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

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&lt;p class="1Body"&gt;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.&lt;/p&gt;

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Mechanical Engineering Research; Vol. 5, No. 2; 2015

ISSN 1927-0607 E-ISSN 1927-0615
Published by Canadian Center of Science and Education

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

Xiao-Song Wang 1

1 Institute of Mechanical and Power Engineering, Henan Polytechnic University, No. 2001, Century Avenue,
Jiaozuo, Henan 454003, China

Correspondence: Xiao-Song Wang, Institute of Mechanical and Power Engineering, Henan Polytechnic
University, No. 2001, Century Avenue, Jiaozuo, Henan 454003, China. Tel: 86-391-3987543. E-mail:
wangxs2016@163.com

Received: June 27, 2015  Accepted: July 24, 2015  Online Published: August 7, 2015

doi:10.5539/mer.v5n2p17     URL: http://dx.doi.org/10.5539/mer.v5n2p17

Abstract

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.

Keywords : contact angle, wetting, Young’s equation, Wenzel equation, surface tension, curvature effect,
cylindrical droplet

1. 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 q _Y_ for the wetting of spherical droplets on rough and
chemically homogeneous substrates is determined by the following equation (Young, 1805)

q _Y_ = s _SG_ -s

cos

_Y_ = _SG_ _SL_

= s _SG_ -s _SL_ (1.1)

_LG_

where q _Y_ is the contact angle, s _LG_ is the surface tension of the liquid‐vapor interface corresponds to the choice of
the surface of tension as a dividing surface, s _SG_ is the surface free energy per unit area of the solid‐vapor
interface, s _SL_ 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).

www.ccsenet.org/mer Mechanical Engineering Research Vol. 5, No. 2; 2015

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.

2. 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

Figure 1 an illustration of hydrophilic wetting of a cylindrical droplet on a rough substrate.

F = F + FL G + FLG + F + F + FSL SG SLG (2.1)
where F is the total Helmholtz free energy, FL, FG, FLG, FSL, FSG and FSLG are the Helmholtz free energies of
the seven parts respectively.

The Helmholtz free energies of these seven parts can be written as

F = -pL LV + L m L NL
(2.2)

F = -pG GV + G m G NG
(2.3)

FLG = s LGA LG + m LG N LG
(2.4)

F = SL s SLASL + m SL NSL
(2.5)

F = SG s SGASG + m SG NSG
(2.6)

FSLG = L k SLG + m SLG NSLG
(2.7)

where _pL_ and _G_ _VL_ and _VG_ are the
_p_ are the pressures of the liquid phase and the vapor phase respectively,
volumes of the liquid phase and the vapor phase respectively, m is the chemical potential of the six subsystems,

_N_ _L_, _NG_, _N_ _LG_, _N_ _SL_, _N_ _SG_ and _N_ _SLG_ 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, _ALG_, _ASL_ and _ASG_ are the surface area of the liquid‐vapor interface, the solid‐liquid interface, the
solid‐vapor interface respectively, s _LG_, s _SL_ and s _SG_ are the surface tensions of the liquid‐vapor interface, the

www.ccsenet.org/mer Mechanical Engineering Research Vol. 5, No. 2; 2015

solid‐liquid interface, the solid‐vapor interface respectively, _LSLG_ 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

L G ) ( q- sin q cos q ) _R_ 2

F= -(p - pL G ) ( q- sin q cos q )

q- sin q cos q )

_R_ 2 _L_

- pGV + N + Nt m L m

V + N + N m L m

G t m L m G

+ s LG 2 q RL+ N m _SL_ + N m

s LG 2 q RL+ N m _SL_ + N m

LG q m _SL_ m LG

(2.8)

_SL_

+ ( _rs_ s SL - s SG )2Rsin

_L_

_rs_

SG t m SG

_s_

_rs_

s SL - s SG )2Rsin q

SL s SG

+ _rs_ s SGAt + N m

_s_ s SGAt + N m

+ 2 _rL_ L k + m NSLG,

_rL_

SLG

_L_

k + m

where R is the radius of the cylindrical liquid droplet, q is the contact angle, L is the length of the

cylindrical liquid droplet, At is the total surface area, _r_ _s_ is the surface roughness factor, _r_ _L_ is the line roughness
factor.

3. 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)

_F_

æ¶ _F_ ö
çç ÷÷ =
è¶qø _T_, m

(3.)

qø _T_, m

_T_,

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)

t
## W = å (F -i m i Ni ), (3.2)

## = å (F -i m i Ni ),

i=1

where t is the number of subsystems of the system, F i is the Helmholtz free energy of the i‐th subsystem, m i
is the chemical potential of the the i-th subsystem, Ni is the mole numbers of molecule of the the i-th subsystem.

Putting Equation (3.2) into Equation (3.1), the thermodynamic equilibrium condition becomes (Nijmeijer, Bruin,
Woerkom, & Bakker, 1992)

æ ¶W ö
çç ÷÷ =
è¶qø _T_, m

qø _T_, m

_T_,

Putting Equation (2.8) into Equation (3.2), the total grand potential W of the above system is

(3.3)

(3.4)

=-(p - pL G ) ( q- sin q cos q ) _R_ 2

=-(p - pL G ) ( q- sin q cos q )

W =-(p - pL G ) ( q

q- sin q cos q )

_R_ 2 _L_

- pGV + t s LG 2 q RL

s LG 2 q

G t s LG

+ ( _rs_ s SL - s SG )2Rsin

_L_

_rs_

_s_

s SL - s SG )2Rsin q

SL s SG

_rs_ s SGA t +2 _rL_

_s_ s SGA t +2 _rL_ L k .

_s_ s SG t _L_

SG t

A t +2 _rL_ L k .

Putting Equation (3.4) into Equation (3.3), we have

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-(p - pL G ) æçç¶ _f_ 1 ö÷÷ - _f_ 1 æçç¶ (p - pL G )

_f_ 1 ö÷÷ - _f_ 1

-(p - pL G ) çç¶ 1 ÷÷ - _f_ 1 çç¶ L G ÷÷

è¶qø _T_, m è ¶q ø _T_,

1 ç

qø _T_, m è ¶q

_T_, m è ¶q ø _T_,

, m è ø _T_, m

s LG ççæ¶ _f_ 2 ÷÷ö + _f_ 2 ççæ¶s LG

è¶qø _T_, m è ¶q

LG çç¶ 2 ÷÷ + _f_ 2 çç¶s LG ÷÷

è¶qø _T_, m è ¶q ø _T_,

_f_ 2 ÷÷ö + _f_

_T_, m è ¶q ø _T_,

, m è ø _T_, m

_f_ 3 ö÷ + _f_

+ _rs_ ( s SG - s SL ) æçç¶ _f_ 3 ö÷÷ + _f_ 3 _rs_ æçç¶ ( s SG - s SL )

s SG - s SL ) æçç¶ _f_ 3 ö÷÷ + _f_ 3 _rs_ æçç¶ ( s SG - s

è¶qø _T_, m è ¶q

SG - s SL ) çç¶ 3 ÷÷ + _f_ 3 _rs_ çç¶ s SG s SL ÷÷

è¶qø _T_, m è ¶q ø _T_,

_rs_ ( s SG - s SL ) çç 3 ÷÷ + _f_ 3 _rs_

_s_ s SG s SL ç ÷ + 3 _s_ ç ÷

è¶qø _T_, m è ¶q ø _T_,

(3.5)

, m è ø _T_, m

_L_

_s_ s SG æçç¶ _f_ 4 ö÷÷ + _f_ 4 _rs_ æçç¶s SG

è¶qø _T_, m è ¶q

_L_ k

SG çç¶ 4 ÷÷ + _f_ 4 _rs_ çç¶s SG ÷÷

è¶qø _T_, m è ¶q ø _T_,

G ) æçç¶ _f_ 1 ö÷÷ - _f_ 1 æçç¶ (p - pL G ) ö÷÷

è¶qø _T_, m è ¶q ø _T_, m
ççæ¶ _f_ 2 ÷÷ö + _f_ 2 ççæ¶s LG ÷÷ö
è¶qø _T_, m è ¶q ø _T_, m

SG - s SL ) æçç¶ _f_ 3 ö÷÷ + _f_ 3 _rs_ æçç¶ ( s SG - s SL ) ö÷÷

è¶qø _T_, m è ¶q ø
æçç¶ _f_ 4 ö÷÷ + _f_ 4 _rs_ æçç¶s SG ö÷÷
è¶qø _T_, m è ¶q ø _T_, m

_f_ 4 ö÷÷ + _f_

_rs_ s SG çç 4 ÷÷ + _f_ 4 _rs_

_s_ s SG ç ÷ + 4 _s_ ç ÷

è¶qø _T_, m è ¶q ø _T_

m è ø _T_, m

_rL_

ççæ¶ _f_ 5 ÷÷ö + _f_ 5 _rL_ æçç¶kö÷÷ =
è¶qø _T_, m è¶qø _T_, m

_f_ 5 ÷÷ö + _f_ 5 _rL_

_f_ 5 ÷÷ö + _f_ 5 _rL_ æçç¶k

qø _T_, m è¶q

5 ÷÷ + _f_ 5 _rL_ çç¶k÷÷ = 0,

qø _T_, m è¶qø _T_, m

_T_, m è¶qø _T_

m è ø _T_, m

where

_f_ 1 = ( q- sin q cos q ) _R_ 2 _L_, (3.6)

_f_ 2 = 2 q _RL_, (3.7)

_f_ 3 = 2 _LR_ sin q, (3.8)

_f_ 4 = A t, (3.9)

_f_ 5 = 2 _L_ . (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

pL, 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

æç¶( pL - p _G_ )ö÷ =
ç ÷
è ¶q ø _T_, m

pL - p _G_ )ö÷ = 0,

( pL - p _G_ )

q ø _T_, m

æçç¶s _SL_ ö÷÷ =
è ¶q ø _T_, m

s

_SL_ ÷÷ = 0,

_T_, m

æçç¶s SG ö÷÷ =
è ¶q ø _T_, m

æçç¶s LG ö÷÷ =
è ¶q ø _T_, m

s

SG

_T_, m

s

LG

_T_, m

We have the following results

_f_ 1 ö÷÷ = 2 _R_ 2 _L_

æçç¶ _f_ 1 ö÷÷ =
è¶qø _T_, m

1 ÷ 2 _R_ 2 _L_ sin 2

2 _R_ 2 _L_ sin 2 q,

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

1 ÷÷ = 2 _R_ 2 _L_ sin 2 q,

qø _T_, m

_T_,

, m

_f_ 2 ö÷÷ = 2 _RL_,

æçç¶ _f_ 2 ö÷÷ =
è¶qø _T_, m

2

qø _T_, m

2 _RL_,

_T_,

_f_ 3 ö÷÷ = 2 _RL_

æçç¶ _f_ 3 ö÷÷ =
è¶qø _T_, m

3

2 _RL_ cos q,

3 ÷ = 2 _RL_ cos q,

qø _T_, m

_T_,

, m

_f_

4

æçç¶ _f_ 4 ö÷÷ =
è¶qø _T_, m

æçç¶ _f_ 5 ö÷÷ =
è¶qø _T_, m

qø _T_, m

_T_,

_f_

5

qø _T_, m

_T_,

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Putting Equations (3.11-3.19) into Equation (3.5), we obtain

+s LG + _rs_ ( s _SL_ -s _SG_ )cos q+ _rL_ æçç¶kö÷÷ =

_R_ è¶qø _T_, m _i_

) sin _R_ 2 q+s LG + _rs_ ( s _SL_ -s _SG_ )cos q+ _rL_ æçç¶kö÷÷

_R_ è¶qø _T_, m _i_

q+s LG + _rs_ ( s _SL_ -s _SG_ )cos q+ _rL_ æçç¶k

_R_ è¶q

_T_, m _i_

- (p - pL G ) sin _R_ 2 q+s LG

- (p - pL G ) sin _R_ 2 q+s LG + _rs_ ( s _SL_ -s _SG_ )cos q+ _L_ çç¶k÷÷ = 0.

_R_ è¶qø _T_, m _i_

(3.20)

It is known that a generalized Laplace’s equation of a free cylindrical droplet in vapor can be written as (Wenzel,
1936)

_pL_ - _pG_ = s _RLG_ + êé ~~ê~~ ë _d_ s _dRLG_ úúûù,

(3.21)

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

cos q = _rs_ s _SG_ s- _LG_ s _SL_ + _R_ sin sq LGtan q é ~~ê~~ êë d s dRLG

(3.22)

LG

= _rs_ s _SG_ s- _LG_ s _SL_ + _R_ sin sq LGtan q é ~~ê~~ êë d s dRLG ùúúû

- _rL_ æçç¶kö÷÷ .

s LG R cos qè¶qø _T_, m _i_

q = _rs_ s _SG_ s- _LG_ s _SL_ + _R_ sin sq LGtan q é ~~ê~~ êë d s dR

_R_

_r_

s _SG_ -s

_SG_ _SL_

_s_

s _LG_

_rL_

_rL_ æç¶k

s LG R cos qè¶q

m _i_

_L_

_L_ ç¶k÷ .

ç ÷

R cos qè¶qø _T_, m _i_

_T_, m _i_

LG qè¶qø _T_, m

Equation (3.22) 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

é ~~ê~~ d s LG ùú = 0,
ê dR úû _R_ = _R_

é ~~ê~~ d s LG ùú =
êë dR úû _R_ = _Rs_

d s

_R_ = _Rs_

Where R _s_ is the radius of the surface of tension _M_ _s_

If we choose the surface of tension _M_ _s_ as the dividing surface, then Equation (43) becomes

(3.23)

(3.24)

= _rs_ s _SG_ -s _SL_ - _rL_ æçç¶kö÷÷

s _LG_ s LG R cos qè¶qø

m _i_

_rs_ s _SG_ -s _SL_ - _rL_

q = _rs_ s _SG_ -s _SL_ - _rL_ æçç¶k

s _LG_ s LG R cos qè¶q

_T_, m _i_

s _SG_ -s

_s_ s _SG_ -s _SL_ - _L_

cos q = _rs_ s _SG_ -s _SL_ - _L_ çç¶k÷÷ .

s _LG_ s LG R cos qè¶qø _T_, m _i_

_SG_ _SL_

_LG_

Equation (3.24) 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

q = _r_ _s_ s _SG_ -s

s _LG_

= _r_ _s_ s _SG_ -s _SL_ . (3.25)

cos q = _r_ _s_ s _SG_ -s _SL_ .

Equation (3.25) is the Wenzel equation (Wenzel, 1936).

4. 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

d _RGs_ 2 æççè 1 + 2 d _RGs_ ö÷÷ø

= 2

1 _d_ s _LG_ ( _Rs_ ) d _RGs_ 2 æççè 1 + 2 d _RGs_

s _dRs_ = d _G_ d _G_ 2

_G_ ç 1 d _G_

1 _d_ s _LG_ ( _Rs_ ) _RGs_ 2 ççè 1 + 2 _RGs_ ÷÷ø

s _dRs_ = 1 +d _RGs_ + 2 d _RG_ 2 _s_ 2,

(4.1)

_d_ s _LG_ ( _Rs_ ) _Rs_ 2 çè 2 _Rs_
_dRs_ = d _G_ d _G_ 2

_LG_ _s_ _s_ è _s_

_s_ 1 d _G_ d _G_

+ _G_ +

_G_

_Rs_ 2 _Rs_

_s_ _s_

where s _LG_ ( _Rs_ ) is the surface tension of a cylindrical droplet with radius _Rs_, _Rs_ is the radius of Gibbs’ surface
of tension, d _G_ º _Re_ - _Rs_, _Re_ is the radius corresponding to the equimolar dividing surface.

For sufficiently large droplets, we neglect d _G_ 3 / _Rs_ 3 and d _G_ 2 / _Rs_ 2, and treat _G_ d _T_, Equation (4.1)

d as the constant
can be solved. The result is

_LG_ ( _Rs_ ) = exp æç-d _T_ ö÷,

ç ÷

_R_

= exp æç-d _T_ ö÷
çè _Rs_ ÷ø

(4.2)

s _LG_ ( _Rs_ ) = exp æç-d _T_
s ¥ çè _Rs_

_s_

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where s ¥ is the surface tension of plane surface with radius _Rs_ = ¥, d _T_ º _R_ lim _s_ ®¥

d _T_ º lim d _G_ .
_Rs_ ®¥

d _T_ is usually called the Tolman length in the literatures (Rowlinson & Widom, 1982).

We have the following Taylor’s expansion

_x_ 2 _x_ 3
exp( - _x_ ) = 1 - _x_ + - + ... (4.3)
2! 3!

Applying Equation (4.3), Equation (4.2) can be expanded as

_LG_ ( _Rs_ ) = 1 - d _RT_ + d _T_ 22 + ...

_R_

= 1 - d _T_ + d _T_ 2 + ... (4.4)

s _LG_ s ¥ ( _Rs_ ) = 1 - d _RTs_ + d _RTs_ 22

_s_ _s_

5. Generalized Young’s Equation Considering the Curvature Effects of the Surface Tension

Using Equation (4.2), Equation (3.24) can be written as

æçèç-d _RTs_ ö÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶kqö÷÷ø

d _RTs_ ö÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶kqö÷÷ø _T_, m

s¥ exp æçèç-d _RTs_ ö÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶kq÷÷

_T_ ÷÷ cos q = _rs_ (s _SG_ -s _SL_ ) - _L_

exp çèç- _RTs_ ÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _L_ qççè¶¶kq÷÷ø _T_, m .

(s _SG_ -s _SL_ )

(5.1)

_s_ ø qè¶qø _T_,

Equation (5.1) is a generalized Young’s equation for spherical droplets on rough but chemically heterogeneous
non-deformable substrates considering the curvature effects of the surface tension

s _LG_ . Equation (5.1) is one of the main results of this work.

Applying Equation (4.3), Equation (5.1) becomes

çèæç 1 - d _RTs_ + d _RTs_ 22 + ... ÷øö÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶qkö÷÷ø

d _RTs_ + d _RTs_ 22 + ... ÷øö÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶qkö÷÷ø _T_, m

s¥ çèæç 1 - d _RTs_ + d _RTs_ 22 + ... ÷øö÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶qk

_T_ + d _T_ 2 + ... ÷÷ cos q = _rs_ (s _SG_ -s _SL_ ) - _L_

1 - _RTs_ + _RTs_ 2 + ... ÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _L_ qççè¶¶qk÷÷ø _T_, m .

(s _SG_ -s _SL_ )

(5.2)

_s_ _s_ ø qè¶qø _T_

According to experiments (Adamson, 1990), the curvature effects of the surface tension s _LG_ of macroscopic
cylindrical droplets can be neglected. However, for nano-scale cylindrical droplets, the curvature effects of the
surface tension s _LG_ 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 d _T_ and higher terms in Equation (5.2), then we have

d _T_ and higher terms in Equation (5.2), then we have
_Rs_ 2

_T_
2

æçèç 1 - d _RTs_ ö÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶kqö÷÷ø

d _RTs_ ö÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶kqö÷÷ø _T_, m

s¥ æçèç 1 - d _RTs_ ö÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _rL_ qæççè¶¶kq

_T_ ÷÷ cos q = _rs_ (s _SG_ -s _SL_ ) - _L_

1 - _RTs_ ÷ø÷ cos q = _rs_ (s _SG_ -s _SL_ ) - R cos _L_ qççè¶¶kq÷÷ø _T_, m .

(s _SG_ -s _SL_ )

(5.3)

_s_ ø qè¶qø _T_,

6. 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.

Acknowledgments

This work was supported by the Key Project of Science of the Education Bureau of Henan Province (Grant No.
15A130001) and the Doctor Research Foundation of Henan Polytechnic University (Grant No. 72515‐466).

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