Brownian Coagulation of Aerosols at Low Knudsen Number -f BY GILBERT AND MILTONKERKER A. NICOLAON * Dept. of Chemistry and Institute of Colloid and Surface Science Clarkson College,of Technology Potsdam New York 13676 U.S.A. Receiued 15th January 1973 Dibutyl phthalate aerosols of narrow size distribution have been prepared in a falling-film gener- ator using nitrogen rather than helium as the carrier gas. The Knudsen number in nitrogen is considerably lower so that the Cunningham correction is much less. A large number of coagulation experimentsgive average coagulation times in excellent agreement with theory. However thespread ofthe values ismuchgreater than obtained earlier in helium. The spread of the results reported here may be due to the lower thermal conductivity of nitrogen.Smoluchowski's theory of Brownian coagulation of aerosols in which the motion of the particle is controlled by the Stokes-Einstein diffusivity is valid in the hydrodynamic domain where Kn = L/a+1. (1) The Knudsen number is the ratio of the mean free path 3. of the gas molecules to the radius of the particle a. When this becomes as large as unity the empirical Cunning- ham correction must be ulilized leading to This correction is given by Davies as A = 1.257+0.4. . .exp (-1.10 a/R). (3) The first integral of eqn (2) describes the rate of formation of particles of radius ah by coagulation of particles of radius a with those of radius ah-[. The second integral gives the rate of disappearance of particles of radius ahby coagulation bith other particles.The smallest and largest classes of particles are given by a and a,. The particle concentration of class a is given by n,,' k is the Boltzmann constant Tis the Kelvin temperature and v is the viscosity of the gas. When the Knudsen number is large (Kn>10) the particles can be treated as if they were large molecules and the coagulation can then be described by the kinetic theory of gases. Lai et aL3 have discussed this case recently. There is a transition regime (10>Kn>O.l) for which the fluid mechanics of the particles have not yet t This project has been financed in part with federal funds from the National Science Foundation under grant number GP-33656X. 133 BROWNIAN COAGULATION OF AEROSOLS been reduced to theoretical analy~is.~ Indeed it is the lower end of the transition regime (1 .O> Kn> 0.1) which is treated with the aid of the Cunningham correction.For aerosols suspended in air at atmospheric pressure and 20°C the transition regime corresponds approximately to 1.5 nm <a< 150 nm and it includes in the atmosphere the important class of so-called Aitken nuclei which probably act as condensation and freezing nuclei. Although Smoluchowski's theory was published more than 50 years ago numerical solutions to his non-linear integro-differential equation have been obtained only recently with the advent of electronic digital computers. Virtually all earlier work had been restricted to the initial rate of coagulation of a monodisperse system for which the process follows second-order reaction kinetics where N is the total particle concentration.Experimental studies have lagged even further. These have mainly utilized eqn (4) and typically have attempted to demonstrate merely that the particle concentra- tion follows the second-order rate law hopefully with a rate constant close to that predicted by the theory. Frequently these aerosols were poorly-defined systems and the particle sizes shapes and concentrations were not determined accurately so that the results can hardly be considered definitive. We have recently completed an experimental study of the Brownian coagulation of an aerosol for which Kn = 0.78. The results were in agreement with Smoluchow- ski's theory of Brownian coagulation as modified by the Cunningham correction [i.e.eqn (2)] and we plan to extend this experiment throughout the entire range of Knudsen numbers viz. the free molecule the transition and the hydrodynamic regimes. This paper reports the next step in this programme. It is a study of coagulation at a lower Knudsen number (Kn = 0.20).* This lower value of the Knudsen number was obtained by preparing the dibutyl phthalate (DBP) aerosol in nitrogen rather than in helium as in the earlier work. Thus it was mainly the mean free path of the gaseous medium that was altered rather than the aerosol particle size and this permitted utilization of our previously developed light-scattering technique for monitoring the particle size distribution as well as an aerosol generator similar to the one which had proved so successful in the earlier work.A number of modifi- cations were made in the preparation of the aerosol when nitrogen was utilized as the carrier gas in place of helium. DBP AEROSOLS IN NITROGEN The aerosol generator has been described ear1ier.*-l1 A mixture of the gas and NaCl nuclei flows laminarly down a vertical tube along whose wall flows a film of DBP maintained at an elevated temperature. Aerosol is formed upon cooling to room temperature by condensation of the DBP upon the nuclei. The monodispersity of the aerosol can be significantly improved by evaporating and then recondensing the initial DBP aerosol. This will be termed a regenerated aerosol. Heat transfer and convective diffusion calculations are in agreement with measurements of the temperature distribution and the extent of saturation of the vapour respectively.* This value of the Knudsen number corresponds to the modal value in the size distribution. For the uncoagulated systems the coefficient of variation was about 0.10. However for the coagulated systems the spread of sizes becomes considerably greater. G. A. NICOLAON AND M. KERKER The influence upon the particle size distribution of parameters such as furnace temperature,' DBP boiler temperature,8 flow rate,8 number and size of nuclei,'" and cooling rate,l0 have been discussed. F1LTER.-+ rl t FIG.1.-Filter for collection of aerosol. The major procedural change in this work with nitrogen was to collect the aerosol for gravimetric analysis by filtration rather than by thermal precipitation.There was considerable leakage of aerosol through the thermal precipitator 'when nitrogen was used presumably because of the lower thermal conductivity of nitrogen compared to helium. Fig. 1 depicts the filter. It utilized millipore filters with pore diameters of either 1.2 or 3.0 pm. The results in table 1 verify that the amount of aerosol collected was independent of pore size over the range 0.8-8.0 pm. TABLE 1.-AMOUNTOF AEROSOL COLLECTED FOR DIFFERENT PORE SIZES OF THE MILLIPORE FILTER (CARRIER GAS N2; FLOW RATE 1 l/min ; BOILER TEMPERATURE 110") pore diam./ mass/(mg/l) rnass/(mg/l) /rm (expt 1) (expt 2) 0.8 0.76 0.79 1.2 0.78 0.79 3.O 0.75 0.80 8.o 0.74 0.78 We have also noted even for helium that the mass concentration of aerosol obtained when collection was by thermal precipitation was about 5 % less than that by filtration.This would give higher aerosol number concentrations in the earlier coagulation work with better agreement between theoretical and experimental coagulation times. The percentage saturation of the DBP vapour at the exit of the boiler was calculated using the earlier convective diffusion theory and this is compared in table 2 with the experimental results at two flow-rates. Also listed in this table are new results for helium based upon collection by filtration rather than thermal precipitation. The agreement is excellent particularly since there is significant uncertainty both in BROWNIAN COAGULATlON OF AEROSOLS values for the diffusion constant of DBP as well as the actual temperature of the DBP at the vapour-liquid interface.In this connection a comment on the teinperature of the DBP is in order The elevated temperature of the DBP filiii is niaintaincd by circulating oil at constant temperature through an external jacket. In effect the gas stream is always slightly cooler than the oil so that the temperature of the DBP at the vapour/liquid interfxe TABLE 2.-cOMPARISON OF THE PERCENTAGE SATURA'TION AT THE EXIT OF THE VAPORIZER OBTAINED THFORETICALLY AND F.XPEH1Mt.N'TALI.Y (DBP TEMPERATURE 108") gas flow rate,'(I min) saturation t heorct ical ";saturation u xper iment aI He 1 .o 96 95 He 2.0 84 88 N* 1 .o 74 82 N2 2.0 53 51 is also probably cooler.Thus for the above convective diffusion calculation we have assumed a DBP temperature of 108 in view of the fact that the oil temperature was 110" and the temperature withir the gas stream was 106-107". Acutally the measured concentration of DBP provides a better criterion for the effect ofthe boiler conditions upon the properties of the aerosols than the " boiler temperature " and the former quantity will be utilized henceforth. The coagulation experiments to be described in the next section were carried out (as in the earlier work) with a standard aerosol. The conditions were nitrogen flow rate 1.0l/min; furnace temperature 590°C ; DBP flow rate 25 ml/min ;concentration (by filtration) 0.78mg/l.This aerosol was regenerated in the manner described earlier. We have prepared and analyzed the particle size distribution of several hundred DBP aaosols in both helium and nitrogen over the past three years and these results are summarized in table 3. The second column pertains to the standard aerosol in helium the third column to the regenerated standard aerosol in hetium and the fourth column to the regenerated standard aerosol in nitrogen. The operating condItIons for the helium system are helium flow rate 2.0 l/min ;furnace temperatuw 590'C ; DBP flow rate 25 ml/min ; concentration (by filtration) 0.84 mg/l. The conditions for the nitrogen system have been given above. The low values of the standard deviation of the modal radius and of the coeficient of variance indicate the high degree of reproduoibility obtainabk with this aerosol generator.TABLE 3.-sUMMARY OF PARTTCLE SIZE ANALYSES FOR DBP AEKOS0I.S IN HF.1-IUM ASD IN NlTROCiEN DgP in heliurv DBP in nilrogcn DBP in helium regeocratsd regennaf ed no. of runs 210 100 125 modal radius/prn 0.235 0.236 0.314 std. deviation 0.005 0.006 0.007 coefficient of variation 0.16 0.10 0.41 std. deviation 0.01 0.oi 0.0 1 Just as in the case with helium the effect of regeneration in nitrogen was to give a npxe monodisperse aerosol wi,th the Same modal radius. Also even a narrower sizc.distribution was obtained if only the aerosol near the axis or near the wall is G. A. NICOLAON AND M. KERKER sampled. Indeed in such a case the aerosol is about as monodisperse as the well- known Dow polystyrene latexes.The effects are illustrated in fig 2 and 3. in fig. 2 the angular distribution of the polarization ratio of the scattered light is plotted against scattering angle for (A) a standard aerosol which has not been regenerated (B) for a regenerated standard aerosol and (C)for the latter aerosol which has been sampled from within the axial region. The polarization ratio is the ratio of the polarized radiance whose electric vector is parallel to the scattering plane relative to the polarized radiance whose electric vector is perpendicular. The procedure for inverting these data to obtain the logarithmic particle size distribution is described el~ewhere.~ The corresponding size distributions are plotted in fig.3. The modal value of the radius is 0.240pm for each aerosol but the breadth parameters (which correspond closely to the co- efficient of variation) are 0.16 0.10 and 0.04 respectively. Although these particular examples were selected from results with helium similar effects were obtained with nitrogen. I 1 46.1 70 I00 I30 6 Fw. 2.-PolarizatiOn ratio against scattering angle for (A) standard DBP in helium ; (B) the same which has undergone evaporation and condensation; (C) the same as (B) which has been sampled from the axial region. COAGULATION OF DBP AEROSOLS IN NITROGEN The coaguhtion experiment was similar to that described earlier The initial size distribution of the regenerated standard DBP aerosol in .nitrogen was determined BROWNIAN COAGULATION OF AEROSOLS by light scattering early in the life history of the aerosol.This must occur prior to appreciable coagulation since inversion of the light-scattering data with the aid of the Mie-Lorenz functions is accurate only if the distribution is narrow and is unimodal. 3 8 ct x radiuslpm FIG.3.-Size distribution corresponding to aerosols (A) (B) and (C). whose polarization ratio against scattering angle is plotted in fig. 2. The modal value of the radius is 0.240pm. The breadth parameters uo = 0.16 0.10 and 0.04 respectively. Light-scattering data were then also obtained at later times after passage through hold-up tubes of various volumes. The size distribution of the aerosol was calculated as a function of time using the initial size distribution and Smoluchowski’s theory of Brownian coagulation.Theoretical light-scattering results corresponding to the distribution for the coagulated systems were calculated and these were then compared with the experimental light-scattering data. The flow chart for the calculation is outlined in fig. 4. Presumably if the experimental data can be fitted to a calculated result the coagula- tion mechanism proceeds in accordance with Smoluchowski’s model. Furthermore if the experimental and calculated time scales agree there is no potential barrier to coalescence upon collision. If the experimental time is greater than the calculated time the collision efficiency is less than unity and the potential barrier can be cal- culated.12 If the experimental time is less the aerosol is coagulating faster than predicted by Brownian diffusion so that other mechanisms must be involved.One improvement in the procedure used in this work was to make 4 to 6 light- G. A. NICOLAON AND M. KERKER scattering analyses of the aerosol over an extended period of time prior to obtaining any coagulation data and using the average of the values for the size distribution. The values obtained were usually similar attesting to the stability of the aerosol generator. However on occasion there would be a small deviation and if this transient value had been used to characterize the initial aerosol the calculated size distribution of the coagulated aerosol would have been significantly different.LIGHT SCATTERING DA~A LIGHT SCATTERING DATA FOR INITIAL AEROSOL FINAL AEROSOL -FOR 1 1 I. INVERSION Of DATA 4 COMPARISON 1 t INITIAL SIZE DISTRIBUTION CALCULATED LtGHT SCATTERING RESULTS &* I q I M. I I \ / 2 BROWNIAN COAGULATION 3 LIGHT SCATTERING CALCULATION 6'\ / CALCULATION 1 FfWL SIZE DlSfRlBUtlONl FIG.4.-Flow chart for coagulation calculation. Average hold-up time is t' ; average calculated time is t". Another procedural change was to utilize for the aerosol number concentration the value based upon a gravimetric analysis of each particular run. In the earlier work an average value for many runs had been used and the concentration was not determined for each particular run. The aerosol was collected for weighing by TABLE 4.-vARIATION OF NUMBER CONCENTRATION OVER A PERIOD OF 12 DAYS modal value breadth mass of radius parameter.concentrationI number concentration no. omlm 00 (mg/l) no./(10-6 cm-3 1 0.308 0.10 0.75 5.5 2 0.309 0.11 0.79 5.6 3 0.315 0.12 0.76 5.0 4 0.311 0.11 0.72 5.0 5 0.313 0.11 0.74 5.1 6 0.317 0.12 0.77 5.0 7 0.313 0.12 0.78 5.3 8 0.314 0.11 0.80 5.4 9 0.311 0.10 0.79 5.6 10 0.320 0.12 0.79 5.O 11 0.308 0.10 0.79 5.8 12 0.311 0.11 0.75 5.2 filtration rather than by thermal precipitation. The extent of variation in the number concentration for a series of runs carried out over a period of 12 days is shown in table 4. The day-to-day variation was sufficient to introduce a significant error in the number concentration of the initial aerosol had the average value been used.On the other hand the concentration was stable over the course of a day run. BROWNIAN COAGULATION OF AEROSOLS The results are shown in table 9. The experimental timCs are average residence times obtained from the volume of the hold-up tube and the htv rate. The third column termed quasi-static time is the dlculated time with the assumption that each fluid particle spends the same amount ofi-time in the hold-up tube. Actually the aerosol is in Poiseuille flow with a pambolic.velocity profile. The aerosol near 1 the wali is moving much mote slowly than the aerosol abng the axis of the tube and is therefore undergoing coagulation for the longer time. The results in the last column were obtained by the procedure outlined earlier for transforming quasi- static to Poiseuille times The angular distribution of the polarizatioo'ratio which is used in these experimmts to monitor the coagulation goes through a sequence of states for the Poiseuille flow calculation which is similar to that for the quasi-static calculation except that the former proceeds more slowly.This permits preparation of a calibration curve. Then the calculation is carried out according to the scheme of fig. 4 assuming quasi-static flow varying tl untit the calculated fight-scattering results best fit the experimental data. Finally the corresponding Poiseuillc time is obtained from the calibration curve. The fourthscolumn in table 5 gives the standard deviation for the quasi-static times.TABLE 5.-cOMPARImN OF EXP6RIMENTAL AND CALCULATED COAGULATION TIMES (SECONDS) exptl t imc no. of runs quasi-Statictime ad Poi~uillc t imc 82 32 67 27 84 154 105 116 49 145 235 23 196 41 245 327 24 26s 61 340 The miterioa for best fit was the minimum value of tk deviation measure given by ( 130" where p(0) and p'(0) are the measured and calculated values of the polarization ratio at each of the 19 angIes obtained between 40"and 130" zt 5" intervals. Fig. 5 illustrates a typical example of the fit of the calculated results to the experimental data which are plotted as points. The curve represents the angular distribution of the polarization rates calculated for that coagulation time for the initial aerosol which minimizes the deviation measure (eqn (5)) For this cxample the experimental hold-up time is 154 as.The catculated result gives a hold-up time of 169 s with a Corresponding deviation measure' of 0.10. ' 6 * Although the average values of the coagulation time in table 5 agree well with the experimental coagulation times there is a considerdble spread in the individual values as indicated by the standard devations. In ordeb to determine whether this spread arose from the accuracy in fitting the experimental and calculated results all runs were eliminated for which the deviation measure'was greater. than 0.25.. The redts are shown in table 6. Theagreement between the experimentaland calculated coigula- tion times is not affected significantly (it is slightly poorer) nor is the spread of the results any narrower.Accordingly the light-scattering analysis does not appear to be a factor in accounting for the spread of the.coagulation time. There is another factor which; may account fgr.t,hese,resultg. We hare observed occasionally sporadic convective " storms ** in these aerosols in nitrogen particularly in the bold-up tubes in contrast to #thequiescent appearance under illumination of the helium system. Ths,tendewy,rnay be-due to the low heat conductance (and 0.A. NlGOLAON AND M. KERKER sharper temperature gradieats) of .nitragen and the randomization d the results which was not,encountered in the earlier work with helium &t be caused by this effect. The subsequent mi>cing,would tad to makc cbe system deviate from the condition of Poiseuille flow and inore resemble the well-mixed system which we have e Frc;.5.-Angular distribution of polarization ratio. Points are measured values for experimental time of 1% s. Curvc corresponds to calculated time(169 s) which best fi6 these values (D= 0.10). called the quasi-static model. One would expect then that the coagulation time calculated according to Poisetdle flow would be an upper limit and that the average values would be lower. Any barrier to coalescence corresponding to a coalescence efficiency of less than unity would lengthen the coagulation time. A combination of these two effects could account for thc observed results. TABLE 6.-cOMPARlSON OF EXPERlMENTAL AND CALCULATED COAGULATION TIMES (SECONDS).RUNSWITH DEVIATIOS MEASURES GREATER THAN 0.25 ELIMINATED exp1l t irnc no. o rnns quas,i-statictime s.d. Poiseuille time 82 20 71 22 89 154 59 127 47 I58 235 21 203 35 254 327 24 268 61 340 A question has been raised about the stability of the aerosol to evaporation during its passage through the coagulation tube. There is the possibility because of the Kelvin effect that material might distill to the walls or from the smaller to the larger particles or both of these effects might occur. The possibility of distillation to the walls has been checked repeatedly by collecting and weighing the aerosol prior to entrance and upon emergence from the coagluation tube and we were unable to detect any hold-up. Furthermore the light-scattering data can only be interpreted by an increase in average particle size which would not be the case were distillation to the walls to occur to a significant cxtcnt.Distillation from the smaller to the larger particles-for which the driving force is much less than distillation to the walls- BROWNIAN COAGULATION OF AEROSOLS would have the effect of shifting the size distribution to a greater average size just as coagulation. However since the effects which we have observed are accounted for by coagulation which must proceed in any case it seems highly unlikely that evaporation of these particles occurs to any significant extent in the course of these experiments. We have noted the much greater variation of these experimental results in nitrogen compared to earlier results in helium and have attributed this to convective storms in the nitrogen system due to its lower heat conductance.We doubt if this effect can be attributed to the possibility of evaporation of the dibutyl phthalate particles. The average particle size in the helium work was smaller (a = 0.24pm compared to 0.31 pm) and hence the Kelvin effect was greater. Furthermore the rate of evaporation is greater in helium than in nitrogen. Yet the kinetics of the process is accounted for in both cases by coagulation ;the agreement in helium was even more striking than in nitrogen. M. Smoluchowski 2.phys. Chem. (Lpg) 1917,92 129. C. N. Davies Proc. Phys. Sac. 1945 57 259. F. S. Lai S. K. Friedlander J. Pich and G.M. Hidy J. Colloid Interface Sci. 1972 39 395. G. M. Hidy and J. R. Brock The Dynamics of Aerocolloicial Systems (Pergamon New York 1970). H. L. Green and W. R. Lane Particulate Clouds (D. Van Nostrand New York 1957). G. Nicolaon M.Kerker D. D. Cooke and M. Matijevic J. Colloid Interface Sci. 1972 38 460. M. Kerker The Scattering of Light and other Electromagnetic Radiation (Academic Press New York 1969). * G. Nicolaon D. D. Cooke M. Kerker and E. Matijevic,J. Colloid Interface Sci. 1970,34,534. G. Nicolaon D. D. Cooke E. J. Davis M. Kerker and E. Matijevic J. Colloid Interface Sci. 1971,35,490. lo G. Nicolaon and M. Kerker J. Colloid Interface Sci. 1973 42 to be published. E. J. Davis and G. Nicolaon J. Colloid Interface Sci. 1971 37 768. I* N. A. Fuchs The Mechanics of Aerosols (Macmillan New York 1971).