MATHEMATICAL STUDY OF LAPLACE AND YOUNG EQUATIONS IN THE CASE OF THE CONTACT BETWEEN A DROP AND FIBRE
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1 MATHEMATICA STUDY O APACE AND YOUNG EQUATIONS IN THE CASE O THE CONTACT BETWEEN A DROP AND IBRE T. Hmie nd A. Bid Institut de Cimie des Sufces et Intefces I.C.S.I.-C.N.R.S.-UPR 969 5, Rue Jen Stcky - B.P Muouse Cedex - nce Univesité de Hute-Asce,, Rue des èes umièes Muouse Cedex - nce SUMMARY : Tis ppe constitutes mtemtic contiution on te teoy of cpiity in pticu cse of te contct of dop on cyindic fies, y giving new fomution of te pce nd Young equtions in tt cse nd poving te effect of te gvity foce on te ccution of te contct nge of dop on fie nd te sufce tension of te soid.. Te mecnic equiiium conditions wic descie te stte of nce coss te seption sufce o ong te contct ine wee deived s foce nce etions etween pessue, sufce tension nd te geometic spe of te sufce o te contct ine; y using te cssic metod of te gnge mutipies nd te vition teoy. Ou esuts e imited to cpiy systems wic e eite in xisymmetic geomety o wose sufces my e descied y non-pmetic sufce function of te fom u ux,z. In mny pctic cses, te geomety of te soid is typicy xisymmetic. A typic cse of xisymmetic soid, wic ws not mtemticy vey studied in itetue, is tt of iquid dop in xisymmetic position in contct wit oizont o vetic igid cyindic sufce. KEYWORDS : ie, contct nge, temodynmics, pce nd Young equtions, dop, sufce enegy, gnge mutipies. INTRODUCTION Wetting of soid sufce y iquid is of vit impotnce in mny industi pocess, nd moe pticuy te wettiity of fies pesents mjo inteest in mny industi ppictions s dying nd potection of syntetic fies, textie industies, cosmetoogy industy nd composite mtei eotion. Mny mtemtic studies on te contct of iquid on pne soid sufce o contine sufce [-5]. Howeve, we did not find ny seious mtemtic study wit cyindic sufce, ike fie. We popose in tis ppe to study te mtemtic modeing of te contct etween n xisymmetic dop nd cyindic fie in oizont o vetic geometic position, y using te cssic metod of te gnge mutipies nd te vition teoy [5-7].
2 CASE O A HORIZONTA CYINDRICA IBRE In mny pctic cses, te geomety of te soid is typicy xisymmetic. A typic cse of n xisymmetic soid, vey studied nd commented in itetue, is tt of contine ounding iquid - fuid system. Anote cse is tt of iquid dop in xisymmetic position in contct wit oizont igid cyindic sufce, wee te contct ine is geney cice wit specific ine fee enegy,, tt is equ to constnt ine tension, σ figue. u u u s u ig. : Contct etween iquid dop nd cyindic sufce in xisymmetic equiiium. Wee u is sufce cuve septing etween te iquid nd te soid, u s te engt of te cyindic sufce ving dimete equ to. Te iquid dop ving n xisymmetic geomety intesects te -xis in ± nd te u-xis in ± u. We suppose tt tee is no cemic ection etween te iquid nd te soid, nd te contct nge of te wetting iquid is constnt. We peviousy sowed [8] tt te tot enegy is composed y contiutions etive to diffeent egions of uk, nd dividing sufces nd ines. Te most impotnt enegetic contiution fo te system is te intinsic enegy of uk of evey pse ccteized y v P. Anote fom of enegy pesent in te system is te gvittion potenti enegy tt is function of te position into te system. Te enegetic contiution of te exteio fied is otined y integtion ove te tot usefu voume of te system incuding ot iquidfuid oundies nd te tee pse contct ines. iny, if te quntity of te iquid pesent in te system is finite, we ve to considete its mss o its voume s constint intoduced in tis system. In te cse wee te system is incompessie, metod we-know nd vey used is tt of te gnge mutipies, λ, wic mutipies te voume V to fom new tem of enegy. Accoding to te vitu wok pincipe, te tot fee enegy, E t, s to e cnceed fo cetin vition tt s not viote te constints of te system. o n incompessie system, te voume constint is unique. Two mecnic equiiium conditions cn e distinguised : te fist condition is otined f ong te iquid-fuid intefce, A nd te second one is eized ong te tee pse contct ine.
3 ee enegy of te dop If E is te fee enegy of te iquid pse nd te specific fee enegy o te fee enegy density of te iquid, ten E cn e expessed y te foowing integ see figue : Ε u 4π du d wee et u u If we suppose tt te contiution of te intinsic inten enegy of te uk does not depend on tt of te fee enegy of te exten fied, we cn considete tt is independent fom te u-coodinte, ten simpe integtion of eds to : Ε 4 π d u ee enegy of te fuid pse f Te fee enegy Ε of te fuid pse gs o vpo in ou study wi e given y : f f f mx f Ε 4π [ u us ] d 4π u d 4π us d wee mx is te supeio imit of te exten oundy of te system. ee enegies of ine nd sufce egions o soid-iquid intefce : Ε o iquid-fuid intefce s s s s da 4π du A u u 4π d 4 f f f du Ε f da 4π A d d 5 o soid-fuid intefce sf u sf sf s sf us u Ε s da 4π du 4π A u d 6 nd fo te contct ine, we ve : sf sf Ε 4 π d o 7 Te expessions of iquid pse V nd fuid pse V f voumes e espectivey given y : V mx mx d 4π u d 4π us u f 4π u u Effect of te exteio foces Te sme types of pevious expessions wee otined wit te ction of te gvittion fied. As exmpe, we give Ε gvit., coesponding to te contiution of te gvity of te iquid dop, is given y : u g ρ du d s d g u Ε gvit. 4π 4π ρ d 9 wee ρ is te uk density of te iquid pse. 8
4 Study of te vition poem Te tot fee enegy of te system is given y : f f s sf Ε Ε t sf f s f sf sf gvit. gvit. gvit. gvit. gvit. gvit. Te mecnic equiiium conditions wi e otined y using te fct tt te integ of te tot fee enegy is constnt. Te vition poem cn e witten ee s : δ [ Εt λv ] wit V 4π u d wee λ is te gnge mutipie. Te poem cn e so witten s :. mx 4 δ d d d d wee nd depend on, u nd u et u,,,, If we ppy te vition ccuus to te fist integ, we otin : u : d u d u we otin : f f f f f ρ g ρ g ρ ρ g λ 4 f f f R R R Wee te pincip dii of cuvtue of te iquid-fuid intefce e given y : u nd f 5 R f R Te oundy conditions in ow to otin te pce eqution : pce eqution Ccutions ed to pce eqution : f f w P f f 6 R R Te pessue vition is given y : f f f P g ρ ρ g f 7 R nd f f f w ρ g 8 Eqution 6 epesents te pce eqution in te cse of iquid dop in xisymmetic geomety on oizont cyindic fie. Young eqution Te tnsvesity condition ppied t te tee pse contct point, ows to otin te Young eqution.
5 f f w sin θ ρ g u s sf sf w w w 9 u wee w i i ρ i g wee i { f, sf, sf, sf, f } Eqution 9 epesents te geneized Young eqution in te cse of dop suounding oizont cyindic sufce. Pticu cses - If te effect of te gvittion is negected, ten eqution 9 ecomes : sf s sf f sin θ u u - utemoe, if we suppose tt te soid sufce is pne,, we otin : sf s sf f f u - If we negect te tee pse tension ine, ten eqution ecomes: sf s cos θ f Eqution is excty te cssic Young eqution. CASE O A VERTICA CYINDRICA IBRE If te position of te pevious cyindic sufce is vetic, we otin te foowing esuts : pce eqution f f w u P f f R R f f f f P ρ g ρ ρ g u f 4 R Eqution de Young Tnsvesity condition eds to : f f w u sin θ ρ g u f s sf sf ρ ρ w w w gu 5 u
6 Pticu cses - If te effect of te gvittion of te contct sufces nd ines is negected, we wi ve : f sin θ u - In te cse of pne sufce, we otin : ρ f sf s sf ρ gu u 6 sf s f sf ρ ρ f gu f 7 u - If we negect te tee pse tension ine, we otin : sf s f ρ ρ gu f 8 If we compe te two cses of oizont nd vetic cyindic sufces, we oseve tt in te second cse vetic one tee is n impotnt infuence of te gvittion of te iquid nd of te soid on te contct nge of te dop on te soid. EQUIIBRIUM O A SOID-IQUID -IQUID -UID SYSTEM Anote cse ws studied is tt of te contct gz-iquid-iquid-cyindic sufce ig.. Tee sufces cn e distinguised : soid, iquid nd iquid oi, fo exmpe in pesence of fuid f gs. We so distinguis etween fou sufce intefces : soid-fuid sf, soid-iquid s, iquid-oi nd oi-fuid f, nd two contct ines : soid-iquid-fuid sf nd iquid-oi-fuid f. We studied te cse wee oi nd iquid e in contct wit te fie. iquid sufce e ccteized y u nd oi sufce y v. v c u u u s v u ig.. Contct etween, iquid, iquid oi nd cyindic fie
7 Vition poem Te tot fee enegy is given y te foowing expession : Ε t Ε f s s sf f s sf gvit. f s s sf f s sf gvit. gvit. gvit. gvit. gvit. gvit. gvit. gvit. gvit. 9 Te vition poem wi e witten s : [ Ε λ V λ V ] δ t Ccutions give : mx c c δ d d d d d d wit [ f ] u λ w s w s s [ u w w ] u, u w w [ ], λ f f [ ] v w v v, v w w [ ], λ u, u w [ ] v sf s sf w w w [ ] [ ][ ] u u w u, λ 4 f [ ] f v vw w v u [ w ] u f v, v w w 4 [ ], λ 5 5, λ 6 f w u s 6 7 Te oundy conditions e given y : u u et u, ou. 8 v u v et v c, v c ou. v c 9 Vition ccuus ppied espectivey to, nd to ow to otin : w ρ g w w λ λ R R R 4 f f f w ρ g w w λ f f f R R R 4 w ρ g w λ R R R 4
8 pce equtions Appying te oundy conditions in c nd espectivey to te equtions 4 nd 4, we otin : ρ g w w λ 44 f w c f f λ ρ g w c w c c 45 nd pce equtions wi e given y : w P R R 46 f f w P f f R R 47 wee P ρ g ρ R 48 f f f f P ρ g ρ ρ c f R c 49 Young equtions Using te foowing tnsvesity conditions u en en 5 we otin te fist Young eqution f f w sin θ ρ g u s sf sf w w w 5 u nd te second Young eqution : w H H v sin θ H ρ g H v s sf sf w w w 5 v f f wee θ H is te contct nge of te oi dop wit te fie.
9 Pticu cses - In te cse of pne sufce, we ve : sf s sf f f u 5 sf s sf H f f v 54 - If u v, i.e., te two tee pse tension ines s nd sf e te sme, te nge β etween te two intefces iquid-oi nd oi-fuid in te point, u wi e given y β θ H - θ. CONCUSIONS In tis study, we gve new mtemtic fomution of te pce nd Young equtions in te tee foowing cses : - contct etween iquid dop nd oizont cyindic sufce - contct of te dop wit vetic cyindic sufce - contct etween two iquids nd oizont cyindic sufce. REERENCES. J.W. Gis, Coected woks of J.W. Gis, Vo., Ye Univesity Pess, New Hven, Conn., 98.. J.W. Gis, Te scientific ppes of J.W. Gis, Vo., Dove, New Yok, 96.. I. Mue, Temodynmics, Pitmn dvnced puising pogm, ondon, C. Woods, Te temodynmics of fuid systems, Cendon pess, oxfod, A.W. Neumnn nd J.K. Spet, Appied sufce temodynmics, Mce Dekke, New Yok, J.C.C. Nitsce, ectues on minim sufces, Vo., Cmidge Univesity Pess, New yok, M.M. ipscutz, Scum's Outine seies : Teoy nd poems of diffeenti geomety, Mc Gw-Hi, Toonto, T. Hmie nd A. Bid, New mtemtic fomution of pce nd Young eqution in te cse of te contct etween dop nd cyindic fie, Intention Confeence n Advnced Composites ICAC 98, Hugd, Egypt, Deceme 5-8, 998, pp. 9-.
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