameter
determination failed. All stars with a missing RV value and possible errors in
the catalogue are marked via the vflag column in our catalogue.

Figure 4: Comparison of the surface gravity with literature values. The y-axis
represents the values published by Stassun et al. (2019); Paegert et al.
(2021), and the black dashed line represents equality.

Figure 5: Comparison of the metallicity with literature values. The y-axis
represents the values published by Stassun et al. (2019); Paegert et al.
(2021), and the black dashed line represents equality. Figure 6: Wavelength
bands used for normalization and rectification of the spectra. The black lines
show three examples of PHOENIX spectra with different stellar parameters.

###  2.2 Extraction of $R_{\mathrm{HK}}^{\prime}$

We followed the recipe of Perdelwitz et al. (2021), namely the computation of
a grid of fluxes in six bands from PHOENIX models (Husser et al., 2013), with
varying $T_{\mathrm{eff}}$, $\log{g}$, and $\mathrm{[Fe/H]}$, where the grid
step size follows that of the PHOENIX grid (see Table 1 in Husser et al.,
2013). We adapted the edges of one bandpass ($k_{1}$) with respect to those
used in Perdelwitz et al. (2021) in order to minimize the number of absorption
lines and hence the influence of errors caused by the determination of stellar
metallicity. Tab. 2 lists the updated bandpasses, and fig. 6 displays the
bandpasses for three example model spectra. In addition to the parameters
given by the PHOENIX grid, the spectra were artificially broadened with
velocities in the range 1-200 km s-1 in order to account for stellar rotation.
The measured spectra were then rectified in the regions of the Ca ii H$\&$K
lines and normalized with regard to flux, the photospheric contribution in the
lines was subtracted, and the resulting chromospheric excess was normalized
with the bolometric flux, yielding $R_{\mathrm{HK}}^{\prime}$. As mentioned in
Sect. 2.1, for some stars in the stellar parameter catalogue we were not able
to determine $v\sin{i}$. During data reduction, $v\sin{i}$ was set to 2 km s-1
for these targets. While the time series of these stars can be used to
determine rotation periods or activity cycles, $R_{\mathrm{HK}}^{\prime}$ of
these stars may be biased, which is why these entries are marked with the
$vflag$ in both the stellar parameter and the time series table.

Table 2: Updated wavelength bands used for rectification and flux extraction.
The changes with respect to Perdelwitz et al. (2021) are highlighted.

Band |  Wavelength range [$\lambda_{\rm min}$, $\lambda_{\rm max}$]a
---|---
$K$ |  [$\lambda_{K}-0.6$ Å, $\lambda_{K}+0.6$ Å]
$H$ |  [$\lambda_{H}-0.6$ Å, $\lambda_{H}+0.6$ Å]
$\mathbf{k_{1}}$ |  [$\lambda_{K}\mathbf{-2.3}$ Å, $\lambda_{K}\mathbf{-1.8}$ Å]
$k_{2}$ |  [$\lambda_{K}+2.5$ Å, $\lambda_{K}+3.0$ Å]
$h_{1}$ |  [$\lambda_{H}-1.75$ Å, $\lambda_{H}-1.25$ Å]
$h_{2}$ |  [$\lambda_{H}+2.0$ Å, $\lambda_{H}+2.5$ Å]
444 $a$$a$footnotetext: Central wavelengths: $\lambda_{K}$ = 3933.66 Å and
$\lambda_{H}$ = 3968.47 Å.

We determined $R_{\mathrm{HK}}^{\prime}$ for two sets of spectra:

(i): Individual measurements. To provide time series for individual targets,
these values and their uncertainties are included as additional columns in the
updated HARPS-RVBank. Since the purpose of the single-measurement catalogue is
the study of the time series of individual stars, $\Delta
R_{\mathrm{HK}}^{\prime}$ was derived purely based on the uncertainty as to
the flux.

(ii): Co-added templates. To obtain an averaged, high-S/N value per target,
the resulting $R_{\mathrm{HK}}^{\prime}$ are listed in the catalogue of
stellar parameters described in Sec. 2.1. In order to allow for a comparison
of the average activity levels of different stars, here, the uncertainties of
the stellar parameters were included in the Monte Carlo approach for the error
estimation.

##  3 Results and discussion

Our analysis resulted in a total of 195 387 $R_{\mathrm{HK}}^{\prime}$
measurements, or $\sim 77\%$ of the entire HARPS RVBank. In the following, we
give a short discussion of the relationship between $R_{\mathrm{HK}}^{\prime}$
and RV jitter, and compare our findings to previous results. Table 3 gives an
overview of the statistics of the parameter derivation.

Table 3:  Overview of the number of spectra, and the source of the stellar parameters used for the derivation of $R_{\mathrm{HK}}^{\prime}$. Sample | number of spectra
---|---
HARPS spectra that passed our quality control | 252 615
$R_{\mathrm{HK}}^{\prime}$ derived with SPECIES parameters | 158 507
$R_{\mathrm{HK}}^{\prime}$ derived with TIC parameters | 36 880
Total | 195 387

###  3.1 RV jitter as a function of activity and spectral type

The number of stars of spectral types F, G, and K in the sample is large
enough to derive lower limits on the RV jitter as a function of the spectral
type and activity level. Figure 7 shows the RV jitter $\sigma_{RV}$ as a
function of $R_{\mathrm{HK}}^{\prime}$ derived from the co-added spectra for
all main sequence stars of spectral types F, G, or K in the sample with at
least 20 individual measurements. Here, we also excluded binaries with an RV
jitter of $\sigma_{RV}>200$ m s-1. For each spectral type, the sample was
subdivided into activity levels with step sizes of $0.25$ in
log($R_{\mathrm{HK}}^{\prime}$), and the $10^{th}$ percentile of all RVs was
adopted as the lower limit for each bin.

We derived the best-fit linear approximation for the lower limit of RV jitter
of F-type stars to be

$\log{\sigma_{RV}\mathrm{[m\,s}^{-1}\mathrm{]}}=0.81\times\log{R_{\mathrm{HK}}^{\prime}}+4.55,$
(1)

to be

$\log{\sigma_{RV}\mathrm{[m\,s}^{-1}\mathrm{]}}=0.91\times\log{R_{\mathrm{HK}}^{\prime}}+4.91$
(2)

for G-type ones, and

$\log{\sigma_{RV}\mathrm{[m\,s}^{-1}\mathrm{]}}=0.57\times\log{R_{\mathrm{HK}}^{\prime}}+3.04$
(3)

for K dwarfs.

Figure 7: RV jitter as a function of activity. The colour coding denotes the
spectral type, and the dotted, solid, and dashed line correspond to the lower
RV jitter for spectral types F, G, and K, respectively.

###  3.2 Comparison to previous publications

Gomes da Silva et al. (2021) recently published a catalogue of chromospheric
activity for a sample of 1674 FGK stars based on HARPS data. Since the
underlying spectra are entirely included in our dataset, we compared the
median $R_{\mathrm{HK}}^{\prime}$ from their catalogue to that derived from
our template spectra. Fig. 8 shows that there are two types of systematic
offset: (i) Cooler stars appear more active and (ii) most inactive stars
exhibit lower $R_{\mathrm{HK}}^{\prime}$ in our catalogue, both relative to
the values given by Gomes da Silva et al. (2021). While identifying the source
of these systematics is beyond the scope of this paper, we tentatively
conclude that it originates in the different methods for the extraction of
$R_{\mathrm{HK}}^{\prime}$. Our approach takes into account all stellar
parameters including metallicity and rotational velocity, whereas the standard
method of deriving $R_{\mathrm{HK}}^{\prime}$, as carried out in Gomes da
Silva et al. (2021), is the extraction of the Mount Wilson $S$ index, which is
subsequently converted to the excess chromospheric emission using a conversion
based solely on the stellar colour ($B-V$).

Neglecting some stellar parameters in extracting $R_{\mathrm{HK}}^{\prime}$
may be an oversimplification. For example, Schröder et al. (2009) found that
fast stellar rotation leads to the Ca ii H$\&$K lines to be filled in, thus
increasing the flux as determined by the classical Mount Wilson method.
Furthermore, it has been found that $R_{\mathrm{HK}}^{\prime}$ is dependent on
the stellar metallicity (Saar & Testa, 2012). We conclude that any derivation
of $R_{\mathrm{HK}}^{\prime}$ via a conversion factor that depends only on the
stellar colour will likely be biased. Furthermore, as described in Perdelwitz
et al. (2021), the method of using narrow bands close to the cores of the Ca
ii lines to rectify and normalize the spectra, while resulting in larger
statistical errors compared to the classical approach, is less prone to
systematics caused by irregularities in the order-merging or normalization of
the pipeline-reduced spectra.

##  4 Summary and conclusions

We present an update to the HARPS-RVBank with newly calculated RVs, nightly
zero points, and an addition of $R_{\mathrm{HK}}^{\prime}$ for 77% of all
spectra. Table 4 shows an example of five (out of $252\,615$) rows of the full
catalogue. For 3 230 stars, we were able to derive stellar parameters using
co-added template spectra, which are available as a separate table. The
updated RVs will enable the detection of new planet candidates in the same
manner as the previous version of the HARPS-RVBank (e.g. Feng et al., 2020;
Sreenivas et al., 2021).

Figure 8: Comparison of our median $R_{\mathrm{HK}}^{\prime}$ values to those
derived by Gomes da Silva et al. (2021). The colour coding denotes B-V.

The addition of the relative chromospheric emission in the Ca ii H&K lines,
$R_{\mathrm{HK}}^{\prime}$, provides an additional tool for the study of
stellar magnetic activity and the identification of false-positive planet
detections caused by stellar rotation. The updated HARPS-RVBank has already
been used in several works on planet detection (Trifonov et al., 2022),
stellar activity jitter (Kossakowski et al., 2022), and magnetic activity
cycles (Fuhrmeister et al., 2022). While our $R_{\mathrm{HK}}^{\prime}$ values
show colour-dependent systematic deviations compared to those derived in the
classical way, that is, via the S index, we conclude that this is caused by
the simplification made in the conversion with a factor that is solely a
function of the stellar colour $B-V$.

Table 4: Updated HARPS-RVBank. The dots represent the rest of the columns from the previous version (Trifonov et al., 2020). The full table is available on the CDS. The vflag and mflag columns denote stars for which rotational velocity or metallicity could not be determined.  Name | RA | Dec | BJD | … | $R_{\mathrm{HK}}^{\prime}$ | $\Delta R_{\mathrm{HK}}^{\prime}$ | $vflag$ | $mflag$
---|---|---|---|---|---|---|---|---
|  [∘] |  [∘] | [d] |  |  |  |  |
BD+012494 | 163.0321195 | 0.49329 | 2458249.6510659 | … | 1.60E-5 | 1.15E-6 | 0 | 0
BD+012494 | 163.0321195 | 0.49329 | 2458251.5921099 | … | 1.43E-5 | 9.92E-7 | 0 | 0
BD+012494 | 163.0321195 | 0.49329 | 2458262.4807279 | … | 1.70E-5 | 1.04E-6 | 0 | 0
BD+012494 | 163.0321195 | 0.49329 | 2458263.543513 | … | 6.60E-6 | 1.11E-6 | 0 | 0

###### Acknowledgements.

V.P. acknowledges funding through the Kimmel prize fellowship of the Center
for Earth and Planetary Science at the Weizmann Institute of Science, and the
Dean’s fellowship of the faculty of chemistry (WIS). T.T. acknowledge support
by the BNSF programme ’VIHREN-2021’ project No. KP-06-DV-5/15.12.2021. T.T.
acknowledges support by the DFG Research Unit FOR 2544 ’Blue Planets around
Red Stars’ project No. KU 3625/2-1. This work was also funded by the Israel
Science Foundation through grant No. 1404/22.
Based on observations made with ESO Telescopes at the La Silla Paranal
Observatory under programme IDs 0100.C-0097, 0100.C-0111, 0100.C-0414,
0100.C-0474, 0100.C-0487, 0100.C-0708, 0100.C-0746, 0100.C-0750, 0100.C-0808,
0100.C-0836, 0100.C-0847, 0100.C-0884, 0100.C-0888, 0100.D-0176, 0100.D-0273,
0100.D-0339, 0100.D-0444, 0100.D-0535, 0100.D-0776, 0101.C-0106, 0101.C-0232,
0101.C-0274, 0101.C-0275, 0101.C-0379, 0101.C-0407, 0101.C-0497, 0101.C-0510,
0101.C-0516, 0101.C-0623, 0101.C-0788, 0101.C-0829, 0101.C-0889, 0101.D-0091,
0101.D-0465, 0101.D-0494, 0101.D-0697, 0102.A-0697, 0102.C-0171, 0102.C-0319,
0102.C-0338, 0102.C-0414, 0102.C-0451, 0102.C-0525, 0102.C-0558, 0102.C-0584,
0102.C-0618, 0102.C-0812, 0102.D-0119, 0102.D-0281, 0102.D-0483, 0102.D-0596,
0103.C-0206, 0103.C-0240, 0103.C-0432, 0103.C-0442, 0103.C-0472, 0103.C-0548,
0103.C-0719, 0103.C-0759, 0103.C-0785, 0103.C-0874, 0103.D-0445, 0104.C-0090,
0104.C-0358, 0104.C-0413, 0104.C-0418, 0104.C-0588, 0104.C-0849, 0104.C-0863,
072.A-0244, 072.C-0096, 072.C-0488, 072.C-0513, 072.C-0636, 072.D-0286,
072.D-0419, 072.D-0707, 073.A-0041, 073.C-0733, 073.C-0784, 073.D-0038,
073.D-0136, 073.D-0527, 073.D-0578, 073.D-0590, 074.C-0012, 074.C-0037,
074.C-0061, 074.C-0102, 074.C-0221, 074.C-0364, 074.D-0131, 074.D-0380,
075.C-0087, 075.C-0140, 075.C-0202, 075.C-0234, 075.C-0332, 075.C-0689,
075.C-0710, 075.C-0756, 075.D-0194, 075.D-0600, 075.D-0614, 075.D-0760,
075.D-0800, 076.C-0010, 076.C-0073, 076.C-0155, 076.C-0279, 076.C-0429,
076.C-0878, 076.D-0103, 076.D-0130, 076.D-0158, 076.D-0207, 077.C-0012,
077.C-0080, 077.C-0101, 077.C-0295, 077.C-0364, 077.C-0513, 077.C-0530,
077.D-0085, 077.D-0498, 077.D-0633, 077.D-0720, 078.C-0037, 078.C-0044,
078.C-0133, 078.C-0209, 078.C-0233, 078.C-0403, 078.C-0510, 078.C-0751,
078.C-0833, 078.D-0067, 078.D-0071, 078.D-0245, 078.D-0299, 078.D-0492,
079.C-0046, 079.C-0127, 079.C-0170, 079.C-0329, 079.C-0463, 079.C-0657,
079.C-0681, 079.C-0828, 079.C-0898, 079.C-0927, 079.D-0009, 079.D-0075,
079.D-0118, 079.D-0160, 079.D-0462, 079.D-0466, 080.C-0032, 080.C-0071,
080.C-0581, 080.C-0664, 080.C-0712, 080.D-0047, 080.D-0086, 080.D-0151,
080.D-0318, 080.D-0347, 080.D-0408, 081.C-0034, 081.C-0119, 081.C-0148,
081.C-0211, 081.C-0388, 081.C-0774, 081.C-0779, 081.C-0802, 081.C-0842,
081.D-0008, 081.D-0065, 081.D-0066, 081.D-0109, 081.D-0531, 081.D-0610,
081.D-0870, 082.B-0610, 082.C-0040, 082.C-0212, 082.C-0308, 082.C-0312,
082.C-0315, 082.C-0333, 082.C-0357, 082.C-0390, 082.C-0412, 082.C-0427,
082.C-0608, 082.C-0718, 082.D-0499, 082.D-0833, 083.C-0186, 083.C-0413,
083.C-0627, 083.C-0794, 083.C-1001, 083.D-0040, 083.D-0549, 083.D-0668,
083.D-1000, 084.C-0185, 084.C-0228, 084.C-0229, 084.C-1024, 084.C-1039,
084.D-0338, 084.D-0591, 085.C-0019, 085.C-0063, 085.C-0318, 085.C-0393,
085.C-0614, 085.D-0296, 085.D-0395, 086.C-0145, 086.C-0230, 086.C-0284,
086.C-0448, 086.D-0078, 086.D-0240, 086.D-0657, 087.C-0012, 087.C-0368,
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090.C-0131, 090.C-0395, 090.C-0421, 090.C-0540, 090.C-0849, 090.D-0256,
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091.C-0844, 091.C-0853, 091.C-0866, 091.C-0936, 091.D-0469, 091.D-0759,
091.D-0836, 092.C-0282, 092.C-0427, 092.C-0454, 092.C-0579, 092.C-0715,
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094.C-0297, 094.C-0322, 094.C-0428, 094.C-0790, 094.C-0797, 094.C-0894,
094.C-0901, 094.C-0946, 094.D-0056, 094.D-0274, 094.D-0596, 094.D-0704,
095.C-0040, 095.C-0105, 095.C-0367, 095.C-0551, 095.C-0718, 095.C-0799,
095.C-0947, 095.D-0026, 095.D-0155, 095.D-0194, 095.D-0269, 095.D-0717,
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096.D-0064, 096.D-0072, 096.D-0257, 096.D-0402, 097.C-0021, 097.C-0090,
097.C-0277, 097.C-0390, 097.C-0434, 097.C-0561, 097.C-0571, 097.C-0864,
097.C-0948, 097.C-1025, 097.D-0120, 097.D-0150, 097.D-0156, 097.D-0420,
098.C-0042, 098.C-0269, 098.C-0292, 098.C-0304, 098.C-0366, 098.C-0440,
098.C-0446, 098.C-0518, 098.C-0645, 098.C-0739, 098.C-0820, 098.C-0860,
098.D-0187, 099.C-0081, 099.C-0093, 099.C-0138, 099.C-0205, 099.C-0303,
099.C-0304, 099.C-0334, 099.C-0374, 099.C-0458, 099.C-0491, 099.C-0599,
099.C-0798, 099.C-0880, 099.C-0898, 099.D-0236, 099.D-0380, 105.2045.001,
105.2045.002, 105.207T.001, 105.208G.001, 105.20AK.002, 105.20AZ.001,
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105.20L8.002, 105.20MP.001, 105.20N0.001, 105.20NV.001, 105.20PH.001,
106.20Z1.001, 106.20Z1.002, 106.212H.001, 106.212H.007, 106.212H.009,
106.212H.010, 106.215E.001, 106.215E.002, 106.215E.004, 106.216H.001,
106.21DB.001, 106.21DH.001, 106.21ER.001, 106.21GB.003, 106.21MA.001,
106.21PJ.001, 106.21PJ.002, 106.21R4.001, 106.21TJ.001, 106.21TJ.008,
107.22R1.001, 107.22UN.001, 108.21XB.001, 108.21YY.001, 108.21YY.002,
108.21YY.004, 108.21YY.005, 108.222V.001, 108.2271.001, 108.2271.002,
108.2271.003, 108.229Z.001, 108.229Z.002, 108.22A8.001, 108.22CE.001,
108.22E7.001, 108.22KV.001, 108.22KV.002, 108.22KV.003, 108.22KV.004,
108.22KV.005, 108.22L8.001, 108.22LE.001, 108.22LR.001, 108.23MM.004,
108.23MM.005, 109.230J.001, 109.2317.001, 109.233Q.001, 109.2374.001,
109.2392.001, 109.239V.001, 109.23J8.001, 110.23XW.001, 110.23XW.002,
110.23YQ.001, 110.241K.001, 110.241K.002, 110.242T.001, 110.2434.001,
110.2438.001, 110.2460.001, 110.248C.001, 110.24BB.001, 110.24C8.001,
110.24C8.002, 110.24D8.001, 1101.C-0557, 1101.C-0721, 1102.C-0249,
1102.C-0339, 1102.C-0923, 1102.D-0954, 111.24PJ.002, 111.24UR.001,
111.24ZQ.001, 111.24ZW.001, 111.250B.001, 111.254A.001, 111.254E.001,
111.254R.002, 111.255C.001, 111.255C.003, 111.255C.004, 111.263V.001,
111.264X.001, 180.C-0886, 182.D-0356, 183.C-0437, 183.C-0972, 183.D-0729,
184.C-0639, 184.C-0815, 185.D-0056, 187.D-0917, 188.C-0265, 188.C-0779,
190.C-0027, 190.D-0237, 191.C-0505, 191.C-0873, 191.D-0255, 192.C-0224,
192.C-0852, 196.C-0042, 196.C-1006, 198.C-0169, 198.C-0836, 198.C-0838,
2101.C-5015, 276.C-5009, 281.D-5052, 281.D-5053, 281.D-5054, 282.C-5034,
282.C-5036, 282.D-5006, 283.C-5017, 283.C-5022, 288.C-5010, 289.C-5053,
289.D-5015, 292.C-5004, 295.C-5031, 295.C-5035, 297.C-5051, 495.L-0963,
60.A-9036, 60.A-9109, 60.A-9501, 60.A-9700, 60.A-9709.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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# Critical condition for electrowetting-induced detachment of a droplet from a
curved surface

Ke Xiao and Chen-Xu Wu∗ Department of Physics, School of Physical Science and
Technology, Xiamen University, Xiamen 361005, People’s Republic of China
<EMAIL_ADDRESS>
(August 29, 2024)

###### Abstract

Based on energy conservation, we derive a critical condition theoretically for
electrowetting-induced droplet detachment from a hydrophobic curved surface.
Phase diagrams are constructed in terms of droplet volume, viscosity,
Ohnesorge number, friction coefficient at contact line, surface curvature,
surface wettability and electrowetting number. The deduced critical condition
offers a general and quantitative prediction on when the detachment occurs, a
criterion enabling us to gain more insights into how to accurately manipulate
the electrowetting-induced detachment of an aqueous droplet on a curved
surface. The results obtained in this paper also imply that the detachable
regimes of the phase diagrams can be enlarged through increasing droplet
volume and surface curvature, and reducing liquid viscosity, friction
coefficient, Ohnesorge number and wettability of substrates.

Directly harnessing detachment and interfacial geometric shape of droplet on a
flat substrate has sparked an increasing amount of interest because of its
significance in fundamental scientific understanding and engineering and
technological applications, including self-cleaning K.M.Wisdom2013 ; L.Yao2014
, anti-icing/dew Q.L.Zhang2013 ; J.B.Boreyko2009 , inkjet/soft printing
T.Boland2006 ; L.B.Zhang2015 , fast response displays R.A.Hayes2003 , fast
optical imaging C.L.Hao2014 , optical devices S.Kuiper2004 ; J.Heikenfeld2005
, and novel digital microfluidic devices K.Choi2012 ; S.JunLee2012 ;
J.Hong2015 ; P.G.Zhu2017 . Except for spreading and evaporation when a sessile
droplet initially rests on solid substrate, it is also able to depart from the
substrate with some external stimulation. Generally, there are various
approaches to generating detachment of droplet from the substrate, such as
electrowetting (EW)-induced detachment S.JunLee2012 ; S.J.Lee2014 ;
A.Cavalli2016 ; Z.T.Wang2017 ; B.Traipattanakul2017 ; Q.Vo2019 ; K.X.Zhang2019
; C.T.Burkhart2020 ; Q.G.Wang2020 , coalescence-induced jumping Y.Nam2013 ;
R.Enright2014 ; T.Mouterde2017 ; H.Vahabi2018 , impact-caused bouncing
Y.H.Yeong2014 ; B.Y.Zhao2017 ; A.Bordbar2018 ; M.Abolghasemibizaki2019 ;
Olinka2020 , acoustic wave actuated bouncing M.H.Biroun2020 ;
M.H.Biroun2020PRApp , light-triggered bouncing W.Li2020 , and Laplace pressure
driven jumping X.Yan2020 . Among them, EW has become a prevalent technique due
to its advantages of easy fabrication and manipulation, adaption to various
geometries, short response time, little power consumption and high
reversibility F.Mugele2005 ; W.C.Nelson2012 . It is widely known that if an
electric voltage is applied between the droplet and the substrate, a droplet
settling on a flat substrate will alter its apparent wettability (contact
angle), a phenomenon referred to electrowetting-on-dielectric (EWOD)
F.Mugele2005 ; L.Q.Chen2014 . Over the past two decades, numerous efforts
using experiment S.JunLee2012 ; S.J.Lee2014 ; A.Cavalli2016 ; Z.T.Wang2017 ;
B.Traipattanakul2017 ; Q.Vo2019 ; C.T.Burkhart2020 ; Q.G.Wang2020 ,
theoretical modeling P.Birbarah2015 ; A.Cavalli2016 ; K.X.Zhang2019 , and
numerical simulation K.A.Raman2016 ; A.Merdasi2019june ; A.Merdasi2019oct
have been devoted to getting a better understanding of the dynamics of droplet
detachment induced by EW.

Experimentally, it has been found that a variety of factors, such as viscosity
Q.Vo2019 ; Q.Vo2018PRE ; Q.G.Wang2020 , surface wettability A.Cavalli2016 ,
contact line friction Q.Vo2019 ; Q.Vo2018PRE ; Q.G.Wang2020 , and droplet
volume C.T.Burkhart2020 , play a prime role in determining the dynamics of
droplet detachment from a flat solid substrate induced by EW. For example, by
taking surface energy difference, viscous dissipation and stored energy into
account, a critical condition is proposed to determine the detachable and
nondetachable regimes Q.Vo2019 . Besides, a critical equation for threshold
value of applied electric voltage which triggers droplet detachment is built
and further confirmed by experimental data Q.G.Wang2020 . In addition,
theoretical modeling and numerical simulation also provide a useful complement
to understanding the underlying mechanism of droplet detachment from a flat
substrate. By combining experiments and numerical simulations, the process of
EW-induced ejection of a droplet from a flat substrate has been quantitatively
analyzed, and the effect of applied voltage and intrinsic contact angle on the
energy conversion efficiency have been further discussed A.Cavalli2016 . A
similar investigation reveals that the droplet detachment velocity can be
predicted accurately by studying the dynamic process of ejection
quantitatively K.X.Zhang2019 .

Despite of the fact that dynamic process of EW-induced droplet detachment has
been widely studied either via experimental or theoretical approaches, the
efforts are all focused on flat solid substrates. Numerous standing questions
still remain open concerning the detachment mechanism of an aqueous droplet on
a curved surface induced by the same EW effect. A detailed investigation of
the relations between the threshold of the applied electric voltage and the
system parameters, e.g., droplet volume, liquid properties, surface curvature,
and Young’s contact angle etc., on curved surfaces, is needed to broaden the
current understanding of droplet detachment.

In this paper, we perform a systematic theoretical study on EW-induced droplet
detachment from a curved surface by building expressions for interfacial
energy, energy of dissipation, and accumulated energy. A criterion equation
for EW-induced droplet detachment, which is able to return to that for a flat
surface, is theoretically derived. To further explore the underlying
dependence of the threshold of the applied electric voltage on droplet volume,
liquid properties, surface curvature, and Young’s contact angle, phase
diagrams are also constructed.

We begin our investigation by considering a water droplet placed on a curved
solid substrate (curvature $1/R$) consisting of an insulating surface layer
(thickness $d$) on top (purple) and an electrode underneath, as shown in Fig.
1(I). The apparent contact angle $\theta_{\rm Y}$ at the equilibrium state
satisfies Young’s equation in the absence of voltage, $\gamma_{\rm
sm}-\gamma_{\rm ls}=\gamma_{\rm lm}{\rm cos}\theta_{\rm Y}$, where
$\gamma_{\rm sm}$ and $\gamma_{\rm ls}$ are interfacial tensions of the solid-
medium and liquid-solid interfaces, respectively.

Figure 1: (Color online) Schematic pictures of different scenarios of
electrically actuated droplet detachment on a convex surface. Stages I-II
schematically illustrate a droplet spreading on a curved surface due to the EW
effect. Recoiling and detachment of a droplet after the switch-off of the
applied voltage on a curved surface are illustrated as stages III-V.

When an external voltage $U$ is applied between a droplet and a flat
substrate, the accumulation of free charges near the electrode causes a
reduction of the local liquid-solid surface tension $\gamma_{\rm ls}$ and
subsequently induces the spreading of the droplet, yielding another
equilibrium state, with its new equilibrium contact angle $\theta_{\rm e}$
described by the well-known Young-Lippmann equation Lippmann1875 ;
F.Mugele2005  ${\rm cos}\leavevmode\nobreak\ \theta_{\rm e}={\rm
cos}\leavevmode\nobreak\ \theta_{\rm Y}+\varepsilon_{0}\varepsilon
U^{2}/2d\gamma_{\rm lm}$. For a spherical curved surface, as shown in Fig.
1(II), this equation has to be replaced by Y.Wang2012

$\displaystyle{\rm cos}\leavevmode\nobreak\ \theta_{\rm e}={\rm
cos}\leavevmode\nobreak\ \theta_{\rm Y}+\frac{\varepsilon_{0}\varepsilon
U^{2}}{2d\gamma_{\rm lm}}\cdot\frac{1}{1\pm\kappa}={\rm
cos}\leavevmode\nobreak\ \theta_{\rm Y}+\eta,$ (1)

where $\kappa=d/R$ represents the effect of surface curvature, and
$\varepsilon_{0}$, $\varepsilon$, $\gamma_{\rm lm}$ and $\eta$ are the
dielectric permittivity in vacuum, the relative dielectric constant, the
interfacial tension between the droplet and its surrounding medium above the
substrate, and the dimensionless EW number, respectively. We use the sign
$\pm$ to distinguish the effect of convex and concave surfaces here and
throughout the rest of this paper, with $+$ standing for convex surface and
$-$ for concave one. Once the applied electric voltage is switched off [Fig.
1(III)], the droplet will keep its current shape as that of stage II as its
transient state, but with a sudden increase of the local liquid-solid surface
tension owing to the short time scale of discharge process of the droplet-
electrode capacitor, which is much faster than the relaxation time of the
droplet A.Cavalli2016 ; K.X.Zhang2019 . As a consequence, the restoration of
liquid-solid interfacial tension after the switch-off of external voltage
increases the surface energy of the droplet-substrate system instantly,
leading to a total free interfacial energy expressed as

$\displaystyle E_{\rm s}^{\rm III}=\gamma_{\rm lm}A_{\rm
c}\biggl{(}\frac{1+{\cos}\Delta\theta}{1+{\cos}\theta}-{\cos}\theta_{\rm
Y}\biggr{)}.$ (2)

where $\Delta\theta=\mid\theta-\theta_{\rm e}\mid$, with $\theta$ the apparent
contact angle, and $A_{\rm c}=2\pi R^{2}(1-{\rm cos}\Delta\theta)$ is the
contact area under the spherical droplet cap. Here it is necessary to note
that Young’s equation has been used in deriving the above equation.

Subsequently, as a result of recovery effect, the droplet undergoes a
recoiling stage [Fig. 1(IV)], during which the flow of droplet will induce a
viscous dissipation typically comprising of contributions in the bulk droplet,
near the substrate, and at the vicinity of the contact line Q.Vo2018SP ;
Q.Vo2018PRE ; Q.Vo2019 . As a result, the viscous dissipation for the bulk
flow and that near the substrate can be estimated via $E_{\rm vis}^{\rm
b,s}=\int_{0}^{\tau}\int_{\rm V}\Phi dVdt$ S.Chandra1991 . Meanwhile, the
energy of dissipation contributed by the contact line can be approximately
calculated as

$\displaystyle E_{\rm vis}^{\rm c}=\gamma_{\rm lm}A_{\rm c}\frac{\lambda
OhR\Delta\theta\eta^{1/2}}{\pi\mu R_{0}},$ (3)

where $\lambda$, $\mu$, $R_{0}$, and $Oh=\mu/\sqrt{\rho\gamma_{\rm lm}R_{0}}$
are, respectively, the friction coefficient at contact line, the viscosity of
the droplet, the initial radius of the droplet, and the Ohnesorge number,
which describes the relative importance of viscosity and surface tension. Here
parameter $\rho$ in $Oh$ denotes the density of the droplet. Strictly
speaking, the total viscous dissipation is the sum of $E_{\rm vis}^{\rm b,s}$
and $E_{\rm vis}^{\rm c}$. However, as the contribution at the vicinity of the
contact line dominates the whole dissipation process in the system Q.Vo2019 ,
we therefore merely consider the contribution made by the contact line. In
addition, apart from the viscous dissipation, the pinning and subsequently the
stretching of the liquid interface at the contact line vicinity during the
recoiling process also lead to an accumulation of energy Q.Vo2019  estimated
as

$\displaystyle E_{\rm stored}=\gamma_{\rm lm}A_{\rm c}\frac{\pi{\rm
sin}^{2}\theta_{\rm r}}{\ln(L\sigma^{-1})},$ (4)

where $\theta_{\rm r}$, $L$, and $\sigma$ are the receding contact angle, the
macroscopic cutoff length, and the topological defect size of the substrate,
respectively. Here the expression on a spherical surface $\theta_{\rm
r}=\theta_{\rm r,0}\pm\Delta\theta$, instead, is used for the receding contact
angle C.W.Extrand2008 , with $\theta_{\rm r,0}$ the droplet’s intrinsic
receding contact angle on curved surface. Finally, at the end of the
retracting stage, if the total free interfacial energy of stage III is
sufficient to overcome the energy barrier hindering droplet jumping, the
droplet will detach from the substrate, as illustrated in Fig. 1(V),
corresponding to a total energy written as

$\displaystyle E_{\rm s}^{\rm V}=4\pi R_{0}^{2}\gamma_{\rm lm}+E_{\rm k},$ (5)

where the first term $E_{\rm s}^{\rm V}=4\pi R_{0}^{2}\gamma_{\rm lm}$ and the
second one are the surface energy and the kinetic energy of the droplet,
respectively. $E_{\rm k}>0$ means detachable and $E_{\rm k}<0$ implies
nondetachable, and the critical condition $E_{\rm k}=0$ for the detachment
corresponds to

$\displaystyle\Delta E_{\rm s}=E_{\rm s}^{\rm III}-E_{\rm s}^{\rm V}=E_{\rm
vis}^{\rm c}+E_{\rm stored},$ (6)

where $\Delta E_{\rm s}$ denotes the surface energy difference between stages
III and V. Here we assume that the droplet volume is small, corresponding to
an initial radius smaller than the capillary length $l_{\rm
c}=\sqrt{\gamma_{\rm lm}/\rho g}\sim 1.9\leavevmode\nobreak\ {\rm mm}$. Thus
the gravitational effect can be neglected, and the droplet at stages III and V
can be treated as a spherical one in this paper. In addition, the Marangoni
and the evaporation effects of the aqueous droplet are excluded as well,
thereby a reasonable assumption is made that the volume of the droplet $V_{0}$
is conserved during the whole process, corresponding to a constraint

$\displaystyle
V_{0}=\frac{\pi}{3}R^{3}\frac{{\sin}^{3}\Delta\theta}{{\sin}^{3}\theta}f(\theta)-\biggl{[}\pm\frac{\pi}{3}R^{3}f(\Delta\theta)\biggr{]}=\frac{4\pi}{3}R_{0}^{3},$
(7)

where $f(\theta)=2-3{\cos}\theta+{\cos}^{3}\theta$ is a dimensionless
function. Here it is reasonable to assume that $\Delta\theta$ is a small
quantity when $R\gg R_{0}$, hence we can do a Taylor expansion with respect to
$\Delta\theta$ for Eq. (7), leading to

$\displaystyle\Delta\theta=\biggl{[}\frac{4}{f(\theta_{\rm
e})}\biggr{]}^{1/3}\frac{R_{0}}{R}{\sin}\theta_{\rm e}.$ (8)

In order to investigate the influence of surface curvature on the critical
detachment condition of the droplet, we substitute Eqs. (2), (3), (4), and (5)
into Eq. (6) and do the same Taylor expansion for both sides. By using Eq.
(8), the critical condition for the detachment of droplet on a curved surface
can be derived as

$\displaystyle\Gamma-\Omega-\frac{\pi{\sin}^{2}\theta_{\rm
r,0}}{\ln(L\sigma^{-1})}=\pm\biggl{[}\frac{\pi{\sin}(2\theta_{\rm
r,0})}{\ln(L\sigma^{-1})}-\frac{2{\sin}\theta_{\rm e}}{(1+{\cos}\theta_{\rm
e})^{2}}\biggr{]}\Delta\theta,$ (9)

where $\Gamma=2/(1+{\cos}\theta_{\rm e})-4\bigl{[}4/f(\theta_{\rm
e})\bigr{]}^{-2/3}{\sin}^{-2}\theta_{\rm e}-{\cos}\theta_{\rm Y}$ represents
contribution of excessive surface energy, and $\Omega=(\lambda
Oh/\pi\mu)\eta^{1/2}\bigl{[}4/f(\theta_{\rm e})\bigr{]}^{1/3}{\sin}\theta_{\rm
e}$ denotes the energy of viscous dissipation, both depending on surface
curvature due to Eq. (1). Intriguingly, the critical detachment condition of
droplet on a curved surface Eq. (9) returns exactly to the one on a flat
surface Q.Vo2019  if the surface curvature $1/R$ approaches zero.

Figure 2: (Color online) Phase diagrams for the detachment of droplet in the
projection planes of (a) electrowetting number $\eta$ and droplet radius
$R_{0}$ with droplet viscosity $\mu$=1.0 mPa s; (b) electrowetting number
$\eta$ and droplet viscosity $\mu$ with droplet volume
$V_{0}=1.0\leavevmode\nobreak\ {\rm\mu L}$, where the Young’s contact angle is
set as $\theta_{\rm Y}=160^{\circ}$, and the curve lines denote the critical
detachment conditions for different surface radii ($\infty$, $10^{5}$,
$10^{3}$, and $10^{2}\leavevmode\nobreak\ {\rm mm}$). Phase diagrams in
($\eta$, $R_{0}$) plane for surface radius $R=10^{2}\leavevmode\nobreak\ {\rm
mm}$, where the curve lines denote the critical droplet detachment conditions
for (c) different droplet viscosities (1.0, 5.0, and 10.0 mPa s) with
$\theta_{\rm Y}=160^{\circ}$, and (d) different surface wettabilities
($\theta_{\rm Y}=140^{\circ},150^{\circ}$, and $160^{\circ}$) with droplet
viscosity $\mu$=1.0 mPa s.

Our calculation is carried out by using the same values of parameters as those
in Ref. Q.Vo2019 , i.e., $\gamma_{\rm lm}=37.2\leavevmode\nobreak\ {\rm
mN\cdot m^{-1}}$, $\rho=1.003\leavevmode\nobreak\ {\rm g\cdot cm^{-3}}$,
$\varepsilon=1.93$, $d=2.2\leavevmode\nobreak\ {\rm\mu m}$, $\theta_{\rm
r,0}=121.7^{\circ}$, $\sigma=464\leavevmode\nobreak\ {\rm nm}$ and
$\lambda=C(\mu\mu_{0})^{1/2}$, where $C$ and $\mu_{0}$ (viscosity of the
surrounding medium of droplet) are fixed as 34 and 1.8 mPa s. In order to
reveal the effect of droplet volume (which has been converted to the radius of
droplet $R_{0}$), liquid viscosity, and EW number $\eta$ on droplet
detachment, we constructed phase diagrams in $\eta-R_{0}$ (EW number versus
radius) and $\eta-\mu$ (EW number versus viscosity) space, as demonstrated in
Fig. 2. It is found that all phase diagrams are divided into two regimes,
namely nondetachable phase regime and detachable phase regime separated by a
coexisting line representing the critical condition for a droplet to detach
from a curved surface, above which the excess surface energy will be converted
into the kinetic energy of the detached droplet. A specific case of our model
corresponding to a droplet on a flat surface($R\rightarrow\infty$) is also
shown (white circle) in Figs. 2(a) and 2(b). On the one hand, for a fixed
surface radius, i.e., $R=10^{3}\leavevmode\nobreak\ {\rm mm}$ (red critical
line in Figs. 2(a) and 2(b)), increasing droplet radius $R_{0}$ or decreasing
droplet viscosity $\mu$ clearly reduces the detachment threshold value of
voltage applied, indicating that, for a small droplet with high viscosity, a
large excessive surface energy is needed to overcome the viscous dissipation
before detachment, which correspondingly requires a higher electric voltage
applied. On the other hand, Figs. 2(a) and 2(b) also show that surface
curvature plays a significant role in deciding the threshold value of the
electric voltage applied for the detachment to occur. If we increase the
surface curvature to $10^{-2}$ mm-1, but still small compared to that of the
droplet $1/R_{0}$, the threshold value of the applied electric voltage for
detachment significantly decreases for convex surface as compared with that of
flat surface, as shown by the blue curve line in Figs. 2(a) and 2(b). However,
a higher threshold value of the applied electric voltage is required for the
occurrence of droplet detachment on concave surfaces, as depicted by the black
line in Figs. 2(a) and 2(b).

Figure 3: (Color online) Phase diagrams of the applied electric voltage $U$ vs
(a) Ohnesorge number $Oh$ and vs (b) friction coefficient $\lambda$, with
droplet volume $V_{0}$=1.0 $\mu$L and the Young’s contact angle $\theta_{\rm
Y}=160^{\circ}$, where the curve lines denote the critical droplet detachment
conditions for different surface radii ($\infty$, $10^{4}$, and
$10^{2}\leavevmode\nobreak\ {\rm mm}$). Phase diagrams $U$ vs $Oh$ for surface
radius $R=10^{2}\leavevmode\nobreak\ {\rm mm}$, where the curve lines denote
the critical droplet detachment conditions for (c) different droplet volumes
(1.0, 2.5, 5.0, and 10.0 $\mu{\rm L}$) with $\theta_{\rm Y}=160^{\circ}$, and
(d) different surface wettabilities ($\theta_{\rm Y}=140^{\circ},150^{\circ}$,
and $160^{\circ}$) with droplet volume $V_{0}$=1.0 $\mu$L.

Meanwhile, the critical curve lines in $\eta-R_{0}$ space for different values
of droplet viscosity and Young’s contact angle are presented in Figs. 2(c) and
2(d), where it is found that detachable regime is remarkably shrunk as we
increase the droplet viscosity and reduce the surface wettability, a result in
agreement with the conclusion drawn from Fig. 2(b).

As it is clearly seen that, according to Eq. 3, the energy of viscous
dissipation is directly related to Ohnesorge number and the friction
coefficient at contact line during the recoiling process, it is meaningful to
probe the influence of Ohnesorge number $Oh$ and friction coefficient
$\lambda$ on the critical voltage value. Here accordingly we plot the phase
diagram of detachment projected in $U-Oh$ (applied electric voltage versus
Ohnesorge number) plane and $U-\lambda$ (applied electric voltage versus
friction coefficient) plane, as shown in Fig. 3. For a fixed set of the
parameters (i.e., surface radius $R=\infty$, droplet volume
$V_{0}=1.0\leavevmode\nobreak\ \mu{\rm L}$, and Young’s contact angle
$\theta_{\rm Y}=160^{\circ}$), Figs. 3(a) and 3(b) demonstrate that a higher
electric voltage is required to detach the droplet from the substrate with the
increase of Ohnesorge number and the friction coefficient. Here it is
worthwhile to note that within some fixed parameters range, droplet detachment
is unable to occur at large values of $Oh$ and $\lambda$ Q.Vo2019  because the
excessive surface energy in this case is largely dissipated, except for a
small detachable regime in parameter space restricted at the left top corner
of the figure where $Oh$, and $\lambda$ are small. Such a feature does not
depend on the curvature of the substrate when the surface curvature is not
small enough, i.e. the red curve with $R=10^{4}\leavevmode\nobreak\ {\rm mm}$
overlaps that for flat surface (white circle with $R\rightarrow\infty$), as
shown in Figs. 3(a) and 3(b). However, for the black (concave) and blue
(convex) curves with $R=10^{2}\leavevmode\nobreak\ {\rm mm}$, we still see
deviations to different directions from that of Figs. 2(a) and 2(b),
indicating that it is more difficult to detach a droplet from a concave
substrate in comparison with a convex one. Furthermore, the critical curve
line in $U-Oh$ space under different droplet volumes (1.0, 2.5, 5.0, and 10.0
$\mu$L) and Young’s contact angles ($140^{\circ}$, $150^{\circ}$, and
$160^{\circ}$) are plotted in Figs. 3(c) and 3(d), where the variation of the
detachable regime once

Figure 4: (Color online) Phase diagrams show detachable and nondetachable
regimes in the parameter space spanned by the applied electric voltage $U$ and
the surface curvature $R^{-1}$ with droplet viscosity $\mu$=1.0 mPa s, where
the curve lines denote the critical droplet detachment conditions for (a)
different droplet volumes (1.0, 2.5, 5.0, and 10.0 $\mu{\rm L}$) with
$\theta_{\rm Y}=160^{\circ}$, and (b) different surface wettabilities
($\theta_{\rm Y}=140^{\circ},150^{\circ}$, and $160^{\circ}$) with droplet
volume $V_{0}=5.0\leavevmode\nobreak\ \mu{\rm L}$. Similar phase diagrams
spanned by the applied electric voltage $U$ and the Young’s contact angle
$\theta_{\rm Y}$ with droplet viscosity $\mu$=1.0 mPa s, where the curve lines
denote the critical droplet detachment conditions for (c) different surface
radii ($\infty$, $10^{3}$, and $10^{2}\leavevmode\nobreak\ {\rm mm}$) with
droplet volume $V_{0}=5.0\leavevmode\nobreak\ \mu{\rm L}$, and (d) different
droplet volumes (1.0, 2.5, 5.0, and 10.0 $\mu{\rm L}$) with surface radius
$R=10^{2}\leavevmode\nobreak\ {\rm mm}$.

again verifies that smaller droplet and less hydrophobic surface have a
negative effect on droplet detachment, and thus require a higher critical
electric voltage to induce detachment.

Finally, in order to understand how surface curvature and surface wettability
affect the critical value of applied electric voltage, phase diagrams of
detachment in $U-R^{-1}$ (applied voltage versus surface curvature) space and
$U-\theta_{\rm Y}$ (applied voltage versus Young’s contact angle) space are
constructed, as shown in Fig. 4. Interestingly, the critical curve lines for
concave surface (blue dashed line) and convex surface (blue solid line) in
Figs. 4(a) and 4(b) show almost opposite behaviors in response to the
variation of surface curvature, that is, the critical electric voltage
required for detachment decreases with the increase of $R^{-1}$ for convex
surface, while higher critical electric voltage for detachment is needed as we
increase $R^{-1}$ for concave surface. Meanwhile,the detachable regimes in
parameter space are also apparently broadened with the increase of droplet
volume and Young’s contact angle [see Figs. 4(a) and 4(b)]. In addition, the
critical curve in $U-\theta_{\rm Y}$ space for different surface radii
($\infty,10^{3}$, and $10^{2}$ mm) and droplet volumes (1.0, 2.5, 5.0, and
10.0 $\mu$L) are presented in Figs. 4(c) and 4(d), where we can deduce that
the decrease of wettability, namely increasing Young’s contact angle
(hydrophobicity), will lead to a decrease of the threshold value of the
electric voltage applied to the droplet for detachment. It is also found from
the figure that the red curve for curved surface ($R=10^{3}$ mm) overlaps the
circle curve for flat surface ($R\rightarrow\infty$), indicating that the
surface curvature is not large enough to bring about a change in critical
electric voltage for the detachment to occur. However, the detachable regime
is dramatically widened (narrowed down) for convex (concave) surface at
$R=10^{2}$ mm (still large enough as compared with $R_{0}$) if compared with
that for flat surface, as shown by the blue line and black line in Fig. 4(c).
Furthermore, the variation tendency of the critical curves in $U-\theta_{\rm
Y}$ space as shown in Fig. 4(d) denotes that the detachable regime is also
affected by the size of droplet, or more specifically, the detachable regime
is noticeably broadened with the increase of droplet volume. As a result, Fig.
4 confirms that large droplet and superhydrophobicity of surface both play a
positive role in inducing a droplet detachment.

Consequently, we argue that larger droplet volume, lower droplet viscosity,
Ohnesorge number, the friction coefficient, large (convex surface) or small
(concave surface) surface curvature and high hydrophobicity favors droplet
detachment. Since whether the detachment occurs or not largely relies on the
competition among the three types of energy, namely the excessive surface
energy, the energy of viscous dissipation, and the accumulated energy at the
vicinity of the contact line. The droplet detachment only occurs under the
condition that the excessive surface energy is sufficient to overcome the
energy barrier, namely the sum of viscous dissipation and the accumulated
energy around the contact line. In order to examine the validity of our
theoretical model, it is necessary to compare our theoretical predictions with
experimental results and other theoretical analyses. For example, it has been
reported both experimentally A.Cavalli2016  and theoretically K.X.Zhang2019
that increasing Young’s contact angle gives rise to an increase in EW-induced
velocity of the detached droplet, a result in good agreement with our present
arguments. By employing high density ratio-based lattice Boltzmann method,
Raman et al. K.A.Raman2016  have examined the effect of Ohnesorge number on
the jumping velocity of the detached droplet and demonstrated that increasing
Ohnesorge number leads to a decrease of droplet jumping velocity, which is
also in accordance with our conclusions. In addition, it has been shown, by
providing an analytical method to study EW-induced jumping of droplet on flat
hydrophobic substrates, that jumping motion of droplet can be enhanced via
increasing Young’s contact angle or decreasing Ohnesorge number K.X.Zhang2019
, a conclusion which can be made and has been included by our theoretical
model. What’s more, Wang et al. Q.G.Wang2020  showed via experiments that
higher threshold voltage is needed to induce detachment of droplet with larger
friction coefficient and smaller volume. These results all support the
theoretical model we present in this paper.

In summary, we investigate the detachment of an aqueous droplet induced by EW
effect on a curved surface, and derive a general criterion equation for the
detachment to occur. According to the obtained phase diagrams consisting of
detachable and nondetachable regimes, we show that the direct droplet
detachment can be triggered by applying an electric voltage (corresponding to
EW number) beyond a critical value and then switching it off. It is found
that, by judging the critical curve of voltage applied, it is easier (more
difficult) for a droplet on a convex (concave) surface to detach if compared
with one on a flat surface. The dependence of the threshold value of the
applied voltage on droplet volume, liquid properties, surface curvature, and
Young’s contact angle is discussed. In particular, the results here suggest
that the threshold value of the applied electric voltage decrease with the
increase of droplet volume, surface curvature (convex surface) and Young’s
contact angle, or the decrease of droplet viscosity, Ohnesorge number,
friction coefficient and surface curvature (concave surface). Therefore, it is
possible to harness the detachment of a droplet triggered by switching off the
electric voltage applied, by tuning the volume and the viscosity of the
droplet, and the wettability and the curvature of the substrate. Here it
should be noted that the gravitational effect and the oscillation of the
droplet are both ignored in our study.

###### Acknowledgements.

This work was funded by the National Science Foundation of China under Grant
No. 11974292 and No. 11947401.

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