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Calculation of the fractal dimension of aggregates
Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203 www.elsevier.com/locate/colsurfa
Calculation of the fractal dimension of aggregates Lech Gmachowski Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland Received 13 August 2001; accepted 6 June 2002
Abstract Simple determination of the fractal dimension as the exponent in the mass-radius relation is possible only when the structure of growing aggregates is scale-invariant. If it is not the case, this method produces artifacts even if the slope in a log /log plot is nearly constant. It is shown that the difference between the fractal dimension and that determined from the slope is significant for diffusion limited particle /cluster aggregates. The fractal dimension of a given aggregate can be determined from its mass and radius of gyration using the proposed fractal dimension dependence of the structural coefficient, the prefactor in mass-radius relation. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Fractal dimension; Mass-radius relation; Structural coefficient; Aggregates; Particle /cluster aggregation
1. Introduction Fractal dimension of an aggregate determines its hydrodynamic size and hence the translational friction coefficient, which characterizes the transport properties such as sedimentation and diffusion coefficients. Proper characterization of aggregates requires the experimental determination or calculation of the fractal dimension. Let us consider an aggregate covered with the sets of spheres of decreasing size [1]. If the aggregate has a self-similar structure, the number of spheres needed to cover the aggregate times an adequately chosen power of their radius N(r) × rD does not depend on the radius of covering spheres and expresses a quantity proportional to the Hausdorff measure [2] E-mail address: [email protected] (L. Gmachowski).
MH 8N(r) × rD
(1)
Since the number of spheres will decrease as a negative power of their size, the Hausdorff measure can serve to determine the exponent D , which is the fractal dimension. The determination of fractal dimension using the Hausdorff measure is possible for sufficiently large aggregates, for which the fractal structure can be detected by geometrical analysis of the arrangement of constituent particles, as discussed by Gruy [3]. Under a certain number of constituents, the fractal properties of aggregates are manifested via the dependence of their dynamic properties on the number of primary particles, which was analyzed by Takayasu and Galembeck [4] to confirm the validity of the fractal description of oligomers down to the monomer.
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L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
The procedure with the Hausdorff measure can be performed in the range of radii from the radius R of the sphere circumscribed on the aggregate down to the radius R1 of the smallest sphere containing one primary particle, for which the object still has the fractal character. From Eq. (1) one gets 1 × RD i × RD 1
(2)
where i is the number of the constituent particles in an aggregate. Let us introduce the hydrodynamic radius of aggregate r , which is the radius of an impermeable sphere of the same mass having the same dynamic properties. It is proportional to the radius of aggregate R and converges to the primary particle radius a for the number of constituent particles equal to unity [4,5]. Hence a R1
r
(3)
R
With Eq. (2) one gets the equation i
D r R
×
D R a
(4)
which is the size dependence of the aggregate mass, known as the mass-radius relation. It describes not only one fractal object of a given number of constituent particles and fractal dimension, for which the Hausdorff measure can be determined. It is a general dependence if the fractal dimension dependence of the normalized hydrodynamic radius is known. It makes it possible to calculate the fractal dimension of an aggregate from its mass and size. It is effective for both large aggregates and oligomers [6]. Eq. (4) can be also used to analyze aggregates of different mass and the same fractal dimension, assuming that the structure of growing aggregate is scale-invariant. In such cases the mass-radius relation represent the proportionality between the number of primary particles i and a power of the radius. Introducing the radius of gyration Rg ; often used to analyze the aggregate structure, which is proportional [7] to the aggregate radius
R Rg
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D D
(5)
one gets the proportionality in the form i 8RD g
(6)
which makes it possible to determine the fractal dimension D as the slope of the straight line representing the mass-radius relation in a log /log plot. If the aggregates are not scale-invariant, a prefactor kg appears, known as the structural coefficient, which is not a numerical constant. The structural coefficient changes with the structure of growing aggregate, represented by the fractal dimension. The fractal dimension is no longer the slope in a log /log plot, even if the line is nearly straight. The complete form of the massradius relation reads D Rg i kg (7) a when the aggregate size is represented by the radius of gyration. The purpose of this paper is to present a method which makes it possible to calculate the fractal dimension of an aggregate from its mass and size. This method decides whether the structure of growing aggregate is scale-invariant and when the method of the slope of the straight line representing the mass-radius relation in a log /log plot can be used to determine the fractal dimension.
2. Fractal dimension of growing aggregate In cluster /cluster aggregation (CCA), when two identical clusters of any mass take part in each aggregation act, the fractal dimension of a growing aggregate is practically constant at each stage of the process [8], due to scale-invariance. For other types of aggregation this may not be the case. Mandelbrot [9] established that the structure of aggregates growing by particle /cluster aggregation (PCA) sequence are definitely not scaleinvariant, becoming increasingly compact. For such aggregates the fractal dimension stabilizes
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
for large mass. The majority of aggregation phenomena, however, involve collisions of different clusters and primary particles with different sequences, producing aggregates of fractal dimension changing with the aggregate mass. Let us analyze the logarithmic form of the mass-radius relation R ln i ln kg D ln g (8) a By differentiating d ln i d ln kg d ln D d × R R R d ln D d ln g d ln g d ln g a a a R D ln g a D R d ln kg d ln D D ln g × R a d ln D d ln g a
(9)
one gets the slope of the tangent to the mass-radius relation D R d ln kg Dslope D ln g a d ln D ×
d ln D R d ln g a
199
gets from Eqs. (4), (7) and (11) ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D D 2 1:56 1:728 0:228 kg 2 D=2 2D × D
(12)
This dependence is depicted in Fig. 1, showing the increase of structural coefficient with the fractal dimension. The corresponding derivative in Eq. (10) is thus positive. If the fractal dimension grows as the aggregation proceeds, the remaining derivative is also positive. This means that the slope of the tangent to the mass-radius relation overestimates the fractal dimension.
3. Simulation of diffusion-limited particle /cluster aggregation To simulate a process of aggregation, the aggregation act equation is used, derived [5,10] using the concept of the collision radii of the approaching aggregates. Here a short derivation is presented. The collision radius of an aggregate is taken as its contribution to the radius of the
(10)
To estimate the slope, let as consider the massradius relation [5] based on the aggregate radius and the hydrodynamic radius (Eq. (4)) with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r D 2 1:56 1:728 0:228 (11) R 2 which, apart from the previous verification by the structure of aggregates composed of any number of constituents [6] and aggregation kinetics [5,10,11], has been recently confirmed by obtaining the same values of fractal dimension of aggregates by a method connecting the aggregate structure with its settling behavior and by the light scattering technique [12]. Employing the relation (5) between the radius and radius of gyration, one
Fig. 1. Fractal dimension dependence of the structural coefficient.
200
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
resulting aggregate. R Rci Rcj
Rci ri
ri
Rcj rj
rj
(13)
The collision radius normalized by the hydrodynamic radius is dependent on the character of the lines followed by the colliding particles and aggregates. So it is constant for a given mechanism of aggregation, independently of aggregation sequence. Hence R
Rc (ri rj ) r
(14)
The effectiveness of such formulation of the aggregation act equation has been confirmed [5] by different sequences of aggregation occurring in spaces of different dimensions. With the use of the mass-radius relation (4) for the constituents and resulting aggregate D r ri ai1=Di ; RD aD (i j) (15) R one gets the following equation describing an act of aggregation D D Rc D r a (ij) (ai1=Di aj 1=Dj )D [i j R r D r R D (i1=Di j 1=Dj )D (16) r Rc For an advanced stage of PCA the growing aggregate is much larger than the added primary particle, and the fractal dimension is close to the asymptotic value. The denominator of the fraction is equal to the numerator calculated for the asymptotic value of fractal dimension, which is described by a mean-field theory [13]. For threedimensional space it reads D 2
1 3 2 2 Dw
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 1:56 1:228 0:228 Rc 2 Dw r
(17)
where Dw is the particle trajectory fractal dimension. Substituting in Eq. (11) one gets
(18)
So the equation describing an aggregation act in three-dimensional space has the following form [14] ij
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 2 1:56 1:728 0:228 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 0:228 1:56 1:228 2 Dw
D
(i1=Di j 1=Dj )D (19) It makes it possible to calculate the fractal dimension of a new aggregate provided the masses and fractal dimensions of the joining clusters are known. The aggregation act equation gives a constant value of fractal dimension for hierarchical CCA, when two identical CCA clusters of any mass take part in each aggregation act, equal to that calculated for aggregate of two primary particles. This is in accordance with the self-similar structure of such aggregates. For diffusion-limited mechanism, Dw is 2. For particle /cluster sequence the mass of second cluster j is taken 1. Starting from i /1, the fractal dimension Di1 of growing aggregate was calculated after each addition of primary particle using the following equation (i 1)1=Di1 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 1:080 1:56 1:728 i1 0:228 2 (i1=Di 1)
(20)
The results are presented in Fig. 2. Then the values of normalized radius of gyration were calculated from Eqs. (7) and (12), corresponding to increasing mass and fractal dimension of aggregate. The results are depicted in Fig. 3. Besides, an equation approximating the mass-radius relation obtained Brasil at al. [15] in
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
201
method dealt with the consecutive addition of the primary particle to the growing aggregate, assuming uniform distributions of the point and orientation of attachment, such that Eq. (7) was satisfied for prescribed values of D and kg :/
4. Discussion and conclusions
Fig. 2. Aggregation number dependence of the fractal dimension of aggregate growing by diffusion-limited P /C mechanism.
Fig. 3. Mass-radius relation for diffusion-limited P /C aggregates (solid line). The dotted line represents constant slope of 2.5. Thick short line is a graphical representation of the massradius relation for numerically simulated aggregates [15].
simulations of P /C aggregates 2:75 Rg i 0:51 × a
(21)
is also represented in Fig. 3. The simulation
The presented model made it possible to simulate diffusion-limited PCA. It is shown in Fig. 2, how the fractal dimension increases with the aggregation number from a value characteristic for hierarchical CCA to the asymptotic value of 2.5. The slope of the calculated mass-radius relation, drawn in Fig. 3, increases initially and then decreases to the asymptotic value of the fractal dimension, according to the tendency observed by Tolman and Meakin [16]. They become close to each other for very large aggregation numbers. The slope of the mass-radius relation for lower aggregation numbers changes slowly, so the considerable parts of the line can be regarded to have nearly constant slope. This slope, however does not correspond to the fractal dimension. The line represented by Eq. (21) is very close to the corresponding part of the mass-radius relation in both the slope and location. Moreover, the results of Brasil at al. [15] agree with those reported in the literature (see Wu and Friedlander [17]) for diffusion-limited aggregates, obtained by both P /C and C /C sequences. Therefore the method used to obtain Eq. (21) may be regarded as modeling diffusion-limited P /C aggregates. The range of aggregation number, in which Eq. (21) approximates the mass-radius relation, is from about 10 to 1000. According to the line depicted in Fig. 2, this is the range of maximum increase of fractal dimension, which changes from 1.9 to 2.3. Nevertheless, the slope of the corresponding part of mass-radius relation drawn in Fig. 3, which is very close to the straight line representing Eq. (21), is about 2.8. Nearly constant slope of the massradius relation is thus accompanied by sudden increase in fractal dimension. Moreover, the slope is much higher than the fractal dimension and even than its asymptotic value.
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Fig. 4. Aggregation number dependencies of fractal dimensions for P /C diffusion-limited aggregation (DLA) and for P /C aggregates simulated by Brasil et al. [15].
The fractal dimension of aggregates following Eq. (21) can be calculated without the PCA-model presented. This was done by determining the values of normalized radius of gyration for chosen aggregation numbers from Eq. (21), and then by calculation of the fractal dimension from Eqs. (7) and (12). As shown in Fig. 4, the calculated values are close to that for P /C diffusion-limited aggregates but far from the exponent 2.75 in Eq. (21). So if the structure of a growing aggregate is not scale-invariant, the method of calculation of the fractal dimension based on Eqs. (7) and (12) seems to be the only reliable one, except for that based on the Hausdorff measure, given by Eq. (1). The method usually used, assumes the scale-invariance of the structure of growing aggregate and approximates a sector of mass-radius relation, drawn in a log /log plot, by a straight line, the slope of which is identified with the fractal dimension. This method, however, produces artifacts, giving highly overestimated values of fractal dimension for P /C aggregates if not large enough, as has been shown in this paper. If the proposed method gives one value of fractal dimension for a growing aggregate, this means that slope of the mass-radius relation is equal to the fractal dimension, according to Eq. (10). The both methods are equivalent for similarity growth of aggregates.
The proposed method of calculation of the fractal dimension of aggregates makes it possible to model the structure of growing aggregate or to interpret the results obtained in simulations or experiment. It decides if the structure of growing aggregate is scale-invariant and hence whether the method of the slope of the straight line representing the mass-radius relation in a log /log plot can be used.
Appendix A: Nomenclature a primary particle radius (m) D fractal dimension (/) /D slope/ slope of the tangent to the mass-radius relation Dw particle trajectory fractal dimension (/) i number of primary particles in an aggregate (/) j number of primary particles in an aggregate (/) /k / structural coefficient, prefactor in massg radius relation ( /) /M / Hausdorff measure (/mD )/ H /N(r)/ number of spheres covering the aggregate (/)
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
r R Rc /R / g /R / 1 r/
/
hydrodynamic (dynamic) radius of aggregate (m) radius of aggregate (m) collision radius (m) aggregate radius of gyration (m) radius of smallest covering sphere containing one primary particle (m) radius of covering sphere (m)
References [1] [2] [3] [4]
J. Feder, Fractals, Plemum Press, New York, 1988. F. Hausdorff, Math. Annal. 79 (1919) 157. F. Gruy, J. Colloid Interface Sci. 237 (2001) 28. M.M. Takayasu, F. Galembeck, J. Colloid Interface Sci. 202 (1998) 84. [5] L. Gmachowski, Colloids Surfaces A: Physicochem. Eng. Aspects 170 (2000) 209.
203
[6] L. Gmachowski, Eur. Biophys. J. Biophy. 30 (2001) 453. [7] W. Hess, H.L. Frisch, R. Klein, Z. Phys. B-Condensed Matter 64 (1986) 65. [8] M. Kolb, R. Botet, R. Jullien, Phys. Rev. Lett. 51 (1983) 2026. [9] B.B. Mandelbrot, Phys. A 191 (1992) 95. [10] L. Gmachowski, Colloids Surfaces A: Physicochem. Eng. Aspects 176 (2001) 151. [11] L. Gmachowski, Colloids Surfaces A: Physicochem. Eng. Aspects 207 (2002) 271. [12] P. Tang, J. Greenwood, J.A. Raper, J. Colloid Interface Sci. 247 (2002) 210. [13] K. Honda, H. Toyoki, M. Matsushita, J. Phys. Soc. Jpn. 55 (1986) 707. [14] L. Gmachowski, Chem. Ing. Tech. 73 (2001) 87. [15] A.M. Brasil, T.L. Farias, M.G. Carvalho, U.O. Koylu, Aerosol Sci. 32 (2001) 489. [16] S. Tolman, P. Meakin, Phys. Rev. A 40 (1989) 428. [17] M. Wu, S.K. Friedlander, J. Colloid Interface Sci. 159 (1993) 246.
Calculation of the fractal dimension of aggregates Lech Gmachowski Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland Received 13 August 2001; accepted 6 June 2002
Abstract Simple determination of the fractal dimension as the exponent in the mass-radius relation is possible only when the structure of growing aggregates is scale-invariant. If it is not the case, this method produces artifacts even if the slope in a log /log plot is nearly constant. It is shown that the difference between the fractal dimension and that determined from the slope is significant for diffusion limited particle /cluster aggregates. The fractal dimension of a given aggregate can be determined from its mass and radius of gyration using the proposed fractal dimension dependence of the structural coefficient, the prefactor in mass-radius relation. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Fractal dimension; Mass-radius relation; Structural coefficient; Aggregates; Particle /cluster aggregation
1. Introduction Fractal dimension of an aggregate determines its hydrodynamic size and hence the translational friction coefficient, which characterizes the transport properties such as sedimentation and diffusion coefficients. Proper characterization of aggregates requires the experimental determination or calculation of the fractal dimension. Let us consider an aggregate covered with the sets of spheres of decreasing size [1]. If the aggregate has a self-similar structure, the number of spheres needed to cover the aggregate times an adequately chosen power of their radius N(r) × rD does not depend on the radius of covering spheres and expresses a quantity proportional to the Hausdorff measure [2] E-mail address: [email protected] (L. Gmachowski).
MH 8N(r) × rD
(1)
Since the number of spheres will decrease as a negative power of their size, the Hausdorff measure can serve to determine the exponent D , which is the fractal dimension. The determination of fractal dimension using the Hausdorff measure is possible for sufficiently large aggregates, for which the fractal structure can be detected by geometrical analysis of the arrangement of constituent particles, as discussed by Gruy [3]. Under a certain number of constituents, the fractal properties of aggregates are manifested via the dependence of their dynamic properties on the number of primary particles, which was analyzed by Takayasu and Galembeck [4] to confirm the validity of the fractal description of oligomers down to the monomer.
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 2 7 8 - 9
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L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
The procedure with the Hausdorff measure can be performed in the range of radii from the radius R of the sphere circumscribed on the aggregate down to the radius R1 of the smallest sphere containing one primary particle, for which the object still has the fractal character. From Eq. (1) one gets 1 × RD i × RD 1
(2)
where i is the number of the constituent particles in an aggregate. Let us introduce the hydrodynamic radius of aggregate r , which is the radius of an impermeable sphere of the same mass having the same dynamic properties. It is proportional to the radius of aggregate R and converges to the primary particle radius a for the number of constituent particles equal to unity [4,5]. Hence a R1
r
(3)
R
With Eq. (2) one gets the equation i
D r R
×
D R a
(4)
which is the size dependence of the aggregate mass, known as the mass-radius relation. It describes not only one fractal object of a given number of constituent particles and fractal dimension, for which the Hausdorff measure can be determined. It is a general dependence if the fractal dimension dependence of the normalized hydrodynamic radius is known. It makes it possible to calculate the fractal dimension of an aggregate from its mass and size. It is effective for both large aggregates and oligomers [6]. Eq. (4) can be also used to analyze aggregates of different mass and the same fractal dimension, assuming that the structure of growing aggregate is scale-invariant. In such cases the mass-radius relation represent the proportionality between the number of primary particles i and a power of the radius. Introducing the radius of gyration Rg ; often used to analyze the aggregate structure, which is proportional [7] to the aggregate radius
R Rg
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D D
(5)
one gets the proportionality in the form i 8RD g
(6)
which makes it possible to determine the fractal dimension D as the slope of the straight line representing the mass-radius relation in a log /log plot. If the aggregates are not scale-invariant, a prefactor kg appears, known as the structural coefficient, which is not a numerical constant. The structural coefficient changes with the structure of growing aggregate, represented by the fractal dimension. The fractal dimension is no longer the slope in a log /log plot, even if the line is nearly straight. The complete form of the massradius relation reads D Rg i kg (7) a when the aggregate size is represented by the radius of gyration. The purpose of this paper is to present a method which makes it possible to calculate the fractal dimension of an aggregate from its mass and size. This method decides whether the structure of growing aggregate is scale-invariant and when the method of the slope of the straight line representing the mass-radius relation in a log /log plot can be used to determine the fractal dimension.
2. Fractal dimension of growing aggregate In cluster /cluster aggregation (CCA), when two identical clusters of any mass take part in each aggregation act, the fractal dimension of a growing aggregate is practically constant at each stage of the process [8], due to scale-invariance. For other types of aggregation this may not be the case. Mandelbrot [9] established that the structure of aggregates growing by particle /cluster aggregation (PCA) sequence are definitely not scaleinvariant, becoming increasingly compact. For such aggregates the fractal dimension stabilizes
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
for large mass. The majority of aggregation phenomena, however, involve collisions of different clusters and primary particles with different sequences, producing aggregates of fractal dimension changing with the aggregate mass. Let us analyze the logarithmic form of the mass-radius relation R ln i ln kg D ln g (8) a By differentiating d ln i d ln kg d ln D d × R R R d ln D d ln g d ln g d ln g a a a R D ln g a D R d ln kg d ln D D ln g × R a d ln D d ln g a
(9)
one gets the slope of the tangent to the mass-radius relation D R d ln kg Dslope D ln g a d ln D ×
d ln D R d ln g a
199
gets from Eqs. (4), (7) and (11) ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D D 2 1:56 1:728 0:228 kg 2 D=2 2D × D
(12)
This dependence is depicted in Fig. 1, showing the increase of structural coefficient with the fractal dimension. The corresponding derivative in Eq. (10) is thus positive. If the fractal dimension grows as the aggregation proceeds, the remaining derivative is also positive. This means that the slope of the tangent to the mass-radius relation overestimates the fractal dimension.
3. Simulation of diffusion-limited particle /cluster aggregation To simulate a process of aggregation, the aggregation act equation is used, derived [5,10] using the concept of the collision radii of the approaching aggregates. Here a short derivation is presented. The collision radius of an aggregate is taken as its contribution to the radius of the
(10)
To estimate the slope, let as consider the massradius relation [5] based on the aggregate radius and the hydrodynamic radius (Eq. (4)) with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r D 2 1:56 1:728 0:228 (11) R 2 which, apart from the previous verification by the structure of aggregates composed of any number of constituents [6] and aggregation kinetics [5,10,11], has been recently confirmed by obtaining the same values of fractal dimension of aggregates by a method connecting the aggregate structure with its settling behavior and by the light scattering technique [12]. Employing the relation (5) between the radius and radius of gyration, one
Fig. 1. Fractal dimension dependence of the structural coefficient.
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L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
resulting aggregate. R Rci Rcj
Rci ri
ri
Rcj rj
rj
(13)
The collision radius normalized by the hydrodynamic radius is dependent on the character of the lines followed by the colliding particles and aggregates. So it is constant for a given mechanism of aggregation, independently of aggregation sequence. Hence R
Rc (ri rj ) r
(14)
The effectiveness of such formulation of the aggregation act equation has been confirmed [5] by different sequences of aggregation occurring in spaces of different dimensions. With the use of the mass-radius relation (4) for the constituents and resulting aggregate D r ri ai1=Di ; RD aD (i j) (15) R one gets the following equation describing an act of aggregation D D Rc D r a (ij) (ai1=Di aj 1=Dj )D [i j R r D r R D (i1=Di j 1=Dj )D (16) r Rc For an advanced stage of PCA the growing aggregate is much larger than the added primary particle, and the fractal dimension is close to the asymptotic value. The denominator of the fraction is equal to the numerator calculated for the asymptotic value of fractal dimension, which is described by a mean-field theory [13]. For threedimensional space it reads D 2
1 3 2 2 Dw
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 1:56 1:228 0:228 Rc 2 Dw r
(17)
where Dw is the particle trajectory fractal dimension. Substituting in Eq. (11) one gets
(18)
So the equation describing an aggregation act in three-dimensional space has the following form [14] ij
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 2 1:56 1:728 0:228 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 0:228 1:56 1:228 2 Dw
D
(i1=Di j 1=Dj )D (19) It makes it possible to calculate the fractal dimension of a new aggregate provided the masses and fractal dimensions of the joining clusters are known. The aggregation act equation gives a constant value of fractal dimension for hierarchical CCA, when two identical CCA clusters of any mass take part in each aggregation act, equal to that calculated for aggregate of two primary particles. This is in accordance with the self-similar structure of such aggregates. For diffusion-limited mechanism, Dw is 2. For particle /cluster sequence the mass of second cluster j is taken 1. Starting from i /1, the fractal dimension Di1 of growing aggregate was calculated after each addition of primary particle using the following equation (i 1)1=Di1 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 1:080 1:56 1:728 i1 0:228 2 (i1=Di 1)
(20)
The results are presented in Fig. 2. Then the values of normalized radius of gyration were calculated from Eqs. (7) and (12), corresponding to increasing mass and fractal dimension of aggregate. The results are depicted in Fig. 3. Besides, an equation approximating the mass-radius relation obtained Brasil at al. [15] in
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 211 (2002) 197 /203
201
method dealt with the consecutive addition of the primary particle to the growing aggregate, assuming uniform distributions of the point and orientation of attachment, such that Eq. (7) was satisfied for prescribed values of D and kg :/
4. Discussion and conclusions
Fig. 2. Aggregation number dependence of the fractal dimension of aggregate growing by diffusion-limited P /C mechanism.
Fig. 3. Mass-radius relation for diffusion-limited P /C aggregates (solid line). The dotted line represents constant slope of 2.5. Thick short line is a graphical representation of the massradius relation for numerically simulated aggregates [15].
simulations of P /C aggregates 2:75 Rg i 0:51 × a
(21)
is also represented in Fig. 3. The simulation
The presented model made it possible to simulate diffusion-limited PCA. It is shown in Fig. 2, how the fractal dimension increases with the aggregation number from a value characteristic for hierarchical CCA to the asymptotic value of 2.5. The slope of the calculated mass-radius relation, drawn in Fig. 3, increases initially and then decreases to the asymptotic value of the fractal dimension, according to the tendency observed by Tolman and Meakin [16]. They become close to each other for very large aggregation numbers. The slope of the mass-radius relation for lower aggregation numbers changes slowly, so the considerable parts of the line can be regarded to have nearly constant slope. This slope, however does not correspond to the fractal dimension. The line represented by Eq. (21) is very close to the corresponding part of the mass-radius relation in both the slope and location. Moreover, the results of Brasil at al. [15] agree with those reported in the literature (see Wu and Friedlander [17]) for diffusion-limited aggregates, obtained by both P /C and C /C sequences. Therefore the method used to obtain Eq. (21) may be regarded as modeling diffusion-limited P /C aggregates. The range of aggregation number, in which Eq. (21) approximates the mass-radius relation, is from about 10 to 1000. According to the line depicted in Fig. 2, this is the range of maximum increase of fractal dimension, which changes from 1.9 to 2.3. Nevertheless, the slope of the corresponding part of mass-radius relation drawn in Fig. 3, which is very close to the straight line representing Eq. (21), is about 2.8. Nearly constant slope of the massradius relation is thus accompanied by sudden increase in fractal dimension. Moreover, the slope is much higher than the fractal dimension and even than its asymptotic value.
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Fig. 4. Aggregation number dependencies of fractal dimensions for P /C diffusion-limited aggregation (DLA) and for P /C aggregates simulated by Brasil et al. [15].
The fractal dimension of aggregates following Eq. (21) can be calculated without the PCA-model presented. This was done by determining the values of normalized radius of gyration for chosen aggregation numbers from Eq. (21), and then by calculation of the fractal dimension from Eqs. (7) and (12). As shown in Fig. 4, the calculated values are close to that for P /C diffusion-limited aggregates but far from the exponent 2.75 in Eq. (21). So if the structure of a growing aggregate is not scale-invariant, the method of calculation of the fractal dimension based on Eqs. (7) and (12) seems to be the only reliable one, except for that based on the Hausdorff measure, given by Eq. (1). The method usually used, assumes the scale-invariance of the structure of growing aggregate and approximates a sector of mass-radius relation, drawn in a log /log plot, by a straight line, the slope of which is identified with the fractal dimension. This method, however, produces artifacts, giving highly overestimated values of fractal dimension for P /C aggregates if not large enough, as has been shown in this paper. If the proposed method gives one value of fractal dimension for a growing aggregate, this means that slope of the mass-radius relation is equal to the fractal dimension, according to Eq. (10). The both methods are equivalent for similarity growth of aggregates.
The proposed method of calculation of the fractal dimension of aggregates makes it possible to model the structure of growing aggregate or to interpret the results obtained in simulations or experiment. It decides if the structure of growing aggregate is scale-invariant and hence whether the method of the slope of the straight line representing the mass-radius relation in a log /log plot can be used.
Appendix A: Nomenclature a primary particle radius (m) D fractal dimension (/) /D slope/ slope of the tangent to the mass-radius relation Dw particle trajectory fractal dimension (/) i number of primary particles in an aggregate (/) j number of primary particles in an aggregate (/) /k / structural coefficient, prefactor in massg radius relation ( /) /M / Hausdorff measure (/mD )/ H /N(r)/ number of spheres covering the aggregate (/)
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r R Rc /R / g /R / 1 r/
/
hydrodynamic (dynamic) radius of aggregate (m) radius of aggregate (m) collision radius (m) aggregate radius of gyration (m) radius of smallest covering sphere containing one primary particle (m) radius of covering sphere (m)
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