- Email: [email protected]
25.P.27 Determination of the power-law prefactor ϵ for fractal agglomerates
J, Aerosol Sci., Vol. 25, Suppl. I, pp. $403-$4~, 1994
Copyril~ht~)1994 Elsevier Science lad
25 P 27
Printed in Great Britain. All rishts reserved 0021-8502/94 $7.00 + 0.00
DETERMINATION OF THE POWER-LAW PREFACTOR FOR FRACTAL AGGLOMERATES S. Nyeki and I. Colbeck, Inst. Env. Research, Chemistry Dept., University of Essex, Colchester/Essex, CO4 3SQ, UK. KEYWORDS Fractal dimension, carbonaceous aerosol, aerosol restructuring INTRODUCTION The mass fractal dimension D of agglomerates may be derived using the well-known power-law equation (Mountain and Mulholland, 1988):N = U(rc/ro)D....... (1) where N is the number of primary spherules, r o is the radius of gyration, r o is the radius of a primary spherule and e is a dimensionless prefactor. The calculation of D is a straightforward task when a In-In plot of N versus (rJro) is performed for an ensemble of particulates. However uncertainty still lies in value of the prefactor which is influenced by the definition of the diameter, the fractal dimension and the formation mechanism. Few experimental values of 8 in the transition/continuum regime have been reported. Wiltzius (1987) found 6 = 1 (continuum regime) and Sorenson et al. (1992) gave ~ = 1.56 (transition/continuum regime), while more recent models have used e = 1.0 for the free molecular regime (Wu and Friedlander, 1992) and 6 = 0.9 for the transition regime (Rogak and Flagan, 1993). Wu and Friedlander (1993) in a recent review found that ~ - 1, for the radius of gyration and mobility diameter in the continuum regime, although values were somewhat variable and dependent on the details of the coagulation mechanism. In a recent method using a modified Millikan cell the fractal dimension (D) of single in-situ butane soot agglomerates (continuum regime), in 3 dimensional space, was measured (Nyeki and Colbeck, 1994). Simultaneous measurement of the volume equivalent diameter d w (by a photoemission technique) and the aerodynamic diameter dAE (by sedimentation) allowed D to be deduced by assuming that the mobility equivalent diameter, d ~ = 2r G and ~ = 1. Other derived parameters included the number of primary spherules in the agglomerate and the dynamic shape factor. Several stages of agglomerate restructuring to a more spherical morphology, induced by reversal of the DC electric field, were observed and at each stage D was measured (Nyeki and Colbeck, 1993). The value o l D was found in the range 1.87-2.19 where the lower limit represented minimal or no restructuring and the upper limit the value of D beyond which further restructuring was not found. RESULTS If an agglomerate at various stages of restructuring is considered then N remains constant while ~ and D are unknown and may be determined on a ln-ln plot. However due to the different degrees of agglomerate restructuring ~ is only given for an average D over a large range. Therefore by assuming that similar values of D exhibit similar degrees of restructuring Table 1 illustrates groupings of similar D, which have been predetermined using 8 = 1. Linear AS 25 S. I-B8
$403
8404
S. NYEKI and I. COLBECK
regression analysis, Fig. 1 suggests a high correlation o f > 0.98 in all cases where 6 ranges from 0.88- 1.24. Table 1. Linear regression analysis on groups of similar D for single, restructured agglomerates allowing the determination of the prefactor e. Range of D Average Nos. of Regression Analysis values D Observations Prefactor Average Correlation for e : 1 for c = 1 c D Coefficient 1.87- 1.92
1.90
4
0.88
1.92
0.99
1.97 - 2.03
2.00
4
1.24
1.97
0.99
2.06 - 2.11
2.08
6
1.20
2.08
0.99
2.14 - 2.19
2.17
4
0.99
2.17
0.98
Despite the narrow range of initial D values the small data set should be interpreted cautiously, illustrated in the proximity of points in Fig. 1. Furthermore e is sensitive to the value of D determined from regression analysis although values compare favourably to those assuming e = 1. For an estimated error in D of+/- 0.03, e varies in the range 0.73 - 1.04 respectively for the first D group. In conclusion results for e are similar to those previously reported (Wu and Friedlander, 1993) but do not allow any trend or constancy in e to be determined hence requiring further investigations. Fig. 1. Ln(N) vs. In (din/do) for single agglomerates grouped into the indicated ranges of D.
//
/
,/ ,-2
~7 ,/
TO
//; //
v ¢/ /// ,~/ "
D=1.87-1.92 D=1.97-2.03 D=2.06-2.11 D=2.14-2.19
0
.)
(:}
7
ln(dME/do) REFERENCES Nyeki S. and Colbeck I. (1993) J. Aerosol Sci. 24, $547 - $548. Nyek/S. and Colbeck I. (1994) J. Aerosol Sci., in print. Rogak S.N. and Flagan R.C. (1993) Aerosol Sci. Tech. 18, 25 - 47. Sorenson C.M., Cai J. and Lu N. (1992) Appl. Optics 31, 6457 - 6557. Wiltzius P. (1987)Phys. Rev. Letters 58, 7 1 0 - 713. Wu M.K. and Frio]lander S.K. (1992) J. Aerosol Sci. 24, 273 - 282. Wu M.K. and Fricdlander S.K. (1993) J. Colloid Int. Sci. 159, 246 - 248.
Copyril~ht~)1994 Elsevier Science lad
25 P 27
Printed in Great Britain. All rishts reserved 0021-8502/94 $7.00 + 0.00
DETERMINATION OF THE POWER-LAW PREFACTOR FOR FRACTAL AGGLOMERATES S. Nyeki and I. Colbeck, Inst. Env. Research, Chemistry Dept., University of Essex, Colchester/Essex, CO4 3SQ, UK. KEYWORDS Fractal dimension, carbonaceous aerosol, aerosol restructuring INTRODUCTION The mass fractal dimension D of agglomerates may be derived using the well-known power-law equation (Mountain and Mulholland, 1988):N = U(rc/ro)D....... (1) where N is the number of primary spherules, r o is the radius of gyration, r o is the radius of a primary spherule and e is a dimensionless prefactor. The calculation of D is a straightforward task when a In-In plot of N versus (rJro) is performed for an ensemble of particulates. However uncertainty still lies in value of the prefactor which is influenced by the definition of the diameter, the fractal dimension and the formation mechanism. Few experimental values of 8 in the transition/continuum regime have been reported. Wiltzius (1987) found 6 = 1 (continuum regime) and Sorenson et al. (1992) gave ~ = 1.56 (transition/continuum regime), while more recent models have used e = 1.0 for the free molecular regime (Wu and Friedlander, 1992) and 6 = 0.9 for the transition regime (Rogak and Flagan, 1993). Wu and Friedlander (1993) in a recent review found that ~ - 1, for the radius of gyration and mobility diameter in the continuum regime, although values were somewhat variable and dependent on the details of the coagulation mechanism. In a recent method using a modified Millikan cell the fractal dimension (D) of single in-situ butane soot agglomerates (continuum regime), in 3 dimensional space, was measured (Nyeki and Colbeck, 1994). Simultaneous measurement of the volume equivalent diameter d w (by a photoemission technique) and the aerodynamic diameter dAE (by sedimentation) allowed D to be deduced by assuming that the mobility equivalent diameter, d ~ = 2r G and ~ = 1. Other derived parameters included the number of primary spherules in the agglomerate and the dynamic shape factor. Several stages of agglomerate restructuring to a more spherical morphology, induced by reversal of the DC electric field, were observed and at each stage D was measured (Nyeki and Colbeck, 1993). The value o l D was found in the range 1.87-2.19 where the lower limit represented minimal or no restructuring and the upper limit the value of D beyond which further restructuring was not found. RESULTS If an agglomerate at various stages of restructuring is considered then N remains constant while ~ and D are unknown and may be determined on a ln-ln plot. However due to the different degrees of agglomerate restructuring ~ is only given for an average D over a large range. Therefore by assuming that similar values of D exhibit similar degrees of restructuring Table 1 illustrates groupings of similar D, which have been predetermined using 8 = 1. Linear AS 25 S. I-B8
$403
8404
S. NYEKI and I. COLBECK
regression analysis, Fig. 1 suggests a high correlation o f > 0.98 in all cases where 6 ranges from 0.88- 1.24. Table 1. Linear regression analysis on groups of similar D for single, restructured agglomerates allowing the determination of the prefactor e. Range of D Average Nos. of Regression Analysis values D Observations Prefactor Average Correlation for e : 1 for c = 1 c D Coefficient 1.87- 1.92
1.90
4
0.88
1.92
0.99
1.97 - 2.03
2.00
4
1.24
1.97
0.99
2.06 - 2.11
2.08
6
1.20
2.08
0.99
2.14 - 2.19
2.17
4
0.99
2.17
0.98
Despite the narrow range of initial D values the small data set should be interpreted cautiously, illustrated in the proximity of points in Fig. 1. Furthermore e is sensitive to the value of D determined from regression analysis although values compare favourably to those assuming e = 1. For an estimated error in D of+/- 0.03, e varies in the range 0.73 - 1.04 respectively for the first D group. In conclusion results for e are similar to those previously reported (Wu and Friedlander, 1993) but do not allow any trend or constancy in e to be determined hence requiring further investigations. Fig. 1. Ln(N) vs. In (din/do) for single agglomerates grouped into the indicated ranges of D.
//
/
,/ ,-2
~7 ,/
TO
//; //
v ¢/ /// ,~/ "
D=1.87-1.92 D=1.97-2.03 D=2.06-2.11 D=2.14-2.19
0
.)
(:}
7
ln(dME/do) REFERENCES Nyeki S. and Colbeck I. (1993) J. Aerosol Sci. 24, $547 - $548. Nyek/S. and Colbeck I. (1994) J. Aerosol Sci., in print. Rogak S.N. and Flagan R.C. (1993) Aerosol Sci. Tech. 18, 25 - 47. Sorenson C.M., Cai J. and Lu N. (1992) Appl. Optics 31, 6457 - 6557. Wiltzius P. (1987)Phys. Rev. Letters 58, 7 1 0 - 713. Wu M.K. and Frio]lander S.K. (1992) J. Aerosol Sci. 24, 273 - 282. Wu M.K. and Fricdlander S.K. (1993) J. Colloid Int. Sci. 159, 246 - 248.