1274
Langmuir 1991, 7, 1274-1280
Fractal Properties of Soot Agglomerates Paul A. Bonczyk and Robert J. Hall' United Technologies Research Center, East Hartford, Connecticut 06108 Received October 11, 1990. I n Final Form: January 4, 1991 Soot particle sizing and number density measurement by laser light scattering have been reformulated to take into account nonspherical shape effecta. By assuminga morphologyconsistingof chainsof monomer spheroids, and the concepts of fractal theory, it has been possible to determine the structural parameters of such clusters from extinction and multiangle scattering measurements. This permits more realistic calculations of soot particle radiative properties and puts soot sizing/number density measurements on a better basis. Experiments performed in an ethylene-air slot burner yield credible cluster parameters using this approach. Introduction
Growth models such as DLA (diffusion limited aggregation)" or DLCA (diffusion limited cluster aggregation)12 predict such structures, and information about the growth process can be inferred from the fractal dimension. For fractal clusters with N primaryspheroids, the radius of gyration, R,, can be shown to be related to the primary spheroid radius a by13J4
Soot formation during the combustion of hydrocarbon fuels has several significant and potentially negative consequences. These include, but are not limited to, such issues as excessive heat transfer within gas turbines and pollution of the air we breathe. In order to assess these concerns quantitatively, values are required for the parameters which characterize the soot morphology. Since 2lI2Rg= aaN('ID) (1) the morphology is in general complex, the task at hand traditionally has been difficult. In many past studies, where a = 1and D is the fractal dimension, typically nonincluding number density and size measurement by laser integer. For most particulates, values of D around 2 are light scattering,' the particulates were assumed to have typical. This can be seen to be associated with a tenuous spherical shapes, with extinction and scattering given by or sparse structure through the intracluster density the well-known Mie theory.2 While there are certainly relationshipl3 situations where the assumption of sphericity is ~ a l i d , ~ ? ~ it is more typical to observe ex situ evidence of the agglomeration of spheroidal monomers in electron micrographs of collected soot.6 When, as is often the case, which is intended to apply on length scales intermediate the inferred effective radius from light scattering (typically between a and R,. The lower the value of D, the more 50-100 nm) greatly exceeds the monomer radius observed tenuous the structure is seen to be. An important formal in micrographs (typically 5-40 nm), one can reasonably restatement of eq 2 is that the two-point density-density assume that the flame particulates exist as aggregates. As correlation function (the monomer pair distribution funcwe show here, the theoretical tools now exist which make tion) it possible to bypass the inappropriate Mie theory in such situations and substitute for it a proper theory based on p 2 ( i ) = S p ( Y ) p ( Y i') d37 (3) the concept of a mass fractal.
+
Theory of Fractal Cluster Scattering Soot particulates seem to belong to a class of objects called fractals. These are objects that vary irregularly in a geometric sense from one to another but have certain statistical similarities; they are said to be "self-similar" and "scale invariant" over certain length scales. The concept of a mass fractal dimension makes it possible to concisely characterize, in a statistical sense, properties of the mass distribution such as the radius of gyration. Examples of mass fractal objects are gold colloids,6s0ot,~+3 metallic smokes,g and "fumed" silica or silica soot.10 (1)Bonczyk, P. A. Combust. Flame 1983,51,219. (2)Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969. (3)Bockhorn, H.; Fetting, F.; Heddrich, A.; Wannemacher, G. In Twentieth Symposium (International)on Combustion;The Combustion Institute: Pittsburgh, PA, 1984,p 979. (4)Harris, S.J.; Weiner, A. M. Combust. Sci. Technol. 1983,32,267. ( 5 ) Bonczyk, P. A.; Sangiovanni, J. J. Combust. Sci. Technol. 1984,36, 12.5 ---.
(6) Weitz, D. A.; Huang, J. S.; Lin, M. Y.; Sung, J. Phys. Reo. Lett.
1985,54,1416. (7)Megaridis, C. M.; Dobbins, R. A. Combust. Sci. Technol. 1990,71, 95.
0743-7463f 91f 2407-1274$02.50f 0
has a power-law form for values of r in the stated range. Specifically
-
p2(r) r-f3-D)h(r/Rg) (4) for clusters embedded in three-dimensional space. In eq 4, h is a cutoff function important for values of r approaching R,. This is usually of exponential or Gaussian form.13-15 Fractal cluster parameters can be obtained by visual, structural examination of electron micrograph images. However, with such ex situ measurements of samples, there is always the question whether the in situ properties are (8)Samson, R. J.; Mulholland, G. W.; Gentry, J. W. Langmuir 1987, 3,272. (9)Forrest, S.R.;Witten,T. A. J.Phy8.A: Math. Gen.1979,12,L109. (10)Hurd, A. J.; Schaefer, D. W.; Martin, J. G. Phys. Reo. A: Gen. Phys. 1987,35,2361. (11)Witten, T. A.; Sander, L. M. Phy8. Reo. Lett. 1981,47,1400. (12)Botet, R.;Jullien, R.; Kolb, M. J . Phys. A: Math. Gen. 1984,17, L75.
(13)Berry, M. V.;Percivai, I. C. Opt. Acta 1986,33,577. (14)Mountain, R.D.;Mulholland, G. W. In Kinetics of Aggregation and Gelation; Family, F., Landau, D. P., Eds.; North-Holland Amsterdam, 19W,p 83. (15)Mountain, R. D.; Mulholland, G. W. Langmuir l988,4,1321.
0 1991 American Chemical Society
Fractal Properties of Soot Agglomerates
Langmuir, Vol. 7, No. 6, 1991 1275
A, A 1 aprtwer
To lock h
L. L1 Ionsor
detoctor
M mkror NDF noutral d.culty flltor P polarizer
Figure 1. Schematic diagram of the apparatus.
would need to be corrected for. Neither multiple scattering effect has been thought to be important in the work to be described, however, for reasons which will be discussed. If the primary monomers are in the Rayleigh range (the cluster itself need not be so) and consequently scattering weakly, then theory's gives an additional important result; namely, that in the absence of multiple scattering the absorption coefficient, a,,for the cluster is just the sum of the monomer absorption coefficients. This assumes a dipole monomer susceptibility alone, with no coupling effects induced by the proximity of the other monomers. Thus, if there are n monodisperse N clusters per unit volume
= n N 8?r2 - p 3h(3) m2m2+2 where m is the complex index of refraction. The absorption can be equated with extinction if the measurement is made at a wavelength where scattering is small. In general, however, it can be determined from the relationship CY,
truly represented. For silica soot, to give an example, the fractal dimension obtained from in situ light scattering is less than that obtained by examination of samples.18 Also, a model is needed to compute radiative coefficients when these values are in hand. Light scattering measurements provide a way of determining the structural parameters without the need to be concerned over sampling perturbation of the cluster structures, and, if a scattering model is available, a self-consistent treatment is possible. For singly scattered waves, the scattering cross section and density autocorrelation are Fourier transform pairs"
The magnitude of the change Q in the incident wave vector due to scattering is given by 9 = 2ki sin (812) =
$sin (812)
(6)
where ki is the incident wave vector, X is the incident wavelength, and 8 is the scattering angle. The reciprocal of q is a measure of the resolution or length scale probed by the scattering. One important consequence of eq 5 which will be utilized in the analysisand experiments to follow is that the forward scattering intensity is proportional to the squared mass of the particulate; thus, if the particle consists of N monomer spheroids, there will be an W coherent enhancement of the forward cross section relative to its monomer value. The asymptotic limits expected for the cross section are18
-
~ - ~ ( 1q2a2/3 + ...) (R,> l / q
> a)
(7b)
where for large 9 there is a surface fractal dimension, De, related to surface roughness. We will present data from another source which permits a determination of D,,. It is possible to correct for multiple scattering (intraparticle) by using the self-consistentfield approach.lgSuch multiple scatteringwould be expected to become important for very large and dense clusters (D> 2) and for strongly scattering monomers. Interparticle multiple scattering for large optical paths is also something that in principle (16) H d ,A. J.; Flower, W.L.J. Colloid Interface Sci. 1988,122,178. (17) Beme, B. J.; Pecora, R.Dynamic Light Scattering, Wiley New York, 1976. (18) Freltoft, T.; Kjems, J. K.; Sinha, S. K. Phys. Reu. B: Condens. Matter 1986,33,269. (19) Nelson, J. J. Mod. Opt. 1989,36, 1031.
(9)
where aeis the scattering coefficient (n da/da) and fie is a unit vector in the direction of scattering. Thus, multiangle scattering supplemented by an absorption coefficient measurement provides a way of determining all of the cluster structural and density parameters. The foregoing theory suggests the following approach, essentially that given in ref 15. (1)Determine R, from small q scattering. (2) Deduce the fractal dimension D from the asymptotic slope of the log scattering-log q plot if accessible experimentally, or least squares.fit a model p&). (3) Use an optical absorption measurement to deduce nNa3. (4) Make an absolute intensity or cross section measurementin the near forward direction, using, for example, calibration by room air Rayleigh scattering. This will yield a value for nWa6 from the relationship
Note that the cluster number density results from division of the square of the number derived in step 3 by the number derived in step 4 and that the result does not depend on the fractal dimension or on any assumptions about the aggregate shape. It would be valid for isolated monomers, assuming that they are in the Rayleigh range with respect to the incident light. The fact that the monomer radius a appears raised to relatively high powers means that this quantity may be subject to smaller measurement error than the other quantities; experimental errors or uncertainties in the complex index of refraction which goes into the overall constants in steps 3 and 4 will have a mitigated impact on the error in a because of this. VV scattering is assumed, and polarization factors have been suppressed. This theoretical-experimental approach assumes monodispersity in Nand a. It is theoretically possible to take into account polydispersity, and this will be discussed in the section concerning analysis of the experimental data.
Experimental Section Apparatus. The apparatus used to carry out the measurements is shown schematically in Figure 1. The Coherent INNOVA 90-4 argon ion laser has 1.7 W of nominal continuous
1276 Langmuir, Vol. 7, No. 6, 1991 wave (CW)power at 514.5 nm. It serves primarily as a source of intense light for soot particulate and molecular Rayleigh scattering. The Rayleigh scattering providea an accuratelyknown intensity to which soot signals are referenced for the purpose of determining their required absolute intensities. Since the Rayleigh signals are weak, the full power of the laser is required to observethem. On the other hand, the much stronger soot signals are observed with the laser attenuated to about 0.5 mW of power by the NDF in Figure 1. The ion laser is linearly polarized in a direction perpendicular to the plane of scattering, and the resulting light intensities are analyzed in this same direction using the polarizer, P. The Jodon HN-1OG infrared He-Ne laser in Figure 1is used exclusivelyto measure soot extinction;it has a nominal C W output of 3 mW at 1150nm. In the approach taken here, we require that the extinction be dominated by absorption, or equivalently that the near forward scattering contribution to the extinction be negligible. By use of an infrared as opposed to a visible wavelength, as we have done, the preceding requirement is best approximated. It will be seen that this assumption of negligibly small relative scattering is borne out by the experimental data. Scattering or extinction measurements are done alternately by properly inserting a translatable reflecting mirror in the path of one laser beam or the other. For scattering, the 514.5-nm radiation is focused into the flame with a plano-convex lens, L1, having a 40-cm focal length. The scattered light is collected by optics mounted to a rotatable platform, and then is sent via an optical fiber to a 1 nm wide narrow-band 514.5-nm filter and photomultiplier. The resulting electrical signal, modulated at the 400 Hz frequency of the light chopper, is synchronously convertedto a dc output by the lock-indetector. On the platform, the two 15 cm focal length lenses, L, are symmetrically located with respect to the burner center and, removed 30 cm from it, the 0.50 mm diameter aperture, Al. These optics and the lens, LI,define, a t 90",a cylindrical sample volume having a 0.2 mm diameter and 0.5 mm length. Since the platform rotates in steps of 4" from 0 to 180°, the sample length is in fact a l/sin 8 function of the scattering angle, 8, which must be accounted for properly in the data reduction. For extinction measurements, the 514.5nm beam is blocked and the 1150-nm beam is passed through the flame and detected by an infrared sensitive type 7102 photomultiplier; synchronous signal detection by a second lock-in detector is used in this case as well. The rectangular burner in Figure 1 has been described previously.20 It has a central fuel slot, two adjacent symmetrically positioned oxidant slots, and a shroud. The sooting diffusion flame is fueled by 229 cmS/min ethylene and 5.7 L/min air. These flows are such that the diffusion flame is highly overventilated and, hence, its two flame sheets are folded inward toward a common apex. This flame, whose end-on view closelyresembles that of a candle flame, is roughly 40 mm in height and 5 mm wide at its base. Like a candle flame, the soot is distributed nonuniformly, peaking near temperature maxima. As such, the light scattering and extinction depend strongly on measurement position. This dependence is, however, least pronounced a t the flame center, near which the measurements reported here were taken. Specifically, probing was done at the horizontal center of the flame for four different vertical positions. The variation of soot volume fraction with height is given in Table 11;the corresponding horizontal variation is like that given in Figure 4c of ref 20. Finally, the soot formed in this and other diffusion flames is a function of other parameters such as fuel type and flow rate. Such variations have not been examined here, but there is no reason to believe that these other cases cannot be handled by the fractal approach used here. Approach. In order to test a fractal description of the soot in our flame, measurement of the parameters n,(da/dQ), and a, from scattering and extinction, respectively,are required. These are defined and determined as follows. For soot, the scattered intensity, I,, is given by I, = I,O&(da/dQ),Qlt, where I," is the incident 514.5-nm laser intensity, n, the soot number density, (da/dQ), the differential scattering cross section, Q the light collection solid angle, 1 the angular dependent optical sample length, and c the light collection efficiency. In principle, the ~~
(20) Bonczyk, P. A. Combust. Sei. Technol. 1988,59,143.
Bonczyk and Hall
Figure2. Scattering of 514.5-nm laser light by soot particulates at 28-mm height in an ethylene/air diffusion flame. evaluation of s(da/dQ), from experiment is straightforward once I, is measured and I," and Qk are assigned their respective numerical values. In practice, the precise evaluation of Qle is not easy, especially since 1 is a function of scattering angle. Fortunately this difficulty may be overcome by making auxiliary Rayleigh scattering measurements. In this case, the scattered (da/dQ)RQk, R where IR" is the intensity, IR, is given by IR = I R " ~ incident laser intensity, n~ the number density of Rayleigh scatterers (room air a t 20 "C), (du/dQ)R the Rayleigh differential scattering cross section, and Qlc the same as above. By use of the equations above for I, and IR, it follows immediately that s(da/dQ),= (I,/IR)(IR0/I,")nR(da/dQ)R. Since n~ and (du/dQ)R are known,2l the left side of the preceding equation can be evaluated without reference to Qlc. Further, since scattered intensities occur in the ratio (I,/IR) it is not even necessary to calibrate the type 8575 photomultiplier in Figure 1. The parameter a,above is the optical extinction coefficient. It is determined from I = IOexp(-a&), where I is the transmitted light intensity, IOthe incident 1150-nmlaser intensity, and L the extinction path length. Since the flame in this work is only approximately two-dimensional, there is a variation of L with height. For the four measurement heights of 17,20,25, and 28 mm, the L values were 34.7,32.3,28.3, and 25.9 mm, respectively. The L values were determined by observing the appearance and disappearance of 90" scattered light as the burner was translated in a direction orthogonal to the viewing axis. In addition to the procedures described above, it was necessary to correct for the attentuation of I, as scattered light passed between the center of the flame and its boundary. Again, this correction varied with scattering angle. Details of how this correction was applied are omitted here except to point out that extinction measurements a t 514.5 nm were made for it and that the corrections ranged from 1/2 to 11%depending on viewing angle and measurement height. These values should be small enough to justify neglect of multiple scattering effects in the theory used to reduce the data. Figure 2 showsthe soot scattering at 514.5 nm at the height of 28 mm above the burner surface.
Interpretation of Experimental Data Summaries of the data taken in the ethylene-air slot burner are shown in Figure 3 and in Table I. Referring to Figure 3, which displays the log-log plots of n(da/dQ) vs q, it is clear that accurate extrapolation to exact forward scattering can be performed, and that, with the exception of the near Rayleigh 17 mm data, the curvature in the (21) Namer,R.; Schefer,R. W.; Chan,M. LawrenceBerkeleyLaboratory Report No. LBL-10655; Lawrence Berkeley Laboratory, University of California: Berkeley, CA, 1980.
Langmuir, Vol. 7, No. 6,1991 1277
Fractal Properties of Soot Agglomerates ,.,
2 +
0.90.8 -
Height above bumer surface, mm
S
2 I -
F 'C
HeighMO mm
1-
z
-3
-
0.6 -
-3.25-
0.5-
8 12, -3.50s
0.4 -
0.3 0.2 -
-3.75-
0.1 -
I
-1 7
-
0.7
,
,
0.25
I
0.50
0.75
I
I
1.25
1
0
I
I
I
I
I
I
1.50
Log@) q (microns-') Figure 3. Scattered intensity vs q for four different vertical positions in the flame. 1.1
Table I. Data Summary 17" 20" 2.05 X 1W 6.3 X lo-'
v i n ds/dn
25" 28" 1.05 X 1od 4.24 X lo-'
srl) (514.5 nm) a. (cm-l) (cm-l
0.118 0.114 0.139 0.068 (1150 nm) a. (cm-l) 0.0507 0.059 0.056 0.0288 1.48 X lod 3.76 X 1od 5.00 X lod 1.77 X 1WS (514.5 nm) n dfi (da/dn) (cm-l) (514.5 nm) w (albedo) 1.25 X 10-9 2.61 X 1W2 3.60 x le2 2.60 x 10-2
(d,.
near forward direction will lead to well-defined R,. However, it is unclear whether the available optical source wavelength and particulate size have provided sufficient access to the asymptotic regime of eq 7b, which would have permitted unambiguous determination of D . The high R d range is limited, as seen. To obtain D , therefore, use will be made of least-squares fitting model p&), as discussed, or to fitting a function like the Fisher-Burford form.22 Table I presents, in addition to the extrapolated forward scattering data, the measured extinction coefficientsat 514.5 and 1150nm. Taking theFigure 3scattering data and performing the angular integral over all scattering directions, as in eq 9, gives the integrated scattering information shown, from which it is possible to calculate the particle albedos given. It is clear that scattering is relatively small even a t 514.5 nm, and is most certainly entirely negligible a t 1150nm. The absorption coefficients at the two wavelengths have roughly the expected inverse wavelength dependence, with some index of refraction dispersion perhaps responsible for departures from this. In any event, our calculated results do not depend strongly on which of the two absorption coefficients we use, provided that we take into account the differing wavelengths. The same value of index of refraction is used at both wavelengths. While scattering is relatively small for the soot generated in this flame, it will not necessarily be negligible for all soot; clusters with larger monomers and/ or larger values of N could well have significant albedos. Extraction of R, and D from the data involved several different approaches. The first of these was a polynomial fit in q2 to the multiangle scattering data. This gives R, from the q 2 coefficient of the expansion. The second was least-squares fitting the Fisher-Burford form I ( q ) / l ( O )= (1+ 2q2R,2/3D)-D/2
(11)
which yielded R, and D . The third was least-squares fitting a model density function using eqs 4 and 5 and the relationship ~
~
~~
(22) Fisher, M. E.; Burford, R. J. Phys. Rev. 1967,156,583.
l0,
.r P +
8
KJ
1 .-0
6 z
HeighMS mm
a
0.9 0.80.7
-
0.6 -
0.50.4
-
0.30.2
-
0.1 -
I
1
0
5
10
15
20
25
,
30
q (microns-')
Figure 5. Exponential cutoff fit to scattered intensity vs q at 25 mm height: R, = 106.3 nm; D = 1.5240.
with the r integration performed numerically. Three cutoff functions were employed; these were the exponential form of Berry and Percival,13 the modified Gaussian form of Mountain and Mulholland,15and the expression given by Hurd and Flower.'G The 17" data were not analyzed in this part of the data reduction because the clusters a t that height seem to be so small that a fractal interpretation may be inappropriate, and attempting to fit these formulas to a Rayleigh object would be subject to error. With regard to the R, determination, all of the approaches gave reasonably similar results. The FisherBurford fits and those resulting from use of the BerryPercival cutoff function also were similar, with fitted fractal dimensions of 1.4-1.55. The Mountain-Mulholland and Hurd-Flower cutoffs gave inferred fractal dimensions of one or less. Attempts to fit a straight line to the high-q regime of the Figure 3 28-mm curve suggest values of D closer to 1.4, but the number cannot be specified with precision because of the narrow q range and noise in the scattering at the backward scattering angles. We do not know whether the peculiar inferred fractal dimensions from the refs 15 and 16 cutoff functions are the result of the stated limited range of the data or of deficiencies in the cutoffs themselves. While these results would seem to argue against a steep cutoff, the limited asymptotic range precludes any such conclusions at this time. Figures 4-6 show the results of fitting the Berry-Percival density function to our data, together with the best fit R, and D . Because the latter values were supported by the FisherBurford fits, which were also excellent, and by the q2
1278 Langmuir, Vol. 7, No.6, 1991
Bonczyk and Hall Table 11. D = 1.51
._. 14
Heightas mm
\
height, mm 17 20 25 28 fv
z
0.2
98 225
n,cm*
15.1 14.4 10.4
4.19 X 1Olo 1.887 X 1Olo 1.016X 1Olo 5.66 X 1Og
Re,69.4 106 132.3
fV0
1.405 X 1O-e 1.644 X 10s 1.567X 10s 7.43 X lW7
is volume fraction.
I I
I
I
polynomial fitting, they were used in the subsequent data reduction. There is no consistent relationship between our in situ fractal dimensions and those determined ex situ by others. For example, the values 1.62 and 1.74' and 1.46-2.0023*u(from carbon black samples) are generally high relative to us, but 1.5-1.6 from Samson et aL8 is reasonably near. It should be pointed out that such comparisons are somewhat suspect in any case since the fractal dimension may be dependent on the specific combustion environment in which soot is formed. For further data reduction, we pick a nominal value of D = 1.51, equal to the average of the values inferred from the fits shown in Figures 4-6,and close to the Hurd-Flower silica soot value. Before proceeding to the calculation of a, N,and n,it is worthwhile to discussother data bearing on the question of soot fractal dimension. We know of no other multiangle laser scattering data on soot analyzed on a fractal basis, but small-angle X-ray scattering data (SAXS)have been so analyzed and reported? If a log-log plot is made of the butadiene slot burner SAXS data reported there, one finds that the data cover mainly the Porod regime (eq 74 with a slope of -4.06. This implies a surface fractal dimension of about 2,meaning that the monomer surfaces are relatively smooth. This will be of relevance to soot oxidation rate analysis, for example. The smallest q values seem to cover the transition from the mass fractal to the Porod regime. The slope between the two smallest scattering angles is about -1.5; the count levels are very high, so noise is probably not a factor, but with so few points, it is hard to draw any conclusions. With R, and D determined, the parameters a, N,and n follow in a straightforward way if the index of refraction is known. We have chosen the Senftleben and BenedicP value of m = 1.94 + 0.66i. As will be discussed, there is considerable sensitivity of the predicted results to index, but a recent summary of the fieldn lends support to our choice. Assuming that R, and D have been determined from least-squares fitting like that shown in Figures 4-6, the data reduction proceeds as follows. First, n is simply determined by dividing the square of step 3 by step 4,as discussed. The monomer radius a is then determined by eliminating n from step 3 and expressing N in terms of R,, D, and a. The cluster monomer population is then deduced by elimination of n and a from step 3. Table I1summarizes (23) Bounat, X.; Oberlin, A. Carbon 1988,26,100. (24) Ehrburher-Dolle, R; Tence, M. Carbon 1990,28,448. (25) England, W. A. Combust. Sci. Technol. 1986,46,83. (26)Senftleben, H.; Benedict, E. Ann. Phys. (Leipzig) 1918,54,65. (27) Vaglieco, B. M.; Beretta, F.; DAlesaio, A. Combust. Flame 1990, 79,259.
48
%,nm
-
0.1 0
N
El Figure 7. Transmission electron micrograph of Boot collected at tip of ethylene/air flame.
the predictions as a function of height above the burner surface. The evolution of the cluster number density with height suggests agglomerating spheroids, and this is borne out by the variation of the predicted N. Although we cannot verify the values of N so obtained, they certainly seem reasonable based on electron micrographs from other sooting flame^.^ The predicted values of the monomer diameter start at about 15 nm at 20mm and reach a value of about 10nm at the greatest height investigated. These are also plausible numbers and, as will be shown, are consistent with samples taken near the top of the flame. The monotonic decrease of spherule radius seems to suggest oxidative erosion of the primary spheroids rather than breakup of the clusters, at least over the range investigated. This qualitative variation does not seem to be sensitive to uncertain parameters like index of refraction or fractal dimension. To test the reasonablenessof the inferred monomer sizes, we collected soot on a metal disk in the region of the luminous flame tip. This was removed, and examined under an electron microscope, as shown in Figure 7. The removal presumably resulted in compaction of the aggregates, preventing any estimate of N. However, the observed diameter of the primary spheroids, 4 to 10 nm, is quite consistent with the value of 10.4 nm measured at the largest height probed by laser. The physical collection was performed even higher in the flame, so somewhat smaller spheroids are expected. This encouraging agreement is probably the most significant result thus far; to our knowledge, there has been no other determination of soot cluster primary spheroid size by laser light scattering. Ultimately, comparisons of inferred monomer sizes with samples taken at the same location will be necessary. It is not easy to judge from the photograph whether there is in fact significant polydispersity in monomer size. The relative smallness of D and a, together with predicted N
Fractal F'roperties of Soot Agglomerates Table 111. D = 1.78 height, m m
N
2a, nm
n, cm-9
262 736 2427
8.6 7.4 4.7
0.8 -
same ae Table I1
0.7 0.6
0.5
Table IV. Height = 25 m m ( D = 1.51) 2a, nm 14.4 21.9
n, cm-8
R,, nm
(28) Dalzell, W. H.; Sarofim, A. F. ASME J. Heat Transfer 1969,91, 100.
A
0.3 0.2 0.1 -
values that are not particularly large, means that intraparticle multiple scattering is indeed probably small.13Jg There is considerable sensitivity of certain of the inferred parameters to the fractal dimension, as exhibited in Table I11 for an assumed D = 1.78. Comparing with Table 11, the values of N are seen to be especially sensitive, and there is considerable sensitivity in the monomer radius. It cannot be said that these numbers are unreasonable, either, so it is clear that determination of the proper in situ fractal dimension for soot aggregates should be a priority task. As we have stated, the limited high q range of our data means that our inferred range of 1.4-1.55 is not definitive. Table IV gives some indication of the predicted sensitivity to assumed index of refraction. Use of the Dalzell-Sarofima value instead of the SenftlebenBenedict value is seen to result in factors of 2 changes in certain of the parameters. While the recent summary2' would seem to lend weight to the latter number, the proper index of refraction for soot remains an important question. As noted, we have used a value of unity for the constant a in eq 1 at the suggestion of ref 13. If one reduces the 25-mm data with a value for a of 0.8, for example, the inferred values of N and a change to 193 and 11.5 nm, respectively (compare Table 11). Choosing a value of 1.2 for this parameter changes N to 56 and a to 17.4 nm. This is more sensitivity than one would like to this somewhat uncertain parameter, indicating a need for more research into its proper value, possibly through structural examination of samples. It is interesting to interpret our experimental data in terms of an "equivalent sphere" model for the scattering using conventional Mie scattering theory. This is normally done in one of two ways: one uses ratios of scattering intensities at two angles; the other is using extinction and scattering at one angle. The results of reducing the 25mm data using these two approaches are shown in Figures 8 and 9. In Figure 8, the Mie scattering function is fitted to the measured intensities at 20' and 160°, with the inferred hard sphere radius, R, and radius of gyration. The predicted multiangle pattern from this sphere is then given by the triangles, and the overall agreement with what was actually observed is seen to be only fair. The agreement is better at the lower heights where the clusters are smaller, and much worse a t 28 mm where the clusters are much bigger. Figure 9 displays the results for fitting a Mie scattering function to the 90' scattering and the extinction, another approach in widespread use. The inferred sphere radius of gyration and the overall scattering pattern differ significantly from those observed. It is also possible to make estimates of polydispersity effects on these results and, in principle, to estimate the width of the distribution from the data. We do not expect a high degree of polydispersity in the monomer sizes,B
- Observed Scattering Rg-106 nm
-
0.4 -
f"
1.016 X 1010 106 1.567 X lo4 52b 4.622 X 109 106 1.327 X lo4 a m = 1.94 + 0.66i (ref 23). m = 1.57 + 0.56i (ref 25). 98"
I
1-
fv
0.9 -
17 20 25 28
N
,
1.1
R,, nm
Langmuir, Vol. 7, No. 6,1991 1279
from sphericd Mie theory and 1(20)/1(160) k74.8 nm (RgS8 nm)
I
0
5
10
20
15
25
30
q (microns-')
Figure 8. Mie fit to scattered intensity vs q at 25 mm height. R, = 106 nm is from Berry-Percival. R = 74.8 nm is from Mie theory and the ratio of scattered intensities at 20 and 160O. R, = 58 nm is the radius of gyration of the sphere of radius R where R, = ( 3 / 5 Y 2 R (from R,2 = SoRr2dm/SoRdm).
2
0.20.1
0
FM6.8 nm (Rg30.7 nm)
-
I
I
I
I
I
0
Figure 9. Alternate Mie fit to scattered intensity vs q at 25 mm height: R = 26.8 nm is from Mie theory and the ratio of the scattered intensity at 90° to extinction. Other radii are defined as in the Figure 8 caption. and we have not included this effect here. Rather, we have assumed polydispersity in the N distribution and show how it changes the predicted cluster parameters. A scheme for estimating the width of the size distribution is also given. If a ZOLD (zero order logarithmic distribution) is assumed? then one has as parameters N,the average cluster number of spheroids, and UO,the width of the distribution. It is not difficult to show that the following substitutions occur:
- - -
nNa3 nPas
n (N)a3
n(P)as
nNa3
(13a)
nPeaoaas
(13b) Since the measured R, is a weighted average of the form
-
It can be shown that
a ~ l l D
,fl1/Deao~l/@+1.6/D)
(15) This still leaves the width of the distribution unknown, but if one picks a value it is then possible to carry through (29) Dobbins, R. A. Presentation (unpublished) at the Combustion Institute Eastern Section Meeting, Albany, NY, 1989. See ale0 Prado, G.; Jagoda, J.; Neoh, K.; Lahaye, J. in Eighteenth Symposium (Znternational) on Combustion; T h e Combustion Institute Pittsburgh, PA, 1981; p 1127.
1280 Langmuir, Vol. 7, No. 6,1991 Table V. D = 1.51,uo = 0.3 height, (mm) N 2a, nm n, cm-3 R,, nm f, 17 4.58 X 1O1O 20 35 11 16.3 2.065 X 1Olo same as Table I1 25 73 22 15.5 1.11 X 1Olo 28 166* 51 11.1 6.2 X lo9
the data reduction as before without a new piece of experimental data. For soot, values of a0 = 0.3 appropriate to a diameter distribution are suggested in the literat ~ r e , ~ Obut * ~we * emphasize that when we use such a number here it is in a loose sense intended only to give a feeling for the magnitude of the effect. Table V shows how the predicted parameters change (compare Table 11) for a predicted N distribution uo of 0.3. The changes are not drastic, particularly with regard to the monomer sizes. To obtain uofrom the data, a new piece of experimental data is needed. This might in principle be provided by some of the higher order coefficients in the near forward scattering intensity expression I(q)
where one can the moments
c
-
bo + b2q2+ b4q4 + ... (16) that the coefficients correspond to
2
The moments of r can be performed if one assumes an interspherule probability distribution such as that given in ref 15. One can then show that, leaving out numerical constants
resulting in
The coefficient b4can be obtained from the q2polynomial fit discussed earlier, and thus one has an additional piece of data needed because of the additional unknown, UO. This approach is being pursued a t present but has not yet given consistent results a t all heights in the flame. It is (30)Prado, G.;Lahaye, J. In Particulate Carbon Formation During Combustion;Siegla, D. C., Smith, G. W., Eds.; Plenum: New York, 1981; p 143. (31)Wersborg, B. L.;Howard, J. B.; Williams, G. C. In Fourteenth Symposium (International)on Combustion;The Combustion Institute: Pittsburgh, PA, 1973;p 929. (32)Guinier,A.;Fournet, G.Small AngleScatteringofX-Rays,Wiley: New York, 1955.
Bonczyk and Hall relatively easy to make estimates of the effect a selfpreserving size d i ~ t r i b u t i o would n ~ ~ have on the calculated cluster parameters, using the N moments given in Hurd and Flower.16 This changes the substitution eqs 13-15 in straightforward ways. Doing so for the free molecular regime results for each height in a predicted number density increase of about a factor of 2, an increase in the monomer size by about a third, and a marked reduction in N by about a factor of 5.
Conclusions A theoretical analysis of fractal cluster scattering and absorption coefficients has suggested experimental approaches which make possible in situ determination of soot cluster structural parameters. The approach has been tested in experiments performed in an ethylene-air slot burner and found to have considerable promise. Plausible values of cluster number density, monomer number, and monomer size are obtained from our measurements, although our fractal dimension determination is subject to some uncertainty. Certain cluster parameters are shown to have an extreme sensitivity to fractal dimension. Measurement of the in situ soot fractal dimension should be a priority task, perhaps with UV laser or neutron sources. This work presents an approach to revamping laser light scattering particulate sizingdiagnostics for those situations where clusters are expected. Soot scattering is negligible relative to absorption in the burner investigated, and volume fraction is the sole parameter governing radiative transfer. Thus, for soot radiative transfer calculations, there is no longer any need to employ unrealistic morphological models of soot particles. Note Added in Proof: After we submitted this manuscript for review, we became aware of a paper by Dobbins, Santoro, and Semerjian (ref 34) that takes a similar approach to interpretation of soot scattering and extinction data. The main difference with our work is that they fixed the primary spheroid size from sampling measurements, but their inferred N values are very similar to ours. One of these authors also pointed out an interesting feature of our work and theirs; namely, that the product of nN, which gives the total monomer density, should be approximately constant high in the flame if one accepts the picture of soot nucleation, surface growth, and coalescence having occurred lower in the flame. One can see from Table I1that this is approximately so. It remains true when a self-preserving form is assumed for the N distribution. Acknowledgment. The authors thank Heidi Hollick and Kathi Wicks for their help in preparing the manuscript. (33)Friedlander, S. K.Smoke, Dust, and Haze; Wiley: New York, 1977;Chapter 7. (34)Dobbins, R.A.;Santoro, R. J.; Semerjian, H. G. In Twenty-third Symposium (International)on Combustion:The Combustion Institute P-itGburgh, PA, in press.
