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

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1 Mechanical Engineering Reearch; Vol. 6o. 1; 16 ISSN E-ISSN Publihed by Canadian Center of Science and Education Contact Angle for Spherical Nanodroplet in Cylindrical Cavity with Quadratic Curve eneratrix Ai-Jun Hu 1 & Bao-Zhan v 1 1 School of Mechanical and Power Engineering, Henan Polytechnic Univerity, Henan 454, China Correpondence: Ai-Jun Hu, School of Mechanical and Power Engineering, Henan Polytechnic Univerity, Henan 454, China. ajhu17@16.com Received: April 1, 16 Accepted: April 6, 16 Online Publihed: May 17, 16 doi:1.559/mer.v6n1p96 UR: Abtract Wetting of a pherical nanodroplet in mooth and homogeneou cylinder urface rotated by quadratic curve wa tudied by method of thermodynamic. The olid-liquid-vapor ytem wa eparated into ix part uing ibb method of dividing urface. The ytem free energy wa calculated. A generalied Young equation for the equilibrium contact angle i propoed taking the line tenion effect into conideration. On the bai of ome aumption, thi generalied Young equation i the ame a the claical Young equation. Keyword: thermodynamic; nanodroplet; line tenion; quadratic curve; contact angle; cylinder 1. Introduction The wetting of olid urface i very familiar in variou natural and technological device (Quéré, de enne, Brochard-Wyart, & Reiinger, 4; Prabhu, Fernade, & Kumar, 9; Waghmare & Mitra, 1; Promraka & Chen, 1; Raufate & Cox, 1; Snoeyink, Barman, & Chritopher, 15). The contact angle for a liquid nanodroplet on a ubtrate urface i uually ued to decribe the wetting characteritic. The Young equation predicted the equilibrium contact angle (Young, 185) - coqy (1) where q Y i the contact angle,,, repreent the thermodynamic urface tenion of olid-vapor interface, olid-liquid interface and liquid-vapor interface, repectively. Equation (1) i applicable for a liquid nanodroplet on an ideal mooth and homogeneou olid ubtrate urface. But, it didn t taking the influence of the line tenion into conideration. ibb firtly preented the line tenion a a urface thermodynamic concept (ibb, 1961). ibb believed the triple phae contact line bear ignificant role in triple phae ytem. From that time, the line tenion impact on the contact angle were tudied by many reearcher. The ignificance of the line tenion in three-phae ytem wa dicued in relation to contact angle meaurement by Jarolaw (Drelich, 1996). A.Amirfali found the drop ie dependence of contact angle for high-energy ytem yield a poitive line tenion (Amirfali, Chatain, & Neumann, 1998). On the bai of the Irving-Kirkwood tre tenor of tatitical mechanic, line tenion at a curved edge of a olid i direct calculated by Anatoly (Ruanov & Brodkaya, 14). Although, the thermodynamic line tenion till remain diputable. It i thought the line tenion i reponible for the many wetting phenomena. When taking the line tenion effect into conideration, the Young equation hould be modified. For a nanodroplet in an inclined olid cone urface of revolution, Dongqing i preented the following equation of equilibrium contact angle (i, 1996) k co b coq coqy - () R where R i the radiu of the triple phae contact line, k i the correponding line tenion, b i the angle of inclination of the olid urface at the triple phae contact line. In Equation (), Dongqing i uppoed the line tenion i contant and the cone urface hape from the rotation by line. When a nanodroplet within the cone urface of rotation by a nonlinear curve, the contact angle hould be different from the Equation (). 96

2 Mechanical Engineering Reearch Vol. 6o. 1; 16 According to the practical ituation, the line tenion can be variable. Ruanov etablihed a modified Young equation (Ruanov, Shchekin, & Tatyanenko, 4) utiliing ibb method of dividing urface k 1 é dk coq coq Y - - () R ê ëdr ú û where the line tenion derivative é dk ê ëdr ú û expre the variation of the k due to the variation of the radiu R of the contact line. Equation () i uitable for the flat urface. For a nanodroplet on curved urface, the Young equation hould be different with Equation (). In previou reference, wetting of nanodroplet in cavity urface rotated by a quadratic curve ha eldom been tudied. In thi work, taking the line tenion effect into conideration, wetting of a pherical nanodroplet in a mooth and homogeneou cylindrical cavity rotated by a quadratic curve i tudied. A generalied Young equation of contact angle for nanodroplet in a mooth and homogeneou cylinder cavity urface rotated by quadratic curve i obtained utiliing the method of dividing urface of ibb baed on thermodynamic.. Calculation of the total Helmholt free energy x In thi tudy, we aume the cylindrical cavity i created by uing the quadratic function - 1 a a generatrix to rotate around axi. The illutration of the wetting i hown in Figure 1, in which the droplet wa cut by a plane parallel to the axi. b i the angle between the ubtrate urface tangent and the principal plane of the three-phae contact line. a i the angle between the liquid nanodroplet urface tangent and the local principal plane of contact line. In thi tudy, the nanodroplet gravity i neglected. The equilibrium hape of nanodroplet in cylindrical cavity above the principal plane of the three-phae contact line bear the hape of a pherical part. Figure 1. Wetting of a pherical nanodroplet within a mooth and homogeneou cylinder olid rotated by x a - around axi b 1 On the bai of the method of ibb dividing urface, we eparated the olid-liquid-vapor ytem into ix portion, liquid phae, vapor phae, olid-liquid interface, olid-vapor interface, liquid-vapor interface, and the three phae contact line. The whole Helmholt free energy F of the ytem i decribed by the following expreion F F + F + F + F + F + F (4) where F, F, F, F, F and F indicate the free energie of ix portion, repectively. We have the free energie of ix portion (Young, 185; Rowlinon & Widom, 1) F - pv + m N (5) 97

3 Mechanical Engineering Reearch Vol. 6o. 1; 16 F - pv + mn (6) F A + m N (7) F A + m N (8) F A + mn (9) F k + m N (1) where p and V denote the preure and volume of liquid phae, repectively. p and V denote the preure and volume of vapor phae, repectively. m denote the chemical potential of liquid phae, vapor phae, liquid-vapor interface, olid-liquid interface, olid-vapor interface and the three-phae contact line, repectively. N denote the mole number of molecule, repectively. A, A, A denote the urface area of liquid-vapor interface, olid-liquid interface and olid-vapor interface, repectively. k i the line tenion. i the length of the triple phae contact line. The volume V of liquid phae i H p V ò p x + R ( + co a)(1 -co a) (11) where R and H denote the radiu of pherical cap of the liquid and height from local principal plane of the contact line to the bottom of the cylinder. The whole volume V t of the ytem i given by Vt V + V (1) The urface area A of the liquid-vapor interface ha the form A p R (1- co a ) (1) The urface area A of the olid-liquid interface yield H 1 ( ) A ò p x + x (14) The entire urface area A t of the olid-liquid and olid-vapor interface i At A + A (15) The length of the three-phae contact line can be written a p Rina (16) According to above equation, we have the following equation of the free energy é H p F - p px + R ( + co a)(1 - co a) + m N (17) ìï é H p ü ï F -p ívt - px + R ( + co a)(1 - co a) ý+ m N ïî ïþ (18) H 1 ( ) F é px x ê + ú+ m N ëò (19) û H F { At - é px x } ê + ú + m N ëò () û F pr (1- co a) + mn (1) F pkrina + m N () Subtituting the Equation (17-) into Equation (4), we obtained the whole Helmholt free energy in the following from é H p F -( p - p) ò px + R ( + co a)(1 -co a) H - pv t + pr(1- co a) + ( - ) ò px 1 + ( x) () + At + pkrina + mn + m N + m N + m N + m N + m N. Derivation of eneralied Young Equation The grand thermodynamic potential W of the olid-liquid-vapor ytem i 98

4 Mechanical Engineering Reearch Vol. 6o. 1; 16 W F -å m N (4) where the mark i denote the amount of ubytem of the ytem. Putting Equation () into Equation (4), we have the following expreion é H p W-( p - p) ò px + R ( + co a)(1 -co a) H - pv t + pr(1- co a) + ( - ) ò px 1 + ( x) (5) + At + pkrina The grand thermodynamic potential W, the urface tenion and do not depend upon the notional variation of the radiu R of the nanodroplet, the following contraint can be obtained (Ruanov, Shchekin, & Tatyanenko, 4) édw (6) dr éd é d, (7) dr dr Subtituting Equation (5) into Equation (6), we have i édv é d é da é da -( p - p) + A + + ( - ) dr dr dr dr édk éd + + k dr dr The dividing urface poition of liquid-vapor interface of a pherical liquid nanodroplet in cylindrical cavity hould be part of concentric and conformal pherical urface. So, we have the following expreion H - Rcoa cont ( Rin a ) H - 1 (9) R Rina () and 4 da co a ( abrin a) - -b R(in a) dr 4 Rin a ( abrin a) - + b R inacoa (1) dh b R dr 4 ( abrin a) - + b R coa () dr co b dr in( a + b ) () According to Eq.(11, 1, 14, 16) and Eq.(1-), we obtained the following equation édv p R (1-co a ) dr (4) éda p Rina co( a + b) 4 pr(1- co a) + dr ú û in( a + b) (5) éda p Rina dr ú û in( a + b ) (6) éd p cob dr ú û in( a + b ) (7) On the bai of the well-known aplace equation (Rowlinon & Widom, 1; Ono, Kondo, & Flügge, 196) of a free pherical liquid droplet in vapor, the following equation i obtained i i (8) 99

5 Mechanical Engineering Reearch Vol. 6o. 1; 16 éd p - p + (8) R ê ë dr ú û It can be ued for the nanodroplet in thi tudy. In order to implify calculation, we tudy only the cae that i greater than ero. So, according to the illutration in Figure 1, the following relation were obtained tan b br ina ( abr in a) - 4 q a + b Subtituting Eq.(4-8) into Equation (8) and uing Equation (9), we have the following equation - kco b inq édk coq - - ê Rina ê ëdr Uing Eq.(), Equation (4) can be rewritten a 4 4 k ar -a 1 ar -a é dk coq coqy R ( a + b ) R - a ( a + b ) R -a ê ëdr ú û (9) (4) (41) Equation (41) i the generalied Young equation for the contact angle of the pherical nanodroplet in cylindrical x cavity rotated by quadratic function - 1. It i valid for any dividing urface between liquid and vapor phae. 4 ar-a If b i very very mall, then tend to one, the generalied Young equation Equation (41) i 4 ( a + b ) R -a the ame a equation () etablihed by Ruanov. If line tenion effect are negligible, Equation (41) change to the claical Young equation (1). 4. Concluion x In thi work, the wetting of a pherical nanodroplet in cylindrical cavity rotated by quadratic function - 1 i invetigated on the bai of ibb method of dividing urface. Conidering the effect of line tenion, a generalied Young equation for the contact angle of a pherical nanodroplet in cylindrical cavity rotated by quadratic function i propoed. Thi generalied Young equation change to the Ruanov equation and the claical Young equation under certain aumption. Acknowledgment Thi work wa upported by the Project of Natural Science of the Education of Henan Province (rant No. 11A466), and the Doctor Reearch Foundation of Henan Polytechnic Univerity (rant No. B9-16). Reference Amirfali, A., Chatain, D., & Neumann, A. W. (1998). Drop ie dependence of contact angle for liquid tin on ilica urface: line tenion and it correlation with olid liquid interfacial tenion. Colloid and Surface A: Phyicochemical and Engineering Apect, 14(), Drelich, J. (1996). The ignificance and magnitude of the line tenion in three-phae (olid-liquid-fluid) ytem. Colloid and Surface A: Phyicochemical and Engineering Apect, 116(1), ibb, J. W. (1961). The cientific paper of J. Willard ibb (Vol. 1). New York: Dover. i, D. (1996). Drop ie dependence of contact angle and line tenion of olid-liquid ytem. Colloid and Surface A: Phyicochemical and Engineering Apect, 116(1), 1-. Ono, S., Kondo, S., & Flügge, S. (196). Encyclopedia of Phyic. Molecular Theory of Surface Tenion in iquid, Springer, Berlin, 14. Prabhu, K. N., Fernade, P., & Kumar,. (9). Effect of ubtrate urface roughne on wetting behaviour of vegetable oil. Material & Deign, (),

6 Mechanical Engineering Reearch Vol. 6o. 1; 16 Promraka, A., & Chen,. J. (1). Modeling contact angle hyterei of a liquid droplet itting on a coine wave-like pattern urface. Journal of colloid and interface cience, 84(1), Quéré, D., de enne, P.., Brochard-Wyart, F., & Reiinger, A. (4). Capillarity and Wetting Phenomena: Drop, Bubble, Pearl, Wave. New York: Springer-Verlag. Raufate, C., & Cox, S. (1). Deformation of a free interface pierced by a tilted cylinder: variation of the contact angle. Colloid and Surface A: Phyicochemical and Engineering Apect, 48, Rowlinon, J. S., & Widom, B. (1). Molecular theory of capillarity. Courier Corporation. Ruanov, A. I., & Brodkaya, E. N. (14). ine tenion at curved edge of a molecular olid. Colloid and Surface A: Phyicochemical and Engineering Apect, 448, Ruanov, A. I., Shchekin, A. K., & Tatyanenko, D. V. (4). The line tenion and the generalied Young equation: the choice of dividing urface. Colloid and Surface A: Phyicochemical and Engineering Apect, 5(1), Snoeyink, C., Barman, S., & Chritopher,. F. (15). Contact Angle Ditribution of Particle at Fluid Interface. angmuir, 1(), Waghmare, P. R., & Mitra, S. K. (1). Contact angle hyterei of microbead upenion. angmuir, 6(), Young, T. (185). An eay on the coheion of fluid. Philoophical Tranaction of the Royal Society of ondon, 95, Copyright Copyright for thi article i retained by the author(), with firt publication right granted to the journal. Thi i an open-acce article ditributed under the term and condition of the Creative Common Attribution licene ( 11