Air Flow Basics (Ideal Case) The
image at the right, from Millsaps and Pohlhausen, [J. Aeronautical
Sci., (1952) 120-126] shows a schematic of the ideal airflow
field above an infinitely large spinning disk. At the surface of the disk
there is a "no-slip" condition so the contacting air must be exactly co-rotating
--- hence the flow vectors pointing essentially tangentially to any point
at a given radius (and proportional to the distance from the center). At
moderate distances from the surface a centripetal acceleration must be
provided by the viscous effects; this condition is thus maintained only
when some outward radial air flow is also occurring. This outward flow
is balanced by some minor downdraft over the entire wafer. This is a steady
state configuration and does not include inertial effects included in the
"spin-up" stages. This air flow pattern also only hold true so long as
the flow is laminar. A "boundary layer" of uniform thickness thus exists
over the entire surface area of the spinning wafer: it is through this
boundary layer that evaporating solvent must diffuse. Because the boundary
layer is constant in thickness over the wafer then the evaporation rate
as a function of position is predicted to also be constant.
Air Flow Complications
The steady flow field described above is limited to cases where the flow
is laminar and where it is "steady". In fact, except for very large wafers,
most spinning conditions DO satisfy the constraint of having laminar flow.
However, there can be un-steady oscillating instabilities in the boundary
layer near the surface of the wafer. These form spiral shaped waves or
rolls that are called "Ekman" spirals. Wahal, et al [Applied Physics
Letters 62 (1993) 2584-6] have experimentally observed
Ekman spirals (shown in the figure at right) for nominally laminar conditions
in spin coating. They claim that these instabilities can lead to coating
thickness variations, but have not explained WHY that would be the case.
The
image at the right, from Millsaps and Pohlhausen, [J. Aeronautical
Sci., (1952) 120-126] shows a schematic of the ideal airflow
field above an infinitely large spinning disk. At the surface of the disk
there is a "no-slip" condition so the contacting air must be exactly co-rotating
--- hence the flow vectors pointing essentially tangentially to any point
at a given radius (and proportional to the distance from the center). At
moderate distances from the surface a centripetal acceleration must be
provided by the viscous effects; this condition is thus maintained only
when some outward radial air flow is also occurring. This outward flow
is balanced by some minor downdraft over the entire wafer. This is a steady
state configuration and does not include inertial effects included in the
"spin-up" stages. This air flow pattern also only hold true so long as
the flow is laminar. A "boundary layer" of uniform thickness thus exists
over the entire surface area of the spinning wafer: it is through this
boundary layer that evaporating solvent must diffuse. Because the boundary
layer is constant in thickness over the wafer then the evaporation rate
as a function of position is predicted to also be constant.
The steady flow field described above is limited to cases where the flow
is laminar and where it is "steady". In fact, except for very large wafers,
most spinning conditions DO satisfy the constraint of having laminar flow.
However, there can be un-steady oscillating instabilities in the boundary
layer near the surface of the wafer. These form spiral shaped waves or
rolls that are called "Ekman" spirals. Wahal, et al [Applied Physics
Letters 62 (1993) 2584-6] have experimentally observed
Ekman spirals (shown in the figure at right) for nominally laminar conditions
in spin coating. They claim that these instabilities can lead to coating
thickness variations, but have not explained WHY that would be the case.