For one such collision, a new particle, "k", is formed of volume v k = v i + v j. The rate of formation of "k" particles is, N ij

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1 Coagulation Theory and The Smoluchowski Equation: Coagulation is defined as growth of particles by collisions among particles. It is usually associated with dense, 3-d growth, in contrast to aggregation which leads to low dimension, branched structures. The process of coagulation usually is associated with sintering and coalescence of particles by Gibbs-Thompson maturing and through strong van der Waals bonding of the particles. Then the result of coagulation is usually considered a solid 3-d particle. Agglomeration is similar to coagulation but, for the most part, does not include coalescence of the particles so that an agglomerate can be "broken down" in to the units of which it is made while a coagulated particle can not be broken down exactly into the subparticles from which it grew. In aerosol and other gas phase growth the time rate of change of particle size distribution is of primary interest. In a general sense we can consider two types of particles, with volumes, v i and v j. The rate of collision between i and j particles is N ij. For spherical particles, each diameter of particles has a unique volume. The collision of two particles leads immediately to the growth of a new particle with the summed volume of the two contributing particles. The concentration of particles "i" is n i and that of "j" is n j. We can define the collision frequency function,, by, N ij = (v i, v j, T, P, etc.) n i n j For one such collision, a new particle, "k", is formed of volume v k = v i + v j. The rate of formation of "k" particles is, N ij i+ j = k The rate of loss of particles "k" due to collision with other particles is, Then, i= N ik dn k = N ij N ik i + j= k i= = β( v i,v j ) n i n j n k β v i,v k i + j= k i= n i This is the Smoluchowski equation (97). The equation converts the problem of the determining the evolution of a spectrum of particle sizes to the problem of determining the collision frequency function,. The functional form for depends on the transport mechanisms at play in the system. The equation is based on instantaneous bonding of particles after collision.

2 Determination of for Brownian Motion of Particles: For nanoparticles (particles smaller than µm) Brownian motion governs the collision frequency. For a spherical particle of radius a i with a coordinate system fixed on its center, and particles, a j, surrounding it and subjected to Brownian motion, the a j particles diffuse to the surface of a i. a i is a perfect sink since the particles bond by van der Waals interactions. We consider that n j = at a i +a j. Fick s second law for the spherical particle, a i, is given by, t = D r δn δr r r We have at r = a i +a j, n j = for all t, and for r > a i +a j, at time t =, n j = n. Fick s second law can be rewritten in terms of the dimensionless parameters, x = r ( a a i j ) w = n n r a i a j n and a i + a j = n n n ( x +) so t = ( x) w t x r = a i a j w = ( x ) r n r + n n n x r = n n ( a i + a j )n + ( x) n r w r = ( a i +a j )n r + ( x) n n r x n r r and w x = n n ( x + ) n n x, and, w x = n x ( x+) n n x and w t = x+ ( x+) r ( δnδr) ( = D = D x+ ) n n t n r r n r r + r = D ( x+) a i a ( j ) n xa ( i a j )+( a i a j ) x + ( a a i j) n = D ( x+ ) n x n ( x+) x + x = D x+ n D n x n x = D w ( a i +a j ) x The boundary conditions become x = ; w = for all t and x > ;t =, w =. This is a -d diffusion equation so the solution is,

3 w = erf x a + a i j Dt or n n n = a i +a j r erf r a i + a ( ( j )) ( Dt) The steady state solution is obtained at t =>, w=>, so n n n a i + a j r Fick s first law can be used to calculate the flux of a j particles to a i (at r = a i + a j ), J = D j r r= a i +a j and the rate of collision, F(t), is given by, F(t) = 4πD r j r r= a i + a j From the expression for n above, F(t) = 4πD( a i + a j )n + a i + a j ( πdt) At steady state, t >> (a i + a j ) /D, F(t) = 4πD( a i + a j )n which is minimum value of the time function since j particles have, for the most part diffused away or coagulated. We must also consider Brownian diffusion of the i particle which was fixed in the previous assessment. The Einstein equation gives the relative displacement of i and j particles, D ij = ( x i x j ) t ( = x i) t ( + x x i j) t + ( x j ) t ( = x i) t + ( x j ) t = D i + D j since the two particles are not correlated in motion, i.e. the motion is random. The collision frequency function, (v i, v j ), ( v i,v j ) = 4π ( D i + D j )( a i + a j ) 3

4 for steady state. For.µm particles the time constant deciding steady state is about x -3 sec allowing for this approximation. The Stokes-Einstein equation gives D =kt/(3πµd p ) for the continuum regime, so ( v i,v j ) = kt 3µ /3 v + i v j /3 v /3 /3 ( i + v i ) Continuum Range In the free molecular regime the functions described in the previous section are used for D and, ( v i,v j ) = 3 6kT 4π ρ p + v i v j / / v /3 /3 ( i + v i ) Free Molecular Range which is based on the kinetic theory of gasses. An interpolation function can be found in Fuchs, NA Mechanics of Aerosols, Pergamon Press, NY (964). This function shows a maximum collision frequency for Kn 5. Collision frequency is highest for particles with widely separated sizes, favoring monomer cluster growth for instance. Development of a Particle Size Distribution by Coagualtion: For a monodisperse system undergoing coagulation, initially v i = v j, = K = 8kT 3µ Continuum Range, Monodisperse This can be inserted in to the Smoluchowski equation to yield, k = K i + j= k n i n j Kn k n i i= The total number of particles per unit volume, N =Σn i. The Smoluchowski equation can be written for the rate of change for the total particle number per unit volume, N = K k= i + j= k n i n j KN = K N so, = N ( t = ) + ( t /τ ) N N = t = + KN ( t = )t / 4

5 The discrete size distribution can be obtained by considering the solution to the kinetic growth equations for a series of sizes, for k =, and for k =, dn = Kn N or n = N t = + t/τ dn ( t /τ ) 3 = Kn Kn N or n = N t = + t /τ In general, ( t /τ) k k+ Continuum range, Constant Collision Frequency from n k = N t = + t /τ Monodisperse System where τ = /(K N ()) = 3µ/(4kT N ()). For k > the distribution has a maximum in time as shown in the figure from Smoluchowski (in Friedlander s book) below. For higher k the peak occurs at longer times. Since particles are growing, the total number decays with time. This model is appropriate for small t/τ, where the particle size distribution is close to monodisperse. This might be applicable to a batch system, or a plug flow reactor. At long times the curves shown for particle size as a function of time begin to show a similar shape, although shifted in time and k. This is the source of a self-preserving shape to the size distribution. That is, the size distribution, when scaled with dimensionless numbers, retains the same shape, so is called self-preserving. For the result of the Smoluchowski equation for constant collision frequency the self-preserving distribution has the form, N k ~e -η where η = k/<k> and,k.=n ()/N (t). We can rearrange the size distribution as, n k N () N = ( t /τ) k ( + t /τ ) k n k N () N = ( t /τ) k ( + t /τ ) = + τ k t k = + τ t t /τ ( k )τ / t For large t/τ, 5

6 Friedlander SK, Smoke Dust and Haze, and + τ t t / τ n k N () N since τ = <k>-, η = (k-)/(<k>-). Then the asymptotic limit of the size distribution is selfpreserving. Similarity Solutions for Particle Size Distributions due to Coagulation (Self Preserving): If it is assumed that the fraction of particles in a given size range is a function of the particle volume normalized by the average particle volume then, e = e η n( v,t)n ()dv =ψ v v d N < v > < v > 6

7 where <v> = φ/n, and ψ is a function describing the particle size distribution which is not time dependent. Since n is the particle number per unit volume both sides of this equation are dimensionless. Since d(v/<v>) = dv/<v> = (N /φ)dv, = N n v,t φ ψ η N = ndv, φ = nvdv, n(v) when v or. where η = v/<v> = N v/φ, for n p (d p, t) can also be obtained, An expression = N 4 / 3 n d d p,t φ /3 ψ d η d where η d = d p (N /φ) /3. For Brownian coagulation we have, ( v i,v j ) = kt 3µ /3 v + i v j /3 v /3 /3 ( i + v i ) Continuum Range For a continuous distribution function the coagulation equation is, t = v which becomes, N dn β ( < v >,(v <v>) )n<v> nv <v> d<v> βv,<v> nv n<v> d<v> ψ + η dψ dη = kt η 3µ ψ ( η )ψ η η kt 3µ ψ η Continuous Distribution ψ η η /3 + ( η η ) /3 [ ] [ ] η /3 + η /3 when the similarity form is substituted (4 th equation above). η /3+ η η η /3+ η /3 d η For a continuous distribution the change in total number concentration with time is, /3 d η dn = βv,v nv n( v)dvd v Continuous Distribution 7

8 By substitution of the similarity distribution for n(v,t) and the continuum definition of β, we obtain, dn = kt 3µ ( + ab )N where a = η /3 ψdη and b = η /3 ψdη Continuum Regime This has the same functionality as Brownian coagulation from a monodisperse system, dn = K N. This expression can be used in the continuous distribution function above (long function) and an ordinary intergrodifferential equation for ψ in terms of η results. This means that the similarity solution for Brownian coagulation is a possible particular solution. The solution can be found using the integral definitions of N and φ given above. The function is solved for the upper and lower ends of the distribution and then the complete distribution can be obtained numerically using these limits. The numerical solution is shown below (from Friedlander s book) Shown below is a comparison of experimental measurements on coagulation in tobacco smoke with the predictions of the figure above. 8

9 Similarity Solution in the Free Molecular Regime: Generally, when the collision frequency function, β(v, <v>), is a homogeneous function of v, it can be transformed to an ordinary integrodifferential equation. If an ordinary integrodifferential equation can be made it must also satisfy the integral boundary conditions and constraints. For the free molecular range we have, ( v i,v j ) = 3 6kT 4π ρ p + v i v j / / v /3 /3 ( i + v i ) Free Molecular Range This is a homogeneous function of order /6 in particle volume. The similarity transformation can be made and a numerical solution can be found. The rate of change of the total number of particles is given by, dn = α 3 4π /6 6kT ρ p / φ /6 N /6 where a is an integral function of ψ with a value of about An example of numerical simulation (Monte Carlo) versus experimental observation for particle size distribution is shown below from Friedlander s text. 9

10 The self-preserving distribution can be approximated by a log-normal distribution function with σ g =.44 for the continuum regime and σ g =.46 for the free molecular regime. If the initial distribution of particles is also a log-normal distribution, the time to reach the self-preserving distribution (SPD), t sp, is a function of the initial distribution, σ g,

11 3 t sp = τ f 4π /6 6kT ρ p / v /6 N Free Molecular which is due to Vemury, S and Pratsinis, SE (995)J. Aerosol Sci., 75. For nm particles at 8 K the SPD is reached in on the order of microsecond. The figure below from Vemury (Friedlander) shows this behavior. The dip corresponds to the SPD geometric standard deviation. For initially monodisperse distributions the time to reach SPD was also calculated by Vemury/Pratsinis, 3 t sp = 5 4π /6 6kT ρ p / v /6 N t sp = 3 ktn 3µ Free Molecular Range Continuum Range