Fluid Flow Basics (Ideal
Case)
The starting point for much of the spin coating modeling was published
by Emslie, Bonner, and Peck [J. Appl. Phys. 29 (1958) 858-862]
(hereafter referred to as EBP). Their seminal treatment is based on assuming
that flow has reached a stable condition where the centrifugal and viscous
forces are just in balance (this is also the basis for most other modeling
work - note that this does NOT apply to the first stage of spin-up and
excess solvent explusion). When the centrifugal and viscous forces are
in balance, this equation must be satisfied:
where z and r define a cylindrical coordinate system aligned
with the axis of substrate rotation, v is the fluid velocity in the radial
direction (a function of depth), and where ris
the fluid density, w is the rotation
rate in radians per second, and h is
the viscosity in poise. With appropriate flow and velocity boundary conditions,
and considering a film that is initially uniform, the film thickness as
a function of time, h(t), was found to be:
where ho is the film thickness at time zero (but
not physically meaningful because of the first stage of unstable solution
expulsion at early time), and K is a system constant defined as:
These equations are strictly valid only when K is constant.
However, for spin coating of sol-gel or other complex solutions this may
not hold true during all stages of spinning. Both viscosity and density
are expected to increase as evaporation progresses, so caution must be
used when applying these equations. In their analysis, EBP also showed
that for early stages of fluid thinning (before evaporation becomes important),
the thinning rate would be defined as:
At longer times, solvent evaporation becomes an important
contribution. Meyerhofer was the first to estimate the effect of this on
final coating thickness [J. Appl. Phys. 49 (1978) 3993-3997].
A quite reasonable approximation is that evaporation is a constant throughout
spinning, as long as the rotation rate is held constant (see below). Therefore,
he simply added a constant evaporation term to the equation above. So,
the governing differential equation became:
where "e" is the evaporation rate [ml/s/cm2] (this
is effectively the contribution to the interface velocity that is driven
by the evaporation process alone).
Instead of solving this equation explicitly, Meyerhofer assumed that early
stages were entirely flow dominated, while later stages would be
entirely
evaporation dominated. He set the transition point at the condition where
the evaporation rate and the viscous flow rate became equal. This can be
thought of as the fluid-dynamical "set" point of the coating process. When
these assumptions are made, the final coating thickness, hf,
is predicted to be:
where co is the solids concentration in the solution.
When the physically applicable dependence of the evaporation rate on spin-speed
was factored in, this was successful in matching the regular exponents
for the dependence of final film thickness with spin speed. Research has
shown that the evaporation rate should be constant over the entire substrate
and depend on rotation rate according to:
where the proportionality constant, C, must be determined
for the specific experimental conditions. This square root dependence arises
from the rate-limiting-step being diffusion through a vapor boundary layer
above the spinning disk. It should be noted that this results when airflow
above the spinning substrate is laminar.
Fluid Flow Complications
The flow behavior described above ignores several effects that are important
for many coating solutions. As noted above, the evaporation step is critical
in defining what the final coating thickness will be. But, evaporation
occur -- by necessity -- from the top surface, and only some of the solution
components are volatile enough to evaporate to any substantial degree.
Thus, there will necessarily be an enrichment of the non-volatile components
in the surface layer of the coating solution during the spinning process. The
figure at right illustrates that concept. One of the key consequences is
that this surface layer will very likely have a higher viscosity than the
unmodified starting solution (this may simply be due to the higher concentration,
but might also occur because of cross-linking effects, etc). With a higher
viscosity, it will then impede the flow characteristics set out above,
making it a difficult differential equation to solve directly. And, this
surface layer may have the secondary result of reducing the evaporation
rate. So both the evaporation and flow processes are coupled through the
behavior of the "skin" that develops on the top of the outwardly flowing
solution during spin coating.
Another important effect is that some solutions are not "Newtonian" in
their viscosity/shear-rate relationships. Some solutions change viscosity
depending on what shear rate is used, thus depending on distance from the
center, the shear rate will be different and thus the flow behavior. This
can give radial thickness variation that varies rather smoothly in a radial
sense, as pointed out by Britten and Thomas [J. Appl. Phys. 71
(1992) 972-979].
The
figure at right illustrates that concept. One of the key consequences is
that this surface layer will very likely have a higher viscosity than the
unmodified starting solution (this may simply be due to the higher concentration,
but might also occur because of cross-linking effects, etc). With a higher
viscosity, it will then impede the flow characteristics set out above,
making it a difficult differential equation to solve directly. And, this
surface layer may have the secondary result of reducing the evaporation
rate. So both the evaporation and flow processes are coupled through the
behavior of the "skin" that develops on the top of the outwardly flowing
solution during spin coating.