Coagulation of Carbon Particles in Premixed Flames BY J. B. HOWARD AND G. C. WILLIAMS B. L. WERSBORG Fuels Research Laboratory Dept. of Chemical Engineering Massachusetts Institute of Technology Cambridge Massachusetts Received 4th December 1972 The size distribution number concentration and fraction charged of carbon particles at successive stages of formation in a low pressure flat flame were measured using molecular beam sampling involving electrical beam deflection and electron microscopy of beam deposits and an optical absorp- tion technique. Observed cluster-type structure within roughly spherical particles and decreasing particle number concentration following rapid nucleation indicate the particles do indeed coagulate during growth. Particle size and number concentration data confirm this conclusion although the experimental coagulation rate exceeds by a factor of about 10the kinetic theory collision rate approxi- mately adjusted for electrostatic forces based upon the measured extent ofparticle charging and Van der Waals attraction.Calculations based upon extrapolation of the experimental coagulation rate constant into the flame region of significant particle nucleation and surface reaction indicate that the particles nucleated first can grow predominantly by surface growth to a volume mean diameter of about 100 8,and that the number of primary particles per spherical unit within the final chainlike clusters is of order 10. Thus crystallites the number of which is of order lo3per spherical unit do not represent former particles.Although the chemistry of dispersed carbon formation in flames has been the object of many investigations coagulation of the particles has received little attention. Coagulation is known to occur in the final stages of carbon formation but its role during particle nucleation and growth by surface reaction has remained obscure. The nucleation step would involve coagulation if nucleation amounts to a continuous transition from large hydrocarbon molecules to small soot particles of continuously decreasing number concentration as advocated by Homann.' BartholomC and Sachsse and Fenimore and Jones assume that carbon particles can grow by coagula-tion to a size too large to permit their burnout. If at least some of the carbon particles are charged coagulation may be influenced by interparticle electrostatic forces.This influence and the importance of ionic nucleation have been discussed previo~sly.~-~ Recently,' the following quantitative connection between particle growth by surface reaction and coagulation was derived by relating the rate of change of the volume mean particle radius a3 to particle number concentration n,rate of appearance Nu of the smallest particles of radius ao and surface growth rate Si of particles in the ith class having area mean radius af The surface growth rate at a given flame position may be assumed to be independent of particle size. Also and coagulation rate fiC are related by Nu = dn/dt+N,. (2) 109 COAGULATION OF CARBON PARTICLES Denoting the area mean radius as a, eqn (1) then becomes These equations show that the rate of change in particle radius is in general different from the surface growth rate and they provide a means for distinguishing between the contributions of coagulation surface growth and nucleation.The objective of the present investigation was to measure the rate of carbon particle coagulation under well known flame conditions and to assess quantitatively the role of coagulation during particle nucleation and growth. To these ends particle size distribution number concentration and fraction of particles charged were measured at different stages of carbon formation in a unidimensional low pressure flame using molecular beam sampling electron microscopy and optical absorption.These and other experimental results are compared with the theoretical coagulation rate based upon kinetic gas theory and a simple account of interparticle Van der Waals and electrostatic forces. The sampling and analysis of carbon particles from flames using multistage molecular beam sampling and electron microscopy were improved until 15A diameter particles could be included in the measurements. Details of the apparatus and experi- mental techniques were described earlier.' In general a flat premixed acetylene- oxygen flame maintained at 20Torr equivalence ratio 3 and cold gas velocity of 50cm/s on a 7cmdiameter burner was probed along its centre line at different heights above the burner using a quartz nozzle expanding into the beam system.By opening a shutter for an adjusted length of time the sample beam was admitted to the detection stage operated at 8 x Torr in which particles were deposited directly onto electron microscope grids. A second sample was taken under identical conditions except that an appropriate electric field was applied across the beam to deflect all charged particles. The difference between the deposit intensities in the two cases permitted calculation of the fraction of particles charged. The intensity was obtained together with particles size and size distribution by a particle count and diameter measurement on electron micrographs. The intensity measurements gave relative particle number concentrations from which approximate absolute values were obtained by calculating a calibration factor from the beam geometry traced by soot deposits.The absolute value of particle number concentration was also determined by optical absorption measurements * in a second flame produced under identical conditions on a 10 cm diameter burner. The attenuation signal was obtained using a laser source (A = 6328A) and two photomultiplier detectors. Temperature profiles were obtained using Si0,-coated thermocouples. The absorption technique requires knowledge of the extinction cross section of young carbon particles which because of their substantial hydrogen content differ optically from the older carbon particles for which optical properties are available. This information was obtained using the molecular beam system to collect particle samples on glass slides and adjacent electron microscope grids.Subsequent measurement of the attenuation of the laser beam by the slide deposits and of particle size distribution and number per unit area in the deposits using the electron microscope grids permitted calculation of the extinction cross section. Under the flame conditions here studied visual observation of flame luminosity indicates that carbon formation starts at 1.5 cm height above the burner. Electron micrographs of representative carbon particles collected at different heights above PLATE1.-Carbon particles at different growth stages (a,b and c at 4cm 6 cm and 7 cm height above the burner of a 20 Torr C2H2-02 flame magnification 90 000 x ;d at onset of chaining in a 1 atm C3Hs-02 flame magnification 158 000 x ).To face page 1111 J. B. HOWARD B. L. WERSBORG AND G. C. WILLIAMS the burner are shown in plate 1. Up to 4 cm the particles are mainly spherical (a) whereas chainlike clusters become noticeable at 6 cm (b) and predominant at 7 cm (c). In (d) are shown enlarged particles from a propane-oxygen flame at 1 atm. height above burner /cm FIG.1.-Mean particle diameter [volume mean diameter (a),number mean diameter (a)]. 1o.c m I g o\ 2 \ 7.5 .-+I *E 8 0 8 5.c B E CI .-U 2.5 a a 01234567 height above burnerlcm FIG.2.-Particle number concentration [all particles (A),uncharged particles ( a)]. Their shape which is representative of that which prevails just before chaining becomes predominant may be interpreted as evidence for the fact that carbon particles will collide at all stages of formation but the earliest collisions are hidden by large simultaneous surface growth which tends to fill in the boundaries between particles.COAGULATION OF CARBON PARTICLES This behaviour leads to a gradual transition from roughly spherical clusters to the familar chainlike clusters. Thus the spherically appearing units within the chains may not be used to calculate surface growth rates or the number of nuclei as these units themselves are generally composed of several primary units. This interpreta- tion is supported by ultrahigh resolution electron micrographs on which may be recognized particle domains commonly known as crystallites arranged around different growth centres within one " spherical " unit.The growth centres appear to have nucleated independently and grown for some time as separate particles and to have coagulated while surface growth possibly accompanied by some migration was sufficiently rapid approximately to even out the cluster surface. In the determination of particle size distribution and number concentration from electron micrographs clusters were counted as single particles and non-spherical clusters were assigned the diameter of the volume equivalent sphere. By analyzing about 100 particles per micrograph measurement error was kept below 10 % for diameter and 20 % for relative number concentration. Owing to the difficulty of seeing the smallest particles under the electron microscope error in the case of absolute number concentration undoubtedly exceeds 20 %.The size distributions thus measured progress from approximately Gaussian at 2 cm height above the burner to approximately lognormal at 7 cm. The relative standard deviation is about 0.2 and independent of mean particle size so long as the particles are spherical but it rapidly approaches about 0.5 with the appearance of chainlike clusters. Mean particle size and number concentration at different heights above the burner are shown in fig. 1 and 2. From a small concentration at 2 cm particle number increases rapidly at first peaks shortly after 3 cm and then decreases. This behaviour is assumed to reflect the opposing effects of nucleation and coagulation the number increasing effect of nucleation being dominant at first but subsequently negligible in comparison with the decreasing effect of coagulation.In view of the error in these data the exact position of the particle number peak should be located by more accurate measurements. The absolute number concentrations measured by absorption (not shown) at 4 and 5 cm height above the burner differ by less than 20 % from those determined by electron microscopy which is within the error limit of the latter technique. The absorption values at 3cm height above the burner are however twice as large. The difference could be due to absorption by large gas phase hydrocarbons or to incomplete detection of small particles on the micrographs.A similar decrease in particle number with increasing height above the burner was observed before by Bonne Homann and Wagner.lo Their remarkable study does not use the concept of a volume equivalent particle diameter for chainlike clusters. Thus their number concentration measurements describe the number of approximately spherical units within a chainlike cluster which stays nearly constant in the tail of the flame. The measured concentration of neutral particles is given by the dashed line in fig. 2. The difference between these results and the corresponding total concentra- tion values is the concentration of positively and negatively charged particles. The polarity of the charge and charge per particle where studied by reducing the deflecting field strength in a stepwise manner and collecting the particles on grids adjacent to the deposition area of neutral particles.This measurement gave intensity problems which prohibited a reliable count of particle density on the grids due to the difficulty of identification. Only chainlike clusters could reliably be distinguished from back- ground grain and contamination ; these particles were exclusively positively charged. The results do indicate however that a1 charged particles under the flame conditions J. B. HOWARb B. L. WERSBORG AND G. C. WILLIAMS studied are predominantly of positive polarity with one charge per particle but a few negative particles and a few particles with two charges cannot be exchded. Our present charge measurements which use a Faraday cup instead of the electron microscope grids indicate that carbon particles under different flame conditions can be predominantly neutral positively charged or negatively charged.The charging state is a strong function of flame temperature which in turn is influenced by the cold gas velocity and gas composition. Experimental coagulation rate constants were obtained by assuming a mono- disperse system and expressing the coagulation rate in the form of the Smoluchowski l1 equation N = $Kn2 (4) where K is the coagulation rate constant. Combining eqn (2) and (4) gives d(l/n)/dt = K/2-N,/n2 (5) which shows there exists a linear relationship between n-l and t when the rate of nucleation is sufficiently small the slope of the line being K/2.Inverse particle ~ A' <,.a&' ' __.--..I/__i !23456789 height above burner/cm FIG.3.-Gas volume per particle at different growth stages [electron microscope data of this study ( x ) ; calculation from electron microscope data of Homann and Wagner (A); calculation from absorption data of Bonne and Wagner (e)]. number plots are shown in fig. 3 for the electron microscopy data of both this study and that of Homann and Wagner l2 and absorption data of Bonne and Wagner.13 Particle number concentrations of Bonne Homann and Wagner were calculated from their values of soot mass fraction and particle diameter. Differences between the present data and those of the previous studies are due in part to the facts that the previous workers used an equivalence ratio of 3.5 and Bonne and Wagner in the calibration of their absorption measurements did not allow for change in the extinc- tion cross section of young carbon particles with change in particle composition.COAGULATION OF CARBON PARTICLES The approximately straight part of the curves downstream of about 4cm height above the burner indicates a predominant coagulation of carbon particles to larger clusters; twice the slope of each line is the experimental coagulation rate constant for the flame conditions and zones represented. Additional information is derived from eqn (3) which in the region where surface growth and nucleation are negligible compared with coagulation reduces to d[ln(n)]/3d[ln(a3)] = -1. (6) Values of the characteristic ratio identified by eqn (6) at different heights above the burner are shown in fig.4. The numerical evaluation of particle number concentra-tion extrapoIates the values of this study to 8 cm height above the burner and shows that the characteristic ratio is scattered due to measuring errors. Its average is almost exactly -1 indicative of predominant particle coagulation downstream of 4.5 cm height above the burner. The characteristic ratio calculated from the absorp-tion measurements of Bonne and Wagner l3 is -1 from 4 to 6 cm height above the burner. In this region the particles are approximately spherical and their size and number concentration change predominantly by coagulation. Although coagulation dominates also between 7 and 9 cm the characteristic ratio nevertheless increases in this region due to the use of the diameter of the spherical units within chains - 1.4 -"-2 -!.L3%5b7.33 height above burnerlcm FIG.4.-Characteristic ratio for particle coagulation [electron microscope data of this study ( x ) ; calculation from electron microscopedata of Homann and Wagner (A);calculationfrom absorption data of Bonne and Wagner (O)].which remained nearly constant instead of the volume equivalent diameter of the cluster. Similar to the values obtained by electron microscopy in the present study the deviations from -1 in the region from 2 to 3 cm are due to surface growth and nucleation influences. The values obtained for the study of Homann and Wagner l2 are rather scattered which may be due to the fact that particles less than 40 diameter J.B. HOWARD B. L. WERSBORG AND G. C. WILLIAMS were not included in the electron microscope analysis. Thus experimental coagula-tion rate constants are only obtained from the measurements of Bonne and Wagner between about 4 and 7 cm height above the burner and from the present data down-stream of the 4.5 cm position. The values found are given below in terms of the kinetic theory collision model. Carbon particles in low pressure flames are under free molecular flow conditions and therefore constitute a highly dispersed aeros01.l~ If the size distribution is approximated as monodisperse the coagulation rate constant is K = 16a2y(zkT/m)) (7) where a is the mean particle radius y is a correction factor accounting for inter-particle forces kis the Boltzmann constant T is temperature and m is particle mass here estimated by assuming particle density to be 2 g/cm3.Experimental y factors shown in fig. 5 were obtained from eqn (7) using K values calculated as described above from line slopes in fig. 3 and a gas velocity of 3 m/s at 2000 K. Values for all flame positions studied are presented for completeness but the only values sufficiently free of nucleation effects so as to reflect coagulation behaviour alone are those in the regions described above as yielding coagulation rate constants. In these regions y appears approximately constant indicating little or no net particle size and tempera-ture influences The y values from this study are approximately 29 and very similar to those of the absorption study which are around 21.n A W a, Y 20 20- C .-. /d c.’ I ‘a 1 M ---__ .-I----____ ----__A-A-8 A A 8 /.// I / 2 0 1234567a9 height above burnerlcm FIG.5.-Experimental coagulation rate factor (y) [electron microscope data of this study (x ) ; calculation from electron microscope data of Homann and Wagner (A);calculation from absorption data of Bonne and Wagner (O)]. In an attempt to explain why the observed coagulation rate is much larger than the kinetic theory collision rate the contributions of Van der Waals and electrostatic forces were examined by calculating theoretical y values for simplified cases described presently.Cloud shielding and diffusional effects are negligible under the experi-mental conditions and the particles are again assumed to be uniform in size. The effects of gas-particle collisions on the energy and monentum of two interacting COAGULATION OF CARBON PARTICLES particles are neglected. The contribution of Van der Waals forces to the potential of two spherical particles of radius a with distance r between centres is E = -(H/12)[z-'+(z-1)-'+2 In (1 -z-')] (8) where H is the Hainaker constant related to the London-Van der Waals constant OL by H = n2g2u,g being the number of atoms per unit volume in the particles and z = (r/2a)2. The electrostatic contribution to the potential using Maxwell's l6 method of electric images is Ee = (kee2/2a>[(s1I$- '-1)(Q +Q:) +2( a/r>siz$-'QiQ2I (9) where k = 9 x 109Jm C-2 e = 1.6~1O'l9C $ = S -(a/r)2S:2 a3 s, = (i-e2)c eyi-e2ni+2) m=O /J=[(1 +8)a/rI2,8=y-(y2-l)f y=r2/2az-l and eel and eQ2 are the particle charges.The equivalent equations for unequal particle radii are given elsewhereO6 If both particles are charged with the same polarity opposing electrostatic repul- sion and Van der Waals attraction lead to a positive maximum in the potential E(r,) = E +E at r = rm where rm> 2a. Since collision is then limited to particles whose initial kinetic energy re€ative to axes moving with the mass centre exceeds E(r,) it is reasonable to assume that y = exp [-E(r,)/kT]. (10) In all other cases including both particles charged but with opposite polarity only one particle charged and both particles neutral the interparticle forces are attractive and from classical analysis of the two particle encounter,17 the collision cross section is increased by the factor y = (1 +&/a)2[1-2E(&)/pv;] (11) where 2~ is the minimum separation distance to within which the particles' surfaces may approach without resulting in collision E(E)is the value of E,+& when r = 2a+2~, p is the reduced mass here given by m/2,and u is the relative velocity of the particles.Since the average initial kinetic energy of the pair relative to axes moving with their mass centre is 2kT,'* u is taken as 4kT/m and pu; becomes 2kT. Therefore y = (1 +&/a)2[1-E(&)/AT]. (12) The proper value of E is that value for which y is rninim~rn.'~ If neither particle is charged eqn (8) and (1 2) give y = z,C1+(~/12k~){z,' +(zm-l)-' +21n (I-Z;'))] (13) where z, which is the value of z at r = 2a+2~,is the root of the equation (3-22,)/(2,-1)'-2 In (1-2;') = 12kT/H.(14) If one particle is charged or if both particles are charged but of opposite polarity E is found by numerical or graphical minimization of y in eqn (12) using eqn (8) and (9). J. B. HOWARD B. L. WERSBORG AND G. C.'WILLIAMS Values of y calculated as described above for different particle sizes states of charging and H values are shown in fig. 6. The value of N for carbon particles in flames is not known but it should be within the range 10-20-10-L8 If neither J. particle is charged y is independent of particle size and equal to 2.75 for the largest H here considered.If one particle carries one charge y is increased by image forces particle diameter 2a/A FIG.6.4ncrease in collision cross section of two equal sized particles by Van der Waals and electro- static attraction [Ql = -Q2 = 1 (solid); Ql = 0,Q2 = k1 (broken) ; Q1= Q2= 0 (dashed) ; Hamaker constant = lO-'O(a) 10-19(b),10-18(c) J; T = 1800 K). but this effect compared with that of Van der Waals forces alone is substantial only for small particles and small values of H. It appears that the y resulting from image forces cannot be substantially larger than 3 if the particles have predominantly only one charge. If both particles carry one charge of opposite polarity y is of order 10 for small particles.However if ambipolar charging predominates repulsion between particles of the same polarity must also be considered and the net effect on coagulation rate may be small. In view of the above calculations and the observed condition of 0 or 1 positive charge per particle it appears that Van der Waals and electrostatic forces together may account for a factor of only 2 or 3 increase in the coagulation rate constant under the conditions studied. The situation may however be quite different for other flame conditions in which particle ionization is more pronounced and even under the present conditions the detected charge must be regarded as one cause for the chained appearance of the final carbon particles. Nevertheless a factor of order 10 increase in y remains to be explained.Other possible causes meriting study include polydispersity and deviations from the assumed sphericat shape. 2o Our calculations not reported here show that polydispersity exerts less than a 20 % increase if the particles are assumed to be neutral hut the actual effect undoubtedly COAGULATION OF CARBON PARTICLES exceeds this prediction since y for small charged particles colliding with large neutral particles can be significant. The possibility that substantial numbers of particles could be decomposed or burnt out seems unlikely for the fuel rich conditions employed. The inability to predict the coagulation rate constant in the flame region in which it can be measured i.e. in the region where nucleation has practically ceased prevents confident prediction of y values in the region of significant nucleation.In spite of this shortcoming the cumulative number concentration of carbon particles may be calculated by integrating eqn (2) numerically using the experimentally determined value of y. This calculation is quite insensitive to the extrapolation of y since the coagulation rate constant is most important in the zone of carbon formation where it can be determined experimentally or just upstream of this zone in a region where the extrapolation will not be far from the experimentally determined value. In the early stages of carbon formation particle coagulation seems to be unimportant due to the small particle number concentration.In this range the change in particle number is approximately equal to the nucleation rate.’ The cumulative number concentration calculated with a constant y = 28.7 is shown in fig. 7 as a function of height above burner and there compared with the experimental particle number concentration. It is apparent that the contribution of coagulation to particle growth is small up to about 3 cm height above the burner. This approximation implies that the particles nucleated first grow predominantly by surface growth to a volume mean diameter of around loOA. The ratio of cumulative number concentration to the prevailing number concentration gives the cumulative number of particles appearing under the electron microscope per prevailing particle. This ratio (fig. 7) is believed to be a good approximation to the average number of nuclei or original 01234567 height above burner/cm FIG.7.-Instantaneous and accumulative particle number concentration and average number of nuclei per particle at different growth stages [accumulative particle number concentration ( x ) ; instantaneous particle number concentration (A);ratio of accumulative to instantaneous particle number concentration (011.J. B. HOWARD B. L. WERSBORG AND G. C. WILLIAMS particles in each prevailing particle because the concentration of unobservable particles should be small owing to the good lower limit of visibility (15 A) and the rapid growth of young particles by surface reaction. Surface growth of carbon particles becomes quite small after 4 cm height above the burner which position coincides with the onset of predominant chain formation.' Thus the number of nuclei per spherical unit in a chainlike particle is about equal to that calculated between 4 and 5 cm height above the burner.This ratio is of order 10 and is substantially smaller than the number of crystallites per spherical unit which is of order lo3. According to this result crystallites in carbon particles do not represent former particles. Thus the structure of these particle domains may be used to locate zones of predominant surface growth which in turn identify nuclei within spherical units. We are grateful to Project SQUID whose support under contract N00014-67-A- 0226-0005 NR-098-039 made this work possible. K. H.Homann Angew. Chem. Int. Ed. 1968 7,414. E. BartholomC and H. Sachsse 2. Elektrochem. 1949 53 326. C. P. Fenimore and G. W. Jones Combustion Flame 1969 13 303. E. R. Place and F. J. Weinberg Eleoenth Symp. (Znt.) on Combustion,(The Combustion Inst. Pittsburgh 1967) p. 245. J. B. Howard Twerfth Symp. (Int.) on Combustion (The Combustion Inst. Pittsburgh 1969) p. 877. R. T. Ball and J. B. Howard Thirteenth Symp. (Int.) on Combustion (The Combustion Inst. 'B.Pittsburgh 1971) p. 353. L. Wersborg J. B. Howard and G. C. Williams Fourteenth Symp. (Znt.) on Combustion (The Combustion Inst. Pittsburgh 1973 p. 929). L. Fox S.M. thesis (1972 Massachusetts Inst. Tech. Cambridge Massachusetts). F. A. Heckman personal communication 1971 Cabot Corp.Billerica Massachusetts. lo U.Bonne K.H. Homann and H. Gg. Wagner Tenth Symp. (Int.) on Combustion (The Com- bustion Inst. Pittsburgh 1965) p. 503. l1 M.von Smoluchowski 2.phys. Chem. 1917,92,129. l2 K. H. Homann and H. Gg. Wagner Ber. Bunsenges. phys. Chem. 1965,69,20. l3 U. Bonne and H. Gg. Wagner Ber. Bunsenges phys. Chem. 1965 69 35. l4 B. L.Wersborg Sc.D thesis (1972 Massachusetts Inst. Tech. Cambridge Massachusetts). H. C. Hamaker Physica 1937,4 1058. l6 J. C. Maxwell A Treatise on Electricity and Magnetism (Dover Publications Inc. New York 1954 republication of 3rd ed. of 1891) vol. 1 chap. 11 pp. 244-283. l7 J. 0. Hirschfelder C. F. Curtiss and R. B. Bird Molecular Theory of Gases and Liquids (John Wiley and Sons,Inc. New York 1954) chap. 1 pp.45-51. S. Chapman and T. G. Cowling 77ze Mathematical Theory of Non-Uniform Gases (Cambridge University Press Cambridge 2nd ed. 1962) chap. 5,p. 93. l9 N. A. Fuchs and A. G. Sutugin J. Colloid Sci. 1965,20,492. 'O G. Zebel Aerosol Science ed. C. N. Davis (Academic Press New York 1966) chap. 2 p. 31.