The following article has been submitted to The Journal of Chemical Physics.
©2021 Yasutaka Yamaguchi. This article is distributed under a Creative Commons
Attribution (CC BY) License.

# Curvature dependence of the interfacial tensions around nanoscale cylinder:
Young’s equation still holds

Keitaro Watanabe  present affiliation: Department of Management of Industry
and Technology, Osaka University  Department of Mechanical Engineering, Osaka
University, 2-1 Yamadaoka, Suita 565-0871, Japan  Hiroki Kusudo
<EMAIL_ADDRESS>Department of Mechanical Engineering, Osaka
University, 2-1 Yamadaoka, Suita 565-0871, Japan  Carlos Bistafa
<EMAIL_ADDRESS>Department of Mechanical Engineering,
Osaka University, 2-1 Yamadaoka, Suita 565-0871, Japan  Takeshi Omori
<EMAIL_ADDRESS>Deptartment of Mechanical Engineering, Osaka City
University, 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan  Yasutaka
Yamaguchi<EMAIL_ADDRESS>Department of Mechanical
Engineering, Osaka University, 2-1 Yamadaoka, Suita 565-0871, Japan  Water
Frontier Research Center (WaTUS), Research Institute for Science & Technology,
Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601,
Japan

###### Abstract

By extending the theoretical framework derived in our previous study [Y.
Imaizumi et al., J. Chem. Phys. 153, 034701 (2020)], we successfully
calculated the solid-liquid (SL) and solid-vapor (SV) interfacial tensions of
a simple Lennard-Jones fluid around solid cylinders with nanometer-scale
diameters from single equilibrium molecular dynamics (MD) systems, in which a
solid cylinder was vertically immersed into a liquid pool. The SL and SV
interfacial tensions $\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}$ and
$\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}}$ relative to that for bare solid
surface $\gamma_{\mathrm{S0}}$, respectively were obtained by simple force
balance relations on fluid-containing control volumes set around the bottom
and top of the solid cylinder, which are subject to the fluid stress and the
force from the solid. The theoretical contact angle calculated by Young’s
equation using these interfacial tensions agreed well with the apparent
contact angle estimated by the analytical solution fitted to the meniscus
shape, showing that Young’s equation holds even for the menisci around solids
with nanoscale curvature. We have also found that the curvature effect on the
contact angle was surprisingly small while it was indeed large on the local
forces exerted on the solid cylinder near the contact line. In addition, the
present results showed that the curvature dependence of the SL and SV
interfacial free energies, which are the interfacial tensions, is different
from that of the corresponding interfacial potential energies.

##  I Introduction

As we see cap-shaped liquid droplets on solid surfaces almost everyday,
wetting behavior is one of the most common physical phenomena in human life,
and is also a topic of interest in various scientific and engineering fields.
de Gennes (1985); Ono and Kondo (1960); Rowlinson and Widom (1982); Schimmele,
Naplórkowski, and Dietrich (2007); Drelich  _et al._ (2019) By introducing the
concept of interfacial tensions and contact angle $\theta$, wetting is usually
described by Young’s equation Young (1805) given by

$\gamma_{\mathrm{SL}}-\gamma_{\mathrm{SV}}+\gamma_{\mathrm{LV}}\cos\theta=0,$
(1)

where $\gamma_{\mathrm{SL}}$, $\gamma_{\mathrm{SV}}$ and
$\gamma_{\mathrm{LV}}$ denote solid-liquid (SL), solid-vapor (SV) and liquid-
vapor (LV) interfacial tensions, respectively. Young’s equation (1) was first
proposed based on the wall-tangential force balance of interfacial tensions
exerted on the contact line (CL) in 1805 – before the establishment of
thermodynamics Gao and McCarthy (2009) –, now it is often explained from a
thermodynamic point of view instead of the mechanical force balance. de Gennes
(1985) Practically, the contact angle is used as a common measure of
wettability. Various models have been proposed to capture the details of the
CL, such as introducing the concept of precursor film de Gennes (1985);
Popescu  _et al._ (2012) or the microscopic contact angle, White (1977) or
considering the effects of line tension due to the contact-line curvature in
Eq. (1). Boruvka and Neumann (1977); Marmur (1997) However, it is difficult to
experimentally validate these models mainly because measuring the interfacial
tensions $\gamma_{\mathrm{SL}}$ and $\gamma_{\mathrm{SV}}$, which include the
solid phase, is not trivial. Kumikov and Khokonov (1983); Tyson and Miller
(1977)

Wetting plays a key role especially in the nanoscale with a large surface to
volume ratio, and from a microscopic point of view, Kirkwood and Buff (1949)
first provided the theoretical framework of surface tension based on the
statistical mechanics. Recent development of molecular simulation methods
including molecular dynamics (MD) and Monte Carlo (MC) advanced the
microscopic understanding of the interfaces, and the calculation of the
surface tension based on Bakker’s equation, Bakker (1928); Ono and Kondo
(1960); Rowlinson and Widom (1982) which describes the relation between the
stress integral around the liquid-vapor or liquid-gas interface and the
surface tension, is used as a standard approach. Allen and Tildesley (1987) In
addition, MD or MC studies about microscopic wetting have been conducted,
ranging from simply evaluating the apparent contact angle, e.g., from the
average droplet shape, to quantitatively extract the SL and SV interfacial
tensions through a mechanical manner and/or a thermodynamic manner. Tang and
Harris (1995); Gloor  _et al._ (2005); Ingebrigtsen and Toxvaerd (2007); Das
and Binder (2010); Weijs  _et al._ (2011); Seveno, Blake, and de Coninck
(2013); Nijmeijer and van Leeuwen (1990); Nijmeijer  _et al._ (1990); Nishida
_et al._ (2014); Imaizumi  _et al._ (2020); Surblys  _et al._ (2014);
Yamaguchi  _et al._ (2019); Kusudo, Omori, and Yamaguchi (2019); Leroy, Dos
Santos, and Müller-Plathe (2009); Leroy and Müller-Plathe (2010, 2015); Kanduč
and Netz (2017); Kanduč (2017); Surblys  _et al._ (2018); Bistafa  _et al._
(2021); Grzelak and Errington (2008); Lau  _et al._ (2015); Kumar and
Errington (2014); Ardham  _et al._ (2015); Jiang, Müller-Plathe, and
Panagiotopoulos (2017); Ravipati  _et al._ (2018); Omori  _et al._ (2019);
Bey, Coasne, and Picard (2020) For the mechanical approach – called the
mechanical route –, Bakker’s equation was extended to describe the connection
between the stress integral through the SL or SV interface and the
corresponding interfacial tension (See Appendix A). Nijmeijer and van Leeuwen
(1990); Nijmeijer  _et al._ (1990); Nishida  _et al._ (2014); Imaizumi  _et
al._ (2020); Surblys  _et al._ (2014); Yamaguchi  _et al._ (2019); Kusudo,
Omori, and Yamaguchi (2019) On the other hand for the latter approach – called
the thermodynamic route –, the SL and SV interfacial tensions were interpreted
as the interfacial free energy per interfacial area. For instance, by the
thermodynamic integration method, the SL interfacial energy was evaluated as
the free energy difference from a reference system, in which the SL interface
was substituted by bare solid and liquid surfaces quasi-statically under
constant number of particles $N$, temperature $T$ and pressure $p$
($NpT$-ensemble) or volume $V$ ($NVT$-ensemble) condition. Surblys  _et al._
(2014); Yamaguchi  _et al._ (2019); Kusudo, Omori, and Yamaguchi (2019);
Leroy, Dos Santos, and Müller-Plathe (2009); Leroy and Müller-Plathe (2010,
2015); Kanduč and Netz (2017); Kanduč (2017); Surblys  _et al._ (2018);
Bistafa  _et al._ (2021) These studies indicated that the apparent contact
angle of the meniscus or droplet obtained in the simulations agreed well with
the one predicted by Young’s equation (1) in case the solid surfaces are flat
and smooth so that the CL pinning may not be induced. Yamaguchi  _et al._
(2019); Kusudo, Omori, and Yamaguchi (2019)

Considering the potential applications of nano-wetting, e.g., a flow in a
confined space such as a nanofoam Rode  _et al._ (1999) or a carbon nanotube,
Falk  _et al._ (2014) the solid surfaces can have a nanoscale radius of
curvature, and the interfacial tensions should depend on the curvature.
Regarding the water wetting on carbon nanotubes as a solid with a nanoscale
curvature, unique wetting behavior Homma  _et al._ (2013) and a strong
diameter dependence of the capillary force were experimentally reported.
Imadate and Hirahara (2018) For the liquid-vapor interface, Tolman (1949)
first formulated the size effect of droplet surface tension with a lengthscale
called the “Tolman length,” Blokhuis and Kuipers (2006); Tumram, Kesava Rao,
and Ananth (2017); Elliott (2021) and MD or MC simulations have been carried
out as well. Yaguchi, Yano, and Fujikawa (2010); Das and Binder (2011); Lau
_et al._ (2015); Cheng and Ceriotti (2018); Rehner and Gross (2018); Gao  _et
al._ (2021) Indeed, the LV interfacial tension can be extracted using a strict
definition of the interface position, e.g., based on the force and momentum
balances, Yaguchi, Yano, and Fujikawa (2010) and the difference between the
pressures inside and outside the droplet based on the Young-Laplace equation.
On the other hand, the calculation of the SL and SV interfacial tension on a
curved solid surface is not trivial. Cheng and Ceriotti (2018); Montero de
Hijes  _et al._ (2019) For instance, via thermodynamic routes, if we suppose
calculation systems for the thermodynamic integration to calculate the SL
interfacial tension on a cylindrical solid surface with a nanoscale radius,
then the three interface areas, i.e., the radii of the target solid-fluid
interface, the reference bare solid and bare liquid interfaces should be all
different, and this difference would become critical for the evaluation of the
desired interfacial tension when the cylinder radius is comparable to the
radius differences. Another possibility is via mechanical routes, and it is
technically possible to calculate the stress distribution in the cylindrical
or spherical coordinates, although the calculation cost significantly incrases
to obtain the distribution, and indeed, precise calculation in these
coordinate systems is not implemented into the MD packages such as LAMMPS
Plimpton (1995) or GROMACS GRO (2008), and the implementation into in-house
codes is also rather complicated. Thompson  _et al._ (1984); Yaguchi, Yano,
and Fujikawa (2010)

Going back to the relation between the SL or SV interfacial tension and the
fluid stress in the interface via the mechanical routes, what we need is not
the stress distribution but the stress integral. Considering this feature, in
our previous study, Imaizumi  _et al._ (2020) we provided a theoretical
framework to extract the SL and SV interfacial tensions from a single MD
simulation by using the local forces and the local interaction potential
exerted on a quasi-two-dimensional (2D) flat and smooth solid plate immersed
into a liquid pool of a simple liquid, called the Wilhelmy plate, and verified
through the comparison between the MD results and the interfacial works of
adhesion obtained by the thermodynamic integration (TI). This modified
Wilhelmy method is advantageous because it does not require computationally
demanding calculations such as the local stress distributions and the
thermodynamic integration which needs averaging at each discrete states along
the integration path.

In this study, we extracted the SL and SV interfacial tensions of a simple
Lennard-Jones fluid around solid cylinders with nanometer-scale radii by
applying the modified Wilhelmy equations derived in our previous study
Imaizumi  _et al._ (2020) to investigate the curvature effect on the SL and SV
interfacial tensions. From the results, we also examined whether Young’s
equation holds even for menisci around solids with nanoscale curvature.
Finally, we discuss the difference in the curvature dependence between the SL
and SV interfacial tensions (free energy) and the SL and SV interaction
potential energies (part of internal energy).

##  II Method

###  II.1 MD Simulation

Figure 1:  (a) A bird-eye view and (b) top and (c) side views of equilibrium
molecular dynamics (MD) simulation systems of hollow solid cylinders dipped
into a liquid pool of a simple Lennard-Jones (LJ) fluid. (d) Side view of the
system with a hollow solid plate [see Ref. 26 for details].

In this study, we employed equilibrium MD simulation systems of a quasi-
axisymmetric meniscus on a hollow cylinder dipped into a liquid pool of a
simple fluid as shown in Fig. 1. Except the boundary condition in the lateral
directions, the basic setup is similar to our previous study of quasi-2D
meniscus formed on a hollow rectangular solid plate. Imaizumi  _et al._ (2020)
Generic particles interacting through a LJ potential were adopted as the fluid
particles. The 12-6 LJ potential given by

$\Phi^{\mathrm{LJ}}(r_{ij})=4\epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}+c_{2}^{\mathrm{LJ}}\left(\frac{r_{ij}}{r_{\mathrm{c}}}\right)^{2}+c_{0}^{\mathrm{LJ}}\right],$
(2)

was used for the interaction between fluid particles, where $r_{ij}$ is the
distance between the particles $i$ at position $\mbox{\boldmath$r$}_{i}$ and
$j$ at $\mbox{\boldmath$r$}_{j}$, while $\epsilon$ and $\sigma$ denote the LJ
energy and length parameters, respectively. This LJ interaction was truncated
at a cut-off distance of $r_{\mathrm{c}}=3.5\sigma$ and quadratic functions
were added so that the potential and interaction force smoothly vanished at
$r_{\mathrm{c}}$. The constant values of $c_{2}^{\mathrm{LJ}}$ and
$c_{0}^{\mathrm{LJ}}$ were given in our previous study. Nishida  _et al._
(2014) Hereafter, fluid and solid particles are denoted by ‘f’ and ‘s’,
respectively and corresponding combinations are indicated by subscripts.

Three solid cylinders in contact with the fluid were prepared by using the
geometrical configuration of carbon nanotubes (CNTs) with their chiral indices
of (20,0), (40,0) and (80,0), where the solid particles were fixed on the
coordinate with the positions of hexagonal periodic structure with an inter-
particle distance $r_{\mathrm{ss}}$ of 0.141 nm. Note that the solid surface
is considered to be smooth with $r_{\mathrm{ss}}$ much smaller than
$\sigma_{\mathrm{ff}}$ and $\sigma_{\mathrm{sf}}$, and pinning is not induced
on this surface. The corresponding radii $R_{\mathrm{s}}$ of the cylinder are
0.777, 1.55 and 3.11 nm, respectively. The central axis of the cylinders is
set on the $z$-axis, i.e., the zigzag edge of the honeycomb structure was set
parallel to the $xy$-plane. The top and bottom parts of the cylinders are
covered by locating additional solid particles as the lid to prevent fluid
particles from entering into the cylinder. Note that the structure of these
lids does not have direct effect on the simulation results as indicated in our
previous study. Imaizumi  _et al._ (2020)

The solid-fluid (SF) interaction, which denotes SL or SV interaction, was also
expressed by the LJ potential in Eq. (2), where the length parameter
$\sigma_{\mathrm{sf}}$ was given by the Lorentz mixing rule, while the energy
parameter $\epsilon_{\mathrm{sf}}$ was changed in a parametric manner by
multiplying a SF interaction coefficient $\eta$ to the base value
$\epsilon^{0}_{\mathrm{sf}}=\sqrt{\epsilon_{\mathrm{ff}}\epsilon_{\mathrm{ss}}}$
as

$\epsilon_{\mathrm{sf}}=\eta\epsilon^{0}_{\mathrm{sf}}.$ (3)

This parameter $\eta$ expressed the wettability, i.e., $\eta$ and the contact
angle of a hemi-cylindrically shaped equilibrium droplet on a homogeneous flat
solid surface had a one-to-one correspondence Nishida  _et al._ (2014);
Yamaguchi  _et al._ (2019); Kusudo, Omori, and Yamaguchi (2019), and we set
the parameter $\eta$ between 0.03 and 0.15 so that the corresponding cosine of
the contact angle $\cos\theta$ is from $-0.9$ to $0.9$. The definition of the
contact angle is described later in Sec. III. Note that due to the fact that
the solid-solid inter-particle distance $r_{\mathrm{ss}}$ shown in Table 1
were relatively small compared to the LJ length parameters
$\sigma_{\mathrm{ff}}$ and $\sigma_{\mathrm{fs}}$, the surface is considered
to be very smooth, and the wall-tangential force from the solid on the fluid,
which induces pinning of the CL, is negligible. Yamaguchi  _et al._ (2019);
Kusudo, Omori, and Yamaguchi (2019)

We set a horizontal potential wall on the bottom (floor) of the calculation
cell fixed at $z=z_{\mathrm{flr}}$ about 5.3 nm below the bottom of the solid
plate, which interacted only with the fluid particles with a one-dimensional
potential field $\Phi_{\mathrm{flr}}^{\mathrm{1D}}$ as the function of the
distance from the wall given by

$\displaystyle\Phi_{\mathrm{flr}}^{\mathrm{1D}}(z^{\prime}_{i})=4\pi\rho_{n}\epsilon^{\mathrm{flr}}_{\mathrm{sf}}\sigma_{\mathrm{sf}}^{2}\left[\frac{1}{5}\left(\frac{\sigma_{\mathrm{sf}}}{z^{\prime}_{i}}\right)^{10}\\!\\!\\!\\!-\frac{1}{2}\left(\frac{\sigma_{\mathrm{sf}}}{z^{\prime}_{i}}\right)^{4}+c_{2}^{\mathrm{flr}}\left(\frac{z^{\prime}_{i}}{z_{\mathrm{c}}^{\mathrm{flr}}}\right)^{2}+c_{1}^{\mathrm{flr}}\left(\frac{z^{\prime}_{i}}{z_{\mathrm{c}}^{\mathrm{flr}}}\right)+c_{0}^{\mathrm{flr}}\right],$
(4) $\displaystyle z^{\prime}_{i}\equiv z_{i}-z_{\mathrm{flr}}.$ (5)

where $z_{i}$ is the $z$-position of fluid particle $i$, and
$\epsilon^{\mathrm{flr}}_{\mathrm{sf}}$ is set at
$0.09\epsilon^{0}_{\mathrm{sf}}$. This potential wall mimicked a mean
potential field created by a single layer of solid particles with a uniform
area number density $\rho_{n}$. Similar to Eq. (2), this potential field in
Eq. (4) was truncated at a cut-off distance of
$z_{\mathrm{c}}^{\mathrm{flr}}=3.5\sigma_{\mathrm{sf}}$ and a quadratic
function was added so that the potential and interaction force smoothly
vanished at $z_{\mathrm{c}}^{\mathrm{flr}}$. As shown in Fig. 1, fluid
particles were rather strongly attracted on this plane because this roughly
corresponded to a solid wall showing complete wetting. With this setup, the
liquid pool was stably kept even when the liquid pressure is low with a highly
wettable solid plate, and sufficient liquid bulk region was formed between
this wall and the bottom of the cylinder. Furthermore, we set another
horizontal potential wall on the top (ceiling) of the calculation cell fixed
at $z=z_{\mathrm{ceil}}$ about 4.7 nm above the top of the solid plate
exerting a repulsive potential field $\Phi_{\mathrm{ceil}}^{\mathrm{1D}}$ on
the fluid particles given by

$\displaystyle\Phi_{\mathrm{ceil}}^{\mathrm{1D}}(z^{\prime\prime}_{i})=4\pi\rho_{n}\epsilon^{\mathrm{ceil}}_{\mathrm{sf}}\sigma_{\mathrm{sf}}^{2}\left[\frac{1}{5}\left(\frac{\sigma_{\mathrm{sf}}}{z^{\prime\prime}_{i}}\right)^{10}+c_{2}^{\mathrm{ceil}}\left(\frac{z^{\prime\prime}_{i}}{z_{\mathrm{c}}^{\mathrm{ceil}}}\right)^{2}+c_{1}^{\mathrm{ceil}}\left(\frac{z^{\prime\prime}_{i}}{z_{\mathrm{c}}^{\mathrm{ceil}}}\right)+c_{0}^{\mathrm{ceil}}\right],$
(6) $\displaystyle z^{\prime\prime}_{i}\equiv z_{\mathrm{ceil}}-z_{i},$ (7)

where $\epsilon^{\mathrm{ceil}}_{\mathrm{sf}}$ is equal to
$\epsilon^{\mathrm{flr}}_{\mathrm{sf}}$ while a cut-off distance of
$z_{\mathrm{c}}^{\mathrm{ceil}}=\sigma_{\mathrm{sf}}$ is set to express a
repulsive potential wall.

In addition to these bottom and top potential walls, we also set another
cylindrical side potential wall with its axis on the $z$-axis and with a
radius $R_{\mathrm{sid}}$, which exerts a one-dimensional potential field on
the fluid particles $\Phi_{\mathrm{sid}}^{\mathrm{1D}}(r^{xy}_{i})$ as the
function of the horizontal distance from the wall given by

$\displaystyle\Phi_{\mathrm{sid}}^{\mathrm{1D}}(r^{xy}_{i})=4\pi\rho_{n}\epsilon^{\mathrm{sid}}_{\mathrm{sf}}\sigma_{\mathrm{sf}}^{2}\left[\frac{1}{5}\left(\frac{\sigma_{\mathrm{sf}}}{r^{xy}_{i}}\right)^{10}\\!\\!\\!\\!-\frac{1}{2}\left(\frac{\sigma_{\mathrm{sf}}}{r^{xy}_{i}}\right)^{4}+c_{2}^{\mathrm{flr}}\left(\frac{r^{xy}_{i}}{r_{\mathrm{c}}^{\mathrm{sid}}}\right)^{2}+c_{1}^{\mathrm{flr}}\left(\frac{r^{xy}_{i}}{r_{\mathrm{c}}^{\mathrm{sid}}}\right)+c_{0}^{\mathrm{sid}}\right],$
(8) $\displaystyle r^{xy}_{i}\equiv
R_{\mathrm{sid}}-\sqrt{x_{i}^{2}+y_{i}^{2}},$ (9)

where $r^{xy}_{i}$ is the lateral distance of fluid particle $i$ from the side
potential wall. Note that this side wall was adopted to achieve a quasi-
axisymmetric 2D-meniscus instead of applying the periodic boundary condition
in the horizontal $x$\- and $y$-directions as in our previous study with a
solid plate. The parameter $\epsilon^{\mathrm{sid}}_{\mathrm{sf}}$ was set at
0.0975$\epsilon^{0}_{\mathrm{sf}}$ so that the resulting contact angle at the
side wall may be roughly 90 degrees.

The temperature of the system was maintained at a constant temperature of
$T_{\mathrm{c}}$ at 90 K, which is above the triple point temperature Mastny
and de Pablo (2007), by applying velocity rescaling to the fluid particles
within 0.8 nm from the floor wall regarding the velocity components in the
$x$\- and $y$-directions. Note that this region was sufficiently away from the
bottom of the solid plate and no direct thermostating was imposed on the
region near the solid plate, so that this temperature control had no effects
on the present results.

With this setting, an axisymmetric quasi-2D LJ liquid with a meniscus-shaped
LV interface and the CL parallel to the $xy$-plane was formed as an
equilibrium state as exemplified in Fig. 1, where a liquid bulk with an
isotropic density distribution existed above the bottom wall by choosing a
proper number of fluid particles $N_{\mathrm{f}}$ as shown in Fig. 2. We
checked that the temperature was constant in the whole system after the
equilibration run described below. The velocity Verlet method was applied for
the integration of the Newtonian equation of motion with a time increment of 5
fs for all systems. The simulation parameters are summarized in Table 1 with
the corresponding non-dimensional ones, which are normalized by the
corresponding standard values based on $\epsilon_{\mathrm{ff}}$,
$\sigma_{\mathrm{ff}}$ and $m_{\mathrm{f}}$.

The physical properties of each equilibrium system with various $\eta$ values
were calculated as the time average of 20 ns which followed an equilibration
run of more than 10 ns.

Table 1:  Simulation parameters and their corresponding non-dimensional values.  property | value | unit | non-dim. value
---|---|---|---
$\sigma_{\mathrm{ff}}$ | 0.340 | nm | 1
$\sigma_{\mathrm{sf}}$ | 0.357 | nm | 1.05
$r_{\mathrm{ss}}$ | 0.141 | nm | 0.415
$\epsilon_{\mathrm{ff}}$ | $1.67\times 10^{-21}$ | J | 1
$\epsilon^{0}_{\mathrm{sf}}$ | $1.96\times 10^{-21}$ | J | 1.18
$\epsilon^{\mathrm{flr}}_{\mathrm{sf}}$ | $0.176\times 10^{-21}$ | J | 0.106
$\epsilon^{\mathrm{ceil}}_{\mathrm{sf}}$ | $0.176\times 10^{-21}$ | J | 0.106
$\epsilon^{\mathrm{sid}}_{\mathrm{sf}}$ | $0.192\times 10^{-21}$ | J | 0.115
$\epsilon_{\mathrm{sf}}$ | $\eta\times\epsilon^{0}_{\mathrm{sf}}$ |  |
$\eta$ | 0.03 – 0.15 | - | -
$m_{\mathrm{f}}$ | $6.64\times 10^{-26}$ | kg | 1
$R_{\mathrm{s}}$ | 0.777 – 3.11 | nm | 2.29 – 9.15
$T_{\mathrm{c}}$ | 90 | K | 0.703
$N_{\mathrm{f}}$ | 53778 - 60834 | - | -

##  III Results and discussion

###  III.1 Apparent contact angle

Figure 2:  (a) Half side snapshot, (b) distribution of the time-averaged fluid
density, and (c) distributions of the time-averaged downward force density
acting on the solid plate and solid-fluid (SF) potential energy density per
solid surface area for the system with the solid radius $R_{\mathrm{s}}=1.55$
nm and a SF interaction parameter $\eta=0.15$.

Similar to our previous study, Imaizumi  _et al._ (2020) we calculated the
distribution of force exerted from the fluid on the solid particles by
dividing the system into equal-sized bins in the $z$-direction, where a bin
height of $\delta z=$ 0.2115 nm was used, considering the periodicity of the
CNT structure. We defined the average force density
$\mathrm{d}\xi_{z}/\mathrm{d}z$ as the time-averaged total downward (in
$-z$-direction) force from the fluid on the solid particles in each bin
divided by the solid bin area $2\pi R_{\mathrm{s}}\delta z$. Except at the top
and bottom of the solid plate, $\mathrm{d}\xi_{z}/\mathrm{d}z$ corresponds to
the downward force per surface area. We also calculated the average SF
potential energy per area $u_{\mathrm{sf}}$, which was obtained by the same
procedure but substituting the downward force by the SF potential energy.

Figure 2 shows a half side-snapshot and the distribution of time-averaged
fluid density $\rho$ around the solid cylinder for the system with solid-fluid
interaction parameter $\eta=0.15$. The time-averaged distributions of the
downward force acting on the solid plate $\mathrm{d}\xi_{z}/\mathrm{d}z$ and
the SF potential energy $u_{\mathrm{sf}}$ are also displayed in the right
panel. We briefly summarize two essential features below, which are
qualitatively the same as in our previous study. Imaizumi  _et al._ (2020) 1)
Multi-layered adsorption layers were formed around the solid cylinder and the
bottom and side potential walls, and liquid bulk with a homogeneous density is
observed away from the cylinder, potential walls and the LV interface. 2) The
total downward force as the integral of $\mathrm{d}\xi_{z}/\mathrm{d}z$ can be
clearly separated into three local parts, i.e., $\xi_{z}^{\mathrm{top}}$
around the top, $\xi_{z}^{\mathrm{cl}}$ around the contact line, and
$\xi_{z}^{\mathrm{bot}}$ around the bottom. As indicated in Fig. 2 (c),
$\xi_{z}^{\mathrm{top}}$ and $\xi_{z}^{\mathrm{cl}}$ are positive, i.e.,
downward forces, and $\xi_{z}^{\mathrm{bot}}$ is negative, i.e., an upward
force. Note that the distributions of $\mathrm{d}\xi_{z}/\mathrm{d}z$ and
$u_{\mathrm{sf}}$ around the top and bottom had less physical meaning because
they included the top and bottom solid lids in the bin, and these parts for
$u_{\mathrm{sf}}$ are not displayed in the figure. However, the local integral
of $\mathrm{d}\xi_{z}/\mathrm{d}z$ indeed gave the physical information about
the force around the top and bottom parts. Also, note that $\xi_{z}$ has the
same dimension as the surface tension of force per length.

As exemplified in the density distribution in Fig. 2 (b), we evaluated the
contact angles for the plate and cylinder systems with different solid-fluid
interaction parameter to examine the curvature effects. For the plate system,
we followed the same procedure used to determine the apparent contact angle as
our previous study: Imaizumi  _et al._ (2020) the LV interface was defined as
the least-squares fitting circle on the density contour of $\rho=$400 kg/m3 at
the LV interface at height $z(x)$ excluding the region in the adsorption
layers near the solid, Nishida  _et al._ (2014); Yamaguchi  _et al._ (2019);
Kusudo, Omori, and Yamaguchi (2019); Imaizumi  _et al._ (2020) considering
that with a constant liquid-vapor interfacial tension $\gamma_{\mathrm{LV}}$
the static force balance is written as

$\frac{\mathrm{d}}{\mathrm{d}x}\left[\sin{\psi(x)}\right]=\frac{p_{\mathrm{V}}^{\mathrm{blk}}-p_{\mathrm{L}}^{\mathrm{blk}}}{\gamma_{\mathrm{LV}}},\quad\tan{\psi(x)}=\frac{\mathrm{d}z(x)}{\mathrm{d}x},$
(10)

where $\psi(x)$, $p_{\mathrm{V}}^{\mathrm{blk}}$ and
$p_{\mathrm{L}}^{\mathrm{blk}}$ denote the angle from the $x$-direction and
bulk pressure values in the vapor and liquid bulk, respectively. The solution
of Eq. (10) results in a constant curvature with constant
$\gamma_{\mathrm{LV}}$, $p_{\mathrm{V}}^{\mathrm{blk}}$ and
$p_{\mathrm{L}}^{\mathrm{blk}}$. Then, by setting the solid-fluid (SF)
interface position $r=r_{\mathrm{SF}}$ as the limit that the fluid could
reach, which can be easily estimated from the density distribution, we defined
the apparent contact angle $\theta_{\mathrm{app}}$ by the angle between the
extrapolation of the cylindrical LV-interface and SF-interface plane at
$r=r_{\mathrm{SF}}$. This definition provided a mechanical description
consistent with Young’s equation. Yamaguchi  _et al._ (2019)

Similarly, the contact angles on the cylinders were evaluated using the
analytical formula of the macroscopic meniscus shape. For an axisymmetric
equilibrium meniscus around a $z$-centered cylinder with neglecting gravity,
it follows for the meniscus height $z(r)$ given as a unique function of the
radial position $r\equiv\sqrt{x^{2}+y^{2}}$ that

$\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}\left[r\sin{\psi(r)}\right]=\frac{p_{\mathrm{V}}^{\mathrm{blk}}-p_{\mathrm{L}}^{\mathrm{blk}}}{\gamma_{\mathrm{LV}}},\quad\tan{\psi(r)}=\frac{\mathrm{d}z(r)}{\mathrm{d}r},$
(11)

where $\psi(r)$ denote the angle from the $r$-direction. We evaluated the
three constant values in this differential equation (11) from MD simulations:
$\gamma_{\mathrm{LV}}$ was obtained from a MD system with planer LV interfaces
by a standard mechanical process, Surblys  _et al._ (2014) whereas
$p_{\mathrm{V}}^{\mathrm{blk}}$ and $p_{\mathrm{L}}^{\mathrm{blk}}$ were
evaluated as the force per area on the top and bottom potential walls of the
present cylinder system, respectively, both excluding the region near the side
wall. Thus, by fitting the density contour of $\rho=$400 kg/m3 at the LV
interface excluding the region in the adsorption layers formed near the solid
surface and also excluding that near the side potential wall, a numerical
solution of the 2nd-order ordinary differential equation (11) can be obtained
for each system with different cylinder radius $R_{\mathrm{s}}$ and solid-
fluid interaction parameter $\eta$. As shown by the dotted black line in Fig.
2, the meniscus shape is well reproduced in this system. We determined the
contact angle as the angle between the extrapolated solution of the meniscus
shape and the SF-interface position in the $rz$-plane including the liquid
side. We assume a concave control volume (CV) shown in red color in Fig. 2 (b)
around the bottom of the cylinder to calculate the SL interfacial tension
below. The bottom face and the side cylindrical face of the CV are in the
liquid bulk, where the former is sufficiently away from both the bottom of the
system and bottom of the cylinder, whereas the latter at $r=r_{\mathrm{CV}}$
is away from the side boundary. Analogously, the top face is at the height
$z=z_{\mathrm{SL}}$ sufficiently away from both the bottom of the cylinder and
the contact line. On the other hand, the concave faces are set at the solid-
liquid boundary with its inner side face at $r=r_{\mathrm{SF}}$.

Figure 3:  (a) Density distributions around the (i) solid cylinder of radius
$R_{\mathrm{s}}=0.777$ nm and (ii) plate, and (b) relation between the cosine
of contact angle and solid-fluid interaction parameter for different cylinder
radii $R_{\mathrm{s}}$ and plate.

The density distributions around the solid cylinder with the smallest radius
$R_{\mathrm{s}}=0.777$ nm and the plate, and the relation between the SF
interaction coefficient $\eta$ and cosine of the contact angle
$\cos\theta_{\mathrm{app}}$ are shown in Fig. 3. For the latter, we displayed
the error bars and guide lines only for the plate and for the cylinder systems
with $R_{\mathrm{s}}=0.777$ nm, for better visualization: the error bars for
the systems with other radii were comparable to those for
$R_{\mathrm{s}}=0.777$ nm. As seen in Fig. 3 (a), the apparent meniscus shapes
of the cylinder and plate are different, indicating that different force
balances described by Eqs. (10) and (11), should be adopted to properly
evaluate the contact angle from the meniscus shape. With the increase of
$\eta$, the solid became more wettable, i.e., $\cos\theta_{\mathrm{app}}$
increased, and the cylinder with the smallest Radius $R_{\mathrm{s}}=0.777$ nm
were less wettable, i.e., had smaller $\cos\theta_{\mathrm{app}}$ than the
plate for all $\eta$ values tested. However; we should stress that the radius
dependence of the contact angle seen in Fig. 3 (b) was unexpectedly small even
with the smallest cylinder with its radius $R_{\mathrm{s}}$ that is comparable
to $\sigma_{\mathrm{ff}}$ or $\sigma_{\mathrm{sf}}$. We discuss the reason in
the following with the comparison between the apparent contact angle
$\theta_{\mathrm{app}}$ and the contact angle predicted by Young’s equation
(1) using the interfacial tensions obtained by the local forces.

###  III.2 Curvature dependence of the force around the contact line and the
solid-liquid and solid-vapor interfacial tensions

Figure 4:  Schematic of the extraction of the $z$-direction force from the
solid on the fluid in the red control volume around the bottom considering the
force distribution in Fig. 2.

We further investigate the curvature dependence of wetting behavior with the
calculations of $\xi_{z}^{\mathrm{cl}}$ and the interfacial tensions. We start
from the extraction of the upward force $F_{z}^{\mathrm{bot}}$ exerted from
the solid on the liquid in the red control volume (CV) around the bottom
illustrated in the top-left panel of Fig. 4. Note that we evaluate the upward
force $F_{z}$ on the corresponding liquid from the solid as follows: the
positive direction for $F_{z}$ is $+z$-direction and is opposite to that for
$\xi_{z}$ (force per length) in the $-z$-direction on the corresponding solid
from the liquid. The top face of the CV at $z=z_{\mathrm{SL}}$ is sufficiently
away from both the bottom of the solid and the contact line, where the liquid
density $\rho$ near the solid is constant in the $z$-direction satisfying

$\frac{\partial\rho}{\partial
z}=0\quad\mathrm{for}\quad\left(r-R_{\mathrm{s}}\right)^{2}+(z-z_{\mathrm{SL}})^{2}\leq(r_{\mathrm{c}}^{\mathrm{sf}})^{2},$
(12)

with $r_{\mathrm{c}}^{\mathrm{sf}}$ the cut-off distance for solid-fluid
interaction. The force of present interest $F_{z}^{\mathrm{bot}}$ comes from
the neighbouring solid within the cutoff range, i.e., from the red-dotted and
black-dotted solid parts as indicated by the blue arrows (top-left panel). On
the other hand, with the condition in Eq. (12), the sum of the upward forces
on the liquid parts in the red-solid and black-solid lines from the red-dotted
solid part (bottom-left panel) is $2\pi R_{\mathrm{s}}\xi_{z}^{\mathrm{bot}}$,
which is the reaction force on the solid around the bottom indicated in Fig.
2. From the comparison of the arrows regarding the two, $F_{z}^{\mathrm{bot}}$
is obtained by adding the missing force and subtracting the unnecessary force
as in the bottom panel as

$\displaystyle F_{z}^{\mathrm{bot}}$ $\displaystyle=2\pi
R_{\mathrm{s}}\xi_{z}^{\mathrm{bot}}+F_{z}^{\mathrm{diag(SL)}}-(-F_{z}^{\mathrm{diag(SL)}})$
$\displaystyle=2\pi
R_{\mathrm{s}}\xi_{z}^{\mathrm{bot}}+2F_{z}^{\mathrm{diag(SL)}}$ (13)

where the two “diagonal” forces denoted by $F_{z}^{\mathrm{diag(SL)}}$ and
$-F_{z}^{\mathrm{diag(SL)}}$ have an opposite sign with the same absolute
value due to the symmetry under the condition in Eq. (12).

The value of unknown $F_{z}^{\mathrm{diag(SL)}}$ now must be determined.
Although this $F_{z}^{\mathrm{diag(SL)}}$ can be obtained directly by MD
simulations based on the definition, in the special case where the solid is so
smooth compared to the length-scale of solid-fluid inter-particle interaction
that the density can be considered constant independent of the position as the
present solid with the graphene geometry, $F_{z}^{\mathrm{diag(SL)}}$ can be
analytically expressed by

$F_{z}^{\mathrm{diag(SL)}}=-\pi R_{\mathrm{s}}u_{\mathrm{SL}}.$ (14)

The detailed derivation is described in Appendix B. Similarly, the diagonal
force $F_{z}^{\mathrm{diag(SV)}}$ on the vapor below a plane
$z=z_{\mathrm{SV}}$ from the solid above the plane can also be analytically
formulated by

$F_{z}^{\mathrm{diag(SV)}}=-\pi R_{\mathrm{s}}u_{\mathrm{SV}}.$ (15)

By considering a force balance similar to that illustrated in Fig. 4 and by
also assuming that the solid is smooth and the fluid particles are not pinned
around the contact line, the downward force on the solid
$\xi_{z}^{\mathrm{cl}}$ per length is analytically given by (see Appendix C
for details)

$\xi_{z}^{\mathrm{cl}}=-u_{\mathrm{SL}}+u_{\mathrm{SV}}=(-u_{\mathrm{SL}})-(-u_{\mathrm{SV}}),$
(16)

where the final equality shows that $\xi_{z}^{\mathrm{cl}}$ is the difference
in the magnitude between the two interfacial potential energy densities,
considering that $u_{\mathrm{SL}}$ and $u_{\mathrm{SV}}$ are both negative as
exemplified in Fig. 2.

Figure 5:  (Left) dependence SL and SV potential energy densities
$-u_{\mathrm{SL}}$ and $-u_{\mathrm{SV}}$ on the solid-fluid interaction
parameter $\eta$ for solid plate and solid cylinders with different radii
$R_{\mathrm{s}}$. (Right) Comparison between the downward force
$\xi_{z}^{\mathrm{cl}}$ on the solid around the CL and the difference of the
potential energy density $-u_{\mathrm{SL}}+u_{\mathrm{SV}}$.

Figure 5 shows the dependence of SL and SV potential energy densities
$-u_{\mathrm{SL}}$ and $-u_{\mathrm{SV}}$ on the solid-fluid interaction
parameter $\eta$ for solid plate and solid cylinders with different radii
$R_{\mathrm{s}}$, and the comparison between the downward force
$\xi_{z}^{\mathrm{cl}}$ on the solid around the CL and the difference of
potential energy density $-u_{\mathrm{SL}}+u_{\mathrm{SV}}$. As easily
expected, $-u_{\mathrm{SL}}$ and $-u_{\mathrm{SV}}$ increased with the
increase of $\eta$ as shown in the left panel; however, $-u_{\mathrm{SL}}$ and
$-u_{\mathrm{SV}}$ showed opposite dependence on $R_{\mathrm{s}}$:
$-u_{\mathrm{SL}}$ was larger for smaller cylinder radius $R_{\mathrm{s}}$
whereas $-u_{\mathrm{SV}}$ was smaller. We will discuss this interesting
difference later. In the right panel of Fig. 5, a very good agreement between
$\xi_{z}^{\mathrm{cl}}$ and $-u_{\mathrm{SL}}+u_{\mathrm{SV}}$ are observed
for the whole range of $\eta$ with different radii $R_{\mathrm{s}}$ in the
right panel. This indicates that the force from the solid on the liquid in the
CV around the bottom shown as a red concave in Fig. 4 can be properly
evaluated by Eq. (14) because the present system with solid particles located
at the position of graphene was supposed to be sufficiently smooth to meet the
condition assumed in the analytical derivation in Appendices B and C. In
addition, due to the opposite radius dependence of $-u_{\mathrm{SL}}$ and
$-u_{\mathrm{SV}}$ shown in the left panel, the difference of
$\xi_{z}^{\mathrm{cl}}$ for the smallest radius and that for the plate was as
large as about $10\times 10^{-3}$ N/m, which is comparable to
$\gamma_{\mathrm{LV}}$. In addition, $\xi_{z}^{\mathrm{cl}}$ was much larger
than the difference of $-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})$ and
$-(\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}})$ shown later.

We now examine the relative SL interfacial tension by using the total static
force balance including the force $F_{z}^{\mathrm{bot}}$ on the red-solid CV
in Fig. 4:

$2\pi\int_{r_{\mathrm{SF}}}^{r_{\mathrm{CV}}}\mathrm{d}r\>\tau_{zz}(r,z_{\mathrm{SL}})r+\pi
r_{\mathrm{CV}}^{2}p_{\mathrm{L}}^{\mathrm{blk}}+F_{z}^{\mathrm{bot}}=0,$ (17)

where the 1st and 2nd terms of the LHS are the fluid stress integral on the
top face and pressure from the bottom, respectively. Note that side boundary
is set at the bulk satisfying $\tau_{rz}=0$, and also that the fluid stress is
zero on the faces around the concave under the present stress definition.
Rowlinson and Widom (1982); Yamaguchi  _et al._ (2019); Imaizumi  _et al._
(2020); Kusudo, Omori, and Yamaguchi (2021) The first term of the LHS is
related to the relative SL interfacial tension
$\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}$ with extended Bakker’s equation
applied for a cylindrical SL interface by (see Appendix A)

$\displaystyle\gamma_{\text{SL}}-\gamma_{\text{S0}}$
$\displaystyle=\frac{1}{r_{\mathrm{SF}}}\left[\int_{r_{\mathrm{SF}}}^{r_{\mathrm{CV}}}\mathrm{d}r\>\tau_{zz}(r,z_{\mathrm{SL}})r+\frac{(r_{\mathrm{CV}}^{2}-r_{\mathrm{SF}}^{2})p_{\mathrm{L}}^{\mathrm{blk}}}{2}\right].$
(18)

Thus, from Eqs. (17), (13) and (18),
$\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}$ results in

$\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}=-\frac{R_{\mathrm{s}}}{r_{\mathrm{SF}}}\xi_{z}^{\mathrm{bot}}-\frac{r_{\mathrm{SF}}}{2}p_{\mathrm{L}}^{\mathrm{blk}}-\frac{F_{z}^{\mathrm{diag(SL)}}}{\pi
r_{\mathrm{SF}}},$ (19)

where the pressure $p_{\mathrm{L}}^{\mathrm{blk}}$ was measured in MD systems
as the normal force per area exerted on the bottom potential wall for $r\leq
r_{\mathrm{CV}}$. Under the condition that Eq. (14) holds for
$F_{z}^{\mathrm{diag(SL)}}$ as in the present systems Eq. (13) is further
rewritten by

$F_{z}^{\mathrm{bot}}=2\pi
R_{\mathrm{s}}\left(\xi_{z}^{\mathrm{bot}}-u_{\mathrm{SL}}\right),$ (20)

and Eq. (19) writes

$\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}=-\frac{R_{\mathrm{s}}}{r_{\mathrm{SF}}}\xi_{z}^{\mathrm{bot}}-\frac{r_{\mathrm{SF}}}{2}p_{\mathrm{L}}^{\mathrm{blk}}+\frac{R_{\mathrm{s}}}{r_{\mathrm{SF}}}u_{\mathrm{SL}}.$
(21)

Similarly, the relative solid-vapor interfacial tension
$\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}}$ writes

$\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}}=\frac{R_{\mathrm{s}}}{r_{\mathrm{SF}}}\xi_{z}^{\mathrm{top}}-\frac{r_{\mathrm{SF}}}{2}p_{\mathrm{V}}^{\mathrm{blk}}+\frac{R_{\mathrm{s}}}{r_{\mathrm{SF}}}u_{\mathrm{SV}}.$
(22)

Note that Eqs. (21) and (22) are equivalent to the derivation for the quasi-2D
Wilhelmy plate Imaizumi  _et al._ (2020) except the point that the radii of
the solid surface area $R_{\mathrm{s}}$ and solid-fluid interface area
$r_{\mathrm{SF}}$ are different for the present Wilhelmy-cylinders. Also note
that the meniscus shape including the contact angle does not explicitly appear
in Eqs. (21) and (22).

Figure 6: Dependence of the SL and SV interfacial tensions on the radius of
solid cylinders with different wettability.

Figure 6 shows the relative SL and SV interfacial tensions calculated by Eq.
(21) and (22) for different solid radii $R_{\mathrm{s}}$ with various
wettability parameter $\eta$. For a direct comparison between the two
interfacial tensions, the values
$-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})+\gamma_{\mathrm{LV}}$ and
$-(\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}})$ are shown with the left
vertical axis, which correspond to the works of adhesion:

$W_{\mathrm{SL}}\equiv-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})+\gamma_{\mathrm{LV}}$
(23)

and

$W_{\mathrm{SV}}\equiv-(\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}}),$ (24)

where the value of $\gamma_{\mathrm{LV}}=9.79\times 10^{-3}$ N/m obtained in
our previous study Imaizumi  _et al._ (2020) was used. For a flat interface,
they are defined as the minimum works needed to strip the liquid and vapor off
the flat solid surface, respectively under constant temperature and pressure
condition. Yamaguchi  _et al._ (2019); Bistafa  _et al._ (2021) Note that both
works of adhesion $W_{\mathrm{SL}}$ and $W_{\mathrm{SV}}$ are positive, and we
will discuss about them later from a viewpoint of the free energy as well as
the curvature effects. With the decrease of the radius $R_{\mathrm{s}}$, both
$-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})$ and
$-(\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}})$ became smaller, and the
dependence was more remarkable for larger $\eta$ value. For the smallest
cylinder with $R_{\mathrm{s}}=0.777$ nm,
$-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})$ was about $2\times 10^{-3}$ N/m
smaller than that of the flat plate. However,
$-(\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}})$ was also reduced with the
decrease of $R_{\mathrm{s}}$, and this resulted in the rather small dependence
of the contact angle on the radius shown in Fig. 3.

Figure 7:  Comparison between the apparent contact angle
$\cos\theta_{\mathrm{app}}$ and that estimated by Young’s equation
$\cos\theta_{\mathrm{Y}}$ using the interfacial tensions obtained by the
mechanical route. The value of $\eta$ ranges from $0.03$ to $0.15$ for each
cylinder radius $R_{\mathrm{s}}$ and plate.

###  III.3 Applicability of Young’s equation

Using the relative interfacial tensions
$\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}$ and
$\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}}$ obtained in the previous
subsection, we examined whether Young’s equation holds for the present system
with a curved solid surface. Figure 7 shows the comparison between the
apparent contact angle cosine $\cos\theta_{\mathrm{app}}$ in Fig. 3 determined
from the meniscus shape and that estimated by Young’s equation (1) defined by

$\displaystyle\cos\theta_{\mathrm{Y}}$
$\displaystyle=\frac{\gamma_{\mathrm{SV}}-\gamma_{\mathrm{SL}}}{\gamma_{\mathrm{LV}}}\equiv\frac{(\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}})-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})}{\gamma_{\mathrm{LV}}}$
$\displaystyle=\frac{W_{\mathrm{SL}}-W_{\mathrm{SV}}}{\gamma_{\mathrm{LV}}}-1$
(25)

using the interfacial tensions obtained above via the mechanical route. Note
that $\gamma_{\mathrm{LV}}$ was set constant considering that its curvature
dependence appeared only for a radius of curvature smaller than about
$3\sigma_{\mathrm{ff}}$ for the LJ fluid, Yaguchi, Yano, and Fujikawa (2010)
which is smaller than that in the present study. In addition, it has been
shown that $\gamma_{\mathrm{LV}}$ consistent with Young’s equation should be
defined at a position excluding the adsorption layers around the SL interface,
Yamaguchi  _et al._ (2019) at which the radius of curvature of LV interface is
sufficiently large and the curvature effect is negligible. For the whole range
of $\eta$ values and radii $R_{\mathrm{s}}$ tested,
$\cos\theta_{\mathrm{app}}$ and $\cos\theta_{\mathrm{Y}}$ agreed very well,
and this indicates that Young’s equation holds for the present systems with
curved solid surfaces without pinning if the solid-related relative
interfacial tensions $\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}$ and
$\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}}$ are properly evaluated via a
mechanical route.

###  III.4 Discussion

Figure 8:  Comparison of the fluid-density around a solid-liquid interface
between the (a) cylinder with $R_{\mathrm{s}}=0.777$ nm and (b) flat plate,
with $\eta=0.15$. Positions of the solid particles are shown with small black
and gray circles, and the distance range of $r_{\mathrm{sf}}$ from a solid
particle satisfying interaction potential
$\Phi_{\mathrm{sf}}(r_{\mathrm{sf}})\leq-\epsilon_{\mathrm{sf}}/2$ is depicted
by magenta circles.

A question arises: why was the curvature dependence of the relative
interfacial tensions $\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}$ and
$\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}}$ shown in Fig. 6 not so large
compared to the contact line force
$\xi_{z}^{\mathrm{cl}}=-u_{\mathrm{SL}}+u_{\mathrm{SV}}$ seen in Fig. 5? More
specifically, why did $-u_{\mathrm{SL}}$ in the left panel of Fig. 5 and
$-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})$ in Fig. 6 show opposite
dependence on $R_{\mathrm{s}}$? To examine the curvature dependence of
$-u_{\mathrm{SL}}$, we evaluated the density field around the solid-liquid
interface, where we have carried out an additional simulation with a solid
cylinder with a chiral index (60,0) ($R_{\mathrm{s}}=2.33$ nm). Figure 8 shows
the comparison of the fluid-density around a solid-liquid interface between
the (a) cylinder with $R_{\mathrm{s}}=0.777$ nm and (b) flat plate, with
$\eta=0.15$. Positions of the solid particles are shown with small black and
gray circles, where the particles with the same color are at the same height
$z$, and the distance range of $r_{\mathrm{sf}}$ from a black solid particle
satisfying interaction potential
$\Phi_{\mathrm{sf}}(r_{\mathrm{rf}})\leq-\epsilon_{\mathrm{sf}}/2$, as
indicated by the blue potential graph, is depicted by magenta circles. As seen
in this figure, high density region in red corresponding to the fluid first
adsorption layer are included in this distance range more for
$R_{\mathrm{s}}=0.777$ nm, and this results in the higher $-u_{\mathrm{SL}}$
for smaller $R_{\mathrm{s}}$ indicated in the left panel of Fig. 5 because the
fluid particles in this distance range have the main contribution. Similarly,
one can expect smaller $-u_{\mathrm{SV}}$ for smaller $R_{\mathrm{s}}$ because
the fluid particles at the SV interface are simply adsorbed on the solid
surface as a single particle, i.e., these fluid particles do not form
adsorption layers as those around the SL interface observed in Fig. 2, and
such single particle is subject to weaker interaction potential from the
curved solid than from the flat solid.

Figure 9:  Solid-liquid interfacial potential energy per area
$-u_{\mathrm{SL}}$, the work of adhesion $W_{\mathrm{SL}}$, and the entropy
loss $-T\Delta s_{\mathrm{SL}}$ calculated by Eq. (26) for various cylinder
radius $R_{\mathrm{s}}$ with a solid-fluid interaction parameter $\eta=0.15$.

Different from the average SL interaction potential $-u_{\mathrm{SL}}$, the
relative interfacial tension is considered the interfacial free energy per
area, i.e., $\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}$ is related to the SL
work of adhesion $W_{\mathrm{SL}}$ in Eq. (23), and it consists of internal
energy and entropy terms: Surblys  _et al._ (2018); Bistafa  _et al._ (2021)

$W_{\mathrm{SL}}=-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})+\gamma_{\mathrm{LV}}=-u_{\mathrm{SL}}-(-T\Delta
s_{\mathrm{SL}}),$ (26)

where $-T\Delta s_{\mathrm{SL}}$ is due to the entropy “loss” $-\Delta
s_{\mathrm{SL}}$ induced by the density increase in the adsorption layers of
the SL interface. Figure 9 shows the solid-liquid interfacial potential energy
per area $-u_{\mathrm{SL}}$, the work of adhesion $W_{\mathrm{SL}}$ and the
entropy loss $-T\Delta s_{\mathrm{SL}}$ in Eq. (26) for various cylinder
radius $R_{\mathrm{s}}$ with $\eta=0.15$. Note that the additional data for a
cylinder radius $R_{\mathrm{s}}=2.33$ nm were also shown, and the radius for
the plate was set as $R_{\mathrm{s}}=\infty$. A larger entropy loss exceeding
the potential gain in $-u_{\mathrm{SL}}$ resulted in the opposite curvature
dependence between the SL interfacial potential energy $-u_{\mathrm{SL}}$ and
the relative interfacial tension
$-(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})$ seen in Figs. 5 and 6. In
addition, it is indicated that the curvature effect could remain even for a
relatively large cylinder radius $R_{\mathrm{s}}$ larger than about
$10\sigma_{\mathrm{ff}}$, which is much larger than the radius dependence
range of $\gamma_{\mathrm{LV}}$ reported for the LJ droplets. Yaguchi, Yano,
and Fujikawa (2010)

##  IV CONCLUDING REMARKS

In this study, we successfully extracted the SL and SV interfacial tensions of
a simple Lennard-Jones fluid around a solid cylinder with a nanometer-scale
diameter by extending the theoretical nano-Wilhelmy equations for a quasi-two-
dimensional flat solid plate from our previous study. The SL and SV
interfacial tensions were calculated from the integral of the normal-stress in
the wall-tangential direction by considering the mechanical force balances on
control volumes set around the bottom and top of the solid cylinder subject to
the fluid stress and external force from the solid, where the local force on
the solid around the contact line expressed by these external forces agreed
well with the analytical expression. The theoretical contact angle calculated
by Young’s equation using these interfacial tensions agreed well with the
apparent contact angle estimated by the analytical solution fitted to the
meniscus shape, showing that Young’s equation holds even for menisci around
solids with nanoscale curvature if the interfacial tensions are properly
evaluated. It was also shown that the curvature dependence of the SL and SV
interfacial tensions as the free energy was different from that of the
corresponding interaction potential energies as a part of the internal energy,
which explains the weak curvature dependence of the contact angle in the
present results.

The accurate calculation of the interfacial tensions on curved surfaces could
explain the unique wetting behavior of water on carbon nanotubes. Homma  _et
al._ (2013); Imadate and Hirahara (2018) In addition, it should enable the
exploration of the Tolman equation for the solid-related interfaces. Related
to this, an interesting future target is the interfacial tensions inside the
curved interface from a mechanical route, which should enable the analysis of
nano-confinement effects or nanoscale capillary as well, e.g., in carbon
nanotubes.

###### Acknowledgements.

T.O., H.K. and Y.Y. were supported by JSPS KAKENHI grant (Nos. JP18K03929,
JP20J20251 and JP18K03978), Japan, respectively. Y.Y. was also supported by
JST CREST grant (No. JPMJCR18I1), Japan.

DATA AVAILABILITY

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

AUTHOR DECLARATIONS
Conflict of Interest

The authors have no conflicts to disclose.

##  Appendix A Extended Bakker’s equation for cylinder

Figure 10:  Thought experiment for the connection between relative solid-
liquid interfacial tension and pressure distribution around the solid
cylinder.

We formulate the relative solid-liquid (SL) interfacial tension relative to
the solid-vacuum (S0) $\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}}$ between
cylindrical solid surface and liquid through the thought experiment shown in
Fig. 10. The side piston normal to the cylinder axis $z$ is in contact only
with the liquid, i.e., the inner radius is at $r_{\mathrm{SL}}$ set at the
limit radius nearest to the solid that the fluid particle can reach, whereas
the outer radius $r_{\mathrm{L}}^{\mathrm{blk}}$ is at liquid bulk
sufficiently away from the SL interface. The axis-normal stress defined by

$\tau_{zz}(r)\equiv-p_{z}(r)$ (27)

is a function of radial position $r$, and it satisfies

$p_{z}(r)=p_{\mathrm{L}}^{\mathrm{blk}}=\text{const.}\quad(r\geq
r_{\mathrm{L}}^{\mathrm{blk}}).$ (28)

On the other hand, the top piston is set at the liquid bulk, on which
homogeneous pressure identical to the bulk pressure
$p_{\mathrm{L}}^{\mathrm{blk}}$ is exerted. We suppose a virtual infinitesimal
displacement $\delta z$ of the side piston with a simultaneous downward
displacement of the top piston so that the liquid volume may not change. If
the operation is quasi-static at a constant temperature, the change of the
Helmholtz free energy $\delta F$ is equal to the work $\delta W$ exerted on
the system given by

$\displaystyle\delta F=\delta W$
$\displaystyle=p_{\mathrm{L}}^{\mathrm{blk}}\delta V-2\pi\delta
z\int_{r_{\mathrm{SL}}}^{r_{\mathrm{L}}^{\mathrm{blk}}}\mathrm{d}r\>p_{z}(r)r$
(29)

where the volume increase and decrease $\delta V$ due to the motions of side
and top pistons, respectively writes

$\delta V=2\pi\delta
z\int_{r_{\mathrm{SL}}}^{r_{\mathrm{L}}^{\mathrm{blk}}}\mathrm{d}r\>r=\pi\left[(r_{\mathrm{L}}^{\mathrm{blk}})^{2}-r_{\mathrm{SL}}^{2}\right]\delta
z$ (30)

The free energy change $\delta F$ in Eq. (29) is uniquely defined as long as
$r_{\mathrm{L}}^{\mathrm{blk}}$ is set in the bulk satisfying Eq. (28). From a
macroscopic point of view, the SL interface is increased and S0 interface is
reduced with this operation. By assuming that the SL interface is at
$r_{\mathrm{SL}}$, it follows that

$\delta F=2\pi
r_{\mathrm{SL}}(\gamma_{\mathrm{SL}}-\gamma_{\mathrm{S0}})\delta z$ (31)

By equating Eqs. (29) and (31), the following relation is derived as extended
Bakker’s equation for a cylindrical SL interface:

$\displaystyle\gamma_{\text{SL}}-\gamma_{\text{S0}}=\frac{1}{r_{\mathrm{SL}}}\left[\int_{r_{\mathrm{SL}}}^{r_{\mathrm{L}}^{\mathrm{blk}}}\mathrm{d}r\>\tau_{zz}(r)r+\frac{p_{\mathrm{L}}^{\mathrm{blk}}\left[(r_{\mathrm{L}}^{\mathrm{blk}})^{2}-r_{\mathrm{SL}}^{2}\right]}{2}\right],$
(32)

where Eq. (27) is used as well. Equation (32) means that the relative SL
interfacial tension is obtained with the stress integral, bulk pressure and
the interface position.

The relative solid-vapor (SV) interfacial tension is expressed as well by

$\displaystyle\gamma_{\mathrm{SV}}-\gamma_{\mathrm{S0}}$
$\displaystyle=\frac{1}{r_{\mathrm{SV}}}\left[\int_{r_{\mathrm{SV}}}^{r_{\mathrm{V}}^{\mathrm{blk}}}\mathrm{d}r\>\tau_{zz}(r)r+\frac{p_{\mathrm{V}}^{\mathrm{blk}}\left[(r_{\mathrm{V}}^{\mathrm{blk}})^{2}-r_{\mathrm{SV}}^{2}\right]}{2}\right].$
(33)

##  Appendix B Extraction of the interaction force between the liquid and
solid across a $z$-normal plane at a SL interface.

Figure 11:  Region for the double integral of the mean field to calculate the
interaction between the solid cylinder above $z_{\mathrm{s}}=z^{\mathrm{SL}}$
and liquid below $z_{\mathrm{f}}=z^{\mathrm{SL}}$ (top-left triangle in solid
blue line). The geometrical relation is shown in the inset. The cutoff
distance $z_{\mathrm{c}}$ for $|z_{\mathrm{f}}-z_{\mathrm{s}}|$ depends on the
relative radial position $r_{\mathrm{f}}-R_{\mathrm{s}}$.

We derive the interaction force $F_{\mathrm{z}}^{\mathrm{diag(SL)}}$
exemplified in the bottom panels of Fig. 4, namely, the interaction force
between the solid above a $z$-normal plane $z=z_{\mathrm{SL}}$ and liquid
below that plane. The plane $z=z_{\mathrm{SL}}$ is at a height around which
the fluid density is independent of the position $z$ because the plane is
sufficiently away from the contact line and the bottom of the cylinder.

Taking into account that the solid is supposed to be smooth for the fluid
particles because the interparticle distance parameters $\sigma_{\mathrm{ff}}$
and $\sigma_{\mathrm{sf}}$ are sufficiently large compared to
$r_{\mathrm{ss}}$ between solid particles, the solid-liquid interaction force
can be analytically modeled by assuming the mean fields of the fluid and
solid. The mean number density per volume
$\rho_{V}^{\mathrm{f}}(r_{\mathrm{f}})\ (=\rho/m_{\mathrm{f}})$ of the fluid
is given as a function of the radial position $r_{\mathrm{f}}$ of the fluid,
whereas a constant mean number density per area $\rho_{A}^{\mathrm{s}}$ of the
solid at $r=R_{\mathrm{s}}$ is used considering the present system with a
solid cylinder of zero-thickness without volume; however, the following
derivation can easily be extended for a system with a solid with a volume and
density per volume in the range $r\leq R_{\mathrm{s}}$.

We start from the potential energy on a solid particle at position
$(R_{\mathrm{s}},\vartheta_{\mathrm{s}},z_{\mathrm{s}})$ due to a fluid
particle at $(r_{\mathrm{f}},\vartheta_{\mathrm{f}},z_{\mathrm{f}})$ both in
the cylindrical $r\vartheta z$-coordinate, given by Eq. (2). Assuming that the
fluid particles are homogeneously distributed in the azimuth direction
$\vartheta$ with a number density
$\rho_{V}^{\mathrm{f}}(r_{\mathrm{f}},z_{\mathrm{f}})$ per volume, the mean
potential field from an infinitesimal fluid volume segment of
$\mathrm{d}z_{\mathrm{f}}\times 2\pi r_{\mathrm{f}}\mathrm{d}r_{\mathrm{f}}$
on a solid particle is defined by using
$\rho_{V}^{\mathrm{f}}(z_{\mathrm{f}},r_{\mathrm{f}})$ and the mean local
potential $\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$ as
$\rho_{V}^{\mathrm{f}}(z_{\mathrm{f}},x_{\mathrm{f}})\mathrm{d}z_{\mathrm{f}}\mathrm{d}r_{\mathrm{f}}\cdot\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$,
where $\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$ is given
by

$\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})\equiv\int_{0}^{2\pi}\Phi_{\mathrm{LJ}}\left(r_{\mathrm{sf}}\right)r_{\mathrm{f}}\mathrm{d}\vartheta_{\mathrm{f}}$
(34)

where we define

$r_{\mathrm{sf}}=\sqrt{(r_{\mathrm{f}}\cos\vartheta_{\mathrm{f}}-R_{\mathrm{s}}\cos\vartheta_{\mathrm{s}})^{2}+(r_{\mathrm{f}}\sin\vartheta_{\mathrm{f}}-R_{\mathrm{s}}\sin\vartheta_{\mathrm{s}})^{2}+z{{}^{\prime}}_{\mathrm{f}}^{2}},\quad$
(35)

and

$z{{}^{\prime}}_{\mathrm{f}}\equiv
z_{\mathrm{f}}-z_{\mathrm{s}}\equiv-z{{}^{\prime}}_{\mathrm{s}},\quad
r{{}^{\prime}}_{\mathrm{f}}\equiv r_{\mathrm{f}}-R_{\mathrm{s}}.$ (36)

Then, the local force in the $z$-direction
$f_{z}^{\mathrm{s}}(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})\mathrm{d}z_{\mathrm{f}}\mathrm{d}r_{\mathrm{f}}\mathrm{d}z_{\mathrm{s}}$
exerted on the solid cylinder in
$[z_{\mathrm{s}},z_{\mathrm{s}}+\mathrm{d}z_{\mathrm{s}}]$ with an area number
density of the solid particles $\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})$ from
the present fluid volume-segment is given by:

$\displaystyle
f_{z}^{\mathrm{s}}(z_{\mathrm{s}},z_{\mathrm{f}},r_{\mathrm{f}})\mathrm{d}z_{\mathrm{f}}\mathrm{d}r_{\mathrm{f}}\mathrm{d}z_{\mathrm{s}}$
$\displaystyle=-\rho_{V}^{\mathrm{f}}(z_{\mathrm{f}},r_{\mathrm{f}})\frac{\partial\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})}{\partial
z_{\mathrm{s}}}\mathrm{d}z_{\mathrm{f}}\mathrm{d}r_{\mathrm{f}}\cdot 2\pi
R_{\mathrm{s}}\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})\mathrm{d}z_{\mathrm{s}}$
(37)

where

$f_{z}^{\mathrm{s}}(z_{\mathrm{s}},z_{\mathrm{f}},r_{\mathrm{f}})=-2\pi
R_{\mathrm{s}}\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})\rho_{V}^{\mathrm{f}}(z_{\mathrm{f}},r_{\mathrm{f}})\frac{\partial\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})}{\partial
z_{\mathrm{s}}}$ (38)

denotes the axial force density on the solid given as a function of
$z_{\mathrm{s}}$, $z_{\mathrm{f}}$ and $r_{\mathrm{f}}$.

Since $\Phi_{\mathrm{LJ}}(r_{\mathrm{sf}})$ is truncated at the cutoff
distance $r_{\mathrm{c}}$ in the present case,

$\displaystyle\phi\left(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}}\right)=0,\quad\frac{\partial\phi\left(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}}\right)}{\partial
z_{\mathrm{s}}}=0$ (39)
$\displaystyle\mathrm{for}\quad|z{{}^{\prime}}_{\mathrm{f}}|\geq\sqrt{r_{\mathrm{c}}^{2}-r{{}^{\prime}}_{\mathrm{f}}^{2}}\equiv
z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})\quad\mathrm{or}\quad
r{{}^{\prime}}_{\mathrm{f}}\geq r_{\mathrm{c}}$

holds, where $z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})$ as a function of
$r{{}^{\prime}}_{\mathrm{f}}$ denotes the cutoff with respect to
$z{{}^{\prime}}_{\mathrm{f}}$. This cutoff is not critical as long as
$\phi\left(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}}\right)$
quickly vanishes with the increase of $r$, but we continue the derivation
including the cutoff for simplicity. With the definition of $r_{\mathrm{SF}}$
as the limit that the fluid could reach, it follows that

$\rho_{V}^{\mathrm{f}}=0\quad\mathrm{for}\quad
r_{\mathrm{f}}<r_{\mathrm{SF}}.$ (40)

In addition, considering that
$\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$ is an even
function with respect to $z{{}^{\prime}}_{\mathrm{f}}$, i.e.,

$\phi\left(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}}\right)=\phi(-z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}}),$
(41)

it follows for the mean local potential $\phi$ that

$\frac{\partial\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})}{\partial
z_{\mathrm{s}}}=-\frac{\partial\phi(-z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})}{\partial
z_{\mathrm{s}}},$ (42)

and

$\frac{\partial\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})}{\partial
z_{\mathrm{s}}}=-\frac{\partial\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})}{\partial
z_{\mathrm{f}}},$ (43)

where Eq. (36) is applied for the latter. This corresponds to the action-
reaction relation between solid and fluid particles under a simple two-body
interaction, i.e.,

$f_{z}^{\mathrm{f}}(z_{\mathrm{s}},z_{\mathrm{f}},r_{\mathrm{f}})=-f_{z}^{\mathrm{s}}(z_{\mathrm{s}},z_{\mathrm{f}},r_{\mathrm{f}})=-2\pi
R_{\mathrm{s}}\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})\rho_{V}^{\mathrm{f}}(z_{\mathrm{f}},r_{\mathrm{f}})\frac{\partial\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})}{\partial
z_{\mathrm{f}}}$ (44)

holds for the tangential force density on the fluid $f_{z}^{\mathrm{f}}$.

Based on these properties, we now derive the analytical expression of the
force exerted on the solid above a $z$-normal plane $z=z_{\mathrm{SL}}$ from
the liquid below that plane, i.e., the force of interest
$F_{z}^{\mathrm{diag(SL)}}$ given by

$\displaystyle F_{z}^{\mathrm{diag(SL)}}$
$\displaystyle=\int_{0}^{r_{\mathrm{c}}}\mathrm{d}r{{}^{\prime}}_{\mathrm{f}}\int_{-z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\int_{-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}\mathrm{d}z{{}^{\prime}}_{\mathrm{s}}f_{z}^{\mathrm{f}}$
$\displaystyle=-\int_{0}^{r_{\mathrm{c}}}\mathrm{d}r{{}^{\prime}}_{\mathrm{f}}\left[\int_{-z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\left(\int_{-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}\mathrm{d}z{{}^{\prime}}_{\mathrm{s}}f_{z}^{\mathrm{s}}\right)\right],$
(45)

where the double integral in the square brackets corresponds to the top-left
region in Fig. 11. Let the density $\rho_{V}^{\mathrm{f}}$ for
$z_{\mathrm{SL}}-z_{\mathrm{c}}<z_{\mathrm{f}}<z_{\mathrm{SL}}+z_{\mathrm{c}}$
be given as a unique function of $r_{\mathrm{f}}$ by

$\rho_{V}^{\mathrm{f}}(z_{\mathrm{f}},r_{\mathrm{f}})=\rho_{V}^{\mathrm{f(SL)}}(r_{\mathrm{f}}).$
(46)

Then, it follows for the double integral in the square brackets in Eq. (45)
that

$\displaystyle\int_{-z_{\mathrm{c}}}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\left(\int_{-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}}\mathrm{d}z{{}^{\prime}}_{\mathrm{s}}f_{z}^{\mathrm{s}}\right)$
$\displaystyle=-2\pi
R_{\mathrm{s}}\int_{-z_{\mathrm{c}}}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\left[\int_{-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}}\mathrm{d}z{{}^{\prime}}_{\mathrm{s}}\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})\frac{\partial\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})}{\partial
z_{\mathrm{s}}}\right]$ $\displaystyle=-2\pi
R_{\mathrm{s}}\int_{-z_{\mathrm{c}}}^{0}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\left\\{\left[\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})\right]_{z{{}^{\prime}}_{\mathrm{s}}=-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}}-\int_{-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}}\mathrm{d}z{{}^{\prime}}_{\mathrm{s}}\frac{d\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})}{dz_{\mathrm{s}}}\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})\right\\}$
$\displaystyle=2\pi
R_{\mathrm{s}}\int_{-z_{\mathrm{c}}}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\left[\rho_{A}^{\mathrm{s}}(-z{{}^{\prime}}_{\mathrm{f}})\phi(-z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})+\int_{-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}}\mathrm{d}z{{}^{\prime}}_{\mathrm{s}}\frac{d\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})}{dz_{\mathrm{s}}}\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})\right],$
$\displaystyle=2\pi
R_{\mathrm{s}}\int_{-z_{\mathrm{c}}}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\rho_{A}^{\mathrm{s}}(-z{{}^{\prime}}_{\mathrm{f}})\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$
$\displaystyle+2\pi
R_{\mathrm{s}}\int_{-z_{\mathrm{c}}}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\left[\int_{-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}}\mathrm{d}z{{}^{\prime}}_{\mathrm{s}}\frac{d\rho_{A}^{\mathrm{s}}(z_{\mathrm{s}})}{dz_{\mathrm{s}}}\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})\right]$
(47)

where $\phi(z_{\mathrm{c}},r{{}^{\prime}}_{\mathrm{f}})=0$ and Eq. (41) are
used for the 4th equality.

With an additional assumption of

$\rho_{A}^{\mathrm{s}}=\mathrm{const.},$ (48)

the 2nd term of the right-most HS becomes zero, and it follows

$\displaystyle\int_{-z_{\mathrm{c}}}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\left(\int_{-z{{}^{\prime}}_{\mathrm{f}}}^{z_{\mathrm{c}}}\mathrm{d}z{{}^{\prime}}_{\mathrm{s}}f_{z}^{\mathrm{s}}\right)$
$\displaystyle=2\pi
R_{\mathrm{s}}\rho_{A}^{\mathrm{s}}\int_{-z_{\mathrm{c}}}^{0}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$
$\displaystyle=\pi
R_{\mathrm{s}}\rho_{A}^{\mathrm{s}}\int_{-z_{\mathrm{c}}}^{z_{\mathrm{c}}}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$
(49)

considering that $\phi$ is an even function with respect to
$z{{}^{\prime}}_{\mathrm{f}}$ for the second equality. By inserting Eq. (49)
into Eq. (45), it follows

$F_{z}^{\mathrm{diag(SL)}}=-\pi
R_{\mathrm{s}}\rho_{A}^{\mathrm{s}}\int_{0}^{r_{\mathrm{c}}}\mathrm{d}r{{}^{\prime}}_{\mathrm{f}}\int_{-z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}^{z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$
(50)

Indeed, the RHS of Eq. (50) can be expressed using the following SL potential
energy density $u_{\mathrm{SL}}$ given by

$u_{\mathrm{SL}}\equiv\rho_{A}^{\mathrm{s}}\int_{0}^{r_{\mathrm{c}}}\mathrm{d}r{{}^{\prime}}_{\mathrm{f}}\int_{-z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}^{z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\rho_{V}^{\mathrm{f(SL)}}(r{{}^{\prime}}_{\mathrm{f}})\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}})$
(51)

which represents the SL potential energy per solid area at the SL interface
away both from the CL and from the bottom of the solid plate. With
$u_{\mathrm{SL}}$, Eq. (50) writes

$F_{z}^{\mathrm{diag(SL)}}=-\pi R_{\mathrm{s}}u_{\mathrm{SL}}$ (52)

which corresponds to Eq. (14) in the main text.

Similar to $F_{\mathrm{z}}^{\mathrm{diag(SL)}}$, the interaction force
$F_{\mathrm{z}}^{\mathrm{diag(SV)}}$ between the solid above a $z$-normal
plane $z=z_{\mathrm{SV}}$ and the vapor below that plane writes

$F_{z}^{\mathrm{diag(SV)}}=-\pi R_{\mathrm{s}}u_{\mathrm{SV}},$ (53)

where the SV potential energy density $u_{\mathrm{SV}}$ is given by

$u_{\mathrm{SV}}\equiv\rho_{A}^{\mathrm{s}}\int_{0}^{r_{\mathrm{c}}}\mathrm{d}r{{}^{\prime}}_{\mathrm{f}}\,\rho_{V}^{\mathrm{f(SV)}}(r{{}^{\prime}}_{\mathrm{f}})\int_{-z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}^{z_{\mathrm{c}}(r{{}^{\prime}}_{\mathrm{f}})}\mathrm{d}z{{}^{\prime}}_{\mathrm{f}}\,\phi(z{{}^{\prime}}_{\mathrm{f}},r{{}^{\prime}}_{\mathrm{f}}).$
(54)

##  Appendix C Extraction of the force exerted on the solid around the contact
line.

Figure 12:  Schematic of the extraction of the $z$-direction force on the
solid around the contact line from the fluid.

We can formulate the downward force on the solid around the contact line $2\pi
R_{\mathrm{s}}\xi_{z}^{\mathrm{cl}}$ from the fluid as the reaction force,
i.e., the upward force on the liquid from the solid, from the force balance
similar to that displayed in Fig. 4. Let $F_{z}^{\mathrm{cl}}$ be the force on
the liquid around the contact line, $2\pi R_{\mathrm{s}}\xi_{z}^{\mathrm{cl}}$
can be obtained by adding
$F_{z}^{\mathrm{diag(SL)}}-F_{z}^{\mathrm{diag(SV)}}$ and subtracting
$-F_{z}^{\mathrm{diag(SL)}}+F_{z}^{\mathrm{diag(SV)}}$ as

$2\pi
R_{\mathrm{s}}\xi_{z}^{\mathrm{cl}}=F_{z}^{\mathrm{cl}}+2\left(F_{z}^{\mathrm{diag(SL)}}-F_{z}^{\mathrm{diag(SV)}}\right)$
(55)

as illustrated in Fig. 12. In case the solid surface is smooth and flat and no
pinning is induced,

$F_{z}^{\mathrm{cl}}=0$ (56)

holds because the average surface lateral force on each fluid particles from
the solid is zero. Yamaguchi  _et al._ (2019); Kusudo, Omori, and Yamaguchi
(2019); Imaizumi  _et al._ (2020); Bistafa  _et al._ (2021) This condition is
applicable to the present systems. By inserting Eqs. (14), (15) and (56) into
Eq. (55),

$\xi_{z}^{\mathrm{cl}}=-u_{\mathrm{SL}}+u_{\mathrm{SV}}=(-u_{\mathrm{SL}})-(-u_{\mathrm{SV}})$
(57)

is derived as the analytical expression of $\xi_{z}^{\mathrm{cl}}$ in Eq. (16)
in the main text, where the final expression is to emphasize that the
potential energy densities $u_{\mathrm{SL}}$ and $u_{\mathrm{SV}}$ are both
negative.

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