Relation between coordination number and fractal dimension of aerosol agglomerates

Pergamon

J. Aerosol Sci. VoL 28, Suppl. 1, pp. $765-$766, 1997 ©1997 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0021-S502/97 $17.00+0.00

PII:S0021-8502(97)00429-1

RELATION BETWEEN COORDINATION NUMBER AND FRACTAL DIMENSION OF AEROSOL AGGLOMERATES Alfred P. Weber 1'2 and Sheldon K. Friedlander I lChemical Engineering Department, University of California, Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90024, USA. 2 Institut flir Mechanische Verfahrenstechnik und Mechanik der Universit~it Karlsruhe (TH), Kaiserstr. 12, 76128 Karlsruhe, Germany KEYWORDS coordination number, agglomerates, fractal dimension, restructuring The average number of contacts of a particle in an agglomerated structure, in the following defined as coordination number, c~, is of key importance for many physical properties of agglomerates such as mechanical stability or electrical conductivity. For agglomerates formed in certain growth processes there may be, on average, a correlation between cN and the fractal dimension, D~-• This correlation was investigated in computer simulations and experiments. Using Monte Carlo type simulations, clusters of primary particles have been generated with diffusion limited aggregation (DLA), chemically limited aggregation (CLA) and ballistic aggregation. It was found that the lattice type (cubic/hexagonal) for the random walk did affected c N but not D r , while with decreasing sticking probability c~ and D r increased. A lower sticking probabiliiy results in a higher mobility of the diffusing species within the agglomerates leading to denser clusters and more nearest neighbors. The coordination number is also influenced by the number of primary particles per cluster, N, as shown in Fig. 1. For N < 50, c N shows a steep increase with N. At larger values of N, c~ approaches a asymptotic value which depends on the agglomeration mechanism. However, the asymptotic behavior is observed for DLA (D[ = 2.5) and ballistic (D¢ = 3) agglomeration. The constant fraction of singly bonded primary partacles for a growing cluster indicates that the cluster growth does not with increasing preference occur at the agglomerate tips. 4 3,6

~,

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~) q

2,8 2,6 2,4 ~

>~ I

eDLA (with singly bonded) IIDLA (without singly bonded) &BALLISTIC (with singly bonded) OBALLISTIC (without singly bonded)

2.2 2 0

100

200

300

n u m b e r o f particles

Fig. 1:

400

500

In c l u s t e r

Influence of cluster size on coordination number for DLA and ballistic particle-cluster growth.

For the experimental studies, agglomerates of silver particles generated by spark discharge (Weber e t al., 1996) or by an evaporation-condensation process (Weber and Friedlander, 1997) were used. The fractal dimension was determined with in situ methods. The coordination number was obtained from the analysis of TEM micrographs. For agglomerates with D t > 2.4 this counting method failed due to shadowing effects. However, for agglomerates with D: = 3 a

$765

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Abstracts of the 1997 European Aerosol Conference

general, purely geometrical correlation between coordination number and solid volume fraction (-- 1-porosity) was employed to obtain cN from density measurements. In the process of restructuring of nanometer agglomerates, as induced by tempering, two restructuring pathways were identified (Fig. 2). Agglomerates with low fractal dimension become more compact when heated resulting in higher fractal dimension and higher coordination number as indicated by an arrow in Fig. 2. Each point represents a certain structure on the restructuring pathway. The kinetics of restructuring depend on the time needed to move along the curve. On the other hand, agglomerates with D e = 3, undergo internal rearrangement towards a more compact structure leading to higher values of c N. In both cases, the driving force for restructuring is the minimization of the free energy. Experimental and simulation results for the correlation between c~ and D r are summarized in Fig. 2 including one point from Dimon et al. (1986) who measured fractar dimension and coordination number with small angle x-ray scattering for chemically limited particle-cluster aggregation of gold particles. 12

_

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I

@silver

agglomerates

o silver

agglomerates

with

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• simulation

8

¢: ¢: O (= c .m 0 0

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(CLA)

,I g o l d a g g l o m e r a t e s ( D i m o n et el., 1986) 6

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X simulation

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(OLA)

(ballistlc)

~

~

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oT

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1

J

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, 2

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fractal

Fig. 2:

,

2,5

dimension

Relation between coordination number and fractal dimension including experimental and simulation results.

The chemically limited particle-cluster aggregation shows results qualitatively similar to the restructuring pathway starting from low density clusters. When the sticking probability goes to zero, the fractal dimension approaches 3 and a steep increase of the coordination number occurs. This similarity may be understood in the way that for low sticking probabilities the particles are highly mobile within the cluster which is similar to the situation in the restructuring process when the particles overcome the activation energy for sliding or rolling. However, restructuring is driven by the minimization of the free energy of the cluster (neglecting entropy changes) which is not implemented in the CLA model. REFERENCES Dimon, P., Sinha, S.K., Weitz, D.A., Safinya, C.R., Smith, G.S., Varady, W.A., Lindsay, H.M. (1986), Structure of aggregated gold colloids, Phys. Rev. Lett. 57, 595-598. Weber, A.P., Baltensperger, U., G~iggeler, H.W., Schmidt-Ott, A. (1996), In situ characterization and structure modification of agglomerated aerosol particles, J. Aerosol Sci. 27(6), 915-929. Weber, A.P., Friedlander, S.K. (1997) In situ determination of the activation energy for restructuring of nanometer aerosol agglomerates, J, Aerosol Sci. 28(2), 179-192.