Post-Tanner stages of droplet spreading: the energy balance approach revisited
aa r X i v : . [ c ond - m a t . s o f t ] S e p Post-Tanner stages of droplet spreading: the energybalan e approa h revisitedS Me hkov , A-M Cazabat and G Oshanin , Laboratoire de Physique Théorique de la Matière Condensée, Université Pierreet Marie Curie, 4 pla e Jussieu, 75252 Paris Cedex 5 Fran e Laboratoire de Physique Statistique, E ole Normale Supérieure, 75252 ParisCedex 5 Fran e Laboratory J.-V. Pon elet (UMI CNRS 2615), Independent University ofMos ow, Bolshoy Vlasyevskiy Pereulok 11, 119002 Mos ow, RussiaE-mail: me hkovlptm .jussieu.fr, anne-marie. azabatlps.ens.fr,oshaninlptm .jussieu.frAbstra t. The spreading of a ir ular liquid drop on a solid substrate an bedes ribed by the time evolution of its base radius R ( t ) . In omplete wettingthe quasistationary regime (far away from initial and (cid:28)nal transients) typi allyobeys the so- alled Tanner law, with R ∼ t α T , α T = 1 / . Late-time spreadingmay di(cid:27)er signi(cid:28) antly from the Tanner law: in some ases the drop does notthin down to a mole ular (cid:28)lm and instead rea hes an equilibrium pan ake-likeshape; in other situations, as revealed by re ent experiments with spontaneouslyspreading nemati rystals, the growth of the base radius a elerates after theTanner stage. Here we demonstrate that these two seemingly on(cid:29)i ting trends an be re on iled within a suitably revisited energy balan e approa h, by takinginto a ount the line tension ontribution to the driving for e of spreading: apositive line tension is responsible for the formation of pan ake-like stru tures,whereas a negative line tension tends to lengthen the onta t line and indu esan a elerated spreading (a transition to a faster power law for R ( t ) than in theTanner stage).PACS numbers: 68.08.B Keywords: spreading, the Tanner law, line tensionSubmitted to: J. Phys.: Condens. Matterost-Tanner stages of droplet spreading: the energy balan e approa h revisited 21. Introdu tionThe spreading of a liquid on a solid surfa e is a ompli ated pro ess where manyfa tors ome into play, not ne essarily known and not always ontrollable: the kineti behavior may be strongly in(cid:29)uen ed or even dominated by the volatility and vis osityof the liquid (or by other rheologi al parameters if the liquid is non-Newtonian),by the presen e of impurities in the bulk phases ( hemi al ontaminants, surfa tants,polymers, et ), by the roughness or texture of the surfa e, or by its rystalline stru tureand hemi al omposition. In onsequen e, the spreading of thin (cid:28)lms is generallydependent upon the details of the stru tures and intera tions in the o-existing phases.By ontrast, for thi ker (cid:28)lms and drops one expe ts a less spe i(cid:28) behavior, des ribedby universal laws. However, although the prominent features of the spreading ofma ros opi drops are relatively well understood [1, 2, 3℄, a omprehensive theoreti alframework in whi h all of the experimentally observed phenomena harmoniously (cid:28)ndtheir pla e is still la king at present.Here we are on erned with a standard textbook problem - the spontaneousspreading of a non-volatile drop on an ideal, (cid:29)at, lean, horizontal, homogeneoussolid surfa e (see (cid:28)gure 1). The drop has a ma ros opi size but is su(cid:30) ientlysmall that we an ompletely dis ard the e(cid:27)e ts of gravity. The spreading parameter S = σ SG − σ SL − σ is positive, so that the drop tends to over as mu h of the solidsurfa e as possible to shield it against the gas phase; σ SG , σ SL and σ are the interfa ialtensions of the solid/gas, solid/liquid and liquid/gas interfa es, respe tively.We fo us on the following three known features of spontaneous spreading: • The base radius R ( t ) of a ir ular drop grows during spreading and, atintermediate times t (far from initial and (cid:28)nal transients), typi ally obeys a powerlaw R ( t ) ∼ t α T with α T = 1 / [4, 5℄, known as the Tanner law. • The (cid:28)nal stage of spreading of non-volatile droplets is not always a mole ular(cid:28)lm. Sometimes a (cid:29)at, bounded stru ture is rea hed instead - a so- alled pan ake(see [1℄ and [6, 7, 8℄). • An a eleration of the spreading pro ess (an apparent transition from Tanner'spower law to a faster one) has been observed for spontaneously spreading nemati liquid rystals [9, 10℄. Experiments revealed an algebrai growth R ( t ) ∼ t α with α nearly twi e as large as the exponent α T hara terizing the Tanner law: α = 0 . [9℄ and α = 0 . [10℄.At present time, it is well understood why the radius R ( t ) of a spontaneously spreading ir ular drop grows in proportion to t / . This law has been derived analyti ally[4, 5, 11℄ and on(cid:28)rmed experimentally on many a ounts [5, 12, 13, 14℄. Thefundamental argument is that the hydrodynami s in the bulk of a drop are drivenby apillary for es alone, whi h dire tly yields R ∼ t / for a self-similar bulk, inthe lubri ation approximation [4, 5℄. Alternatively, the trend an be regarded asa ompetition between the hydrodynami dissipation (primarily in the onta t lineregion of the drop) and an unbalan ed apillary for e [1, 2, 3, 11℄. Note that thelaw is rather universal - in the sense that the observed exponent / is most oftenindependent of the pre ise nature of the spreading liquid - and has been observednot only for simple liquids, but for oils, polymeri liquids, liquid metals and nemati liquid rystals. For non-Newtonian liquids some deviations from the Tanner law anbe observed, attributable to their spe i(cid:28) rheologi al properties and hen e, spe i(cid:28) features of the hydrodynami dissipation in the bulk.ost-Tanner stages of droplet spreading: the energy balan e approa h revisited 3 microscopicprecursormicroscopicprecursor microscopiccontact linemicroscopic contact linemacroscopiccontact linefilm (foot)mesoscopicmesoscopic film (drop)macroscopic liquid drop tangent at inflection pointsolid substratesolid substrate(a)(b) R t ( )
R t ( ) θ ( ) t Figure 1. a) A sket h of a ma ros opi liquid droplet spreading on a solidsubstrate. R ( t ) and θ ( t ) denote the base radius and the onta t angle of thema ros opi part of the droplet, respe tively, as inferred from the in(cid:29)e tion pointat the apparent onta t line. b) A sket h of a liquid droplet at a late spreadingtime, in the situation where the (cid:28)nal stage of spreading is a mesos opi pan ake.An in(cid:29)exion point may still exist, but the relevant R ( t ) now orresponds to thelimit between the mi ros opi and mesos opi regions; the orresponding θ ( t ) isunde(cid:28)ned.The reason why in some ases a spontaneously spreading droplet attains anequilibrium pan ake-like form is also lear. Su h (cid:29)at pan ake-like stru tures aresometimes more favorable energeti ally than mole ular (cid:28)lms: this o urs when short-range intera tions promote dewetting, even though the overall situation is that of omplete wetting. Theoreti ally, su h stru tures have been predi ted and analyzed in[1℄ and [6, 7℄. They were also observed experimentally (see, e.g., [15℄). A key featureof the spreading pro ess as it rea hes a pan ake shape - as a transient following theTanner stage - is that the base radius of the drop tends to a stationary value (see (cid:28)gure1b). It must be noted, though, that the shape of su h a drop di(cid:27)ers onsiderably fromthat of a apillary ap; the geometri al meaning of R ( t ) is also quite di(cid:27)erent.As opposed to the Tanner law and the emergen e of mesos opi pan akes, thephysi al origin of the a elerated spreading observed in [9, 10℄ has yet to be lari(cid:28)ed.Also, the latter trend is apparently in on(cid:29)i t with the notion that a Tanner stagemust be terminated by the onset of either a mole ular (cid:28)lm or a mesos opi pan ake.Our motivation in this paper is to identify the essential fa tors of the spreadingkineti s that might have been disregarded so far, and thus to a hieve a more omplete qualitative understanding of the standard textbook problem. We fo us morespe i(cid:28) ally on the spreading dynami s that may develop after the Tanner stage andtry to a ount for the two apparently on(cid:29)i ting trends, possibly by introdu ing newnotions or me hanisms. As a (cid:28)rst step in this dire tion, we perform an analysis withinthe lassi al framework of an energy balan e approa h [1℄. This approa h is knownto provide a qualitative derivation of the Tanner law, apturing only a few essentialfeatures of the phenomenon, in a physi ally transparent fashion. As noted in [16℄,the energy-based equations are fun tionally equivalent to the standard hydrodynami approa hes used in the literature but are lighter in terms of analyti al al ulations andost-Tanner stages of droplet spreading: the energy balan e approa h revisited 4assumptions involved (as ompared to, e.g., phenomenologi al boundary onditions,should the thi kness pro(cid:28)le be des ribed by a di(cid:27)erential equation).Within this approa h, the Tanner law is obtained by balan ing the rate of energydissipation in the spreading ma ros opi droplet and the driving for e of spreading,whi h is taken equal to the unbalan ed Young for e. We point out that the analysis ofde Gennes [1℄ disregards the line tension ontribution to the driving for e of spreading[17, 18℄. Typi ally, the line tension τ is very small - only − to − N (see, e.g.,[19, 20, 21℄) - and it is legitimate to negle t it when dealing with large, essentially apillary droplets. Nonetheless we show that, when su h a ontribution is taken intoa ount, a onsistent, non- on(cid:29)i ting pi ture emerges, with the following trends for adrop (supposedly well-approximated by a spheri al ap): • At su(cid:30) iently early spreading times the e(cid:27)e t of τ is negligible and the Tannerstage holds. • At long times and for negative values of τ , the spreading pro ess rosses overto a signi(cid:28) antly faster power law than Tanner's R ∼ t / (as observed for thenemati droplets [9, 10℄). • At long times and for positive values of τ , the growth of R ( t ) slows down andterminates at a (cid:28)nite value R ( ∞ ) . This latter trend is indi ative of the emergen eof pan akes.Therefore, the approa h presented in this paper resolves a seemingly ontroversialbehavior of spreading pro esses, provided that a properly de(cid:28)ned line tension τ istaken into a ount. We note, however, that this approa h is justi(cid:28)ed only in the aseof ma ros opi drops, for whi h both the surfa e tension and the line tension are validnotions.The paper is outlined as follows. In se tion 2 we (cid:28)rst present the derivation ofthe Tanner law in terms of the energy balan e approa h. Then, in se tion 3, we revisitthe standard pi ture by analyzing di(cid:27)erent fa tors whi h may in(cid:29)uen e the spreadingkineti s, espe ially the notion of line tension. Finally, in se tion 4 we dis uss possiblelimitations of our approa h.2. Energy balan e approa hTo lay the basis of our analysis, we start with the derivation of the Tanner law withinthe framework of the energy balan e approa h, originally presented by de Gennesin his 1985 review paper [1℄. Additional details, dis ussions and appli ations of thisapproa h an be found in [14, 16℄. Further on, in se tion 3, we will dis uss a oupleof additional fa tors whi h are missing from the seminal approa h, but may a ountfor the abnormal spreading behavior observed after the Tanner stage.In (cid:28)gure 1a we sket hed a typi al on(cid:28)guration for a ma ros opi liquid dropletspreading spontaneously on a solid substrate. The drop an be (cid:16)divided(cid:17) into thefollowing three regions: a (cid:16)ma ros opi (cid:17) bulk, a (cid:16)mesos opi (cid:17) (cid:28)lm (a region that iswithin the range of surfa e for es), and a (cid:16)mi ros opi (cid:17) pre ursor, the thi kness ofwhi h amounts to several mole ular diameters. Note that (cid:28)gure 1 is s hemati andthe relative sizes of these three regions are not up to s ale. For brevity, we willhen eforth not use the adje tives (cid:16)ma ros opi (cid:17), (cid:16)mesos opi (cid:17) and (cid:16)mi ros opi (cid:17); wewill instead refer to the three regions as the bulk droplet, the (cid:28)lm and the pre ursor.We note (cid:28)rst that the edge of the pre ursor spreads well ahead of the (cid:28)lm andthe bulk droplet: the radius of the pre ursor grows as √ t [22, 23, 24, 25, 26, 27, 28℄ost-Tanner stages of droplet spreading: the energy balan e approa h revisited 5and the (cid:28)lm plays the role of a reservoir feeding the pre ursor; this pi ture is valid aslong as the said reservoir is far from being exhausted. Thus the pre ursor is de oupledfrom the rest of the drop and its only role in the pro ess an be seen as lubri atingthe substrate for spreading of the (cid:28)lm and the bulk drop.A fundamental assumption of this approa h is that the length s ales of the bulkand the (cid:28)lm are well separated, i.e., that the bulk of the drop is mu h wider and tallerthan the (cid:28)lm. Thus the bulk an be adequately approximated by a spheri al ap withthe base radius R ( t ) , onta t angle θ ( t ) , and nearly onstant volume V ( V = π R θ for su(cid:30) iently small θ ), i.e., the bulk is in equilibrium at onstant volume V and aninstantaneous base radius R ( t ) .Then, we an de(cid:28)ne an instantaneous free energy πR ( t ) F ( t ) along with aninstantaneous rate of energy dissipation πR ( t ) W ( t ) , and propose that the evolutionof these two quantities obeys the standard relationship of the me hani s of dissipativesystems: ( W macro + W meso + W micro ) R = − U d ( R F )d R , (1)where U = d R/ d t is the instantaneous velo ity of the apparent onta t line. Theright-hand-side (r.h.s.) of (1) is the rate of hange of the free energy of the system. Itis equivalent to the power of the driving for e d F/ d R applied to the moving onta tline, and is balan ed by the total dissipation that o urs in the system, i.e., the left-hand-side (l.h.s.) of (1). The terms W macro , W meso and W micro are the dissipationrates in the bulk, (cid:28)lm and pre ursor, respe tively, divided by the length πR of theapparent onta t line. Keep in mind that F ( t ) , W ( t ) and the driving for e d F/ d R arealso, by de(cid:28)nition, redu ed by the length πR ( t ) .Next we spe ify the r.h.s. of (1) [1℄: − R d ( R F )d R ≃ S + σ (1 − cos θ ) . (2)Equation (2) takes into a ount the surfa e energies of the three interfa es meeting atthe ma ros opi onta t line and determines their variation with respe t to R ( t ) at onstant spheri al ap volume. Note that the result is equivalent to a straightforwardappli ation of the Young law; in fa t, the r.h.s. of (2) is typi ally referred to as an(cid:16)unbalan ed Young for e(cid:17).As for the l.h.s. of (2), we an formally de ompose the dissipation a ording tothe regions outlined in (cid:28)gure 1a, as follows: W macro orresponds to hydrodynami dissipation in the bulk drop, where vis ous (cid:29)ows are driven by the apillary pressure; W meso orresponds to hydrodynami dissipation in the (cid:28)lm, where vis ous (cid:29)ows aredriven by the disjoining pressure; W micro orresponds to fri tion at the mi ros opi s ale, both at the edge of the (cid:28)lm and in the mole ular pre ursor.The dissipation in the bulk drop is well-approximated by that in a wedge, and isof the form W macro ≃ η U g ( θ ) ln (cid:12)(cid:12)(cid:12)(cid:12) x max x min (cid:12)(cid:12)(cid:12)(cid:12) , (3)where η is the vis osity, x max and x min are e(cid:27)e tive uto(cid:27) lengths for the integrationover the droplet height and g ( θ ) is a known fun tion of the instantaneous onta tangle. A salient feature is that the leading asymptoti behavior of g ( θ ) when θ → is g ( θ ) ≃ /θ , whi h means that W macro exhibits an unbounded growth as θ → .In the following we shall use the notation κ = 3 ln (cid:12)(cid:12)(cid:12) x max x min (cid:12)(cid:12)(cid:12) : this is a slow-varying,ost-Tanner stages of droplet spreading: the energy balan e approa h revisited 6empiri al quantity, whi h varies only slightly as the droplet spreads and introdu esminor, logarithmi orre tions to the power laws; experimental data suggest that agood hoi e is a nearly- onstant κ ≈ [13℄.We now ome to the dissipation in the (cid:28)lm. A striking result of Hervet and deGennes [11℄ is that the omplete wetting regime is hara terized by W meso ≃ S U, (4)whi h means that the dissipation within the (cid:28)lm ompensates exa tly the (cid:28)rst termon the r.h.s. of (2), rendering the rate of spreading independent of S . This result wasobtained for non-retarded van der Waals substrate for es, but an be generalized.Finally, the form of the dissipation term W micro was dis ussed by Blake andHaynes [29, 30℄: it was found that W micro ∼ ζU at leading order in U , where ζ is a onstant fri tion oe(cid:30) ient. Note that in the ase of omplete wetting ζ is dependenton the thi kness of the pre ursor.Thus, provided that S is onsumed entirely in the (cid:28)lm, the dynami al behaviorresults from a ompetition between the two remaining dissipation hannels, W macro and W micro . As θ → , W micro is θ -independent, whereas W macro ∼ /θ and thus learly dominates at long spreading times. Dissipation at the mi ros opi onta t linemay dominate (e.g., for low vis osity liquids) at intermediate times, but ultimatelyhydrodynami dissipation in the ore drop will take over [1, 14℄.Consequently, negle ting W micro as ompared to W macro , one (cid:28)nds in the small- θ limit that (1) adopts the following form: θ ≈ κ Ca , (5)whi h is a fundamental relation between the velo ity of the moving onta t line andthe instantaneous value of the onta t angle ( Ca = η U /σ is known as the apillarynumber). It was derived analyti ally by Voinov [4℄ and by Tanner [5℄ using a di(cid:27)erentapproa h (within the lubri ation approximation).Now taking into a ount that the volume V ≈ π R θ of the bulk drop remainsnearly onstant during spreading, we obtain ˙ R = 64 π V (cid:16) κη σ (cid:17) − R − , (6)from whi h the Tanner laws R ∼ t / and θ ∼ t − / ensue trivially. The behaviordes ribed by (5) and (6) has been observed experimentally in [5, 12, 13℄.3. Energy balan e approa h revisitedEquations (2), (5) and (6), under the assumption of well separated length s ales ofthe bulk and (cid:28)lm, predi t an unbounded growth of R . This is an ideal spreadingbehaviour, through whi h the droplet virtually thins down to a mole ular (cid:28)lm. As wehave already remarked, this is not always the ase, as the spreading may terminatewith the appearan e of equilibrium pan ake-like stru tures [1, 6, 7℄. In the lattersituation, the Tanner law learly des ribes an intermediate stage, and the transitionto a pan ake must be des ribed by some kind of rossover in terms of R ( t ) : ourintuition is that R will tend to a (cid:28)nite value although it is not lear whether thede(cid:28)nition of R will remain onsistent with (cid:28)gure 1a.An opposite trend was revealed by re ent experimental studies fo using on thespontaneous spreading of nemati liquid rystals ( yanobiphenyl 5CB) on hydrophili ost-Tanner stages of droplet spreading: the energy balan e approa h revisited 7[9℄ or hydrophobi [10℄ substrates: after a transient Tanner stage, an (cid:16)a eleration(cid:17) ofthe spreading pro ess has been observed. The data suggest that the base radius R , asinferred from the in(cid:29)e tion point of the thi kness pro(cid:28)le, grows algebrai ally but withan exponent whi h is substantially larger than α T = 0 . . In [9℄ it was shown that theTanner law rosses over to R ∼ t α with α ≈ . . Later it was realized (see (cid:28)gure 6in [10℄) that the Tanner relation in (5) does not hold for late stages of spreading: forsmall θ and Ca the best (cid:28)t to the experimental data follows θ ∼ Ca . rather than thatpredi ted by (5). The latter relation, together with the volume onservation ondition R θ ∼ V yields R ∼ t α with α ≈ . . A more thorough analysis of the behaviordepi ted in (cid:28)gure 5 in [10℄ suggests that a tually the reported law R ∼ t / is only apart of a rossover from the Tanner stage to an even faster growth law. Experimentaldata in [10℄ span time s ales ranging from a se ond to two hours, and at the end ofthe experiment the trend is learly rather R ∼ t / than R ∼ t / , and possibly stilla elerating. Consequently, although there is no on lusive eviden e on the pre isevalue of the exponent α hara terizing the a elerated spreading regime, it is lear that α is signi(cid:28) antly larger than the Tanner exponent and thus the physi al me hanismresponsible for the late, post-Tanner stages of spreading might be di(cid:27)erent from theone des ribed in se tion 2.3.1. First guess: shear thinningWe noti e (cid:28)rst that the a elerated spreading was observed for nemati liquid rystals.Nemati rystals are known to have a non-Newtonian, shear-thinning rheology (see,e.g., [31, 32℄). Shear thinning a(cid:27)e ts the (cid:29)ow pattern, whi h ne essarily modi(cid:28)esthe spreading dynami s. Thus our (cid:28)rst idea is to revisit the l.h.s. of (1) and, morespe i(cid:28) ally, the term W macro in (3). The expression (4) for W meso is also queried inthe ase of shear thinning.A detailed analysis of the onta t line dynami s within the framework of thethin (cid:28)lm model shows that the hara teristi shear rates in the apillary wedge andin the (cid:28)lm de rease as the onta t line velo ity de reases [33℄. In onsequen e, for aspontaneously spreading droplet of a non-Newtonian, shear-thinning (cid:29)uid the e(cid:27)e tivevis osity will in rease with time resulting in a spreading law of the form R ∼ t α with α < / . Numeri al simulations arried out in [34℄ on(cid:28)rm that α < / for shear-thinning (cid:29)uids and that α > / for shear-thi kening (cid:29)uids. Hen e thedominant e(cid:27)e t from shear thinning is that the spontaneous spreading of a non-Newtonian (cid:29)uid is generally slower than predi ted by the Tanner law and an notexplain the experimentally observed a eleration of the spreading pro ess.3.2. Se ond guess: line tensionWe now turn our attention to the r.h.s. of (1) and noti e that the unbalan ed Youngfor e - whi h is also the r.h.s. of (2) - is, in fa t, a mere approximation of the a tualdriving for e of spreading. In general, the total free energy of the liquid/solid system an be de omposed into bulk, surfa e, line, and point ontributions (see, e.g., [17, 18℄).Thus, the driving for e (2), as a derivative of the total free energy, should also ontainall these ontributions. This pi ture, of ourse, is meaningful only in the ase ofma ros opi drops, for whi h both the surfa e tension and the line tension are wellde(cid:28)ned.As a matter of fa t, the spheri al ap adequately des ribes the pro(cid:28)le ofost-Tanner stages of droplet spreading: the energy balan e approa h revisited 8the bulk drop, but the pro(cid:28)le of the mesos opi (cid:28)lm deviates from it, su h thatthe quasistationary free energy πR F must in lude a orre tion term, whi h isa umulated in the vi inity of the apparent onta t line and an be seen as a lineenergy τ multiplied by the apparent perimeter πR [35℄. The expression (2) takesinto a ount the surfa e energies of the three ma ros opi interfa es meeting at theapparent onta t line, but does not in lude the line tension ontribution.The idea that the line tension may have an appre iable impa t on the globalbehavior is not new. As an ex ess quantity, τ an be positive or negative, as noti edalready by Gibbs [36℄. Negative line tension, for example, an signi(cid:28) antly redu e thework required to reate a nu leus (100 Angstrom in diameter) of a new phase on solidor liquid substrates [37℄. Conversely, positive line tension an explain the stability ofNewtonian bla k (cid:28)lms towards rupture [38℄. For liquid droplets of nanometer size,negative (resp. positive) line tension an promote spreading (resp. dewetting) evenif the ma ros opi spreading parameter S is negative (resp. positive) [35℄. A reviewof di(cid:27)erent phenomena aused by line tension e(cid:27)e ts and some on eptual aspe ts ofline tension an be found in [17, 18℄.Evaluation of the ontribution due to the line tension τ involves many deli ateissues (e.g., a proper de(cid:28)nition of the e(cid:27)e tive interfa e potential used in the model, ora proper onvention when hoosing the Gibbs dividing interfa e) and, in general, is amore omplex problem than the al ulation of the surfa e tension - essentially be ausemore phases meet at the onta t line than at an interfa e (see, e.g., [17, 18℄ for amore thorough dis ussion). The problem is already di(cid:30) ult in equilibrium situations(e.g., partial wetting, with S < ), and learly be omes even more omplex whenone onsiders spontaneous spreading, sin e here one has to a ount for the temporalevolution of the droplet thi kness pro(cid:28)les.The onsideration of these subtle points is beyond the s ope of the presentapproa h. For our purposes it will be su(cid:30) ient to resort to a re ently proposedphenomenologi al generalization of (2) in terms of non-equilibrium thermodynami s:as shown in [39℄, the for e applied to the apparent onta t line of a droplet an bewritten down as f τ = S + σ (1 − cos θ ) − τR , (7)whi h di(cid:27)ers from the expression in (2) by an additional term a ounting for the ontribution of the onta t line tension τ to the driving for e of spreading. Thede(cid:28)nition of f τ as a generalized Young for e is valid both in omplete and partialwetting, and is onsistent with the so- alled modi(cid:28)ed Young equation f τ = 0 , obtainedat equilibrium by an appropriate generalization of Gibbs lassi al theory of apillarity[35, 40℄.At this point we must stress that the expression (7) for f τ is formally valid onlyif τ is onstant. If we assume a power-law behaviour for τ ∼ R β , then the modi(cid:28)edYoung for e be omes S + σ (1 − cos θ ) − τR (1 + β ) . Here we argue that it is not likelyfor τ to vanish at long t and large R , and thus β ≥ . Thereby (7) essentially holdsfor τ ∼ R β , up to a numeri al fa tor (1 + β ) > applied to the line tension term, i.e.,the last term on the r.h.s. of (7). This onsideration has little impa t on the followingqualitative argument, but will be relevant to quantitative impli ations.ost-Tanner stages of droplet spreading: the energy balan e approa h revisited 93.3. Line tension e(cid:27)e ts on spreadingSuppose now that (1) holds; that the dissipation W meso in the (cid:28)lm obeys (4); thatthe dissipation W macro is given by (3); but that the driving for e of spreading is nowdetermined by (7), i.e., that the line tension ontribution is taken into a ount . Then(5) is repla ed with the following: κ ηθ ˙ R = 12 σθ − τR . (8)Sin e τ is typi ally very small, one naturally (cid:28)nds that the surfa e tension ontribution will dominate at small and intermediate times, again giving rise to theTanner law R ∼ t / . On the other hand, as R grows to su(cid:30) iently high values,the line tension ontribution will inevitably take over and be ome a dominant drivingfor e, provided that τ /R de ays slower than the apillary term σθ .In the latter regime, the behavior is ru ially dependent on the sign of τ : • If τ is positive and tends to a onstant value, whi h is physi ally plausible, then(8) predi ts that spreading will terminate at a (cid:28)nite value of R , for whi h the (cid:28)rstand the se ond terms on the r.h.s. of (7) be ome equal to ea h other. One mayinterpret this as an indi ation of the formation of a pan ake. However, during thea tual transition to a pan ake, a drop would not retain the shape of a apillary ap (see (cid:28)gure 1b), whi h somewhat hallenges this predi tion. • If τ is negative and the se ond term in (8) dominates, we (cid:28)nd the followingpost-Tanner behavior: R ∼ (cid:18) − Z t τ dt (cid:19) / . (9)This spreading law is qualitatively di(cid:27)erent from R ∼ t / . In the frameworkoutlined in se tion 2, the driving for e of spreading is asso iated with the surfa etension. By ontrast, during the late stages of spreading, the droplet be omes(cid:29)atter and an be viewed as (cid:16)quasi two-dimensional(cid:17). It is then not surprising thatthe line tension τ should govern the spreading pro ess, provided that | τ | ≫ σRθ .It is tempting to obtain oarse quantitative results from (9) and from the ondition | τ | ≫ σRθ . In parti ular, if we assume that τ is a negative onstant, then (9) predi ts R ∼ t / , whi h agrees with previously reported experimental results [9, 10℄. Lookingat (cid:28)gure 5 in [10℄ we an estimate the value of τ from the hara teristi base radiusat the apparent rossover between the Tanner stage and the a elerated spreadingregime: this yields τ ≈ − − N. This value is an order of magnitude higher thanpreviously reported values of the line tension in the partial wetting situations, butit must be noted that many experimental measurements of τ have been performedfor simple liquids; in the ase of nemati liquid rystals an elasti ontribution tothe e(cid:27)e tive interfa e potential (a onsequen e of the an horing properties) may yieldsubstantially higher values of τ .However, upon a loser examination of the ase of negative τ , there is no goodreason to expe t that τ should approa h a onstant value. One rather expe ts thatthe (cid:28)lm region will be ome progressively more pronoun ed and the drop pro(cid:28)le willsigni(cid:28) antly deviate from a spheri al ap-like shape; hen e τ , a fun tional of lo aldroplet thi kness, will grow as a fun tion of time (in terms of its absolute value).Consequently, at late spreading stages, one may expe t a growth of R ( t ) that is fasterthan R ∼ t / .ost-Tanner stages of droplet spreading: the energy balan e approa h revisited 104. Con lusionsWe have presented in this paper both the lassi al energy balan e approa h - asdeveloped by de Gennes - and a revised version of it, whi h in orporates a line tension ontribution to the driving for e of spreading. The revisited framework was motivatedby apparently ontradi tory trends at long spreading times for ma ros opi dropletsin omplete wetting. By taking line tension into a ount, we have omplemented the lassi al framework with the following twofold interpretation: • A positive line tension - essentially a (cid:16) ollar(cid:17) around a spreading droplet - stopsspreading and is responsible for the formation of mesos opi pan akes. • A negative line tension - whi h tends to lengthen the apparent onta t line -governs the late stages of spreading, resulting in the a eleration of this pro ess.We must now voi e a few words of aution on erning our approa h, whi h seemsintuitive but may have several short omings due to its simpli ity.First we admit that the formal de(cid:28)nition of line tension τ in partial wetting - afun tional of an equilibrium pro(cid:28)le, and an integral of the e(cid:27)e tive interfa e potential- annot be easily generalized to omplete wetting and thus remained undeterminedwithin our analysis. The ase of onstant, positive τ is plausible, but negative τ ismore likely to grow as a fun tion of time and base radius, a growth whi h we areunable to spe ify. We merely expe t that, as in the partial wetting ase, τ is a ertainfun tional of the mesos opi thi kness pro(cid:28)le, rather than an independent, arbitraryquantity (see, e.g., [41℄).We have already stated that in the ase of positive τ our predi tion of theemergen e of pan akes is indi ative at best: indeed, in the pro ess of rea hing thestationary shape of a pan ake, a droplet gradually deviates from the spheri al apshape assumed by se tions 2 and 3; this deviation entails orre tions that our approa hdoes not a ount for. A similar word of aution exists for negative τ and arises from athorough omparison with experiment. Indeed, a remarkable feature of nemati 5CBdroplets observed in [9, 10℄ is that the reported a elerating phase is a ompanied bythe development of a large (cid:16)foot(cid:17) (essentially the drop adopts a bell shape similar to(cid:28)gure 1a, without exaggeration). This large foot is a warning against the appli abilityof energy balan e as developed in se tions 2 and 3: the key assumption of a spheri al ap of onstant volume V cap = π R θ is less and less valid as the mesos opi (cid:28)lm drainsliquid from the ma ros opi droplet. In other terms, the separation of ma ros opi and mesos opi length s ales (both verti al and lateral), in the experimental layoutthat we are trying to des ribe, may be more pre arious than what was assumed in ouranalyti al framework.In the light of these short omings, our agenda is to develop a more robustapproa h des ribing late stages of droplet spreading, based on the seminal approa h byTanner des ribing the time evolution of the thi kness pro(cid:28)le of whole droplets withinthe thin (cid:28)lm approximation [4, 5℄. In the latter framework, s ale separation is not anissue, and the fundamental result is that droplets gradually turn into di(cid:27)usive (cid:28)lms inthe sense of Derjaguin [42℄. Results have already been obtained for the spe i(cid:28) aseof nemati droplets, and we shall present them in a ompanion paper [43℄.We are also looking forward to developing the notion of a dynami line tension inspreading pro esses, through a detailed study of the hydrodynami wedge as in [1, 44℄.It is important to note that nemati 5CB droplets exhibit both of the non-trivial typesof post-Tanner behaviour detailed in this paper: R ( t ) rosses over from R ∼ t . toost-Tanner stages of droplet spreading: the energy balan e approa h revisited 11faster power laws (at this point the drop looks like (cid:28)gure 1a), but eventually thespreading terminates with a mesos opi pan ake ((cid:28)gure 1b). In terms of τ , this meansthat the physi ally relevant line tension is negative during the a elerating phase, butlater be omes positive. As (cid:28)gure 1 suggests, the geometri al properties of the dropletare quite di(cid:27)erent in both regimes, and it is not lear whether a onsistent de(cid:28)nitionof ττ