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1. Nanotechnology: Overview of Aerosol Manufacture of Nanoparticles Prof. Sotiris E. Pratsinis Particle Technology Laboratory Department of Mechanical and Process Engineering, ETH Zürich, Switzerland www.ptl.ethz.ch Sponsored by Swiss National Science Foundation and Swiss Commission for Technology and Innovation 1 Nanoparticles 1 - 100 nm (at least into two dimensions) Remember, the thickness (diameter) of a human hair is 50,000 - 100,000 nm! 2 3 The Melting Point Decreases with Decreasing Nanoparticle Size Au Melting Point, K Melting Point, K Particle diameter, nm Bi Peppiatt, Proc. Roy. Soc. A 345, 1642 (1975) Particle diameter, Å Buffat and Borel, Phys. Rev. A 13, 2287 (1976) 4 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! 5 Comparison of wet- & dry-technology Dry-technology (aerosol): Wet-technology: •Mix precursor •Dry flame conversion •Filtration •Milling •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 6 7 AEROSOL MANUFACTURING OF NANOPARTICLES Wegner, Pratsinis Chem. Eng. Sci. 58, 4581-9 (2003). Product Particles Process Coagulation Surface Coalescence Growth Volume t/y Value $/y Carbon black 8M 8B Flame, CxHy X X Titania 2M 4B Flame, TiCl4 X ? Fumed Silica 0.2 M 2B Flame, SiCl4 X - Zinc Oxide 0.6 M 0.7B Hot –Wall, Zn X X Filamentary Ni 0.04M ~0.1B Hot-Wall, Ni(CO)4 X X ~0.3B Hot-Wall, Spray… X X Fe, Pt, Zn2SiO4/Mn ~0.02M 8 A rough analogy to flame aerosol reactors … just well attached to the ground ! 9 10 Attic red-figure hydria, 430-420 BC, Abdera Archeological Museum, Greece Attic black-figure amphora, 540-530 BC, Museum of Cycladic Art, Athens, Greece 11 12 13 14 15 16 Prof. Gael W. Ulrich Dept. of Chemical Engineering University of New Hampshire 17 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 18 19 20 Applications Paints Niche Fields: Catalysts, Sensors, Photocatalysts, Cosmetics etc. Plastics coatings TiO2 (titania) Paper coatings 21 22 23 24 25 Limitations of science in the ‘70s Understanding of particle formation has little impact on 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 relationships 4. Too complex fluid mechanics (reactive systems). Industrial reactors were still treated as "black boxes" Design and operation were dominated by empiricism. 26 27 28 29 30 Thermophoretic Sampling (Dobbins and Megaridis Megaridis,, 1987)) TEM Image Analysis Average primary particle diameter, nm . 60 16 ± 4 nm 50 40 30 20 TiO2 data by thermophoretic sampling 10 0 0 1 2 3 Distance from the burner, cm 4 31 Sintering rate of particle area a da 1 = − (a − a s ) dt τ Koch, Friedlander, J. Colloid Interface Sci. 140, 419 (1990) 32 Number Concentration dN 1 2 = − βN dt 2 Agglomerate area da 1 dN 1 =− − a (a − a s ) τ dt N dt Collision diameter dc = dp (v / v p )1/Df 33 Kruis, Kusters, Pratsinis, Scarlett, Aerosol Sci. Technol. 19, 514 (1993) Concentration, f(c) Design of Equipment a = 1.0 c = 0.01 c = 0.1 Ra te , b = 0.3 Predictions within 3% of product SSA (Gutsch, 1997) Co f(b olin ) g b = 3.0 Temperature, f(a) a = 1.4 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. 34 N 234 nano-structure black Niedermeier, Messer, Fröhlich (TR814.1E) 35 TODAY Aerosol 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 particles High value functional nanoparticles with sophisticated composition and structure. Manufacture these nanoparticles without going through the Edisonian cycle of the past. Health effects of nanoparticles. 36 Novel Processes and Uses of Flame-made Nanoparticles Prof. Sotiris E. Pratsinis Particle Technology Laboratory Department of Mechanical and Process Engineering, ETH Zürich, Switzerland www.ptl.ethz.ch Sponsored by the Swiss National Science Foundation and Swiss Commission for Technology and Innovation 1 Flame-made particles Pros Challenges High purity Easy collection No liquid waste Proven scale-up No moving parts Agglomerates Size control Multicomponent ceramic/ceramic metal/ceramic 2 Experimental set-up for TiO2 production Hood Pump Filter Assembly NaOH Solution Flame Reactor FIC Methane Air Argon / TiCl 4 FIC Methane Air FIC Argon TiCl 4 Pratsinis, Zhu, Vemury, Powder 3 Technol. 86, 87-93 (1996) Mixing of reactant gases ==> product size-shape Pratsinis, Zhu, Vemury, Powder Technol. 86, 87-93 (1996); Johannessen, Pratsinis, Livberg, ibid., 118, 242-250 (2001). CH4 CH4 Air Air Air CH4 Air TiCl4 TiCl4 TiCl4 TiCl4 CH 4 4 S. Vemury, S.E. Pratsinis, L. Kibbey, J. Mater. Res. 12, 1031-1042 (1997). Electrically Assisted Synthesis of Nanoparticles U.S. Patent 5,861,132 (1999) Precision Size Control by Charging Kammler, PhD thesis, ETH #14622 (2002) 5 w/o electric field HAB HAB Evolution of TiO2 particle growth with and w/o external electric fields Filter 20 cm Filter 20 cm 10 cm 200 nm 2 cm HAB 10 cm with electric field 200 nm 5 cm 0 kV/cm 5 cm 2 kV/cm 0.5 cm 0.5 cm 6 Kammler, Pratsinis, Morrison, Jr., Hemmerling, Combust. Flame 124, 369 (2002) Dental n-Composites: flame-made silicas in a dimethylacrylate matrix (50:50) with ETH non-aggl. SiO2 SSA = 35 m2/g with OX50 (Degussa) SSA = 50 m2/g Müller, Vital, Kammler, Pratsinis, Beaucage, Burtscher, 7 Powder Technol. 140, 40-48 (2004). Precision Synthesis by Nozzle Quenching Wegner, Stark, Pratsinis, Mater. Lett. 55, 318 (2002) 8 Reduction of Agglomeration 6 L/min O2 flow rate TS in front of nozzle (BND = 1.5 cm) No nozzle Product powder Nozzle BND = 1.5 cm 9 BET-equivalent Particle Diameter, nm Control of TiO2 size, color & crystallinity 30 O2 / Ti decreases 50 40 30 2 L/min O2 3 4 5 6 25 20 20 15 10 10 5 0 0 1 2 3 4 Burner - Nozzle Distance, cm 5 0 Filter 10 No Nozzle V2O5/TiO2: Catalytic Removal of NOx In Exhaust Gases by SCR with NH3 NOx removed / % 100 flamemade 50 0 100 150 wet-phasemade 200 250 Process temperature / °C Stark, Wegner, Pratsinis, Baiker, J. Catal. 197, 182 (2001) 11 Pilot unit for flame synthesis of C/SiO2 , 0.5 m and now catalysts: Baghouse filter (2.5 m tall) V2O5 /TiO2 and TiO2 /SiO2 Kammler, Mueller, Senn, Pratsinis, AIChE J. 47, 1533 (2001) 12 Comparison to conventional DeNOx catalyst @ U. Essen (Prof. Cramer) NO removed / % 100 Fixed bed pilot-scale test reactor, 2.4 cm/sec 80 60 18 wt% VOx/TiO2, flame-made, 100 g/h 40 20 20 wt% VOx/TiO2, impregnated 0 160 200 240 Reactor Temperature / °C 280 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 •W. J. Stark, A. Baiker, S. E. Pratsinis, Part. Part. Sys. Charact. 19, 306-311 (2002) 13 225 150 g/h 2 Specific surface area / m g -1 Epoxidation Catalysts: TiO2 /SiO2 200 175 150 125 100 0 2 4 Titania content / wt% Hydrogen/air flame, burner diameter 19 mm, 0.73 m3 H2/h; 5.2 m3 air/h •W. J. Stark, S. E. Pratsinis, A. Baiker, J. Catal., 203, 516 (2001) and Ind. Eng. Chem. Res., 41, 4921 (2002) 14 TiO2/SiO2 epoxidation catalysts Industrially (Shell, Enichem, Arco), several Mt/y : C3 ⇒ propene ⇒ propene oxide ⇒ polymers, surfactants Selectivity / % 100 peroxide usage olefin usage 90 OH OH TBHP (5) O Ti/silica 80 1 2 70 60 50 Shell Aerogel Enichem 6 g/h 150 g/h •W. J. Stark, H. K. Kammler, R. Strobel, D. Günther, A. Baiker, S. E. Pratsinis, 500 g/h Ind. Eng. Chem. Res., 41, 4921 (2002) Flame-made 15 X-ray Absorption Near Edge Spectroscopy In-situ XANES: -Geometry of the active site -Water content -Degree of hydration H 4, 5, 6 - coordinated Ti 1 % Ti, wet-phase Si Si Si tetrahedral Ti 1.3 % Ti, flame Si OH Si Si O O Ti O O Si Si H H HO O O Ti O Si O O Si Si Si O Ti O Si O O Si Si Si O H O Ti O Si O O Si Si O O Ti O O Si •J. D. Grunwaldt, C. Beck, W. J. Stark, A. Hagen, A. Baiker, Phys. Chem. Chem. Phys., 4, 3514 (2002). 16 Selectivity 100 Olefin selectivity / % Peroxide selectivity / % 100 80 60 40 Co Cr Mn Fe 20 80 60 40 Co Cr Mn Fe 20 10 100 1000 Content / ppm 10 100 1000 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) 17 Flame Spray Pyrolysis 0.1 µm Al2O3, ZnO, CeO2, ZrO2 ZnO/SiO2 , BaTiO3 Au, Pt on TiO2, SiO2, Al2O3 2 cm Bi2O3 Solid Hollow Mädler, Pratsinis, J.Am.Ceram.Soc. 85, 1713 (2002) Varistors Sensors Catalysts 18 Flame spray pyrolysis Strobel, Stark, Mädler, Pratsinis, Baiker, J. Catal. 213, 296-304 (2003) Spray flame producing Pt/Al2O3 19 Enantioselective hydrogenation of ethyl pyruvate by FSP-made Pt/Al2O3 Conversion / % 100 FSP-made Synthesis of chiral pharmaceuticals. 75 50 Engelhard (E4759) 25 0 0 50 100 150 200 Time / min Enantiomeric excess (ee) / % 100 75 50 E4759 FSP-made 25 0 0 50 100 Time / min 150 200 20 Strobel, Stark, Mädler, Pratsinis, Baiker, J. Catal. 213, 296-304 (2003) Open structure enhances activity 0.8 3 Pore volume / cm g -1 Nonporous, dense particles ÎBetter accessibility ÎHigh surface without trapped Pt in micropores. E4759 0.6 flame made 0.4 0.2 0 ÎMaximum use of expensive platinum reduces costs 1 10 100 Pore diameter / nm Strobel, Stark, Mädler, Pratsinis, Baiker, J. Catal. 213, 296-304 (2003) 21 Rapid Synthesis of Stable ZnO Quantum Dots (1 - 5 nm) Absorbance (scaled), a.u. Blue Shift of Absorption Spectra Mädler, Stark, Pratsinis, J. Appl. Phys. 92, 6537-40 (2002) dXRD= 3 nm 300 350 4 12 nm 400 5 nm 380 450 λl Bulk l 1/2 , nm Wavelength, λ Wavelength , nm Tani, Mädler, Pratsinis, J. Mater. Sci. 37, 4627-4632 (2002) 50 nm 370 Quantum Size Effect 360 350 Flame made (FSP) this work FSP powder thermally treated (2h @ 600°C) Wet-made, Meulenkamp (1998) 340 0 5 10 ZnO diameter (dXRD), nm 15 22 Average particle size, nm Angular, rough, edgy-like n-CeO2 16 as-prepared 14 20 nm 12 Catalysts Fuel Cells Polishing 10 8 6 4 dXRD 2 0 dBET 100 nm 0 2 4 6 8 Oxygen flow rate, l/min Mädler, Stark, Pratsinis, J. Mater Res. 17, 1356 (2002). annealed for 2h @ 900°C 23 Hollow particles by emulsion-fed FSP Al2O3-N Tani, Watanabe, Takatori and Pratsinis, J. Am. Ceram. Soc., 86, 898 (2003). pump Al2O3-Cl FIC Dispersing Air or O2 FIC Emulsion Air FIC Al2O3-H-N H2 FIC Air Main Flame Al2O3-N-O Filter Supporting Flame Hood Vacuum Pump 1 µm 400 nm 24 TiO2 ZnO ZrO2 Fe2O3 Y2O3 CeO2 1 µm 400 nm Emulsion-fed FSP.Tani, Watanabe, Takatori and Pratsinis, J. Am. Ceram. Soc., 86, 898 (2003). 25 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 26 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. 27 ETHZ, Particle Technology Laboratory R. Müller O. Wilhelm S. Tsantilis R. Jossen W.J. Stark J. Kim L. Mädler K. Wegner S.E. Pratsinis 28 H.K. Kammler T. Tani S. Veith 2. Selected Fundamentals of Aerosol Formation Prof. Sotiris E. Pratsinis Particle Technology Laboratory Department of Mechanical and Process Engineering, ETH Zürich, Switzerland www.ptl.ethz.ch Sponsored by Swiss National Science Foundation and Swiss Commission for Technology and Innovation ETH Zurich Pratsinis 2004 1 Particle Dynamics Coagulation Fragmentation Convection in Shrinking by evaporation or dissolution Growth by condensation or chemical reaction Convection out Diffusion ETH Zurich Pratsinis 2004 Settling 2 Theory: Population Balance Equation ∂n ∂t + ∇ ⋅n u = ∇ ⋅ D∇n convection diffusion + ∂ dv n ∂v dt growth − ∇ ⋅cn external force ∞ 1v ~ ~ ~ ~ ~ v )n(v )n(~ v )d~ v + ∫ β(v, v − v )n(v )n(v − v )dv − ∫ β(v, ~ 20 0 coagulation ∞ − S(v )n(v ) + ∫ γ (v, ~ v )Sn(~ v )d~ v v fragmentation u = gas velocity vector D c β = particle diffusivity S = fragmentation rate γ u x , u y , uz ∇ ⋅ n u = u ∇n + n ⋅ ∇ u { = velocity of particles of size v (e.g. settling) 0 continuity = coagulation rate = fragment size distribution ETH Zurich Pratsinis 2004 3 2. Fundamentals of Particle Formation 2.0 Books Smoke, Dust and Haze, S.K. Friedlander, Oxford, 2nd edition, 2000 Aerosol Processing of Materials, T.Kodas M. Hampden-Smith, Wiley, 1999 Aerosol Technology, W. Hinds, Wiley, 2nd Edition, 2000. 2.1 Coagulation Atmospheric 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 theory b) field forces ETH Zurich Pratsinis 2004 4 2.1.1 Collision frequency function Assume that collisions occur between two clouds of partices of volume vi and vj: vj vi vk The number of collisions per unit time and unit volume is: ( ) Pij = β vi , v j ni n j Where the collision frequency is the rate of collisions per particle per unit volume. This function depends on temperature, 5 ETH Zurich Pratsinis 2004 pressure and particle size. The birth of particles of size k=(i+j) is given by: 1 Pij ∑ 2 i+ j= k The factor ½ is included to correct for double counting. The loss of particles of size k by collision with all other particles is: ∞ ∑ Pik i=1 ETH Zurich Pratsinis 2004 6 Then the net rate of change in particle concentration is: dnk 1 = ∑ Pij − ∑ Pik dt 2 ∞ 1 = ∑ β(v i , v j )ni n j − nk ∑ β(v i , v k )ni 2 i + j =k i=1 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 ETH Zurich Pratsinis 2004 7 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 nj away 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 ETH Zurich i. Pratsinis 2004 of particle aj ai+aj ai 8 Let us drop the subscript j for convenience and write a balance for particles of size aj. For spherical symmetry: ∂ n D ∂ 2 ∂ n = 2 r ∂t r ∂r ∂r With boundary conditions: r = ai + aj : r→∞ : t=0 : ETH Zurich Pratsinis 2004 n=0 n = n0 n = n0 ∀r 9 The solution of this equation is: ( ai + a j 2 n( r, t) = n0 1 − 1− r π r − ai + a j 2 Dt ∫ 0 [ ai + a j r − ai + a j = n 0 1 − erfc r 2 Dt ETH Zurich Pratsinis 2004 ) − z2 e dz ] 10 Now calculate the rate at which particles arrive at the surface ( ) 2 ( F = 4 π ai + a j J a + a = 4 π ai + a j i j ) 2 ∂ n D ∂ r r =a +a i j ai + aj = 4 π ai + aj )Dn0 1 + πD t ( For t >> 0 (dP=1µm t>10s or dp=0.1µm t>0.01s): F By definition β = , so: F = 4 π ai + aj Dn0 n0 ( ETH Zurich Pratsinis 2004 ) (1) 11 Now consider that the sphere ai is in Brownian motion. Then we introduce the diffusion coefficient describing the relative motion of the two particles: D = Dij xi − x j ) ( = 2 Einstein equation 2t 0 Dij = ETH Zurich Pratsinis 2004 xi2 2t − 2xixj 2t + xj2 2t = Di + D j (2) 12 Then the collision frequency function becomes from (1) & (2): ( ( ( β vi , v j ) = 4 π Di + D j ) ai + aj ) where k BT D= f k B T 1 1 d P,i d P, j + β = 4π + 3πµ d P,i d P, j 2 2 2k B T 1 1 13 13 = + 1 3 vi + v j 1 3 3µ v vj i ( ) This is the collision frequency function in the continuum limit ( dP >> λ ). ETH Zurich Pratsinis 2004 13 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: 1 dN = − β( v1, v1)N2 2 dt where the collision frequency function is: ( ) 2k B T 1 1 1 3 1 3 8k B T β(v1 , v1 ) = + v1 + v1 = 1 3 1 3 3µ v 3µ v 1 1 Then dN β 2 = − N and integration gives: N = dt 2 ETH Zurich Pratsinis 2004 N0 βN0 1+ t 2 14 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. N = 0.9 : For N0 N = 0.5 : For N0 N = 0.1 : For N0 ETH Zurich Pratsinis 2004 N0 − 1 2 N t= . s ≈ 15 βN0 t ≈ 14 s t ≈ 125 s 15 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: c= 8 kB T π m1 The total rate at which molecules strike a surface dS is ETH Zurich Pratsinis 2004 1 e( s) = Nc dS 4 16 For a sphere of radius a2 colliding with particles (molecules) of equivalent spherical radius a1 F = e( s) = 1 1 8k T Nc S = N 4 π a2 = π Nc a2 π m1 4 4 where a=a1+a2 is the collision radius. Now if the sphere also moves then the number of collisions increases as: F = π Nc12 a2 = π N c12 + c22 a2 8 kB T 1 1 F = β fm N = π N + π ρP v1 v 2 β fm 3 = 4π 16 12 3 4π 6 kB T 1 1 + ρP v1 v 2 12 ( 23 ( v11 3 v11 3 + + ) 13 2 v2 ) 13 2 v2 This is the collision frequency function for dP << λ . ETH Zurich Pratsinis 2004 17 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 shape that 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 2004 18 Size distribution of an aging free molecule aerosol generated by exposing filtered laboratory air in 90 m3 polyethylene bag to solar radiation. ETH Zurich Pratsinis 2004 Size distribution as on left side, plotted in the self-preserving form. The curve is based on the data. 19 According to this, the particle volume v becomes nondimensional by dividing by the average volume concentration where V is the aerosol volumetric concentration [mp3/mG3]=[-] and N the number concentration respectively: v N⋅v η= = v V And the particle size distribution is defined in a nondimensional form as: V ψ(η) = n (v ) N2 ETH Zurich Pratsinis 2004 20 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: ∂n + ∇ ⋅ vi n = 0 ∂t With boundary conditions: at dP = dP∗ t=0 ETH Zurich Pratsinis 2004 ni vi∗ = I∗ n = n0 ( dP ) nucleation initial distribution 21 The goal is to determine: 1. the critical diameter for particle formation which is dictated by thermodynamics 2. the growth rate that is determined by thermodynamics and transport 3. the nucleation rate which is determined by thermodynamics and kinetic theory by physical (e.g.cooling) or chemical (e.g. reactions) driving forces ETH Zurich Pratsinis 2004 22 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 radius dP at equilibrium (Seinfeld, 1986) If the interface was flat which is, for example, the tabulated equilibrium 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: ∆G = Gembryo system − Gpure vapor (1) ETH Zurich Pratsinis 2004 23 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 n = nT − g . Then the above equation is written as: ∆G = nG v + gGl + πdP2 σ − n T G v (2) where GV and Gl are the free energies of a molecule in a liquid and vapor phases and σ is the surface energy ∆G = g(Gl − Gv ) + π dP2 σ π dP3 2 = G − G + π d ( l v) P σ (3) 6 vl π dP3 Noting that g vl = 6 Where vl is the volume occupied by a molecule in the liquid phase (equivalent sphere in liquid phase). 24 ETH Zurich Pratsinis 2004 Before we go further let’s evaluate the difference in Gibbs free energy: dG = VdP then dG = (vl - vv) dP But vl << vv then dG = - vv dP According to ideal gas law vv = kBT/P Then Gv − Gl = −k B T PA ∫ PA 0 dP PA = −k B T ln = −k B T ln S P PA 0 Where S is the saturation ratio. ETH Zurich Pratsinis 2004 25 Now equation 3 becomes: ∆G = − π dP3 k B T ln S 6vl 14 4244 3 + volume free energy of an embryo Now plot ∆G as a function of dP π dP2 σ 123 surface free energy ∆G droplet at equilibrium with surrounding vapor S <1 S>1 dP S <1 monotonic increase in ∆G S > 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 26 ETH Zurich Pratsinis 2004 important. dP∗ At ∂ ∆G =0 ∂ dP ⇒ ∗ dP 4 σ vl = k B T ln S 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. ETH Zurich Pratsinis 2004 27 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 28 ETH or Zurich Pratsinis phase. 2004 (liquid solid) 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. ETH Zurich Pratsinis 2004 29 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: droplet ∂C D ∂ 2 ∂C = 2 r = 0 ∂ t r ∂r ∂r dP ETH Zurich Pratsinis 2004 vapor (1) D = vapor diffusivity C = vapor concentration (moles/cm3) molecules 30 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: d C−C = 1− 2r C −C d ∞ P (2) d Then the rate of condensation F is: ∂C dP 2 = D(C∞ − Cd )0 + F = D πd P 2 2(d P 2 ) ∂r r = d P 2 ETH Zurich Pratsinis 2004 = 2D(C∞ − Cd )πd P (3) 31 And the rate of particle volume growth is: 3 ( dv d π d P 6 ) FMW 2D(C ∞ − C d )MWπd P = = = dt ρP dt ρP where MW and ρP are the molecular weight and density of the condensing material So the diameter growth rate is (molecules/cm2): dd P 4D(C ∞ − C d )MW = dt ρP d P ETH Zurich Pratsinis 2004 (4) 32 2.3.2 Mass transfer to a particle surface (free molecule) The collision rate per unit area is: N AV Cc z= 4 (5) where c and m1 are the molecular velocity and mass and NAV the Avogadro number 12 so z becomes ETH Zurich Pratsinis 2004 N AV (C∞ − Cd ) 8k B T z= 4 πm1 (6) 33 Then the rate of condensation F to particle surface is: 12 F = z ⋅ area / N AV k BT = 2πm1 πd 2P (C∞ − Cd ) (7) And the rate of particle volume growth is: 12 dv FMW k B T = = dt ρP 2πm1 πd 2P MW (8) (C ∞ − C d ) ρP So the diameter growth rate is: 12 dd P 2MW k BT = ρP 2πm1 dt ETH Zurich Pratsinis 2004 (C∞ − Cd ) (9) 34 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: dd P 4D(C ∞ − C d )MW 1 + Kn (10) = 1 + 1.71Kn + 1.33Kn 2 dt ρP d P where the Knudsen number is Kn= 2λ/dP This is called the Fuchs effect. ETH Zurich Pratsinis 2004 35 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)) ETH Zurich Pratsinis 2004 36 (adapted from Hinds (1982)) ETH Zurich Pratsinis 2004 37 ETH Zurich Pratsinis 2004 (adapted from Hinds (1982)) 38