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J. Fluid Mech. (2006), vol. 559, pp. 429–450.
c
¸ 2006 Cambridge University Press
doi:10.1017/S0022112006000425 Printed in the United Kingdom
429
Surfactant eﬀects in the Landau–Levich problem
By R. KRECHETNI KOV
1
AND G. M. HOMSY
2
1
California Institute of Technology, Pasadena, CA 91125, USA
2
University of California, Santa Barbara, CA 93111, USA
(Received 6 June 2004 and in revised form 14 December 2005)
In this work we stud

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J. Fluid Mech.
(2006),
vol.
559,
pp.
429–450.
c
2006 Cambridge University Pressdoi:10.1017/S0022112006000425 Printed in the United Kingdom
429
Surfactant eﬀects in the Landau–Levich problem
By R. KRECHETNIKOV
1
AND
G. M. HOMSY
2
1
California Institute of Technology, Pasadena, CA 91125, USA
2
University of California, Santa Barbara, CA 93111, USA
(Received
6 June 2004 and in revised form 14 December 2005)
In this work we study the classical Landau–Levich problem of dip-coating. While inthe clean interface case and in the limit of low capillary numbers it admits an asym-ptotic solution, its full study has not been conducted. With the help of an eﬃcientnumerical algorithm, based on a boundary-integral formulation and the appropriateset of interfacial and inﬂow boundary conditions, we ﬁrst study the ﬁlm thicknessbehaviour for a clean interface problem. Next, the same algorithm allows us to inves-tigate the response of this system to the presence of soluble surface active matter,which leads to clariﬁcation of its role in the ﬂow dynamics. The main conclusion is thatpure hydrodynamical modelling of surfactant eﬀects predicts ﬁlm thinning andtherefore is not suﬃcient to explain the ﬁlm thickening observed in many experiments.
1. Introduction
While many diﬀerent processes are employed in coating applications, as discussed inthe comprehensive book by Kistler & Schweizer (1997), the simplest – ﬁlm depositionby withdrawing a substrate from a bath with solution – remains one of the most clearand fundamentally important coating processes. The basic problem is to understandthe dependence of the ﬁlm thickness on the withdrawal speed
U
, the accelerationdue to gravity
g
, the size of the domain, and the physical properties of the ﬂuid, i.e.ﬂuid density
ρ
, viscosity
µ
and surface tension
σ
. This question was ﬁrst answeredby Landau & Levich (1942) for dip coating from an inﬁnite bath in the low-capillary-number limit (when surface forces dominate viscous ones). Their analysis,now recognized as a matched asymptotic expansion combined with a lubricationapproximation, hinges on the geometrical matching of the constant curvature of thestatic meniscus to the zero curvature in the thin-ﬁlm region through a transitionregion. This yields the classical result,
h
∞
= 0
.
945
l
c
Ca
2
/
3
,
Ca
=
µU σ
(1.1)where the relevant length scale is the capillary length
l
c
=
√
σ/ρg
. The solution (1.1)was successively improved to account for gravity corrections by White & Tallmadge(1965), but with an incorrect approximation to the normal stress, which was correctedby Spiers, Subbaraman & Wilkinson (1974). Wilson (1982) put the
ad hoc
treatmentsby previous authors on the basis of systematic perturbation theory. For a history of the dip-coating problem, see Tallmadge (1967) for ﬂat substrates, Qu´er´e (1999) onﬁbre coating, Weinstein & Ruschak (2004) on general coating ﬂows, and Ruckenstein(2002) for a short summary of
ad hoc
understanding of coating physics. From theexperimental side, the law (1.1) was suggested by Morey (1940) even before theanalysis of Landau & Levich (1942) was published (Of course, the power 2
/
3 was notknown at that time and a general power law was tested using the data). This work,
430
R. Krechetnikov and G. M. Homsy
(
a
) (
b
) (
c
)
Figure 1.
Landau–Levich problem. (
a
) Static wetting. (
b
) Dynamic wetting. (
c
) Far ﬁeld.
as well as many others (e.g. Deryagin & Titievskaya 1945) conducted to test andverify the classical law and its generalizations used diﬀerent variations of gravimetricmeasurements to determine the ﬁlm thickness.Consider a perfectly wetting substrate and an inﬁnite bath of a perfectly wettingliquid as depicted in ﬁgure 1(
a
). This corresponds to what we refer to as the
Landau–Levich problem
, which is an ideal situation, since in reality the bath is always of ﬁnitesize. When the plate is at rest, the shape of the
static meniscus
interface is well-knownand corresponds to the balance of surface and gravity forces. When the plate startsmoving up with speed
U
, the singularity at the contact line is resolved by the pullingup of the interface in this zone by viscous forces and ﬁlm entrainment (ﬁgure 1
b
).Assuming that both Reynolds number, based on
l
c
and the speed of withdrawal,and the capillary number are small, the local ﬂow in the meniscus region is creepingand, when viewed in the far ﬁeld, the ﬂow is as shown in ﬁgure 1(
c
). The meniscusshape stays undistorted (static) except for the small region in which all three forcesdue to viscosity, surface tension and gravity are in dynamic equilibrium. Therefore,this region is called the
dynamic meniscus
region. (In this paper we understand thedynamic meniscus to be the region inﬂuenced by dynamic eﬀects, such as viscous andMarangoni stresses, i.e. the notion is applicable to the surfactant case as well. Forfurther discussion, see Krechetnikov & Homsy 2005). The extent of this region (alongthe plate dimension)
l
is determined by the balance of viscous and capillary stressesand by matching the curvature in the static and dynamic meniscus regions,
µU h
2
∞
∼
σ/l
c
l,h
∞
l
2
∼
l
−
1
c
.
Therefore,
l
∼
(
h
∞
l
c
)
1
/
2
,
h
∞
obeys Landau–Levich law (1.1), and the appropriatescalings for the coordinates
x
, velocity ﬁeld
v
and pressure
p
, are
x
=
l
c
x
,
v
=
U
v
, p
=
√
ρgσp.
(1.2)The ﬂow topology in the Landau–Levich problem (ﬁgure 2
a
), suggests another point of view: the ﬂow in the corner between the moving solid boundary and the free interfaceas in ﬁgure 1(
c
) admits a local self-similar solution (2.7) discussed below, which corres-ponds to a reversed solution of a ﬂat plate drawn into a viscous ﬂuid found by Moﬀatt(1964). This self-similarity, of course, breaks down at the corner and is resolved by theabove mentioned balance of viscous and capillary forces resulting in ﬁlm entrainment,which necessitates a sink-type solution superimposed onto Moﬀatt’s solution.
Landau–Levich problem
431
–5–4 –3 –2–10 –10 –505
–1 –0.50 –2 –101(
a
) (
b
)
Figure 2.
Flow ﬁeld: streamlines patterns at moderate
Ca
. (
a
) Landau–Levich ﬂow.(
b
) Bretherton ﬂow.
Therefore, one may consider the Landau–Levich problem as belonging to the classof interfacial singularities that includes Taylor cones, tip-streaming, etc. The physicalsigniﬁcance of this 2
/
3-law is also revealed in a close connection to other problems,such as ﬁlm deposition in tubes by a penetrating bubble, ﬁbre coating, and a freelysuspended ﬁlm drawn vertically from a reservoir (known as Frankel’s law). The ﬁrstof these problems we refer to as the Bretherton problem, after the asymptotic solutionin Bretherton (1961), which is a brother of the Landau–Levich problem, as can beappreciated from ﬁgure 2(
b
). However, the fundamental diﬀerence in geometry of these two problems (cf. ﬁgure 2) – tube versus semi-inﬁnite bath – diﬀerentiates themeven in the clean interface case. The distinction in geometry leads to a diﬀerence inthe ﬂow topology – one stagnation point in the Landau–Levich case versus two inthe Bretherton problem – which has crucial implications for the surfactant interfacecase treated here.The experimental studies of surfactant eﬀects, which are of interest here, are due toGroenveld (1970) and Krechetnikov & Homsy (2005) for ﬂat substrates; Bretherton(1961) for coating the inner walls of circular tubes; and Carroll & Lucassen (1973),Ramdane & Qu´er´e (1997), Qu´er´e & de Ryck (1998) and Shen
et al.
(2002) on ﬁbrecoating. Historically, in the classical Bretherton problem (cf. Fairbrother & Stubbs1935; Taylor 1961), a disagreement was found between the theory by Bretherton (1961)and experiments for
Ca
<
5
×
10
−
3
(which were carried out by Bretherton himself).In this range, the theory underpredicts the measured values of ﬁlm thickness. Thissystematic deviation is usually attributed to the presence of surfactants and, assuggested by Schwartz, Princen & Kiss (1986), is due to accumulation of surfactantin the rear of a ﬁnite-length bubble. The common feature of the whole body of experimental observations of the thickening phenomena is the occurrence of very thinﬁlms,
h
∞
6
20
µ
m, for which the deviation is detected.The more thoroughly studied Bretherton problem reveals some controversy con-cerning the eﬀects of surfactants. The discrepancy between the theory and experimentsby Bretherton (1961) for low capillary numbers was tackled by Ginley & Radke (1989)for the case of insoluble surfactant, but with constant bulk concentration (which can
432
R. Krechetnikov and G. M. Homsy
be justiﬁed for thick enough ﬁlms, as will be discussed in
§
2.3). Their analysis leads tothinning of the ﬁlm contrary to the known experiments. Alternatively, the discrepancywas ascribed by Ratulowski & Chang (1990) to the variation in the bulk concentration.This issue was also studied by Wassmuth, Laidlaw & Coombe (1993) who returnedto the assumption of Ginley & Radke (1989) of constant bulk concentration andconstructed a ﬁnite-diﬀerence solution. The conclusion of their work is that ‘strongdeviations from the clean-surface. . . can be either positive or negative depending onthe sign of the local surfactant gradient’. Unfortunately, the srcin of this variationin sign and connection between the physics and mathematics of interface equationswere not uncovered. In the same vein is the work by Ghadiali & Gaver (2003), whoextended the previous numerical analysis to the case of variable bulk concentrationusing a dual reciprocity boundary-integral method. The extensive numerical dataagain support the ﬁndings of Wassmuth
et al.
(1993) concerning the inﬂuence of surfactants on the ﬁlm thickness, i.e. either thickening or thinning can occur.An analogous (even though less rich) body of contradictory literature is observedfor the Landau–Levich problem as well. In one of the srcinal works on theoreticalexplanations of the thickening eﬀect of surface active substances, Groenveld (1970)speculated that the Marangoni eﬀect ‘will cause the ﬂuid at the interface in themeniscus region to ﬂow upwards during withdrawal’, so that there is no longera stagnation point at the interface (it is moved to the interior). Later, with noreference to Groenveld (1970), the same simple picture was used by Park (1991) in alubrication analysis modelling the eﬀect of insoluble surfactant on withdrawal. Thekey assumption made by Park (1991) is that the dynamic meniscus is restricted tothe same scale
l
∼
l
c
Ca
1
/
3
as the transition region in the clean interface case, whilethe rest of the meniscus is both static and uniformly covered by surfactant. The sameassumption has been used in other studies of both Landau–Levich and Brethertonproblems for trace as well as elevated amounts of surfactant, as done by Stebe &Barth`es-Biesel (1995). As we will show, the meniscus is dynamic in the presence of surfactants on the scale
l
∼
l
c
and the stagnation point sits at the interface, thusinvalidating all the assumptions made by Groenveld (1970) and Park (1991). The factthat the meniscus is dynamic everywhere, i.e. on the scale
∼
l
c
, prevents application of standard perturbation techniques and local analyses, and necessitates a full nonlinearstudy of an elliptic boundary-value problem.A closer look at the surfactant dynamics reveals some further diﬃculties in theattribution of ﬁlm thickening to Marangoni stresses, at least within the current stateof knowledge. The theoretical attempts to justify this eﬀect of thickening throughMarangoni stresses in a frame of macroscopic equations has been done for traceamounts of surfactant by Ratulowski & Chang (1990) for the Bretherton problemwith soluble surfactant in the diﬀusion-limited case and by Park (1991) for theLandau–Levich problem with insoluble surfactant. The common feature of thesetheories is a lubrication approximation, that is the assumption that the meniscus isdynamic on the same scale as the transition region in the clean interface case, namely
l
∼
l
c
Ca
1
/
3
. Simple estimates show that this is not always the case. Consider a typicalexperiment with an aqueous solution of SDS (sodium dodecyl sulfate, the most studiedsurfactant) at bulk concentrations
C
∼
1CMC (8.3mM). When withdrawal speeds areof the order of 1 cms
−
1
, the characteristic time of interface stretching based on thewhole meniscus size,
∼
l
c
, is
∼
0
.
2s. At the same time, the diﬀusion length is estimatedas a ratio of the interfacial
Γ
and bulk
C
concentrations, i.e.
η
d
∼
Γ/C
∼
1
µ
m, andthe diﬀusion time is
t
diﬀ
∼
η
2
d
/D
∼
10
−
3
s versus the adsorption time of 0
.
2s deducedfrom the kinetic properties of SDS. Since the withdrawal time is of the same order

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