Lecture Notes ni Physics detidE yb .J ,srelhE ,nehcnQM .K Z~rich Hepp, .R ,nehcn~M ,nhahneppiK .A .H ,rell~mnedieW grebledieH dna .J ,ztrattiZ n16K gniganaM Editor: .W Beiglb6ck, grebledieH 105 Dynamics and Instability of Fluid Interfaces Proceedings of a meeting, held at the Technical University of Denmark, Lyngby, May 1978 Edited byT. S. Serensen galreV-regnirpS nilreB Heidelberg New kroY 1979 Editor Torben Smith Serensen Fysisk-Kemisk Institut DTH 206 DK-2800 Lyngby ISBN 3-540-09524-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09524-1 Springer-Verlag New York Heidelberg Berlin yrarbiL in Publication Cataloging of Congress ataD Main entry title: Dynamics under dna instability of fluid .secafretni ; in physics notes (Lecture .v Bibliography: 105) .p index. Includes .1 Surfaces --)scisyhP( .sessergnoC .2 Surface .sessergnoC--yrtsimehc 3. .sessergnoC--sdiulF .I ,nesneroS 1945- Smith, Torben .II Title: III. Series. Fluid .secafretni OC173.4.$94D96 541'.3453 79-16836 NBSI 1-42590-7'83-0 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210 EDITORS PREFACE The present volume is the result of a meeting on the topic "Dynamics and Instability of Fluid Interfaces" held at the Technical Univer- sity of Denmark, Lyngby (near Copenhagen), in May 1978. The participants were distinguished researchers in the field. They came from such diverse places as Belgium, USA, DDR, Brazil, Bulgaria, Spain, Sweeden and Denmark. If one should describe the atmosphere at the meeting in just one word, this word could be: fascination. We were all deeply fascinated by the range and the scope of the physico-chemical problems covered by the lectures at the meeting. This diversity is very well represented in the content of the present volume. There are contributions on Marangoni effects during liquid-liquid mass transfer of significance for the chemical engineering of liquid-liquid extraction and for the illumination of surface chemical dissipative structures. There are other papers on the formation, thinning, instability and rupture of thin liquid films of utmost importance for foam stability, wetting phenomena and phenomena connected with the biolo- gical bilipid cell membrane. Still other papers deal with hydrodynamic, interfacial instabilities enforced by mechanical, chemical and electrical constraints. With the present industrial interest in electrically induced spontaneous emulsification, theoretical studies of this kind are strongly needed. Viscoelastic properties of surface layers of surfactants and macro- molecules are also dealt with, and another paper tries to reconcile the theories on Marangoni instability with the theories of longitudinal inter- facial waves. Thermally induced Marangoni instabilities and the influence of non-Boussinesquian effects and the Soret effect is the topic of another contribution. Finally, a paper deals with instability phenomena of spheri- cal drops with surface chemical reactions and transfer of surfactants as a model for "kicking drops", spontaneous emulsification and certain cyto- kinetic phenomena, such as the division of the biological cell. VI The common denominator for all the contributions is the simul- taneous use of concepts from surface chemistry and physics and from hydro- dynamics. Theoretic and experimental work is equally represented and even united in many of the papers. It is my hope that this volume will be of value for physicists, physico-chemists, chemical engineers, surface scientists and biologists. I am greatful to the Technical University of Denmark for the supply of meeting facilities, to Undervisnlngsministeriets Internationa- le Kontor and Det Naturvidenskabelige Forskningsr~d for economical support in the form of travel grants and to the participants for their most interesting contributions. Finally, the editor of "Lecture Notes in Physics" Dr. .W BeiglbSck as well as the Springer-Verlag should be thanked for the assistance made in the publication of this congress report. Torben Smith SCrensen 15-12 1978 Fysisk-Kemisk Institut Technical University of Denmark. TABLE OF CONTENT Instabilities induced by mass transfer, low surface tension and gravity at isothermal and deformable fluid interfaces T. S. S~RENSEN Dissipative structures and nonlinear kinetics of the Marangoni-instability H. LINDE , P.SCHWARTZ and H. WILKE 57 Formation and thinning of liquid films I. B. IVANOV and R. K. JAIN 120 Instability and rupture of thin liquid films R. K. JAIN , I. B. IVANOV, C. MALDARELLI and E. RUCKENSTEIN i~o Mechanical, chemical}and electrical constraints and hydrodynamic interfacial instability A. SANFELD, A. STEINCHEN, M. HENNENBERG, P. M. BISCH, D. VAN LAMSWEERDE-GALLEZ and W. DALLE-VEDOVE 168 Dynamic effects of surfactants and macromolecules on interfacial viscoelastic properties D. T. WASAN, N. F. DJABBARAH, M. K. VORA and S. T. SHAH 205 Interfacial instability and longitudinal waves in llquid-liquid systems M. HENNENBERG, P. M. BISCH, M. VIGNES-ADLER and A. SANFELD 229 Diffusion, reaction and convection at an interface: A few aspects of the stability problem M. G. VELARDE 260 Instability of a spherical drop with surface chemical reactions and transfer of surfactants T. S. S~RENSEN and M. HENNENBERG 276 INSTABILITIES INDUCED BY MASS TRANSFER, LOW SURFACE TENSION AND GRAVITY AT ISOTHERMAL AND DEFORMABLE FLUID INTERFACES. by Torben Smith S~rensen Fysisk-Kemisk Institut, Technical University of Denmark, DK 2800, Lyngby, Denmark 1. Introduction The beautiful hexagonal convection pattern observed by B~- nard 1'2 in shallow pools of liquids heated from below inspired Rayleigh 3 to perform his classical stability analysis of buo- yancy-driven convection flow. We now know that the driving force for the B~nard cells is not the buoyancy, but rather temperature induced gradients in surface tensionS'5. For example, Block g ob- served also B~nard cells in shallow pools ~ooZed from below, and B~nard cells were observed in experiments performed by as- tronauts in the Apollo 14 and Apollo 17 spacecrafts where the gravitational acceleration was l0 -6 G. 7'8 Nevertheless, the identification between B~nard's observations and Raylelgh's ana- lysis is very persistent even in standard reference texts on hy- drodynamic and thermodynamic stability 9'10 Lateral variations in surface tension can be generated by as well fluctuations in temperature as f luctuatio~ in the adsorp- tion of surfactive material to the interface. The first linear stability analysis taking surface tension gradients as the dri~ ll ing forces were performed independently by Pearson (heat and mass transfer) and by Sternling and Scriven 12 (mass transfer of surfactant), although qualitative statements that surface ten- sion is the driving force in many instability phenomena near surfaces were made already in the previous century by James Thomson 13, Marangonl l~ and Rayleigh 15 In the analysis of Pearson as well as in the paper of Stern- ling and Scriven (S.&S.)the interface is assumed to be flat du- ring convection. But it is well known that the interface de- forms itself during convection. B~nard reported (ref.2 .p.120) that there was a depression of the surface where liquid was streaming towards the surface and an elevation of the surface above downf&ow regions. This is characteristic for surface ten- sion-driven convection whereas the converse is true for buoyancy convection, see ref. h P.339 • The same type of interfacial de- pression above upflow regions was observed by Linde, Pfaff and Zirke116 in connection with transfer of ethanol from water to air, although it seemed to be a characteristic of the "seconda- ry" roll cells rather than of the smaller "primary" roll cells. Theoretically, the effects due to deformable interfaces were 4 studied by Scriven and Sternling in connection with heat trans- fer through a liquid-air boundary. They considered only statio- nary states of marginal stability, however, and excluded gravi- ty effects (e.g. Rayleigh convection). The primary effect due to surface curvature is to introduce the absolute value of the surface tension through the law of Laplace for the overpresure behind a convex interface. K.A.Smith 17 studied the influence of gravity waves on the surface in heat transfer instabilities, but still did not consider the competition between surface ten- sion gradients and buoyancy. As well S. & S. as Smith seem to agree that the effects due to deformation on the neutral stabi- lity of stationary disturbances only show up at very large wave- lengths. The first authors maintain, however, that there does not exist a critical Marangoni-number below which all perturba- tions are stable, whereas Smith claims that the findings of S. & S. are due to their neglect of gravity in the equations of 18 surface dynamics. Palmer and Berg have studied the stabili- sing effects of surfactants on thermal instabilities in shallow pools of dilute binary solutions heated from below. They consi- d@r as well purely surface tension-driven flows as convection with competition between Marangoni-mechanisms and Rayleigh in- stability. They also discuss as well stationary as oscillatory instabilities, but they do not discuss the effect of simulta- neous mass transfer through the interface. In a recent series of papers 19-22, S~rensen, Hennenberg and their coworkers have carried out stability analysis on isother- mal, deformable interfaces with transfer of surfactive substan- ces as an outgrow of research in the role of chemical surface reactions in connection with motile events near cell membranes carried out by the Sanfeld-group 23-25. They include gravity in the surface dynamics and analyse as well stationary as oscilla- tory instabilities with values of the parameters representative for the mass transfer of acetic acid from ethylene glycol to ethyl acetate, a system studied experimentally by Orell and Westwater 26. Calculations were made for as well linear concen- tration profiles as for more realistic exponential profiles. It is a callenge for the future to try to work together the theoretical results of S. & S. , Smith, Palmer and Berg and the S~rensen-Hennenberg-Sanfeld group with the experimental find- ings in the monumental work of Linde and coworkers on liquid 16,27-h0 motion i~ connection with mass- and heat transfer Also, P.Schwartz and H.Wilke from Linde's group in Berlin have con- structed computer models being able to follow the instabilities into the non-llnear realm, but without allowing for deformation of the interface 41-42. In the present paper my task will be of a much more limited scope, however. The model describing the dynamics of an isothermal, deformable fluid interface will be introduced in § 2. The characteristic eqation for a deformable - originally plane - fluid interface with transfer of surfactant is shown in § 3. Some special solutions to the characteristic equation in the vicinity of critical wavenumbers for stationary instabilities are discussed in § 4, and possible interactions between the Marangoni-lnstability and the gravity-driven Ray- leigh-Taylor instability are considered in § 5. In § 6 I give a summary of the assumptions behind and the results of S. & S.'s 1959-theory. In order to compare our results for exponential concentration profiles with S. & S.'s theory, a "modified S. & S. theory for exponential profiles" is constructed in § 7. Computer studies and comparisons between the models are re- ported in § 8 where especially the ability of low surface ten- sions to destabilise certain oscillatory modes will be discuss- ed. Such a destabilisation might have significance for sponta- neous emulsification and perhaps even for the formation of mi- croemulsions in the process of tertiary oil recovery by surfac- 43 tant-cosurfactant-polymer slugs In § 9 a series of graphs for the determination of regions of stability and instability in terms of suitable dimensionless numbers is given for the re- alistic case of exponential concentration profiles (being an 4 approximation to errorfunction profiles). An illustration of, how the graphs may be used, is given in § l0 in case of the ethylene glycol (EG) - ethyl acetate (EA) - acetic acid (HAc) system studied by 0rell and Westwater 26, Finally, a discussion of what to do in the future is included in § ll. 2. Dynamics a,,,,fo Newt,onian fluid interface ~/co.vecT,o. ~ /BULK OIFFUS|O. DNA (SURFACE UETFtlC) DIFFUS|ON DNA NOITCEVNOC Fig. 1. Illustration of interracial mass balance. In fig. 1 we have illustrated the physics behind the balance of surfaetant adsorbed to the interface. The interracial con- centration F varies due to convection at the interface, due to dilatation or compression of the interface (deformational change of the surface metric), due to surface diffusion and due to dif- ferences in the diffusion to or from the adjacent bulk solution. In symbols we have (see ref. 19 pp.50-52): 2V 8F ~ T-~ + ~a r +--sV - (r...sv) Ds s r = ~ {D~ c} . Mn (1) where a is the local value of the surface metric determinant and index s denotes "surface". The velocity in interfacial (in general Gausslan) coordinates is denoted s v and the vector ope- rators in the surface will in general be given by the tensorial description connected with the surface coordinates (see the pa- per by Scriven hh and the book by Aris 45. A stands for a differ- ence between a property in fluid 2 and in fluid 1 when crossing the interface between 1 and 2. We assume constant diffusion co- efficients, very small deformation of the interface 6z and very small fluctuations in the interfacial concentration 8F. Hereby linearised we obtain the interfacial mass balance (Fo is the unperturbed surface mass density). a~r = _ (A__) r - r v .v + o v2~r + ~t 2a o o~s ~s s s o (2) d2c ~c I + D °-9---~ I ~z} A {D ~ z=o dz 2 z= -o The last term on the r.h.s, containing the curvature of the un- perturbed concentration profile was forgotten in some of our publications 19"20'22, but the mistake was corrected in ref. 21. This term was shown to be important for the destabilisation of oscillatory modes at low surface tensions. The term due to the change in the metric (number one at r.h.s, of (2)) will only contain second order terms in the surface deflection in case of small deformations of a plane interface, but in case of small spherical deformations of a interface, that term has to be ta- ken into account (see ref. 25, P.198). For small deformations of a plane interface we have also the particular simple rela- tions V2 = ~2 + 22 (3) s ~x 2 ~y2 ~v ~v V v = x + y (4) M s ~s 8x ~y where v and v are components of the bulk velocity adjacent to x y the interface.