Contact Angle for Spherical Nanodroplet in Cylindrical Cavity with Quadratic Curve Generatrix

Wetting of a spherical nanodroplet in smooth and homogeneous cylinder surface rotated by quadratic curve was studied by methods of thermodynamics. The solid-liquid-vapor system was separated into six parts using Gibbs method of dividing surface. The system free energy was calculated. A generalized Young equation for the equilibrium contact angle is proposed taking the line tension effects into consideration. On the basis of some assumptions, this generalized Young equation is the same as the classical Young’s equation.


Introduction
The wetting of solid surfaces is very familiar in various natural and technological devices (Quéré, de Gennes, Brochard-Wyart, & Reisinger, 2004;Prabhu, Fernades, & Kumar, 2009;Waghmare & Mitra, 2010;Promraksa & Chen, 2012;Raufaste & Cox, 2013;Snoeyink, Barman, & Christopher, 2015).The contact angle for a liquid nanodroplet on a substrate surface is usually used to describe the wetting characteristic.The Young' equation predicted the equilibrium contact angle (Young, 1805) cos LG s s q s -= ( 1 ) where Y q is the contact angle, SG s , SL s , LG s represent the thermodynamic surface tension of solid-vapor interface, solid-liquid interface and liquid-vapor interface, respectively.Equation ( 1) is applicable for a liquid nanodroplet on an ideal smooth and homogeneous solid substrate surface.But, it didn't taking the influences of the line tension into consideration.
Gibbs firstly presented the line tension as a surface thermodynamics concept (Gibbs, 1961).Gibbs believed the triple phase contact line bears significant role in triple phase systems.From that time, the line tension impacts on the contact angle were studied by many researchers.The significance of the line tension in three-phase systems was discussed in relation to contact angle measurements by Jaroslaw (Drelich, 1996).A.Amirfazli found the drop size dependence of contact angles for high-energy system yields a positive line tension (Amirfazli, Chatain, & Neumann, 1998).On the basis of the Irving-Kirkwood stress tensor of statistical mechanics, line tension at a curved edge of a solid is direct calculated by Anatoly (Rusanov & Brodskaya, 2014).Although, the thermodynamics line tension still remains disputable.It is thought the line tension is responsible for the many wetting phenomena.When taking the line tension effects into consideration, the Young equation should be modified.For a nanodroplet in an inclined solid cone surface of revolution, Dongqing Li presented the following equation of equilibrium contact angle (Li, 1996) cos cos cos where L R is the radius of the triple phase contact line, k is the corresponding line tension, L b is the angle of inclination of the solid surface at the triple phase contact line.In Equation (2), Dongqing Li supposed the line tension is constant and the cone surface shapes from the rotation by line.When a nanodroplet within the cone surface of rotation by a nonlinear curve, the contact angle should be different from the Equation (2).
According to the practical situation, the line tension can be variable.Rusanov established a modified Young's equation (Rusanov, Shchekin, & Tatyanenko, 2004) utilizing Gibbs method of dividing surfaces where the line tension derivative In previous references, wetting of nanodroplet in cavity surface rotated by a quadratic curve has seldom been studied.In this work, taking the line tension effects into consideration, wetting of a spherical nanodroplet in a smooth and homogeneous cylindrical cavity rotated by a quadratic curve is studied.A generalized Young equation of contact angle for nanodroplet in a smooth and homogeneous cylinder cavity surface rotated by quadratic curve is obtained utilizing the methods of dividing surfaces of Gibbs based on thermodynamics.

Calculation of the total Helmholtz free energy
In this study, we assume the cylindrical cavity is created by using the quadratic function  On the basis of the method of Gibbs's dividing surface, we separated the solid-liquid-vapor system into six portions, liquid phase, vapor phase, solid-liquid interface, solid-vapor interface, liquid-vapor interface, and the three phase contact line.The whole Helmholtz free energy F of the system is described by the following expression LG F and SLG F indicate the free energies of six portions, respectively.
We have the free energies of six portions (Young, 1805;Rowlinson & Widom, 2013) LG A , SL A , SG A denote the surface areas of liquid-vapor interface, solid-liquid interface and solid-vapor interface, respectively.k is the line tension.SLG L is the length of the triple phase contact line.
The volume L V of liquid phase is where R and H denote the radius of spherical cap of the liquid and height from local principal plane of the contact line to the bottom of the cylinder.
The whole volume t V of the system is given by (2 cos )(1 cos ) 3  4), we obtained the whole Helmholtz free energy in the following from

Derivation of Generalized Young Equation
The grand thermodynamic potential W of the solid-liquid-vapor system is where the mark i denotes the amount of subsystems of the system.
Putting Equation ( 23) into Equation ( 24), we have the following expression The grand thermodynamic potential W , the surface tensions SL s and SG s do not depend upon the notional variation of the radius R of the nanodroplet, the following constraints can be obtained (Rusanov, Shchekin, & Tatyanenko, 2004) Substituting Equation ( 25) into Equation ( 26), we have The dividing surface positions of liquid-vapor interface of a spherical liquid nanodroplet in cylindrical cavity should be parts of concentric and conformal spherical surfaces.So, we have the following expressions According to Eqs. (11,13,14,16) On the basis of the well-known Laplace' equation (Rowlinson & Widom, 2013;Ono, Kondo, & Flügge, 1960) of a free spherical liquid droplet in vapor, the following equation is obtained It can be used for the nanodroplet in this study.
In order to simplify calculation, we study only the case that z is greater than zero.So, according to the illustration in Figure 1, the following relations were obtained Using Eqs.(33), Equation ( 40) can be rewritten as Equation ( 41) is the generalized Young's equation for the contact angle of the spherical nanodroplet in cylindrical cavity rotated by quadratic function

Conclusion
In this work, the wetting of a spherical nanodroplet in cylindrical cavity rotated by quadratic function of the k due to the variation of the radius L R of the contact line.Equation (3) is suitable for the flat surfaces.For a nanodroplet on curved surface, the Young equation should be different with Equation (3).
around z axis.The illustration of the wetting is shown in Figure1, in which the droplet was cut by a plane parallel to the z axis.b is the angle between the substrate surfaces tangent and the principal plane of the three-phase contact line.a is the angle between the liquid nanodroplet surface tangent and the local principal plane of contact line.In this study, the nanodroplet gravity is neglected.The equilibrium shape of nanodroplet in cylindrical cavity above the principal plane of the three-phase contact line bears the shape of a spherical part.

Figure 1 .
Figure 1.Wetting of a spherical nanodroplet within a smooth and homogeneous cylinder solid rotated by 2 2 2 2 1 x z a b -= around z axis 12)The surface area LG A of the liquid-vapor interface has the form is valid for any dividing surfaces between liquid and vapor phases., the generalized Young's equation Equation (41) is the same as equation (3) established by Rusanov.If line tension effects are negligible, Equation (41) changes to the classical Young equation (1).
basis of Gibbs's method of dividing surface.Considering the effects of line tension, a generalized Young equation for the contact angle of a spherical nanodroplet in cylindrical cavity rotated by quadratic function is proposed.This generalized Young equation changes to the Rusanov's equation and the classical Young's equation under certain assumption.