PHYSICS OF FLUIDS 24, 052103 (2012)
Landau-Levich flow visualization: Revealing the flow
topology responsible for the film thickening phenomena
H. C. Mayer and R. Krechetnikov
Department of Mechanical Engineering, University of California, Santa Barbara,
California 93106, USA
(Received 31 May 2011; accepted 28 February 2012; published online 4 May 2012)
An extensive body of experimental work has proven the validity of the analysis of
Landau and Levich, who were the first to determine theoretically the thickness of
the film deposited by the withdrawal of a flat substrate from a bath of liquid with
a clean interface. However, there are a number of experimental investigations that
have shown that surfactants in the liquid may result in a thickening of the deposited
film. Marangoni phenomena have usually been considered responsible for this
effect. However, some careful experiments and numerical simulations reported in the
literature seemed to rule out this view as the cause of the observed behavior. Despite
all these studies and the number of reports of film thickening, an experimental study
of the flow field close to the coated substrate in the presence of surfactants has never
been undertaken. In this paper we will present a set of flow visualization experiments
on coating of a planar substrate in the range of capillary numbers 10
−4
Ca 10
−3
for sodium dodecyl sulfate solutions with bulk concentrations of 0.25 CMC ≤ C
≤ 5.0 CMC (critical micelle concentration). It was evident during experiments that the
flow field near the meniscus region exhibits patterns that can only be explained with
a stagnation point residing in the bulk and not at the interface. As opposed to patterns
with an interfacial stagnation point, the observed flow fields allow for the increase
in film thickness due to the presence of surfactants compared to the clean interface
case.
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2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4703924]
I. INTRODUCTION
Dip coating, in which a substrate is withdrawn from a bath, is perhaps the oldest and simplest
form of liquid film deposition.
1
Although employed in industry to coat objects of various shapes and
sizes, scientific investigation of dip coating has typically made use of flat and cylindrical substrates.
For the case of Newtonian liquids the thickness of the film
h
∞
is a function of fluid properties
(i.e., surface tension σ , viscosity μ, and density ρ), gravity g, withdrawal velocity U, and substrate
curvature r. Dimensional analysis provides a relationship for the relevant parameters of the form
2
h
∞
/l
c
= f (Ca, Re, Go), (1)
where l
c
=
√
σ/ρg is the capillary length, Ca = μU/σ is the capillary number, Re = ρU h
∞
/μ is
the Reynolds number, and Go = r/l
c
is the Goucher number. The question of film thickness for the
vertical withdrawal of a flat substrate, Go 1, in the limit of small capillary number with no inertia
effects was first addressed theoretically by Landau and Levich
3
who arrived at a relationship of the
form
h
∞
= 0.945 l
c
Ca
2/3
. (2)
This law holds only for clean interfaces and implies the corresponding flow field near the substrate
4
shown in Figure 1(a). The distinguishing feature of this flow field is a surface stagnation point located
3
h
∞
from the substrate.
5
The first experimental confirmation of the Landau-Levich expression can
be attributed to Deryagin and Titievskaya,
6
who coated tubes of negligible curvature using oil for
2 × 10
−5
< Ca < 10
1
. The coating of thin wires, the formation of thin films on the inside of
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2012 American Institute of Physics24, 052103-1