[email protected] Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt 1 Bruce Mayer, PE...

[email protected] • Applied_Math-Physics_Vapor-Gen_Transient_Behavior.ppt1

Bruce Mayer, PE Chabot College Engineering

Chabot Engineering

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Semiconductor Machine-Tool Semiconductor Machine-Tool Chemical Delivery Chp3Chemical Delivery Chp3

Bubblers-323Bubblers-323



The Following Presentation Lead to an American Institute of Physics (AIP) Publication in 2001



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)



Patent 5 078 922



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.



CONCEPTUAL Degree of Saturation vs Liquid Level

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%

0 20 40 60 80 100 120

Liquid Level (Arbitrary Units)

De

gre

e o

f S

atu

rati

on

file = Vap_Prss.xls

With a 2.2” Liq Level does the WJ bubbler operate HERE? Or HERE?



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



WJ-1999 Bubbler Test; t = 0

Carrier N2 Flow Rate

in slpm

Bubble

6.35 mm

Sparger Tube

Water Surface



WJ-1999 Bubbler Test; vr,f

Bubble

6.35 mm

Sparger Tube

Water Surface

Sparger Tube

Bubble

t =

0

t = 33.3m

s

9.7 mm

3.7 mm

QN2 = 1 slpm



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 Hole-Size, 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



IntraBubble Vapor Mass TransportPartial Differential Equation

Assume Bubble Diffusion Physics at right

Assume Diffusion of vapor obeys the Fick Eqn

r

trCDtrF v

vv

,,

– Whereo Fv the molar flux in the r-direction in kmol/m2s

o Dv the (assumed constant) vapor diffusivity in N2 in m2/s

o Cv the molar concentration of the vapor in kmol/m3

o r the radial coordinate in the bubble in m



Bubble Sat PDEcont.-1

Molar Flux INTO the Bubble Control Volume

2

,,, 4 rr

CDAFn

r

vvinsidesurfoutvoutv

Molar Flux OUT of the Bubble Control Volume

2,,, 4 drr

r

CDAFn

drr

vvoutsideCVsurfinvinv

STORAGE Rate of Vapor in the BubbleControl Volume

drrt

Cn

r

vstorv

2, 4



Setting: Influx − Outflux = Storage Rate


t

C

D

1

r

Cr

rr

1

t

Cdrr

r

Cdrr

r

Crdr2D v

v

v22

v2

r

2v

22

r

vv

This is the 1-Dimensional Diffusion Equation in Spherical CoOrdinates

t

P

Dr

Pr

rrv

v

v

11 2

2

Now use Perfect Gas Theory to Convert to Vapor Pressure Formulation

Taylor series expansion in Appendix-A of JVST-A 2001 paper; Perfect Gas conversion in Appendix-B



Comments on the PDE Linear & Homogeneous 2nd order in r (need two Boundary Conditions) 1st Order in t (need one Initial Condition)

BC1: Assume Equilibrium at Bubble Edge


timeallforPtrP satvov .),( BC2: By Symmetry have No diffusion at r = 0

timeallforr

P

r

v 00

011 2

2

t

C

Dr

Cr

rrv

v

v



IC: At t=0 bubble is 0% Saturated (trivial IC)


Define the Degree of NonSaturation (a.k.a. Complementary Degree of Sat) c

rallforrPv 0)0,( NonDimensionalize

orr 2ov rtD

satvvv PP ,

satv

vsatv

satv

v

satv

satvvc P

PP

P

P

P

P

,

,

,,

,1



PDE Summary

P a r a m e t e r P r o b l e m F o r m u l a t i o n

P v ( r , t ) v ( , )

c ( , )

P D E 1 12

2

r r rPr D

Pt

v

v

v

12

2

v v

12

2

c c

B C - 1 P r t Pv o v s a t( , ) . v ( , )1 1 c ( , )1 0 B C - 2 . 1 P r t f i n i t ev ( , ) v f i n i t e( , ) c f i n i t e( , ) B C - 2 . 2

Pr

v

r

0

0

v

0

0

c

0

0

I C P rv ( , )0 0 v ( , ) 0 0 c ( , ) 0 1




Bubble Sat PDE Solution

Non-Dim Solution for c

1

1 22sin12,

n

nn

c en

n

Dimensional Solution for v

1

1 222sin121,

n

rtDn

o

on

vove

rrn

rrntr

See next Slide for Graphical Representation of This (really cool) Solution



Liquid Source Vapor Bubble Saturation Transient

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0 2.3 2.5

Radial Position Inside Bubble, r (mm)

Va

po

r S

atu

rati

on

Fra

cti

on

, v

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)

file = BubPv(t)1.xls

• Bubble Diameter = 5 mm

• D for TEOS in N2 = 0.05 cm2/s Increasing Time

P r t

n r rn r r

evn o

o

n Dt r

n

o,sin

1 2 1 1

1

2 2 2

1st 100 Terms of Summation



Bubble Size Determination Perform Force Balance as shown below

Bubble Breaks free when Buoyant Force just barely exceeds the Surface Tension Force



Bubble Size Determinationcont.-1

The Buoyant Force

33 3434 olgloB rggrF – Where

o FB the the buoyant force in newtons

o g the acceleration of gravity, 9.8 m/s2

o l the density of the liquid in kg/m3 (936 kg/m3 for TEOS)

o 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



Bubble Size Determinationcont.-2

The Surface Tension Force

– Whereo F 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 the liquid surface tension in N/m (0.022 N/m, the value of ethanol at 30 °C)

hDF

Thus the Bubble Radius Equation31

4

3

l

ho g

Dr



Rising-BubbleLiquid 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



Bubble Residence Time, rcont.-1

The Drag Force

222rloDD vrCF

– Whereo 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

dt

dvmamFFF r

BrBDBy




– Whereo 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

logloB rrm 33 3434

Thus the Expression for Bubble Acceleration

o

rDr

rrrr r

vCgv

dy

dv

dtdy

dy

dv

dt

dva

8

3 2



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




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.


dtvHdydtdy

vr

r

H

r

00

Implicitly evaluate vr(t) at any arbitrary time, tA using ODE

dt

r

tvCgdtatvdv

AAAr t

o

rDt

rAr

tv

r

0

2

00 8

3)(




Using the “H” and “vr(tA)” Equations

Almost Done. Find CD in Idelchik Text Ref.

A

t

Do

rH

dtdtCr

tvgHdy

r A

0 0

2

0 8

3

21313 2

491.0

2

565.4299.23

Re

491.0

Re

565.4Re

99.23

ororor

Drvrvrv

C




Collapse constant expressions into “K” Terms

This eqn can be solved numerically as described in JVST ppr, eqns 2529

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

A

t

o

rrrH

dtdtr

vKvKvKgHdy

r A

0 0

233

3521

0

375.0




Example Calc: ro = 1 mm, = 7.4x10-7 m2/s

TimeStep, n

a(m/sq-s)

Re del-v(m/s)

v(m/s)

del-y(m)

H(m)

Time(ms)

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



Degree of Saturation We (finally) have all the tools to determine the

degree of saturation, Sv, for the rising bubble

dre

rrn

rrndrrrS

oorv

o r

n

rDn

o

onr

rvov

01

1

0

222sin121,)(

Conceptually

),,,,,( HDDSS lhvvv Note

Dh and H are DESIGN-controlled

Well known liquid properties = l

Poorly Characterized Liquid properties = Dv, ,



Estimate Properties for TEOS, Etc.

Degree of Saturationcont.-1

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

Chemical Temperature(K)

Mw(kg/kmol)

l

(kg/m3)Dv

(m2/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



Degree of Saturationcont.-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”)

Actual ro = 1.5-2 mm

vr,f = 9.7mm/33.3ms = 0.29 m/s (0.65 mph)

Fully Humidified



Validation Testing



TEOS Liquid Source Vapor Bubble Saturation v. Liquid Height

0%

20%

40%

60%

80%

100%

0.0 0.2 0.4 0.6 0.8 1.0

Liquid Level Inside Bubbler, y (inch)

Inte

gra

ted

Bu

bb

le S

atu

rati

on

, S v

0

30

60

90

120

150

Bu

bb

le R

ise

Tim

e,

r (m

s)

Integrated Saturation (%)

Rise Time (ms)

file = Sv(t)_01.xls

• Bubble Diameter = 1.999 mm

• Dv for TEOS in N2 = 0.0727 cm2/s

• Kinematic viscosity,, = 0.00511 cm2/s

99% Saturation after 67 ms, or 0.46"

Linear portion of r curve indicatesterminal velocity of ~0.23 m/s



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

Degree of Saturation - Conclusions

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