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

Transcription

1 Mechanical Engineering Reearch; Vol 5, No ; 015 ISSN E-ISSN Publihed by Canadian Center of Science and Education Derivation of eneralized Young Euation for Wetting of Cylindrical Droplet on Rough Solid Surface Xiao-Song Wang 1 1 Intitute of Mechanical and Power Engineering, Henan Polytechnic Univerity, No 001, Century Avenue, Jiaozuo, Henan , China Correpondence: Xiao-Song Wang, Intitute of Mechanical and Power Engineering, Henan Polytechnic Univerity, No 001, Century Avenue, Jiaozuo, Henan , China el: wangx016@163com Received: June 7, 015 Accepted: July 4, 015 Online Publihed: Augut 7, 015 doi:105539/merv5np17 UR: Abtract he urface tenion depend on the radiu of curvature of the liuid-vapor interface or nano-cale wetting phenomena of cylindrical droplet, we hould conider the curvature effect of the urface tenion and the line tenion However, previou work have not analyzed the influence of the curvature effect of the urface tenion In thi paper, we dicu the influence of the curvature effect of the urface tenion on the contact angle baed the Kim-ee-Han-Park euation he hydrophilic wetting of cylindrical droplet on rough and chemically homogeneou non-deformable ubtrate were tudied by method of thermodynamic A generalized Young euation for wetting of cylindrical droplet on chemically homogeneou and rough non-deformable ubtrate wa derived baed on the thermodynamic euilibrium condition hi euation reduce to the Wenzel euation if we ignore the influence of line tenion or contact angle of cylindrical droplet with ufficiently large radii, a generalized Young euation were derived conidering the curvature effect of the urface tenion Keyword: contact angle, wetting, Young euation, Wenzel euation, urface tenion, curvature effect, cylindrical droplet 1 Introduction Wetting phenomena are common in olid-liuid-ga ytem, for intance, wetting of liuid droplet on olid urface, adheive, lubricant and capillary penetration in to porou media (Adamon, 1990; enne, Brochard-Wyart, & Quere, 004) Wetting abilitie are important in many indutrial application, for example, the wetting abilitie of electrolyte on electrode play a key role in improving the pecific energy denity of upercapacitor (Kim, Koo, ee, & Braun, 014) and lithium-ion batterie (Pfleging & Proella, 014) In 1805, homa Young argued that the contact angle for the wetting of pherical droplet on rough and Y chemically homogeneou ubtrate i determined by the following euation (Young, 1805) - S co = (11) Y where i the contact angle, i the urface tenion of the liuid vapor interface correpond to the choice of Y the urface of tenion a a dividing urface, i the urface free energy per unit area of the olid vapor interface, i the urface free energy per unit area of the olid liuid interface S Now, Euation (11) i called the Young euation he Young euation Euation (11) i widely applied to macrocopic capillary phenomena (Pfleging & Proella, 014; Xiao-Song et al, 014) In 1878, ibb for the firt time gave a theoretical derivation of the Young euation Euation (11) baed on the theory of thermodynamic (ibb, 198) Since thenany theoretical reearch work have been carried out (Pfleging & Proella, 014) he urface tenion depend on the radiu of curvature of the liuid-vapor interface or nano-cale wetting phenomena of cylindrical droplet, we hould conider the curvature effect of the urface tenion and the line tenion However, previou work have not analyzed the influence of the curvature effect of the urface tenion 17

2 wwwccenetorg/mer Mechanical Engineering Reearch Vol 5, No ; 015 he purpoe of thi paper i to preent a theoretical tudy the contact angle of cylindrical droplet on hydrophilic ubtrate and dicu the influence of the curvature effect of the urface tenion on the contact angle he Helmholtz ree Energy for Wetting of a Cylindrical Droplet on Rough Subtrate Conider a ingle-component cylindrical liuid droplet in contact with chemically homogeneou and rough ubtrate If the contact angle le than 90, then we ay that the olid urface i hydrophilic If the contact angle larger than 90, then we ay that the olid urface i hydrophobic We only conider the wetting of a cylindrical droplet on hydrophilic olid urface An illutration of hydrophilic wetting i hown in igure 1 Introducing ibb concept of dividing urface and the concept of dividing line (ibb, 198; Ono & Kondo, 1960; Rowlinon & Widom, 198), the above olid-liuid-vapor ytem can be divided into ix ubytem, ie liuid phae, vapor phae, the liuid-vapor interface, the olid-liuid interface, the olid-vapor interface and the three-phae contact line herefore, the total Helmholtz free energy of the ytem i the um of the Helmholtz free energie of thee even part hu, we have igure 1 an illutration of hydrophilic wetting of a cylindrical droplet on a rough ubtrate = S + + S (1) where i the total Helmholtz free energy,,,, S, and S are the Helmholtz free energie of the even part repectively he Helmholtz free energie of thee even part can be written a = -pv + mn () = -pv + mn (3) = A + m N (4) S = SA S + msns (5) = A + mn (6) S = ks + ms NS (7) where p and p are the preure of the liuid phae and the vapor phae repectively, V and V are the volume of the liuid phae and the vapor phae repectively i the chemical potential of the ix ubytem, N, N, N, NS, Nand NS are the mole number of molecule of the liuid phae, the vapor phae, the liuid vapor interface, the olid liuid interface, the olid vapor interface and the three phae contact line repectively, A, AS and A are the urface area of the liuid vapor interface, the olid liuid interface, the olid vapor interface repectively,, S and are the urface tenion of the liuid vapor interface, the 18

3 wwwccenetorg/mer Mechanical Engineering Reearch Vol 5, No ; 015 olid liuid interface, the olid vapor interface repectively, contact line, k i the line tenion S i the value of the length of the three phae In order to calculate the geometrical uantitie in the above euation, we may introduce the following aumption: Aumption 1: Suppoe the euilibrium hape of a droplet on a rough and homogeneou olid ubtrate i a part of a cylinder which wa cut by a plane parallel to the axi of the cylinder Baed on Aumption 1, the total Helmholtz free energy of the ytem i = -(p - p ) ( - in co ) R - pv t+ mn + mn + R+ mn S + mn (8) + r ( S - )Rin + ra t + mn + r k+ mn, S where R i the radiu of the cylindrical liuid droplet, i the contact angle, i the length of the cylindrical liuid droplet, At i the total urface area, r i the urface roughne factor, r i the line roughne factor 3 eneralized Young Euation baed on hermodynamic Euilibrium Condition he purpoe of thi ection i to derive a generalized Young euation for cylindrical droplet on rough and homogeneou olid by method of thermodynamic According to ibb concept of dividing urface (Ono & Kondo, 1960), we can chooe an arbitrary conformal urface a a dividing urface Now, we uppoe that the radiu R of the dividing urface ha already been choen according to ome fixed condition he contact angle now become variable he thermodynamic euilibrium condition at a fixed temperature of an open ytem i (Nijmeijer, Bruin, Woerkom, & Bakker, 199) ç = 0, (3) ø where the ubcript and m tand for fixed temperature and fixed chemical potential m It i convenient to introduce the concept of grand potential to treat an open ytem he definition of the grand potential of a ytem i (Rowlinon & Widom, 198) i=1 i i i t W= å ( -m N ), (3) where t i the number of ubytem of the ytem, i i the Helmholtz free energy of the i th ubytem i i the chemical potential of the the i-th ubytem, N i i the mole number of molecule of the the i-th ubytem Putting Euation (3) into Euation (31), the thermodynamic euilibrium condition become (Nijmeijer, Bruin, Woerkom, & Bakker, 199) W ç = 0 (33) ø Putting Euation (8) into Euation (3), the total grand potential W of the above ytem i W=-(p - p ) ( -in co ) - p V + R t + r ( - )Rin + r S t A +r k R (34) Putting Euation (34) into Euation (33), we have 19

4 wwwccenetorg/mer Mechanical Engineering Reearch Vol 5, No ; 015 (p - p ) -(p - p ) + 1 ç f1 - ç ø è ø ç f + ç ø è ø ( - ) + r ( - ) ç + f ç + r 3 S S ç 3r ç ø è ø, m 4 ç f4r + ç ø è ø æ 5 k + rk ç + f5r ç = 0, ø ø (35) where f1 = ( - in co ) R, (36) f = R, (37) f3 = Rin, (38) f 4 = At, (39) f5 = (310) According to ibb concept of dividing urface (ibb, 1961; Rowlinon & Widom, 198), we can chooe an arbitrary conformal urface a a dividing urface Now, the radiu R of the dividing urface ha already been choen according to ome fixed condition At the fixed temperature and fixed chemical potential, the preure p, p and urface tenion will not influence the contact angle hu, in order to implify Euation (35), we introduce the following aumption Aumption : Suppoe the following euation are valid for the wetting of cylindrical droplet on rough and chemically homogeneou non deformable ubtrate ( p -p ) ç 0, = è ø (311) S ç 0, = è ø (31) ç 0, = è ø (313) ç 0 = è ø (314) We have the following reult 1 ç = Rin, ø (315) ç = R, ø (316) 3 ç = R co, ø (317) 4 ç = 0, ø (318) 5 ç ø = 0, (319) 0

5 wwwccenetorg/mer Mechanical Engineering Reearch Vol 5, No ; 015 Putting Euation ( ) into Euation (35), we obtain k - (p - p ) Rin + + r( S - )co + ç = 0 (30) R ø, mi It i known that a generalized aplace euation of a free cylindrical droplet in vapor can be written a (Wenzel, 1936) é d ù p - p = +, (31) R ê ë dr ú û Where the differential in uare bracket[ ] denote the change reulted from a mathematical variation of the poition of thi dividing urface by the amount dr in the ame phyical ytem under the ame fixed phyical tate Applying Euation (31), Euation (30) become - in tan d S R é ù co = r + ê dr ë úû (3) k - ç Rco ø Euation (3) i a generalized Young euation for wetting of a cylindrical droplet on chemically homogeneou and rough non deformable ubtrate Euation (43) i the main reult of thi work ollowing ibb (Kim, ee, Han, & Park, 006; Nijmeijer, Bruin, Woerkom, & Bakker, 199), we introduce the concept of urface of tenion M a follow, mi éd ù êë dr úû ê ú = R= R 0, (33) Where R i the radiu of the urface of tenion If we chooe the urface of tenion M M a the dividing urface, then Euation (43) become - S k co = r - ç Rco ø, mi (34) Euation (34) i a ueful generalization of the Young euation Euation (11) for wetting of cylindrical droplet If we neglect the econd term on the right ide of Euation (34), we have S co r - = (35) Euation (35) i the Wenzel euation (Wenzel, 1936) 4 Brief Review of Curvature Effect of Surface enion of Cylindrical Droplet In 006, Kim, ee, Han, and Park (006) obtained the following euation for the curvature effect of urface tenion of cylindrical droplet d æ d 1 1 d ç + ø, d d ( R ) R R = è dr 1+ + R R (41) where ( R) i the urface tenion of a cylindrical droplet with radiu R, R i the radiu of ibb urface of tenion, d º Re - R, R e i the radiu correponding to the euimolar dividing urface 3 3 or ufficiently large droplet, we neglect d / R and d / R, and treat d a the contant d, Euation (41) can be olved he reult i ( R ) æ d = exp, ç - R è ø (4) 1

6 wwwccenetorg/mer Mechanical Engineering Reearch Vol 5, No ; 015 where i the urface tenion of plane urface with radiu R =, d º lim d R d i uually called the olman length in the literature (Rowlinon & Widom, 198) We have the following aylor expanion 3 x x exp( - x) = 1 - x+ - + (43)! 3! Applying Euation (43), Euation (4) can be expanded a ( R ) d d = (44) R R 5 eneralized Young Euation Conidering the Curvature Effect of the Surface enion Uing Euation (4), Euation (34) can be written a æ d k exp ç - co r( S) R = - - ç è ø Rco ø (51) Euation (51) i a generalized Young euation for pherical droplet on rough but chemically heterogeneou non-deformable ubtrate conidering the curvature effect of the urface tenion Euation (51) i one of the main reult of thi work Applying Euation (43), Euation (51) become æ 1 d d co r ( S) k ç R R = - - ç è ø Rco ø (5) According to experiment (Adamon, 1990), the curvature effect of the urface tenion of macrocopic cylindrical droplet can be neglected However, for nano-cale cylindrical droplet, the curvature effect of the urface tenion are important (Wenzel, 1936; Rowlinon & Widom, 198; Adamon, 1990; Nijmeijer, Bruin, Woerkom, & Bakker, 199; enne, Brochard-Wyart, & Quere, 004;) or droplet with ufficiently large radii, if we neglect d and higher term in Euation (5), then we have 6 Concluion R æ d k 1 ç - co r( S) R = - - ç è ø Rco ø he hydrophilic wetting of cylindrical droplet on rough and chemically homogeneou non-deformable ubtrate were tudied by method of thermodynamic A generalized Young euation for wetting of cylindrical droplet on chemically homogeneou and rough non-deformable ubtrate wa derived baed on the thermodynamic euilibrium condition hi euation reduce to the Wenzel euation if we ignore the influence of line tenion It i known that the urface tenion depend on the radiu of curvature of the liuid-vapor interface or nano-cale wetting phenomena, we hould conider the curvature effect of the urface tenion and the line tenion However, previou work have not analyzed the influence of the curvature effect of the urface tenion Baed the Kim-ee-Han-Park euation, we dicu the influence of the curvature effect of the urface tenion on the contact angle or contact angle of cylindrical droplet with ufficiently large radii, a generalized Young euation were derived conidering the curvature effect of the urface tenion Acknowledgment hi work wa upported by the Key Project of Science of the Education Bureau of Henan Province (rant No 15A130001) and the Doctor Reearch oundation of Henan Polytechnic Univerity (rant No ) Reference Adamon, A W (1990) Phyical Chemitry of Surface New York: John Wiley Son enne, P d, Brochard-Wyart,, & Quere, D (004) Capillarity and wetting phenomena: drop, bubble, pearl wave New York: Springer-Verlag (53)

7 wwwccenetorg/mer Mechanical Engineering Reearch Vol 5, No ; 015 ibb, J W (198) he Collected Work of J Willard ibb (vol 1, hermodynamic) New Haven: Yale Univ Pre ibb, J W (1961) he Scientific Paper of J W ibb (Vol 1) New York: Dover Kim, B, ee, J S, Han, M, & Park, S (006) A molecular dynamic tudy on tability and thermophyical propertie of nanocale liuid thread Nanocale and Microcale hermophyical Engineering 10, 83 Kim, S K, Koo, H J, ee, A, & Braun, P V (014) Selective Wetting-Induced Micro-Electrode Patterning for lexible Micro-Supercapacitor Advanced Material, 6, Nijmeijer, M J P, Bruin, C, Woerkom, A B V, & Bakker, A (199) Molecular dynamic of the urface tenion of a drop J Phy Chem, 96, Ono, S, & Kondo, S (1960) Molecular heory of Surface enion in iuid In S lugge (Ed), Encyclopedia of Phyic (volume 10) Berlin: Springer-Verlag Pfleging, W, & Proella, J (014) A new approach for rapid electrolyte wetting in tape cat electrode for lithium-ion batterie Journal of Material Chemitry A,, Rowlinon, J S, & Widom, B (198) Molecular heory of Capillarity Oxford: Clarendon Pre Wenzel, R N (1936) Reitance of olid urface to wetting by water Indutrial & Engineering Chemitry, 8(8), Xiao-Song, W, Shu-Wen, C, ong, Z, Sheng-Hua, X, Zhi-Wei, S, & Ru-Zeng, Z (014) A eneralized Young' Euation for Contact Angle of Droplet on Homogeneou and Rough Subtrate J Adheion Sci ech, 8, Young, (1805) An eay on the coheion of fluid Philo ran Roy Soc ondon, 65 Copyright Copyright for thi article i retained by the author(), with firt publication right granted to the journal hi i an open-acce article ditributed under the term and condition of the Creative Common Attribution licene ( 3