Journal of Colloid and Interfae Siene 338 (9) 93 Contents lists available at SieneDiret Journal of Colloid and Interfae Siene www.elsevier.om/loate/jis Contat angle of a emisperial bubble: An analytial approa M.A.C. Teixeira a, P.I.C. Teixeira b,, * a University of Lisbon, CGUL, IDL, Edifíio C8, Campo Grande, P-749-6 Lisbon, Portugal b Instituto Superior de Engenaria de Lisboa, ua Conseleiro Emídio Navarro, P-95-6 Lisbon, Portugal Centro de Físia Teória e Computaional, University of Lisbon, Avenida Professor Gama Pinto, P-649-3 Lisbon, Portugal artile info abstrat Artile istory: eeived 4 Mar 9 Aepted 7 May 9 Available online June 9 Tis paper is dediated to te memory of M.A. Fortes, wo initiated te work, but did not live to see its results Keywords: Foams and emulsions Contat lines We ave alulated te equilibrium sape of te axially symmetri Plateau border along wi a sperial bubble ontats a flat wall, by analytially integrating Laplae s equation in te presene of gravity, in te limit of small Plateau border sies. Tis metod as te advantage tat it provides losed-form expressions for te positions and orientations of te Plateau border surfaes. esults are in very good overall agreement wit tose obtained from a numerial solution proedure, and are onsistent wit experimental data. In partiular we find tat te effet of gravity on Plateau border sape is relatively small for typial bubble sies, leading to a widening of te Plateau border for sessile bubbles and to a narrowing for pendant bubbles. Te ontat angle of te bubble is found to depend even more weakly on gravity. Ó 9 Elsevier In. All rigts reserved.. Introdution A liquid foam is an assembly of gas bubbles bounded by liquid films. Foams are enountered in many pratial appliations su as beverages, toiletries, leaning produts, fire figting, oil reovery, mixture frationation, te manufature of ellular materials, and ore purifiation by flotation []. Te beaviour of a foam wit a low-visosity liquid pase (e.g., an aqueous foam or a metal foam, as opposed to a polymeri foam) is dominated by surfae tension. Su foams tus serve as models for systems in wi te interfaial area (in tree dimensions (3D)) or te perimeter (in two dimensions (D)) is minimised at equilibrium. In te limit of a perfetly dry foam, su as may be obtained after drainage of most of its liquid ontent due to gravity, te films an be approximated as surfaes of ero tikness endowed wit a ontratile tendeny tat is desribed by a film tension, denoted (a free energy per unit lengt of a D film, or per unit area of a 3D film, wi is twie tat of te liquid vapour interfae, ). At equilibrium a dry foam satisfies Plateau s laws []: films of onstant mean urvature meet in triple lines at p=3 angles; te triple lines meet in fourfold verties at te tetraedral angles; and te different pressures in te bubbles equilibrate te ontratile fores on te films. Te energy of su a foam is just te energy of its films. In atual moderately dry foams (liquid ontent below about 5%), we may still neglet te film tiknesses (of order nm), but te * Corresponding autor. Address: Centro de Físia Teória e Computaional, University of Lisbon, Avenida Professor Gama Pinto, P-649-3 Lisbon, Portugal. E-mail address: piteixeira@ii.f.ul.pt (P.I.C. Teixeira). triple lines are deorated wit regions of triangular ross-setion alled Plateau borders (of widt of order. mm) were most of te liquid resides. In addition, were a foam meets a onfining surfae tere are wall Plateau borders. Tese are bounded by two liquid surfaes of tension and one solid surfae (te wall) of tension WL (te wall liquid interfaial tension). Wall Plateau borders affet bot te statis and te dynamis of foams: not only do tey ontribute to te total foam energy, tey also exert onsiderable drag on te walls in foam flow experiments. In perfetly dry foams te film ontat angle at a wall is p=. In D wet foams, te (irular) film prolongations into a wall PB also meet te wall at p= [3]. However, tis appears not to be te ase in 3D wet foams in ontat wit walls: deviations from p= ave been reported for a single bubble on a wet porous substrate [4], e.g., / 85 (measured inside te bubble and extrapolated to te substrate surfae see Fig. ) for a bubble of radius ¼ :4 mm. Te reason is tat te Plateau border possesses urvature in te oriontal diretion, due to te axial symmetry of te bubble. Altoug generally weaker tan tat existing in te vertial diretion, tis urvature depends on bubble and Plateau border sie, and modifies te ontat angle of te bubble. In an earlier paper [5] te Plateau border sapes and te apparent ontat angle of a single bubble at a wall were alulated by numerial integration of te appropriate Laplae equation. Te usual disparity of sales between te two urvatures, owever, suggests te use of perturbation metods, wi would allow greater insigt into te pysial meanism involved. Taking into aount tat, in most pratial situations, te eigt of te Plateau border is onsiderably smaller tan te radius of te bubble, in -9797/$ - see front matter Ó 9 Elsevier In. All rigts reserved. doi:.6/j.jis.9.5.6
94 M.A.C. Teixeira, P.I.C. Teixeira / Journal of Colloid and Interfae Siene 338 (9) 93 θ tis paper we develop an approximate analytial solution for te Plateau border sape in powers of =. Tis solution will also be used to study te relatively weak gravity effets on te Plateau border (i.e., te differene between sessile and pendant bubbles). Gravity is important in many aspets of foam resear, e.g., drainage [], and assumes ever greater relevane as it beomes inreasingly possible to arry out experiments in mirogravity environments, e.g., aboard te International Spae Station. Te analytial approa used in te present study as te advantage of allowing a better ontrol of input onditions, and onsequently an easier and quiker exploration of parameter spae, tan te numerial model used in [5] (were gravity effets were not addressed). Tis paper is organised as follows: in Setion we desribe our model, te Laplae equation for te Plateau border around a single sperial bubble at a flat wall, and obtain losed-form expressions for te inner and outer Plateau border surfaes. Our results for te apparent ontat angle, extrapolated ontat angle and Plateau border sape are disussed in Setion 3: we ompare results at different orders in = wit tose from numerial solution of te Laplae equation, for different ontat angles of te liquid on te substrate and varying gravity strengts. Comparison is also made wit wat is, to our knowledge, te only existing set of experimental results for tis system [4]. Finally, Setion 4 ontains some onluding remarks.. Teoretial model x( =) θ π θ φ Substrate Fig.. Semati diagram of te model problem. A bubble, and its assoiated Plateau border, are sown. is te radius of te bubble. is te eigt of te Plateau border. and p are te ontat angles of te Plateau border surfaes as tey interset te substrate. is te apparent ontat angle of te bubble (angle of te film at te top of te Plateau border). / is te extrapolated ontat angle of te bubble in te absene of a Plateau border. is te eigt and x is te radial distane of te film to its axis of symmetry (te axis). Te starting point is Laplae s equation for an axisymmetri geometry, wi may be written: " þ dx # 3= " d x d d þ þ dx # d ¼ Dp x ; ðþ were is te eigt and x is te distane between te film surfae and its axis of symmetry (ere assumed to be te axis). Dp is te pressure differene aross te film surfae (inner minus outer) and is te surfae tension of te fluid under onsideration. Defining ot ¼ dx=d; is te angle, measured on a vertial plane ontaining te axis, between te film diretion and te oriontal diretion. Te ontat angle at wi te outer surfae of te Plateau border intersets te substrate at te bottom of te bubble will be alled ¼ ð ¼ Þ, wile te orresponding ontat angle of te inner surfae is, of ourse, p. Finally, te apparent ontat angle of te bubble, defined as te angle of te film at te top of te Plateau border, were bot surfaes are tangent, is alled ¼ ð ¼ Þ (see Fig. ). eplaing te dependent variable x by in Eq. (), tat equation takes te form d d þ sin þ sin ¼ Dp ot d : ðþ Applying Eq. () at te inner and outer surfaes of te Plateau border yields d d þ sin þ sin ¼ p i p b ðinnerþ; ð3þ ot d d d sin þ sin ¼ p o p b ðouterþ; ð4þ ot d were p i ; p o and p b are te pressures inside te bubble, outside te bubble, and inside te Plateau border, respetively. Te pressure inside te Plateau border is assumed to be in ydrostati equilibrium, su tat p b ¼ p b qg; were p b is te pressure at te bottom of te Plateau border, g is te aeleration of gravity and q is te density of te fluid under onsideration. Additionally, it sould be noted tat te pressure differene between te inside and te outside of te bubble is given by p i p o ¼ 4 : In teir present form, Eqs. (3) and (4) annot be solved analytially. In order to make tis possible, tey are first inverted, so tat beomes te dependent variable and te independent variable. To do tis, it is neessary to ange te variable of integration of te integral from to, on noting tat, at te surfaes bounding te Plateau border, is a monotoni funtion of : Z ot d ¼ Z ot d d d: Wen tis is done, and Eq. (5) is also taken into aount, Eqs. (3) and (4) take te form p i p b þ qg sin sin þ d ot dd ¼ sin ðinnerþ; d ð8þ d p o p b þ qg þ sin sin þ d ot dd ¼ sin ðouterþ: d ð9þ d At tis point, it is useful to make te variables of tis problem dimensionless, so tat te orders of magnitude of te various terms beome learer. is dimensionless by nature, and takes values of O(). Sine tese equations are going to be integrated over te Plateau border eigt, wi is, a normalised eigt is defined as ¼ =. Ten, Eqs. (8) and (9) beome 3 ðp i p b Þ þ 4 qg sin 5 sin ot d d ¼ sin ðinnerþ; d ðþ ðp o p b Þ þ 4 qg þ sin þ sin sin þ sin ot d d d d d 3 5 ð5þ ð6þ ð7þ d ¼ sin ðouterþ: d ðþ Tese equations are subjet to te lower boundary ondition ð ¼ p Þ¼ (at te inner surfae) or ð ¼ Þ¼ (at te outer surfae) and te upper boundary ondition ð ¼ Þ¼ (at bot surfaes).
M.A.C. Teixeira, P.I.C. Teixeira / Journal of Colloid and Interfae Siene 338 (9) 93 95 Noting tat = is small, wen te fator involving te integral is expanded in powers of = up to first order, Eqs. () and () an be written approximately as " ðp i p b Þ þ qg sin Z # ot d sin sin d d d ¼ sin ðinnerþ; ðþ " d ðp o p b Þ þ qg þ sin Z # ot d sin sin d d d ¼ sin ðouterþ: ð3þ d In Eqs. () and (3), it is lear tat only te first term in square brakets is of O(), to balane te rigt-and side of tese equations, wile te tird term, related to te oriontal urvature of te Plateau border, is of Oð=Þ. Te last term, related to te fat tat te inner surfae of te Plateau border is loser to te axis of symmetry tan te outer surfae, is of Oð=Þ. Finally, te seond term, related to gravity effets, is also of Oð=Þ. In fat, weter tis term is of seond order or not depends on te value of te dimensionless parameter qg =, but in te ases tat will be addressed bubbles of radius 3 mm, as in [4] tis parameter is indeed of O(), as required. For tese values of and using ¼ 33:6 3 Jm (see [4]) gives qg = :3 :6. Te problem is takled by expanding bot te dimensionless eigt ðþ and te dimensionless inner and outer pressures in power series of =, as follows: ¼ þ þ þ; ð4þ ðp i p b Þ ¼ p þ ðp o p b Þ ¼ p þ p i þ p o þ p i þ; p o þ: ð5þ ð6þ Altoug a similar power series solution for ould presumably be used to solve te equations before inversion (i.e., Eqs. (3) and (4)), alulations would ertainly be less straigtforward, beause appears as te argument of sine and osine funtions. Additionally, tis would neessarily impose an a-priori dependene of on = (quadrati for a seond-order expansion) wi, as will be seen, is not supported by te numerial simulations. Te solution proedure will first be desribed for te inner surfae of te Plateau border... Inner surfae One Eqs. (4) and (5) are inserted into Eq. (), tree equations result, valid at erot-, first- and seond-order in =, respetively: p d ¼ sin ; d ð7þ p d d þ p i sin d sin d ¼ ; p d d þ þ p i sin d sin d p i þ qg þ sin sin Z ot d d d d d ¼ : ð8þ ð9þ Tese equations must be solved subjet to te boundary onditions ð ¼ p Þ¼ ð ¼ p Þ¼ ð ¼ p Þ¼ and ð ¼ Þ ¼, ð ¼ Þ¼ ð ¼ Þ¼. Integrating Eqs. (7) (9) between ¼ p and ¼ using tese boundary onditions yields te following expansion oeffiients of te inner pressure: p ¼ os þ os ; ðþ p i ¼ p þ sin os sin os p sin ; ðþ p i ¼ qg þ p i p p i p : ðþ In fat, as will be seen later, one does not need to find expliit expressions for ; and in order to obtain a relation between, and =. However, tis is neessary for plotting te atual sapes of te Plateau border surfaes. To obtain su expressions, Eqs. (7) (9) must be integrated instead between ¼ p and a generi, wit te result ¼ os þ os ; ð3þ p ¼ p i ðos þ os Þ p p sin ð p þ sin os sin os Þ; ð4þ ¼ qg ðos þ os Þ p i p 3 p p i ðos þ os p 3 Þ þ p i p þ sin os sin os : ð5þ p 3 p 3 sin In order to speify te inner surfae of te Plateau border, it is also neessary to know its oriontal position. An equation analogous to Eq. () an be obtained if we define x ¼ x= and Dx ¼ x ð ¼ Þ x ðþ (were x is te oriontal oordinate of te inner surfae). From tese definitions it follows tat, altoug x is of Oð=Þ; Dx is of O(). If we note tat dx =d ¼ ot, ten it an be sown from Eq. () tat " ðp i p b Þ þ qg sin þ # Dx sin sin ddx d ¼ os: ð6þ If Dx is expanded in a power series of =, as follows: Dx ¼ Dx þ Dx þ Dx þ; ð7þ and Eqs. (4) and (5) are also taken into aount, tree equations for Dx ; Dx and Dx are obtained: p ddx d ¼ os; d þ p i sin ddx sin d ¼ ; p i sin ddx sin d þ p ddx p ddx d þ ddx d ¼ : p i þqg sin Dx sin ð8þ ð9þ ð3þ Te solutions to tese equations satisfying te boundary onditions Dx ð ¼ Þ¼Dx ð ¼ Þ¼Dx ð ¼ Þ¼ (wi result from te definition of Dx and Eq. (7)), are: Dx ¼ sin sin ; ð3þ p Dx ¼ ðsin sin Þ p i ðsin sinþ; ð3þ p p sin Dx ¼ p i sin sin þ p i p 3 p 3 sin p 3 ðsin sinþ qg p 3 ð þsin os sinosþ: p i p qg os p 3 Tis ompletely speifies te inner surfae of te Plateau border. ð33þ
96 M.A.C. Teixeira, P.I.C. Teixeira / Journal of Colloid and Interfae Siene 338 (9) 93.. Outer surfae Now te same proedure must be followed for te outer surfae of te Plateau border. Hene, from Eqs. (4), (6) and (3), tree equations result, again valid at erot-, first- and seond-order in =, respetively: p d d ¼ sin; p d d þ p o þ sin d sin d ¼ ; p d d d þ d þ p o þ sin sin p o þqg sin sin Z ð34þ ð35þ ot d d d d d ¼ : ð36þ Tese equations are subjet to te boundary onditions ð ¼ Þ¼ ð ¼ Þ¼ ð ¼ Þ¼ and ð ¼ Þ¼, ð ¼ Þ¼ ð ¼ Þ¼. Integrating tem between ¼ and ¼ gives p ¼ os os ; ð37þ p o ¼ sin os þ sin os p sin ; ð38þ p o ¼ qg p o p p o : ð39þ p On te oter and, integrating Eqs. (34) (36) between ¼ and a generi yields ¼ os os ; ð4þ p ¼ p o ðos p os Þ p sin ð sin os þ sin os Þ; ð4þ ¼ qg ðos os Þ p o p o ðos p 3 p p 3 os Þ þ p o þ sin os þ sin os : ð4þ p 3 p 3 sin An equation analogous to Eq. (6) may be obtained for te oriontal displaement of te outer surfae, Dx ¼ x ð ¼ Þ x ðþ: " ðp o p b Þ þ qg þ sin þ # Dx ddx sin sin d ¼ os: ð43þ Expanding Dx in powers of =, as in Eq. (7), te following tree equations are obtained from Eq. (43): p ddx ¼ os ; ð44þ d p ddx d þ p o þ sin ddx ¼ ; ð45þ sin d p ddx d þ p o þ sin ddx sin d þ p o þ qg þ sin Dx sin ddx ¼ : ð46þ d Subjet to te same boundary onditions as enuniated before for te inner surfae, tese equations ave te solutions: Dx ¼ sin sin ; ð47þ p Dx ¼ p sin ðsin sin Þþ p o ðsin p sin Þ; Dx ¼ þ p o sin sin p 3 p 3 sin p o p 3 p o qg p os p 3 ðsin sin Þ qg ð þ sin os sin os Þ: p 3 ð48þ ð49þ Tis speifies te outer surfae ompletely. It sould be noted tat, by design of te solutions, te upper vertex of te Plateau border, were te inner and outer surfaes meet, is loated at Dx ¼ and ¼..3. elation between, and = Te relation between te inner and outer pressures (due to te bubble urvature) allows us to relate ; and =. Te dimensionless version of Eq. (6) is ðp i p o Þ ¼ 4 : ð5þ Tis equation sows tat, altoug te pressure differenes between te inside of te Plateau border and te inside or te outside of te bubble are of erot order in =, te differene between tese pressure differenes is only of first order. Tis, of ourse, is onsistent wit te assumed disparity of sales between te urvatures of te Plateau border surfaes and of te bubble. Eq. (5), ombined wit Eqs. (5) and (6), gives te following equation for = (aurate to seond order in =), ðp i p o Þ þðp i p o 4Þ þ p p ¼ ; ð5þ wi enables us to find = as a funtion of and, troug te solution qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4 p i þ p o ð4 p þ i p o Þ 4ðp i p o Þðp p Þ ðp i p o Þ ð5þ (were te pressures are speified by Eqs. () () and (37) (39)). Tis formula provides a relation between =; and aurate to seond order in =. A relation aurate to first-order may be obtained by retaining only te erot- and first-order terms in Eq. (5), wi gives: ¼ p p 4 p þ : ð53þ i p o Finally, a erot-order approximation an be obtained by negleting bot te first- and te seond-order terms in Eq. (5), yielding: p ¼ p ) os ¼ ) ¼ 9 : ð54þ Note tat in Eq. (5) te pysially meaningful root, i.e., te one tat redues to te first-order approximation as p i and p o tend to ero, as been seleted. Tis an be eked by rewriting Eq. (5) as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4 p i þ p o Þ 4ðp p Þðp i p o Þ ¼ ð4 p i þp o Þ ðp i p o Þ ð55þ
M.A.C. Teixeira, P.I.C. Teixeira / Journal of Colloid and Interfae Siene 338 (9) 93 97 and expanding te term inside te square brakets in a Taylor series. Note also tat te proedure desribed above is not part of te formal power series solution to te problem (were te oeffiients multiplying suessive powers of = would ave to be set to ero to satisfy Eq. (5)). In fat, tese oeffiients ave been determined previously in Eqs. () () and (37) (39), so Eq. (5) is not valid for any arbitrary =, but rater defines = as a funtion of and. Eq. (5) and its versions trunated at lower orders, wi give rise to te solutions of Eqs. (5), (53) or (54), may be viewed as akin to trunating te final result of a perturbation expansion solution at te required order. In Eq. (5), and unlike wat is usual in perturbation expansions, wat is determined is te small parameter = as a funtion of oter problem variables, instead of te oter way round. Tis was done for simpliity but we believe te proedure to be onsistent, as te ensuing results will sow, despite te fat tat it is not very standard. Obviously, solutions analogous to Eq. (5) would be eiter mu lengtier or impossible to obtain analytially using tis approa if te perturbation expansion was extended to tird or iger order. 3. esults 3.. Apparent ontat angle of te bubble Fig. a and b sows te variation of te apparent ontat angle of te bubble wit = for tree values of (te ontat angle between te Plateau border surfaes and te substrate). esults in Fig. a are for ero gravity and tose in Fig. b for two values of te gravity parameter (positive and negative, orresponding to sessile and pendant bubbles, respetively). In Fig. a, te erot-order solution, Eq. (54), is oinident wit te upper oriontal axis, te upper set of lines orrespond to te first-order solution, Eq. (53), and te lower set of lines to te seond-order solution, Eq. (5). Te symbols are numerial data obtained by te metod of [5], were te approximation = is not made. In Fig. b te filled symbols orrespond to qg = ¼ and te open symbols to qg = ¼. Te positive value is witin te range of values taken by qg = in te experiments of odrigues et al. [4], as mentioned previously. It sould be noted first of all tat gravity as no effet on te analytial relation between =; and, at least at te urrent order of approximation. Tis is a onsequene of te fat tat only te differene p i p o appears in Eq. (5), and aording to Eqs. () and (39), tis differene does not depend on gravity, beause te terms involving gravity anel exatly. On te oter and, te erot-order solution, wi only takes into aount te vertial urvature of te Plateau border, is trivial and a bad approximation: is not predited to depend on = or on. Te first-order approximation, were te oriontal urvature of te Plateau border is taken into aount, produes a reasonable predition for, up to = ¼ :3, wit te orret dependene on. Finally, te seond-order approximation, were apart from te pysial effets mentioned above, te oriontal tikness of te Plateau border is taken into aount, yields te best preditions. Tese are quite aurate up to = :45. Te analytial solutions neverteless diverge onsiderably from te numerial results above tese limits, as expeted. In Fig. b, te numerial results sow tat, ontrary to wat te analytial model predits, tere is a dependene of on gravity, altoug tis is relatively weak. For a given value of =, positive gravity (orresponding to a sessile bubble) dereases, wile negative gravity (orresponding to a pendant bubble) inreases it. θ (º) θ (º) θ (º) 9 8 7 6 5 4 9 8 7 6 5 (a), θ =º, θ =º, θ =º....3.4.5.6.7 / (b),, θ =º,, θ =º,, θ =º 4....3.4.5.6 / 9 85 8 75 (), exp. data st order nd order 7..5..5..5 / Fig.. Variation of te apparent ontat angle of te bubble wit te dimensionless eigt of te Plateau border. Lines: analytial model, symbols: numerial or experimental data. (a) esults witout gravity. Upper set of lines: st-order approximation, lower set of lines: nd-order approximation. Te t-order approximation oinides wit te upper oriontal axis. Symbols: numerial model. (b) esults wit gravity from nd-order approximation. Filled symbols: qg = ¼, open symbols: qg = ¼, bot from numerial model. () Comparison wit experimental data (symbols) for ¼. Te point at ð:; 74 Þ does not follow te general trend and may not be meaningful. Fig. ompares te first-order and seond-order analytial solutions and experimental data [4]. Note tat tis omparison is arried out over a mu smaller range of = tan in Fig. a and
98 M.A.C. Teixeira, P.I.C. Teixeira / Journal of Colloid and Interfae Siene 338 (9) 93 b. Experimentally, te bubbles are formed by blowing air troug a ole in a porous glass substrate; te latter an be eiter dry, if only a few drops of surfatant solution are plaed on it, or wet, if it is onneted to te surfatant solution reservoir and terefore overed wit a ontinuous liquid film [6]. Filled symbols are for bubbles on a dry substrate, wereas open symbols are for bubbles on a wet substrate. A ontat angle of ¼ was assumed in eiter ase, wi seemed te most sensible oie, sine te bubble was in ontat wit te fluid in bot ases. Te error bars were omputed taking into aount tat te measurement error of is.5 mm, and te error of is [4]. Te measurement error of was estimated as.8 mm [6]. It an be seen tat, wile te agreement wit te seond-order analytial predition is not very good (it is outside te error bars), te data approximately follow a straigt line (wi is onsistent wit te analytial results at any order of approximation for tis range of =). However, te analytial results onsiderably overestimate. Te disrepanies migt be attributed to a systemati measurement error, sine te value of was estimated visually from te optial opaity of te Plateau border. It is plausible tat te upper region of te Plateau border (wi determines ) is very tin, and tus its upper limits are diffiult to detet from mere visual inspetion. Tis ould aount for an underestimation of (and tus also of =), altoug tere may of ourse be oter soures of error. We empasise tat tese are te only experimental data tat we know of wi are relevant to te present problem. 3.. Extrapolated ontat angle of te bubble Te extrapolated ontat angle of te bubble / is te angle at wi te bubble would interset te substrate if te bubble s emisperial sape extended for <. Tis angle is defined, by te use of standard trigonometry, as os os / ¼ : ð56þ Like ; / does not depend on gravity at te urrent order of approximation. Owing to te way in wi it is defined, / is onsiderably loser to 9 tan, wit a narrower range of variation, and its relative error in te analytial approximation developed is tus expeted to be larger. In Fig. 3 te variation of / wit = is presented for tree values of, for ero gravity (Fig. 3a), and for positive and negative gravity (Fig. 3b). Lines and symbols ave te same meanings as in Fig. a and b. Taking into aount Eq. (56), it is lear tat te erot-order value of / is equal to te orresponding value of, namely / ¼ 9, and terefore does not depend on or on =. In Fig. 3a, it an be seen tat te first-order approximation for / as reasonable auray up to = ¼ :, and te seond-order approximation is aurate up to about = ¼ :4. Bot first-order and seond-order approximations diverge strongly from te numerial results for larger values of =, but, again, tis sould be expeted. In Fig. 3b, it an be seen tat te dependene of / on = inluding gravity beomes inaurate at approximately te same values of = as in te ero-gravity ase. However, disrepanies are sligtly more marked for negative gravity (wi is onsistent wit Fig. b). For positive gravity, / is somewat smaller tan in Fig. 3a, staying loser to te predition of te seond-order approximation. In te ase of negative gravity, on te ontrary, / is larger, and so departs more strongly from te seond-order result. Qualitatively, tis ontrasting beaviour of a sessile and a pendant bubble (i.e., positive and negative gravity, respetively) appears intuitive, sine a pendant bubble sould portrude more from te substrate due to its own weigt. φ (º) φ (º) 9 88 86 84 8, θ =º, θ =º, θ =º (a) 8....3.4.5.6.7 / 9 88 86 84 8 (b),, θ =º,, θ =º,, θ =º....3.4.5.6 / Fig. 3. Variation of te extrapolated ontat angle of te bubble wit te dimensionless eigt of te Plateau border. Lines: analytial model, symbols: numerial model. (a) esults witout gravity. Upper set of lines: st-order approximation, lower set of lines: nd-order approximation. Te t-order approximation oinides wit te upper oriontal axis. (b) esults wit gravity from nd-order approximation. Filled symbols: qg = ¼, open symbols: qg = ¼. 3.3. Profiles of te Plateau border surfaes Te solutions for Dx and provided by Eqs. (3) (33) and (3) (5) for te inner surfae and by Eqs. (47) (49) and (4) (4) for te outer surfae are studied next for a bubble wit = :4. Tis large value of = is osen so tat te effet of te seond-order orretions is learly visible, and on te oter and so tat we are still rougly witin te limits of appliability of te teory (as suggested by Figs. and 3). Fig. 4a sows te erot-, first- and seond-order solutions for te sape of te Plateau border (wit te bubble to te rigt). It is assumed tat ¼ and ¼ 6 (for te first- and seond-order solutions), wi orresponds to = ¼ :48 aurate to seond-order in = (and = ¼ :436 aurate to first-order). Also sown in Fig. 4a is a solution intermediate between te erot-order and te first-order solutions, wi we ave alled /t-order solution. Tis is equal to te erot-order solution, exept tat we ave presribed ¼ 6 (as in iger-order solutions), wi is inonsistent at erot-order (as Eq. (54) sows). Te purpose of tis is to understand more learly te pysial proesses involved. In Fig. 4a it an be seen tat in te erot-order solution ¼ 9 (as it must be) and te inner and outer surfaes of te Pla-
M.A.C. Teixeira, P.I.C. Teixeira / Journal of Colloid and Interfae Siene 338 (9) 93 99. (a) '.8.6.4 t order /t order st order nd order.. -.6 -. -.8 -.4..4.8 Δx'. (b) '.8.6.4 θ =º θ =º θ =º.. -. -.6 -. -.8 -.4..4 Δx'. () '.8.6.4 ρg /γ= ρg /γ= ρg /γ=-.. -. -.6 -. -.8 -.4..4.8 Fig. 4. Sapes of te outer (on te left) and inner (on te rigt) surfaes of te Plateau border for a bubble wit ¼ 6 (exept for te t-order solution) = :4, near te limits of validity of te teory. Lines: analytial model, symbols: numerial model. (a) t-, /t-, st- and nd-order solutions for ¼, witout gravity. (b) ndorder solution for ¼ ; ¼ and ¼, witout gravity. () nd-order solution for ¼ and tree values of qg =. Δx' teau border are symmetri ars of irle. elative to te erot-order solution, in te /t-order solution te inner and outer surfaes of te Plateau border are translated to te left, due to te fat tat 9. To onsistently be able to meet te substrate at te imposed value of, te inner surfae must inrease its urvature onsiderably, wereas te outer surfae must derease it. In te first-order solution, te additional effet of te oriontal urvature of te bubble is taken into aount. Tis urvature as te same sign as te urvature of te inner surfae, but te opposite sign to te urvature of te outer surfae, inreasing furter te latter, but dereasing te former (for an approximately onstant pressure inside te Plateau border). Tis effet is espeially important wen te surfaes are nearly vertial (i.e., near te top), wile it is mu weaker wen tey are nearly oriontal (i.e., near te bottom). Tis explains te sligt translation of te surfaes to te left from te /t- to te first-order solution. Tis effet is also stronger in te outer surfae tan in te inner surfae, beause te latter as larger urvature, and tus te relative effet of tis orretion is smaller. Tat explains te larger translation of te outer surfae, and tus te sligt widening of te Plateau border from te /t- to te first-order solution. It is wort noting tat altoug te anges on going from te erot-order to te /t-order solution and from te /t-order solution to te first-order solution are bot of first-order (te former being assoiated wit te vertial bubble urvature and te latter wit te oriontal bubble urvature) in pratie te former effet is mu larger tan te latter. Canges on going from te first-order to te seond-order solution are more subtle and diffiult to interpret. Tese anges are due to te differene between distanes of te inner and outer surfaes to te axis of symmetry (as was mentioned earlier), but tis effet is not easy to understand in terms of urvature. It an neverteless be noted tat te seond-order solution is mu loser to te first-order solution tan te latter is to te erot-order solution. Tis suggests tat te power series for Dx and are asymptoti. In Fig. 4b and, only results for te seond-order solutions (te most aurate) are sown. Fig. 4b displays te Plateau border surfaes for ¼ 6 and ¼ ; ;. Tis orresponds to = ¼ :44; = ¼ :48 and = ¼ :47, respetively, aurate
M.A.C. Teixeira, P.I.C. Teixeira / Journal of Colloid and Interfae Siene 338 (9) 93 (a) (b) Fig. 5. Semati diagram illustrating te approximate sape of te Plateau border of a sessile bubble wit (a) weak gravity (b) strong gravity. Te onvex urvature at te bottom in (b) is due to te inrease in pressure assoiated wit ydrostati equilibrium. to seond-order. Te Plateau border widens as dereases (i.e., as te fluid wets te solid inreasingly better), being onsiderably larger for ¼ tan for ¼.InFig. 4b, agreement between te analytial and numerial solutions is exellent. Finally, in Fig. 4, te dependene of te solutions on te gravity parameter is sown, again for a Plateau border wit ¼ and ¼ 6, orresponding to = ¼ :48. Te effet of gravity is peraps as expeted, leading to a widening of te Plateau border for positive qg = (sessile bubble) and a narrowing for negative qg = (pendant bubble). Tis beaviour an be interpreted using ydrostati equilibrium. For a positive g, te pressure is lower in te upper part of te Plateau border tan in te lower part. Tis leads to an inrease in te urvature of te inner and outer surfaes near te top, and a orresponding redution near te bottom. It is straigtforward to onlude tat tis orresponds to a widening. Te onverse appens for negative g. Te displaement of te outer surfae of te Plateau border due to te variation of eiter or g = is mu larger tan te displaement of te inner surfae. Tis may be due, again, to te fat tat te urvature of te inner surfae is onsiderably larger, and tus less affeted by iger order effets. Anoter possible ause is tat te inner surfae is loser to vertial, and tus a smaller oriontal translation is neessary for it to adjust to te variation of te parameters. Te agreement between analytial and numerial results, altoug qualitatively orret, is not so good for non-ero gravity. It migt be tat te values of qg = onsidered are too ig, altoug tis sould not be te ase, sine tey are of O(). We used tese values so tat te differenes between te various urves in Fig. 4 were suffiiently lear (and also beause te positive value is ompatible wit te experiments of [4], as remarked above). It sould be realled tat gravity is treated ere as a seond-order effet, so wen it beomes too large it would probably be better treated at lower order. But tat approa, if feasible, would ertainly inrease te matematial omplexity furter. A more fundamental matematial reason for te worse performane of te analytial model inluding gravity may be te assumption (made in te desription of te teoretial model, in Setion ) tat is a monotoni funtion of. Wen gravity effets are relatively large, te numerial simulations sow tat te urvature of te lower part of te Plateau border in a sessile bubble beomes weak and may even ange its sign due to te ydrostati inrease of te pressure (not sown). In tis ase, stops being a monotoni funtion of, and tus a one-to-one relation eases to exist between tese two variables (see Fig. 5). 4. Disussion It appears tat, for an aurate predition of te ontat angle of relatively small bubbles (i.e., bubbles wit suffiiently large =), it is ruial to onsider at least a seond-order approximation in our analytial, perturbation expansion, model. Tis pysially orresponds to taking into aount not only te oriontal urvature of te Plateau border (due to te speriity of te bubble), but also te differene in distanes to te axis of symmetry of te bubble from te inner and te outer surfaes of te Plateau border. Altoug gravity as a onsiderable impat on te sape of te Plateau border, wi is aptured by te present teory, its effet on te relation between ; = and appears to be weak in most situations. Tese differing impats of gravity on different aspets of Plateau border geometry seem to be orroborated by Fig. b and Fig. 4. Sine bubbles wit relatively ig = are generally small, and in tat ase qg = is generally low, te oie of treating gravity as a seond-order effet may not be too inappropriate (it is in partiular adequate for bubbles su as tose studied experimentally by odrigues et al. [4]). However, in ases were gravity is more important, te alulations would need to be reformulated, for example wit gravity onsidered at lower order in te perturbation expansion applied to te equations. Su alulations are, owever, likely to be even more involved tan tose presented ere. Numerial results also suggest tat, wen gravity effets are suffiiently strong, te urvature of te Plateau border surfaes may ange sign loally (see Fig. 5). Tis would plae a mu more definite limit on te range of appliability of te present alulations, sine tey rely on te existene of a one-to-one relation between and. Witin teir limits of validity, te analytial alulations developed in te present paper ave te advantage of providing losedform expressions for te sape of te Plateau border, and an expliit relation between ; and =. Tis is a onsiderable improvement over te numerial model used to address a related problem [5], were numerial solutions must be integrated from te bottom of one of te Plateau border surfaes, and it is not possible to impose, for example, te oordinates of te upper vertex of te Plateau border. Tis limitation of te numerial model means tat te ontat angle of one of te Plateau border surfaes wit te substrate must be adjusted by trial and error. Besides addressing te effet of gravity in a simple way, te analytial model proposed ere terefore allows a mu more exaustive exploration of te parameter spae tan te numerial model. Aknowledgments We tank B. Saramago for illuminating disussions and an anonymous referee for insigtful omments, wi ave onsiderably improved tis paper. Tis work was supported by FCT under Projet AWAE/PTDC/CTE-ATM/655/6. eferenes [] D. Weaire, S. Hutler, Pysis of Foams, Oxford University Press, Oxford, 999. [] J.A.F. Plateau, Statique expérimentale et téorique des Liquides soumis aux seules Fores Moléulaires, Gautier Villars, Paris, 873. [3] M.A. Fortes, P.I.C. Teixeira, Pilos. Mag. Lett. 85 (5). [4] J.F. odrigues, B. Saramago, M.A. Fortes, J. Colloid Interfae Si. 39 () 577. [5] P.I.C. Teixeira, M.A. Fortes, Pys. ev. E 75 (7) 44. [6] B. Saramago, private ommuniation, 8.