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1 Auho's pesonal copy Advances in Colloid and Ineface Science 65 (2) 7 9 Conens liss available a ScienceDiec Advances in Colloid and Ineface Science jounal homepage: Theoy of non-equilibium foce measuemens involving defomable dops and bubbles Deek Y.C. Chan a,b,c,, Eve Klaseboe b, Rogeio Manica b a Paiculae Fluids Pocessing Cene, Depamen of Mahemaics and Saisics, Univesiy of Melboune, Pakville, 3, Ausalia b Insiue of High Pefomance Compuing, Fusionopolis Way, 38632, Singapoe c Depamen of Mahemaics, Naional Univesiy of Singapoe, 7543, Singapoe aicle info absac Available online 24 Decembe 2 Keywods: Young Laplace Sokes Reynolds Coalescence Film dainage AFM foce measuemens Defomable dops and bubbles Ove he pas decade, diec foce measuemens using he Aomic Foce Micoscope (AFM) have been exended o sudy non-equilibium ineacions. Pehaps he moe scienifically ineesing and echnically challenging of such sudies involved defomable dops and bubbles in elaive moion. The scienificinees sems fom he ich complexiy ha aises fom he combinaion of sepaaion dependen suface foces such as Van de Waals, elecical double laye and seic ineacions wih velociy dependen foces fom hydodynamic ineacions. Moeove he effecs of hese foces also depend on he defomaions of he sufaces of he dops and bubbles ha ale local condiions on he nanomee scale, wih defomaions ha can exend ove micomees. Because of incompessibiliy, effecs of such defomaions ae songly influenced by small changes of he sizes of he dops and bubbles ha may be in he millimee ange. Ou focus is on ineacions beween emulsion dops and bubbles a aound μm sizeange.aheypical velociies in dynamic foce measuemens wih he AFM whichspanheangeofbownianvelociiesofsuch emulsions, he aio of hydodynamic foce o suface ension foce, as chaaceized by he capillay numbe, is ~ 6 o smalle, which poses challenges o modeling using diec numeical simulaions. Howeve, he qualiaive and quaniaive feaues of he dynamic foces beween ineacing dops and bubbles ae sensiive o he deailed space and ime-dependen defomaions. I is his dynamic coupling beween foces and defomaions ha equies a deailed quaniaive heoeical famewok o help inepe expeimenal measuemens. Theoies ha do no ea foces and defomaions in a consisen way simply will no have much pedicive powe. The echnical challenges of undeaking foce measuemens ae subsanial. These ange fom geneaing dop and bubble of he appopiae size ange o conolling he physicochemical envionmen o finding he opimal and quanifiable way o place and secue he dops and bubbles in he AFM o make epoducible measuemens. I is pehaps no supise ha i is only ecenly ha diec measuemens of non-equilibium foces beween wo dops o wo bubbles colliding in a conolled manne have been possible. This eview coves he developmen of a consisen heoy o descibe non-equilibium foce measuemens involving defomable dops and bubbles. Pedicions of his model ae also esed on dynamic film dainage expeimens involving defomable dops and bubbles ha use vey diffeen echniques o he AFM o demonsae ha i is capable of poviding accuae quaniaive pedicions of boh dynamic foces and dynamic defomaions. In he low capillay numbe egime of inees, we obseve ha he dynamic behavio of all expeimenal esuls eviewed hee ae consisen wih he angenially immobile hydodynamic bounday condiion a liquid liquid o liquid gas inefaces. The mos likely explanaion fo his obsevaion is he pesence of ace amouns of suface-acive species ha ae esponsible fo aesing inefacial flow. Cown Copyigh 2 Published by Elsevie B.V. All ighs eseved. Conens. Inoducion Backgound and moivaions Pespecive and scope Coesponding auho. Paiculae Fluids Pocessing Cene, Depamen of Mahemaics and Saisics, Univesiy of Melboune, Pakville, 3, Ausalia. Tel.: ; fax: addess: D.Chan@unimelb.edu.au (D.Y.C. Chan) /$ see fon mae. Cown Copyigh 2 Published by Elsevie B.V. All ighs eseved. doi:.6/j.cis.2.2.

2 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) Dop and bubble defomaions Augmened Young Laplace equaion Special film shapes: Dimple, pimple, wimple and ipple Equaion fo hin film defomaions Film beween a dop and a spheical paicle Film beween wo dops Dop shape ouside ineacion zone A pinned conac line A consan conac angle Bubble compessibiliy Maching soluions fo he foce-displacemen fomula Dop sphee ineacion Dop dop ineacion Hydodynamic ineacions Sokes Reynolds lubicaion heoy Non-defoming inefaces Tangenially immobile inefaces Navie slip inefaces Two dops wih mobile inefaces Two bubbles: Cheses Hofman model Bubbles wih suface-acive species Sefan Reynolds Fla Film Model Neo Fla Film models Sokes Reynolds Young Laplace model Govening equaions and bounday condiions Scaled equaions fo compuaions Ineacion unde given displacemen funcion Ineacion unde consan foce Peubaion soluions Axisymmeic dops Dops in he micofluidic Hele Shaw cell geomey Foce-displacemen fomula fo AFM expeimens Numeical algoihm Compaisons wih expeimens Dynamic defomaions Opposing poubean dops Bubble agains quaz plae Mecuy dop agains mica plae Dynamic foce measuemens Dop sphee ineacion Dop dop ineacion Bubble bubble ineacion Conclusion Acknowledgmens Refeences Inoducion.. Backgound and moivaions Sudies of non-equilibium ineacions involving defomable dops and bubbles pedaed he fomulaion of he Dejaguin Landau Vewey Ovebeek heoy of colloidal sabiliy [,2] wih he sudies of Dejaguin and Kussakov [3] on ime-dependen behavio of a ising bubble owads a fla plae unde buoyancy foce. Subsequen non-equilibium sudies concenaed on he dainage phenomena of he liquid film beween defomable menisci [4]. Ealy diec measuemens of non-equilibium foces wee based on he Suface Foces Appaaus o measue he ime-dependen appoach beween wo coss-cylindes of mica down o nanomee sepaaions in aqueous [5] and non-aqueous liquids [6]. Foces unde condiions of seady sae oscillaions of he mica sufaces wee also sudied in he conex of examining he possible vaiaions in fluid viscosiies of nanomee hick confined liquid films [7] and he lubicaing popeies of adsobed polymes [8]. Wih he adven of he aomic foce micoscope, inees coninued in he hydodynamic ineacion involving solid sphees in he ens of micomee size ange. Alhough much inees was geneaed by epos of hydodynamic bounday slip a he solid liquid ineface [9], paiculaly in he conex of micofluidic applicaions [], ecen epeaed measuemens sugges ha insumenal aifacs ae likely o be esponsible fo such obsevaions a smooh well defined sufaces [ 3]. In he fis applicaions of he aomic foce micoscope o measue equilibium foces involving defomable bubbles, he defomaional esponse of he bubble was eaed as a Hookean sping [4,5]. In subsequen equilibium sudies involving dops, he Young Laplace equaion was used o accoun fo he dop defomaional behavio [6,7]. In consideing non-equilibium ineacions, he ime-dependen foce beween, fo insance, wo appoaching defomable dops a any insan, does no only depend on he insananeous shapes and sepaaion beween he dops, bu also on iniial condiions ha deemine he dop shape and inefacial velociies. In addiion, flow of he coninuous fluid phase also conibues o he hydodynamic ineacion. Theefoe appopiae expeimenal daa needs o be ecoded o povide iniial and bounday condiions fo heoeical analysis.

3 Auho's pesonal copy 72 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) 7 9 Anohe challenge in dealing wih defomable bodies is he issue of having muliple lengh scales of vey diffeen magniudes. Fo nondefoming bodies, only he geomey aound he ineacion zone beween sufaces in close poximiy needs o be specified in ode o deemine he ineacion foce, using say, he Dejaguin appoximaion [8,9]. Fo defomable bodies, on he ohe hand, a mechanical equaion of sae connecs local vaiaions of he ineacion foce and he local geomey [2]. If he defomable maeial is incompessible, his coupling beween foces and geomeies can exend ove lengh scales of many odes of magniude. Fo example, he ineacion beween millimee size dops o bubbles acoss films of nanomee hickness can cause defomaions exending ove ens o hundeds of micomees. Fuhemoe, he incompessibiliy of dops and o a good appoximaion, bubbles, means ha small changes on he scale of he size of dops o bubbles can have significan effecs on he scale of hin films. As a consequence, appoaches based on diec numeical simulaions ae unlikely o have sufficien esoluion and pecision o span such a lage ange of lengh scales. The inuiion we obain in sudying dops and bubbles whee hei defomaional esponse o applied foces can be well chaaceized is also valuable in analyzing ineacions involving ohe ypes of sof defomable bodies ha may have elasic o viscoelasic esponses..2. Pespecive and scope Thee ae hee key elemens ha mus be included in modeling ime-dependen ineacions involving defomable dops and bubbles: (a) A descipion of how dops/bubbles defom unde he influence of sesses aising fom hydodynamic flow and disjoining pessue fom suface foces, (b) A descipion of he flow of he inevening fluid wihin he hin film confined by he defomable sufaces of dops o bubbles, and (c) A consideaion of suface o colloidal foces ha will vay wih local defomaions of he inefaces in close poximiy. All such facos deemine collision sabiliy o coalescence and mus of couse be eaed in a self-consisen way. Fo insance, he defomed inefaces of he dops o bubbles deemine he boundaies of he hin film whee he inevening fluid mus flow duing ineacion. Howeve, such flow will geneae pessue pofiles wihin he film ha will in un deemine he shape of he inefaces. Fo insance, he applicaion of he Sefan Reynolds Fla Film Model [2,22] o model film dainage in which he dop inefaces ae assumed o be plane paallel, immediaely gives ise o inenal inconsisencies ha equie subsequen coecion. Indeed he use of his model, in spie of is inabiliy o give quaniaive ageemen wih even he simples expeimens, has in ou view disoed ou undesanding of non-equilibium ineacions beween defomable dops and bubbles. A he ypical velociies in dynamic foce measuemen wih he AFM which also span he ange of Bownian velociies of such emulsions (~ μm size) o a velociies used o sudy dynamic film dainage in mm size dops and bubbles he aio of hydodynamic foce o suface ension foce, as chaaceized by he capillay numbe, Ca μv/σ is ~ 6 o smalle. Hee μ is he viscosiy of he coninuous phase, V a chaaceisic velociy and σ he inefacial ension. Unde such condiions he heoy in his eview eas hydodynamic ineacions in he low Reynolds numbe o Sokes flow egime elevan o many measuemens of non-equilibium foces using he aomic foce micoscope as well as diec obsevaions of imedependen defomaions of dops and bubbles ha ae undegoing ineacions. Flow in he hin film apped by he defoming dops (o bubbles) is consideed in he lubicaion appoximaion because he film hickness is small compaed o he laeal dimension of he film. Defomaions of he inefaces of he dops o bubbles ae deemined by he combinaion of capillay foces, hydodynamic and disjoining pessue. Spaial and empoal evoluion of he defomaions of he inefaces of he dops o bubbles as a esul of ineacion ae modeled in deail wheeas defomaions of he es of he dops o bubbles ae eaed analyically o povide bounday condiions ha eflec how he ineacing dops o bubbles ae diven ogehe. An alenae appoach o ea hydodynamic ineacions is o solve he complee Sokes flow equaions using diec numeical simulaions [23 26]. Such mehods have been used o sudy dop coalescence fo capillay numbes in he ange: Ca ~.., whee he compuaional ime fo a collision encoune can ake ove h of CPU ime. Howeve, fo capillay numbes of inees in film defomaion and foce measuemen expeimens consideed hee whee Ca ~ 6, a diec numeical appoach has ye o be aemped. In conas, he model we ouline hee akes advanage of he simplificaions affoded by he special chaaceisics of film dainage and AFM foce measuemen expeimens wheeby we can undeake calculaion of a ypical dainage o foce un in aound min on a noebook compue. In ode o focus on he key physical pinciples, we shall only conside non-equilibium ineacion beween defomable dops and bubbles fo which hee is axial symmey. As we shall see in Secion 5, his is elevan o a numbe of diffeen expeimens ha measue non-equilibium foces and defomaions of dops and bubbles. The equaions ha goven he defomaion of dops and bubbles will be developed in Secion 2. In paicula, deails of how o obain bounday condiions using he asympoic analyic soluions fo he dop shape ouside he film will be given. Diffeen models fo hydodynamic ineacions, including he familia Sefan Reynolds Fla Film Model, will be discussed in Secion 3. The Sokes Reynolds Young Laplace model fo descibing non-equilibium ineacions beween defomable dops and bubbles, incopoaing he developmen in Secions 2 and 3 will be sudied using peubaion analysis in Secion 4 and deailed implemenaion of obus numeical soluions of he equaions will also be given. In Secion 5, pedicions of he Sokes Reynolds Young Laplace model ae compaed wih expeimens ha measue dynamic defomaions and dynamic foces o illusae he uiliy of he model. This eview is heefoe aimed a eades who ae familia wih esablished heoies of suface foces and disjoining pessues a he level of he Dejaguin Landau Vewey Ovebeek (DLVO) model. In addiion, he eade should have some familiaiy wih he basic opeaions and limiaions of he aomic foce micoscope when i is used o measue foces beween igid sufaces. 2. Dop and bubble defomaions In his secion, we develop he equaions ha goven he defomaion of a dop o bubble as a esul of exenal foces aising fom he ineacion wih anohe paicle o dop. Since in mos cases, he behavio of dops and bubbles is vey simila, we will hencefoh use he em dops o denoe boh dops and bubbles unless specified ohewise explicily. Abou 2 yeas ago in 85, he Biish physician Thomas Young [27] gave an analysis of he shape of a defomable fluid ineface unde he acion of capillay foces wihou using any equaions. The Fench asonome Piee-Simon Laplace [28] consideed he same poblem using a foce balance mehod in he nomal and angenial diecion o he fluid ineface. Cal Fiedich Gauss [29] gave an analysis of he poblem in ems of he pinciple of minimizaion of inefacial aea unde he acion of inefacial ension o enegy. Sicly speaking such an appoach is no applicable unde nonequilibium condiions. In he pesence of hydodynamic ineacion ha is of inees hee, we can esimae he ime scale equied fo a

4 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) dop, unde capillay foces, o adjus is shape in esponse o exenal peubaions. Capillay waves of velociy c and wavelengh λ on a spheical dop wih inefacial ension σ o obey he dispesion elaion [3]: c 2 =2πσ o /[λ (ρ d +ρ e )], wih ρ d and ρ e being he densiies of he dispesed and coninuous phases. Taking λ ~ μm which is an uppe limi of he size of he defomaion zone of small dops, gives c ~ m/s, which is much fase han he chaaceisic appoach velociies of dops consideed hee. In ohe wods, in he pesence of hydodynamic ineacions, we make he easonable assumpion ha a dop can adjus is shape insananeously o accommodae changes in he hydodynamic pessue. Thus we can add he effecs of he hydodynamic pessue, p o effecs due o equilibium disjoining pessue aising fom suface foces. 2.. Augmened Young Laplace equaion Conside a sessile dop on a subsae, immesed in a coninuous medium as shown in Fig.. I is defomed by ineacions due o suface foces beween iself and a solid paicle locaed a a disance D fom he subsae. If he ange of he suface foces is small compaed o he dimensions of he dop and he paicle, he defomaion will be confined o a small ineacion zone of adius a aound he apex of he dop. Wihin he Dejaguin appoximaion he dop paicle ineacion is given in ems of an ineacion fee enegy pe uni aea, E(h)o he disjoining pessue Π(h) de(h)/dh. These quaniies ae assumed o be known funcions of he film hickness h(,) aound he axisymmeic dop which may change slowly wih ime,. We assume he dop has consan inefacial ension o suface enegy pe uni aea, σ o. The equilibium defomaion of a dop can be obained by minimizing he Helmholz suface enegy of he sysem ha can be wien in ems of he dop heigh z(,) [29]. This mehod has been used o deive he equaion fo an equilibium dop unde exenal foces [3 35]. The suface enegy minimizaion also gives he Young Dupé condiion: σ o cos θ+σ L =σ S fo he equilibium conac angle θ a he base of he dop ha is fa fom he ineacion zone a he apex of he dop (see Fig. ). Howeve, o descibe dynamic defomaions fo which he pinciple of enegy minimizaion would no sicly apply, we can adop a quasi-saic foce balance appoach ha is a genealizaion of he mehod due o Laplace. Conside an aea elemen of he ineface of an axisymmeic dop (Fig. ) whee suface ension foces ac on he peimee of he elemen along he ineface and he pessue diffeence acoss he ineface acs in he diecion of he suface nomal. In he pola diecion, he suface ension foces along a longiude on wo opposing sides of he aea elemen of lengh (φ) dα and (φ+dφ) dα ae: F σ pola = σ o ½ðφÞdαŠðφÞ + σ o ½ðφ + dφþdαšðφ + dφþ ð2::þ whee σ o is he inefacial ension, n and ae he ouwad uni nomal veco and uni angen veco especively, and he angles α and φ ae defined in Fig.. Using he explici expessions fo hese uni vecos: n = sinφ ˆ + cosφ ẑ and = cosφ ˆ sinφ ẑ, we expand Eq. (2..) o fis ode in dα and dφ o give F σ pola = σ o d dφ dφ dα σ o dφdα n ð2::2þ Similaly, in he azimuhal diecion, we have he uni vecos: ˆα = sinα ˆx + cosα ŷ, ˆ = cosα ˆx + sinα ŷ = sinφ n + cosφ,wih d ˆα = dα = ˆ,(Fig. ) so he suface ension foces along a laiude on wo opposing sides of he aea elemen of lengh ds ae o linea ode in he change in azimuhal angle dα: F σ azimuh = f σ o ˆα + σ o ˆα ðα+ dαþgds = σ o ˆdα ds ð2::3þ = σ o sinφ dα ds n σ o cosφ dα ds The nomal foce due o he pessue diffeence acoss he ineface is: F nomal = fp in ðp ou + p + ΠÞgds dα n ð2::4þ whee p in is he inenal pessue of he dop, p ou is he ambien pessue ouside he dop, p is he hydodynamic pessue and Π is he disjoining pessue. The sign convenion is ha p and Π ae posiive if hey ac in he diecion opposie o he ouwad suface uni nomal veco, n. We obain he augmened Young Laplace equaion by equaing he nomal componens of he foces in Eqs. (2..2) (2..4), and using dφ/ds=(d/ds)(dφ/d) =cosφ (dφ/d) dφ p in ðp ou + p + ΠÞ = σ o ds + sin φ ð2::5aþ 2σ o ðp + ΠÞ = σ R o cos φ dφ L d + sin φ ð2::5bþ = σ o d d ð sin φ Þ ð2::5cþ Fig.. Uppe: Schemaic of an axisymmeic sessile dop in a coninuous phase, defomed aound is apex wihin a small ineacion zone of adius, a due o ineacion wih a solid paicle wih adius, R s. Lowe: An illusaion of foces acing on a suface elemen of he axisymmeic sessile dop along he pola and azimuhal diecions. = σ o! z +z 2 = 2 ð2::5dþ

5 Auho's pesonal copy 74 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) 7 9 whee z z/ anφ, see Fig.. Fo lae analysis, i is convenien o define he Laplace adius, R L by p in p ou 2σ o R L ð2::6þ Equaing he angenial componens of Eqs. (2..2) and (2..3) simply gives he ideniy d=cosφ ds (see Fig. ). I is ionic ha key equaions in he heoy of capillaiy have been named afe Thomas Young because in his wiing, he managed o avoid he use of mahemaical noaions and equaions alogehe Special film shapes: Dimple, pimple, wimple and ipple We can deduce a numbe of geneal esuls concening dop defomaion by aking he fis inegal of Eq. (2..5c) wih espec o o give σ o sin φ = σ o 2 R L 2π Φð; Þ ð2:2:aþ Φð; Þ 2π p ; + Π ; d ð2:2:bþ whee he funcion Φ(,) is elaed o he oal foce, F() F ðþ=2π p ; + Π ; d Φð ; Þ ð2:2:2þ acing on he dop due o he hydodynamic pessue, p, and he disjoining pessue, Π. Fo a, boh p and Π ae negligible and Eq. (2.2.a) fo he dop shape ouside he ineacion zone can be wien as sin φ = 2 F ðþ ; fo NN a: R L 2πσ o ð2:2:3þ The above esul does no depend on he exac values of p and Π in he ineacion zone, ba. Thus he dop shape ouside he ineacion zone only depends on he oal foce F(). A vey simila esul was obained some wo cenuies ago in Caesian fom [36,37]. Eq. (2.2.3) povides he necessay bounday condiion fo he numeical poblem of solving fo ime defomaions of he film, down o nanomee hickness in dainage and dynamic foce expeimens (see Secion 4.). The em in backes on he RHS of Eq. (2..5a) is he sum of he local cuvaues a he suface. Using Eq. (2.2.3) we can idenify hese cuvaues ouside he ineacion zone o be K dφ ds = + F ðþ R L 2 2πσ o K 2 sin φ = F ðþ R L 2 2πσ o ð2:2:4aþ ð2:2:4bþ whee he hydodynamic pessue, p and he disjoining pessue, Π ae boh negligible. We noe ha he sum of he cuvaues (K +K 2 ) is independen of he foce as expeced fom Eq. (2..5a). We can now undesand he physical oigin of vaious film shapes obseved and discussed in he lieaue. These shapes: he pimple [38], he dimple [39], he wimple [4] and he ipple [4] simply eflec he numbe of imes he slope of he dop suface: z/ anφ, becomes zeo o changes sign. Fom Eq. (2.2.a) we see ha he sign of he slope is conolled by he quaniy: [(2πσ o 2 /R L ) Φ(,)] sinφ z/, ha is only deemined by he behavio of he Young Laplace equaion. This is he eason why dimple fomaion, fo insance, appeas in heoies ha have vey diffeen models fo film dainage and suface mobiliy condiions [42 48] Equaion fo hin film defomaions In his secion we develop he govening equaions fo he hickness of he film beween a defomable dop and a spheical paicle and beween wo defomable dops in ems of he Young Laplace equaion of he pevious secion. In AFM foce measuemen expeimens, dop defomaion is localized in an ineacion zone aound he apex of he dop. Boh he size of his zone and he exen of he defomaion ae small compaed o he dop adius. This allows us o focus on he popeies of he hin film. How his film is elaed o he es of he dop ouside he ineacion zone will be consideed in Secion Film beween a dop and a spheical paicle As dop defomaions ae confined o a small ineacion zone wihin he adius ~a aound he apex, we seek a descipion of he shape of he film wih hickness, h(,) in his egion. This film hickness is equied o calculae he disjoining pessue, Π(h) and o specify he film bounday in which hydodynamic flow akes place. The geomeic elaion beween h(,) and he dop shape z(,) follows fom Fig. : z; ð Þ = D ðþ+ 2 = ð2r s Þ hð; Þ: ð2:3:þ In he absence of a song aacion beween he dop and he paicle, and ceainly when he ineacion is epulsive, we can expec he vaiaion in he dop shape would be small on he scale of he dop size, ha is, z z/ wihin he ineacion zone. In his case, we only need o eain he linea em in z in Eq. (2..5d). On combining Eq. (2.3.) wih he lineaized fom of Eq. (2..5d) we have he following esul fo he equaion of he film hickness, h(,) beween a dop and a spheical paicle of adius, R s ha is valid in he ineacion zone wihin he adius ~a σ o h = 2σ o R ds Π p; b e a Dop sphee ð2:3:2aþ R ds R L + R s ð2:3:2bþ In he pesence of a song aacion beween he dop and he paicle, he dop shape, z(,) may exhibi a cusp wih an associaed lage gadien. Howeve, in dynamic ineacions involving defomable dops, his siuaion only occus duing he sho ime ineval jus pio o he las sage of a coalescence even. Theefoe as we shall see in bubble bubble coalescence expeimens in Secion 5.2.3, he fac ha z/ may no be saisfied fo he sho ime jus pio o coalescence does no affec he abiliy of he heoy o pedic he fom of he dynamic foce leading up o coalescence and he acual coalescence ime wih good accuacy. In geneal, Eq. (2.3.2a) has o be solved numeically when he disjoining pessue, Π and he hydodynamic pessue, p ae given. We sess ha Eq. (2.3.2a) fo he film hickness is obained fom a lineaizaion of he augmened Young Laplace equaion, Eq. (2..3) and he geomeic condiion Eq. (2.3.). Alhough he fis em on he igh hand side of Eq. (2.3.2a):(2σ o /R ds ) has he dimensions of pessue, i is no he Laplace pessue of he dop. This is ofen a poin of confusion in less han igoous deivaions of his esul [49].

6 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) Eq. (2.3.2a) can be inegaed wice o give h; ð Þ = hð; Þ + 2 log 2R ds 2πσ o 2R dso Φð; Þ ð2:3:3aþ 2 z 2 Ξð; + 2πσ o Þ 2π Ξð; Þ Dop sphee p ; + Π ; log d 2R dso ð2:3:3bþ whee h(,), he film hickness a =, is a consan of inegaion ha is o be deemined. The chaaceisic lengh scale of his dop sphee poblem is R dso (/R o +/R s ), and i is used o scale he logaihmic ems in Eqs. (2.3.3a) (2.3.3b). In he egion Na, he film hickness, h will become sufficienly lage fo Π and p o be negligible so ha he value of in he funcions Φ(,) and Ξ(,) may be eplaced by infiniy o yield he geneal oue asympoic fom of he film hickness h; ð Þ hð; Þ + 2 F ðþ 2R ds 2πσ o log 2R dso + HR ð dso ; Þ; N a Dop sphee ð2:3:4þ Wihin he ineacion zone defined by ~a, in which he slopes: z i /=z i, (i=, 2), we can lineaize Eq. (2.3.6) and use he geomeic elaion (Fig. 2) z ð; D o2 o Fig. 2. Schemaic diagam of he axisymmeic defomaion wihin a small ineacion zone of adius, a aound he apex of wo dissimila dops ineacing in a coninuous phase. Þ + z 2 ð; Þ = D ðþ hð; Þ ð2:3:7þ a h z HR; ð Þ p ; + Π ; log σ o 2R d ð2:3:5þ o deive he equaion fo he film hickness, h(,) beween wo dops σ 2 h = 2σ R ðπ + pþ; b e a Dop dop ð2:3:8þ Eq. (2.3.4) is he limiing fom of he soluion of he film shape Eq. (2.3.2a) beween a dop wih consan inefacial ension and a sphee. Alhough he consans h(,) and H(R dso,) ae as ye unknown a his sage, his soluion can be mached o he soluion valid ouside he ineacion zone (see Secion 2.4) and povide us wih he appopiae bounday condiions fo he numeical soluion of Eq. (2.3.2a). A esul ha is simila o Eq. (2.3.4) has been obained by Yiansios and Davis [23] who used a scaling analysis of he poblem of wo defomable dops appoaching unde a consan buoyancy foce. In Secion 2.5, we will see ha he logaihmic behavio fo Na in Eq. (2.3.4) will mach up wih he inne asympoic behavio of he oue soluion of he augmened Young Laplace equaion Eq. (2.2.3) ha will be developed in he nex secion. This esul is analogous o Hooke's Law fo a linea sping whee he foce exeed on he sping can be deduced fom he exension. Fo a defomable dop, he oal foce, F() acing on i is encoded in is geomeic shape ouside he ineacion zone. This logaihmic limiing fom of he film hickness has been obseved expeimenally [5]. Resuls fo he special case of a dop ineacing wih a fla solid suface can be obained fom Eqs. (2.3.3a), (2.3.3b) (2.3.5) in he limi of he sphee of infinie adius: R s Film beween wo dops Fo wo ineacing dops denoed by i = o 2, wih dop shape, z i (,) and Laplace adii, R Li, locaed on subsaes a he sepaaion, D apa (Fig. 2), he augmened Young Laplace equaion fo each dop is σ oi! z i +z 2 =2 i = 2σ oi R Li Π p; i = o 2 Dop dop ð2:3:6þ The consans R and σ ae defined by R + 2 R L R L2 and σ 2 + σ o σ o2 ð2:3:9þ and ae someimes called he equivalen adius and equivalen inefacial ension. A faco (/2) on he lef hand side of Eq. (2.3.8) appeas fo his dop dop case, when compaed o Eq. (2.3.2a). Eq. (2.3.8) can be inegaed wice o give h; ð Þ hð; Þ + 2 R 2 F ðþ log 2πσ 2Ro +2H R o ; ; N a Dop dop ð2:3:þ again a scale faco R o 2/(/R o +/R o2 ), defined in ems of he unpeubed adii of he wo dops is used. An exa faco of 2 appeas in he logaihm em in Eq. (2.3.) when compaed o he dop paicle esul in Eq. (2.3.4). Again he unknown quaniies h(,) and H R o ; in Eq. (2.3.) can be found by fis solving Eq. (2.2.3) fo he dop shape ouside he ineacion zone in Secion 2.4 and hen maching hese o he inne soluions in Secion Dop shape ouside ineacion zone We sudy he shape of he dop ouside he ineacion zone, Na, by saing wih he soluion of he Young Laplace equaion expessed in Eq. (2.2.3). To simplify he noaion in his secion, we define he quaniies G ðþ F ðþ and R R 2πσ L ð2:4:þ o

7 Auho's pesonal copy 76 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) 7 9 whee G() has dimension of lengh and is a naual lengh scale fo he size of he ineacion zone. Thus Eq. (2.2.3) can be expessed in he simplified noaion sin φ = 2 R G ðþ; fo NN a: ð2:4:2þ Appoaching he apex of he dop fom ouside owads he ineacion zone, fom Eq. (2.4.2) one obseves ha he angen angle, φ, when (GR) /2. Thus we can idenify a [FR/(2πσ o )] /2 as he adius of he ineacion zone. We can inegae he ideniy: dz/d= anφ (Fig. ), using he esul = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R sin φ + sin 2 φ +4G=R ð2:4:3þ obained fom solving Eq. (2.4.2), o give he heigh, z() of he dop ouside he ineacion zone φ z ðþ= θ dz d d dφ dφ = φ 2 R B sin φ C + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia dφ θ sin 2 φ +4G = R ð2:4:4þ This esul can be expessed in ems of ellipic inegals [32,36,37]. Howeve, we will follow a diffeen appoach. Since we ae only ineesed in he esul fo small foces ha can be expessed as (G/R)=F/(2πσ o R), he deviaions of he conac angle: θ=θ o + δθ and he Laplace adius R=R o +δr fom hei unpeubed values, θ o and R o, will be small. Theefoe, we expand he lowe limi of inegaion, θ and he inegand in Eq. (2.4.4) o linea ode in δθ, δr and G ha esuls in 8 9 φ < φ = z ðþ= R sin φ dφ + fr o sinθ o g δθ + : sin φ d φ ; G θ o θ o ð2:4:5þ = R cos φj φ θ o + fr o sinθ o g δθ + cos φ log G 2 cos φ The shape of z() nea he apex of he dop can be found by using he appoximaions found fom Eq. (2.4.2) and valid as φ : j φ θ o volume of he dop, V d is (inegaing he oue soluion, neglecing he small eo of he ineacion zone): zðþ V d = π 2 dz = π 2 dz d d dφ dφ θ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 = πr3 θ sin φ + sin 2 φ +4G=R 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin φ dφ sin 2 φ +4G = R ð2:4:8þ Again expanding his esul o linea ode in δθ, δr and G he consan volume consain can be expessed as δv d = V d δθ + θ o n = πr 3 o sin 3 θ o V d R o δr + o n n o + πr 2 oð cosθ o Þ G = V d G G o o δθ + πr 2 o ð cosθ o Þ 2 ð2+ cosθ o Þ δr ð2:4:9þ This povides a fis elaion beween δθ, δr and G. A second elaionship beween hese hee vaiables mus oiginae fom he way he dop esponds o he exenal foce. Two possibiliies ae ha eihe he conac line emains fixed (pinned conac line o =consan while θ o changes, see Fig. ) o he conac angle emains consan (θ o =consan, while changes). These wo possibiliies will be exploed in he nex subsecions A pinned conac line Fo a conac line pinned a he adial posiion on he subsae =R o sinθ o wih φ=θ=θ o +δθ, (Fig. ) we have fom Eq. (2.4.2) 2 R sin ð θ o + δθ Þ R G = ð2:4:þ and o linea ode in δθ, δr R R o and G his povides a second elaion beween δθ, δr and G n o sin 2 θ o δr + fr o sin θ o cos θ o g δθ + G = ð2:4:þ Thus he consan volume consain, Eq. (2.4.9) and he pinned conac line condiion, Eq. (2.4.) enable us o expess δr and δθ in ems of G sin φ φ + ::: = R G ð2:4:6aþ δr = ð cos θ o Þ G; δθ = G: ð2:4:2þ R o sin θ o cos φ 2 φ2 + ::: = 2 2R 2 + G R + O G2 ð2:4:6bþ Subsiuing hese esuls ino Eq. (2.4.5) and eaining only linea ems in δθ, δr and G gives z ðþ R o ð cos θ o 2R + f cos θ og δr + fr o sin θ o g δθ + log ++ 2R o 2 log + cos θ o G cos θ o Þ 2 ð2:4:7þ This is he fom of he dop shape z() as one appoaches he ineacion zone fom he ouside: a (GR) /2, φ. The las ems ha ae independen of epesen changes o he dop heigh oiginaing fom he influence of he exenally applied foce. In geneal, he vaious peubaion ems δθ, δr and G do no vay independenly. Fo example, if he dop mainains consan volume as i defoms hee will be a condiion ha elaes hese ems. The The dop shape nea he ineacion zone: φ, a in Eq. (2.4.7) can now be expessed eniely in ems of he foce, F, using Eq. (2.4.): z ðþ= R o ð cos θ o + F ðþ log 2R L 2πσ 2R o Þ log + cos θ o cosθ o ð2:4:3þ The fis em on he RHS is he heigh of he undefomed dop, he second em is a local quadaic coecion and he las em expesses he defomaion due o he exenal foce, F() unde a consan volume consain and a pinned hee phase conac line a he base of he dop A consan conac angle Fo defomaions a consan conac angle, θ o, i is obvious ha δθ= in Eq. (2.4.9) and he esuling elaion beween δr and G is δr = ð cosθ o Þð2+ cosθ o Þ G ð2:4:4þ

8 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) so fom Eqs. (2.4.7) and (2.4.), he dop shape nea he ineacion zone: φ, a fo defomaion unde a consan conac angle is: z ðþ= R o ð cos θ o + F ðþ 2R L 2πσ V log + 2R o 2+ cos θ o Þ log + cos θ o cos θ o ð2:4:5þ This esul diffes fom ha in Eq. (2.4.3) in ha he defomaion due o he exenal foce, F() is unde a consan volume consain and a fixed conac angle a he base of he dop. In summay, he oue shape of a defomed dop, z(,) as one appoaches he ineacion zone, a fom above has he fom z; ð Þ R o ð cosθ o Þ 2 2R L + F ðþ 2πσ o log 2R o + Bðθ o Þ ; a ð2:4:6aþ whee he funcion B(θ) depends on whehe he dop defoms wih a pinned hee phase conac line a posiion o wih a consan conac angle θ o + cos θ + log pinned 2 cos θ BðÞ V θ + + cos θ log 2 cos θ 2 + cos θ consan θ ð2:4:6bþ Eqs. (2.4.) o (2.4.6a), (2.4.6b) wee deived peviously by Bados [5] using a moe complicaed appoach. The advanage of he cuen appoach is ha i is moe physically anspaen and compac, wihou he need o eso o he use of ellipic inegals. Also, hee is no need o ea acue and obuse angles sepaaely as was done in Bados' wok. Befoe we mach his esul o he oue limiing fom of he film hickness deived in Secion 2.3, we examine he applicabiliy of he consan volume consain fo he case of a defoming bubble Bubble compessibiliy Assume ha he inenal pessue, P in of he bubble conaining N molecules in a volume, V b obeys he ideal gas equaion: P in V b =NkT, whee k is Bolzmann's consan. If he Young Laplace equaion: P in =P ou +2σ o /R o is valid, he volume is hen: V b ðr o Þ = NkT ð2:4:7þ P ou + 2σ o R o The vaiaion of he bubble volume wih egad o he change in he Laplace adius, δr, is δv b = V b δr = V bðr o Þ 2σ = R o δr R o R o P ou +2σ = R o ð2:4:8þ If he ems ha conain cosθ o in Eq. (2.4.9) ae of ode uniy, he magniude of he coefficien of δr in Eq. (2.4.9) will be compaable o ha of he coefficien of δr in Eq. (2.4.8) when (2σ/R o )~P ou ~ Ba. Thus fo a bubble in wae, bubble compessibiliy conibues abou a 2% effec assuming a bubble adius of 7 μm Maching soluions fo he foce-displacemen fomula We now mach he limiing foms of he soluions of he film equaion jus beyond he ineacion zone deived in Secion 2.3 o he limiing fom of he soluion of he dop shape ouside he ineacion zone as one appoaches he film fom he ouside obained in Secion Dop sphee ineacion A he film bounday, ~a, we use he geomeic condiion in Eq. (2.3.) o mach he film soluion given by Eq. (2.3.4) fo he dop sphee ineacion wih he dop shape soluion given by Eqs. (2.4.6a) (2.4.6b). The consan [h(,)+h(r dso,)] in Eq. (2.3.4) can be eliminaed o give h; ð Þ D ðþ R o ð cosθ o Þ + 2 F ðþ log 2R ds 2πσ o 2R o + Bðθ o Þ ; e a ð2:5:þ The film adius fo he dop sphee ineacion, a ds can be aken o be he posiion whee h/=, and fom Eq. (2.5.), we find a ds F ðþr = 2 dso ð2:5:2þ 2πσ o whee, cf Eq. (2.3.2b) R dso R o + R s ð2:5:3þ is defined in ems of he undefomed dop adius, R o and he sphee adius, R s. On evaluaing Eq. (2.5.) a =a ds,wefind ΔDðÞ D ðþ R o ð cos θ o Þ hða ds ; Þ F ðþ log F ðþr dso 4πσ o 8πσ o R 2 o +2Bðθ o Þ ð2:5:4þ An impoan obsevaion is ha his non-linea foce-displacemen elaionship follows he Young Laplace equaion and implies ha a dop o bubble does no behave as a Hookean sping unde defomaion as ofen assumed [4,52]. This esul is also independen of he deails of he epulsive disjoining pessue which deemines he magniude of he film hickness h(a ds,) ha in pacice is small compaed o he displacemen D. Consequenly, fo epulsive ineacions, measuing he saic o equilibium foce-displacemen elaionship will povide infomaion abou he inefacial ension σ o, he dop adius R o and he conac angle θ o. Howeve, he esul will be insensiive o he deailed fom of he epulsive disjoining pessue. Fo foce measuemens using he aomic foce micoscope, Eq. (2.5.4) eplaces he consan compliance condiion as had conac is no longe a valid concep when defomable dops ae involved Dop dop ineacion Again a he film bounday, ~a, we use he geomeic condiion in Eq. (2.3.7) o mach he film soluion given by Eq. (2.3.) fo he dop dop ineacion wih he dop shape given by Eqs. (2.4.6a) (2.4.6b). The consan [h(,)+2 H(R o,)] in Eq. (2.3.) can be eliminaed o give h; ð Þ D ðþ R o ð cos θ o Þ R o2 ð cos θ o2 Þ + 2 R F ðþ log + Bðθ 2πσ o 2R o Þ o F ðþ log + Bðθ 2πσ o2 2R o2 Þ ; e a o2 ð2:5:5þ The film adius fo he dop dop ineacion, a dd can be aken o be he posiion whee h/=, hus fom Eq. (2.5.5) a dd! F ðþr = 2 o ð2:5:6þ 2πσ

9 Auho's pesonal copy 78 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) 7 9 whee, cf Eq. (2.3.9) + Ro 2 R o R o2 ð2:5:7þ is defined in ems of he undefomed dop adii, R o and R o2.on evaluaing Eq. (2.5.5) a =a dd,wefind ΔDðÞ D ðþ R o ð cos θ o Þ R o2 ð cos θ o2 Þ hða dd ; Þ (! ) F F ðþr log o 4πσ o 8πσR 2 o +2Bðθ o Þ (! ) + F F ðþr log o +2Bðθ 4πσ o2 8πσR 2 o2 Þ F 2πσ o2 ð2:5:8þ which is he foce, F vs displacemen, D elaion (see Fig. 2) fo he ineacion beween wo dissimila defomable dops. The inefacial ensions, σ o and σ o2, he conac angles, θ o and θ o2 and he undefomed dop adii, R o and R o2 can all be measued independenly. To ecapiulae, he heoeical fomulaion in his wok is valid when viscous foces ae small compaed o suface ension foces, ha is, in he egime of small capillay numbe, and he ineacion foce is small as measued by he condiion: F/(2πσ o R). Unde such cicumsances, he size of he ineacion zone, a beween he dops is also small compaed o he dimensions of he dop, R. 3. Hydodynamic ineacions In non-equilibium foce measuemen expeimens, he elaive moion beween he ineacing dops o beween he dop and solid paicle will geneae hydodynamic ineacions ha aise fom he flow of he coninuous phase. In expeimens using he aomic foce micoscope (AFM) o he suface foces appaaus (SFA), he sepaaion beween he ineacing bodies (h b μm) is small compaed o he dimensions of he dops o paicles (R ~ μm), so flow in he hin film beween ineacing bodies povides he dominan conibuion o he measued non-equilibium foce. While he ypical dive velociies used in AFM expeimens (V ~ 5 μm/s) span he ange of Bownian velociies of he dops o paicles, he Reynolds numbe, Re in wae, is small (Re= ρrv/ μ b 2 ) so ha a descipion based on Sokes flow is appopiae. Also he ypical capillay numbe, Ca, he aio of viscous foces o suface ension foces, is small (Ca=μV / σ ~ 6 ) and jusifies he use of he augmened Young Laplace model o descibe dop defomaions. A eview of vaious eamens of hydodynamic ineacions will be he focus of his secion. 3.. Sokes Reynolds lubicaion heoy The descipion of fluid flow beween hin defomable films in he low Reynolds numbe egime is as follows. Hydodynamic flow in he defomable hin film apped by he dop can be descibed using he lubicaion heoy [53,54]. Wihin his axisymmeic film compised of a Newonian liquid wih shea viscosiy μ, he dominan velociy componen, u(,z,) is in he adial -diecion and he pessue, p only vaies in he -diecion. The velociy field is given by he adial componen of he Sokes equaions μ 2 u; ð z; Þ z 2 = p; ð Þ ð3::þ Inegaion of he coninuiy equaion fom z= oh(,) ogehe wih he kinemaics condiion on he film suface gives he geneal evoluion equaion of he film hickness hð; Þ h; ð Þ u; ð z; ÞdzA ð3::2þ Eq. (3..) can be inegaed wih espec o z wice o find u(,z,). In ode o do so, hydodynamic bounday condiions a he film suface (z= and z=h(,)) mus be specified. Subsiuing his soluion ino Eq. (3..2), gives an equaion elaing h(,) and p(,), ha ogehe wih Eqs. (2.3.2a) o (2.3.8) povides a complee descipion of he evoluion of he film (boh spaial and empoal). Diffeen ypes of hydodynamic bounday condiions a he sufaces of solids, dops and bubbles have been poposed. The appopiae choice will be guided by expeimenal condiions. Fo compleeness, we summaize he film evoluion equaions coesponding o diffeen bounday condiions. The assumpion a a solid liquid ineface is o equie he fluid velociy a he ineface o be he same as he velociy of he solid suface. This is efeed o as he no-slip bounday condiion. The analogous condiion a a fluid fluid ineface is he angenially immobile bounday condiion in which he fluid velociy a such inefaces is also specified even hough he inefaces can defom. This condiion is egaded o be appopiae a inefaces populaed by suface-acive molecules ha can aes inefacial flow. A ideal clean liquid liquid o liquid gas boundaies, he fully mobile condiion wheeby one assumes he coninuiy of he angenial componens of he fluid velociy and of he angenial shea sess is expeced o hold. This means i becomes necessay o mach hydodynamic flow inside and ouside he dops. In lubicaion flow, his gives ise o an addiional inegal equaion involving he inefacial velociy ha has o be solved (see Secion 3.5). The case of wo ineacing bubbles is of special inees because of is ubiquious elevance in many aeas of applicaion. Theoeically, i is also a singula case in ha if he fully mobile bounday condiion is applied a he bubble inefaces, he hin film lubicaion Eq. (3..) will only admi a consan plug flow soluion ha povides no infomaion abou he hydodynamic pessue. This led Cheses and Hofman [45] o include ineia effecs in fomulaing hei lubicaion model. Alhough hei numeical calculaions also included he effecs of bubble defomaion, fo non-defoming bubbles his model acually yields an infinie foce beween he bubbles (see Secion 3.6). Such conadicoy esuls sems fom he fac ha he lubicaion fomulaion is no valid when ineial effecs ae dominan [54]. The effecs of a non-unifom disibuion of suface-acive molecules being pesen a an ineface will povide a suface ension gadien along he suface ha can oppose he angenial hydodynamic sess [55]. Howeve, in such a model, i becomes necessay o conside convecion and diffusion of such suface-acive molecules duing ineacion (see Secion 3.7). Finally, he Sefan Reynolds Fla Film Model [2,22] has been used fo a long ime o descibe dop defomaion and associaed film dainage. In is oiginal fom i is aacive because simple analyic soluions ae available (see Secion 3.8). Howeve, he unknown geomeic paamees and inheen conadicoy assumpions of he model mean ha i lacks pedicive capabiliy. This model has spawned a numbe of modificaions, which we call collecively Neo Fla Film models ha involved addiional assumpions and paamees wih inceasing mahemaical complexiy (see Secion 3.9). In spie of such developmens, he abiliy of his model in pedicing expeimenal esuls is limied.

10 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) Non-defoming inefaces The film hickness beween wo non-defoming solid sphees wih adii R a and R b, is given by h; ð Þ = h o ðþ+ 2 ; R H + R H 2 R a R b ð3:2:þ whee he funcion h o () specifies how he sphees ae moved elaive o each ohe. The soluion of he Sokes Eq. (3..) wih he no-slip bounday condiion: u= a z= and z=h(,) is u; ð z; Þ = p 2μ zh z ð Þ: ð3:2:2þ Combining Eqs. (3..2), (3.2.) and (3.2.2), gives he film hinning equaion dh o ðþ = d 2μ p; ð h 3 p This can be inegaed o give Þ = 6μ dh o ðþ d ð3:2:3þ sds ½hs; ð ÞŠ = 3μR H dh o ðþ 3 2 d h o ðþ+ 2 2 =R H ð3:2:4þ The pessue is defined o be zeo ouside he film and fom Eq. (3.2.4) we see he pessue decays as / 4 as. The hydodynamic foce, F() acing beween he sphees, in ems of he sepaaion, h o () and he elaive velociy, dh o ()/d, is F ðþ=2π pð; Þd = 6πμR2 H h o ðþ dh o ðþ d ð3:2:5þ I is posiive fo a epulsive foce as he sphees appoach wih dh o ()/db and i scales wih he squae of he dop adius, R H 2. If he sphees ae diven ogehe unde a consan exenal foce, F ex (N fo he sphees being pushed ogehe) agains hydodynamic epulsion, he sepaaion will decease exponenially wih ime accoding o h o ðþ= h o ðþexpð = τ SS Þ ð3:2:6þ wih chaaceisic decay ime fo film dainage beween he solid sphees τ SS 6πμR 2 H = F ex ð3:2:7þ In his model, he sphees only come ino conac as. If an aacive non-eaded Van de Waals foce wih Hamake consan, A expessed in he Dejaguin appoximaion A F VdW ðhþ = ðπr H Þ 2πh 2 ð3:2:8þ pulls he sphees ogehe agains he hydodynamic epulsive foce, he sepaaion hen vaies wih ime accoding o h o ðþ= h o ðþð =τ VdW Þ = 2 ð3:2:9þ 3.3. Tangenially immobile inefaces Fo axisymmeic flow in he adial diecion in a film wih angenially immobile boundaies, he bounday condiions ae: u= a z= and z=h(,) so he velociy has he same fom as Eq. (3.2.2). Howeve, when his is used in Eq. (3..2) we now have Sokes Reynolds equaion fo he film hickness, h(,) h = 2μ h 3 p ð3:3:þ ha has o be solved simulaneously wih he Young Laplace equaion in eihe Eq. (2.3.2a) o (2.3.8) o obain he hydodynamic pessue, p(,). The ime-dependen foce, F() is hen found using Eq. (2.2.2). The numeical algoihm of he soluion of he Sokes Reynolds Young Laplace equaions will be discussed in Secion Navie slip inefaces In he Navie slip model [56], he angenial componen of he fluid velociy a an ineface is aken o be popoional o he shea sess wih a consan of popoionaliy called he slip lengh. I was posulaed ha his condiion is appopiae fo hydophobic sufaces [57]. Bu he magniude of he slip lengh equied o mach expeimens was unealisically lage in excess of μm. Thee wee also ealie epos of slip obseved in dynamic foce measuemen beween a solid paicle and a fla suface using he aomic foce micoscope [58 6]. Bu subsequen efined measuemens evealed ha he slip phenomenon depended on he ype of he foce sensing canileve used in he expeimen [ 3,3]. Fo compleeness, we give he film dainage equaion beween wo solid sufaces ha obey he Navie slip bounday condiion wih he possibiliy of diffeen slip lenghs b o a z= and b h a z=h. The esuling Sokes Reynolds equaion hen has he fom h = 2μ h 3 p + 4μ " # ðb o + b h Þh 3 +4b o b h h 2 h + b o + b h! p ð3:4:þ I conains an addiional em compaed o he angenially immobile model of Eq. (3.3.). A classical no-slip condiion will be obained by seing boh slip lenghs o zeo. If one suface (say h ) belongs o ha of an ideal bubble wih zeo viscosiy, whose suface canno susain any angenial shea sess, one can assume he limi b h. Howeve, as we shall see in Secion 3.6, we canno obain he esul fo film dainage beween he sufaces of wo such ideal bubbles fom Eq. (3.4.) Two dops wih mobile inefaces If he inefaces of he ineacing dops wih inenal viscosiy μ d, canno susain angenial sess, he soluion of Eq. (3..) will equie he velociy U(,) of he ineface o be non-zeo. Based on he coninuiy of he angenial sess acoss he ineface, he following se of coupled equaions mus be used insead of Eq. (3.3.) [23,6,62]: h = 2μ h 3 p ð h U Þ mobile dops ð3:5:aþ whee he coalescence ime, defined by h o (τ VdW )=, is τ VdW 36πμR H ½h o ðþš 2 = A: ð3:2:þ 2μ d U; ð Þ = ϕð; ρ Þ hðρ; Þ p ð ρ; Þ ρ dρ ð3:5:bþ

11 Auho's pesonal copy 8 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) 7 9 ϕð; ρþ = ρ π 2π cos θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dθ ð3:5:cþ 2 + ρ 2 2ρ cos θ The dimensionless numbe: m (μ/μ d )(R/h) /2 [23,6] deemines he behavio of he soluion of he above equaions. Fo m, he inefaces end owads angenial immobiliy, while fo m, he dainage aes ae much fase and he dynamics of he sysem ae mainly deemined by he inenal flow of he dops [63]. Ohe fomulaions fo he dainage beween dops wih mobile inefaces exis ha may appea simple [64,65], bu unfounaely he eamen of he flow field inside he dop is no coec. The coec accoun of mobile inefaces equies he use of an inegal equaion o couple he inefacial velociy o he shea sess [6] Two bubbles: Cheses Hofman model If he ineacing bubbles have fully mobile sufaces ha canno susain shea sess, he soluion o Eq. (3.2.2) ha is valid fo low Reynolds numbes, would be a consan plug flow velociy ha povides no infomaion on he pessue disibuion. This led Cheses and Hofman [45] o conside he ineacion beween wo idenical defomable bubbles wih mobile sufaces a high Reynolds numbes when viscosiy effecs of he fluid may be negleced. The Reynolds lubicaion eamen was kep, while only ineial effecs wee included in hei model. Eq. (3..) was eplaced by he equaion fo he plug flow velociy U(,) in he adial diecion of he film: U + U U = p ρ wih ρ he liquid densiy. The coninuiy equaion now becomes ð3:6:þ h = ð h UÞ ð3:6:2þ Assuming an iniial film pofile of h(,)=h o (=)+ 2 /R o wih iniial bubble adii, R o hese equaions wee solved numeically ogehe wih he Young Laplace Eq. (2.3.8). The disjoining pessue was se o zeo (Π=). The bubbles wee assumed o appoach wih consan velociy, V. If he bubbles do no defom, we will have h o ()=h ini V, and Eq. (3.6.2) can be inegaed o give he plug flow velociy U; ð Þ = V ð3:6:3þ 2 h o ðþ+ 2 = R o The pessue, p can hen be found fom Eq. (3.6.) h i p; ð Þ = ρv 2 R 2 2h o o ðþ+ 2 = R o 8 h o ðþ+ 2 2 ð3:6:4þ =R o Howeve, he esulan foce, F() beween he bubbles, found by inegaing his pessue ðþ=2π F p ; d ð3:6:5þ is infinie because he pessue in Eq. (3.6.4) does no decay fas enough as fo he inegal in Eq. (3.6.5) o convege. The eason is ha he lubicaion equaion failed o mach o he full Navie Sokes equaion ouside he hin film when only ineia effecs ae consideed [54]. Cheses and Hofman [45] consideed defomable bubbles fo which only numeical soluion of Eqs. (3.6.) and (3.6.2) wee given. They concluded ha film upue always occus a he coalescence ime CH ρvr 2 o = σ: ð3:6:6þ wihou he need fo any aacive suface foces. This ime is measued fom he momen a which he bubbles would have ouched had defomaion been absen. In conas, he models discussed Secions , all equie he pesence of aacive suface foces [66] in ode fo film upue o occu a a finie ime. Fo 5 μm bubbles aveling a 5 μm/s, he pediced coalescence ime of 9 s is much shoe han any expeimenal obsevaions. Wihou fuhe invesigaion, i is no clea if he divegence poblem associaed wih he slow decay of he pessue field can be avoided by suface defomabiliy. In expeimens on ising bubbles, he zeo shea sess condiion has been obseved only when exeme cae has been aken o deionise and clean he wae [67 72]. Howeve, when he same bubbles ise owads a solid suface such as a iania plae, hei ae of appoach suggesed ha he bounday condiion a he bubble suface is again a angenially immobile condiion [73,74]. The pecise physical eason fo his behavio has ye o be esablished alhough ace impuiies ha oiginae fom he iania plae o accumulaed duing bubble ise may be implicaed. Diec measuemens of bubble bubble ineacions using he aomic foce micoscope sugges ha ace suface-acive impuiies in he sysem ae sufficien o ende he bubbles o exhibi angenially immobile inefaces even hough exeme cae has been aken o avoid such impuiies (see Secion 5). Thus i may be difficul in pacice o achieve and mainain he level of cleanliness o guaanee he zeo shea sess condiion a bubble sufaces Bubbles wih suface-acive species The pesence of mobile suface-acive molecules o anspo pocesses associaed wih chemical eacions, maeial anspo o empeaue gadiens can give ise o suface ension gadiens. The coesponding bounday condiion a such inefaces will be he coninuiy of he angenial componens of he fluid velociies and a jump in he angenial shea sess acoss he ineface balanced by he suface ension gadien [55]. These follow fom kinemaic and angenial foce balance consideaions. In addiion, effecs such as inefacial viscosiy and suface elasiciy have been poposed as being impoan when sufacans ae pesen a inefaces. Theoeical consideaions of hese effecs involve he inoducion of model paamees ha ae difficul o deemine independenly. While such eamens modified he Sokes Reynolds dainage equaion, he coesponding effecs on he deivaion of he Young Laplace equaion had no been consideed [75]. Consequenly, a balanced appoach is no available. Howeve as discussed in Secion 3.6, hee is expeimenal evidence ha he pesence of ace amouns of suface-acive species would ende an ineface o be angenially immobile so ha he complex effecs of suface-acive species ha have been posulaed heoeically do no feaue in dynamic foce measuemens Sefan Reynolds Fla Film Model The complexiy of he Sokes Reynolds Young Laplace model has led o he developmen of a numbe of appoximae heoies aimed a descibing film dainage dynamics based on he Sefan Reynolds Fla Film Model [2,22]. Howeve, all hese models fail o pedic quaniaive ageemen when compaed wih expeimens [76]. In spie of he inheen inconsisencies of he oiginal model, numeous exensions and modificaions have been poposed wih new addiional feaues. These exensions and modificaions ae no

12 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) always well jusified fom a physics poin of view [77] and ae no capable of poducing he esuls solving he Sokes Reynolds Young Laplace equaions. We eview hese key issues in his secion. Conside he Sefan Reynolds Fla Film Model of a dop (o bubble) appoaching a fla solid suface. Insead of using he Young Laplace equaion o accoun fo he defomaion of he dop, he defomaion is assumed o have he shape of a cicula fla disk wih adius, a (see Fig. 3). Howeve, he dimension of he disk adius, a is no known a pioi. Assume ha he dop appoaches he suface due o an exenal ime-dependen foce, F ex. The fluid in he film has a adial velociy, u(,z,) (assuming axial symmey and Sokes flow) and he film is now of unifom hickness, h(). A z= (he locaion of he solid suface), he bounday condiion u(,,) = is se. If he suface of he dop a z=h, is angenially immobile, he condiion u(,h,)= applies. If on he ohe hand he suface of he dop is assumed o have zeo shea sess, hen he condiion u(,z,)/z= mus be applied a z=h. The pessue pofile can be obained fom Eqs. (3..) and (3..2) wih he above bounday condiions as p; ð Þ = p o 3μ dh a 2 2 ; b b a ð3:8:þ βh 3 d The pessue in he coninuous phase, Na, is epesened by p o, and μ is is viscosiy. The consan β is aken o be β= if he fla suface of he dop is angenially immobile and β=4 if he fla suface of he dop is fully mobile. This pessue pofile gives ise o he hydodynamic foce (posiive fo epulsion beween he dop and he fla suface) a F hydo =2π ðp p o Þd = 3πμa4 2βh 3 dh d ð3:8:2þ The quadaic pessue pofile in Eq. (3.8.) has a maximum a = fo an appoaching dop wih dh/db and decays monoonically o p o a he oue film egion (=a). I pesens us wih an immediae inconsisency. Since he defomed ineface of he dop is assumed o be a fla disk a he ouse, hen accoding o he Young Laplace equaion, he pessue on eihe side of such a fla ineface mus be equal. Theefoe he quadaic pessue disibuion given by Eq. (3.8.) wihin he fla film is inconsisen wih he unifom pessue inside he dop. If we ignoe his inconsisency fo he ime being and assume hee is a disjoining pessue, Π(h), acing on he film wih aea (πa 2 ), a quasi-saic foce balance on he dop in he z diecion gives: F ex + F hydo + πa 2 ΠðhÞ = ð3:8:3þ The exenal foce F ex is aken o be posiive if i acs o push he dop owads he fla suface. This has been he saing poin of modeling he sabiliy of daining films unde he acion of Van de Waals foces [78]. A geneal pedicion of he Sefan Reynolds model is ha a epulsive disjoining pessue will slow down he ae of dainage, dh/d, accoding o Eq. (3.8.3). As we shall see in Secion 5, his is opposie o expeimenal obsevaions and pedicions of he full soluion of he Sokes Reynolds Young Laplace equaions Neo Fla Film models Numeous expeimenal sudies since he oiginal expeimen of Dejaguin and Kussakov [3] have demonsaed ha eal films ae no fla. To accommodae such expeimenal evidence, hee have been a numbe of aemps o develop coecions o he Reynolds Fla Film Model. All of hese aemps involve simplificaions of he Sokes Reynolds paial diffeenial equaion given in Eq. (3.3.) by making a se of assumpions ha involve mahemaical elaions beween he acual film hickness, h(,) and he aveage film hickness, h av () defined by h av ðþ 2 a a 2 h; ð Þd ð3:9:þ Hee, he Reynolds film adius, a will also be ime-dependen accoding o Eq. (3.8.3) if he exenal foce is no consan. The following assumpions have been poposed: (a) he quasi-seady assumpion [75,79]: hð; Þ dhav ðþ d hð; Þ h av ðþ (b) he homogeneous assumpion [79]: ð3:9:2þ hð; Þ h av ðþ h; ð Þ h av ðþ ð3:9:3þ Fex z R (c) he assumpion of small deviaions fom he aveage film hickness [79]: ½h; ð Þ h av ðþ Š 2 bb½h av ðþ Š 2 ð3:9:4þ (d) he aveage hickness decays exponenially wih ime, [79] h av ðþ= h av ðþe b ð3:9:5þ h Fig. 3. Schemaic diagam of he Sefan Reynolds Fla Film Model fo a dop o bubble wih adius R, appoaching a fla solid suface. The adius of he fla film egion is a. a u whee b and h av () ae consans o be found by fiing o expeimenal daa. The assumpions given in Eqs. (3.9.) (3.9.5) ae hen used o educe he Sokes Reynolds Young Laplace coupled paial diffeenial equaions fo he film hickness, h(,) and he hydodynamic pessue, p(,): h = 2μ h 3 p ð3:9:6þ

13 Auho's pesonal copy 82 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) 7 9 σ 2 h = 2σ ΠðhÞ p R L ð3:9:7þ soluions of he govening equaions ha will faciliae quaniaive compaisons beween heoy and expeimens ae also discussed. fo he case of wo ineacing idenical dops wih consan inefacial ension, σ and Laplace adius, R L, o a single linea odinay diffeenial equaion [79] σ 2 ΔΔh ð ; Þ + Π ð h avþ h av Δh; ð Δ ; V ðþ dh avðþ d Þ + 2μVðÞ h 4 avðþ h; ð Þ = ð3:9:8aþ ð3:9:8bþ A numbe of papes and eviews have been devoed o obain appoximae soluions o analyze he behavio of his appoximae linea equaion [49,8 82]. Howeve, he assumpions in Eqs. (3.9.2) (3.9.5) have always been acceped wihou jusificaion and ceainly have no been esed by quaniaive compaison wih numeical soluions of he oiginal Eqs. (3.9.6) and (3.9.7). Fo insance, he inepeaion of he deivaive (h/h av )ineq.(3.9.2) and he jusificaion of Eq. (3.9.3) have no been esablished, and Eq. (3.9.5) is ceainly no coec if a sable film can be fomed. Thus in spie of numeous papes devoed o his appoach o analyze film dainage, sabiliy and coalescence is domain of validiy is sill vey much an open quesion and appeas o be a souce of confusion [77]. 4. Sokes Reynolds Young Laplace model In his secion we summaize applicaions of he Sokes Reynolds Young Laplace (SRYL) model o diec foce measuemens involving defomable dops. In paicula, we give explici foms of he govening equaions and bounday condiions ha need o be solved. We also ouline peubaion soluions ha can eveal he key physics of he poblem even hough he esuls ae only applicable fo weak ineacions. In a foce measuemen expeimen wih he aomic foce micoscope (AFM), he foce, F beween wo objecs is obained fom he deflecion, S of a canileve wih calibaed sping consan, K: F=KS, by vaying he posiion X of he canileve (Fig. 4). The absolue sepaaion, h beween ineacing igid objecs is infeed fom he so-called consan compliance egion when he wo objec ae in had conac when he condiion: ΔS=ΔX holds. As a esul, Fvs ΔX daa can be conveed o F vs h infomaion. Fo ineacions involving defomable dops, he concep of had conac no longe exiss as he dops can defom. Howeve, we can deive analyic fomulae fo ineacing dop sphee and dop dop configuaions o eplace he consan compliance condiion fo ineacing igid objecs. Deails of implemenaions of numeical z (,) 2 h(,) z (,) dop, sphee o bubble dop o bubble K D() X() Fig. 4. The geomey of he aomic foce micoscope in which he disances D and X ae defined. S 4.. Govening equaions and bounday condiions Fo he Sokes Reynolds film dainage equaion, we will conside in deail only he case fo which he angenially immobile bounday condiion holds a he film boundaies. This is guided by he fac ha all expeimenal esuls ha we have gaheed in his eview ae found o be consisen wih his condiion (see Secion 5). In he domain bb max ha conains all deailed infomaion abou ineacions involving he defomable film we solve he Sokes Reynolds equaion (see Eq. (3.3.)) ha elaes he ae of he change of he film hickness, h(,) o he hydodynamic pessue, p(,): h = 2μ h 3 p ð4::þ Fo numeical calculaions, he choice of max will be discussed in Secion 4.5. Fo he axisymmeic ineacions consideed hee, we apply he symmey condiions: h/==p/ a =. As appoaches max, we see fom Eqs. (2.3.4) and (2.3.) ha he film hickness, h inceases wih a quadaic dependence in which will esul in a pessue ha decays like 4, see Eq. (3.2.3). This asympoic pessue behavio can be implemened as he condiion: (p/)+4p= a = max. The ime-dependen foce can be calculaed using Eq. (2.2.2). In a numeical implemenaion, we use F ðþ 2π max p ; + Π ; d +2π p ; ð Þ d ð4::2þ since he disjoining pessue Π is sho-anged and is negligible fo N max, and he second inegal can be evaluaed analyically using he 4 dependence of he pessue expessed in he fom: p(,) p( max,) ( max /) 4,fo N max. The Young Laplace equaion povides a second equaion beween he film hickness, h(,) and he pessue, p(,). As discussed in Secion 2, his equaion akes on diffeen foms fo dop sphee o dop dop ineacions. To simplify he noaion, we define a consan n n = ; Dop Sphee 2; Dop Dop max ð4::3þ We only conside dops wih consan inefacial ensions fo which he Young Laplace equaion can be wien as σ n n h = 2σ n R n Π p ð4::4þ wih he equivalen suface ension and equivalen adius defined as: σ n = σ o ; Dop Sphee 2 ð=σ o +=σ o2 Þ ; Dop Dop ( ð R n = =R o +=R s Þ ; Dop Sphee 2 ð=r o +=R o2 Þ ; Dop Dop ð4::5þ ð4::6þ whee he dops have consan inefacial ensions: σ o, σ o and σ o2, undefomed adii: R o, R o and R o2, and he solid sphee has adius: R s.the ineacion beween a dop and a fla plae coesponds o he limi R s. As long as he defomaions of he dops ae small compaed o he dop size, a condiion ha is well saisfied, we can appoximae he Laplace adii by he undefomed adii of he dops in Eq. (4..4).

14 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) The iniial film hickness is aken o be ha beween unpeubed dops and has he fom: h; ð Þ = h ini + n2 2R n ð4::7þ To solve he Sokes Reynolds Young Laplace Eqs. (4..) and (4..4) we need one moe bounday condiion. In non-equilibium foce measuemens using he AFM, he displacemen, X() of he end of he foce sensing canileve (Fig. 4) deemines he ype of collision ha he dops will expeience and povides he final bounday condiion a max. The ineacion foce, F beween a defomable dop and a sphee is elaed o he canileve deflecion, S=F/ K whee K is he canileve sping consan. The final bounday condiion can be found by diffeeniaing he oue asympoic fom of he film hickness, Eq. (2.5.), wih espec o ime, and use he geomeic elaion D=S+X (see Fig. 4) o give, a = max hð max ; Þ = dxðþ + d K dfðþ d dfðþ log 2πσ o d max 2R o + Bðθ o Þ Dop Sphee ð4::8þ The funcion B(θ) is defined by Eq. (2.4.6b). We noe ha in pevious wok [48,62], only he fis em on he RHS of Eq. (4..8) is used in he so-called consan velociy bounday condiion. The second em on he RHS of Eq. (4..8) accouns fo he defomaion of he dop ouside he film as well as he effec of he deflecion of he AFM canileve. Fo he ineacion beween wo dops we diffeeniae he coesponding asympoic fomula in Eq. (2.5.5) o give he equied bounday condiion a = max hð max ; Þ = dxðþ + dfðþ dfðþ log max d K d 2πσ o d 2R o dfðþ log max + Bðθ 2πσ o2 d 2R o2 Þ Dop Dop o2 + Bðθ o Þ ð4::9þ The las 3 ems on he RHS of Eq. (4..9) ae conibuions o he bounday condiion due o deflecion of he canileve and defomaions of he dops ouside he ineacion zone. We have now specified all he govening equaions and appopiae bounday condiions ha ae necessay o model non-equilibium foce measuemen using he AFM Scaled equaions fo compuaions Wih appopiae scaling, he Sokes Reynolds Young Laplace equaions have a geneal fom fom which geneal feaues of he soluion can be exaced. To illusae he appoach, we conside ineacions in a foce measuemen expeimen wih he aomic foce micoscope whee he displacemen funcion, X() of he end of he foce-sensing canileve is specified as a funcion of ime and he foce will vay. We also conside ineacions a consan foce whee he elaive velociy will vay wih ime Ineacion unde given displacemen funcion Le V be a chaaceisic value of he piezo dive velociy, dx()/d of he aomic foce micoscope. The aio of viscous foces o suface ension foces is chaaceized by he capillay numbe Ca μv/σ n. Fo he expeimenal sysems consideed hee, Ca~ 6. Non-dimensionalizaion of he SRYL equaions leads o a univesal fom of he sysem of equaions wih he following scaling paamees [83]: film hickness: h, z~ca /2 R n, adial coodinae: ~Ca /4 R n, ime: ~Ca /2 R n /V, pessue: p~σ n /R n and foce: F~Ca /2 σ n R n. The Sokes Reynolds equaion ha descibes film dainage beween wo dops wih immobile inefaces, Eq. (4..), becomes (using aseisks fo dimensionless vaiables), h = 2 h 3 p while he Young Laplace Eq. (4..4) becomes n h =2 Π p wih Π (R n /σ n )Π. The iniial condiion in Eq. (4..7) becomes ð4:2:þ ð4:2:2þ h ; = h o + n ð Þ 2 : ð4:2:3þ 2 Apa fom he scaled disjoining pessue, hese equaions conain no paamees. The bounday condiion a max, given by Eqs. (4..8) o (4..9), has only a weak logaihmic dependence on he capillay numbe Ca h max ; h max ; = dx d 2π = dx df (! ) d log Ca = 4 R n max + Bðθ 2 R o Þ Dop Sphee o ð4:2:4þ d σ n df (! ) 2π σ o d log Ca = 4 R n max + Bðθ 2 R o Þ o σ n df (! ) 2π σ o2 d log Ca = 4 R n max + Bðθ 2 R o2 Þ Dop Dop o2 ð4:2:5þ whee dx()/d V dx ( )/d. Fo he case of wo idenical dops, numeical soluions of hese equaions in he absence of a disjoining pessue, Π=, and fo a consan appoach velociy: dx()/d= V [84] evealed ha he film pofile will fis exhibi a dimple when he cenal sepaaion eaches he value h=; ð Þ = cr n Ca = 2 h dimple ð4:2:6þ In Eq. (4.2.6), c is a consan anging fom abou.3 fo Ca~ o abou.5 fo Ca~ 4. Eoneously, in Manica e al. [84] and Chan e al. [85] he coesponding consan quoed fo h dimple was fo a dop agains a solid plae, namely, ~.4 fo Ca~ and~.7 fo Ca~ 4. The maximum shea sess fo he above dop dop case is abou τ max ~.5Ca /4 σ n /R n and occus a he im posiion, im ~3 Ca /4 R n.i akes abou ~5 Ca /2 R n /V fo he hickness a he im o dain fom h dimple o half his value Ineacion unde consan foce The consan foce case coesponds o df/d=. This can be modeled by choosing a convenien consan velociy dx()/d= V and monio he foce, F() unil i eaches he desied value, F o,a some ime = o, and hen se dx()/d= fo N o. Assuming he disjoining pessue is negligible (Π~), a univesal se of equaions

15 Auho's pesonal copy 84 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) 7 9 conaining no paamees a all [83] appeas again using he scaling paamees: film hickness: h, z~f o /σ n, adial coodinae: ~(F o R n /σ n ) /2, ime: ~μr n 2 /F o and pessue: p~σ/r n Fo his case, he dimple in he film will fis appea a he sepaaion h=; ð Þ =:8ðF o = σ n Þ h dimple ð4:2:7þ The maximum shea sess of τ max ~.25 (F o σ n /R n 3 ) /2 is eached jus afe he ime of dimple fomaion and is locaed aound he im egion of he film. As film dainage poceeds, he shea sess gadually diminishes. Fo N2 μr n 2 /F o, he im adius eaches a consan value im ~.375 (F o R n /σ n ) /2. The im educes o ½ h dimple aound ~ μr n 2 /F o and o abou. h dimple a ~ μr n 2 /F o exhibiing an asympoic ime dependence of 2/3 [23] Peubaion soluions The Sokes Reynolds Young Laplace (SRYL) Eqs., (4..) and (4..4) fom a pai of coupled paial diffeenial equaions ha can only be solved numeically. Noneheless, some of he key physics of non-equilibium ineacions involving defomable dops can be exaced by consideing a peubaion soluion of he SRYL equaions. Such soluions have been found fo axisymmeic ineacions beween dops diven ogehe unde consan foce [23] and also fo dop dop ineacions in he Hele Shaw micofluidic channel geomey [9]. We emak ha alhough he deivaion of he diffeen foms of he augmened Young Laplace equaion, Eqs. (2.3.2a) and (2.3.8), aleady involved a lineaizaion in he dop shape, he esuling equaions ae sill non-linea funcions of he film hickness, h because of he pesence of he hydodynamic pessue, p and disjoining pessue, Π(h). In addiion, he Sokes Reynolds equaion ha descibes film hinning, Eq. (3.3.) is also non-linea in h Axisymmeic dops Fis we summaize esuls fo he case of wo idenical axisymmeic dops wih inefacial ension, σ o and undefomed adius, R o. Boh dops ae assumed o es on fla subsaes wih conac angle, θ and he subsaes ae diven a elaive velociy V() N fo sepaaing dops. We seek soluions fo he film hickness and pessue of he fom h; ð p; ð Þ h o ð; Þ + h ð; Þ ð4:3:þ Þ p o ð; Þ + p ð; Þ ð4:3:2þ by choosing he paabolic pofile h o (,) H()+ 2 /R o as he efeence shape. The peubaions h (,) and p (,) ae found by subsiuing Eqs. (4.3.) and (4.3.2) ino (4..) and (4..4) and eaining only linea ems in h and p. The soluion, in he absence of a disjoining pessue is [85] h; ð Þ = H ðþ+ 2 = R o +! (! ) 3μR2 ov ðþ log H = R o 4σ o H ðþ 4R o +2BðÞ θ ð4:3:3þ wih H()=H o + V(τ) dτ. This peubaion soluion is valid when he film capillay numbe, Ca f ~(μv o /σ o )(R o /H o ) 2, wih V o being he chaaceisic velociy. The em in baces in Eq. (4.3.3) is negaive so ha he defomaion h (,) and he paabolic pofile h o (,) have opposie signs. This has wo physical implicaions. Fo appoaching dops coesponding o V() b, he peubaion will cause he cenal sepaaion o hicken and his is he physical oigin of dimple fomaion. The peubaion soluion pedics a ciical cenal film hickness h dimple ~α Ca /2 R o a which dimple fomaion will occu a consan velociy. Howeve, when compaed o numeical soluions of he SRYL equaions, he consan α is oo lage by an ode of magniude. This is pehaps no supising since dimple fomaion acually occus a sepaaions whee non-defoming dops would have ovelapped, a egime beyond he validiy of he fis ode peubaion esul above. Fo sepaaing dops coesponding o V() N and he cenal sepaaion, H() inceasing wih ime, he peubaion h (,) will iniially be negaive and deceases he cenal film hickness. Fo sufficienly lage film capillay numbe Ca f, he iniial decease in cenal film hickness can bing he sepaaion down o he ange whee he de-sabilizing influence of Van de Waals aacion can ake hold and iniiae coalescence. Such coalescence on sepaaion phenomenon has been obseved expeimenally [25,86 9]. The defomaion behavio unde consan foce condiions (e.g. due o buoyancy) has also been sudied by peubaion mehods. The esul shows ha he cenal sepaaion evolves wih an exponenial dependence on ime unde a consan exenal foce, F ex [23,85]: h i H ðþ= H o exp 2F ex = 3πμR 2 o ð4:3:4þ Dops in he micofluidic Hele Shaw cell geomey The film hickness, h(x,) and he pessue, p(x,) fo wo ineacing pancake-shaped dops in he Hele Shaw cell geomey ae deemined by he following coupled equaions in he absence of a disjoining pessue [9] h = p 3μ x h3 x σ o 2 h x 2 = σ o R o p ð4:3:5þ ð4:3:6þ in which he spaial coodinae x is ansvese o he axis of symmey. Again, he angenially immobile hydodynamic bounday condiion has been assumed a he dop ineface. The simple fom of he SRYL equaion in he Hele Shaw geomey means ha he asympoic analysis discussed in Secion 2 can be caied ou elaively easily [92]. The naual peubaion paamee is he capillay numbe in Hele Shaw geomey: Ca HS ~(μv o /σ o )(R o /H o ) 3/2 which diffes fom ha of he axisymmeic case by a 3/2-powe dependence on he aspec aio (R o /H o ). Howeve, he effecs of he peubaion em due o defomaions of appoaching o sepaaion dops ae qualiaively he same as in he axisymmeic case Foce-displacemen fomula fo AFM expeimens In AFM foce measuemen expeimens involving igid bodies, he absolue sepaaion can be infeed fom he consan compliance egions of he foce vs canileve displacemen esponse ha occus when he wo ineacing bodies come ino had conac. Fo ineacions involving defomable dops, had conac does no occu as he dops can defom. In place of his limiing behavio, we can use he geomeic elaion in he AFM fo a dop sphee ineacion (Fig. 4): D()=X()+S(), whee he canileve deflecion S()=F()/K is elaed o he foce, F and canileve sping consan, K by Hooke's law. Thus using he esul in Eq. (2.5.4) we have he esul

16 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) ha eplaces he consan compliance condiion fo he dop sphee ineacion Δ X ðþ F ðþ log F 4πσ o 8πσ o ðþr dso R 2 o +2Bðθ o Þ 4πσ o K Dop Sphee ð4:4:þ The funcion B(θ) isdefined ealie in Eq. (2.4.6b). A low diving velociies, small dx/d, his elaion is valid fo he enie foce ange. A highe diving velociies, his elaion is valid when he foce is sufficienly lage. In Secion 5, we will see an example of how his esul can be applied in pacice. Similaly fo AFM foce measuemens beween wo dissimila defomable dops, we use he esul in Eq. (2.5.8) o give (! ) (! ) ΔXðÞ F ðþr log o +2Bðθ 4πσ o 8πσR 2 o Þ + F F ðþr log o 4πσ o o2 8πσR 2 o2 +2Bðθ o2 Þ F 2πσ + Dop Dop 2πσ K ð4:4:2þ This is a genealizaion of a simila esul given ealie fo dops wih he same inefacial ension [93]. Unfounaely ha esul conained a ypogaphical eo. Boh Eqs. (4.4.) and (4.4.2) ae vey useful fo checking he lage foce limi of expeimenal foce vs canileve esuls as well as fo checking numeical soluions of he Sokes Reynolds Young Laplace equaions. They ae applicable in he egime when he ineacing dops have been pushed ogehe a he sepaaion below which hey would ovelap if hey had no defomed Numeical algoihm In his secion, we give deails on how o solve he scaled nondimensional Sokes Reynolds Young Laplace equaions deived in Secion 4.2: h = 2 h 3 p h n =2 Π p wih he iniial condiion, h ; = ho n bounday condiions a = h p == and a = max p +4p = h max; = dx ð Þ d + Φ n df max d ð4:5:þ ð4:5:2þ ð4:5:3þ ð4:5:4þ ð4:5:5þ ð4:5:6þ whee Φ n ( max ) he coefficien of df /d in Eq. (4.2.4) fo dop sphee (n =) o Eq. (4.2.5) fo dop dop (n =2) ineacions. The scaled foce is given by Eq. (4..2) afe using he / 4 asympoic fom fo he pessue fo N max : F max 2π p ; + Π ; d + π 2 max p max ; ð4:5:7þ The pessue p can be eliminaed beween Eqs. (4.5.) and (4.5.2) o obain an equaion fo h /. The -deivaives of he esuling equaion can be disceized using cenal diffeencing in [, max ]o obain a se of coupled diffeenial equaions fo H k ( ) h (k Δ, ), k=,.., N and Δ= max /N. The funcion F ( ) is elaed o all he H k ( ) via Eq. (4.5.7) whee he inegal can be evaluaed by Simpson's ule and can be wien as: F = w k gh k ð4:5:8þ k The sysem of coupled fis ode equaions fo he ime deivaive dh k ( )/d H k hen has he fom :: H f :: H B : : f : C :: Φ AB C B H N A dx = d C A :: F k w k gh ð k Þ F whee f k epesen he ems esuling fom he disceizaion in coesponding o each equaion fo dh k ( )/d. This sysem has a singula mass maix and is a diffeenial-algebaic equaion. I can be solved using sandad sofwae ouines such as ODE5S in Malab. Once he diving funcion dx/d is specified, Eq. (4.5.9) can be solved fo N. In acual implemenaions, o obain numeical answes ha ae independen of domain size o 5 significan figues we choose a domain size max =, wih a sep size Δ =.2 which poduces a sysem of 5 equaions. A complee foce cuve can be compued in abou min on a noebook compue. 5. Compaisons wih expeimens We complee his eview by showing how he Sokes Reynolds Young Laplace (SRYL) model can be applied o undesand he dynamic behavio in diffeen ypes of non-equilibium expeimens. We demonsae in Secion 5. ha he SRYL model can make accuae quaniaive pedicions abou he evoluion of he shape of defomable films apped beween dops o bubbles as hey undego non-equilibium ineacions wih solid sufaces o wih ohe dops. In Secion 5.2, we compae non-equilibium foce measuemens involving dops and bubbles using he aomic foce micoscope (AFM) wih pedicions of he SRYL model. Once he SRYL model is shown o be able o give an accuae accoun of ime vaiaions of he nonequilibium foces, we can confidenly use he model o infe he spaial and empoal evoluions of he shape of he film beween ineacing dops and bubbles fo ineacions ha esul in film sabiliy o coalescence. The availabiliy of a quaniaive heoy ovecomes one of he limiaions of he expeimenal mehods based on he AFM, namely ha a pesen, he film pofile beween ineacing dops canno be obseved diecly. In all sysems ha we have consideed, he disjoining pessue conains only conibuions fom Van de Waals and elecical double laye ineacions in he DLVO heoy [,2]. To calculae he Van de Waals ineacion whee i is dominan in bubble bubble coalescence sudies a high sal concenaions, we use he full Lifshiz heoy [94] wih he mos complee dielecic daa available. In expeimens whee he elecical double laye epulsion is dominan, we use he full non-linea Poisson Bolzmann heoy o calculae he disjoining

17 Auho's pesonal copy 86 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) 7 9 pessue. Values of he suface poenials and elecolye concenaions ae aken fom independen expeimenal measuemens. 5.. Dynamic defomaions In his secion, we highligh examples of expeimens ha measue he non-equilibium shapes of dops and bubble duing dop dop, bubble solid and dop solid ineacions. We compae pedicions of he SRYL model wih such non-equilibium expeimens Opposing poubean dops In his expeimen, he ineacion is beween opposing poubean half dops of glyceol ha have emeged fom he ends of wo sealed capillaies (3 mm diamee) in silicone oil 47V3. The dops have an iniial adius of R o =.52 mm, while he inefacial ension fo his sysem is σ o =3 mn/m. The dops ae aached o he capillaies wih a conac angle, θ=9º [48,84]. The dops wee diven ogehe fom es a an iniial lage sepaaion, by mechanically moving one capillay owads he second wih a consan velociy, V=6.7 μm/s. Seing he ime = o he ime when he wo dops would have ouched if hey would no have defomed, he appoach was sopped a sop =27 s. The finges obained wih a lase-induced inefeence paen wee used o measue he film hickness pofile, h(,) as a funcion of posiion and ime. A compaison of he measued and pediced finge paen jus a he ime when he dive sopped is shown in Fig. 5, ogehe wih samples of he film hickness h(,) a ealie imes. Noe ha a dimple was obseved o develop afe 3 s. This sysem is ulimaely unsable in ha he silicone oil film beween he glyceol dops coninues o hin and evenually he dops coalesced. The ime scale of he appoach owads coalescence is consisen wih he angenially immobile hydodynamic bounday condiion a he glyceol-silicone oil ineface Bubble agains quaz plae The evolving shape of a wae film apped beween an expanding bubble ha has been pessed agains an opically fla hydophilic quaz plae has been measued by an opical mehod [76,95,96]. The bubble, iniially 4 μm fom he quaz plae, wih unpeubed adius.6 mm, was expanded fom an oifice wih diamee 2 mm in a facion of a second. The shape of he apped wae film ha subsequenly dained was ecoded. The final equilibium film was sabilized by elecical double laye epulsion beween he quaz suface and he bubble, and depending on he added elecolye concenaion, he dainage pocess ook up o 2 s. In a dainage expeimen, he film hickness a a fixed adial posiion is measued as a funcion of ime wih a esoluion of abou 3 μm. The film pofile h(,) is hen e-consuced fom epeaing such measuemens a diffeen posiions [76,95,96], poving implicily he epoducibiliy of he expeimens. A compaison of he expeimenal and pediced pofiles of he non-equilibium evolving wae film is given in Fig. 6 [97]. Again he ime scale of he dainage pocess is consisen wih a angenially immobile hydodynamic bounday condiion a he bubble suface and a no-slip bounday condiion a he quaz plae Mecuy dop agains mica plae The ime evoluion of he shape of he aqueous elecolye film apped beween a mecuy dop and an appoaching o eceding moleculaly smooh mica plae has been measued by acking inefeence finges of equal chomaic ode [98,99]. The suface poenial of he conducing mecuy dop was conolled independenly o give epulsive o aacive elecical double laye disjoining pessues ha allowed he evoluion of sable and coalescing films o be invesigaed. We highligh he esuls of wo expeimens ha demonsae how an iniially sable film esponds o elecical and mechanical peubaions [87,]. In Fig. 7, we show he collapse of an iniially sable film fomed beween he mica plae and he mecuy dop ha ae boh negaively chaged. The film was sabilized by elecical double laye epulsion. A =, he sign of he poenial of he mecuy dop was changed o posiive so he elecical double laye ineacion became aacive and caused he film o collapse. Fo he mos pa, he collapse pocess eained axial symmey and he mecuy dop jumped ino conac wih he mica plae a he edge of he film whee he elecolye can easily dain fom he film. Howeve, i is eniely possible ha he axial symmey can be boken a he final momen of collapse, duing a ime ha is oo sho o be esolved by he expeimen mm NaCl 2 s 3 5 s mm NaCl 2 s 3 s 5 s s Fig. 5. The pofile of he silicone oil film apped beween wo glyceol dops ha ae diven ogehe unil 27 s. A dimple develops a 3 s a a sepaaion of 5.5 μm. Noe he vey diffeen hoizonal and veical scales. The symbols epesen he expeimenal daa ha wee obained fom he inefeence paens such as he one shown below fo 27 s. The solid lines epesen he numeical soluion. The igh side of he inefeence paen shows a numeical econsucion based on he SRYL heoy, he lef hand side is he obseved expeimenal paen s 2 s 2 Fig. 6. The axisymmeic shape of he daining wae film hickness, h(,) as a funcion of posiion, beween a bubble and a quaz plae in.25 mm and mm NaCl. Expeimens (symbols), heoy (solid lines).

18 Auho's pesonal copy D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) h(, ) (nm) s s. s dop. Such coalescence on sepaaion behavio has now been obseved diecly using he aomic foce micoscope (see Secion 5.2) Dynamic foce measuemens We highligh examples of non-equilibium foce measuemens involving defomable dops [2 7] and bubbles [9,8,9] using he aomic foce micoscope (AFM). The obseved feaues ha aise fom dynamic defomaions ae compaed wih quaniaive pedicions of he SRYL model. 3.2 s Jump in Jump in (µm) Fig. 7. An iniial sable equilibium aqueous elecolye film (. mm KCl) beween a mica plae and a mecuy dop wih suface poenial of he same sign as he mica is made o collapse by changing he polaiy of he conducing mecuy dop a =. The pofile of he collapsing axisymmeic film is shown a = s,.8 s, 2. s and finally a 3.2 s jus befoe he mecuy jumped ino conac wih he mica plae. Expeimens (symbols), heoy (solid lines) Dop sphee ineacion An example of he compaison beween AFM expeimen and heoy fo non-equilibium foces beween a silica mico-sphee and a eadecane emulsion dop in wae wih 5 mm SDS is shown in Fig. 9 [2]. Thee is vey good ageemen beween expeimen and heoy paiculaly in elaion o he deails of he hyseeic loop beween he appoach and eacion banch of he foce-displacemen cuve in elaion o diffeen dive velociies. Fo hese esuls, he pedicions of he analyic foce-fomula, Eq. (4.4.), ae also in good quaniaive ageemen wih expeimens when he foce becomes epulsive. Wha is clea is ha he collapse pocess is pediced wih qualiaive accuacy by he SRYL model. Thee is no evidence of he mechanism of hemal flucuaions ha has been posulaed as being he key mechanism fo iniiaing dop coalescence []. In anohe expeimen, afe he fomaion of an equilibium wae film beween he mica plae and he mecuy dop, sabilized by elecical double laye epulsion, he mica plae was eaced vey apidly fom he mecuy dop. The defomaional esponse of he mecuy dop o such a peubaion, pio o jumping apa, is shown in Fig. 8 ogehe wih pedicions fom he SRYL model. Pio o he mica plae sepaaing compleely fom he mecuy dop, poions of he aqueous film acually became hinne duing he ansiion peiod. This slighly coune-inuiive behavio aises because he defomable mecuy ineface is able o espond o he aacive hydodynamic pessue geneaed in he film when he mica plae was eaced apidly. Fo he esul in Fig. 8, if he disjoining pessue is songly aacive a below ~8 nm i is possible ha he eacion of he mica plae can induce collapse of he aqueous film ahe han being able o sepaae fom he mecuy Dop dop ineacion Examples of he non-equilibium foce as a funcion of ime beween wo decane emulsion dops in 3 mm SDS and mm NaNO 3 ae shown in Fig. [6]. Fo displaying esuls of dynamic foce measuemens i is pefeable o show he explici ime dependence of he measued foce because he canileve displacemen, fo example in Fig. 9, does no conain diec infomaion abou ime. Again he SRYL model is capable of poviding accuaely he ime dependence and vaiaions of he deph of he eacion minimum wih dive speed Bubble bubble ineacion The simple expeimen of diving wo bubbles ogehe o sepaaing hem in a well conolled and chaaceized manne while measuing diecly he dynamic foce beween hem has only been aemped wih quaniaive success ecenly using he aomic foce micoscope [9]. In he expeimen, one bubble was mouned on he subsae and he ohe anchoed a one end of a ecangula 4 3 h(, ) (nm) s.32 s.48 s.64 s Foce (nn) s (µm) Fig. 8. The esponse of he equilibium aqueous elecolye film (. mm KCl) beween a mica plae and a mecuy dop as he mica plae is eaced vey apidly fom he dop. Befoe he mica and he mecuy dop sepaae, pas of he aqueous film became hinne in ansiion. Expeimens (symbols), heoy (solid lines)..2.4 Displacemen X (µm) Fig. 9. The non-equilibium foce beween a mico-sphee and a eadecane dop in 5 mm SDS soluion measued on he AFM as a funcion of canileve displacemen. Fo visual claiy, he zeo foce level fo esuls a μm/s has been displaced veically by nn. Theoeical esuls fom he SRYL model: and fom Eq. (4.4.):, expeimens (symbols).

19 Auho's pesonal copy 88 D.Y.C. Chan e al. / Advances in Colloid and Ineface Science 65 (2) a Sop 5 Foce (nn) 2 2 Appoach 2 Appoach only Time (s) Fig.. The non-equilibium foce, measued on he AFM as a funcion of ime fo diffeen dive velociies beween wo decane dops in an aqueous soluion ha conains 3 mm SDS and mm NaNO 3 soluion. The ime axis is scaled by he oal ime of he collision, oal. Resuls fom he SRYL model:, expeimens (symbols). 2 b Appoach Reac foce-sensing canileve. The ohe end of he canileve was diven owads he subsae a a nominal speed of~5 μm/s. Thee diffeen collision modes wee employed in ode o sudy bubble collision and coalescence phenomena: () Appoach only collision: In his case he canileve is moved a consan speed while he epulsive foce beween he bubbles inceased monoonically wih ime. Fo wo bubbles wih adii 62 μm and 86 μm and iniial sepaaion of 5.5 μm, he measued foce is shown in Fig. a as a funcion of ime. The bubbles coalesced when he foce eached abou 5 nn. (2) Appoach-Sop collision: Hee, he canileve was diven owads he subsae a he same nominal speed bu was hen sopped while he ineacion beween he bubbles evolved owads he final sae. The foce coesponding o such a collision beween bubbles wih adii 67 μm and 85 μm and iniial sepaaion of.65 μm, is also shown in Fig. a. Noe ha afe he canileve sopped, he bubbles coninued o evolve unde an almos consan foce condiion (~27 nn) befoe coalescence evenually occued. (3) Appoach-Reac collision: In he hid mode of collision he canileve was diven owads he subsae a he same nominal speed fo a pedeemined ime ineval and hen eaced a he same speed. The oucome of such a collision depended ahe sensiively on he iniial sepaaion fo he same maximum displacemen of he canileve. The key deeminan is he disance ove which he wo bubbles have been pushed ogehe beyond he poin of conac if hey did no defom. Fo a pai of bubbles wih he same adii (74 μm) a an iniial sepaaion of 2.45 μm, he foce (Fig. b) eached a maximum of abou 8 nn and hen deceased duing he eacion phase, eached a minimum of abou 6nN in magniude befoe sepaaing evenually. No coalescence occued fo his case and he foce cuves ae simila o hose obseved in dop paicle and dop dop expeimens. The expeimen was epeaed wih he same bubbles and he same canileve displacemen funcion, bu saing a a smalle iniial sepaaion of only 2.5 μm. The foce eached a lage maximum value aound 8 nn (Fig. b). Bu insead of eaching a minimum duing he eacion phase as in he pevious Appoach-Reac case, he bubbles coalesced insead duing he sepaaion sage. This coune inuiive behavio is simila o he esuls obseved in Secion 5..3 when he mica plae was eaced apidly fom a poximal mecuy dop sepaaed by a sable film. Foce (nn) Deailed foms of he non-equilibium foce cuves fo diffeen collision modes descibed above ae pediced wih quaniaive accuacy by he SRYL model. Fuhemoe, he same heoy also pediced coecly he coalescence ime unde all hee diffeen collision scenaios [9]. 6. Conclusion Time (s) Fig.. The ime-dependen foce beween wo bubbles (adii 5 μm) duing collision evens a a nominal dive speed of ~5 μm/s in.5 M NaNO 3 aqueous elecolye: expeimens (symbols), heoy (solid lines). Coalescence is indicaed by down aows. The esuls of hee collision poocols ae shown: (a) Appoach only and Appoach-Sop. The lef veical axis epesens he Appoach Sop foce values (maximum ~27 nn), while he igh veical axis epesens he Appoach only case (maximum ~5 nn). The oigin of ime is abiay. (b) Appoach-Reac fo he same bubble pai a diffeen iniial sepaaion, h o. In his eview we have given a deailed developmen of he Sokes Reynolds Young Laplace (SRYL) equaions ha povided a consisen accoun of non-equilibium ineacions beween defomable dops and bubbles which included suface foces, hydodynamic effecs and suface defomaions in an inenally consisen way. Phenomena elevan on he scale of he dop size such as how he dops ae diven ogehe ae imposed as appopiae bounday condiions fo he SRYL equaions ha focus on descibing defomaions on he scale of he hin film beween ineacing dops. The model is applicable in he paamee egime elevan o collision of dops in he μm size ange and elaive appoach velociies ha span Bownian hemal velociies of such emulsion dops. I gives accuae pedicions of he dynamic foce measued using he aomic foce micoscope fo conolled collusions involving defomable