Chabot Engineering
Semiconductor Machine-Tool
Chemical Delivery Chp3
Bubblers-323
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Engineering
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
The Following Presentation Lead to an
American Institute of Physics (AIP)
Publication in 2001
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2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
WJ’s Patented Bubbler

C. C. Collins,
M. A. Richie,
F. F. Walker,
B. C. Goodrich,
L. B. Campbell
“Liquid Source
Bubbler”, United
States Patent
5,078,922 (Jan
1992)
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3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Patent 5 078 922
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4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
WJ Bubbler Design
Schematic diagram of a the WJ chemical vapor
generating bubbler system used in CVD applications.
Note the use of the dilution MFC to maintain constant
mass flow in the output line. An automatic temperature
controller sets the electric heater power level
Cut-away view of a WJ chemical source vapor bubbler.
The bubbler features a total internal volume of 0.95
liters, and a 25 mm thick isothermal mass jacket with
an exterior diameter of 180 mm.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
CONCEPTUAL Degree of Saturation vs Liquid Level
110%
100%
90%
Degree of Saturation
80%
70%
60%
With a 2.2” Liq Level does the WJ bubbler operate
HERE?
Or
HERE?
50%
40%
30%
20%
10%
0%
0
20
40
60
80
100
120
Liquid Level (Arbitrary Units)
file = Vap_Prss.xls
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6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Microscopic Transient Behavior:
Bubble Vapor Saturation




How Well Does the Bubbler “Humidify” the
“Dry” Nitrogen Carrier Gas?
Does the Liquid LEVEL in the Bubbler Affect
this Humidification (degree of Saturation)
What other Factors affect the Degree of
Saturation, and in What Quantity?
What does Bubbling Look like?
 Flow Visualization
–
–
BT98_VRo.ppt
BT_9806c.ppt
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7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
WJ-1999 Bubbler Test; t = 0
Water Surface
Bubble
6.35 mm
Chabot College Engineering
8
Carrier N2
Flow Rate in
slpm
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
WJ-1999 Bubbler Test; vr,f
Water Surface
9.7 mm
Bubble
Bubble
t=0
t = 33.3ms
6.35 mm
3.7 mm
QN2 =
1 slpm
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9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Saturation Problem Partition

The Bubble Saturation Problem Consists of
3 Loosely Coupled Sub-Processes [2]
1. Bubble Saturation as a Function of Bubble Size
and Vapor Diffusivity
2. Bubble Size as Function of Sparger Tube HoleSize, Liquid Density, and Liquid Surface Tension
3. Residence Time of the Bubble in the liquid by
integration the bubble rise-velocity over the liquid
height
[2] B. Mayer, “Liquid Source Bubbler Carrier Gas Vapor-Saturation Transient Analysis”, WJ-SEG
Engineering Library Report, file BM961112.doc, 12Nov96
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10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
IntraBubble Vapor Mass Transport
Partial Differential Equation


Assume Bubble Diffusion
Physics at right
Assume Diffusion of
vapor obeys the Fick Eqn
 Cv r , t 
Fv r , t   Dv 


r


–
Where
o
o
o
o
Fv  the molar flux in the r-direction in kmol/m2s
Dv  the (assumed constant) vapor diffusivity in N2 in m2/s
Cv  the molar concentration of the vapor in kmol/m3
r  the radial coordinate in the bubble in m
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11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Sat PDE
cont.-1

Molar Flux INTO the Bubble Control Volume
nv ,in


Cv
 Fv ,in  ACVsurf ,outside   Dv
r

Molar Flux OUT of the Bubble Control Volume
nv ,out


Cv
2
 Fv ,out Asurf ,inside   Dv
 4r 
r r


STORAGE Rate of
C v
2
n


4

r
dr
Vapor in the Bubble v , stor
t r
Control Volume
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
 4 r  dr  
r  dr

2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Sat PDE
cont.-2

Setting: Influx − Outflux = Storage Rate
2

Cv

Cv  2 Cv 1   2 Cv  1 Cv
2
Dv 2rdr
 r dr 2   r dr
 2 r

r r
r r 
t r r  r  Dv t


This is the 1-Dimensional Diffusion Equation
in Spherical CoOrdinates

Now use Perfect Gas Theory to Convert to
Vapor Pressure Formulation
1   2 Pv  1 Pv
r

2
r r  r  Dv t
Taylor series expansion in Appendix-A of JVST-A 2001 paper; Perfect Gas conversion in Appendix-B
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Sat PDE
cont.-3

1   2 Cv  1 Cv
0

Comments on the PDE 2  r
r r  r  Dv t
 Linear & Homogeneous
 2nd order in r (need two Boundary Conditions)
 1st Order in t (need one Initial Condition)

BC1: Assume Equilibrium at Bubble Edge
Pv ( ro , t )  Pv. sat

BC2: By Symmetry have No diffusion at r = 0
Pv
r
0
for all time
r 0
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for all time
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Sat PDE
cont.-4

IC: At t=0 bubble is 0% Saturated (trivial IC)
Pv ( r,0)  0

NonDimensionalize
  r ro

for all r
  Dv t r
2
o
P v  Pv Pv , sat
Define the Degree of NonSaturation (a.k.a.
Complementary Degree of Sat) Pc
Pv , sat
Pv , sat  Pv
Pv
Pc  1  Pv 


Pv , sat Pv , sat
Pv , sat
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15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Sat PDE
cont.-4

PDE Summary
Parameter
Problem Formulation
Pv(r,t)
Pv(,)
Pc(,)
1   2 Pv  1 Pv
r

r 2 r  r  Dv t
1   2 P v   P v


 2     
1   2 Pc  Pc


 2     
BC-1
Pv ( ro , t )  Pv .sat
Pv (1,  )  1
Pc (1,  )  0
BC-2.1
Pv ( r , t )  finite
P v (  ,  )  finite
P c (  ,  )  finite
Pv
0
r r0
Pv
0
  0
Pc
0
  0
Pv ( r ,0)  0
P v (  ,0)  0
Pc (  ,0)  1
PDE
BC-2.2
IC
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Sat PDE Solution

Non-Dim Solution for Pc

P c  ,   2  1
n 1
n 1

P v r, t   1  2  1
n 1
2
n 1
sin n r ro   n 
e
n r ro
2
2
Dv t ro2
See next Slide for Graphical Representation
of This (really cool) Solution
Chabot College Engineering
17
2
Dimensional Solution for Pv


sin n   n  
e
n
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Liquid Source Vapor Bubble Saturation Transient
1.0
• Bubble Diameter = 5 mm
• D for TEOS in N2 = 0.05 cm2/s
Vapor Saturation Fraction, P v
0.8
Increasing Time
Pv(r,t) (t=0.01 s)
Pv(r,t) (t=0.04 s)
Pv(r,t) (t=0.10 s)
Pv(r,t) (t=0.15 s)
Pv(r,t) (t=0.25 s)
Pv(r,t) (t=0.35 s)
Pv(r,t) (t=0.50 s)
0.6
0.4
0.2

Pv  r , t   1  2  1
n1
n1
sin n r ro   n2 2 Dt ro2
e
n r ro
1st 100 Terms
of Summation
0.0
0.0
0.3
0.5
0.8
1.0
1.3
1.5
1.8
2.0
Radial Position Inside Bubble, r (mm)
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2.3
2.5
file = BubPv(t)1.xls
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Size Determination

Perform Force Balance as shown below

Bubble Breaks free when Buoyant Force just
barely exceeds the Surface Tension Force
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Size Determination
cont.-1

The Buoyant Force
FB  4 3r g  l   g  4 3g r
3
o
–
3
l o
Where
FB  the the buoyant force in newtons
g  the acceleration of gravity, 9.8 m/s2
l  the density of the liquid in kg/m3 (936 kg/m3 for TEOS)
g  the density of the carrier gas in kg/m3 (1.01 kg/m3 for
N2 at 65 °C)
o ro  The outside radius of the bubble in m
o
o
o
o
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Size Determination
cont.-2

The Surface Tension Force
Fs  Dhs
–
Where
o Fs  the surface tension force in newtons
o Dh  the diameter of the vent hole in the sparger tube in
meters (0.508 mm, or 0.02”, from WJ bubbler dwg 986595)
o s  the liquid surface tension in N/m (0.022 N/m, the value
of ethanol at 30 °C)

Thus the Bubble Radius Equation
 3Dhs 
ro  

4
g


l 
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13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Rising-Bubble
Liquid Residence Time

Assume rough Equivalence for
Fluid-Mechanical Drag between:
 light bubble rising through a liquid
 heavy sphere falling through the same liquid

Position-varying
drag forces
determine the
velocity of a
bubble rising in a
liquid
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Residence Time, r
cont.-1

The Drag Force
FD  C D r
2
o
–
 v 2
l
2
r
Where
o FD  the drag force in newtons
o CD  the the coefficient of drag, a dimensionless number
o vr  the rise velocity of the bubble in m/s

Apply Newton’s Law of Motion to Rising Bubble
dvr
 Fy  FB  FD  mB ar  mB dt
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23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Residence Time, r
cont.-2
–
Where
o Fy  the sum of the forces, in the y-direction, acting on
the bubble in newtons
o ar  the rise acceleration of the bubble in m2/s
o mB  the “mass” of bubble in kg

Effective Bubble Mass is the Liquid Displaced
mB  4 3r  l   g  4 3r  l
3
o

3
o
Thus the Expression for Bubble Acceleration
dvr dvr dy dvr
3C D vr2
ar 


vr  g 
dt
dy dt dy
8ro
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Residence Time, r
cont.-3

Comments on Acceleration Equation
 Ordinary Differential Equation (ODE) for vr in
terms of y or t
–
–

NONlinear & NONhomogeneous
1st order in y or t (need one BC or IC)
BC/IC: Assume velocity is ZERO at the
instant the bubble breaks away from the tube
 BC/IC: y = t = 0  vr = 0

Note: the Bubble Reaches Terminal Velocity
 vr,f when: ar = dvr/dt = dvr/dy = 0
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Residence Time, r
cont.-4

r Solution Strategy (see JVST-A paper)
 If we know vr(t) at every instant in time, then
simply integrate vr over liquid height H.
H

dy
vr   dy  H   vr dt
0
dt 0
r

Implicitly evaluate vr(t) at any arbitrary time, tA
using ODE

vr t A 
0
tA
0
0
dvr  vr (t A )   ar dt  
Chabot College Engineering
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tA

3CD vr2 t  
 g 
dt
8ro 

Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Residence Time, r
cont.-4

Using the “H” and “vr(tA)” Equations
H
r
 dy  H  
0

0
 t
 0

A

3v r2 t   
C D  dt  dt A
 g 
8ro

 
Almost Done. Find CD in Idelchik Text Ref.
23.99 4.565 0.491 23.99
4.565
0.491
CD 
 3




13
12
Re
Re
Re  v r 2ro   v r 2ro 
 v r 2ro 

 



     
  
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Residence Time, r
cont.-5

Collapse constant expressions into “K” Terms
r
H
 dy  H  
0


0
 t
 0

A

0.375K1v r  K 2 v r5 3  K 3v r3 2   
 g 
 dt  dt A
ro

 
This eqn can be solved numerically as
described in JVST ppr, eqns 2529
Table on the next slide shows a typical result
 The 2mm diameter bubble reaches a terminal
velocity of 0.214 m/s (0.48 mph)
–
This is consistent with the literature
 Bubble rises the WJ std 2.2” liq Height in 280 ms
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Bubble Residence Time, r
cont.-6

Example Calc: ro = 1 mm,  = 7.4x10-7 m2/s
Time
Step, n
a
(m/sq-s)
Re
del-v
(m/s)
v
(m/s)
del-y
(m)
H
(m)
1
9.8
n/a
0.0098
0.0098
9.8E-06
9.8E-06
1
2
9.715655
26.48649
0.009716
0.019516
1.95E-05
2.93E-05
2
3
9.570822
52.74501
0.009571
0.029086
2.91E-05
5.84E-05
3
4
9.382618
78.6121
0.009383
0.038469
3.85E-05
9.69E-05
4
5
9.159834
103.9705
0.00916
0.047629
4.76E-05
0.000145
5
6
8.909038
128.7268
0.008909
0.056538
5.65E-05
0.000201
6
7
8.635705
152.8053
0.008636
0.065174
6.52E-05
0.000266
7








276
1.55E-08
579.3578
1.55E-11
0.214362
0.000214
0.055772
276
277
1.44E-08
579.3578
1.44E-11
0.214362
0.000214
0.055986
277
2.2” = 0.0559m
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Time
(ms)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Degree of Saturation

We (finally) have all the tools to determine the
degree of saturation, Sv, for the rising bubble
S v ( ro )   P v r, r  dr  
ro
ro
0
0

Conceptually

Note


n 1 sin n r ro   n 


1

2

1
e


n r ro
n 1

2
2
Dv r ro2

 dr

Sv  Sv ( Dv , Dh ,s ,  l , H , )
 Dh and H are DESIGN-controlled
 Well known liquid properties = l
 Poorly Characterized Liquid properties = Dv, s, 
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Degree of Saturation
cont.-1


Estimate Properties for TEOS, Etc.
Chemical
Temperature
(K)
Mw
(kg/kmol)
l
(kg/m3)
Dv
(m2/s)
s
(N/m)

(m2/s)
TEOS
338 (65 °C)
208.3
936
7.27x10-6
0.0240
5.11x10-7
TMB
297 (24 °C)
103.92
915
7.87x10-6
0.0226
6.58x10-7
TMPi
297 (24 °C)
124.08
1005
8.42x10-6
0.0259
19.3x10-7
Ethanol
303 (30 °C)
47.06
789
13.7x10-6
0.0220
12.7x10-7
Water
298 (25 °C)
18.01
998
23.9x10-6
0.073
9.13x10-7
Saturation Safety Factor, N
Chemical
Temperature
(K)
ro
(mm)
vr,f
(m/s)
r,99
(ms)
r,tot
(ms)
N
TEOS
338 (65 °C)
0.9996
0.233
67
257
3.8
TMB
297 (24 °C)
0.9873
0.220
60
273
4.6
TMPi
297 (24 °C)
1.001
0.169
57
343
6.0
Chabot College Engineering
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Degree of Saturation
cont.-2

Validation Testing Performed in Jun98 by
MSWalton, B. Mayer, C. Koehler
 Water used as Benign Surrogate
–

See next slide
Calculated

 ro = 1.45 mm
 vr,f = 0.274 m/s
(0.61 mph)
 Min Saturation height =
6-7mm (0.25”)
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Actual
 ro = 1.5-2 mm
 vr,f = 9.7mm/33.3ms
= 0.29 m/s (0.65 mph)
 Fully Humidified
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Validation Testing
Chabot College Engineering
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
TEOS Liquid Source Vapor Bubble Saturation v. Liquid Height
100%
150
99% Saturation after 67 ms, or 0.46"
120
60%
90
Linear portion of r curve indicates
terminal velocity of ~0.23 m/s
40%
60
20%
30
• Bubble Diameter = 1.999 mm
2
• Dv for TEOS in N2 = 0.0727 cm /s
Integrated Saturation (%)
• Kinematic viscosity,, = 0.00511 cm2/s
Rise Time (ms)
0%
0
0.0
0.2
0.4
0.6
0.8
Liquid Level Inside Bubbler, y (inch)
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Bubble Rise Time, r (ms)
Integrated Bubble Saturation, S
v
80%
1.0
file = Sv(t)_01.xls
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt
Degree of Saturation - Conclusions

The standard WJ bubbler liquid level of 2.2”
more than assures 100% saturation of the N2
carrier gas with the source chemical vapor.
 The 2.2” liquid height results in saturation time
factors of safety of 3.8 for all source chemicals.

The liquid level can drop about 1.5”
(to 0.7” above the sparger tube) before
non-saturation becomes a potential problem
 The 1.5” depth equates to a 460 ml working
volume for post-dep fill applications
Chabot College Engineering
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt