To investigate the wettability of spherical drops in a smooth and homogeneous cylindrical capillary with hemispherical head, based on Gibbs’s method of dividing surface and Rusanov’s concept of dividing line, the contact angle of spherical droplets has been successfully derived considering the effects of the line tension. Additionally, under the condition of ignoring the line tension, the equation describing the contact angle is simplified as the classical Young equation

Sun, Guang-Hua

Zhou, Long

English

To investigate the wettability of spherical drops in a smooth and homogeneous cylindrical capillary with hemispherical head, based on Gibbsâ€™s method of dividing surface and Rusanovâ€™s concept of dividing line, the contact angle of spherical droplets has been successfully derived considering the effects of the line tension. Additionally, under the condition of ignoring the line tension, the equation describing the contact angle is simplified as the classical Young equation

Journal for Foundations and Applications of Physics

Science Front Publishers           Journal for Foundations and Applications of Physics, vol. 5, No. 1 (2018) (sciencefront.org)                                                                                                                      ISSN 2394-3688 23  Contact angle of spherical drops inside a smooth and homogeneous cylindrical capillary  with hemispherical head  Long Zhou1,* and Guang-Hua Sun2  1School of Mechanical and Power Engineering, Henan Polytechnic University, No. 2001, Century Avenue, Jiaozuo, Henan 454003, China 2School of Business Administration, Henan Polytechnic University, No. 2001,  Century Avenue, Jiaozuo, Henan 454003, China  *Corresponding author E-mail: zhoulong@163.com  (Received 11 January 2018, Accepted 05 February 2018, Published 12 February 2018)  Abstract     To investigate the wettability of spherical drops in a smooth and homogeneous cylindrical capillary with hemispherical head, based on Gibbs’s method of dividing surface and Rusanov’s concept of dividing line, the contact angle of spherical droplets has been successfully derived considering the effects of the line tension. Additionally, under the condition of ignoring the line tension, the equation describing the contact angle is simplified as the classical Young equation.  Keywords:  Liquid droplet; Contact angle; Young equation; Line tension; Hemisphere head 1. Introduction In the last two decades, the investigations in wetting and spreading of liquid drops on a solid substrate have increased substantially, particularly since 2010 [1-4]. The contact angle between liquid drops and solids is a crucial parameter that characteristics the wetting performance in many industrial applications and our everyday life [5-9]. Wettability of ideal surfaces is presented by the well-known Young equation [10]     cos SV SLYLVσ σθσ−=  (1) where Yθ is the Young angle, and SVσ , SLσ and LVσ are the surface tensions at the solid/liquid, solid/liquid and liquid/vapor interfaces, separately. In wetting phenomena, the line at which solid, liquid and vapor phases contact each other is called the triple phase line. The characteristic of the triple phase line plays a key role in actual Zhou et al                                    Journal for Foundations and Applications of Physics, vol. 5, No. 1 (2018) 24  wetting applications. The line tension defined as the free energy per unit length of the triple phase line is a crucial parameter in surface wetting. The concept of the line tension was first proposed thermodynamically by Gibbs. W. C. Jensen and D. Q. Li [11] measured the line tension of the capillary rise in a conical tube. A. I. Rusanov et al [12] studied the line tension dependence on the curvature radius of the triple phase line. A. Amirfazli and A. W. Neumann [13] reviewed both theoretically and experimentally the line tension studies. B. V. Toshev [14] presented the thermodynamic theory of thin liquid films considering the impacts of the line tension. B. M. Law et al [15] summarized the effects of the line tension on drops and particles at surfaces. Many scholars have ever growing interest in understanding the wetting and spreading phenomena of liquids on solids due regard to its extensive applications. However, the wetting of spherical drops in a smooth and homogeneous cylindrical capillary with hemisphere head has not been investigated to this day. In this paper, based on Gibbs’s method of dividing surface and Rusanov’s concept of dividing line, the contact angle describing the wetting of spherical drops in a smooth and homogeneous cylindrical capillary with hemisphere head is derived considering the effects of the line tension. Accordingly, the generalized Young equation of the derived contact angle is simplified as the classical Young equation when neglecting the line tension.  2. Calculation of free energies Consider a single-component spherical liquid droplet (phase L) contacting with vapor (phase V) in equilibrium, placed in a smooth and homogeneous cylindrical capillary (phase S) with hemisphere head, as illustrated in Figure 1.  θββRVaporLiquidSolidRLHOA Figure 1. Spherical droplet in a smooth and homogeneous cylindrical capillary with hemisphere head Zhou et al                                    Journal for Foundations and Applications of Physics, vol. 5, No. 1 (2018) 25  For sake of generality, assume that the height of the liquid drop exceeds the hemisphere head in Figure 1, that is, sinH R β> . According to both Gibbs’s theory of dividing surface and Rusanov’s approach of dividing line, the free energy of the solid-liquid-vapor system in Figure 1 is given by [16] L V SL SV LV SLVF F F F F F F= + + + + +  (2) where L L L L LF p V Nµ= − +                            (3) V V V V VF p V Nµ= − +                           (4) SL SL SL SL SLF A Nσ µ= +                          (5) SV SV SV SV SVF A Nσ µ= +                         (6) LV LV LV LV LVF A Nσ µ= +     (7) SLV SLV SLV SLVF kL Nµ= +     (8) where F is the free energy (the single, double, and triple subscripts denote the quantities concerning the corresponding phases, such as the subscript SLV being the triple phase line), p is the pressure,V is the volume, µ is the chemical potential, N is the mole number of molecule,σ is the surface energy of unit area, A is the surface area, k is the line tension, and L is the length of the triple phase line. For simplicity, suppose that the gravity and the other forces or fields are neglected, so the equilibrium shape of the spherical drop in Figure 1 is the combination of both a hemisphere, a cylindricity and a segment. As a result, the drop volume LV appearing in Eq. (3) is expressed as ( ) ( ) ( )23 3 2 2 32 sin sin sin 1 cos 2 cos3 3LV R R H R Rpi piβ pi β β β β= + − + − +          (9) where R is the drop radius, β is the apparent contact angle, and H is the drop height. The entire volume tV of the system is t L VV V V= +  (10) The area LVA of the liquid-vapor interface is ( )22 1 cosLVA Rpi β= −                       (11) The area SLA of the solid-liquid interface is 2 sinSLA RHpi β=  (12) The total area tA of the solid-liquid and solid-vapor interfaces is t SL SVA A A= +  (13) The length of the triple phase line is 2 sinSLVL Rpi β=  (14) Based on the relations stated above, various free energies can be represented, separately, as Zhou et al                                    Journal for Foundations and Applications of Physics, vol. 5, No. 1 (2018) 26  ( ) ( ) ( )23 3 2 2 32 sin sin sin 1 cos 2 cos3 3L LL LF p R R H R RNpi piβ pi β β β βµ = − ⋅ + − + − +  +       (15)   ( )( ) ( )3 3 2 2232sin sin sin31 cos 2 cos3V V tV VF p V R R H RR Npi β pi β βpi β β µ = − ⋅ − + − + + − + +                  (16) 2 sinSL SL SL SLF RH Nσ pi β µ= ⋅ +                      (17) 2 sinSV SV t SV SVF A RH Nσ pi β µ = ⋅ − +               (18) ( )22 1 cosLV LV LV LVF R Nσ pi β µ= ⋅ − +                          (19) 2 sinSLV SLV SLVF k R Npi β µ= ⋅ +                          (20) Consequently, putting Eqs. (15-20) into Eq. (2), one has ( ) ( )( ) ( ) ( )( )3 3 2 223 22sin sin sin31 cos 2 cos 2 1 cos32 sin 2 sinL VV t LVSL SV SV tL L V V LV LV SL SL SV SV SLV SLVF p p R R H RR p V RRH A k RN N N N N Npi β pi β βpi β β σ pi βσ σ pi β σ pi βµ µ µ µ µ µ= − − ⋅ + −+ − + − ⋅ + ⋅ −+ − ⋅ + ⋅ + ⋅+ + + + + + (21)  3. Derivation of contact angle The total potential energy Ω of the system presented in Section 2 is defined by the equation ( )L L V V LV LV SL SL SV SV SLV SLVF N N N N N Nµ µ µ µ µ µΩ = − + + + + +  (22) Using the free energy Eq. (21) and potential energy Eq. (22), the potential energyΩ can be rewritten as ( ) ( )( ) ( ) ( )( )3 3 2 223 22sin sin sin31 cos 2 cos 2 1 cos32 sin 2 sinL VV t LVSL SV SV tp p R R H RR p V RRH A k Rpi β pi β βpi β β σ pi βσ σ pi β σ pi βΩ = − − ⋅ + −+ − + − ⋅ + ⋅ −+ − ⋅ + ⋅ + ⋅ (23) Minimizing the potential energyΩ with respect to the radius R , namely 0ddRΩ =   (24) Due to the surface tensions SLσ and SVσ independent of the dividing surface [12], one obtains Zhou et al                                    Journal for Foundations and Applications of Physics, vol. 5, No. 1 (2018) 27  00SLSVddRddRσσ =   =   (25) By utilizing equations (23-25), the following expression is obtained ( )( ) 0LV LVLL V LV LVSL SLVSL SV SLVd dAdVp p AdR dR dRdA dLdk L kdR dR dRσσσ σ    − − ⋅ + ⋅ + ⋅             + − ⋅ + ⋅ + ⋅ =                     (26) From geometry, the following relations can be easily obtained sinLR R constβ= =  (27) cosH R OA constβ− = =  (28) 2piβ θ= −  (29) Taking the derivation of Eqs. (27-28) with respect to the radius R , one finds that sincosddR Rβ ββ= −  (30) 1cosdHdR β=  (31) 0LdRdR=  (32) Similarly, we take the derivation of Eqs. (9, 11, 12, 14) with respect to the radius R and apply Eqs. (30-31), leading to ( )22 1 cosLdV RdRpi β  = −    (33) ( )2sin4 1 cos 2cosLVdA R RdRβpi β pi β = − − ⋅   (34) 2 sincosSLdA RdRpi ββ =   (35) 0SLVdLdR =   (36) The Laplace equation [16] of the spherical drop in vapor is formulated as 2 LV LVL Vdp pR dRσ σ − = +    (37) Now putting Eqs. (33-37) into Eq. (26), we find cossin SL SVLV LVdkdRσ σ ββσ σ−  = + ⋅    (38)  According to Eq. (29), Eq. (38) can be rewritten as Zhou et al                                    Journal for Foundations and Applications of Physics, vol. 5, No. 1 (2018) 28  sincos SV SLLV LVdkdRσ σ θθσ σ−  = − ⋅    (39) If the familiar Young equation (1) is substituted into Eq. (39), we arrive at a new generalized Young equation that describes the contact angle of spherical drops inside a smooth and homogeneous cylindrical capillary with hemisphere head sincos cos YLVdkdRθθ θσ = − ⋅    (40) In addition, if the effects of the line tension are neglected, Eq. (39) is simplified to the classical Young equation (1).  4. Conclusion In this paper, using Gibbs’s method of dividing surface and Rusanov’s concept of dividing line, the wettability of a spherical droplet in a smooth and homogeneous cylindrical capillary with hemisphere head is studied thermodynamically. Considering the effects of the line tension, the contact angle of spherical droplets in a smooth and homogeneous cylindrical capillary with hemisphere head was successfully derived. When ignoring the effects of the line tension, this generalized Young equation reduces to the classical Young equation.  Acknowledgments This work was supported by the Doctor Research Foundation of Henan Polytechnic University (No. 660407/002).  REFERENCES [1] C. A. Ward, and K. Sefiane, “Adsorption at the solid-liquid interface as the source of contact angle dependence on the curvature of the three-phase line”, Advances in Colloid and Interface Science, 161, 171-180 (2010) [2] M. Soleimani, R. J. Hill, and G. M. Ven, “Bubbles and drops on curved surfaces”, Langmuir, 29, 14168-14177 (2013) [3] B. B. Straumal, A. A. Mazilkin, and B. Baretzky, “Grain boundary complexions and pseudopartial wetting”, Current Opinion in Solid State and Materials Science, 20, 247-256 (2016) [4] E. Bormashenko, “Progress in understanding wetting transitions on rough surfaces”, Advances in Colloid and Interface Science, 222, 92-103 (2015) [5] T. M. Bucher, and H. V. Tafreshi, “Modeling air-water interface in disordered fibrous media with heterogeneous wettabilities”, Colloids and Surfaces A, 461, 323-335 (2014) [6] J. H. 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Contact angle of spherical drops inside a smooth and homogeneous cylindrical capillary  with hemispherical head

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Contact angle of spherical drops inside a smooth and homogeneous cylindrical capillary with hemispherical head

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