On te absence of marginal pincing in tin free films P. D. Howell and H. A. Stone 6 August 00 Abstract Tis paper concerns te drainage of a tin liquid lamella into a Plateau border. Many models for draining soap films assume tat teir gas-liquid interfaces are effectively immobile. Suc models predict a penomenon known as marginal pincing, in wic te film tends to pinc in te margin between te lamella and te Plateau border. We analyse te opposite etreme, in wic te gas-liquid interfaces are assumed to be stress-free. We sow tat no marginal pincing occurs in tis case: te film always approaces a monotonic profile at large times. Introduction Many eperimentalists ave reported convective-like motions associated wit soap films tat ave nonuniform ticknesses at teir borders, a penomenon tat goes by te name of marginal regeneration [9]. Wit a view to understanding te initial stages of tis process, Aradian, Rapaël and de Gennes [] studied marginal pincing, wic refers to te development of a pinced region in te neigbourood of te Plateau border at te edge of a soap film. Under te assumption tat te fluid velocity is zero at te surfactantcovered boundaries of te film, te film tickness was found to satisfy a nonlinear partial differential equation familiar from lubrication teory (e.g. [0]. Similarity solutions were presented by treating te film in tree distinct regions: a flat central film, a pinced region, and a capillary statics region at te Plateau border. In a real soap film, tere is a balance between te advection of surfactant along te interfaces, replenisment from surfactant molecules in solution, and te Marangoni stress resulting from nonuniform surface concentration of surfactant (see [, 5]. Te evolution equation analysed by [] is a limiting case, in wic te free surfaces are so loaded wit surfactant, and te stresses due to even small surfactant gradients so large, tat te free surfaces are effectively tangentially immobile. For eample, soap films made from large Matematical Institute, University of Oford, 9 St Giles, Oford OX LB, UK. Division of Engineering & Applied Sciences, Harvard University, 08 Pierce Hall, Cambridge, MA 08, USA.
(a (b (,t a H Figure : (a Scematic of a tin film of tickness H attaced to a Plateau border of radius a. (b Close-up of te margin between te film and te Plateau border. molecular weigt surfactants or polymers or from mitures of surfactants suc as SDS and dodecanol (e.g. [9] are generally observed to ave rigid surfaces. Higly mobile interfaces are also possible. In tese circumstances, a large bulk concentration of surfactant, generally in te form of micelles, provides a reservoir of monomer tat can maintain te interface at nearly uniform surfactant concentration, tereby eliminating surfactant gradients and stresses (toug te value of te surface tension can be significantly reduced. In tis paper, we analyse tis limit of ig interfacial mobility, for wic te film tickness and fluid velocity satisfy a pair of coupled nonlinear partial differential equations. Suc motions of tin fluid seets also appear in situations suc as curtain coating [, see te appendi by Taylor] and te drawing of glass seets [8]. We find tat, in contrast wit te immobile case, tere is no marginal pincing: te film always approaces a monotonic profile as time increases. We set out te governing equations and boundary conditions in section. In section, we perform a coordinate transformation tat reduces te nonlinear problem to te linear eat equation, wit suitable matcing conditions. A simple travelling-wave solution of te transformed problem is identified in section and sown to be globally attractive in section 5. Te teory is illustrated by sowing te beaviour of typical time-dependent solutions in section 6, and our results are discussed in section 7. Problem Statement We consider te model problem of a semi-infinite film of uniform initial tickness 0 attaced to a Plateau border wit radius of curvature a, as sown scematically in figure. If is te film tickness and is te coordinate measuring distance along te film, ten we require H as, as +. ( a
Tese conditions imply tat te following scales are appropriate in nondimensionalising and : ah = H, =. ( Tus te film as an effective aspect ratio a/h; if a is assumed to be muc larger tan H, ten te dynamics may be described using a long-wave approimation. If one assumes, as in [], tat te surfaces of te film are immobile, ten te drainage timescale is found to be t t i = µa γh, ( were µ and γ are, respectively, te viscosity and surface tension of te liquid comprising te film. In tis case, te evolution of te film tickness is governed by te dimensionless lubrication equation t + ( = 0. ( If, owever, te liquid-air interfaces are stress-free, ten drainage occurs over te timescale t t s-f = µa γ, (5 wic is a factor of H/a smaller tan te immobile timescale t i. In tis case, te film tickness and tangential liquid velocity u satisfy te coupled dimensionless equations [6, 8] t + (u = 0, (6a ( u + = 0, (6b were time as been scaled by t s-f and velocity by ah//t s-f. Te system (6 is to be solved subject to te boundary conditions and te initial condition as, as +, (7 = 0 ( at t = 0. (8 Generic solutions of te so-called lubrication equation (, subject to te boundary conditions (7, were sown by [] to eibit marginal pincing. At large times, te film tickness approaces a non-monotonic profile, attaining its continually decreasing minimum value in te margin between te lamella and te Plateau border. In tis paper we eamine weter or not solutions of te system (6 eibit qualitatively similar beaviour.
Problem transformation Analysis of (6b as approaces reveals tat u must approac a function only of t. Since te system (6 is invariant under a time-dependent translation in, we may coose te coordinate aes suc tat u 0 as. (9 Ten (6b may be integrated once wit respect to, applying te conditions as to obtain u = (, (0 wic may be rewritten as u = ( ( =. ( Now we transform variables from (, t to (ξ, t, were ξ is defined by ( ξ = + (, t d, ( so tat Using (6a, we calculate ξ/ t as follows: ξ t = = = ξ =. ( (, t d = (, t / t (, t ( / u(, t + u(, t (, t d (, t (, t / u(, t (, t + ( u(, t(, t d u(, t d, ( (, t after integrating te second term by parts. Now we use te epression ( to obtain ξ t = ( u +. (5 Tus, te cain rules to transform to te new coordinates are ξ, t t ( u + ξ. (6
As a consequence, equation (6a is transformed to or t = ξ + t = ξ ( ξ Tus, te function φ = / satisfies te linear eat equation, (7 (. (8 ξ φ t = φ ξ. (9 To obtain te boundary conditions for φ we must eamine te beaviour as, tat is as te film approaces te Plateau border. If as a Laurent epansion + B(t + C(t + a (t + a (t + as, (0 were B, C, a, a,... are all arbitrary functions of t, ten we ave ξ = B(t + ( ξ B(t log + D(t + + ( ( ξ D(t ep ( ( φ ep (ξ D(t as ξ, ( were D is a furter arbitrary function of t. Substitution of ( into (9 leads to dd dt = 9 D = D 0 t, (5 were D 0 is determined by te initial conditions. Specifically, te definition ( may be used to epress D 0 in te form ( ( D 0 = + d + d. (6 0 0 Tus te asymptotic beaviour of φ is φ as ξ, (7a φ ( ep (ξ D 0 + 9 t as ξ +, (7b 5
increasing t 5-0 -8-6 - - Figure : Film tickness versus distance, as given by (0 and (, for values of t =,,,. Tis is te travelling-wave solution, propagating rigt to left. along wit te obvious initial condition φ = φ 0 (ξ at t = 0, φ 0 (ξ = [ 0 ((ξ, 0] /. (8 We ave terefore reduced te original nonlinear coupled partial differential equations (6 to te eat equation (9, subject to te boundary and initial conditions (7 (8. Travelling-wave solution Now, we can identify an eact solution of (9 tat also satisfies te matcing conditions (7, namely φ T (ξ, t = + ( ep (ξ D 0 + 9 t. (9 Te film tickness profile corresponding to (9 is recovered via (, t = φ T (ξ, t /, were (0 ξ ( = ξ + φt (ξ, t / dξ ( ( = D 0 8 + π 9 log 6 ( t 9 + + e (ξ D 0/ / +9t/ ( e (ξ D 0 t F ; ; 5 ; e (ξ D 0/ 9t/, ( and F is a ypergeometric function [7, page 065]. Tis solution is plotted in figure. Now, (9 and (0 imply tat is a function only of te group (ξ + t/ wic itself, by (, is a function only of + t/. Tus (9 corresponds to a travelling-wave solution of te original system (6, as suggested by figure. Wit tis insigt, we can obtain te solution more directly as follows. 6
Suppose tat and u are functions only of η = ct, were te constant wavespeed c is to be determined. Ten (6a and te matcing conditions (7 as imply tat (u c = c du dη = c d dη. ( Now we substitute du/dη from ( into ( to obtain c d 5/ dη = d ( d dη dη d dη = c (, ( again using te matcing conditions (7 at. Now te beaviour η / as η gives c ( c =. (5 Tis result agrees wit te propagation speed deduced from (0 and (. Ten, one furter integration of ( leads to te solution in te form ( tan + ( + log + + = (η η 0, (6 + were η = ct and η 0 is te arbitrary initial translation. Tis travelling-wave solution is equivalent to te steady solution of (6 obtained by [], subject to a steady translation. Plots of (6 agree eactly wit figure. Notice tat, in tis solution, is a monotonic increasing function of η, so tat tere is no marginal pinc. 5 Global stability Clearly, te simple travelling wave (6 is not te only solution of (6 satisfying boundary conditions (7: te general solution must depend also on te initial film profile 0 (. We will now sow, owever, tat (6 is globally attractive, so tat any initial condition will ultimately take tis form. To establis tis property, we first eamine more carefully te matcing conditions as + and te film approaces te constant-curvature Plateau border. Wit given asymptotically by (0, (6b gives te corresponding form of u as u A(t B(t C(t + a (t + B(t B(tC(t + as, (7 7
were A is a furter arbitrary function of t. Ordinary differential equations relating tese coefficients are obtained by substituting (0 and (7 into (6a: db dt dc dt da dt da dt = A, (8a = C AB + B, (8b = 0, (8c = a + (A + Ba + (B C, (8d and so fort. By substituting (0 into (, we obtain corrections to te epansions ( and ( as follows B ξ C B log + D + + + a + 5B 6BC + (9 e (ξ D/ B + B C e (ξ D/ a (ξ D + (0 φ e(ξ D/ (B C 8 e e (ξ D/ + a + O ( e (ξ D/ as ξ. Now, (8c implies tat a is constant in time and, by combining (8a and (8b, we find tat te translation-invariant combination B C satisfies Tus te asymptotic beaviour of φ is φ e(ξ D 0/ ( d ( B C = ( B C ( dt ( B C = ( B0 C 0 e t. ( +9t/ (B 0 C 0 8 e (ξ D 0/ +t/ + a + ( as ξ. All te coefficients in ( may be determined in principle from te initial film profile 0 (, since φ(ξ, 0 is related to 0 ( by equation (8. A solution of te eat equation tat satisfies ( and also approaces as ξ is Φ = e(ξ D 0/ +9t/ (B 0 C 0 8 + ( a + + ( a e (ξ D 0/ +t/ ( ξ erf t. (5 8
Tus, if we set φ = Φ + φ, ten φ satisfies φ t = φ ξ, (6a φ 0 as ξ ±, (6b φ = φ 0 (ξ Φ(ξ, 0 at t = 0, (6c wic implies tat φ 0 as t. Tus te solution Φ is globally attractive and, ence, te large-time beaviour of is given by (, t = Φ(ξ, t /, (7 ξ ( = ξ + Φ(ξ, t / dξ. (8 Notice tat Φ reduces to te travelling-wave solution φ T (equation 9 if B 0 C 0 = 0 and a =. As t, te first term in (5 dominates te second ecept were ξ D 0 t, in wic case bot are smaller tan e t/. Te dominant beaviour of Φ at large t is, terefore, Φ e(ξ D 0/ +9t/ Φ Φ + e(ξ D 0/ +9t/ ξ D0 t, ξ D 0 t, ξ D0 t. (9a (9b (9c In oter words, te travelling-wave solution φ T is always te dominant beaviour as t ; te etra eponential and error-function terms in (5 just provide decaying transient effects. 6 Sample solutions We illustrate te global stability beaviour described above in figure. First, figure (a sows te evolution of te film tickness wen B0 C 0 = 0 and a =. Tis initial sape corresponds to an increased slope in te parabola at plus infinity and causes a dip in te profile close to te meniscus. As time increases, toug, te amplitude of te pinc decays rapidly and te travelling-wave solution (propagating from rigt to left is recovered. Second, in figure (b we set B0 C 0 = 0 and a =, corresponding to a decreased initial slope at plus infinity. Te initial profile is noticeably flatter but, again, te travelling-wave solution soon emerges. In figure, we illustrate solutions wit B0 C 0 = 0 and different values of a. Recall tat a is te coefficient of / in te large- epansion of. Not surprisingly, it as a 9
(a 5 (b 5-0 -8-6 - - -0-8 -6 - - Figure : Film tickness versus distance for values of t =,,,. Parameter values (a B 0 C 0 = 0, a =, D 0 = 0; (b B 0 C 0 = 0, a =, D 0 = 0. (a 5 (b 5-0 -8-6 - - -0-8 -6 - - Figure : Film tickness versus distance for values of t =,,,. Parameter values (a B 0 C 0 = 0, a = 5, D 0 = 0; (b B 0 C 0 = 0, a = 5, D 0 = 0. 0
somewat smaller influence on te qualitative beaviour tan does B 0 C 0. In figure (a, a = 5, wic leads to a sligt decrease in te slope at plus infinity and tus a flatter profile. In figure (b, a is set to 5, wic increases te slope at plus infinity and gives rise to a small pinc near te meniscus. In bot cases, te effects decay as t increases and te travelling-wave solution again emerges as te dominant beaviour. 7 Discussion We ave sown tat a stress-free tin liquid film, draining into a Plateau border, tends to a monotonic sape, irrespective of its initial profile. In contrast, a film wose liquid-air interfaces are immobile approaces a non-monotonic profile, wit a pinc in te margin between te film and te Plateau border. Eiter scenario is a limiting case of a true lamella, in wic various pysical effects compete to render te surfaces partially mobile. For eample, Breward & Howell [] consider a lamella stabilised by soluble surfactant and find tat bot monotonic and nonmonotonic profiles are possible. Similarly, Braun et al. [] model te gravity-driven drainage of a lamella into a bat, including insoluble surfactant and surface viscosity, and find tat te film may or may not be monotonic, depending on te rigidity of te interfaces. None of tese autors, toug, was able to give a teoretical criterion to distinguis solutions tat do or do not eibit marginal pincing. We plan to analyse tis transition in te future. We found tat te globally attractive solution of (6 is a travelling wave, in wic te Plateau border (wit radius of curvature a propagates towards te film (of tickness H at dimensional speed c = γ a 8µ H, (50 were γ and µ are te surface tension and viscosity respectively. Tis counterintuitive result is partly a consequence of our coosing a reference frame in wic te film velocity far from te Plateau border is zero. In a typical eperiment, in wic te Plateau border is fied, te film would flow into te Plateau border at speed c. Furtermore, we ave analysed te model problem of a semi-infinite film, wose tickness approaces a constant value H at infinity. Te case of a film wose lengt is finite, altoug significantly longer tan ah, may be analysed using similar metods, wit te film tickness H varying slowly wit time as te film drains (see []. Tis drainage is symptomatic of a qualitative difference in beaviour between our model equations (6 and te lubrication equation (. Te ig-curvature free surfaces cause a large negative capillary pressure in te Plateau border, wic tends to suck fluid in from te tin film. If te surfaces are stress-free, tis suction is transmitted trougout te film, via te two real caracteristics t = const of (6. (Tese caracteristics correspond to te two first integrals of (6 obtained in section. If te free surfaces are immobile, owever, te influence of te Plateau border on te film is confined to te margin between tem, and tis is reflected in te parabolic caracter of (.
Our analysis inges on a coordinate transformation tat converts te nonlinear coupled equations (6 for te film tickness and velocity u into te linear eat equation. Tis transformation may well prove useful in oter applications of tese model equations. Acknowledgements We tank A. Aradian, E. Rapaël and P.-G. de Gennes for useful conversations. HAS tanks te Harvard MRSEC for partial support. References [] A. Aradian, E. Rapaël & P.-G. de Gennes, 00 Marginal pincing in soap films. Europys. Lett. 55(6, 8 80. [] R. J. Braun, S. A. Snow & S. Naire, 00 Models for gravitationally-driven free-film drainage. J. Engng Mat., 8. [] C. J. W. Breward & P. D. Howell, 00 Te drainage of a foam lamella. J. Fluid Mec. 58, 79 06. [] D. R. Brown, 96 A study of te beaviour of a tin seet of moving liquid. J. Fluid Mec. 0, 97 05. [5] J.-M. Comaz, 00 Te dynamics of a viscous soap film wit soluble surfactant. J. Fluid Mec., 87-09. [6] T. Erneu & S. H. Davis, 99 Nonlinear rupture of free films. Pys. Fluids A 5, 7. [7] I. S. Gradsteyn & I. M. Ryzik. 99 Table of Integrals, Series, and Products. Fift Edition. Academic Press. [8] P. D. Howell, 996 Models for tin viscous seets. Euro. J. Appl. Mat. 7,. [9] K. J. Mysels, K. Sinoda & S. Frankel, 959 Soap Films, Studies of Teir Tinning and a Bibliograpy (Pergamon Press, New York. [0] A. Oron, S. H. Davis & S. G. Bankoff, 997 Long-scale evolution of tin liquid films. Rev. Modern Pys. 69, 9 980.