GENERAL DISCUSSION Mr. K. C. Pointon (Shefield University) said Our method is likely to produce pure particles because if the background temperature is sufficiently low they will have crystallized before touching any foreign surface. The shapes and sizes of fall-out particles were examined by scanning electron microscopy. Particles in the clouds are allowed to settle on to a flat tip of highly polished silica held by the substrate probe. The probe is then withdrawn from the generating chamber into the glass manipulation section where a layer of 60 :40 Au-Pd approximately 1008 thick is deposited on the silica surface by vacuum evaporation. This is to protect the fall- out from reaction with the atmosphere during transfer to the electron microscope. Fig. I and 2 are micrographs of fall-out from Zn aerosols produced at a background temperature of 0.74 Tf.The Au-Pd film deposited on the substrate shown in fig.la and Ib was thick and has exfoliated ; most of the Zn particles have adhered to the Au-Pd layer but where particles have remained on the silica surface holes have been plucked in the exfoliated film. The Zn particles are typically 1-5pm diam. and have well-developed crystal faces although the particle shape is sometimes complex. A polyhedral crystat with a hexagonal profile would appear to be typical of the crop of particles formed by condensation in the cloud chamber. Kimoto et a2.l. have observed the same well-defined crystal habit for Zn particles prepared by evaporation in purified argon at low pressure.Their crystallites were equiaxed and the morphology was consistent with the normal c.p.h. lattice of Zn. Kimoto reports that a small amount of oxygen in the argon caused a remarkable deterioration in the crystal habit of the particles; they became very irregular and rough. Most of the particles shown in fig. 1 and 2 are well formed crystallographic- ally which suggests they are pure. If twinkling results from the scattering of light by reflection at crystal faces a spheroidal particle with a large number of flat faces would be expected to twinkle rapidly and rather weakly. This may explain the indistinct flickering observed. It would be difficult to understand however why the Zn particles in the form of hex-agonal prisms clearly visible in fig. 2b did not show a distinct twinkling effect.From the telescopic observation that the Airy images of the particles in a Pb smoke became brighter as the smoke thinned out we believe Pb aerosols must agglomerate considerably at high particle densities. We have yet to test this inter- pretation by examination of Pb fall-out. Fall-out from clouds of Zn aerosols shows that agglomeration was a rare event (fig. 2a 21). This is in keeping with the absence of a brightening effect in this case. Dr. S. C. Graham (Shell Res. Ltd. Chester)said 1do not consider that small metal particles with x<O.1 where x = nd/A would give preferential back scattering at visible wavelengths. As indicated in my reply to Kerker’s question following my paper with Homer the critical terms in the expansion of Qsca as a power series in x are the lowest order contributions to the expressions for the electric and magnetic dipole terms a and bl respectively.Tn the limit as x -+ 0 (munspecified) QbCd = (6/.~2>(1a,12-I-lbll’) where la1I2 = IsI2x6and lbl12 = Is12u2x10. ‘ K. Kimoto Y. Karniya M. Nonoyama and R. Uyeda Jup. J. Appl. Phys. 1963,2,702. K. Kimoto and I. Nihida Jag. J. Appl. Phys. 1967 6 1047. 97 s7-4 GENERAL DISCUSSION Thus the importance of lb112which causes the high back scattering is determined by the ratio For the simple Rayleigh Q,J= 6/x21p12x6)to be accurate to 1 % one must have at least an approximate upper limit on x,viz. xG0.1. But for x = 0.1 the ratio lb112/1a112reaches 1 % when luI2 = i.e.luI2 = 100 so that = 100 or lrn2+2I2 = 9x lo4. 1 1 Assuming that m has the approximate form m = (1/~2)Iml(l-i) which is a good approximation for metals at infra-red and longer wavelengths where such high m values are found (ref. (17)) chap. 14) then Jm2+2I2= lml4+4 = 9 x lo4 so that Iml = 17.3 and m = 12.3-12.3i. If one takes xGO.31 as the limit of the Rayleigh scattering domain for real m then lb112/1a112 reaches for lm2 +212 = 9 x lo2 or m2x30. There are few if any metals with such high values of m2 at visible wavelengths and at x = 0.3 the simple Rayleigh expression will certainly not be accurate to within 1 % even for real m values. Though metals do not have such high values of m at visible wavelengths much higher values are found at longer wavelengths.For example van der Hulst (ref. (17) pp. 268 and 288) quotes an m value of 37 -41i for platinum at A = 10 pm and 236-2361' for silver at A = 30 pm. For larger values of x,particularly 0.5 <x,< 5 different considerations apply and the back scattering efficiencies of metals at visible wavelengths vary in a manner similar to that found for a perfectly reflecting sphere and I am grateful to Kerker for drawing my attention to this possibility. Van der Hulst (ref. (17) chap. 14 especially pp. 284-287) has shown for spheres with m = 3.41 -1.94i (not untypical of the m values of metals at visible wavelengths) that the scattering efficiencylsteradian in the backward direction rises monotonically from x = 0 to a large maximum near x = 1 with a deep minimum at x = 1.65 followed by further less extreme maxima and minima.For m = 00 (and for m = 00 -ico) a similar series of maxima and minima are found at similar x values. How-ever at any given x,the magnitude of the back scattering efficiency is invariably greater for m = 00 than for m = 3.41-1.94i and presumably for any other finite value of m also. Because these maxima and minima which are very unlike the resonances observed for particles with real m are well spread out with respect to the x co-ordinate and because the xvalues at which they occur are insensitive to the value of m it is possible that the variation in back scattering efficiency as a function of x could be used to determine the size of particles in an aerosol. In principle measurements could be made on individual particles or on a number of particles provided that the aerosol was not too polydisperse.For a particle or particles with rapidly changing sizes (co- agulation or condensational growth) one would make observations at a fixed wave- length. For particles of a constant size one would vary x by varying the frequency of the incident radiation. Are such measurements of back scattering efficiencies possible and if so valuable in Pointon's experimental system? Prof. M. Kerker (Clarkson Coll. Techn.,Potsdam) said Has Pointon looked at the small metallic particles with white light and if so what is the colour? It would be interesting to observe whether these particles behave as Rayleigh scatterers or whether (a) magnification 2300 x (b) magnification 5700 x FIG.1.-Fall-out from Zn aerosols at a background temperature of 0.74 Tf.[Tofacepage 98 (n)magnification 2300 x (11) magnification 5700 x FIG.2.-Fall-out from Zn aerosols at a background temperature of 0.74 Tf. GENERAL DISCUSSION as for small perfectly reflecting particles they scatter preferentially in the backward direction. Mr. K. C. Pointon (Shefield University) said In reply to Kerker we have not used white light only green light. Particles that are easily visible with the telescope are 1 pm or larger in diameter. The size of the Airy disc however is not determined by the size of the particle but by the telescopic aperture. Therefore the brightness of the particle determines whether it is visible.The angle between the light incident on the particles and the light received by the telescope has been varied over a sub- stantial range by introducing vitreosil light pipes and angled metallic reflectors into the generating chamber. There is a great improvement in the brightness of particles when the viewing direction is nearly in line with the incident beam. We normally observe the particles against a dark background so that they appear self-luminous. By chance we observed an interesting effect which might relate to Kerker’s question about colour. When Ca vapour falling from the supersaturator crossed the path of green light entering the eye after reflection from the wall of the viewing pipe it extinguished the light. The vapour looked like a stream of black treacle.We do not know whether this effect was the result of atomic absorption or multiple scattering from particles in the very earliest stages of growth. Dr. S. C. Graham (She12 Res. Ltd. Chester) said With regard to the extinction of the stray green light (A = 5461 A) by the Ca vapour or calcium particles surely such extinction is due to absorption rather than scattering as the former is relatively so much greater for small particles than for large ones. Because Pointon’s observations are normally made against a dark background he would not normally observe such absorption. Dr. E. R. BuckIe and Mr. K. C. Pointon (Shefield University) (communicated) In reply to Graham the extinction could still be due to back-scattering.How could light of this wavelength be absorbed by Ca vapour ? The effect is interesting because it may indicate that ultra-fine particles are formed in the vapour immediately after it leaves the supersaturator. Using the telescope we do not see any particles against a dark background at this stage so that they could either be very small strongly back- scattering or both. Dr. S. C. Graham (Shell Research Ltd. Chester) (communicated) In reply to Buckle and Pointon I do indeed think that the “ stream of black treacle ” observed by Pointon was extinction caused by particles and not by calcium vapour. If the particles were sufficiently small (particle circumference/wavelength 50.2) which seems probable as simultaneous scattering was not observed then this extinction would be due to the true absorption of light within the particles rather than to scattering of light away from the forward direction (see Van der Hulst Light Scattering by Small Particles chap.14). I do not agree that the scattering of 5461 8 radiation by calcium particles into the backward hemisphere will ever be substantially greater than scattering into the forward hemisphere. Dr. E. R. Buckle (Shefield University) said I would ask Graham two questions about his paper with Homer. First his analysis of the light scattering data in terms of coagulation kinetics relies as he states on the attainment of a self-preserving size distribution. The mathematical analysis depends on the assumptior? that the size distribution function can be put into a form in which the time and the particulate 100 GENERAL DISCUSSION volume are separable (see eqn (18) of Dunning’s paper).Is there any independent evidence that this is a reasonable assumption on which to base the interpretation of your experiments? The second question is about the slope of the plot in fig. 5. I think it correct to conclude only that this is in keeping with the form of the expression for the classical free-particle collision parameter (eqn (l)) since one obtains fic~t~/~ from eqn (3) without making the assumptions that lead to eqn (1 5). When he tested the experimental value of the coefficient k of t6l5 against the one in eqn (15) there was a discrepancy. A feature of this comparison worries me. The theoretical coefficient kcoag from eqn (15) depends on the hypothesis of a self-preserving distribution but so does the “ experimental ” one k,, = kcalibkgraph because kcalib(eqn (22)) is derived from eqn (16).Therefore the quantities compared both depend on this hypothesis. Is this connected with his difficulty in getting an exact match of the theory with the experimental results ? Dr. S. C. Graham (Shell Res. Ltd. Chester) said With regard to Buckle’s first question when free-molecule or diffusion-controlled (Smoluchowski) coagulation is simulated on a computer by feeding in an initial distribution and then following the 1.75-1.50 -1.25 --1.00 -0.75 2 0.5c 0.25 -0.25 I -o*ml -0.7 5 -~.&--+-210 2f5 20 315 410 415 5fo 515 652 1 FIG.1.-The error function E($(q)) plotted against q.change in number concentration of each particle size using the known expressions for the collision rate density for particles of different sizes it is found that after sufficiently long “ coagulation ” times the computed distributions (expressed in dimensionless form) tend towards the solution of the corresponding integro-differential equation which was derived (unlike the computed distributions) using the sex-preserving hypothesis. To my knowledge no-one has yet proved that the limiting asymptotic form of the computed distributions has to be the corresponding self-preserving size distribution but there exists considerable verification that this is the case particularly for diffusion-controlled coagulation.The closeness of our computed distribution to the self-preserving distribution can be examined by feeding the computed value of t+h designated t,bcomp for each q into GENERAL DISCUSSION eqn (6) of our paper and plotting the resultant error function E($,omp),(Le. the 1.h.s. of eqn (1 6)) as a fupction of q. If $camp is identical to the self-preserving distribution then the error function is the straight line E($comp(q))= 0. The following figure illustrates E($comp(q)) for our own computed distribution as well as for those of Ulrich and of Lai Friedlander et al.,and shows how close our computed distribution is to the self-preserving distribution. With regard to the second question Buckle is correct that the t6/5-dependen~e of the particle volume is a basic feature of free-molecule coagulation and is not special to the self-preserving hypothesis.However we do not observe V(mean volume) directly but rather a scattered light intensity I where This last equation is true for any arbitrary dimensionless size distribution $arb where $arb is defined by Nv t) = (N&/$)$ardq)* Now if $arb were not the self-preserving distribution one would expect s@(q)g2 dq to be time-dependent and thus a graph of log I against log t would not give a linear plot of slope 6/5 even if a graph of log vagainst log t were to give such a plot. There-fore the fact that we do observe experimentally a linear plot of slope 6/5 for log I against log t strongly suggests (a) that l$(q)g2 dg is a constant independent of time and (b)that V,the mean volume varies as t6l5.Incidentally the “ classical ” approximation that every particle has the same volume (which is clearly impossible for a coagulating aerosol) is equivalent to defining ~, a “classical ” size distribution where i+hclass(q)= ~ 3 where 6~is the Dirac delta function. Because at any instant q = 1 it follows that every function of t,hclass is time-independent and thus $class is in a sense selfpreserving. However $class has no special significance and it is not the solution of any physically significant equation. From the above in order to relate I to Vat any instant we have to make an assumption about the particle size distribution given that we have experimental evidence as well as theoretical grounds for believing i$(q)g2 dq to be a constant independent of time.There are no good reasons for assuming that $ = J1, but there are good grounds for believing @ to be the self-preserving size distribution (i.e. the solution to eqn (6)of our paper). In particular the lead particles are formed at extremely high supersaturations so that homogeneous riucleation is rapid and gives a large number of very small particles. These are precisely the conditions that are required to allow the self-preserving size distribution to develop early in the coagula- tion process. We therefore assume that we have the self-preserving size distribution and examine the consistancy of this assumption in particular by comparing the “ experimental ” coagulation rate Constant with the theoretical one and we find that the ratio of the two kcalibkgraphlktheor kc‘cxperimental” lktheor 2.If we made the classical approximation t,hclass(g)= Jl,, this ratio would have the approximate value 2 x I .84 x (1.1 3)6/s x (2.294/2.327)6/5z4. In this last equation we have GENERAL DISCUSSION and finally 2.327 Prof. M. Kerker (Clarkson Coll. Tech. Potsdam) said Has Graham given consideration to the possibility because of the high complex refractive index that these particles might not scatter according to Rayleigh’s limiting equation for a small dielectric particle? It might be useful to carry out a “ Mie ” calculation for comparison with the Rayleigh theory. This might be the source of the discrepancy between the observed and calculated coagulation rates.Dr. S. C. Graham (Shell Research Ltd. Chester) said In reply to Kerker the scattering efficiency of a sphere (Mie theory) is given by Expanding the Mie coefficients a and b in powers of x the first few terms (ref. (17) of our paper pp. 143-4) are al = isx3(1-tx2 +isx3) +- .; bl = ism5+ ...; (2) a2 = isox5+ . . . ; b2 = ...; where The Rayleigh scattering formula refers to the limiting value of Qsca as x+O and mx-d where (2n+l){la,)2+lb,12) + 31a,I2 3 31s12x6. Whilst s t and w assume small limiting values as m-,00 u increases rapidly and without limit so that the expansion of the b as a power series in x cannot be used even as x-0. However for lead at 3 = 488 nm m2 = -10.9-I4li so that lbl12,which has the value 1-8.9-14.1i12x10/(30)2= (0.31)2x10 is still small.For xG0.1 which corresponds to the upper size limit of the lead particles at the end of the shock flow the contribution of this term to Qsca is <1 % and is quite negligible. However the use of the Rayleigh formula does introduce errors >1 % for lead particles with greater than 0.1. Thus la I2 w ls12x6(1-tx2-isx3)(1-t*x2+is*x3) z ls12x6(1-2 Re(t)x2) and for m2 = -10.9-4i 2 Re(t) = 1.35. GENERAL DISCUSSION The term lbi12gives the contribution from the oscillating magnetic dipole and as van der Hulst has shown (ref. (17) p. 160) it is the interference between the induced oscillating electric and magnetic dipoles which gives a very small particle with m = 00 its characteristically high back-scattering efficiency.For light scattered perpendicular to an incident linearly polarized beam and scattered perpendicular to the polarization of the incident beam the relationship between the (differential) scattering cross-sections of a Rayleigh particle and a perfectly reflecting particle (m= 00) of the same size is particularly simple. For a Rayleigh particle this cross-section is given by and for the latter Cdiff sca(B= 4= x/2) = a2x4 (ref. (17) pp. 12 127 and 159). In conclusion the value of m for lead at 488 nm is not large enough to invalidate the use of simple Rayleigh formulae to calculate total and differential scattering cross- sections of particles with sizes satisfying the condition x = nd/;l<O.l.