Stochastic numerics for the gas-phase synthesis of nanoparticles Shraddha Shekar 1 , Alastair J. Smith 1 , Markus Sander 1 , Markus Kraft 1 and Wolfgang Wagner 2 1 Department of Chemical Engineering and Biotechnology University of Cambridge 2 Weierstrass Institute for Applied Analysis and Stochastics, Berlin March 2011 CoMo March 2011 1 / 45 GROUP
Outline 1 Introduction Motivation 2 Model Type space Particle processes Algorithm 3 Numerical study 4 Conclusion CoMo March 2011 2 / 45 GROUP
Motivation What are nanoparticles? Why are they important? Particles sized between 1-100 nm. Both inorganic and organic nanoparticles find wide applications in various fields. Why model nanoparticle systems? To optimise industrial operations and to obtain products of highly specific properties for sensitive applications. To understand the molecular level properties that are difficult to be observed experimentally. CoMo March 2011 3 / 45 GROUP
Motivation II Salient features of the current model: Fully-coupled multidimensional stochastic population balance model. Describing various properties of nanoparticles at an unprecedented level of detail. Tracking properties not only at macroscopic level but also at a molecular level. CoMo March 2011 4 / 45 GROUP
Applications of silica nanoparticles Silica nanoparticles are amorphous and have Si:O = 1:2. Their applications include: Catalysis Bio-medical applications Support material for functional nanoparticles Fillers/Binders Optics CoMo March 2011 5 / 45 GROUP
Physical system To describe the system at a macroscopic level it is essential to understand it at a molecular level. Silica nanoparticles Precursor (TEOS) Flame reactor Industrial Scale P ≥ 1 atm T ≈ 1100 - 1500 K Molecular Scale CoMo March 2011 6 / 45 GROUP
Type space I Each particle is represented as: P q = P q ( p 1 , . . . , p n ( P q ) , C ) . Particle P q consists of n ( P q ) primary particles p i with i ∈ { 1 , . . . , n ( P q ) } and q ∈ { 1 , . . . , N } , where N is the total number of particles in the system. OH O O HO Si Si Si O O O O Si O Si Si OH O O HO p i = p i ( η Si , η O ,η OH ) P q = P q (p 1 ,...,p n(Pq) , C ) Figure: Type Space. CoMo March 2011 7 / 45 GROUP
Type space II Each primary particle p i is represented as: p i = p i ( η Si , η O , η OH ) where η x ( η x ∈ Z with η x ≥ 0) is the number of chemical units of type x ∈ { Si , O , OH } . CoMo March 2011 8 / 45 GROUP
Type space III C is a lower diagonal matrix of dimension n ( P q ) × n ( P q ) storing the common surface between two primary particles: 0 0 0 · · · · · · ... 0 · · · 0 C 21 . . ... ... . . C ( P q ) = . · · · . . . ... . C i 1 · · · C ij . . . . . . . . · · · . · · · . The element C ij of matrix C has the following property: � 0 , if p i and p j are non-neighbouring , C ij = C ij > 0 , if p i and p j are neighbouring. CoMo March 2011 9 / 45 GROUP
Particle processes Particles are transformed by the following processes: Inception Surface reaction Coagulation Sintering Intra-particle reaction CoMo March 2011 10 / 45 GROUP
Particle processes Particles are transformed by the following processes: Inception Surface reaction Coagulation Sintering Intra-particle reaction CoMo March 2011 11 / 45 GROUP
Inception Two molecules in gas phase collide to form a particle consisting of one primary. OH OH OH OH OH OH + OH HO Si HO Si Si OH HO Si Si OH Si -2 H 2 O OH O OH OH HO OH HO HO HO OH HO [primary particle] [monomers] Figure: Inception of primary particles from gas-phase monomers. An inception event increases the number of particles in the system molecule + molecule → P N ( p 1 , C ) , C = 0 . Initial state of primary p 1 given by: p 1 = p 1 ( η Si = 2 , η O = 1 , η OH = 6 ) , CoMo March 2011 12 / 45 GROUP
Inception rate Inception rate for each particle ( P q ) calculated using the free molecular kernel: R inc ( P q ) = 1 2 K fm N A 2 C 2 g , N A is Avogadro’s constant, C g is the gas-phase concentration of the incepting species (Si(OH) 4 ), � π k B T K fm = 4 ( d 2 g ) , m g k B is the Boltzmann constant, T is the system temperature, m g and d g are the mass and diameter respectively of the gas-phase molecule Si(OH) 4 . CoMo March 2011 13 / 45 GROUP
Particle processes Particles are transformed by the following processes: Inception Surface reaction Coagulation Sintering Intra-particle reaction CoMo March 2011 14 / 45 GROUP
Surface reaction Dehydration reaction between gas-phase monomer and particle surface: OH HO Si O HO OH O O +Si(OH) 4 Si Si -H 2 O Si Si O O O O O O Figure: Surface reaction between a particle and a gas-phase molecule. Surface reaction transforms particle as: P q + molecule → P q ( p 1 , ., p i ′ , .., p n ( P q ) , C ′ ) , p ′ i → p i ( η Si + 1 , η O + 1 , η OH + 2 ) . CoMo March 2011 15 / 45 GROUP
Particle rounding due to surface reaction Surface reaction also alters the common surface ( C → C ′ ). Net common surface area of p i changes due to volume addition: 2 σ ∆ s ( p i ) = ( v ( p i ′ ) − v ( p i )) d p ( p i ) , where σ is the surface smoothing factor (0 ≤ σ ≤ 2). C ′ is given by: � 0 , if p i and p j are non-neighbouring , C ′ ij = C ij + ∆ s ( p i ) , if p i and p j are neighbouring. CoMo March 2011 16 / 45 GROUP
Surface reaction rate Surface reaction rate calculated using equation of Arrhenius form: � − E a � R surf ( P q ) = A surf exp η OH ( P q ) N A C g , RT A surf is pre-exponential factor (obtained from collision theory), E a is activation energy, η OH ( P q ) is the total number of –OH sites on particle P q . CoMo March 2011 17 / 45 GROUP
Particle processes Particles are transformed by the following processes: Inception Surface reaction Coagulation Sintering Intra-particle reaction CoMo March 2011 18 / 45 GROUP
Coagulation Two particles collide and stick to each other: P r P s P q + Figure: Coagulation between two particles. Coagulation of particles P q and P r forms new particle P s as: P q + P r → P s ( p 1 , ..., p n ( P q ) , p ( n ( P q )+ 1 ) , ..., p n ( P q )+ n ( P r ) , C ) . CoMo March 2011 19 / 45 GROUP
Coagulation II Primary p i from particle P q and primary p j from P r are assumed to be in point contact. The matrix C ( P s ) is calculated as: . . . C ( P q ) · · · · · · C ij . . . . C ( P s ) = . . . . . C ji . . . C ( P r ) . . . and has dimension n ( P s ) × n ( P s ) , where n ( P s ) = n ( P q ) + n ( P r ) . CoMo March 2011 20 / 45 GROUP
Coagulation rate Coagulation rate between P q and P r calculated using transition coagulation kernel: K sf ( P q , P r ) K fm ( P q , P r ) K tr ( P q , P r ) = K sf ( P q , P r ) + K fm ( P q , P r ) , where the slip-flow kernel is: K sf ( P q , P r ) = 2 k B T � 1 + 1 . 257 Kn ( P q ) + 1 + 1 . 257 Kn ( P r ) � 3 µ d c ( P q ) d c ( P r ) × ( d c ( P q ) + d c ( P r )) , and the free molecular collision kernel is: � 1 � � π k B T 1 1 2 ( d c ( P q ) + d c ( P r )) 2 K fm ( P q , P r ) = 2 . 2 m ( P q ) + 2 m ( P r ) CoMo March 2011 21 / 45 GROUP
Particle processes Particles are transformed by the following processes: Inception Surface reaction Coagulation Sintering Intra-particle reaction CoMo March 2011 22 / 45 GROUP
Sintering Sintering described using viscous-flow model. p k p j p j p i p i No Sintering Partial Sintering Complete Sintering Figure: Evolution of sintering process with time. Sintering between p i and p j of a single particle P q calculated on a primary particle-level. CoMo March 2011 23 / 45 GROUP
Sintering level Sintering level defined to represent degree of sintering between p i and p j : S sph ( p i , p j ) 1 − 2 3 C ij s ( p i , p j ) = . 1 1 − 2 3 S sph ( p i , p j ) is the surface area of a sphere with the same volume as the two primaries. P q conditionally changes depending on the sintering level s ( p i , p j ) . Two types are defined depending upon a threshold (95%): Partial sintering s ( p i , p j ) < 0 . 95 Complete sintering s ( p i , p j ) ≥ 0 . 95 CoMo March 2011 24 / 45 GROUP
Partial sintering Surface areas of primaries are reduced by a finite amount. –OH sites at contact surface react to form Si–O–Si bonds: p j p j Reactions at particle neck OH OH OH OH OH OH OH p i OH OH p i OH OH OH OH OH OH OH OH OH OH OH O OH HO Si OH HO Si HO Si Si OH O Si O Si Si O Si O O OH O OH OH O O OH Si Si Si Si -2H 2 O O OH HO O O O O OH OH OH Si OH Si Si Si OH OH OH OH OH OH OH OH OH OH OH OH OH OH Figure: Dehydration reaction due to sintering. CoMo March 2011 25 / 45 GROUP
Partial sintering II Surface density of –OH sites assumed constant throughout sintering. The change in the internal variables of primaries p i and p j given by: ∆ η OH ( p i ) = ∆ η OH ( p j ) = ρ s ( P q )∆ C ij / 2 , ∆ η O ( p i ) = ∆ η O ( p j ) = − 0 . 5 × ∆ η OH ( p i ) , ∆ η Si ( p i ) = ∆ η Si ( p j ) = 0 . CoMo March 2011 26 / 45 GROUP
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