Derivation of Generalized Young ’ s Equation for Wetting of Cylindrical Droplets on Rough Solid Surface

The surface tension depends on the radius of curvature of the liquid-vapor interface. For nano-scale wetting phenomena of cylindrical droplets, we should consider the curvature effects of the surface tension and the line tension. However, previous works have not analyzed the influence of the curvature effects of the surface tension. In this paper, we discuss the influence of the curvature effects of the surface tension on the contact angles based the Kim-Lee-Han-Park equation. The hydrophilic wetting of cylindrical droplets on rough and chemically homogeneous non-deformable substrates were studied by methods of thermodynamics. A generalized Young’s equation for wetting of cylindrical droplets on chemically homogeneous and rough non-deformable substrates was derived based on the thermodynamic equilibrium conditions. This equation reduces to the Wenzel equation if we ignore the influence of line tension. For contact angles of cylindrical droplets with sufficiently large radii, a generalized Young’s equations were derived considering the curvature effects of the surface tension.


Introduction
Wetting phenomena are common in solid-liquid-gas systems, for instance, wetting of liquid droplets on solid surfaces, adhesives, lubricants and capillary penetration in to porous media (Adamson, 1990;Gennes, Brochard-Wyart, & Quere, 2004).Wetting abilities are important in many industrial applications, for example, the wetting abilities of electrolytes on electrodes plays a key role in improving the specific energy density of supercapacitors (Kim, Koo, Lee, & Braun, 2014) and lithium-ion batteries (Pfleging & Proella, 2014).
In 1805, Thomas Young argued that the contact angle Y q for the wetting of spherical droplets on rough and chemically homogeneous substrates is determined by the following equation (Young, 1805) cos where Y q is the contact angle, LG s is the surface tension of the liquid-vapor interface corresponds to the choice of the surface of tension as a dividing surface, SG s is the surface free energy per unit area of the solid-vapor interface, SL s is the surface free energy per unit area of the solid-liquid interface.Now, Equation (1.1) is called the Young's equation.The Young's equation Equation (1.1) is widely applied to macroscopic capillary phenomena (Pfleging & Proella, 2014;Xiao-Song et al., 2014).
In 1878, Gibbs for the first time gave a theoretical derivation of the Young's equation Equation (1.1) based on the theory of thermodynamics (Gibbs, 1928).Since then, many theoretical research works have been carried out (Pfleging & Proella, 2014).
The surface tension depends on the radius of curvature of the liquid-vapor interface.For nano-scale wetting phenomena of cylindrical droplets, we should consider the curvature effects of the surface tension and the line tension.However, previous works have not analyzed the influence of the curvature effects of the surface tension.
The purpose of this paper is to present a theoretical study the contact angles of cylindrical droplets on hydrophilic substrates and discuss the influence of the curvature effects of the surface tension on the contact angles.

The Helmholtz Free Energy for Wetting of a Cylindrical Droplet on Rough Substrate
Consider a single-component cylindrical liquid droplet in contact with chemically homogeneous and rough substrates.
If the contact angle less than 90 。 , then we say that the solid surface is hydrophilic.If the contact angle larger than 90 。 , then we say that the solid surface is hydrophobic.We only consider the wetting of a cylindrical droplet on hydrophilic solid surface.An illustration of hydrophilic wetting is shown in Figure 1.Introducing Gibbs's concept of dividing surface and the concept of dividing line (Gibbs, 1928;Ono & Kondo, 1960;Rowlinson & Widom, 1982), the above solid-liquid-vapor system can be divided into six subsystems, i.e. liquid phase, vapor phase, the liquid-vapor interface, the solid-liquid interface, the solid-vapor interface and the three-phase contact line.
Therefore, the total Helmholtz free energy F of the system is the sum of the Helmholtz free energies of these seven parts.Thus, we have The Helmholtz free energies of these seven parts can be written as (2.7) where L p and G p are the pressures of the liquid phase and the vapor phase respectively, L V and G V are the volumes of the liquid phase and the vapor phase respectively, m is the chemical potential of the six subsystems, , , , , N N N N N and SLG N are the mole numbers of molecules of the liquid phase, the vapor phase, the liquid-vapor interface, the solid-liquid interface, the solid-vapor interface and the three-phase contact line respectively, , LG A SL A and SG A are the surface area of the liquid-vapor interface, the solid-liquid interface, the solid-vapor interface respectively, , LG s SL s and SG s are the surface tensions of the liquid-vapor interface, the solid-liquid interface, the solid-vapor interface respectively, SLG L is the value of the length of the three-phase contact line, k is the line tension.
In order to calculate the geometrical quantities in the above equations, we may introduce the following assumption: Assumption 1: Suppose the equilibrium shape of a droplet on a rough and homogeneous solid substrate is a part of a cylinder which was cut by a plane parallel to the axis of the cylinder.
Based on Assumption 1, the total Helmholtz free energy F of the system is where R is the radius of the cylindrical liquid droplet, q is the contact angle, L is the length of the cylindrical liquid droplet, t A is the total surface area, s r is the surface roughness factor, L r is the line roughness factor.

Generalized Young's Equations based on Thermodynamic Equilibrium Condition
The purpose of this section is to derive a generalized Young's equations for cylindrical droplets on rough and homogeneous solids by methods of thermodynamics.
According to Gibbs's concept of dividing surface (Ono & Kondo, 1960), we can choose an arbitrary conformal surface as a dividing surface.Now, we suppose that the radius R of the dividing surface has already been chosen according to some fixed conditions.The contact angle q now becomes variable.The thermodynamic equilibrium condition at a fixed temperature T of an open system is (Nijmeijer, Bruin, Woerkom, & Bakker, 1992) , 0, where the subscript T and m stands for fixed temperature T and fixed chemical potential m .
It is convenient to introduce the concept of grand potential to treat an open system.The definition of the grand potential of a system is (Rowlinson & Widom, 1982) where t is the number of subsystems of the system, i F is the Helmholtz free energy of the i-th subsystem, i m is the chemical potential of the the i-th subsystem, i N is the mole numbers of molecule of the the i-th subsystem.
Putting Equation (2.8) into Equation (3.2), the total grand potential W of the above system is (3.10)According to Gibbs's concept of dividing surface (Gibbs, 1961;Rowlinson & Widom, 1982), we can choose an arbitrary conformal surface as a dividing surface.Now, the radius R of the dividing surface has already been chosen according to some fixed conditions.At the fixed temperature T and fixed chemical potential, the pressure L p , p G and surface tension will not influence the contact angle q .Thus, in order to simplify Equation (3.5), we introduce the following assumption.
Assumption 2: Suppose the following equations are valid for the wetting of cylindrical droplets on rough and chemically homogeneous non-deformable substrates ( ) We have the following results Putting ) into Equation (3.5), we obtain It is known that a generalized Laplace's equation of a free cylindrical droplet in vapor can be written as (Wenzel, 1936) , Where the differential in square bracket [ ] denotes the change resulted from a mathematical variation of the position of this dividing surface by the amount dR in the same physical system under the same fixed physical state.
Applying Equation (3.21), Equation (3.20) becomes LG LG ) is a generalized Young's equation for wetting of a cylindrical droplet on chemically homogeneous and rough non-deformable substrates.Equation ( 43) is the main results of this work.
Following Gibbs (Kim, Lee, Han, & Park, 2006;Nijmeijer, Bruin, Woerkom, & Bakker, 1992), we introduce the concept of surface of tension M s as follows Where R s is the radius of the surface of tension  (Wenzel, 1936).

Brief Review of Curvature Effects of Surface Tension of Cylindrical Droplet
In 2006, Kim, Lee, Han, and Park (2006) obtained the following equation for the curvature effects of surface tension of cylindrical droplets According to experiments (Adamson, 1990), the curvature effects of the surface tension LG s of macroscopic cylindrical droplets can be neglected.However, for nano-scale cylindrical droplets, the curvature effects of the surface tension LG s are important (Wenzel, 1936;Rowlinson & Widom, 1982;Adamson, 1990;Nijmeijer, Bruin, Woerkom, & Bakker, 1992;Gennes, Brochard-Wyart, & Quere, 2004;).For droplets with sufficiently large radii, if we neglect (5.3)

Conclusion
The hydrophilic wetting of cylindrical droplets on rough and chemically homogeneous non-deformable substrates were studied by methods of thermodynamics.A generalized Young's equation for wetting of cylindrical droplets on chemically homogeneous and rough non-deformable substrates was derived based on the thermodynamic equilibrium conditions.This equation reduces to the Wenzel equation if we ignore the influence of line tension.It is known that the surface tension depend on the radius of curvature of the liquid-vapor interface.For nano-scale wetting phenomena, we should consider the curvature effects of the surface tension and the line tension.However, previous works have not analyzed the influence of the curvature effects of the surface tension.Based the Kim-Lee-Han-Park equation, we discuss the influence of the curvature effects of the surface tension on the contact angles.For contact angles of cylindrical droplets with sufficiently large radii, a generalized Young's equations were derived considering the curvature effects of the surface tension.

Figure 1
Figure 1 an illustration of hydrophilic wetting of a cylindrical droplet on a rough substrate.L G LG SL SG SLG F = F + F + F + F + F + F (2.1) where F is the total Helmholtz free energy, L F , G F , LG F , SL F , SG F and SLG F are the Helmholtz free energies of the seven parts respectively.
24) is a useful generalization of the Young's equations Equation (1.1) for wetting of cylindrical droplets.If we neglect the second term on the right side of Equation (3.24), we have cos .25) is the Wenzel equation