Aggregate Growth: R =αn 1/ d f

Size: px

Start display at page:

Download "Aggregate Growth: R =αn 1/ d f"

Rodger Shields
6 years ago
Views:

1 Aggregate Growth: Mass-ractal aggregates are partly described by the mass-ractal dimension, d, that deines the relationship between size and mass, R =αn 1/ d where α is the lacunarity constant, R is the aggregate overall size and N is the number o primary particles in an aggregate. The mass-ractal dimension ranges rom 1 to 3 in 3-d space. The growth o aggregates was irst modeled using computer simulations in the 's. Some discussion o the approaches and results orm these simulations will be given below. The Smoluchowski equation can also be modiied or mass-ractal aggregation and this is o the most lexibility since generalized results and conditions can be determined. Additionally, selpreserving distributions have been determined or the Smoluchowski approach that enable a general treatment o mass ractal aggregation. Simulations o Nano-Particle Aggregate Formation: The growth o aggregates can be simulated by making simple assumptions concerning transport o particles to the growing agglomerate and the events which occur when a primary particle or cluster collide with the growing aggregate. Depending on the relative kinetics o these two processes, growth can be diusion limited, similar to the original Smoluchowski equation, or reaction (collision) limited. Additionally, one can consider only monomer/cluster collisions or include cluster/cluster collisions as growth proceeds, such as might be expected in the ree molecular range. I one considers the primary particle as a point object, then the ractal dimension o the growing aggregate should increase orm a 0 dimensional point object to a 1 dimensional linear structure, to a branched chain o dimension on the order o as growth proceeds. The increase in dimension with growth is a natural consequence o the persistence o velocity or nano-particle Brownian motion combined with the random path o the colliding particles. The branch content o the growing aggregate should be a unction o l pa. As growth proceeds, the presence o branches and the convoluted shape o the growing aggregate, reminiscent o the Brownian path o the colliding particles, shields the interior bonding sites rom urther growth. For this reason three-dimensional growth is not possible, except by internal rearrangement. Then we can consider that an asymptotic mass ractal dimension is approached in time that is a unction o the persistence o the colliding particle velocity, l pa, and the degree to which monomers dominate over cluster transport. The latter issue is related to the diusion coeicient in the media and thus, the Knudsen number. In all simulations o mass-ractal aggregation an asymptotic dimension has been observed at long times. This means that the overall density o the aggregate, N/R 3 N (1-3/d) diminishes with time in the asymptotic range since N is a monotonically increasing unction o time. Diusion Limited Aggregation: In the simplest simulations, P. Meakin (1980 s) o diusion limited aggregation aggregates o dimension.5 are produced by ixing an initial primary particle at the origin o a three-dimensional lattice and allowing a random walk o primary particles, one at a time, randomly released rom the edge o the lattice until either collision or loss o the particle rom the ixed volume lattice occur. Despite the simplicity o this model, the dimension.5 is 1

2 repeatedly seen in aggregates produced with low primary particle concentration and high reactivity towards bonding. The persistence o velocity is the same as the primary particle size in this case. The cluster is immobile compared to the primary particle, so this could be considered either continuum transport or ree molecular transport or the monomer and continuum transport or the aggregate. I the concentration o primary particles is higher, and cluster cluster aggregation is allowed the mass-ractal dimension is reduced to 1.8 or diusion limited aggregation simulations. These simulations are conducted by randomly distributing primary particles on a lattice, and allowing all primary particles to collide and orm aggregates which also move by Brownian motion and urther aggregate to orm low dimensional branched structures. The persistence o velocity is constant in these simulations regardless o particle mass, l pa = d p. Additionally, variation in diusion coeicient with particle mass is not accounted or so this simulation might relect ree molecular conditions or both agregates and primary particles. The extreme o persistence o velocity is a ballistic growth simulation where primary particles ollow a linear path to the growing aggregate. For ballistic, monomer-cluster growth, similar to the irst simulation discussed, three-dimensional aggregates are produced. I clustercluster aggregation is allowed, similar to the second simulation above, mass-ractals o dimension 1.95 are produced. For this case the persistence o velocity or monomers and aggregates is larger than the lattice diameter. When persistence o velocity relative to primary particle size and variability o the diusion coeicient with particle size and transport regime are considered in more complicated simulations the results are qualitatively similar to the simple simulations mentioned above. In the ree molecular range Mountain ound values o d o 1.89 to.07 or diusion limited cluster-cluster aggregation, or instance. The aggregates that are produced by computer simulation are not monodisperse in N. Generally, the mass distribution o aggregates reach a sel-preserving orm indicating that an analysis based on the Smoluchowski equation as done or coagulation may be possible or aggregation as was done by Friedlander and is discussed below. Reaction Limited Aggregation Simulations: A sticking probability can be introduced into simulations to account or the probability o a particle or cluster bonding with the growing aggregate on impact. The importance o a stinking probability is that a particle just ater collision that does not stick has a much higher probability to transport by Brownian motion to the same or another local site on the aggregate. Then the colliding particles can probe the surace o the growing aggregate. Additionally, colliding particles can become trapped by high coordination number regions o the aggregate since translation away rom a site bounded by multiple aggregate subunits has a lower probability. The net eect o a sticking probability is to allow a type o local rearrangement towards multiply coordinated bonding. This directly leads to higher dimension aggregates. For primary particlecluster growth with a low sticking probability three-dimensional aggregates are ormed. I cluster-cluster aggregation is allowed a mass-ractal aggregate is ormed with a dimension o.09 which is larger than that observed or diusion limited aggregation (1.8). Other Dimensional Descriptions o Mass Fractal Aggregates:

3 The mass-ractal dimension, together with the aggregate size, and primary particle size yield a partial description o a mass-ractal aggregate that is suicient or the study o some aspects o growth and some properties, especially those related to mass/volume ratios. However, the mass-ractal description is not intended to be a complete description. For instance, given the mass-ractal dimension and size inormation it is not possible to reconstruct a acsimile o the aggregate without making assumptions concerning the branch content. I you are told that an object has a size o 100nm and is composed o 1 nm primary particles with a dimension o the object could range rom the regular object, a disk o thickness 1 nm and diameter 100nm with 10,000 primary particles, to a linear chain o diameter 100nm composed o 1 nm steps on a random path that is composed o 10,000 steps. From the mass-ractal dimension there is no possibility o distinguishing between these objects. Dimensional analysis can be used to describe a variety o eatures beyond the mass-size scaling. For instance, the spectral dimension is used to describe the energy distribution in a mass-ractal object, the chemical dimension is used to describe the reactivity o a mass-ractal structure and the connectivity dimension is used to measure the branch content o a mass-ractal structure. The number o primary dimensions that could be deined or a given object is only limited by the number o observable eatures that exist or the object. A primary dimension relects a undamental scaling eature or an object. One eature o a primary dimension in three dimensional space is that the value is equal to the mass-ractal dimension or regular objects and its value is equal to or less than the mass ractal dimension or ractal objects. Because o the wide range o dimensions that can be deined or various systems, the ield o dimensional analysis based on ractal concepts is challenging to understand. Nonetheless, at least two dimensional values are necessary or even the most rudimentary understanding o aggregation, i.e. in order to distinguish between a disk and a linear random walk. I we consider branching in mass-ractal aggregates, a natural measure o the object is the minimum path or primitive path, or number o steps, required to traverse the object between the urthest points in the aggregate, L. For a disk, L is the diameter o the disk, while or a linear random walk with no branching, L is the degree o aggregation, N. The scaling between L and N yields the connectivity dimension, C, or the aggregate, L C = BNl Where B is a scaling preactor similar to the lacunarity or mass-ractal scaling. For a linear chain L N so C = 1. For a disk L N so C = = d. C is a primary dimension. The number o branches, n br, in a mass-ractal aggregate can be determined rom C since, n br N/L The average coordination number or monomers in the aggregate, c N, is given by, c N = n br /N 1/L The average coordination number can be determined rom simulations and it is ound to reach an asymptote in ractal growth. The mass-ractal and connectivity dimensions are not independent in the sense that branched objects o the same path dimension have a larger mass-ractal dimension. For example a linear, Gaussian chain has a mass ractal dimension o and a connectivity dimension o 1. 3

4 For a randomly branched chain that ollows a Gaussian path the mass ractal dimension is.5 and the connectivity dimension is close to. Application o the Smoluchowski Equation to Mass Fractal Aggregaton: The Smoluchowski equation translates the problem o calculating particle growth rate, δn/δt, to the problem o determining the collision requency unction, β ij. For single particles in the ree molecular range, π β( v i,v j ) = π( d pi + d pj ) kt + 1 m i m j 1 Free Molecular Range where the single underlined part corresponds to the collision cross section or i and j aggregates and the double underlined part corresponds to the average relative velocity between agglomerates. β ij.relects the average hydrodynamic volume swept by the two agglomerates. For the ree molecular regime the aggregate is completely draining and the hydrodynamic volume o the aggregate is N agg v 0 where N agg is the degree o aggregation and v 0 is the volume o a primary particle, N agg = v =α R v 0 a 0 d Free Molecular (Free Draining) R is the hydrodynamic radius o the aggregate or the continuum range, i.e. non-draining range. In the continuum range the aggregate, with respect to diusion, is a sphere like object. So while the velocity depends on a Rouse like model, agg = N agg 0, the aggregate cross section is viewed rom a non-draining hydrodynamic perspective. The reason or this is simple, i the aggregates are rigid structures, then penetration between aggregates is unlikely, especially when d, while, small molecules can easily penetrate the aggregates in the ree molecular range. This picture breaks down when d < since it rapidly becomes possible or aggregates to penetrate the hydrodynamic sphere o each other, or instance the hydrodynamic sphere or a rod is a sphere o diameter L. Two rigid rods can easily penetrate each others hydrodynamic radius by lining up. This is not true or d. Additionally, the projected cross section o an object o d < will depend strongly on the orientation o the object. Again this is not necessarily true or a random mass-ractal object with d. As was shown earlier, the calculation o the diusion coeicient or a mass-ractal aggregate requires, a priori, a value or the mass ractal dimension. The calculation using the Smoluchowski equation yields a dierent result than the typical goal o the computer simulations mentioned above. Most computer simulations seek to determine the mass ractal dimension and perhaps the connectivity dimension as a unction o growth conditions, while the Smoluchowski equation seeks to describe the aggregate growth rate and aggregate size distribution with the ractal dimension as an input parameter. 4

5 The collision cross section, single underlined term above, is determined by the average ractal scaling, N agg = v =α R d Free Molecul, or ractals o dimension between and 3, v 0 a 0 CrossSection R i + R j ( ) i 1 d + j 1 d Substituting or the mass o the aggregates in the irst equation and this expression or the cross section we have, ( ) = 6kT β v i,v j ρ 3 4π λ 6 /d a 1/ d p0 v 1/ d i + v j ( ) v i v j Free Molecular Regime where λ = 1. For the continuum regime the collision kernel is, d β( v i,v j ) = 4π( d pi + d pj )( D i + D j ) Continuum Regime where dpi is the collision diameter or the agglomerates. For ractal aggregates we have, D ~ D 1 /N agg = (d 1 /d agg ) d D 1 Free Molecular Range D ~ D 1 /N agg 1/d = (d 1 / d agg ) D 1 Continuum Range and or large aggregates the continuum regime is used so, β( v i,v j ) = kt ( ) 1 3µ v 1/ d 1/ d i + v j v i 1/ d + v j Sel-Preserving Aggregate Size Distributions: 1/ d Continuum Regime (N>>1000) The collision kernels give above are homogeneous unctions o the colliding aggregate volume, β( αv i,αv j ) =α λ β ( v i,v j ) ; λ = 1 d Free Molecular or the continuum regime λ = 0. Since the kernels are homogeneous the distributions reach an asymptotic sel-preserving distribution. The sel-preserving equation is given by, ( ) = N n v φ ψ η ( ) ; η = v v and v = φ N 5

above, dn dt = 1 acφλ N λ ; c = 6kT ρ 3 4π λ 6 /d a p0 and "a" is an integral unction o the sel-preserving distribution unction above.

6 v is the solids raction o the aggregate, and φ is the volume raction o solids in the system. The particle number density, N, decays with, dn dt = ( ) βv i,v j n i n j dv i dv j using the ree molecular regime collision kernel given above, and the sel-preserving number distribution give above, dn dt = 1 acφλ N λ ; c = 6kT ρ 3 4π λ 6 /d a p0 and "a" is an integral unction o the sel-preserving distribution unction above. Values or a are given by Friedlander rom Monte Carlo calculations, Aggregate sel-preserving distributions solved using a discrete sectional model are shown in the igure above (Vemury and Pratsinis). The time to reach the sel preserving size distribution was also calculated by Vemury and Pratsinis and is shown in the igure below. In both cases the selpreserving limit is reached aster than spherical clusters. 6

7 7

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

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 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