Langmuir 1989,5, 510-518
510
formate structure is seen to develop to approximately the same extent as on clean A1203. A likely site for formate production is therefore an oxygen center that is left exposed during dehydroxylation of the A1203surface.
Acknowledgment. This work was supported by a research grant from IBM Research Laboratory, San Jose, CA. P.B. is grateful for a supported educational leave from Alcoa.
Collisions between Point Masses and Fractal Aggregates Paul Meakin" Central Research and Development Department, E. I. d u Pont de Nemours and Company, Wilmington, Delaware 19880-0356
Bertram Donn Laboratory for Extraterrestrial Physics, NASAIGoddard Space Flight Center, Greenbelt, Maryland 20771
George W. Mulholland National Bureau of Standards, Gaithersburg, Maryland 20899 Received October 11, 1988. I n Final Form: January 3, 1989 Computer simulations have been used to investigate collisions between point masses and fractal aggregates. Three different collision models were used (specular collisions, diffuse collisions, and the equilibrium cosine scattering law) for both transparent (diffusion-limited aggregates, D N 1.80, and ballistic aggregates, D = 1.95, where D is the fractal dimensionality) and opaque aggregates (reaction-limited aggregates, D 2.09, and ballistic aggregates with restructuring, D = 2.12). Our results are consistent with the idea that the collision cross section u is related to aggregate size (s) by u s for D < 2 and u s2JDfor D > 2 in the limits Similarly, the mean number of particle-aggregate contacts (N,) in each collision is consistent with N , const for D < 2 and N , s(1-(2/D)) for D > 2. There may be logarithmic corrections for D < 2. The distribution in the number of contacts can be described in terms of a power law with an exponential cutoff at large contact numbers. The transfer of momentum and angular momentum during collisions was explored. The results for these quantities can be understood in terms of the distributions of the angles between the incident and departing point mass directions.
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00.
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Introduction The transfer of mass, energy, momentum, and angular momentum between small particles and a gaseous environment as well as the transport properties of the particles themselves is a subject of considerable technical importance. The physics of highly dispersed aerosols consisting of single particles with a mean diameter 11OOO 8, has been studied extensively during the past few decades.'I2 In the high Knudsen number (K,) or free molecular limit in which the mean free path of the gas molecules is substantially greater than the particle diameter, the transport properties of the particles and their interactions with the surrounding gas can be understood in terms of single collisions between the dispersed particles and the gas molecules. Under these conditions (K,, >> 1) the dispersed particles behave like giant gas molecules, and their velocity distribution is described by the kinetic theory of gases.3 Using the kinetic theory of gases, Epstein4 showed that in the free molecular limit the friction coefficient F of a spherical particle moving at a constant velocity u through a gas a t rest is given by (1) Fuchs, N. A,; Sutugin, S. G. Highly Dispersed Aerosols; Ann Arbor Science Publications: Ann Arbor, 1970. (2) Hidy, G. M.; Brock, J. R. "The Dynamics of Aerocolloidal Systems"; International Reoiews i n Aerosol Physics and Chemistry; pergamon Press: Oxford, 1970; Vol. 1. (3) Jeans, J. The Dynamic Theory of Gases; Dover: New York, 1954. (4) Epstein, P. S. Phys. Reo. 1924, 23, 710.
0743-7463/89/2405-0510$01.50/0
-
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F = fu where f is the so-called friction coefficient
(1)
f = 4/,6ar2nmv (2) where r is the radius of the sphere, m the mass of a gas molecule, n their number density, v the average speed of the gas molecules, and 6 a numerical factor with a value of 1.0 for specular reflection and a value of 1.44 for diffuse cosine law reflection. The coefficient f is the central quantity in determining the particle's diffusion coefficient and drift velocity. Many highly dispersed aerosols consist of small particles which can join together to form tenuous flocs or aggregates. There is a wide variety of such particles including metals, metal oxides, carbon, and Si02. There is very little information about the transport properties and friction coefficient of such aggregates. One of the few quantitative studies involved the measurement of the dynamic shape factor of propane soot? The dynamic shape factor is the ratio of the friction coefficient of the aggregate to the friction coefficient of a sphere with the same volume as the aggregate. In recent years it has been shown, as a result of both experimental work and computer simulations employing simple nonequilibrium aggregation models (see (5) Friedlander, S. K. Smoke, Dust and Haze; Wiley: New York, 1977. (6) Odumade, 0. A. "Mobilities of Aggregates of Particles"; Ph.D. Thesis, University of Minnesota, 1983.
D 1989 American Chemical Societv
Langmuir, Vol. 5, No. 2, 1989 511
Point Mass and Fractal Aggregate Collisions ref 7-13, for example), that if these aggregates become sufficiently large, they can be described quite well in terms of the concepts of fractal geometry.l4?l5 This means that for sufficiently large cluster sizes (s) the mean cluster radius of gyration REis given byI6 (3)
where the radius of gyration has been averaged over a large ensemble of clusters containing s particles. An effective fractal dimensionality (Do = 1/p) can be obtained from the dependence of R on s for clusters generated in computer simulations. kimilarly, the correlation functions C(rl,r2,...) which describe the cluster geometries have a scale invariant form
Cn(Xr ,Xr2,.. .Xr,) = X-nanCn(r 1,r2,. ..rn)
(4)
The fractal dimensionalities Dnacan be obtained from the exponents a,
Dna = d - an
(5)
where d is the dimensionality of the space in which the aggregates are embedded. The correlation function Cn(rl,r2, ...r,) is given by
Cn(rl,r2, ...rn) = (p(ro)p(ro+rl)... p(ro+rn)) (6) where the averaging is over all origins (ro)and p(r) is the density at position r. For self-similar fractals, D, = Dal = Da2 = ... Since the fractal aggregates used in this work are believed to be self-similar fractals, the symbol “D” will be used for the “all purpose” fractal dimensionality, which is equal to D,, D,, etc. The values reported for D in this work were measured from the dependence of REon s (eq 3) and are effective values for D,. Here the results of computer simulations of the collision of point masses with fractal aggregates consisting of identical spherical particles are presented. We assume that the mass ( m )of the point probe particles is negligible compared to the mass (AI) of the particles in the aggregates ( M >> m). Consequently, the aggregates can be considered to be stationary during the collisions (even for very small particles with a diameter of about 100 A, M / m 2 lo5). The point masses are considered to follow linear (ballistic) paths, and the effects of attractive and repulsive interactions with the aggregate particles are neglected. The main differences between collisions with aggregates and collisions with spherical particles result from the fact that in the former case the point masses may contact the aggregate more than once before they finally leave the region occupied by the aggregate. Because of the apparent complexity of the structure of particle aggregates, there has been little theoretical effort (7) Kinetics of Aggregation and Gelation; Family, F., Landau, D. P., Eds.; North Holland Amsterdam, 1984. (8) On Growth and Form: Fractal and Non-Fractal Patterns in Physics; NATO AS1 Series E100; Stanley, H. E., Ostrowsky, N., Eds.; Martinus Nijhoffi Dordrecht, 1986. (9) Time Dependent Effects in Disordered Materials; NATO AS1 Series B167; Pynn,R., Riste, T., Eds.; Plenum Press: New York, 1987. (10) Meakin, P.In Phose Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.;Academic Press: New York, 1988, Vol. 12, p 335. (11) Jullien, R.;Botet, R. Aggregation and Fractal Aggregates; World Scientific: Singapore, 1987. (12) Meakin, P. Adu. Colloid Interface Sci. 1988,28, 249. (13) Witten, T.A.; Cates, M. E. Science 1987,232, 1607. (14) Mandelbrot, B. B. The Fractal Geometry of Nature; W. H. Freeman: New York, 1982. (15) Feder, J. Fractals; Plenum Press: New York, 1988. (16) Stanley, H.E.J. Phys. A 1977, 10, L221.
to compute the friction coefficient of an aggregate. First-order estimates of the friction coefficient of aggregates in the free molecular and continuum regime were made by Mountain, Mulholland, and Baum17assuming the aggregates to have a fractal structure. Our paper describes a detailed computer simulation study of the interaction of gas molecules with aggregates. The mean free path of the gas is assumed to be much larger than the size of the aggregate in the simulations. Such conditions occur in flames for small agglomerates, exist in the stratosphere, and occurred a t the time of early planetary evolution. From the simulation results, an estimate can be made of the friction coefficient and the dynamic shape factor.
Aggregation Models The clusters used in this work were generated by using several three-dimensional off-lattice models, which are known to give clusters with fractal dimensionalities substantially smaller than 3.0. In all cases the simulations were carried out by using simple but realistic models in which a realistic polydisperse cluster size distribution evolves in a natural way. These models can be used to investigate the kinetics as well as the geometry of colloidal aggregation,*12 but here these models are used only to generate clusters for the collision simulations. Since these models have been described previously, we indicate here only where additional information can be found and the fractal dimensionalities of the clusters generated by these models. The polydisperse off-lattice diffusion-limited cluster-cluster aggregation model is based on the Witten-Sander model for (particle-cluster) diffusion-limited aggregationla and lattice models for diffusion-limited cluster-cluster a g g r e g a t i ~ n . ’ ~ , This ~ ~ model generates clusters with an effective fractal dimensionality (D)of about 1.80.21 The ballistic cluster-cluster aggregation model is based on the earlier models of S ~ t h e r l a n d . ~ ~ ? ~ ~ The simulations were carried out assuming that the cluster velocities are given by the kinetic theory of gases.3 Under these conditions an effective fractal dimensionality of about 1.95 is obtained.24 In addition, simulations were carried out in which one stage of restructuring is included. In this model the clusters are assumed to be rigid, but after one cluster (cluster 1) has contacted another cluster (cluster 21, cluster 1 is rotated about the center of the contacting particle in cluster 2 until a second contact between the two clusters is formed. If possible, cluster 2 is then rotated about the center of the contacting particle in cluster 1 to form a third contact.25 This model generates clusters with an effective fractal dimensionality of about 2.12. Simulations have also been carried out by using an off-lattice model for reaction-limited cluster aggregation. This model is based on the hierarchical lattice models of Jullien et al.26,27and is closely related to the models investigated by Brown and Ball2*and proposed by L e y v r a ~ . ~The ~ model used in this work has been de-
w.;
(17) Mountain, R.D.; Mulholland, G. Baum, H.J. Colloid Interface Sci. 1986, 114, 67. (18) Witten, T.A.; Sander, L. M. Phys. Rev. A 1981, 47, 1400. (19) Meakin, P. Phys. Reu. Lett. 1983,51, 1119. (20) Kolb, M.; Botet, R.; Jullien, R. Phys. Reu. Lett. 1983, 51, 1123. (21) Meakin, P.;Jullien, R. J. Chem. Phys. 1988, 89, 246. (22) Sutherland, D. N. J . Colloid Interface Sci. 1967, 25,373. (23) Sutherland, D. N.:Goodarz-Nia, I. Chem. Eng. Sci. 1971,26,2071. (24) Donn, B.; Meakin, P., unpublished results(25) Jullien, R.; Meakin, P. J. Colloid Interface Sci., in press. (26) Jullien, R.; Kolb, M. J. Phys. A 1984, 17, L639. (27) Kolb, M.;Jullien, R. J. Phys. Lett. 1984,45, L971 (28) Brown, W. D.; Ball, R. C. J . Phys. A 1985, L517. (29) Leyvraz, F.,preprint.
512 Langmuir, Vol. 5, No. 2, 1989
Meakin et ai.
a
C
DIFFUSION-LIMITED CI-CI-3d M = 10.732
.
.
260 DIAMETERS
BALLISTIC CI-CI-3d M = 10.700
.
BALLISTIC CI-CI-3d ONE STAGE RESTRUCTURING S =11186
.
180 DIAMETERS
REACTION LIMITED CI-CI-3d M=16,332
.
L
180 .. DIAMETFRS . .. ....- . -. .
Figure 1. Projection of clusters generated by the four models used in this work. Parts a, b, e, and d show clusters generated hy using the diffusion-limitedaggregation model, the ballistic aggregation model, the ballistic aggregation model with one-stage restructuring, and the reaction-limited aggregation model, respectively. All ofthese simulations were carried out with off-latticepolydisperse models.
scribed in detail by Meakin and Family.”’ The effective fractal dimensionality of the clusters generated in this manner is about 2.09. Figure 1 shows projections of clusters generated by the four models used in this work. Projection Areas In a dense fluid the transport properties of fractal aggregates are determined by the hydrodynamic radius (Eh) through a Stokes-like friction coefficient
f = 65~nRh (7) In this case it has been shown that Rh is proportional t o the radius of gyration?*= The hydrodynamic flow field ~~~
(30) Meakin, P.; Family, F. Phys. Re”. A 1981.36, 5498. (31) Meakin. P.; Family, F. Phys. Reo. A 1988.38, 2110. (32)Meakin, P.; Chen, Z.-Y.; Deuteh, J. M J. Chem. Phys. 1985,8Z,
3786. (33) Heas, W.; Frisch, H. L.; Klein, R.2.Phys. B: Condens. Matter 1986,64,65. (34) Wiltzius, P. Phys. Re”. Lett. 1987,58,110.
does not penetrate into the fractal aggregate (even for D < 2), and the aggregate behaves like a sphere with a radius proportional to s’/D. In the high Knudsen number limit the transport p r o p erties are determined primarily by the number of colliiions betweer. the aggregate and the gas molecules (i.e., on the collision cross section or the area of a projection of the aggregate onto a plane). For the case D < 2 the projection of the cluster does not fill the plane, and consequently we expect that in the asymptotic limit (s m) the area of the projection (u) will he proportional to the number of particles in the cluster. Such a cluster is asymptotically transparent, whereas for D > 2 the cluster would be opaque. In general, for D < 2, one part of the cluster is not “hidden” by another part of the cluster (except for the effects of local correlations, which reduce the area of the projection by a constant factor of order l),and the entire structure can be seen in the projection. Under these
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(35)Chen, Z.-Y.;Meakin, P.; Deutch, J. M. Phys. Reu.Lett. 1987,59, 2121.
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Figure 2. Dependence of In
\I 8.0
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9.0
(u/s)
obtained from the diffusion-limited cluster-cluster aggregation model (D3 1.80),and Figure 2b (bottom) shows results obtained from the ballistic cluster-cluster aggregation model with restructuring (D E 2.10).
conditions the projection and the cluster itself both have the same fractal dimensionality." For finite size clusters we expect significant corrections to the asymptotic behavior (a s), and it is reasonable to attempt to fit the dependence of u on s by a function of the form u(s)
-
As
+ Bs'
(8)
with 6 < 1. To obtain the collision cross sections (projected areas) for clusters generated by using the diffusion-limited aggregation model (D N 1.80), 11 simulations were carried out starting with 50 OOO particles in each simulation. Pairs of clusters were picked randomly from a list of clusters and combined via random walk trajectories. This process was continued until the largest cluster had reached a size (s) of greater than 10 000 particles. This model generates a cluster size distribution quite similar to that expected for diffusion-limited cluster-cluster aggregation. After each cluster (containing s particles) had been generated, the projected area was obtained by projecting the cluster onto a plane and picking 10 s points at random in a circle of radius R, enclosing the projected cluster (here R, is the maximum radius of the cluster measured from its center of mass). If N of these points are in the region containing the projection, then an estimate of the cluster area is given by For each cluster this procedure was repeated for two other projections in directions perpendicular to each other and to the first direction. Figure 2a shows the dependence of In ( u / s ) on In s. Since the fractal dimensionality (D) is less than 2, we expect that in the asymptotic limit (s m) u / s should be independent of s (perhaps u could depend on some power of In s, but not on a power of s). Consequently, the data shown in Figure 2 have been fitted to the form given in
-
0.264s
I
on In s obtained from two cluster-cluster aggregation models. Figure 2a (top) shows results
-
eq 8. For clusters in the size range s = 5-100 A, a nonlinear least-squares fit gave A = 0.3757, B = 0.4098, and 6 = 0.7697. This curve is also shown in Figure 2a. It is apparent that this curve fits the simulation results well for all values of s except for the very largest values of s, for which the statistics are relatively poor due to there being very few very large clusters. For s = 1 (single particles), the curve A s + Bs6 gives a projected area of 0.7855 compared to a value of a14 or 0.7854 expected for a sphere of unit diameter. This does not, of course, prove that in the limit s m u / s approaches a limiting, constant, value (which, if eq 2 is correct, would have a value of A or 0.376). Nevertheless, these results are certainly consistent with the idea that u / s approaches a fixed value in the asymptotic limit. The collision cross sections for clusters formed by ballistic aggregation are of more interest since this aggregation process is expected in the high Knudsen number limit. We3 have previously reported results for the dependence of u on s obtained for clusters generated by using this model. In this case u(s) could be fitted very well by
+ 0.519s0.825
(loa)
for clusters in the size range 5 5 s 5 100 and by u(s) =
0.240s
+ 0.517~~.*~'
(lob)
for 1 I s I 2500. We have also fitted the dependence of u(s) on s by the more general form u(s)
= As7
+ Bs6
(11)
and for clusters containing 1-2500 particles the values A = 0.5535, B = 0.2643, y = 0.9516, and 6 0.6443 were obtained. Although the value obtained for the exponent y is not exactly 1, it is sufficiently close to 1to be consistent with the idea that u s in the large size limit. The dependence of &) on s can also be fit reasonably well (within 5%) by the simple power law u(s) = 0.682~O.~~ for clusters in the size range 7 < s < 2500. This simple power law may be useful for some applications but is not as accurate as eq 10 or 11. For the ballistic aggregation model with restructuring and the reaction-limited aggregation model, the fractal dimensionality (D) is greater than 2. Under these conditions the projection of a random fractal will densely cover the plane, and the projected area is expected to increase with increasing cluster size according to
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(12) in the limit s m. For finite cluster sizes we expect significant corrections to this asymptotic scaling relationship. In these cases u(s) was fitted by the form given in eq 11 with a value of 2/D for the exponent y. Figure 2b shows that for the ballistic aggregation model with restructuring the dependence of u on s can be represented quite well by the two-term power law u(s)
s21D
+
u(s) = 0.270s0.94s 0 . 5 3 7 ~ ~ . ' ~ ~
(13)
Here the exponent y (0.948) was fixed a t a value of 210. The dependence of u on s was also fitted to the form given in eq 11, and all of the parameters were allowed to vary in the least-squares fitting procedure, leading to the result
+
u = 0.462s0.911 0.399~O.~'~
(14)
This gives a better fit to the dependence of u on s for small clusters, but for larger clusters the fit given by eq 13 is as good despite the fact that there is one less adjustable (36)Meakin, P.; Donn, B. Astrophys. J. 1988, 329, L39.
514 Langmuir, Vol. 5, No. 2, 1989 parameter used to obtain the result given in eq 13. These results were obtained from 20 simulations with 200000 particles in each simulation. The simulations were stopped when the largest cluster exceeded a size, s, of 10000 particles. A similar set of simulations was carried out for the reaction-limited aggregation model. In this case 21 simulations were carried out with 200 000 particles per simulation and were stopped when the maximum cluster size exceeded 10000 particles. In these simulations, pairs of clusters are joined rigidly and irreversibly by selecting bonding (contacting) configurations randomly from all possible nonoverlapping bonding configurations. The simulation starts with single particles which, initially, are joined to form dimers. As the simulation proceeds, larger and larger clusters are formed, and a broad distribution of cluster sizes evolves in a “natural” way.2&31 With clusters generated by this simple model, the result u(s) = 0 . 3 5 6 ~ O + . ~0~.~4 4 5 ~ O . ~ ~ ~ (15) was obtained by using the form U ( S ) = As21D+ BsY (16) With the more general form given in eq 11, the result u(s) = 0 . 2 7 8 ~+~ 0.517s0.s33 .~~~ (17) was obtained. In general, our results are consistent with the idea that the asymptotic form for u(s) is given by u(s) s for D e 2 u(s) szlD for D > 2 Although the results obtained from fitting u(s) by the more general form given in eq 11lead to values for the exponent y (y > 6) slightly different from 1.0 and 2/D for D < 2 and D > 2, respectively, the more restricted form given in eq 8 and 16 fits the data almost as well despite the fact that they use one less adjustable parameter.
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Collision Numbers A separate series of simulations was carried out for the ballistic aggregation models (with and without restructuring) and the reaction-limited aggregation model to explore the nature of collisions between these aggregates and point masses in more detail. In these simulations, collisions were carried out for all of the clusters generated during the simulations with masses in the range s1 Is Is2, with s1 = 2 and s p = 1000 or 2000. For each cluster of size s, sllz + 1 collisions were simulated by using three different collision models. In model I the collisions between the point particles and the clusters were assumed to be perfectly elastic. In model I1 the particles are emitted from the surface of a particle (at the position at which they first contacted the particle) in a direction that is biased toward the normal. In this model the probability of emission in an infinitesimal solid angle 6R in a direction of 8 from the normal is given by P(6R) 6R cos (8) (18) This means that the total probability of emission at an angle of 8 to the normal is given by P(8,8+68) 68 cos 8 sin 8 (19) This is the equilibrium cosine scattering law, which has been discussed extensively in the l i t e r a t ~ r e . ~ ,In~ ~ the -~~ third model, the emission of the particle from the surface
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(37) Gaede, W. Ann. Phys. 1913, 41, 289. (38) Clausing, P. Ann. Phys. 1930, 4, 533. (39) Millikan, R. A. Phys. Reu. 1923, 22, 1. (40) Cosma, G. J . Chem. Phys. 1968, 48, 3235. (41) Wenaas, E. P. J . Chem. Phys. 1971,54, 376. (42) Saltsburg, H. Annu. Reu. Phys. Chem. 1973, 24, 493.
Meakin et al.
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is random, so P(6Q) 6R for all directions and P(e,e+Se) 6 8 sin 8. Model I corresponds to specular collision with an accommodation coefficient of zero, while models I1 and I11 correspond to diffuse reflection with an accommodation coefficient of 1. However, as was shown by Gaede3’ and Millikan,39model I11 is inconsistent with the second law of thermodynamics. Model I11 is called “diffuse scattering according to Lenard”43by E p ~ t e i n who , ~ shows how a perpetual motion machine could be constructed by using a combination of specular and model I11 scattering. The justification for the model I1 cosine scattering law for equilibrium scattering has been discussed by W e n a a ~ . ~Inl at least some systems it appears that the collisions between molecules or atoms and small particles can be described by superposition of specular and diffuse cosine law collisions.2144 Both model I and model I1 are consistent with the second law of thermodynamics and are detailed balance models in the sense that in a gas at equilibrium the incoming and outgoing flux of particles are equal in all directions. Here we will refer to models I, 11, and I11 as specular, cosine law diffuse, and random diffuse scattering, respectively. Since a large number of collisions with a large number of clusters must be simulated to reduce statistical uncertainties to acceptable levels and to extend the simulations to reasonably large clusters, quite large amounts of computer time were required for this study. The simulation results presented in this section required several hundreds of hours of CPU time on an IBM 3090 computer. For each cluster point, particles are launched along ballistic trajectories selected at random from all trajectories, which pass within a distance R,, from the center of mass. Here R , is the maximum radius of the cluster measured from the center of mass. After a particle has contacted the cluster, its trajectory is followed by using models I, 11, or I11 until it leaves the vicinity of the cluster. The process is repeated for s1l2 + 1 trajectories, which contact the cluster one or more times have been found. Simulations were carried out with models I, 11, and I11 for clusters generated by using the ballistic aggregation models (with and without restructuring) and the reaction-limited aggregation model. In all cases we measured the mean number of particle-cluster contacts, N,(s), in each contacting trajectory as a function of the cluster size. Here N c ( s )is the mean number of contacts for clusters containing s particles. The distribution in the number of contacts n ( N J was also measured for the largest cluster sizes (clusters in the size range 0 . 9 ~5~s Isz). The quantity n(Nc)is the number of trajectories in which N , contacts occurred. Figure 3 shows the dependence of N , on s obtained from 31 simulations, each starting with 200000 particles, carried out by using the ballistic cluster aggregation model. Figure 3a shows the dependence of In [N,(s)]on In s , and Figure 3b shows the dependence of In [N,(s)]on In (In s) for clusters in the size range 2 5 s 5 2000 particles. These results were obtained by using model I collisions (specular reflection). It is apparent that N,(s) grows quite slowly with increasing s. The results shown in Figure 3a are consistent with a power law growth in the number of contacts
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N,(s) S f (20) However, the effective value for the exponent E is quite small ( E I 0.07), and such small exponents are unlikely. (43) Lenard, P. Ann. Phys. 1920, 61,672. (44) Hinchin, Shepperd In Rarefield Gas Dynamics; Brundin, C., Ed.; Academic Press: New York, 1967.
Langmuir, Vol. 5, No. 2, 1989 515
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(In s) (b, bottom) obtained from ballistic aggregation model clusters with no restructuring (D= 1.95) and model I (specular) collisions with point particles. Consequently, the dependence of In [N,(s)]on In (In s) is displayed in Figure 3b. Figure 3b shows that the data are also consistent with a logarithmic growth in the number of contacts with increasing cluster size
-
(In s)+
(21)
where $ 1 0.4. Quite similar results were obtained from the other models and aggregate structures. For example, Figure 4 shows the dependence of N,(s) on s for the ballistic cluster aggregation model with restructuring (D 2.10) for both model I (specular) and model I11 (random diffuse) scattering. In all cases, models I, 11, and I11 gave very similar results for the dependence of N , on s (almost indistinguishable for the ballistic aggregation model with restructuring and the reaction-limited aggregation model, which generates clusters with essentially the same fractal dimensionalities). The results shown in Figure 4 suggest that the dependence of N,(s) on s can be described by eq 21 for large values of s with a value of about 1.0 for the exponent cp. However, the dependence of N,(s) on s can also be fit quite well by eq 20 with a small value for for these models. For large values of s (s N lOOO), N,(s) is largest for the ballistic aggregation model with restructuring and smallest for the ballistic aggregation model without restructuring. The difference between the Nc(s) curves for ballistic aggregation models with restructuring (D N 2.13) and reaction-limited aggregation ( D N 2.09) is considerably larger than the difference between either of these models and the ballistic aggregation model with no restructuring ( D N 1.95) despite the much larger difference in fractal dimensionality in the latter two cases. Figure 5 shows the distribution of the number of collisions n(N,) obtained by using model I collisions with ballistic aggregates and model I11 collisions with ballistic aggregates generated with restructuring. In both cases, the distributions have been fitted to a power law decay with an exponential cutoff n ( N J = CN;T(e-kNc)
1
2.0
Figure 3. Dependence of In [N,(s)]on In s (a, top) and on In
N,(s)
8
l n (s)
(22)
2
3
4
5
6
7
8
Ln (s)
Figure 4. Dependence of N,(s) on In s obtained from ballistic aggregates (D = 1.95) with specular collisions and ballistic aggregates with restructuring ( D = 2.12) obtained by using model I11 (random diffuse) collisions. 11,
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43700 N,-0.865 e(-0.392 = 7
SIMULATION 5 5 RESULTS 5 4 32 - MODEL I COLLISIONS 1 - BALLISTIC AGGREGATES I
11 I
I
I
I
I
I
I
I
I
I
I
MODEL COLLISIONS BALLISTIC AGGREGATION WITH RESTRUCTURING
015
1.b
1;.
2jO
2j5
3jO
315
l n (Ne)
Figure 5. Distribution of contact numbers (n(N,))obtained from simulations employing ballistic aggregates with specular collisions (a, top) and ballistic aggregates with restructuring obtained by using random diffuse collisions (b, bottom). In both cases clusters in the size range 1800 5 s 5 2000 were used. The simulation data are represented by solid curves, and the results of a least-squares fit to a power law with an exponential cutoff are represented by dashed lines where the simulation results and the least-squares fits can be distinguished.
This simple form seems to fit the data quite well, and the parameters r and k obtained from a least-squares fitting procedure are given in Figure 5. For the other cases, the n(Nc)curves are quite similar to those shown in Figure 5,
516 Langmuir, Vol. 5, No. 2, 1989
Meakin et al.
but in most cases we were not able to obtain stable least-squares fits to the form given in eq 22. However, we believe that more accurate data over a larger range of contact numbers (N,) would enable us to obtain stable fits by using a power law with an exponential cutoff, and it seems likely that n(NJ can be described quite well by this form in all cases. For one other case not shown in Figure 5 (model I11 collisions with unrestructured ballistic aggregates), n(NJ could be fitted quite well by
n(N,)
-
0.10 I I o,09 - MODEL I COLLISIONS
I
,
I
,
I
0.080.070.06
'.Q 0.05
5
004
0 0.02 .
0
3
v
I
0.01 CNc-1.368e-0.215Nc
o.ooo
1
3
2
4
5
6
7
I
I
Ln (SI
The exponent T is very close to that found for model I11 and collisions shown in Figure 5b, but we do not have enough data to establish if there is a unique value of T for each type of collision.
0.401
I
I
I
I
,
,
I
,
I
,
,
2
3
4
5
6
Momentum Transfer
In these simulations we consider only the geometry of the point masses relative to the cluster. The velocities and velocity distributions of the point masses are not considered. However, if we assume that the velocities of the point masses are not changed during the collisions, we can calculate a momentum transfer in terms of the momentum m V1 of the incoming masses. This picture is reasonable for the case of specular collisions but is not realistic for models I1 and 111. For each collision the total momentum transfer 6P was obtained from 6P = IV, - V,I
(23)
where V1 is a unit vector in the direction of the incoming point mass and V, is a unit vector in the director in which the particle finally leaves. Here 6P is measured in units of mlV1l, the momentum of the incoming particle in a frame of reference in which the aggregate is stationary. From the simulations, we obtain the average of 6P (( 6P)) and use 6P to represent the ensemble average momentum transfer. This quantity is equivalent to the quantity 6 in eq 2 for the case of a single particle. Figure 6a shows the dependence of In (6P)on In s obtained for specular collisions with ballistic cluster aggregates (DN 1-95),and parts b and c of Figure 6 show similar results obtained by using model I1 and model I11 collisions, respectively. For single particles, Epstein4 has shown that 6 has a value of 1.0 (In 6 = 0) for model I (specular collisions), 13/9 or 1.444...N (In 6 = 0.3677...) for model I1 (cosine law diffuse collisions), and 4/3 (In 6 = 0.2876 ...) for model I11 (random diffuse collisions). Although we did not measure 6P for single particles, the values obtained from Figure 6 for In (6P)by extrapolating In s to zero are in very good agreement with these rigorous theoretical results. For model I collisions with ballistic aggregates, 6P quickly reached a limiting value of about 1.092. Very similar behavior is found with limiting (large s) values of about 1.175 for ballistic aggregates with restructuring and about 1.101 for reactionlimited aggregates. For both model I1 and I11 collisions, the momentum transfer decreases slowly with increasing cluster size but does not reach a limiting value for clusters in the accessible size range (s I2000). For both models, the range of momentum transfer values is quite small. For example, for model I11 6P = 1.27 for ballistic aggregates, 6P = 1.31 for ballistic aggregates with restructuring, and 6P = 1.27 for reaction-limited aggregates at a cluster size, s, of 1000 particles. For the more realistic model I1 (cosine law diffuse), collisions 6P = 1.36 for both ballistic aggregates and reaction-limited aggregates with a cluster size of 1000 particles.
I
":I' 0.05
0.00
1 7
l n (SI
1
0'30 0.25
-a
2o -
s015C
d
0.10 005 -
MODEL IR COLLISIONS BALLISTIC AGGREGATES
Langmuir, Vol. 5, No. 2, 1989 517
Point Mass and Fractal Aggregate Collisions
800 3.0 -
2.5 I
-
MODEL MODEL II COLLISIONS COLLISIONS BALLISTIC AGGREGATES
-
600
-
0
.-C
-
2.0 -
400
$z
200
-
1
20
'0
40
60
80
8
600I
3.5
I
3,0-
MODEL 11 COLLISIONS BALLISTIC AGGREGATES
I
I
!
1
I
e
100
I
I
I
1
2
300
200
2
3
4
5
I
I
160
I
1
180
, l~400
\. 1350
Id0
1o;
la0
1;o
IO0
o 1;
IlO
160
e 1
140
MODEL II COLLISIONS BALLISTIC AGGREGATION
I
2.5 -
W."
120
6
7
0 180
8
Ln (s)
Figure 7. Dependence of the mean angular momentum transfer (We) on the cluster size for ballistic aggregates with model I (specular; a, top) and model I1 (cosine law diffuse; b, bottom) collisions.
the peak in N ( 6 ) occurs at an angle of about 105'. For models I1 and TI1 there are comparatively few small-angle collisions, and the peak in N ( 6 ) occurs a t significantly larger angles (about 135' for model I1 and about 125' for model 111). These angular distributions are responsible for the differences between models I, 11, and I11 with respect to momentum and angular momentum transfer (I1
> I11 > I).
Parts a and b of Figure 9 show the angular distributions obtained from reaction-limited aggregates with model I (specular) and model I11 (random diffuse) collisions, respectively. The increased fractal dimensionality of the reaction-limited aggregates has little effect on these angular distributions for model I collisions, in accord with the observation that the momentum transfer for the ballistic aggregation and reaction-limited aggregation models is very similar. For model I11 (random diffuse) collisions, the higher fractal dimensionality reaction-limited aggregates have more small-angle collisions, but the peak in N ( 6 ) seems to occur a t slightly larger angles. Figure 9c shows the results obtained by using ballistic aggregates with restructuring and model I collisions. In this case there seems to be distinctly fewer small-angle collisions than for reaction-limited aggregates or ballistic aggregates without restructuring. This observation is in accord with the higher momentum transfers found for ballistic aggregates with restructuring and model I collisions (6P= 1.17) than for the reaction-limited aggregates (6P= 1.10) or ballistic aggregates without restructuring (6P= 1.09). Friction Coefficient The friction coefficient f for a spherical particle moving at a low velocity with respect to a low density gas is given by eq 2. For a more complex structure, eq 2 can be replaced by f = 4((6P)a)nm(u/3)
(26)
The quantities n, m, and u depend on the nature and conditions of the surrounding gas, and the quantities a and
--
io
a'o
60
60
\350 Id0
e Figure 8. Distribution of collision angles (angle between incoming and outgoing particle paths) for ballistic aggregates ( D 1.95) with model I (a, top), model I1 (b, middle), and model I11 (c, bottom)collisions. All of these results were obtained from clusters in the size range 1800 Is I2000 particles.
6P depend on the structure of the object and its interactions with the gas molecules. Here we have used computer simulations to obtain the dependence of both ( a ) and (6P) on the cluster size s, where ( a ) and (6P)are averaged over a large number of collisions. The mean friction force f will depend, according to eq 26, on (a6P);since we do not expect strong correlations between 6P and a, eq 26 can be replaced by the approximation f = 4a(6P)nm(u/3)
(27)
since (6P)is quite insensitive to the cluster size (s) (Figure 6). Our results indicate that the dependence off on s can also be expressed (approximately) as f(s)
-
40(s)(6Pm)nm(u/3)
(28)
The results shown in Figure 6 indicate that for clusters containing more than just a few particles the momentum transfer parameter 6 is within a few percent of its asymptotic (s m) value of 6P,. For a compact (spherical) object, the friction coefficient is proportional to MI3,where M is the mass of the object. Our simulation results indicate that f MIDfor D > 2 and f M for D < 2 in the large aggregate size limit.
-
-
-
Discussion The objective of this work was to develop a better understanding of the interactions between fractal aggregates and gasses in the high Knudsen number limit. Such in-
518 Langmuir, Vol. 5, No. 2, 1989
Meakin et al. 1200
REACTION -LIMITED
4
1000 m 800 .E v)
600
300
z
75
I b
5 ,501
200 L
/
I
41.50
7"
'MODEL I MODEL Il
loot/ ~~~
20
"0
40
60
80
100
120
140
~
160
1I
In ( 5 )
Figure 10. Comparison of m / 4 0 (the number of particles per unit projected area) and the mean number of collisions for ballistic aggregates with model I, model 11, and model I11 collisions. In this figure these quantities are presented as a function of In s, where s is the cluster size.
MODEL IU COLLISIONS REACTION- LIMITED AGGREGATES
0
20
40
60
100
SO
120
140
160
180
9 1000
500 450
400 350 300
z
800 m
600 .5
250 200
400 z
150 100
200
\ v)
0
of its component particles. On the other hand, this effect is compensated by the multiple collisions between the gas molecules and the aggregate. It appears that this compensation is exact for models I and 11. This is a consequence of the fact that under equilibrium conditions the average gas pressure must be the same at all positions, so the number of collisions between the aggregate and the gas must be proportional to the total surface area, irrespective of how the particles are organized in space. Figure 10 shows the dependence of ?rs/4a (sul/u, where u1 is the collision cross section for a single particle) on In s. Also shown in Figure 10 is the dependence of the mean number of contacts (N,) between the point masses and the aggregate during each collision for clusters of size s obtained from simulations carried out using models I, 11, and 111. These results were obtained for ballistic aggregates ( D N 1.95) by using clusters containing up to 2000 particles. The dependence of N , on s is indistinguishable from the dependence of sul/u on s for model I and model I1 collisions. The mean number of collisions is slightly higher for model I11 (diffuse reflection). The results shown in Figure 10 strongly indicate that a/s and N J s ) behave in the same way and must have the same asymptotic behavior. For the "detailed balance" specular scattering (model I) and cosine law (model 11), the results shown in Figure 10 are consistent with the idea that u/s and N,(s) are exactly the same. The results shown in Figures 2,3, and 10 are consistent with a constant value for these quantities in the limits m or a logarithmic dependence on the aggregate size. On the basis of these simulations alone, we cannot distinguish between these two possibilities. However, it does seem to be possible to eliminate a power law dependence of either u / s or N,(s) on s for the asymptotically transparent ballistic aggregates. Similar results were obtained for the reaction-limited aggregation clusters (D 2.09) and the ballistic aggregates with restructuring ( D N 2.12). In these cases the clusters are asymptotically opaque so that u(s) sziDand N,(s)
" J 0 '0
20
40
60
80
e
100
120
140
160
180
Figure 9. Distribution of collision angles for aggregates with fractal dimensionalities greater than 2.0. Parts a (top) and b
(middle)show results for specular and random diffuse collisions, respectively, with reaction-limitedaggregates (D= 2.09). Figure 9c (bottom) shows results for model I collisions and ballistic aggregates with restructuring. In all three cases the distributions N ( 0 ) and N(0)lsin (0) are shown for clusters in the size range 1800 5 s
< 2000 particles.
teractions are important in a number of areas including evolution of the primordial solar n e b ~ l a ,air ~ ~p ~, l~l u~t i o n , ~ and combustion processes.46 An important interaction between highly dispersed particulate systems and gases is the adsorption of gas molecules onto the particle surfaces. In the small accommodation coefficient limit, where many molecule-particle contacts are required before adsorption occurs, the rate of adsorption by particle aggregates is decreased because the collision cross section for the aggregate is smaller than the sum of the cross sections (45) Weidenechilling, S., preprint, 1988. (46) Sampson, R. J.;Mulholland, G. W.; Gentry, J. W. Langmuir 1987, 3, 212.
-
s(l-(2/D))*
-
