GENERAL mscussroiv Dr. E. R. Buckle (University of Shefield) said In introducing his paper Dunning compared the experimental nucleation rates Jexpfor water condensation with those predicted by the so-called "classical "theory Jtheor such that by and large Jexp/Jtheor -lo3. What confidence can be placed on the actual values given for Jexp by the various workers ? Dr. W. J. Dunning (Bristol University) said Prof. J. Clumpner (now at the American University Beirut) has considered all the possible errors (including the so-called "influence coefficients '' of gas dynamics) which may affect the evaluation of Jexp from the Yale nozzle experiments. His conclusions have not yet been published but I understand that for the best work Jexp is known to within a factor of 10 and there is no problem in achieving a factor of lo2.Clearly the precision is adequate to discriminate between theories which predict values of J which differ by factors of 1012- 101 7. Dr. C. S. Kiang (Clark ColZege Atlanta Ga.) said Would Dunning comment on the present state of his eqn (6)? Dr. W. J. Dunning (Bristol University) said In reply to Kiang's question of the present status of my eqn (6) experimenters find it useful to compare their results with classical theory (my eqn (6)) by using a factor rexp to match this theory with their experimental results Jexp = rexp Jclass-In the same way it is useful to compare the rates of nucleation ,Itheor predicted by the different theories Jtheor = rtheor 'Jclass. For example Lothe and Pound's l theory predicts I'L-P-predicts rD -lo4 for water vapour condensation.1017 and my theory The careful work of Wegener Clumpner and Wu on the nucleation and growth of ethanol drops in supersonic flow yielded values of rexp at different stations x along the nozzle (fig. 1). Also illustrated in the figure are the corresponding values of rL and TD; if the classical theory applied rexp would run along the x axis. The value of r, at the onset point is about loi7,the value of rD at this point is about lo5 matching the experimental value of rexp closely. Thus the Lothe-Pound theory predicts nucleation rates which are far too high ; classical theory predicts rates which are rather low In these ethanol experiments the following conditions are favourable.The ethanol was carefully purified the mass-fraction of water vapour in the air was less than The critical nuclei are comparitively large containing 15-20 molecules. The condensation takes place above the triple point so that the nuclei are almost certain to be liquid. In the presence of a carrier gas (eg air) the nuclei and droplets grow under Lothe and Pound J. Chern. Phys. 1962,36,2080. Dunning Adsorption et Croissunce Cristulline (Coll. Int. C.N.R.S. no. 152 Paris) 1965. Wegener Clumpner and Wu Phys. Fluids 1972 15 1869. 42 GENERAL DISCUSSlON isothermal conditions; for pure vapour (e.g. steam) the nuclei and droplets are at a higher temperature than the vapour and nucleation proceeds more slowly. In fig. 2,* the values uniformly calculated of Texpfor ethanol water and benzene are shown plotted against the mass-fraction coo of vapour.For the two latter substances there is more uncertainty about the thermodynamic state of the nuclei and the results for both solid and liquid states are shown; in each case the lower of the pairs is to be preferred for kinetic reasons. Barschdorff Dunning Wu and Wegener have recalculated all experimental results for pure steam and these are all in the range of the hatching at log coo = 0. These results give <lo6 to be compared with l7 = 1 if the classical theory held and with rLP-<rexp lo1' and TD-lo4. 30 10 -20 10 -rL-P -Id" -%x p c rD 10 0 ' I I I I I 0 I 2 x/cm FIG.1.-Nucleation of ethanol in air.Showing the dependence of rexp, FLp rD on x the station after Wegener Clumpner and Wu ref. (3)). Using a shock tube Barschdorff * (unpublished) has investigated the effect of various carrier gases at various mass-fractions on the nucleation rate for water vapour and his results are illustrated in fig. 3; here Jexp/Jtheor = rexp is plotted against coo <lo3. Investigations on other materials have not been so and we find <rexp intensive. Pure nitrogen gives On the other hand experiments on freon and chloroform have given results close to values predicted by the Lothe- Pound theory. * I wish to thank Prof. Wegener and Dr. Barschdofl for allowing me to quote their unpublished results. Dawson Wilson Hill and Russell J. Chem. Phys. 1969 51 5389.GENERAL DISCUSSION U U a **A S 10 -Ah 000 i I no n-tsothermal lo 0.001 0.0 I 0.1 I -5> WO FIG.2.-Nucleation of ethanol water and benzene. rexp as a function of the mass-fraction wo of the substance in air (courtesy of Prof. P. Wegener). A,Ethanol; 0,0 HzO; benzene; 0,., open symbols liquid ; solid symbols crystal. ,O L FIG.3.-Nucleation of H20.reXp as a function of oofor various carrier gases. A Steam; 0, moist air. H20 in argon ; 0,H20in helium ; 0 It must be remembered that " binary "and " hetero-molecular" nucleation theory as discussed by Kiang and Stauffer in this Discussion indicates that impurities in the substance examined would if they were effective always tend to increase the value of rexp-Prof.M. Kerker (Clarkson Coll. Techn. Potsdam) said I wouId ask Dunning what can be said about the form of the size distribution prior to the onset of coagula- GENERAL DISCUSSION tion? We have assumed that the aerosols having a somewhat narrow size distri- bution formed by condensation upon heterogeneous nuclei could be represented by a logarithmic distribution. Dr. W. J. Dunning (Bristol University) said In reply to Kerker the mathematical form of the classical nucleation rate equation seems to be accepted although the values of some of the parameters have been subject to controversy. In a well-defined experiment it is possible to calculate the rates of nucleation and the rates of growth and evaporation of all classes of particles at all stages of the collapse of the super- saturation.Thus the form of the size distribution is ascertainable under such conditions. It is necessary to check by calculation the progress of coagulation since under certain circumstances coagulation may become significant before growth is effectively complete. I have no experience of the Kerker type of aerosol generator and I cannot judge how well defined the thermodynamic variables are. It may be possible to idealize it and calculate the aerosol size distribution to be expected. In this one could assume that Volmer’s theory of heterogeneous nucleation was valid and that an appropriate rate of growth was applicable and at least determine the form of the size distribution thus predicted. Until such studies are made it is not possible to assess whether the logarithmic distribution closely approximates the experimental distribution or not.Dr. E. R. Buckle (Shefield University) (communicated) The size distribution in a spontaneously condensed aerosol will adjust by growth and evaporation towards the stationary form of eqn (1.3) of my paper which resembles Volmer’s distribution (eqn (1.18)). If the conditions change the system will approach a new distribution of similar form. The distribution will be cut off at a size in the region of the critical size in the early stages but eventually will extend to larger sizes as the result of co- agulation. In seeded condensation it may happen that the growing centres are too large at the outset for them to qualify as labile gaseous species and the distributions of my paper would not apply.Prof. C. S. Kiang and Dr. D. Stauffer (Clark College Atlanta Ga.) said Would Dunning give his opinion of the reason why some materials obey classical nucleation theory and some do not. Theoretically the capillarity approximation using the bulk surface tension for small droplets may be inaccurate. Computer simulations of small droplets (“ micro-crystalline calculations ”) do not need such an approximation. Calculations of this type by Burton explain from one model most of the observed deviations from and agreements with the classical nucleation theory. His comparison with classical theory depends on the temperature and the supersaturation which are different for different experiments.(In another preprint Burton proves convincingly the presence of a Lothe-Pound translation-rotation term in such microcrystalline calculations based on vibration frequencies and binding energies of the molecules. This does not yet solve the question whether the capillarity approximation requires too a Lothe-Pound correction since it might already be included in the effective surface tension.) Another (simpler but so far less successful) approach is to make the capillarity approximation to neglect the Lothe-Pound factor but to use a microscopic droplet surface tension different from the measured bulk surface tension. The smaller the J. J. Burton Actu Met. to be published. 0 In this GENERAL DISCUSSION droplet surface tension is compared to the measured bulk surface tension the higher should the nucleation rate be compared with classical theory.We determined this “ microscopic ” surface tension from the two methods,’ (density)/(critical density) and from 1-PVIRT. We found for H20 CH30H and C2H50H good agreement of these microscopic surface tensions with the bulk surface tension. (Experimentally classical nucleation theory works here.) For NH3 C6Hb CHC13 and CFC13 the microscopic surface tensions were much lower than the bulk surface tension (classical nucleation rates are too low here). But unfortunately for hexane heptane octane we also found lower microscopic surface tensions although classical theory works here. Experimentally the possibility cannot be excluded that the water impurities in NH3 lowered the supersaturation necessary for nucleation.2 Our paper on hetero- molecular nucleation indicates that such effects are quite strong for H20 +H2S04.Dr. W. J. Dunning (Bristol University) said In reply to the question of Kiang and Stauffer all possibilities should be considered. With regard to Stauffer’s report on calculations by Burton if we imagine a bulk crystal containing n = km particles to be subdivided into m crystalline nuclei each containing k particles and if E; is the specific surface energy and 0,the surface area of a nucleus the replacement factor qrepwill be 3k-6 1 3n-6 qrep = (&kO-Orn)Ok+ 1 ((bj+qj)-m (fi+&i). j= 1 i= 1 is the surface tension of the bulk crystal ci and yli are the zero-point energies of the ith andjth normal modes of vibration in the nucleus and in the bulk crystal respectively andfi and (bj are their free energies given by 4j = -kT log exp (-hujnj/kT) ni>O with a similar expression for h.The calculation of qrepthus requires a precise evaluation of E; and the frequencies pj of the normal modes in the nucleus and in the bulk crystal. Dr. E. R. Buckle (University of SheBeZd)said The contention in the original paper of Lothe and Pound (ref. (20) of Dunning’s paper) was that the Volmer theory of nucleation is in error in taking no account of the natural chaotic motion of the growing nuclei in the parent phase. I know of no reason for supposing that it was in Volmer’s mind that the embryonic particles were motionless so that it is to be presumed that the particle distribution in his treatment (eqn (1.18) of my paper) was intended to be one of equilibrium involving all thermally accessible energy levels in the various degrees of freedom.In the kinetic formulation of the problem given in ref. (1) of my paper the terms Wi and W of Volmer’s theory are given their molecular interpretations for vapour condensation. The collision terms for the motion of the gaseous clusters appear in the quantity Cs the interfacial free enthalpy. As stated in my paper 5 = WL so that the same terms are contained in 70,of Volmer’s formula. The Brownian motion is therefore taken into the surface tension term and the error in Volmer’s formulation of the reversible work of nucleus formation will reside in the use of a single value of J.Atm. Sci. 1972 28 1222 Appendix B. see J. Chem.Phys. 1972 51 5380 in particular p. 5387. W. J. Dunning in Nucleation ed. A. C. Zettlemoyer (Marcel Dekker New York 1969) p. 35. GENERAL DISCUSSION surface tension for all cluster sizes. As a likely source of error this has long been appreciated and the conclusion is that there is no additional source of error due to the omission of thermal motion terms. It is incorrect to add free enthalpy terms for this motion to Ws as given by Volmer. Dr. W. J. Dunning (Bristol University) said We all recognize the great achieve- ments of Volmer but we should not with hindsight attempt to read intohis work more than is there. For example on p. 97 of his book,' Volmer describes a method due to Gibbs for determining the work of formation of a nucleus of radius r,.One step in this process involves some of the bulk liquid confined in a cylinder by a piston being extruded as a droplet through a small orific in the cylinder wall. The work done in forcing out the droplet is given as This seems to imply that Volmer considered 0,the surface tension to be independent of the radius which it would not be if the Brownian motion is taken into the surface tension term. It seems that both Volmer and Gibbs were tacitly assuming that the droplet stands still on the orifice. In fact the still-attached droplet will be bobbing about on the orifice in Brownian motion and at some stage must be liberated to dance away into the vapour space.The contributions of Kuhrt,2 Lothe and Pound, Dunning and others analyze the consequences of this last stage of liberation. There is a large measure of common agreement but where they differ is in their assessment of the magnitude of the so-called replacement factor. Prof. C. S. Kiang (Clark College Atlanta Ga.) said Dunning mentioned various nucleation processes and their roles in his introductory remarks. From our recent studies we believe that the heteromolecular nucleation (homogeneous or heterogen- eous) process should be included as one of the most important nucleation processes for the fogs and smokes formation in the earth's atmosphere For relative humidity less than 100 % this nucleation process seems to be the only nucleation mechanism with water in the atmosphere.Most of the primary chemical constituents in the atmosphere do not have low vapour pressures and their atmospheric concentrations are measured in p.p.m. or p.p.b. and are not sufficient to allow nucleation in the gas phase. However by chemical reactions or radiation or other energy input secondary products with low volatility can be formed and these reactants may mix with water and undergo heteromolecular nucleation to form new aerosol. For example at room temperature SO2 has vapour pressure around 4atm. In the presence of oxidants (e.g. N02+hv-+NO+0 SO +O-+SO, SO +H,O-+H,SO,) H2S0 can be formed. With only Torr vapour pressure H2S04+ H20 will undergo the heteromolecular nucleation to form aqueous sulphuric acid aerosol. After reacting with NH3 (NH4)2S04 may be formed as stated by Dunning.M. Volmer Kinetik der Phasenbildung (Steinkopff Dresden 1939). F. Kuhrt Z. Physik 1952 131 185 205. J. Lothe and G. M. Pound J. Chem. Phys. 1962 36,2080. W. J. Dunning in Chemistry of the Solid State ed. W. E. Garner (Butterworths 1955) p. 159 ; Adsorption et Croissance Cristalline (Coll. Int. C.N.R.S. Paris 1965) p. 369. see W. J. Dunning Nucleation ed. A. C. Zettlemoyer (Marcel Dekker Inc. New York 1969) pp. 37-47. GENERAL DISCUSSION Dr. E. R. Buckle (Shefield University) said The purpose of my paper is to show that on simple kinetic grounds a particle in a volatile aerosol that is homogeneous and isothermal must grow or evaporate in a way that is dictated by the properties of the whole particulate system.The role of coagulation in particle growth is certainly paramount at low temperatures in very dense clouds of ultra-fine particles although for condensation aerosols the centres of coagulation are particles formed during the previous volatile stage. The properties J that are characteristic of the particle size distribution are found in the growth and evaporation equations of a particle. For instance in eqn (2.15) which gives the net rate of increase of material in a particle the term A(s+l reflects the joint influence of all the particles because its value is related to the size distribution at equilibrium. Now although A(,+ tends to zero for very large clusters the ratio &+l/& is still dependent on J if the processes in the aerosol are occurring naturally.Theoretically therefore the communal effects of the particles are unavoidable. Dr. W. J. Dunning (Bristol University) said Referring to eqn (1)-(5) of Buckle's paper is it possible to assert at this stage of his argument that there is a limiting value of o,as g-' ? If the clusters are bizarre in form how the equilibrium between one such monster and another monster behaves as the monsters increase in size can at present only be conjectured. At the other extreme if the clusters were perfect crystals the equilibrium quotients would oscillate. Dr. E. R. Buckle (Shefield University) said In reply to Dunning if by "monsters " is meant thread-like or branching clusters especially ones without a centre of sym- metry I would expect them to have little chance of formation by homogeneous pro- cesses under the moderate conditions assumed in my paper.They are therefore unlikely to contribute to the formation of stationary states. To prove this it would be necessary to extend the concepts of internal energy fluctuations to include such species. Gaseous ions might interfere in the condensation of heteropolar substances like water to produce elongated structures. For this to have a sensible effect on the condensation process there would have to be very many ions and the gas would be properly regarded as a mixture. Very large clusters are without influence kinetically and I do not visualize the growth of clusters as ever involving the perfect crystalline state. The size at which a pure particle with its inherent structural imperfections is able to assume crystallinity is unknown but it could well be too great for it to have any importance as a rate- determining factor in condensation.Such considerations are important because they bear on a well-known dilemma of " classical " nucleation theory. In the interpretation of condensation by Volmer theory there is always this element of uncertainty about the physical state of the nucleus is it solid or liquid? The uncertainty is embarrassing because macroscopic- ally the surface tension has different values for these states and a decision has to be made as to which to use in the theoretical formula. I doubt whether this difference is really meaningful in nucleation. To overcome this difficulty a theoretical approach is required in which the struc- tures and properties of the condensing particles are related to those of the molecules composing them.As a test of the usefulness of such a theory one would look for its ability to predict the rates of production of the particles and to relate these to measure- able physical properties of the system during condensation. A primitive attempt at such a theory was given in ref. (1) of my paper and Dr. A. A. Pouring and I have been using it in some calculations that should show among other things how sensitive the GENERAL DISCUSSION predictions of the path of condensation are to the molecular details. Whether the comparison of computed and measured values of experimental variables like pressure and temperature will provide a useful check on the adequacy of the cluster models also remains to be seen.Dr. D. Stauffer (Clark College Atlanta Ga.) said Buckle’s eqn (1.9) shows that one can calculate some cluster properties from measured (pressure density temper- ature) relations (equation of state). Fisher’s droplet model gives good agreement with the measured equation of state both near the liquid-gas critical point and at lower temperatures. It also agrees with direct evaluations of cluster concentrations in a two-dimensional m0de1.~ Does Buckle’s approach give an alternative droplet model which also can be tested using the measured equation of state? This would be very valuable as a replacement for the classical nucleation theory.I would also ask what is the relation between his eqn (2.15) and the “ classical ” approach (e.g. eqn (10) of D~nham).~ Dr. E. R. Buckle (Shefield University) said In reply to Kiang and Stauffer I am not optimistic about making comparisons of experimental PV isothermals with theoretical cluster distributions (e.g. my eqn (1.3) and (1.9)) to obtain the properties of the clusters. Except at high pressures for which the cluster models are not intended or at the very high supersaturations which build up in a fast expansion the concentra- tions of clusters fall too rapidly with size. The dependence of the distribution on the external variables should be more apparent experimentally as the system passes through the condensation threshold. During this stage the Volmer minimum in cs and the relaxation hump that follows it move into the region of small cluster sizes and then back out again.The time available for observation is so short that measure- ments with high time-resolution are called for. It seems that the measurements should aim to resolve the sizes also if cluster properties are to be investigated. The results of methods such as bulk scattering of light which give only a mean particle size over a finite time interval would be very difficult to relate to such properties because of the complex form of the relaxing size distribution (the curve of cs against g resembles in shape a PV isotherm of van der Waals at sub-critical temperatures). In the method of Volmer the clusters in an equilibrium store are assumed to grow by molecular collisions and evaporation is ignored.The flux condensation is assumed to be that of eqn (2.12) and is therefore independent of cluster size. Becker and Doering’s modification was to consider also evaporation but again there was no allowance for the possibility that the chance of capture or expulsion of molecules might vary with the size of particle. The theory of my paper shows that the intro- duction of size-dependent accommodation coefficients leads to the conclusion that the net flux on an aerosol particle is dependent on the size distribution and cannot be calculated from the laws for a macroscopic surface. Dr. W. J. Dunning (Bristol Uniuersity) said It seems that the limiting process in Buckle’s eqn (1.5) is not general but ancitipates a Volmer-like model in which the clusters are considered to be compact.Physics 1967 3 255. ’2.Phys. 1970,235 130. Phys. Letters 1972 4QA 345. Phys. Rev.,1972 B6 2777 ; also J. Stat. Phys. 1972 6 49. J. Rec. Atm. 1966 2 331. GENERAL DISCUSSION Dr. E. B. Buckle (University of Shefield) said In a treatment of this kind there is one respect in which the clusters must be regarded as compact. The stepwise processes AgM1 +Al = A, are only amenable to a simple collision theory if the cluster lifetimes are long in comparison with the collision time. High vapour densities are excluded by this and there is therefore some meaning in concepts such as the cluster-gas interface and the surface tension. It is a point of my paper that if clusters are to exist homogeneously at up to indefinitely large sizes the quantity 5 must obey the conditions 5 > 0 AC,/Ag = 0 in the limit as g -+ GO.But this requires w -+ w as g -+ co where a,approaches the constant w from below. These conditions are satisfied if one makes the reasonable assumption that in a growing cluster of iso- tropically bound atoms the nearest-neighbour coordination number k rises to a limit Am. This represents the ordering process that accompanies the differentiation of the condensed phase. With regard to the more diffuse entities only at high vapour densities near the critical point would there be tenuous regions of high local density in significant concentrations. Under such conditions however long-range density fluctuations are already a characteristic of the saturated vapour and the concepts of nucleation theory are no longer useful.Prof. M. Kerker (Clarkson Coll. Techn. Potsdam) said With regard to the paper by Buckle in our experiments on the formation of dibutylphthalate aerosol by cooling a mixture of vapour and NaCl particles (radii ranging from 30 to 100A) we observed that only a small fraction of the particles served as condensation nuc1ei.l Also this number depended strongly upon the rate of cooling ; there were more aerosol particles of smaller average size when the cooling rate was greater. We assumed this was controlled by the rate at which the most effective classes of nuclei could relieve the rapidly increasing supersaturation.Is this observation pertinent to the dynamic processes which Buckle describes in homogeneous nucleation ? Dr. E. R. Buckle (Shefield University) said In reply to Kerker if the particles coagulate while still at temperatures at which they are volatile it confirms that their sizes and individual properties are predetermined by the seed and are not influenced by the kinetic factors that determine size distributions in spontaneously-condensed aerosols. Yet the effect predicted for such an aerosol that the rate of growth of a particle gets slower as its size increases would lead to what Kerker has observed on changing the cooling rate the faster the cooling of the vapour the smaller and more numerous the particles become I believe this is purely coincidental and the effects are characteristic of hetero- geneous nucleation.In experiments on the seeding of water from moist air expanded in the wind tunnel the effectiveness of seed was greatest at low flow rates (i.e. slow cooling) and high humidities. Kerker describes how he makes allowance for the extra time particles have for coagulation if they are moving near the condenser wall. There may also be a connexion between the activity of the NaCl particles as condensa- tion nuclei and their position and speed relative to the wall. The proportion of seed particles that are effective as condensation centres would be expected to be sensitive to the radial velocity and temperature gradients and therefore to the flow speed. Consequently I suspect the reason for what was observed is the non-uniform condi- tions.G. N. Nicholson and M. Kerker J. Colloid Interface Sci. 1973 43 246. E. R. Buckle and A. A. Pouring Nature 1965,208 367. GENERAL DISCUSSION Dr. W. J. Dunning (Bristol University) said With regard to Kiang’s paper how large do the droplets grow before they come to equilibrium with the reduced relative humidity and reduced activity ? The authors have pointed out many possible sources of error and complication. Is the activity of say sulphuric acid derived from the concentration of sulphuric acid in the vapour phase ? Would not this be a composite term including not only H2S04 molecules but other species such as H30HS04? It is assumed that the condition of a nucleus of aqueous sulphuric acid is the same as that of a random sample in the bulk solution.However for bulk solutions some of the sulphuric acid will be dissociated into ions. In nuclei containing say 5 sulphuric acid molecules and perhaps 30 water molecules such dissociation will be inhibited since some of the resultant electrostatic field will stray outside the nucleus where the dielectric constant is low (-1). Further the ions will tend to avoid approaching the surface of the nucleus and to occupy only the middle ; this will affect the entropy of mixing. Such points may need consideration when the theory is developed further. Dr. R. A. Cox (AERE Harwell) said With regard to the heteromolecular nucle- ation of H2S04+H20 vapour mixtures discussed in the paper by Kiang and Stauffer a simple experimental test of the validity of the Flood-Neumann-Doering-Reiss-Doyle theory was suggested by Doyle.His calculations indicated that appreciable nucleation rates would occur at sulphuric acid vapour activities of 10-2-10-3 in air at 50 % relative humidity (RH ;25°C). This corresponded to the equilibrium vapour phase activity above an aqueous solution of H2S04 of composition -75 % wt/wt H2S04. Thus a sulphuric acid solution of this composition should fume in air at 50 % RH. We have measured the concentration of condensation nucleii (CN) in an air-stream passing over concentrated H,S04 solutions of various compositions using a Pollack CN counter. At 72 % wt/wt no CN were detected even at 70 % RH. With 77 % wt/wt H2S04,particles (300-600 ~m-~) were detected at 70 % RH and at 83 % wt/wt the particle counts at 70 % and 5 % RH were 3 x lo5 and lo4 ~m-~ respectively.These experiments are only semi-quantitative since (a) the CN counter does not measure the number of nucleating embryos but only the number of particles which grow to sufficient size to register in the CN counter (225 A). Losses of particles of this size (and smaller) by diffusion to the containing walls will be appreciable. Also (b) equilibrium conditions of H2S04 and H20 vapour were not attained above the liquid surface and the departure from equilibrium cannot be assessed easily. Never-theless these results do indicate that the theoretical predictions are not grossly in error and that heteromolecular nucleation in H2S04+H20 mixtures can occur at an appreciable rate at extremely low partial pressures of H2S04 vapour (i,e.approxi- mately 10-9-10-8 Torr at 50 % RH). Prof. C. S. Kiang and Dr. D. Stauffer (Clark CoZZege Atlanta) said In reply to Dunning 1-6 A is a typical radius of the “critical ” droplets at the saddle point. After this heteromolecular nucleation the droplets grow until the H2S04 gas is (nearly) exhausted which happen for droplet sizes of typically 10-100 A (see ref. (15) for details). The H2S0 activity was determined by thermodynamic relations and specific heat measurements (Giauque et al. ref. (6)) ; the evaluation assumes the presence of only two gaseous species. We do not know the magnitude of the error involved in this approximation. In addition to these dissociation effects which may differ in small J.Chent.Phys. 1961 35 795. GENERAL DISCUSSION droplets from bulk phase dissociation effects one also expects (ref. (2)) an enrichment of one phase near the droplet surface. For small droplets the Gibbs adsorption equation describing this surface enrichment can be seriously wrong in particular if applied to H20+C2H50H heteromolecular nucleation. Computer simulations of small liquid droplets (Monte Carlo molecular dynamics etc.) might eventually answer these questions ; we cannot. In reply to Cox for a 72wt % solution the H2S04 activity is 0.0007 for 77 % it is 0.0044 and for 83 % it is 0.04 according to Giauque et al. (ref. (6)). Dr. D. Stauffer (Clark College Atlanta Ga.) said The vapour pressure of Torr for H,S04 was questioned by Doyle,' who prefers Torr ; which is the better value ? Dr.C. N. Davies (University of Essex) said I agree with Stauffer that there is doubt about the partial pressure of H2S04 above aqueous solutions. It is possible that it could be measured by a technique similar to that described by Frostling.' Droplets as large 8s 2 pm diam. can be kept in suspension with a greatly reduced loss by sedimentation in a cylindrical chamber which is continuously rotated. By holding the atmosphere in the chamber at a constant relative humidity and sampling the aerosol over a long period it might be possible to measure the rate of evaporation of H2S04 as a function of the ambient humidity. Dr. R. G. Picknett (Chem. Defence Est. Porton Down) said I would raise two points about the paper by Davies.The first concerns the evaporation coefficient which must have a value near to unity if the interpolation formula eqn (1.13) is to be applicable. Measurements of this parameter are sparse but a value for water of only 0.04 has been reported and so eqn (1.13) must be employed with caution. Extensive calculations of droplet evaporation have been made by N. L. Cross and myself in which we find that a 70 % difference in droplet lifetime is obtained when evaporation coefficients of 0.04 and 1.0 are used. This is for drops of 5pm radius. My second point concerns the effect of self-cooling on the evaporation rate which Davies correctly states is negligible for the substances other than water in table 2.This effect has also been investigated in the calculations made by N. L. Cross and myself. It is most important for droplet evaporation in vapour-free surroundings and under these conditions it rapidly becomes significant as the vapour pressure increases above 0.1 mbar. If we take a vapour pressure of 1 mbar 10 times as large then a 5pm radius droplet of a typical organic liquid will have the surface temperature depressed by about 0.3"C,i.e. sufficient to cause a material reduction in the rate of evaporation. Dr. C. N. Davies (University of Essex) said In reply to Picknett it is difficult to measure the evaporation coefficient a. Bradley et al.2 gave 0.28 for di-n-butyl phthalate and 0.35 for butyl stearate droplets in air but it was subsequently decided that azl for di-n-butyl phthalate in air hydrogen and fre~n,~ for straight chain hydrocarbons in air,4 and far branched-chain hydrocarbons and a straight-chain fluorocarbon in air.5 For rhombic sulphur it was found that o! = 0.73.6 J.Aerosol Sci. 1970 1 341 ; and 1973 in press. R. S. Bradley M. G. Evans and R. W. Whytlaw Gray Proc. Roy. Suc. A 1946,186,368. J. Birks and R. S. Bradley ibid. 1949 198 226. R. S. Bradley and A. D. Shellard ibid. 1949 198 239. 'R. S. Bradley and G. C. S. Waghorn ibid. 1951 206 65. R. S. Bradley ibid. 1951 205 553. GENERAL DISCUSSION Alty and Mackay concluded that thermal accommodation of water vapour molecules striking a water surface was achieved but that the rate of rc-evaporation of condensing molecules was high at vapour equilibrium so that the rate of evaporation of water molecules from the liquid phase was only 0.034-0.036of the rate of impact from saturated vapour according to kinetic theory.However Jamieson,2 using a dynamic technique found a to be at least 10 times greater than this low value. There has thus been a tendency for a for liquids to increase as experimental techniques improved and recently Jer Ru Maa,3 using a jet tensimeter has obtained results which lead him to conclude that a = 1 for all liquids. Further references are given in table 1.8 of A. G. Amelin’s book Theory of Fog Condensation (Moscow 1972). Picknett’s remarks on evaporation are interesting. For water the tempera- ture of a drop regardless of its size is equal40 that of the ventilated wet bulb thermo- meter.Prof. E.Rosner ( Yale University) said While the evaporation (or condensation) coefficient c1 for the liquids of interest to Davies may indeed be close enough to unity to ensure the utility of eqn (1.12) this assumption would certainly fail for the evapor- ation (or growth) of crystalline aerosols especially when the dominant vapour species do not exist as structural entities in the condensed phase.4 For example low and sharply temperature dependent cc-vaIues have been measured for the individual faces of crystalline solids which (i) dissociate upon sublimation (e.g. AlN(s) Al,O,(s) NH,Cl(s)) or (ii) whose vapours are associated (As(s) and P(s) giving tetramers in the vapour phase). Also of interest in this connection is the fact that for such solids melting is expected to be accompanied by a discontinuous increase (“jump ”) in a (observed for the sublimation of polycrystalline A1203(s) Ga203(s) and 3Al2O3- 2Si02(s)).Since this is probably a general phenomenon holding for phase changes even below the equilibrium transition temperature it is interesting to consider its consequences for the relative evaporation (or grdwth) of aerosols for which the condensed phase may be either crystalline or amorphous. In the usual case for which AHsublim> AH,,,. both high vapour pressure and high a would combine to cause rapid gasification of a liquid aerosol compared to its crystalline counterpart at the same sur- face temperature. For condensational growth from the vapour however these effects would oppose one another causing loss disparity in condensation rates morc than in gasification rates.Finally since compact crystalline aggregates appear to be present (at least as intermediates) during the production of inorganic oxide aerosols in flames we should note that prior to complete sintering such an aggregate (i) would be char- acterized by an effective c1 higher than that corresponding to the surfaces of its crystal- line constituent “ primary ” particles (owing to multiple vapour molecule/solid encounters during escape of condensation from/in the “labyrinth ’7 ; (ii) can lose or gain mass without a corresponding change in the outer dimensions of the aggregate (owing to a change in overall aggregate density p). Dr.C. N. Davies (University of Essex) said Rosner had made some important points. However the association of vapour molecules to form e.g. tetramers will T. Alty and C. A. Mackay ;bid. 1935 149 104. * D. T. Jamieson h’ature 1964 202 583. Jer Ru Maa Ind. Eng. Chem. Fund. 1967 6 504; 1970 8,564; 1970,9 283. This has been reviewed by G. A. Somorjai and J. E. Lester in Progress in Solid State Chemistry H. Reiss ed. (Pergamon Press Oxford 1967) Vol. 4 pp. 1-52. R. P. Burns J. Jason and M. G. Inghram J. Chem. Phys. 1964,40,2739 see also R. P. Burns Ph.D. Dissertation (Dept. Physics Univ. Chicago 1965). GENERAL DISCUSSION reduce the kinetic impact rate by $. so that a corresponding reduction in the rate of vaporization occurs quite apart from any effect due to a.If the vapour is dissociated the impact rates of each type will usually differ so that the values of aneed to adjust themselves accordingly. Prof. M. Kerker (Clarson Coll. Techn. Potsdam) said I would ask Davies how would one transpose this calculation to a heterodisperse aerosol assuming condensa- tion to the wall was minimal and that the process was primarily distillation from smaller to larger particles due to the Kelvin effect? What would the vapour field " seen " by a particular particle be? How would the evaporization rate compare to that for the single particle model? Further questions are (i) how sensitive would the evapor- ation rate be to the evaporation coefficient particularly for small concentration gradients ? (ii) Would one expect that "contamination " might significantly affect evaporation rates? (iii) May one neglect the latent heat effects particularly in an aerosol having a particle concentration of the order of lo6 particles ~m-~ ? Dr.C. N. Davies (University of Essex) said The vapour field " seen " by a particle depends on Kn. When Kn is large the particle " sees " a vacuum roughly up to a distance Kna from its centre. When Kn is very small the concentration gradient outside the particle is such that (4-na" -n*) = a/r where n is the concentration of vapour at radius Y. If the concentration of the aerosol is lo6 particles/cm3 then the volume per particle averages cm3 which is equal to a sphere of radius 62 pm. During quasi-steady evaporation of an isolated particle having a = 0.31 pm the concentration at the surface of a concentric sphere of radius 62 pm would be 99.5 % of the concen- tration at infinity.The evaporation under quasi-steady conditions of such a particle in an aerosol of lo6particles cm3 therefore proceeds at the same rate as it would were the particle isolated. The establishment of quasi-steady conditions requires a time of the order of l/E say s when Kn is large ; for diffusion controlled evaporation the time dependent term is a/ Jnvt. Putting this equal to 0.01 so that the difference between the transient and quasi-steady rates of evaporation is negligible gives t = s. Whatever the value of Kn the time taken to establish quasi-steady conditions is negligible ; the rate of evaporation of a particle in the cloud will be the same as that of an isolated particle.If the coefficient a is very small the rate of evaporation is not controlled by diffusion but by the rate of escape of vapour molecules from the surface even though Kn may be very small. This happens when water droplets are coated with a layer of long chain molecules.1 As a decreases therefore the rate of evaporation becomes less dependent on the concentration gradient and more dependent on a/Kn ; such behaviour is unlikely to occur with drops of organic liquids. Contamination can affect the evaporation of liquid droplets in two ways (i) by lowering the vapour pressure due to the rising concentration of involatile contaminant as evaporation proceeds ; (ii) by forming a film on the surface of aqueous drops which impedes evaporation ; highly specific properties are required in the contaminant for this effect to be large.Suppose the concentration of aerosol is 7.7 x lo-' g/cm3 ; the latent heat of vaporization of DBP is 79 cal/g so that complete evaporation will withdraw from the C. N. Davies Disc. Faraday Soc. 1960 30 144. GENERAL DISCUSSION gas phase 6.1 x cal/cm3. 1 cm3 of air has a heat capacity of 3 x cal/"C. The fall in temperature due to complete evaporation of the disperse phase is thus 3 x 10A4/6.1x lo-' = 4.9"C. Dr. R. G. Picknett (Chem. Defence Est. Porton Down) said Kerker has asked what happens to the evaporation process when a cloud of droplets is present. N. L. Cross and myself have performed calculations for monodisperse aerosols which we hope to publish soon.Provided the mass concentration of aerosol is less than about 1 mg/m3 interaction of the vapour gradients adjacent to neighbouring droplets is negligible and the only effect of the aerosol is to increase the background vapour concentration as the evaporation proceeds thus slowing the process. Prof. M. Kerker (Clarkson Coll. Techn. N. Y.) said We have found our aerosols quite stable to evaporation as pointed out in the discussion on our paper. Since the mass concentration is 1 mg/l. the evaporation must proceed even significantly lower than Cross finds at 1 mg/m. Prof. D. E. Rosner (Yale University) said In quantitative treatments of the evolu- tion of the aerosol size spectrum function n(r x t) under conditions for which the "growth " term d(fn)/ar plays an important role it is common to assume that the individual particle growth rate f is a function of at most the prevailing particle radius r and local environmental variables at position x and time t.For example in the continuum limit (Kn <1) the Maxwell-Smoluchowski equation provides the proportionality fccr-l but it is prudent to recall the assumptions underlying this simple result prior to its formal application to dilute (low volume fraction) aerosols in new situations. In a recent investigation of the vaporization of isolated droplets in the continuum regime conditions leading to the breakdown of the instantaneous focr-l law have been explored with some results of potential significance to aerosol science/technology mentioned here.Briefly for the validity of the f ccr-I law it is not sufficient that droplet and its local environmental conditions undergo negligible fractional changes in the characteristic diffusion time r2/D (where D is the Fick diffusion coefficient for the evaporatingjcondensing vapour). It is also necessary that the velocity of the moving phase boundary be small compared to the character- istic vapour diffusion velocity. This leads to a necessary condition of the form where uV+,is the vapour mass fraction at the droplet surface and coo,, is the vapour mass fraction in the gaseous environment "far " from (i.e. several droplet radii away from) the droplet.2 Ordinarily the density pm of the gaseous environment is much smaller than the density Pdroplet of the droplet itself and the local vapour mass fractions q,, are everywhere much smaller than unity so that this condition for the validity of the quasi-steady approximation (leading to f ccr-') is satisfied.However if one considers a spray of a liquid fuel say at a total pressure level comparable to the D. E. Rosner and W. S. Chang Combustion Sci. Techn. 1973 in press. Under conditions for which the Maxwell-Smoluchowski i' expression is valid the diffusional "driving force "parameter in the absolute value brackets is itself small compared to unity and -i. is linearly proportional to o,,~-Ou,a. However a generalization of the quasi-steady Maxwell-Smoluchowski expression is available for which the i. a r-' dependence is preserved even when the driving force parameter is not small.(See e.g. D. B. Spalding Convective Muss Trunsfer-An Introduction (McGraw-Hill New York 1963)). GENERAL DISCUSSION thermodynamic critical pressure of the fuel then for droplets whose temperature approaches their critical temperature this parameter becomes appreciable and the quasi-steady approximation loses its utility. Interestingly enough this situation is encountered for diesel engine cylinders into which kerosene (pc= 26 atm TCz662 K) is injected as a droplet spray. In such cases a fully transient treatment of the i. function appearing in the aerosol evolution equation is evidently required. D. E. Rosner AZAA J. 1967 5 163.