Generalized Cassie-Baxter equation for wetting of a spherical droplet within a smooth and heterogeneous conical cavity

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1 Science Front Publishers Journal for Foundations and pplications of Physics, vol. 4, No. (017) (sciencefront.org) ISSN Generalized Cassie-axter equation for wetting of a spherical droplet within a smooth and heterogeneous conical cavity ong Zhou 1,*, Guang-Hua Sun, Kai-Hui Zhao 1, Xiao-Song Wang 1 and i-jun Hu 1 1 School of Mechanical and Power Engineering, Henan Polytechnic University, No. 001, Century venue, Jiaozuo, Henan , China School of usiness dministration, Henan Polytechnic University, No. 001, Century venue, Jiaozuo, Henan , China *Corresponding author zhoulong@163.com (Received 1 February 017, ccepted 1 March 017, Published 01 pril 017) bstract Introducing the concepts of both Gibbs s dividing surface and Rusanov s dividing line, the wettability behaviors of spherical drops inside a smooth and heterogeneous conical cavity are studied. new generalized Cassie-axter equation for contact angles including the influences of the line tension is derived thermodynamically. dditionally, various approximate formulae of this generalized Cassie-axter equation are also discussed correspondingly under some assumptions. 017 Science Front Publishers Keywords: Contact angle; Generalized Cassie-axter equation; ine tension; Cone; Dividing surface 1. Introduction Strong understanding the wetting phenomena of liquid droplets on solid surfaces is very important for designing and controlling wettability characteristics in industrial applications and daily lives. Extensive interests have been attracted to investigate the wetting cases of solids by liquids for about three centuries, particularly in recent twenty years. The most fundamental theory about the wetting behaviors is the Young s equation [1] for contact angle θy of liquid droplets on smooth and chemically homogeneous solid surfaces given by, SG S cosθy = (1) where SG, S and are the surface tensions of the solid-vapor, solid-liquid, and liquidvapor interfaces, separately. 31

2 ong Zhou et al Journal for Foundations and pplications of Physics, vol. 4, No. (017) Evidently, the Young equation (1) only applies to smooth and homogeneous cases, but real solid surfaces are usually rough and heterogeneous. Fortunately, the Wenzel equation was developed in the 1930s for the wettability of rough surfaces. Subsequently, in the 1940s, for the wetting of smooth and heterogeneous surfaces, the classical Cassie-axter equation [] for contact angle θ was proposed by cosθ = f cosθ + f cosθ () 1 1 where θ 1 and θ are the contact angles that liquid droplets make with solid surfaces 1 and, f 1 and f are the area fractions of 1 and surfaces below droplets. Despite the theoretical basis for wetting properties of drops is established by the above mentioned Young equation, Wenzel equation, and Cassie-axter equation, all of them ignore the influences of the line tension. large number of investigations [3-10] show that the line tension considerably depends on the curvature radius of the triple phase line. review of earlier studies about the line tension can be obtained in reference [11]. In fact, since Gibbs introduces the concept of the line tension primitively, the classical Young s equation is generalized so as to include the effects of the line tension [6, 9]. For example, the generalized Young s equation proposed by Pethica [3] satisfies k cosθ = cosθ Y (3) R where R is the radius of the three phase line, and k is the corresponding line tension. In 004, through the method of Gibbs s dividing surface, Rusanov [4] also developed a generalized Young s equation k 1 dk cosθ = cosθ Y (4) R dr where the quantity in square bracket stands for the derivation of the line tension k by the radius R of the triple phase line. In terms of the wetting of drops on heterogeneous solid surfaces, many investigators have carried out considerable works. Swain [1] developed a new generalized Young equation for liquid droplets on a rough and heterogeneous solid by a novel minimization method. Fang [13] applied the Neumann-Good parallel strip model to analyze the contact angle hysteresis of drops on a heterogeneous surface. Towles [14] analyzed the influences of the contact angle on the packing, or thickness mismatch contribution to the line tension. Raj [15] thermodynamically studied the effects of the contact line distortion to the wetting of drops on heterogeneous and superhydrophobic surfaces. In a recent paper, Kwon [16] investigated the static and dynamic characteristics of nano-scale drops on chemically heterogeneous surfaces using the method of molecular dynamics simulations. In addition, Zhang [17] discussed the influences of surface heterogeneities on the evaporation of drops from solid surfaces also utilizing molecular dynamics simulations. 3

3 ong Zhou et al Journal for Foundations and pplications of Physics, vol. 4, No. (017) However, up to now, the wetting of spherical droplets inside a smooth and heterogeneous conical cavity has not been performed. In this paper, based on the theories of both Gibbs s dividing surface and Rusanov s dividing line, the wetting of spherical droplets within a smooth and heterogeneous cone is studied. t the same time, a new generalized Cassie-axter equation of drops inside a smooth and heterogeneous conical cavity is derived accordingly. This equation is applicable to random dividing surface between the liquid and vapor phases. dditionally, we yet analyze various simplified expressions of this generalized Cassie-axter equation through some hypotheses.. Calculating the free energy of the system et us now consider a single component spherical droplet (phase ) located inside a smooth and chemically heterogeneous cone (phase S) with its equilibrium vapor (phase G), as illustrated in Figure 1. Vapor γ R R O θ β β iquid α Solid Figure 1. Schematic representation of a spherical droplet in a smooth and chemically heterogeneous conical cavity For the purpose of simplicity, it can be seen from Figure 1 that the cone is composed of two types of different materials and. Consequently, there are two kinds of solid/liquid interfaces, two sorts of solid/vapor interfaces, as well as two classes of triple phase lines. The Young M 33

4 ong Zhou et al Journal for Foundations and pplications of Physics, vol. 4, No. (017) equation describing contact angles ( i 1, ) by, SGi Si cos i θ = of two heterogeneous solids can be together given i θ = (5) where is the surface tension, the triple subscripts denote the quantities with respect to the corresponding interfaces (such as the subscript SGi marks the solid-vapor interface, where i being the label of solid and having two values of 1 and.). Via the theories of both Gibbs s dividing surface and Rusanov s dividing line, the overall solid/liquid/vapor system in Figure 1 includes six parts, i.e., liquid phase, vapor phase, solidliquid interface, solid-vapor interface, liquid-vapor interface, together with the three phase line. Thus, the Helmholtz free energy F of the entire system is F = F + F + F + F + F + F (6) where F, F G, F S, mentioned above. F SG G S SG S, F and F S are the Helmholtz free energies of the six parts The Helmholtz free energies of these six parts can be given respectively by [18], F = p V + µ N (7) F = p V + µ N (8) G G G G G F = + µ N (9) ( 1 1 ) ( 1 1 ) F = f + f + µ N (10) S S S S S S F = f + f + µ N (11) SG SG SG SG SG SG F = g k + g k + µ N (1) S 1 1 S S S where p is the pressure, V is the volume, µ is the chemical potential, N is the mole number of molecule, is the surface area, is the surface tension, k is the line tension, and is the length of the three phase line, f and 1 f are the fractional surface areas occupied by two solidliquid interfaces or two solid-vapor interfaces such that f1 + f = 1, and g 1 and g are the fractional lengths of two three phase lines such that g1 + g = 1. In order to highly calculate the geometrical quantities in the above equations, we assume that the gravity and other forces or fields are ignored simultaneously. Hence, the equilibrium shape of droplets in a conical cavity is the sum of a cone and a segment. The volume V of the liquid phase is π 3 3 cosα π 3 V = R sin β + R ( 1 cos β ) ( + cos β ) (13) 3 3 where R is the radius of the droplet, α is the half cone angle, and β is the apparent contact angle. 34

5 ong Zhou et al Journal for Foundations and pplications of Physics, vol. 4, No. (017) The total volume V t of the system is The surface area The surface area of the liquid-vapor interface is Vt = V + VG (14) S of the solid-liquid interface is π ( β ) = R 1 cos (15) sin β S = π R (16) The total surface area t of the solid-liquid and solid-vapor interfaces is The length of the triple phase line is t = S + SG (17) S = π R sin β (18) pplying the above relations, the free energies of six subsystems can be rewritten by, π cosα π F = p R sin β + R ( 1 cos β ) ( + cos β ) + µ N π cosα π F = p V R sin β + R ( 1 cos β ) ( + cos β ) + µ N G G t G G 1 cos (19) (0) F = π R β + µ N (1) sin β FS = ( f1 S1 + f S ) π R + µ SNS () sin β F f f R N SG = ( 1 SG1 + SG ) t π + µ SG SG (3) F = g1k1 + gk π Rsin β + µ N (4) S S S y putting the above Eqs. (19-4) into Eq. (6), the free energy F of the overall system has the form, π cosα π F = p pg R + R sin β ( 1 cos β ) ( cos β ) p V + R + N + N + N G t π 1 cos β µ µ G G µ sin β + ( f1 S1 + f S ) ( f1 SG1 + f SG ) π R + f + f + g k + g k π Rsin β + 1 SG1 SG t µ N + µ N + µ N S S SG SG S S (5) 35

6 ong Zhou et al Journal for Foundations and pplications of Physics, vol. 4, No. (017) 3. Derivation of a new generalized Cassie-axter equation The grand potential Ω of a system composed of liquid droplets in contact with solid and vapor phases may be expressed as Ω = F µ N (6) where the subscript i stands for the sum of both phases and interfaces of the system. Substituting Eq. (5) into Eq. (6), the grand potential Ω can be rewritten as, i π cosα π Ω = G i i ( p p ) R sin β R ( 1 cos β ) ( cos β ) G t p V + π R 1 cos β sin β + ( f1 S1 + f S ) ( f1 SG1 + f SG ) π R + f + f + g k + g k π Rsin β 1 SG1 SG t 1 1 Minimizing the grand potential Ω with respect to the radius R, we have dω = 0 (8) ecause four surface tensions S1, S, SG1, and SG don t depend on the choice of the dividing surface [6], one obtains d d dr dr S1 S = = 0 d d dr dr y utilizing equations (7-30), we have where, ( + ) SG1 SG = = 0 dx1 d dx ( p pg ) + x + dr dx3 + ( f1 S1 + f S ) ( f1 SG1 + f SG ) dr d g k g k dx + x + ( g k + g k ) = dr π cosα π x R R 3 3 (7) (9) (30) (31) = sin β + 1 cos β + cos β (3) x ( β ) = π (33) R 1 cos sin β x3 = π R x4 = π R sin β (34) (35) 36

7 ong Zhou et al Journal for Foundations and pplications of Physics, vol. 4, No. (017) From Figure 1 we can obtain the following expressions R = Rsin β = cosα R cos β = OM = const γ = β α π θ = γ + In Eqs. (36-37), the derivation with respect to the radius R leads to β sin = R cos d dr ( β α ) ( β α ) 1 37 (36) (37) (38) (39) d = dr cos β α (40) dr dr = cos (41) ( β α ) Using Eqs. (3-35) and Eqs. (39-40), we have ( β α ) β α cos( β α ) cos( β α ) ( β α ) R cos( β α ) dx sin 1 sin cos = π R sin β π R π R 1 cos β cos β dx sin = 4π R 1 cos β π sin β dr dx3 π R sin β = dr cos ( β α ) dx4 π = cos ( β α ) The aplace s equation [18] of a free spherical liquid drop yields d p pg = + R dr y introducing Eqs. (4-46) into Eq. (31), we get sin = f f + ( β α ) ( β α ) ( g k + g k ) SG1 S1 SG S R sin β cos dk1 dk + g1 g + Putting Eq. (38) into Eq. (47) arrives at ( g k + g k ) cosθ = f + f sin β SG1 S1 SG S R sinθ dk g g + dk 1 1 (4) (43) (44) (45) (46) (47) (48)

8 ong Zhou et al Journal for Foundations and pplications of Physics, vol. 4, No. (017) Substituting Eq. (5) into Eq. (48), we have cosθ = f cosθ + f cosθ 1 1 sinθ dk g g ( g k + g k ) 1 1 dk Rsin β Utilizing Eqs. (36, 38, 41), Eq. (49) may be rewritten as g1k1 + gk cosθ = f1 cosθ1 + f cosθ R dk g + g 1 1 dr dr dk (49) (50) Hence, for the spherical droplet in a smooth and heterogeneous conical cavity, Eq. (50) is the new generalized Cassie-axter equation applicable to arbitrary dividing surface between the liquid and vapor phases. π If we suppose thatα =, then = 1, the conical surfaces convert into flat surfaces, Eq. (50) is the generalized Cassie-axter equation suitable for the wetting of droplets on flat surfaces. Then, if we assume that the cone only contains a kind of solid, namely S1 = S, SG1 = SG, k1 = k (51) Then Eq. (50) decreases to the generalized Young equation (4) established by Rusanov. Subsequently, when we assume that the line tension is constant, Eq. (50) changes to the equation (3) developed by Pethica. Finally, if the influences of the line tension are not considered, Eq. (50) reduces to the familiar Young s equation (1). Moreover, if we neglect the effects of the line tension directly in Eq. (50), then Eq. (50) is simplified as the classical Cassie-axter equation (). 4. Conclusion On the basis of the theories of Gibbs s dividing surface and Rusanov s dividing line, we investigated the wetting characteristics of spherical droplets in a smooth and heterogeneous conical cavity by means of thermodynamics. Taking the influences of the line tension into account, a generalized Cassie-axter equation for the contact angle between spherical droplets and the inner surface of a cone, has been derived in terms of the concept of Gibbs s dividing surface. ased on some hypotheses, this generalized Cassie-axter equation decreases to the Cassie-axter equation suitable for planar surfaces, the Rusanov s equation, the Pethica s equation, the Young s equation along with the classical Cassie-axter equation. 38

9 ong Zhou et al Journal for Foundations and pplications of Physics, vol. 4, No. (017) cknowledgments This work was supported by the Doctor Research Foundation of Henan Polytechnic University (No /00). REFERENCES [1] T Young, n essay on the cohesion of fluids, Philos. Trans. Roy. Soc. ondon, 95, (1805) [] D Cassie and S. axter, Wettability of porous surfaces, Trans. Faraday Soc., 40, (1944) [3].. Pethica, The contact angle equilibrium, Journal of Colloid and Interface Science, 6, (1977) [4] I Rusanov, Effect of contact line roughness on contact angle, Mendeleev Commun, 1, (1996) [5] J. uehrle, S. Herminghaus, and F. Mugele, Impact of line tension on the equilibrium shape of liquid droplets on patterned substrates, angmuir, 18, (00) [6]. I. Rusanov,. K. Shchekin, and D. V. Tatyanenko, The line tension and the generalized Young equation: the choice of dividing surface, Colloids and Surfaces, 50, (004) [7] R. Tadmor, ine energy, line tension and drop size, Surface Science, 60, (008). [8] S. Vafaei, D. S. Wen, and T. T. orca, Nanofluid surface wettability through asymptotic contact angle, angmuir, 7, (011) [9] X. S. Wang, S. W. Cui,. Zhou, S. H. Xu, Z. W. Sun, and R. Z. Zhu, generalized Young s equation for contact angles of droplets on homogeneous and rough substrates, Journal of dhesion Science and Technology, 8, (014) [10] M. Iwamatsu, ine-tension effects on heterogeneous nucleation on a spherical substrate and in a spherical cavity, angmuir, 31, (015) [11]. mirfazli and. W. Neumann, Status of the three-phase line tension, dvances in Colloid and Interface Science, 110, (004) [1] P. S. Swain and R. ipowsky, Contact angles on heterogeneous surfaces: a new look at Cassie s and Wenzel s laws, angmuir, 14, (1998) [13] C. P. Fang and J. Drelich, Theoretical contact angles on a nano-heterogeneous surface composed of parallel apolar and polar strips, angmuir, 0, (004) [14] K.. Towles and N. Dan, Coupling between line tension and domain contact angle in heterogeneous membranes, iochimica et iophysica cta, 1778, (008) [15] R. Raj, R. Enright, Y. Y. Zhu, S. dera, and E. N. Wang, Unified model for contact angle hysteresis on heterogeneous and superhydrophobic surfaces, angmuir, 8, (01) [16] T. W. Kwon, M. S. mbrosia, J. Jang, and M. Y. Ha, Dynamic hydrophobicity of heterogeneous pillared surfaces at the nano-scale, Journal of Mechanical Science and Technology, 9, (015) 39

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