PAMM · Proc. Appl. Math. Mech. 17, 653 – 654 (2017) / DOI 10.1002/pamm.201710295
Thin liquid droplet on top of a rotating non-isothermal liquid layer
and Mike Bothe
Strömungsmechanik, Fakultät Bio- und Chemieingenieurwesen, TU Dortmund, Emil-Figge Str. 68, 44227 Dortmund
Wafers are usually coated by using spin-coating, where centrifugal forces are used to spread a droplet on the rotating wafer.
This ﬂow is unstable to the ﬁngering instability, where several segments of the wetting front spread faster than the average,
resulting in several ﬁngers. The liquid ﬂows via the existing ﬁngers while the area in between does not get coated. A precise
experimental investigation is problematic, as the droplet has to be placed quite exactly in the center of the rotation. Replacing
the wafer by a second liquid should lead to a parabolic-shaped free interface and the droplet should center itself due to gravity.
Here, we derive a model for the free interfaces of a thin droplet on top of a rotating liquid by taking gravity, centrifugal forces,
friction, (thermo-)capillary and line forces into account. Additionally, this setup is the simplest example of multiple coating,
where several free interfaces and contact lines inﬂuence each other.
2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Sketch and conservation laws
Cylindrical coordinates in the frame of the rotating system with constant angular velocity
Ω are used (Fig. 1). A rotationally
symmetric droplet (liquid B) rests in the center of the rotation on top of a liquid A and both are surrounded by a gas C. All
ﬂuids are Newtonian with the properties: density ̺
, dynamic viscosity η
, thermal conductivity λ
and heat capacity c
with i = A, B, C. There are three different free interfaces:
divides the ﬂuids B and C, h
A and B, and h
A and C with
the related interfacial tensions
which decrease linearly with increasing temperature. All three free interfaces end
in the contact line a, where the line forces tangential to the interfaces balance each other horizontally and vertically. As the
properties of the gas are much smaller than those of the liquids, the gas is treated to be passive . So there are two sets of
continuity, Navier-Stokes and heat transport equations, each for A and B. The boundary conditions are normal and tangential
stress balances at h
, taking thermo-capillary shear stresses (the Marangoni effect) into account. A kinematic condition
ensures the tangential ﬂow at the interfaces. At the interfaces
there are thermal boundary conditions of third type
and at h
the temperature and heat ﬂux have to be continuous. At the bottom, the temperature is ﬁxed and there is no slip.
Multiple scaling is engaged and using the geometric disparity of a ﬂat droplet leads to the lubrication approximation.
2 Static problem
In the isothermal case, the Navier-Stokes equations reduce to the basic equations of hydrostatic with gravitational and cen-
trifugal forces. The equations can be solved analytically and the solutions are coupled by the normal stress balances, which
is simply the Laplacian pressure drop. Further boundary conditions are: symmetry at the center for h
, the same height at
the contact line for h
, a known volume for both liquids, a contact angle between liquid A and the containment, as well as
the vertical force balance at the contact line. If the position of the contact line a is known, all coefﬁcients can be determined
explicitly. Otherwise, a can be computed using a shooting method to meet the radial force balance at the contact line.
Figure 2 plots the positions of the free interfaces h
without rotation and with a moderate inﬂuence of gravity. The
weight of the droplet lets the droplet sink in the lower liquid A. If the gravitational inﬂuence is increased (c.f. Fig. 3), the
droplet gets more and more into the shape of a cylindrical disk, as the inﬂuence of the curvature pressure drop decreases
relatively. For the highest gravitational inﬂuence, the solution is validated against Archimedes principle. Figure 4 plots the
free interfaces if a moderate centrifugal force is present, which presses the liquid out of the center of rotation. The shape of
the lower liquid A is nearly parabolic, inﬂuenced by the droplet only in a small area near the contact line. If the centrifugal
forces are strong enough, the interface
also gets into a parabolic shape, which leads to a capillary ridge near the contact
line. The presence of such a capillary ridge is the onset of the ﬁngering instability in droplets on top of solid substrates.
Higher centrifugal forces press the droplet into the parabolic shape of the liquid A, which ﬂattens the capillary ridge (Fig. 5).
Therefore the ﬁngering instability should be ampliﬁed most at the angular velocity leading to the highest ridge.
3 Steady problem
If temperature gradients are present, the full system of eight conservation laws and 14 boundary conditions has to be solved.
Unfortunately, even in the limit of the lubrication approximation, no analytical solution can be obtained. But as the temperature
Corresponding author: e-mail email@example.com, phone +49 231 755 2477, fax +49 231 755 3209
In the course of his Bachelor thesis.
2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim