Chapter 7 Fluid Flow in Materials Processing

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Chapter 7 Fluid Flow in Materials Processing
7.1 Overall Balance
7.1.1 Crystal growth Czochralski
In the Czochralski process the single crystal
is pulled from the melt contained in a heated
crucible, as illustrated in Fig. 7.1-1. The
cross-sectional areas of the crystal and the
crucible are As and AL, respectively. The crystal
is pulled at a constant speed V with respect to
the crystal growth laboratory. This speed,
however, is not the actual production speed of
the crystal Vp. As the crystal is being
pulled from the melt, the melt level drop. Let VL
be the speed at the speed at which the melt level
drops. Since the crystal rises at the speed
(VVL) relative to the melt level, the production
rate of the crystal is
7.1-1
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Let the melt be the control volume O, the density
of the crystal ?s, and the density of the melt ?L.
7.1-2
or
7.1-3
Therefore
7.1-4
7.1-5
Substituting Eq. 7.1-4 into Eq. 7.1-1 yields
7.1-5
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7.2 One-dimensional fluid flow
7.2.1 Fiber processing Inviscid melt spinning of
fibers
The inviscid melt spinning process (Section
6.1.5) is used for the preparation of fibers
fromlow-viscosity melts such as molten ceramics
and metals. As shown in Fig. 7.2-1, let pt and ph
be the gas pressures at the top and bottom
chambers, respectively H the melt depth in the
crucible and L and R the length and radius of
the orifice, respectively.
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The pressures at the inlet and outlet of the
orifice, p0 and pL, respectively, are as follows
7.2-1
p0 pL ?gH
and
pL pb
7.2-2
Molten metals and ceramics are Newtonian and
the flow is laminar in the small orifice. Since
the radius of the orifice R is much smaller than
the length L, the end effects are negligible.
Substituting Eqs. 7.2-1and 7.2-2 into the
Hagen-Poisseulle law (i.e., Eq. 1.5-52), the
volume flow rate of the melt through the orifice
is
7.2-3
Substituting the fiber production speed
7.2-4
7.2-5
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The Hagen-Poisseulle law (i.e., Eq. 1.5-52)
Steady-state, laminar flow, imcompressible,
Newtonian fluid In a long, vertical tube. vr
vq 0, vz is independent of q.
?
?
With B.Cs
and
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7.3 Multidimensional fluid flow
7.3.1 Semiconductor Device Fabrication Spin
casting
The spin coating process is used to coat a thin
film of photoresist on the wafer (Section 6.3.2).
A viscous solution containing the photoresist
material is made to form a thin film (on the
order of 1mm) uniform film on the wafer By
spinning the wafer, as illustrated in Fig. 7.3-1.
Assume that the film is an incompressible
Newtonian fluid. Consider the equation of
motion, according to Eq. A of Table 1.5-2.
Since the film is very thin and viscous,
the flow can be considered creeping flow
(Section 1.5.1) with respect to the wafer thus,
all term on the LHS of the equation can be
neglected except for . There is
no radial pressure gradient (i.e.,
) and the flow is axisymmetric . Since the film
is very thin, the viscous force due to the
variation of vr in the r direction is negligible
as compared to that due to the variation of vr in
the z direction. As such, the equation of motion
reduces to
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7.3-1
Since the film is viscous and very thin, it moves
with the wafer as a rigid body in the T direction

7.3-2
7.3-3
7.3-4
7.3-5
7.3-6
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The equation of continuity according to Eq. B
of Table 1.3-1 reduces to
7.3-7
Substituting Eq. 7.3-6 into Eq. 7.3-7
7.3-8
or
7.3-9
On integration from vz0 at z0
7.3-10
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Since
7.3-11
Eq. 7.3-10 becomes
7.3-12
on intergration
7.3-13
or
7.3-14
For very thin films
7.3-15