Scaling-up and bridging scales in process engineering -...

NATIONAL TECHNICAL UNIVERSITY OF ATHENS

Postgraduate Program "Mathematical Modeling in Modern Technologies and Finance“, NTUA, 2 Dec. 2015

Scaling-up and bridging scales in process engineering

Andreas G. Boudouvis Professor & Dean School of Chemical Engineering NTUA, Athens, Greece http://www.chemeng.ntua.gr/dep/boudouvis/

http://www.chemeng.ntua.gr/dep/boudouvis/

A scale-up triumph: Penicillin Production NATIONAL TECHNICAL UNIVERSITY OF ATHENS

Sir Alexander Fleming holding a petri dish with Penicillium notatum culture, 1928 (Left) and inspecting a 15,000 gallon “deep tank” used in penicillin production at a Squibb plant in New Brunswick, NJ, June 1945 (Right).

The project was completed in a very short time 1939: Florey (Oxford University) produces enough penicillin to test it on mice. But, he cannot produce enough for human clinical trials. 1943: A dose of penicillin cost: $20. 1946: A dose of penicillin cost: 55 cents. Submerged fermentation process is still the dominant production technique for penicillin

Scaling-up of penicillin production became a top-priority program of complexity and size rivaling that of the Manhattan Project

Deposition processes

From ordinary life to advanced materials

…coatings, nanomaterials, MEMS…


Metal-organic Chemical Vapor Deposition of Aluminum (Al - MOCVD)

Precursor: DMEAA

Metal-organic CVD

high conformal coverage of complex-in-shape substrates

low deposition temperature

convenient handling of gaseous byproducts

high throughput


Chemical Vapor Deposition: Transport + Reaction

IR Lamps

cooling

INLET

OUTLET susceptor

wafer

surface diffusion

of film precursors adsorption

forced – convection region

transport to surface

+

gas phase reactions

desorption of adsorbed species

surface reaction

CVD reactor

CVD process


H3C

Al

H H H

N

C2H5 CH3

Al H H

H H3C

N

C2H5 CH3

+

Al +3/2H2

Dimethylethylamine alane (DMEAA)

Alane Dimethylethyleamine (DMEA)

In the “test tube” :

A typical chemist’s prospective…

The engineer’s prospective... Scale-up: from the “test tube” to production

CIRIMAT-CNRS, ENSIACET, Toulouse


Showerhead

Wafer

Pump

Trap

P

DMEAA

MFC

T

Τ

N 2

P

MFC

Test tube

Chemical Vapor Deposition: Transport (+ Reaction)

Xenidou et al., Surface Coatings Technology 201, 8868 (2007)

T(Κ)

Temperature

U(m/s)

Velocity U(m/s)

(Aluminum deposition)


CH3 H3C

N

C2H5

CH3 H3C

N

C2H5

H2

Al Al H H

H

Al H H

H

CH3 H3C

Al

H H H

N

C2H5

CH3 H3C

Al

H H H

N

C2H5

CH3 H3C

N

C2H5

+ Al

H H H

Gas-phase reaction

Surface reactions

H2

Yun et al., J. Vacuum Sci. Technol. 16, 419 (1998); Jang et al., Thin Solid Films 333, 137 (1998)

Chemical Vapor Deposition: (Transport +) Reaction NATIONAL TECHNICAL UNIVERSITY OF ATHENS

dimethylethylamine alane (DMEAA) dimethylethylamine (DMEA) + alane (AlH3)

(Aluminum deposition)

The key engineering motivation: Determine “operating windows”

Reactor operating conditions - pressure - temperature - flow rates, …

Film properties - deposition rate - thickness uniformity - film composition, …

Reactor design

high deposition rates thickness uniformity economic use of the reactants

Industrial demands

substrate substrate

Layer thickness control

uniform layer non-uniform layer


The engineering analysis outcome: Reliable Process Design Temperature effect on Aluminum growth rate

Al G

row

th R

ate

(Α/m

in)

0

50

100

150

200

250

300

0 5 10 15 20 25

modelexperiment

0

50

100

150

200

250

300

0 5 10 15 20 25

modelexperiment

0

50

100

150

200

250

300

0 5 10 15 20 25

modelexperiment

T = 160oC T = 200oC

T = 220oC T = 260oC

0

50

100

150

200

250

300

0 5 10 15 20 25

modelexperiment

Xenidou et al., Surface Coatings Technology 201, 8868 (2007)


The goal: Computer-aided process analysis based on first-principles – An enabling tool

The means: Realistic model development – Input from experiment Validation – Comparison with experiment

The benefits: Understanding mechanisms Savings on experimental cost and manpower Improve experimental design Guided experiments


Engineering Analysis

Transport Processes Modeling Summary (N species, single phase)

3 + N Physical (conservation) Laws – 3 + N differential equations

0)(t

=ρ⋅∇+∂ρ∂ v

]ptp[)Tk(]T

tT[cp ∇⋅+

∂∂

+∇⋅∇=∇⋅+∂∂

ρ vv

τ⋅∇+∇−ρ=ρ⋅∇+ρ∂∂ p)()(t

gvvv

Mass:

Momentum:

Energy:

3 + N Unknowns: p, v, T, ρ = ρ(p, T), e.g. ρ=p/RT (ideal gases)

T)(.g.e),( vvv ∇+∇µ=ττ=τ (Newtonian fluids) plus constitutive equations

Boundary and initial conditions


( ) 2i i i i iY D Y R S∇⋅ ρ = ∇ + +vSpecies Equation:

iY

Mathematical model

Numerical approximation/

Code implementation

Partial Differential Equations (conservation laws)

Discretization finite element method finite volume method


Computer-aided Analysis

Algorithms (solvers) High-performance machines

Cost-effective computations

Reliability of solutions

(Validation)

Discretization refinement Comparison with experiments

Governing Equations 2-D Axisymmetric Geometry – Cylindrical coordinates

( ) ( ) ( )1 1 1 2 123

x x rx x r x

u u upr u u r u u r u r gr x r r x r x x r r r x

ρ ρ µ µ ρ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ + = − + − ∇⋅ + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

Momentum Equations

( ) ( ) ( )

( )2

2

1 1 1 1 223

223

xr rx r r r

r

uu upr u u r u u r r ur x r r r r x x r r r r

uu ur r r

θ

ρ ρ µ µ

µµ ρ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = − + + + − ∇⋅ − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

− + ∇ ⋅ +

( ) ( ) 0x r ru u ux r rρ ρ ρ∂ ∂

+ + =∂ ∂

Continuity Equation

( ) ( ) 32

1 1 1 rx r

u u u ur u u r u u r rr x r r x x r r r r r

θ θ θθ θρ ρ µ µ ρ∂ ∂ ∂ ∂ ∂ ∂ + = + − ∂ ∂ ∂ ∂ ∂ ∂

( ) ii i iuY J R Sρ∇⋅ = −∇ ⋅ + +

Species Equation

[ ]( ) i ii

u E p k T h Jρ ∇ ⋅ + = ∇ ⋅ ∇ −

∑

Energy Equation


FLUENT CFD package

Discretization – Finite Volume Method

( ) ( ) CV CV CV

div u dV div dV S dVρ Φ ΦΦ = Γ ∇Φ +∫ ∫ ∫

Integration over each volume of the mesh:

Divergence Theorem: ( )CV A

div a dV n adA= ⋅∫ ∫

( ) A A CV

n u dA n dA S dVρ Φ Φ⋅ Φ = ⋅ Γ ∇Φ +∫ ∫ ∫

Integral Form

convection diffusion sources

( ) ( ) div u div grad Sφρ Φ = Γ Φ +


Versteeg & Malalasekera “Introduction To Computational Fluid Dynamics-The Finite Volume Method”, Longman, 1995

Substitution yields algebraic equations with only center values involved. Subscript NB refers to neighboring cells.

0 0C C NB NBNB

a a SΦ = Φ +∑

CA bΦ =A: Matrix of coeffients ΦC: Unkowns at cell centers b: sources

Assembly of the system to be solved


Discretization (conl’d)

High performance computing (cont’d)

[http://febui.chemeng.ntua.gr/pegasus.htm]


MOCVD: Aluminum deposition NATIONAL TECHNICAL UNIVERSITY OF ATHENS

Collaborative project: CIRIMAT/ENSIACET, Toulouse – NTUA, Athens

Main objectives

optimum process parameters (temperature, flow rates, …)

optimum reactor configuration (showerhead-substrate distance, shower-plate, …)

Teams: Athens: A. Boudouvis, I. Aviziotis, N. Cheimarios, D. Xenidou Toulouse: C. Vahlas, T. Duguet, N. PrudHomme

Xenidou et al., Surface Coatings Technology 201, 8868 (2007); Xenidou et al. J. Electrochemical Soc. 157, D633 (2010)

Typical operating conditions

Parameter Typical value

Ν2 diluent flow rate 305 sccm

Ν2 carrier flow rate 25 sccm

DMEAA bubbler temperature 9 οC

DMEAA flow rate 1.4 sccm

Inlet gas temperature 65 oC

Substrate temperature 200 oC

Walls temperature 25 oC

Total pressure 10 Torr

Deposition time 120 min

Αντλία

P

DMEAA

MFC

T

ΤΘερμοστοιχείο

τύπου SΠαγίδα

συμπύκνωσης

Μανόμετρο

N2

P

MFC

P

Td

Tb

Fd

Fc Tin

Tw

Fp

MOCVD reactor

9 mm

16 mm

20 mm

24 mm

Growth Rate Measurement


Numerical Solution

Reactor discretization

Computational details

• Finite Volume Method • SIMPLEST pressure

correction scheme • Upwind differencing

scheme • TDMA solver • Grid: 33.000 cells (105 x 315 (NX x NZ)) • 120 min CPU time

(2.8GHz Pentium IV/1.GB RAM)

12.7 mm

290mm

20 mm

15mm

83 mm

58 mm

60 mm

110mm

10mm

y

x


ANSYS/FLUENT CFD package

Model predictions at typical operating conditions

T(Κ)

Temperature

Model predictions

− Temperature filed is uniform

above the substrate; this means

that conduction is dominant

compared to convection

− The isotherms follow the shape

of the showerhead, due to heat

transfer through the walls


Model predictions at typical operating conditions

U(m/s)

Model predictions

Velocity

− The recirculation zone may be

attributed to the local pressure

drop

− The recirculation zone will trap the

mixture inside the showerhead and

cause precursor condensation

− It may provide premixing of the

gas mixture, which is beneficially to

the thickness uniformity


0,7

0,85

1

1,15

1,3

0 5 10 15 20 25

Distance in radial direction (mm)

Nor

mal

ized

spe

cies

mas

s fr

actio

ns

N2

DMEAA

H2 DMEA

Model predictions

Chemical species distribution at typical operating conditions


Comparison of experiments and predictions

Arrhenius plot of Al growth from DMEAA

160200220260

1000/T (1/K)

T(oC)

0

50

100

150

200

250

300

1,8 1,9 2,0 2,1 2,2 2,3 2,4

experimentmodel

Gro

wth

Rat

e (Α

/min

)G

row

th R

ate

(Α/m

in) − Growth rate decreases above

200oC, due to DMEAA dissociation

in the gas-phase

− Kinetically-controlled regime

extends below 200oC, while

above 200oC growth takes place

in the transport-controlled regime


Al G

row

th R

ate

(Α/m

in)


0

50

100

150

200

250

300

0 5 10 15 20 25

modelexperiment

0

50

100

150

200

250

300

0 5 10 15 20 25

modelexperiment

0

50

100

150

200

250

300

0 5 10 15 20 25

modelexperiment

T = 160oC T = 200oC

T = 220oC T = 260oC

0

50

100

150

200

250

300

0 5 10 15 20 25

modelexperiment

Comparison of experiments and predictions NATIONAL TECHNICAL UNIVERSITY OF ATHENS

On the showerhead design

design A design B design C

84mm

7mm

10mm

6mm 3mm

50mm

7mm

44mm

101mm


Velocity at typical operating conditions


design A design B design C

Umax = 12.8m/s Umax = 9.6m/s Umax = 6.4m/s


Andreas G. Boudouvis VIMA/ RTRA-STAE @ Toulouse, 4 July 2014


Distribution of the reactants over the substrate

1,38E-02

1,40E-02

1,42E-02

1,44E-02

1,46E-02

1,48E-02

1,50E-02

0 5 10 15 20 25 30 35

Distance in the radial direction of the substrate (mm)

DM

EAA

mas

s fr

actio

n

small

medium

large

design A design B design Cdesign A design B design C

DM

EAA

mas

s fr

actio

ns


Design Δω(%) * A 5.870

B 6.268

C 6.444

* Non-uniformity Δω(%) is calculated through the maximum, minimum and average values:

average

minmax

ωω−ω

=ω∆


Ongoing work

New design

7 mm

1.30 mm

1.5 mm 0.76 mm

10 mm

Actual design

Investigation of the shower-plate

On the showerhead design (ongoing)


Copper CVD: Mixed chemical kinetics

0 expdEk kRT

α = −

[ ] [ ]1 20.07 / , 0.01 /k m s k m s= =

Inhibition effect from H(amd)

Arrhenius plot

( )

( ) ( )( )22

22

1

1 22

d HCu amd

d HH amdCu amd

k k C Cr

k C k C k C

=+ +

( ) 220 HCu amd

Er k C CRT

α

= −

10 1 2066 / , 1.33 10 /E kJ mol k s kmol m sα

− = = ×

Langmuir-Hinshelwood 1-st order

Aviziotis et al., Surf. Coat. Tech. (2014)

Multiscale modeling in CVD

Manipulation of the events in the micro/nano scale

Deposition in a predefined topography Surface nano-morphology

by macro CVD reactor operating conditions

Physical phenomena in micro/nano

5 nm

Hamers et al., Ultramicroscopy (1989)

void

0.2 μm

Kinoshita et al., Jpn. J. Appl. Phys. (2005)

823 K 873 K

A. G. Boudouvis NATIONAL TECHNICAL UNIVERSITY OF ATHENS

Micro- topography corresponding to a boundary

cell @ the wafer

Macro- scale (cm) (Reactor Scale Model)

Micro- scale (μm) (Feature Scale Model)

Coupling (bi-directional exchange of info) of scales Correction of the boundary condition for the species equation.

effective reaction rate: ε effective reactivity factor

Single scale (macro-) computations: Multiscale computations:

Multiscale modeling of CVD: Wafer with micro-topography

cannot use the same models to describe the physical phenomena in macro- & micro- scale Macro- scale Kn < 1 Micro- scale: Kn > 1

reaction rate

,i i i is

eff macrorD Y Mρ γ⋅∇ =nsi i i iD Y M rρ γ⋅∇ =n

,s s

eff macror rε= ⋅sr

Jensen et al., Curr. Opin. Solid St. M. (1998). Cale et al., Comput. Mater. Sci. (2002)

National Technical University of Athens School of Chemical Engineering


Momentum Equations

( ) ( ) 0x r ru u ux r rρ ρ ρ∂ ∂

+ + =∂ ∂

Continuity Equation

( ) ii iuY J Rρ∇⋅ = −∇ ⋅ +

Species Equation

[ ]( ) i ii

u E p k T h Jρ ∇ ⋅ + = ∇ ⋅ ∇ −

∑

Energy Equation

x

r

FLUENT ( ) ( ) ( )1 1 1 2 12

3x x r

x x r xu u upr u u r u u r u r g

r x r r x r x x r r r xρ ρ µ µ ρ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ + = − + − ∇⋅ + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

( ) ( ) ( )

( )2

1 1 1 1 223

223

ρ ρ µ µ

µµ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = − + + + − ∇⋅ − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

− + ∇ ⋅

xr rx r r r

r

uu upr u u r u u r r ur x r r r r x x r r r r

u ur r

Reactor Scale Module (RSM)

7183 cells

boundary condition (surface reactions)

,s

i i i i eff macroD Y v M rρ ⋅∇ =n

Xenidou et al., J. Electrochem. Soc. (2010).

Volumetric reactions


Calculation of the local fluxes and sticking coefficients

, ,( ) ( ) 1 ( ) ( ) ( ) ( , ) ( )ci i direct E i 1 2 N i iS , ,..., Q dA

Α

Γ Γ Γ Γ Γ Γ ′ ′ ′ ′ ′ ′= + − ∫∫x x x x x x x x

i=1,2, …, N

SE,i: Sticking coefficient of species i

Γi,direct (x): direct flux, shadowing effects

Qi(x, x’): geometrical term which incorporates the reemission mechanism of species i

Kokkoris et al., J. Vac. Sci. Technol. A (2004) Osher, S. and R. P. Fedkiw, Springer (2003)

Flux of species i in elementary area on point x :

Reemission

Shadowing

Kn > 1

Feature Scale Module (FSM) : Ballistic transport

+ | | 0, ( , 0 ) ( ), ,t F t q∇ = = = ∈x x xϕ ϕ ϕ Ω

Profile evolution algorithm/Level Set Method

φ: level set function F: normal velocity to the moving boundary F |∇φ| = H: Hamiltonian

www.phietch.org


http://www.phietch.org/

Coupling RSM with FSM

FLUENT

Ballistic model

n =

n +1

Yi , ρ,T

Film growth for Δt

Yes

Level set method

No

Correction of the surface reaction rate term in the BC for the species equation @ A

( )

, '1ns seff m ro

Aicr r dA

A= ∫

sr

( ) ( )( ),

n ns n seff macror rε=

•Boundary condition: ( )

,

nsi i i i eff macroD Y M rρ γ⋅∇ =n

( ) ( )

( )

, ,

, 2

n n

n

s seff macro eff micro

seff macro

r rtol

r

−<

Cheimarios et al., Chem. Eng. Sci. (2010)

( )

( )

,( 1) ( )

,

n

n

seff micron ns

eff macro

r

rε ε+ =

Yi , ρ,T

jε

@si ArΓ →


Case study: Multiscale modeling of Si CVD

Sticking coefficients:

4 2 4

2

,

,

( , , )

1E SiH w H SiH

E SiH

S g T

S

= Γ Γ

= (constant)

Kleijn, J. Electrochem. Soc. (1991)

Volumetric reaction:

SiH4 ↔ SiH2 + Η2

( )40 exp( )V aSiH

Er k f CRT

= −

(Arrhenius type)

SiH4 → Si(s) + 2Η2

SiH2 → Si(s) + Η2

Surface (deposition) reactions:

, 4 2, ,si E i ir S i SiH SiH= Γ =

(Eley-Rideal type)


Results: Coupling RSM with FSM

(thickness top)(thickness bottom)

t

b

dd

Θ =

Conformality (Θ) inside long rectangular trenches Base case

decreasing Tw

decreasing fSiH4 (inlet)

increasing Pop

Effect on conformality by:

Pop= 133 Pa Tw = 1050 K

fSiH4 = 0.1 (inlet)

4

2

4

2

,

,

7

9

3

3.5 10

1

0.

4.18 10

. 10

91

2 9

s

sSiH

E S

E SiH

Si

H

H

i

S

r

S

r

−

−

−= ⋅

=

Θ =

= ⋅

= ⋅dt

db

t = 0s

t = 192s

16 trenches per 32 μm, initial depth = 3 μm, initial width = 1 μm


Conformality (Θ) inside long rectangular trenches

Base case

4

2

4

2

,

,

7

9

3

3.5 10

1

0.

4.18 10

. 10

91

2 9

s

sSiH

E S

E SiH

Si

H

H

i

S

r

S

r

−

−

−= ⋅

=

Θ =

= ⋅

= ⋅

Decreasing Tw

2

4

2

4

5,

11

8

,

9.00 1

2.37 1

1

5.0 10

0

0

1

s

E Si

SiH

sSiH

E SiH

H

r

S

r

S

−

−

−

= ⋅

=

= ⋅

⋅

Θ =

=

Pop= 133 Pa Tw = 1050 K

fSiH4 = 0.1 (inlet)

- Tw = 900 K

-

t = 0s t = 0s

t = 192s t = 2340s

Results: Coupling RSM with FSM (cont’d)




Base case

4

2

4

2

,

,

7

9

3

3.5 10

1

0.

4.18 10

. 10

91

2 9

s

sSiH

E S

E SiH

Si

H

H

i

S

r

S

r

−

−

−= ⋅

=

Θ =

= ⋅

= ⋅

Decreasing fin,SiH4

2

2

4

4

12

2,

9

,

8

1.5 1

9.9

0

5

.0

.6

1

1

0

7

8

0

10

E S

sSiH

sSiH

E SiH

iH

r

S

S

r−

−

−

=

⋅

=

Θ =

⋅

=

=

⋅

- -

fSiH4 = 0.001 (inlet)

t = 0s

t = 192s

t = 0s

t = 15600s

Pop= 133 Pa Tw = 1050 K

fSiH4 = 0.1 (inlet)




Base case

4

2

4

2

,

,

7

9

3

3.5 10

1

0.

4.18 10

. 10

91

2 9

s

sSiH

E S

E SiH

Si

H

H

i

S

r

S

r

−

−

−= ⋅

=

Θ =

= ⋅

= ⋅

Increasing Pop

4

2

4

2

7

7

4,

,

6.0 10

6.66 10

1.19

.85

1

10

0

E S

E SiH

sSiH

sSi

iH

H

S

S

r

r−

−

−

=

=

⋅

Θ =

=

=

⋅

⋅

Pop= 1033 Pa - -

t = 0s

t = 192s t = 185s

t = 0s

Pop= 133 Pa Tw = 1050 K

fSiH4 = 0.1 (inlet)




with micro-topography

no micro-topography

• Effect on the Arrhenius plot



void

0.2 μm

Kinoshita et al. Jpn. J. Appl. Phys. (2005)

823 K 873 K

t = 0s

t = 192s

t = 0s

t = 2340s

900 K 1050 K

43


Model vs Experiment

• Multiscale modeling of MOCVD – Experiments & computations

Ongoing research

Aluminum deposition in rectangular trenches

(courtesy of Dr. C. Vahlas, CIRIMAT/Toulouse)

Challenge Coupling of the three scales (macro-, micro-, nano- )

roughness development in the features



Scaling-up and bridging scales in process engineering -...

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