Measuring line tension: thermodynamic integration during detachment of a molecular dynamics droplet

The contact line (CL) is where solid, liquid and vapor phases meet, and Young's equation describes the macroscopic force balance of the interfacial tensions between these three phases. These interfacial tensions are related to the nanoscale stress inhomogeneity appearing around the interface, and for curved CLs, eg a three-dimensional droplet, another force known as the line tension must be included in Young's equation. The line tension has units of force, acting parallel to the CL, and is required to incorporate the extra stress inhomogeneity around the CL into the force balance. Considering this feature, Bey et al. [J. Chem. Phys. \textbf{152}, 094707 (2020)] reported a mechanical approach to extract the value of line tension $\taul$ from molecular dynamics (MD) simulations. In this study, we show a novel thermodynamics interpretation of the line tension as the free energy per CL length, and based on this interpretation, through MD simulations of a quasi-static detachment process of a quasi-two-dimensional droplet from a solid surface, we obtained the value $\taul$ as a function of the contact angle. The simulation scheme is considered to be an extension of a thermodynamic integration method, previously used to calculate the solid-liquid and solid-vapor interfacial tensions through a detachment process, extended here to the three phase system. The obtained value agreed well with the result by Bey et al. and show the validity of thermodynamic integration at the three-phase interface.

The contact line (CL) is where solid, liquid and vapor phases meet, and Young's equation describes the macroscopic force balance of the interfacial tensions between these three phases.These interfacial tensions are related to the nanoscale stress inhomogeneity appearing around the interface, and for curved CLs, e.g., a threedimensional droplet, another force known as the line tension must be included in Young's equation.The line tension has units of force, acting parallel to the CL, and is required to incorporate the extra stress inhomogeneity around the CL into the force balance.Considering this feature, Bey et al. [J.Chem.Phys.152, 094707 (2020)] reported a mechanical approach to extract the value of line tension τ ℓ from molecular dynamics (MD) simulations.In this study, we show a novel thermodynamics interpretation of the line tension as the free energy per CL length, and based on this interpretation, through MD simulations of a quasi-static detachment process of a quasi-two-dimensional droplet from a solid surface, we obtained the value τ ℓ as a function of the contact angle.The simulation scheme is considered to be an extension of a thermodynamic integration method, previously used to calculate the solid-liquid and solid-vapor interfacial tensions through a detachment process, extended here to the three phase system.The obtained value agreed well with the result by Bey et al.   and show the validity of thermodynamic integration at the three-phase interface.

I. INTRODUCTION
Wetting plays a key role in the behavior of a liquid especially at the nanoscale where the surface-volume ratio is large.In 1805, Young 1 proposed the following equation as the force balance exerted on the contact line (CL) of a liquid in its vapor atmosphere on a flat solid surface: where γ SL , γ SV and γ LV are the solid-liquid (SL), solid-vapor (SV) and liquid-vapor (LV) interfacial tensions, respectively, and θ denotes the contact angle (CA).Equation ( 1) is called Young's equation, and the CA is used as a measure of wettability because it can be easily measured experimentally.4][5][6] In this extended thermodynamic framework, Gibbs formulated surface tension as an excess free energy per unit area of the interface through the definition of the dividing surface.He also introduced the concept of line tension, which we express by τ ℓ here, as an excess free energy per unit length of the CL, i.e., the force tangential to the CL.Boruvka and Neumann 7 included the effect of line tension into Young's equation through the derivation of the variational problem of the equilibrium interface shape as where θ LT is the CA and κ denotes the (principal) curvature of the CL, e.g., κ = 1/r for a 3-dimensional axi-symmetric cap-shaped hemispherical droplet on a flat solid surface with a circular CL of radius r shown in Fig. 1 (b).It follows for Eq. ( 2) that which indicates the dependence of the contact angle θ LT on κ, i.e., a dependence on the size of the droplet.Unlike the surface tension γ LV which must be positive, thermodynamic arguments do not give information about the sign of the line tension, i.e., τ ℓ , which can be either positive or negative. 5,8At present, the equilibrium molecular dynamics (EMD) method can be used to simulate a cap-shaped hemispherical nanoscale liquid droplet on a solid surface.This is a powerful alternative to solving the variational problem in order to obtain the apparent contact angle θ LT and κ for various sized droplets (see also Appendix B).Such geometrical analyses predicted a magnitude of τ ℓ which is around several pN (×10 −12 N), 9,10 indicating that the size effect is negligibly small for ordinary visible droplets.Nevertheless, recent experimental observations of nanometer-sized droplets and bubbles showed that these nanodroplets and nanobubbles have pancake-like flat shape, 11,12 and it was also indicated from MD simulations that such a shape cannot be explained by simple Young's equation (1).
At such scales, line tension indeed may play a key role.
On the other hand, Kirkwood and Buff 13 developed a framework of surface tension from a viewpoint of statistical mechanics.This is a mechanical approach considering the molecular interactions based on microscopic stress description. 14In this molecular scale, an interface is explicitly dealt with as a region with a non-zero thickness where the physical properties change continuously, and the stress is not isotropic even in static equilibrium (see also Fig. 1).The integral of stress anisotropy around the liquid-vapor or liquid-gas interface can be related to the surface tension, a process pioneered by Bakker. 4,5,13,15,16Such a mechanical calculation of surface tension γ LV through Bakker's equation using a quasi-one-dimensional (1D) flat liquid film system is considered a standard MD approach because it is easily realized by using the periodic boundary conditions (PBCs) in the surface-lateral directions. 17Note that only the integral of each principal stress component in the whole system is used for the calculation of γ LV , i.e., one does not need to obtain the stress distributions which is computationally demanding and not straightforward for systems with long-range Coulomb interactions. 18garding wetting including solids, beyond simple evaluation of the apparent contact angle from the shape, a number of MD and Monte Carlo (MC) studies have been done mainly to quantitatively extract the SL and SV interfacial tensions through a thermodynamic and/or a mechanical approach. 9, Espeially related to the latter, called the mechanical route, calculating the local stress distribution is one of the key issues for the understanding of wetting through the connection to macroscopic fluid mechanics. 18From the visualization of the stress field in the molecular scale using a quasi-two-dimensional (2D) system achieved under the PBC, 26,45,46 it has been shown that the stress anisotropy also exists at the SL and SV interfaces with finite thicknesses, and that the CL is a local region where SL, LV and SV interfaces with finite thicknesses meet and has a more complex stress features (see Fig. 1 and also Fig. 8 in Appendix A).The present authors showed that γ SL or γ SV can be obtained by calculating the stress distribution along the direction normal to the solid surface away from the CL region, and proved that the expression was consistent with Young's equation (1)   by considering a control volume (CV) surrounding the CL and by determining the contact angle θ from the extrapolation of the LV interface shape. 29In other words, the force balance on the CV faces away from the CL is considered, and the CL region having complex stress distribution is not explicitly included in Young's equation (1) in this quasi-2D framework.
Related to these studies, in this work we provide one possible intuitive justification about the equilibrium force balance of Young's equation in Eq. ( 1) and modified one in Eq. (2) here.Figure 1 shows the CVs with one face passing through the center of the quasi-2D and 3-dimensional (3D) equilibrium droplets of radius R and contact angle θ or θ LT , respectively.
Both have a LV-interface with uniform curvature.The anisotropic stress features mentioned above are schematized in these figures (see also Fig. 8), and the CL is considered to be a region where three interfaces, each with a finite thickness, meet.Now we think about the equilibrium force balance on the control volumes (CVs) shown in magenta in the center of Fig. 1 with two parallel wall-normal faces; one set across the droplet center and the other set across the solid-vapor interface.This setting is indeed similar to the explanation of the Young-Laplace equation without solid by Berry 47 .The shear stress τ zx , defined as the stress in the x-direction on a face with outward normal in the z-direction, is zero on the top face because it is in the vapor bulk.In addition, when the force from the solid is dealt with as an external force, i.e., not included in stress, the stress integral on the bottom face is zero because no fluid molecules exist below this bottom face. 29,48Thus, under a condition with a flat and smooth solid wall where the external force from the solid which can be assumed to be zero, the force balance on these CVs is expressed by the τ xx components on x-ourwardnormal and −x-outward-normal faces displayed in Figs. 1 (i) and (ii).In the case of the quasi-2D droplet in Fig. 1 (a), by ignoring the thicknesses of the interfaces, the force balance in Figs. 1 (a-i) and (a-ii) is written by where p L and p V denote the pressure values in liquid and vapor bulks, respectively.By inserting the Young-Laplace equation for a cylindrical interface original Young's equation ( 1) is derived, which obviously does not include τ ℓ , i.e., the anisotropic stress on the CL.
On the other hand, in the case of the quasi-3D droplet in Fig. 1 (b), the force balance in Figs. 1 (b-i) and (b-ii) is written by where τ ℓ is assumed to be positive if it gives tensile force on the CL region.By inserting again the Young-Laplace equation for a spherical interface and the following geometrical relation Young's equation including line tension τ ℓ in Eq. ( 2) is derived.
Indeed, Bey et al. 8 proposed a novel approach to extract τ ℓ from similar quasi-2D EMD systems as an extension of the mechanical approach, without the need for the local stress distribution, and examined the dependence of τ ℓ on the contact angle controlled by the solid-fluid interaction strength.They adopted a system with a quasi-2D droplet sandwiched between two parallel walls, i.e., a system with two menisci, and obtained τ ℓ from the stress integral on the face normal to the CLs and geometric information obtained from the interface shape.Note that this stress integral corresponds to that on the y-normal face in Fig. 1 (a) which includes the contribution from line tension.
9][30][31][32][33][34][35][36] Generally, the TI is a method to calculate the relative free energy of a target system as the difference from a reference system by connecting the target and reference systems with a thermodynamically reversible path using a coupling parameter embedded in the system Hamiltonian.As one possible implementation of the TI for the calculation of interfacial tension, Leroy et al. proposed the phantom-wall (PW) method 31,32 and the dry-surface (DS) method 33 described in detail in Sec.II.Briefly, as shown in Fig. 2 (a), a quasi-1D EMD system with a flat SL interface was used as a target system of interest in these methods, and this target system was quasi-statically substituted by a reference system with bare solid (denoted by subscript '0') and liquid surfaces along a thermodynamic path under constant number of particles N, temperature T and volume V in a NV T -ensemble.As a result, the minimum work needed for this change can be estimated as the Helmholtz free energy difference ∆F is directly related to γ SL as: where w SL is called the SL Work of Adhesion (WoA) as the free energy per area S, and γ SL − γ S0 is the interfacial free energy of SL interface relative to that of bare solid surface exposed to vacuum denoted by S0.In the PW method, a virtual wall called the 'phantomwall' interacting only with the fluid is used to strip the liquid off the solid surface by quasistatically lifting up the phantom-wall with assigning the coupling parameter to the position of the phantom wall.This method is advantageous because it is applicable to various kinds of SL combinations with various solid-liquid interaction potential forms. 36,49On the other hand in the DS method, the coupling parameter is assigned to a specific SL interaction parameter.The DS method is powerful in the sense that the SL interfacial tension can be obtained as a semi-continuous function of the SL interaction parameter. 29,35Similarly, the SV interfacial tension γ SV can also be evaluated from the SV work of adhesion w SV using a system shown in Fig. 2 (b), which is also described in detail in Sec.II.
Based on these mechanical and thermodynamic routes, we obtained γ SL or γ SV using a quasi-1D system with a flat SL or SV interface with various solid-fluid combinations, and showed that the contact angle predicted from these values corresponded well with the apparent contact angle of a quasi-2D droplet formed on the solid wall with the same solidfluid interaction parameters. 11,26,28,29,36These studies indicated that the apparent contact angle of the droplet obtained in the MD 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 cannot be induced.
An important and interesting point about these results is that the contact angle of a quasi-2D droplet without including line tension τ ℓ can be estimated from the interfacial tensions γ SL , γ SV and γ LV obtained by mechanical and thermodynamic approaches in quasi-1D systems without having CL.On the other hand, Bey et al. 8 extracted line tension τ ℓ from quasi-2D systems with straight CLs of zero curvature by a mechanical approach considering the stress integral on the face normal to the CL.
In this study, as a thermodynamics approach, we propose an extension of the DS method to extract τ ℓ by evaluating the free energy difference from a reference system as illustrated in Fig. 2 (c) through the quasi-static detachment of a quasi-2D hemi-cylindrical droplet.
The key concept is that we calculate the free energy difference ∆F drop /L per unit depth (L: system depth) given by where the free energy is given as the sum of the energy of two lines 2τ ℓ and interfacial enegies of LV, SL and SV interfaces with lengths s LV , s SL and s SV , respectively.Note again that τ ℓ has a unit of energy/length.In addition, the dependence of τ ℓ on the solid-fluid interaction strength was examined, and was also compared with the result by Bey et al. 8 and that estimated from the size dependence of the contact angle of 3D droplets.

II. METHOD A. MD Simulation Systems
In this study, we employed two types of equilibrium MD simulation systems: (a) quasione-dimensional systems with flat solid-liquid (SL) and solid-vapor (SV) interfaces, and (b) quasi-two-dimensional droplet systems with a hemi-cylindrical droplet on a solid surface as shown in Fig. 3.As the constituent fluid molecules, generic particles with the inter-particle interaction described by the 12-6 LJ potential were used.The 12-6 LJ potential expressed by was adopted for the interaction between fluid particles as a function of the distance r ij between the particle i at position r i and j at r j , with ε and σ being the LJ energy and length parameters, respectively.A cut-off distance of r c = 3.5σ ff was used for this LJ interaction, and by adding quadratic functions the potential and interaction force smoothly approached zero at r c .The values of the constants c LJ 2 and c LJ 0 as functions of r c and σ ff are shown in our previous study. 26The fluid particles are expressed by 'f' and corresponding interactions are denoted by subscripts hereafter.
Both systems in Figs. 3 (a with This potential field corresponds to a mean potential field created by a single layer of uniformly distributed solid particles with an area number density ρ n , which interact with the fluid particles through the LJ potential with the energy and length parameters being ε sf (= ηε 0 sf ) and σ sf , respectively, where the solid-fluid (SF) interaction parameter η was multiplied to the base value of ε 0 sf of 1.29 × 10 −21 J as described below.Similar to Eq. ( 10), this potential field in Eq. ( 11) was truncated at a cut-off distance of z c = 3.5σ sf with a quadratic function with which the potential and interaction force smoothly vanished at z c .
The quasi-1D system in Fig. 3  x-and y-directions for both systems.
The system temperature was maintained at a constant temperature T by using the Nosé-Hoover thermostat with an effective mass Q of 3N f k B T τ 2 with τ = 1.0 × 10 −12 s applied to all fluid particles, where N f is the number of fluid particles.Note that the choice of Q had negligible effect on the results for the present equilibrium systems after sufficient relaxation run.For the temperature, we have chosen T = 100 K, which is between the triple point and critical temperatures. 50Note that the temperature control had no effects on the results since in this study we deal with fully-relaxed equilibrium systems including the detachment processes in Figs. 3 (i)-(iii) explained below.The velocity Verlet method was applied for the integration of the Newtonian equation of motion with a time step of 5 fs for all systems.The simulation parameters are summarized in Table I with the corresponding non-dimensional ones, which are normalized by the corresponding standard values based on ε ff , σ ff and mass m f .The SF interaction parameter η for the top and bottom walls were set at η top and η bot , respectively, and they were changed in a parametric manner except for the top wall in Fig. 3 (b) fixed at η top = 0.001.With the present setup, a hemi-cylindrical droplet was formed on the bottom wall as an equilibrium state in the quasi-2D system in Fig. 3 (b), and as indicated in Fig. 3 (iii), the contact angle θ of the droplet had a one-to-one correspondence with the value of η for the bottom wall at a given temperature, i.e., η expresses the wettability.This is the case for the present solid modeled by a potential field exerting no wall-tangential force on the fluid as an ideally smooth solid surface without inducing pinning of the contact line. 26,29,30Note that in the present quasi-2D systems, effects of the CL curvature can be neglected. 7,9,24,26,29,30,32,51On the other hand, with a narrow lateral size, a quasi-one-dimensional liquid film attached on the bottom wall was formed as an equilibrium state as in Fig. 3 (a) by setting η bot > η top with η bot giving a droplet contact angle θ below 180 degrees.
The physical properties of each equilibrium system with various η values were calculated as the time average of 30 and 50 ns for the quasi-1D and quasi-2D systems, respectively, both of which followed an equilibration run of more than 10 ns.

B. Dry-Surface Method
The thermodynamic integration (TI) is a method to determine the free energy difference of two equilibrium states by connecting them with a quasi-static path through a TI parameter embedded in the system Hamiltonian.Let λ be the TI parameter, and let the target and reference systems correspond to λ = 0 and λ = 1 described by the system Hamiltonian H(Γ, λ) in a constant NV T system as a function of all positions and momenta Γ, i.e., the phase space variable.Then, the difference of the Helmholtz free energy between the two systems writes By using the relation between the Helmholtz free energy F and the configurational partition function Z, it follows for Eq. ( 13) that where the angular brackets denote the ensemble average.If the system Hamiltonian H is analytically differentiable with respect to λ, i.e., ∂H ∂λ can be calculated for each microscopic system, the integrand in the right-most hand side of Eq. ( 13) as the ensemble average with a given λ.Note that in practice, multiple equilibrium MD systems of λ between 0 and 1 are prepared, and ∂H ∂λ is calculated in each system as the time average instead of ensemble average assuming ergodicity.A similar relation can be derived for the Gibbs free energy difference in constant NpT systems. 29,31,32roy and Müller-Plathe 33 proposed the Dry-Surface (DS) scheme as one of the TI methods to calculate the SL interfacial tension through the fluid stripping process from the solid surface by embedding the TI parameter λ into the SF interaction potential.Specifically in the present study, we include the TI parameter λ into the SF interaction in Eq. (11)   expressed by the LJ potential as Then, for a constant NV T system, Eq. ( 13) writes As λ approaches 1, the SF interaction is weakened, and the solid surface becomes 'dry' for λ slightly smaller than 1.This state is denoted by 1 − as the reference system because at λ = 1, the SF repulsion also becomes zero and the fluid particles can freely pass through the solid wall.By considering the relation in Eq. ( 11), and by changing the integration variable in the right-most hand side of Eq. ( 16) where 0 + denotes a value of η ′ slightly larger than zero.For instance, it can be set at η = 0.03 with which the droplet is completely detached from the solid surface as displayed in Fig. 3 (iii).Equation ( 19) means that we get the system trajectory with the corresponding SF interaction parameter η = ξ, whereas we calculate the ensemble average (substituted by the time average) of the total SF interaction potential energy N i=1 Φ sf (z ′ i ; ξ) and divide it by ξ as the integrand to numerically integrate the right-most hand side.A remarkable advantage of the DS method is that Eq. ( 19) is an indefinite integral form of η, i.e., the free energy difference ∆F (η) from the reference system can be obtained as a semi-continuous function of η ∈ [0 + , η max ], where η max is the maximum value of η to be investigated.More concretely, we calculate multiple equilibrium systems with discrete η ′ values between 0 + and η max with a sufficiently small increment dη ′ , and calculate the integrand 19) as the time average in each system, then, ∆F (η) is obtained as the integral from 0 + up to η (≤ η max ) by numerical integration.

A. Works of solid-liquid and solid-vapor adhesion
Figure 2 shows the schematic of the DS method applied to quasi-1D systems to calculate the SL and SV interfacial tensions γ SL and γ SV , respectively as the free energy per area obtained upon quasi-static detachment of the SL and SV interfaces.To calculate the SL interfacial tension, we carried out the DS process of SL detachment as shown in Fig. 2 (a), where η top was kept constant at 0.001 whereas η bot was changed from 0.1 to 0.7 (= η bot max ).Upon this process, the original SL interface is separated into 'dry' solid-vacuum and liquidvacuum interfaces.We denote this vacuum by '0' hereafter.Then, the work of SL adhesion w SL defined as the free energy per surface area S needed for this change is written as where subscript S0 denotes the bare solid surface without liquid or vapor adsorbed on it.
Considering that the effect of the vapor density on the LV surface tension for the temperature range in this study is negligible, γ L0 was approximated by γ LV .The value of γ LV was obtained from a MD system with planar LV interfaces by a standard mechanical process in which the difference between the normal stress components vertical and parallel to the interface was integrated around the LV interface, which resulted in γ LV = 7.47 × 10 −3 N/m at T = 100 K. 11,28 Similarly, the DS process of SV detachment as shown in Fig. 2 (b) was carried out to calculate the SV interfacial tension, where η bot was kept constant at 1.0 whereas η top was changed from 0.1 to 0.7 (= η top max ).This detachment process separates the original SV interface into S0 and V0 interfaces, and the work of SV adhesion w SV is expressed by considering γ V0 ≈ 0. Note that w SL and w SV have the same dimension as the interfacial tensions.As described later, γ SL − γ SV appearing in Young's equation can be calculated from Eqs. ( 20) and ( 21) by eliminating γ S0 .

B. Work of droplet adhesion
We extended the DS method to the quasi-2D droplet systems in Fig. 2 (c) to calculate the line tension τ ℓ .Similar to the quasi-1D DS process, the droplet detachment process was carried out for the quasi-2D system as illustrated in Fig. 2 (c), where the free energy difference ∆F (η) as the numerical integral in Eq. ( 19) was calculated using multiple equilibrium systems with SF interaction η between 0 + and η max .Note that at η = 0 + ≪ 1, the droplet was detached from the bottom wall.Figure 5 shows the work of droplet adhesion W drop defined as the free energy per system depth L y needed to strip off the hemi-cylindrical quasi-2D droplet from the solid surface.Note that W drop has the same dimension as force.
Corresponding time-averaged density distributions around the center of mass of the droplet for several η values are also displayed on the top panel.The qualitative feature of W drop was the same as w SL in Fig. 4, i.e., it increased with the increase of η, and was obtained as a semi-smooth function of η owing to the advantage of the DS method.It was also indicated from the time-averaged density distributions on the top panel that the LV interface away from the solid had a spherical interface with a uniform curvature surface.
We assume that the change of bulk liquid and vapor volumes upon the change of η is negligibly small, i.e., the total free energy of the bulk regions are kept constant and the change of the system free energy is due to the interface and contact line upon the droplet detachment process.Then, W drop is written as where s LV , s SL and s SV denote the lengths of the corresponding interface projected in the xz-plane, and the values with (η) mean that they depend on the SF interaction parameter η.By assuming that γ LV is independent of η, i.e., independent of the curvature of the LV interface for the present droplet size range, 52 and by inserting Eqs. ( 20) and ( 21) into Eq. ( 22), it follows where the simple length relation s SL + s SV = L x with L x being the system size in the xdirection is used, and we set τ ℓ (η) = 0 for η ≈ 0 with the droplet detached from the solid surface.Hence, if the two lengths ∆s LV and s SL are determined as a function of η, then τ ℓ can be obtained using w SL (η), w SV (η) and W drop (η) as However, considering that τ ℓ is small, 8 the results may depend strongly on the definition of the geometric parameters ∆s LV (η) and s SL (η) in Eq. (24).To reduce the statistical error due to the fluctuation, we define these geometric parameters by determining the value of droplet volume V , i.e., the droplet area A ≡ V /L y projected onto the xz-plane, which we assumed to be constant independent of η in Eq. ( 22) so that they are consistent with the works of adhesion w SL (η) and w SV (η) and are semi-smooth functions of η as in Figs. 4 and 5.As a geometrical relation (see Fig. 2 (c)), the droplet area A is given by using the contact angle θ as a function of η and the radius of the 2D-droplet R. By determining A c and its error δA c from the average of the projected droplet area A for various η values (see Supplementary Material), the radius R (θ(η)) is determined by as a function of the contact angle θ(η), and its uncertainty is also estimated using Eq.(26).
By using θ(η) and R (θ(η)) given by Eq. ( 26), the geometric parameters s SL (θ(η)) and ∆s LV (θ(η)) are expressed as functions of θ(η) as well by and ∆s respectively, where s LV (π) = 2 √ πA c is the circumference of a circle with an area A c .Thus, if the contact angle θ(η) is obtained as a semi-smooth function of η, the geomeric parameters in Eqs. ( 27) and ( 29) can be written as semi-smooth functions of η as well, and consequently, τ ℓ in Eq. ( 24) is determined.Now, the agenda is how to obtain θ(η) as a function of η to be consistent with the works of adhesion, and how to determine the constant volume (area in the xz-plane) A c in Eq. ( 26).
For the former, it has been shown in our previous study that the following Young-Dupré equation holds for a droplet on a flat and smooth solid surface: which is rewritten by using Eqs.(20) and (21).The relation between θ and η obtained from Eq. ( 30) using w SV (η) and w SL (η) in Fig. 4 (a) is shown in Fig. 4 (b).The contact angle indeed agreed well with the apparent contact angle, e.g., estimated from the density distributions in the top panel of Fig. 5, since the solid surface in the present study is ideally smooth. 29 the other hand, for the projected area A c , a difficulty exists in the definition of the radius R to determine the volume because the interface is not a surface of discontinuity but a region with a certain thickness at the nanoscale.A possible and common choice is using the Gibbs dividing surface, 53 and another choice as a strict mechanical definition based on the force and momentum balance was also suggested. 52Considering that we assume γ LV to be constant and also that the LV interface has a uniform curvature, we used the arc length, s LV , contact area length, s  solid-fluid interaction coefficient η by Eq. ( 30), meaning that τ ℓ in Eq. ( 24) can be determined as a function of θ or η. Figure 7 shows line tension τ ℓ as a function of the contact angle θ obtained by the present thermodynamic approach, superimposed on the result obtained by Bey et al. 8 from a mechanical approach and that evaluated by Eq. ( 3) from the size dependence of the contact angle as a simple geometrical approach (see Appendix B).Note that a different wall potential form was used in Ref. 8 instead of Eq. ( 11).Corresponding η value is displayed on the top horizontal axis obtained by the relation between η and θ in Fig. 6 (a).The present results agreed well with those obtained by the mechanical approach, although the uncertainty in the present results is large for small contact angles.This is because with the increase of η, both w SL − w SV and s SL become large and resulting error of (w SL − w SV − γ LV )s SL and W drop in the RHS of Eq. ( 24) become large.Due to this limitation, the increase of τ ℓ with the decrease of θ up to positive value indicated in the mechanical result is not obvious.On the other hand, the geometric approach overall has large error bars which is inevitable upon the fitting procedure of cos θ LT -κ to obtain τ ℓ in Eq. ( 29) (see Fig. 9 in Appendix B).
Here, we discuss about the error bars more in detail.At first regarding the geometric method, it is advantageous because of its simplicity, but also because of its simplicity it gives neither mechanical nor thermodynamic explicit insights about line tension.In addition, as indicated by Ravipati et al. 42 accurate calculation of the contact angle from the density distribution may need long averaging time.Regarding the present thermodynamic approach, in addition to the problem of assuming the liquid volume to be constant for wettable cases mentioned above, the error δτ ℓ of τ ℓ depends on the error δA c of A c with and this monotonically increases with the decrease of θ.Therefore, the error increase for the estimation of τ ℓ is basically inevitable for small θ.It is also seen from Eq. ( 31) that the relative error δτ ℓ δAc is proportional to , and it should decrease with the increase of A c , meaning that it can be reduced by using a larger system size.Regarding the computational cost, both the mechanical and present thermodynamic methods do not need local stress calculation which is computationally demanding, and longer time averaging of the ordinary equilibrium MD calculation would reduce the error for both.

IV. CONCLUDING REMARKS
In this study, we reviewed the mechanical interpretation of Young's equation where the line tension is obtained from the microscopic force balance.We then showed a thermodynamics interpretation of the line tension as the free energy per CL length, obtained from the difference between a quasi-two-dimensional hemi-cylindrical droplet on a solid surface and a cylindrical droplet with the same volume.Using this concept, we obtained the value of the line tension τ ℓ through MD simulations of a quasi-static detachment process of a quasi-2D droplet from a solid surface, an extension of the thermodynamic integration method used to calculate the SL and SV interfacial tensions individually.Through the comparison with the results obtained in a mechanical manner, it was shown that the present thermodynamic approach provided a novel way to obtain the line tension.mass of the droplet. 55More concretely, the contact angle θ LT was defined as the angle between the LV interface and the solid-fluid interface plane, where θ LT was obtained by fitting a density contour at ρ = 400 kg/m 3 at the LV interface away from the solid with a spherical surface with a constant curvature. 26,29On the other hand, the solid-fluid interface position was defined as the limit position nearest to the solid that the fluid molecule could reach. 29From the geometric information of the hemi-spherical droplet radius R and the contact angle θ LT , we evaluated the CL curvature κ = 1/Rsinθ LT .By fitting the data with a straight line including the contact angle θ quasi-2D droplet as θ LT at κ = 0, we obtained the values of τ ℓ for various solid-fluid interaction parameters η in Fig. 7.As indicated from the figure, the fitting includes points of 3-dimensional hemi-spherical droplets with κ (= 1/r with r being the CL radius) around 0.2 nm −1 and a point of a quasi-2-dimensional hemicylindrical droplet with κ = 0, and the resulting uncertainty becomes inevitably large for this geometric approach.

FIG. 1 .
FIG. 1.A mechanical interpretation of Young's equation (1) and the line tension modification in Eq. (2), considering the equilibrium force balance on the rectangular control volumes (CVs) depicted in magenta.

FIG. 2 .
FIG. 2. Schematics of the Dry-Surface (DS) method applied to quasi-1D systems to calculate the (a) solid-liquid (SL), and (b) solid-vapor (SV) interfacial tensions γ SL and γ SV , respectively from the change of free energy per area obtained upon quasi-static detachment of the SL and SV interfaces.(c) Schematic of the DS method extended to the quasi-2D droplet systems to calculate the line tension τ ℓ .

FIG. 3 .
FIG. 3. (a) Quasi-one-dimensional (1D) system with liquid and vapor attached on the bottom and top solid surfaces, respectively, and schematics of the (i) solid-liquid detachment, and (ii) solid-vapor detachment processes performed by changing the solid-fluid interaction parameter η bot or η top of the bottom or top surface, respectively.(b) Quasi-two-dimensional (2D) system with a hemi-cylindrical liquid droplet on the solid surface on the bottom, and (iii) schematic of the solid-droplet detachment process in the quasi-2D system.
) and (b) have two solid walls on the bottom and top of the simulation cell shown in light-red.To minimize the number of arbitrary parameters affecting the basic physics of wetting, the solid wall was modeled by a simple one-dimensional potential field interacting with the fluid particles as a function of the distance rather than modeling by a group of solid particles, e.g., those forming a fcc crystal.The interaction Φ sf between the immobile top or bottom solid wall at z = z top s or z = z bot s , respectively and the fluid particle at z = z i was given by (a) contained 2000 fluid particles in a simulation cell of 4 × 4 × 20 nm 3 with the top and bottom walls modeled by Eq. (11) with z top s = 20 nm and z bot s = 0, respectively.On the other hand, the quasi-2D system in Fig. 3 (b) contained 7000 fluid particles in a simulation cell of 40 × 4 × 40 nm 3 with the top and bottom walls at z top s = 40 nm and z bot s = 0.The periodic boundary condition was applied in the wall lateral

FIG. 4 .
FIG. 4. (a) Relation between the work of adhesion and solid-fluid interaction coefficient η for SL and SV interfaces obtained through the DS method, and (b) contact angle cosine estimated by the Young-Dupré equation (30).

Figure 4 (
Figure 4 (a) shows the relation between the work of adhesion and solid-fluid interaction coefficient η for the SL adhesion w SL and the SV adhesion w SV obtained through the DS method.With the increase of η, both works of adhesion w SL and w SV became large, and the work of SV adhesion had non-negligible value for η above about 0.4, where an adsorption layer was formed at the SV interface as observed in Fig.2 (b).Along this quasi-static thermodynamic path, w SL and w SV were obtained as smooth functions of η through the DS scheme based on Eq.(19).We used these values to evaluate the contact angle θ using the Young-Dupré equation as shown in Fig. 4 (b).

FIG. 5 .
FIG. 5. Work of droplet adhesion.Corresponding time-averaged density distributions of around the center of mass of the quasi-2D droplets are shown on the top.The error bars are smaller than the size of symbol similar to those in Fig. 4 (a).

FIG. 6 .
FIG. 6. Geometric parameters s LV and s SL obtained each as a function of η.

Figure 6 2 FIG. 7 .
Figure6shows the geometric parameters s LV and s SL expressed as functions of η.As easily imagined, the contact area s SL increased with the increase of η whereas the LV interface area s LV showed different dependence on η.For both s SL and s LV , the error bars mainly due to the estimation of the area A c were not remarkably large.Now, all the values R(θ), s SL (θ) and ∆s LV (θ) in Eqs.(26), (27) and (29) each as a function of the contact angle θ can be determined for a given θ value, which is directly related to the

TABLE I .
Simulation parameters and their corresponding non-dimensional values.