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1. Nanotechnology: Overview of Aerosol Manufacture of Nanoparticles
Prof. Sotiris E. PratsinisParticle Technology Laboratory
Department of Mechanical and Process Engineering,ETH Zürich, Switzerland
www.ptl.ethz.ch
Sponsored bySwiss National Science Foundation and
Swiss Commission for Technology and Innovation
2
Nanoparticles
1 - 100 nm (at least into two dimensions)
Remember, the thickness (diameter) of a human hair is 50,000 - 100,000 nm!
3
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The Melting Point Decreases with Decreasing Nanoparticle Size
Mel
ting
Poin
t, K
Mel
ting
Poin
t, K
Particle diameter, Å
Particle diameter, nm
Au BiPeppiatt, Proc. Roy. Soc. A 345, 1642 (1975)
Buffat and Borel, Phys. Rev. A 13, 2287 (1976)
5
Applications of Nanoparticles• Large area per gram (adsorbents, membranes)
• Stepped surface at the atomic level (catalysts)
• Easily mix in gases and liquids (reinforcers)
• Superfine particle chains (recording media)
• Easily carried in an organism (new medicine)
• Superplasticity
• Cosmetics that last way into the night ...
Some people believe that nanoparticles are
a new state of matter!
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Comparison of wet- & dry-technologyDry-technology (aerosol):
•Mix precursor•Dry flame conversion•Filtration•Milling
Wet-technology:
•Dissolve•Add precipitation agent•Temperature/Pressure treatment•Filtration•Washing•Drying•Calcination•Milling
Short process chains, very short process time:Reduced costs, green processes
http://www.stanfordmaterials.com/zr.html#info
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AEROSOL MANUFACTURING OF NANOPARTICLES Wegner, Pratsinis Chem. Eng. Sci. 58, 4581-9 (2003).
ProductParticles
Carbon black
Titania
Fumed Silica
Filamentary Ni
Fe, Pt, Zn2SiO4/Mn
Volumet/y
8 M
2 M
0.2 M
0.6 M
0.04M
~0.02M
Value$/y
8 B
4 B
2 B
0.7B
~0.1B
~0.3B
Process
Flame, CxHy
Flame, TiCl4
Flame, SiCl4
Hot –Wall, Zn
Hot-Wall, Ni(CO)4
Hot-Wall, Spray…
CoagulationCoalescence
X
X
X
X
X
X
Surface Growth
X
?
-
X
X
X
Zinc Oxide
9
A rough analogy to flame aerosol reactors
… just well attached to the ground !
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Attic red-figure hydria, 430-420 BC, Abdera Archeological Museum, Greece
Attic black-figure amphora, 540-530 BC,Museum of Cycladic Art, Athens, Greece
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Prof. Gael W. UlrichDept. of Chemical Engineering University of New Hampshire
18
Prof. Ulrich's insightful proposals
1. New particle formation (nucleation) cannot be distinguished from chemical reaction.
2. No surface growth.
3. Turbulence does not affect particle growth.
4. Aggregates or agglomerates form when coagulation is faster than coalescence.
5. The particle size distribution is self-preserving
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Applications
Paints
Paper coatingsPlastics coatings
TiO2 (titania)
Niche Fields:Catalysts, Sensors,
Photocatalysts,
Cosmetics etc.
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Limitations of science in the ‘70sUnderstanding of particle formation has little impacton industrial aerosol reactor design.
Providing a plausible particle synthesis scenario alone was not enough:
1. Probably industrial reactor data could not be duplicated in the laboratory reactors
2. Traditional aerosol instruments were too slow 3. No scale-up relationships4. Too complex fluid mechanics (reactive systems).
Industrial reactors were still treated as "black boxes" Design and operation were dominated by empiricism.
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29
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Thermophoretic Sampling (Dobbins and Megaridis, 1987))Thermophoretic Sampling (Dobbins and Thermophoretic Sampling (Dobbins and MegaridisMegaridis, 1987)), 1987))
0
10
20
30
40
50
60
0 1 2 3 4Distance from the burner, cm
Ave
rage
prim
ary
part
icle
dia
met
er, n
m .
TiO2 data by thermophoreticsampling
Image Analysis
TEM
16 ± 4 nm
32
Sintering rate of particle area a
)a(a1dtda
s−−=τ
Koch, Friedlander, J. Colloid Interface Sci. 140, 419 (1990)
33
2βN21
dtdN −=
)a(aτ1a
dtdN
N1
dtda
s−−−=
f1/Dppc )v(vdd /=
Number Concentration
Agglomerate area
Collision diameter
Kruis, Kusters, Pratsinis, Scarlett, Aerosol Sci. Technol. 19, 514 (1993)
34
c = 0.01 c = 0.1
a = 1.4
a = 1.0
b = 3.0
b = 0.3 Tem
pera
ture
, f(a
)
Coolin
g Rate
,
(b)
Concentration, f(c)
Predictions within 3% of product SSA (Gutsch, 1997)
Design of Equipment
f
H. Mühlenweg, A. Gutsch, A. Schild, C. Becker “Simulation for process and product optimization”, Silica 2001, 2nd International Conference on Silica, Mulhouse, France (2001) and G. Vargas Commercializing Chemical Technology: Realization of Complete Solutions using Chemical Nanotechnology, Lecture at Nanofair, St. Gallen, Switzerland, Sept. 11, 2003.
35
N 234 nano-structure black
Niedermeier, Messer, Fröhlich (TR814.1E)
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TODAYAerosol Scientists and Engineers lead R & D for aerosol manufacture at Degussa, DuPont, Millennium, Cabot etc.
Basic and exploratory research is needed for:On-line control of existing reactors for flexible
manufacture of various particlesHigh value functional nanoparticles with
sophisticated composition and structure.Manufacture these nanoparticles without going
through the Edisonian cycle of the past. Health effects of nanoparticles.
1
Novel Processes and Uses of Flame-made Nanoparticles
Prof. Sotiris E. PratsinisParticle Technology Laboratory
Department of Mechanical and Process Engineering,ETH Zürich, Switzerland
www.ptl.ethz.ch
Sponsored by theSwiss National Science Foundation and Swiss
Commission for Technology and Innovation
2
Flame-made particles
Pros ChallengesHigh purity AgglomeratesEasy collection Size controlNo liquid waste MulticomponentProven scale-up ceramic/ceramicNo moving parts metal/ceramic
3
Experimental set-up for TiO2 production
MethaneAirArgon / TiCl4
FIC
FIC
FIC
Argon
MethaneAir
TiCl4
Flame Reactor
Hood
Filter Assembly
Pump
NaOH Solution
Pratsinis, Zhu, Vemury, Powder Technol. 86, 87-93 (1996)
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Mixing of reactant gases ==> product size-shapePratsinis, Zhu, Vemury, Powder Technol. 86, 87-93 (1996); Johannessen, Pratsinis, Livberg, ibid., 118, 242-250 (2001).
CH4
Air
Air
TiCl4
CH4
CH4
CH4
Air Air
TiCl4TiCl4TiCl4
5
Kammler, PhD thesis, ETH #14622 (2002)
U.S. Patent5,861,132(1999)
Precision Size Control by Charging
Electrically Assisted Synthesisof Nanoparticles
S. Vemury, S.E. Pratsinis, L. Kibbey, J. Mater. Res. 12, 1031-1042 (1997).
6
w/o electric field
200 nmH
AB
200 nm
with electric fieldHABHAB Evolution of TiO2 particle
growth with and w/oexternal electric fields
Filter20 cm
Filter20 cm
2 cm10 cm 10 cm
5 cm 5 cm
0 kV/cm 2 kV/cm
0.5 cm 0.5 cm
Kammler, Pratsinis, Morrison, Jr., Hemmerling, Combust. Flame 124, 369 (2002)
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Dental n-Composites: flame-made silicas in adimethylacrylate matrix (50:50)
with OX50 (Degussa)with ETH non-aggl. SiO2
SSA = 35 m2/g SSA = 50 m2/g
Müller, Vital, Kammler, Pratsinis, Beaucage, Burtscher, Powder Technol. 140, 40-48 (2004).
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Precision Synthesis by Nozzle Quenching
Wegner, Stark, Pratsinis, Mater. Lett. 55, 318 (2002)
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Reduction of Agglomeration6 L/min O2 flow rate
Nozzle BND = 1.5 cm
No nozzleProduct powder
TS in front of nozzle (BND = 1.5 cm)
10Burner - Nozzle Distance, cm
0 1 2 3 4 5BE
T-eq
uiva
lent
Par
ticle
Dia
met
er, n
m
0
5
10
15
20
25
30 2 L/min O23456
0
10
20
30
40
50
FilterNo Nozzle
O2 / Ti decreasesControl of TiO2size, color & crystallinity
11
100 150 200 2500
50
100
made
made
flame-
wet-phase-
NO
x rem
oved
/ %
Process temperature / °CStark, Wegner, Pratsinis, Baiker,J. Catal. 197, 182 (2001)
V2O5/TiO2: Catalytic Removal of NOxIn Exhaust Gases by SCR with NH3
12
0.5
m
Baghouse filter(2.5 m tall)
Pilot unit for flame
synthesis of C/SiO2 ,
and now catalysts:
V2O5 /TiO2 and TiO2 /SiO2
Kammler, Mueller, Senn, Pratsinis, AIChE J. 47, 1533 (2001)
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Comparison to conventional DeNOx catalyst @ U. Essen (Prof. Cramer)
Fixed bed pilot-scale testreactor, 2.4 cm/sec
Gas composition:-400 ppm NO-400 ppm NH3-10 vol% oxygen in nitrogen
Reference:-impregnated Degussa P 25-same V content in both catalysts-specific surface area: 50-55 m2/g
160 200 240 2800
20
40
60
80
100
18 wt% VOx/TiO2, flame-made, 100 g/h
20 wt% VOx/TiO2, impregnated
NO
rem
oved
/ %
Reactor Temperature / °C
•W. J. Stark, A. Baiker, S. E. Pratsinis, Part. Part. Sys. Charact. 19, 306-311 (2002)
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0 2 4100
125
150
175
200
225
Spe
cific
sur
face
are
a / m
2 g-1
Titania content / wt%
Epoxidation Catalysts: TiO2 /SiO2
Hydrogen/air flame, burner diameter 19 mm, 0.73 m3 H2/h; 5.2 m3 air/h
150 g/h
•W. J. Stark, S. E. Pratsinis, A. Baiker, J. Catal., 203, 516 (2001) and Ind. Eng. Chem. Res., 41, 4921 (2002)
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TiO2/SiO2 epoxidation catalystsIndustrially (Shell, Enichem, Arco), several Mt/y :
C3 ⇒ propene ⇒ propene oxide ⇒ polymers, surfactants
50
60
70
80
90
100
6 g/h 150 g/h 500 g/h
Flame-madeEnichemAerogelShell
peroxide usage olefin usage
Sel
ectiv
ity /
%
•W. J. Stark, H. K. Kammler, R. Strobel, D.Günther, A. Baiker, S. E. Pratsinis, Ind. Eng. Chem. Res., 41, 4921 (2002)
OH
1
OH
O
2
TBHP (5)
Ti/silica
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X-ray Absorption Near Edge Spectroscopy
4, 5, 6 - coordinated Ti
tetrahedral Ti
•J. D. Grunwaldt, C. Beck, W. J. Stark, A. Hagen, A. Baiker, Phys. Chem. Chem. Phys., 4, 3514 (2002).
OTi
OOO
SiSi
SiSi
TiO O
OO
OH
Si
SiSi
SiTi
O OOO
O
Si
SiSi
Si
H H
TiO O
OO
OH
Si
SiSi
Si
O
H
H
1.3 % Ti, flame
1 % Ti, wet-phase
OTi
OOO
SiSi
SiSi
In-situ XANES:-Geometry of the active site-Water content-Degree of hydration
17
Selectivity
10 100 100020
40
60
80
100
Co Cr Mn Fe
Per
oxid
e se
lect
ivity
/ %
Content / ppm10 100 1000
20
40
60
80
100
Co Cr Mn FeO
lefin
sel
ectiv
ity /
%
Content / ppm
Even 40 ppm of transition metal strongly reduce selectivity
Good selectivity requires very pure catalysts.
•W. J. Stark, R. Strobel. D. Günther, S. E. Pratsinis, A. Baiker, J. Mater. Chem. 12, 3620-25 (2002)
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HollowHollow
Bi2O3 SolidSolid
0.1 µm
2 cm
Flame Spray PyrolysisAl2O3, ZnO, CeO2, ZrO2ZnO/SiO2 , BaTiO3 Au, Pt on TiO2, SiO2, Al2O3
Mädler, Pratsinis, J.Am.Ceram.Soc. 85, 1713 (2002)VaristorsSensorsCatalysts
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Flame spray pyrolysisStrobel, Stark, Mädler, Pratsinis, Baiker, J. Catal. 213, 296-304 (2003)
Spray flame producing Pt/Al2O3
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Enantioselective hydrogenation of ethyl pyruvate by FSP-made Pt/Al2O3
Synthesis of chiral pharmaceuticals.
Strobel, Stark, Mädler, Pratsinis, Baiker, J. Catal. 213, 296-304 (2003)
0
25
50
75
100
0 50 100 150 200
Time / min
Enan
tiom
eric
exc
ess
(ee)
/ %
E4759
FSP-made
0
25
50
75
100
0 50 100 150 200
Time / min
Con
vers
ion
/ %
Engelhard (E4759)
FSP-made
21
Open structure enhances activity
0
0.2
0.4
0.6
0.8
1 10 100
Pore diameter / nm
Por
e vo
lum
e / c
m3 g
-1
E4759 flame made
Nonporous, dense particles
Better accessibility
High surface without trappedPt in micropores.
Maximum use of expensive platinum reduces costs
Strobel, Stark, Mädler, Pratsinis, Baiker, J. Catal. 213, 296-304 (2003)
22ZnO diameter (dXRD), nm
0 5 10 15
Wav
elen
gth,
l 1/
2 , n
m
340
350
360
370
380l Bulk
Flame made (FSP) this workFSP powder thermally treated (2h @ 600°C)Wet-made, Meulenkamp (1998)
Wavelength , nm300 350 400 450
Abs
orba
nce
(sca
led)
, a.u
.
12 nmdXRD= 3 nm 4
50 nm
Blue Shift of Absorption Spectra
Rapid Synthesis of Stable ZnO Quantum Dots (1 - 5 nm)
Quantum Size Effect
5 nm
λ
λ
Mädler, Stark, Pratsinis, J.Appl. Phys. 92, 6537-40 (2002)
Tani, Mädler, Pratsinis, J. Mater. Sci. 37, 4627-4632 (2002)
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Angular, rough, edgy-like n-CeO2
Mädler, Stark, Pratsinis, J. Mater Res. 17, 1356 (2002).
20 nm
100 nm
as-prepared
annealed for2h @ 900°COxygen flow rate, l/min
0 2 4 6 8
Ave
rage
par
ticle
size
, nm
02468
10121416
dBET
dXRD
CatalystsFuel CellsPolishing
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Tani, Watanabe, Takatori and Pratsinis, J. Am. Ceram. Soc., 86, 898 (2003).
Al2O3-N
Al2O3-N-O
Al2O3-H-N
Al2O3-Cl
400 nm1 µm
SupportingFlame
Filter
Hood
Vacuum Pump
MainFlame
Air
DispersingAir or O2
FIC
FIC
FICAir
H2
FICEmulsion
pump
SupportingFlame
Filter
Hood
Vacuum Pump
MainFlame
Air
DispersingAir or O2
FIC
FIC
FICAir
H2
FICEmulsion
pump
Hollow particles byemulsion-fed FSP
25
CeO2
Fe2O3
ZnO
CeO2
Fe2O3
ZnO
400 nm1 µm
Y2O3
ZrO2
TiO2
400 nm400 nm1 µm1 µm
Y2O3
ZrO2
TiO2
Tani, Watanabe, Takatori and Pratsinis, J. Am. Ceram. Soc., 86, 898 (2003).Emulsion-fed FSP.
26
Conclusions• V2O5 / TiO2: SCR of NOx with NH3
– Purity improves conversion over wet-made ones• TiO2 / SiO2 : Olefin epoxidation:
– improved selectivity– role of transition metal dopants– structure of the active site– pilot-scale production (500 g/h)
• Pt / Al2O3: enantioselective hydrogenation– Open structure improves efficiency
27
Conclusions
• Nanoparticle Technology is a frontier for scientific advances and even, for business opportunities (millionaires are made today!).
• Flame Processing is advantageous for particle manufacture:Unique Structure, Crystallinity and Purity Close control of Particle Size and Morphology
• Functional nanoparticles with tailor-made characteristics are made for catalyst, dental, battery and other materials.
28S.E. PratsinisS.E. Pratsinis
H.K. KammlerH.K. KammlerK. WegnerK. Wegner
S. TsantilisS. Tsantilis
T. T. TaniTani
R. R. MüllerMüller
W.J. StarkW.J. Stark
O. WilhelmO. Wilhelm
J. KimJ. KimR. JossenR. Jossen
L. L. MädlerMädlerS. S. VeithVeith
ETHZ, Particle Technology LaboratoryETHZ, Particle Technology Laboratory
ETH Zurich Pratsinis 20041
2. Selected Fundamentals of Aerosol Formation
Prof. Sotiris E. PratsinisParticle Technology Laboratory
Department of Mechanical and Process Engineering,ETH Zürich, Switzerland
www.ptl.ethz.ch
Sponsored bySwiss National Science Foundation and
Swiss Commission for Technology and Innovation
ETH Zurich Pratsinis 20042
Particle Dynamics
Coagulation
Shrinkingby evaporationor dissolution
Fragmentation
Growth by condensation or chemical reaction
Diffusion Settling
Convection out
Convection in
ETH Zurich Pratsinis 20043
Theory: Population Balance Equation
tn
∂∂
growth
∂∂+
tdvdn
vun⋅∇+ nD∇⋅∇= nc⋅∇−
convection diffusion external force
( ) ( ) ( )∫ −−β+v
v~dv~vnv~nv~v,v~02
1 ( ) ( ) ( )∫∞β−
0v~dv~nvnv~,v
coagulation
fragmentation
( ) ( ) ( ) ( )∫∞
γ+−v
v~dv~nSv~,vvnvS
zyx u,u,u {0unnuun ∇⋅+∇=⋅∇u = gas velocity vector
D = particle diffusivitycontinuity
c = velocity of particles of size v (e.g. settling)β = coagulation rate
S = fragmentation rate
= fragment size distributionγ
ETH Zurich Pratsinis 20044
2. Fundamentals of Particle Formation
2.0 Books
Smoke, Dust and Haze, S.K. Friedlander, Oxford, 2nd edition, 2000Aerosol Processing of Materials, T.Kodas M. Hampden-Smith, Wiley, 1999Aerosol Technology, W. Hinds, Wiley, 2nd Edition, 2000.
2.1 CoagulationAtmospheric processes (air pollution, smog), Plumes, Tailpipe exhaust, Optical fibers for telecommunications, Carbon blacks for tires, Pigments, Enlargement by granulation or flocculation
The theory of coagulation is based on:a) collision theoryb) field forces
ETH Zurich Pratsinis 20045
2.1.1 Collision frequency function
Assume that collisions occur between two clouds of partices of volume vi and vj:
The number of collisions per unit time and unit volume is:
Where the collision frequency is the rate of collisions per particle per unit volume. This function depends on temperature, pressure and particle size.
vj vi vk
( )P v v n nij i j i j= β ,
ETH Zurich Pratsinis 20046
The birth of particles of size k=(i+j) is given by:
The factor ½ is included to correct for double counting.
The loss of particles of size k by collision with all other particles is:
12
Piji j k+ =∑
Piki=
∞∑
1
ETH Zurich Pratsinis 20047
Then the net rate of change in particle concentration is:
This is the basic equation for coagulation that is encountered in many physical phenomena:Granulation, Flocculation etc.
It used to be very intimidating 10 years ago, but not anymore. It can be easily solved.
GOAL: To determine collision frequency function
∑ ∑−= ikijk PP
dtnd
21
( ) ( )∑β−∑ β=∞
==+ 121
iikik
kjijiji nv,vnnnv,v
ETH Zurich Pratsinis 20048
2.1.2 CASE 1: Brownian Coagulation
In a stagnant gas coagulation takes place by diffusion of particles to the surface of each other.
Consider a sphere of radius ai at a fixed point.Particles of radius aj are in Brownian motion and diffuse to the surface of ai:
We would like to calculate the concentration profile njaway from the surface of particle i so we can calculate the flux of particles j to the surface of particle i. This will give the rate of collisions of particles i and j per unit area of particle i.
ai
ai+aj
aj
ETH Zurich Pratsinis 20049
Let us drop the subscript j for convenience and write a balance for particles of size aj.For spherical symmetry:
With boundary conditions:
∂∂
∂∂
∂∂
nt
Dr r
r nr
=
2
2
r a ai j= + : n = 0r → ∞ : n n= 0t = 0 : n n= 0 ∀ r
ETH Zurich Pratsinis 200410
The solution of this equation is:
( )
( )
n r t na a
re dzi j z
r a a
Dti j
, = −+
−
−
− +
∫00
21 1 2 2
π
[ ]
+−+−=
Dt2aar
erfcr
aan jiji10
ETH Zurich Pratsinis 200411
Now calculate the rate at which particles arrive at the surface
For (dP=1µm t>10s or dp=0.1µm t>0.01s):
By definition , so: (1)
( ) ( )F a a J a a D nri j a a i j
r a ai j
i j
= + = +
+
= +4 4
2 2π π
∂∂
( )= + ++
4 10π
πa a Dn
a aDti j
i j
t >> 0
( )F a a Dni j= +4 0πβ = Fn0
ETH Zurich Pratsinis 200412
Now consider that the sphere ai is in Brownian motion. Then we introduce the diffusion coefficient describing the relative motion of the two particles:
Einstein equation
0
(2)
( )D D
x xtij
i j= =−
2
2
D xt
x xt
xt
D Diji i j j
i j= − + = +2 2
22
2 2
ETH Zurich Pratsinis 200413
Then the collision frequency function becomesfrom (1) & (2):
where
This is the collision frequency function in the continuum limit .
( ) ( )( )β πv v D D a ai j i j i j, = + +4f
TkD B=
+
+
πµπ=β
2d
2d
d1
d1
3Tk4 j,Pi,P
j,Pi,P
B
( )31j
31i31
j31
i
B vvv1
v1
3Tk2 +
+
µ=
( )dP >> λ
ETH Zurich Pratsinis 200414
2.1.3 Coagulation of Monodisperse Particles
Assume that all particles have the same size during coagulation. This is a bold assumption but amazingly good and useful. Then, we can describe the rate of change of particle concentration as:
where the collision frequency function is:
Then and integration gives:
( )dNdt
v v N= − 12 1 1
2β ,
( ) ( )µ
=+
+
µ=β
3Tk8vv
v
1
v
13
Tk2v,v B311
31131
131
1
B11
dNdt
N= −β2
2 N NN=
+
00β t1
2
ETH Zurich Pratsinis 200415
This simple expression can be used to estimate the half-life of an aerosol, or the time needed for particles to grow to a certain size by coagulation, or even the significance of coagulation with respect to other processes.
For example, estimate the time needed to reduce the concentration of a monodisperse aerosol to 90%, 50% or 10% of its initial concentration 108 particles/cm3, and initial diameter 100nm, cm3/s.
For N
N00 9= . : t
NNN
s=−
≈2 1
15
0
0β.
For N
N00 5= . : t s≈ 14
For N
N00 1= . : t s≈ 125
ETH Zurich Pratsinis 200416
2.1.4 CASE 2: Coagulation in the free molecule regime
In this case the concept of continuum does not exist anymore so we cannot write the Navier-Stokes equations as we did for case 1.
Instead we rely on the kinetic theory of gases (e.g. N. Davidson, Statistical Mechanics, Ch. 10, McGraw, New York, 1962).
The mean scalar velocity of N gas molecules of mass m1 per cm3 having a Maxwellian distribution is:
The total rate at which molecules strike a surface dS is
c k TmB= 8
1π
( )e s Nc dS= 14
ETH Zurich Pratsinis 200417
For a sphere of radius a2 colliding with particles (molecules) of equivalent spherical radius a1
where a=a1+a2 is the collision radius. Now if the sphere also moves then the number of collisions increases as:
This is the collision frequency function for .dP << λ
( )βπ ρfm
B
P
k Tv v
v v=
+
+3
46 1 11 6
1 2
1 2
11 3
21 3 2
( )F N N k Tv v
v vfmB
P= = +
+β ππρ π
8 1 1 341 2
1 2 2 3
11 3
21 3 2
F Nc a N c c a= = +π π122
12
22 2
( )F e s Nc S N k Tm
a Nc a= = = =14
14
8 41
2 2π
π π
ETH Zurich Pratsinis 200418
2.1.7 Self-Preserving Theory
Observation of natural particle suspensions in gases (atmospheric aerosols) undergoing coagulation indicated that after a long time the particle size distribution attains a shapethat is invariant with time.
More specifically, when the size distribution is scaled by some factor (e.g. average particle size) then the distributions fall on top of each other and are called self-preserving. This was observed first experimentally(e.g. Husar & Whitby, Environ. Sci. Technol. 7:241, 1973):
ETH Zurich Pratsinis 200419
Size distribution of an aging free molecule aerosol generated by exposing filtered laboratory air in 90 m3
polyethylene bag to solar radiation.
Size distribution as on left side, plotted in the self-preserving form. The curve is based on the data.
ETH Zurich Pratsinis 200420
According to this, the particle volume v becomes non-dimensional by dividing by the average volume concentration where V is the aerosol volumetric concentration [mp
3/mG3]=[-]
and N the number concentration respectively:
And the particle size distribution is defined in a non-dimensional form as:
VvN
vv ⋅==η
( ) ( ) 2N
Vvn=ηψ
ETH Zurich Pratsinis 200421
2.2 Particle Formation by Nucleation-Condensation
A phase transition is encountered in many industrial(e.g. crystallization, carbon black production) and environmental (e.g. smog formation) processes
The fundamental equation that describes these processes is:
With boundary conditions:
∂∂nt
v ni+ ∇ ⋅ = 0
at d dP P= ∗ n v Ii i∗ ∗= nucleation
t = 0 ( )n n dP= 0 initial distribution
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The goal is to determine:
1. the critical diameter for particle formation which isdictated by thermodynamics
2. the growth rate that is determined by thermodynamics and transport
3. the nucleation rate which is determined by thermodynamicsand kinetic theory by physical (e.g.cooling) or chemical (e.g.reactions) driving forces
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2.2.1 Critical Particle Size
Key feature: The curved interface
The goal is to derive an expression relating the concentration (vapor pressure) of species A with a particle (droplet) of radiusdP at equilibrium (Seinfeld, 1986)
If the interface was flat which is, for example, the tabulatedequilibrium concentration or vapor pressure at a given temperature and pressure.
Consider the change in Gibbs free energy accompanying the formation of a single drop (embryo) of pure material A of diameter dP containing g molecules of A:
(1)∆G G Gembryo system pure vapor= −
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Now let’s say that the number of molecules in the starting condition of pure vapor is nT. After the embryo forms, the number of vapor molecules remaining is . Then the above equation is written as:
(2)
where GV and Gl are the free energies of a molecule in a liquid and vapor phases and σ is the surface energy
(3)
Noting that
Where vl is the volume occupied by a molecule in the liquid phase (equivalent sphere in liquid phase).
n n gT= −
vTPlv GndgGnGG −σπ++=∆ 2
( ) ( )∆G g G G d dv
G G dl v PP
ll v P= − + = − +π σ π π σ2
32
6
gv dl
P= π 3
6
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Before we go further let’s evaluate the difference in Gibbs freeenergy:
dG = VdP then dG = (vl - vv) dP
But vl << vv then dG = - vv dP
According to ideal gas law vv = kBT/P
Then
Where S is the saturation ratio.
G G k T dPP
k T PP
k T Sv l BP
P
BA
AB
A
A− = − = − = −∫
0 0ln ln
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Now equation 3 becomes:
Now plot ∆G as a function of dP
S <1 monotonic increase in ∆GS > 1 positive and negative contributions at small dP the surface tension dominates and the behavior of ∆G as a function of dP is close to that for S <1. For larger dP the first term becomes important.
∆G
dP
S < 1
dP∗
S > 1
droplet at equilibriumwith surrounding vapor
∆G dv
k T S dP
lB
volume free energy of an embryo
P
surfacefree energy
= − +π π σ3
26
ln1 244 344
123
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At
This is the minimum possible particle size.
This equation relates the equilibrium radius of a droplet of a pure substance to the physical properties of the substance and the saturation ratio of its environment. It is called also the Kelvin equation and the critical diameter is called the Kelvin diameter.
∂∂∆GdP
= 0 ⇒ d vk T SP
l
B
∗ = 4σln
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This equation relates the equilibrium radius of a droplet of a pure substance to the physical properties of the substance and the saturation ratio of its environment. It is called also the Kelvin equation and the critical diameter is called the Kelvin diameter.
The Kelvin equation states that the vapor pressure over a curved interface always exceeds that of the same substance over a flat surface:
See the anchoring of the surface molecules on a flat and a curved surface. Surface molecules are anchored on two molecules on the layer below flat surfaces while on curved interfaces some are anchored on just one!These can easily escape (evaporate) from the condensed (liquid or solid) phase.
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2.3 Particle Growth
The mechanism for particle growth refers to droplet or particle growth from gas (condensation), to crystal growth from solution etc.. In all cases mass should be transported to the particle surface.
In principle, two steps are required, a diffusional step followed by a surface reaction or rearrangement step.In condensation the former is dominant while in crystallization is the latter. In many processes both can be dominant.
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2.3.1 Mass transfer to a particle surface (continuum)
Consider a single droplet growing by condensation without convection at rather dilute conditions. The goal is to determine the flux of mass to its surface. For this the vapor concentration profile around the droplet is needed at steady state:
(1)
D = vapor diffusivityC = vapor concentration
(moles/cm3)
dP
droplet
vapor molecules
0rCr
rrD
tC 2
2 =
∂∂
∂∂=
∂∂
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With boundary conditions:at r = dP/2 C = Cd the equilibrium concentration at the droplet surface
at r = ∞ C = C∞ bulk vapor concentration
Solving the above equation for C as a function of r gives:
(2)
Then the rate of condensation F is:
(3)
r2d1
CCCC P
d
d −=−−
∞
( )( )
2P2
P
Pd
2Pdr
d2d2
d0CCDrCDF π
+−=
∂∂= ∞
=
( ) Pd dCCD2 π−= ∞
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And the rate of particle volume growth is:
where MW and ρP are the molecular weight and density of the condensing material
So the diameter growth rate is (molecules/cm2):
(4)dP = ∞
( ) ( )P
Pd
P
3P dMWCCD2MWF
dt6dd
dtdv
ρπ−=
ρ=π= ∞
( )PPd
MWCCD4dtdd
ρ−
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2.3.2 Mass transfer to a particle surface (free molecule)
The collision rate per unit area is:
(5)
where c and m1 are the molecular velocity and mass and NAV the Avogadro number
so z becomes (6)( ) 21
1
BdAVm
Tk84
CCNz
π
−= ∞
4cCNz AV=
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Then the rate of condensation F to particle surface is:
(7)
And the rate of particle volume growth is:
(8)
So the diameter growth rate is:
(9)
( )d2P
21
1
BAV CCd
m2TkN/areazF −π
π
=⋅= ∞
( )P
d2P
21
1
B
P
MWCCdm2TkMWF
dtdv
ρ−π
π
=ρ
= ∞
( )d
21
1
B
P
P CCm2TkMW2
dtdd −
πρ
= ∞
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2.3.3 Mass transfer to a particle surface (entire spectrum)
For particle growth from the free molecule to continuum regime, the expression for the continuum regime is extended by an interpolation factor:
(10)
where the Knudsen number is Kn= 2λ/dP
This is called the Fuchs effect.
+−= ∞ dP Kn1MWCCD4dd ( )
++ρ 2PP Kn33.1Kn71.11ddt
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The effect of of temperature depression is to reduce the partial pressure of vapor at the droplet surface and slow the rate of evaporation. Similarly a temperature enhancement slows the rate of condensation.
(adapted from Hinds (1982))
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(adapted from Hinds (1982))
ETH Zurich Pratsinis 200438(adapted from Hinds (1982))