Nucleation Growth Ripening and Coagulation in Aerosol Formation BY W. 3. DUNNING School of Chemistry University of Bristol Bristol BS8 1TS Received 4th December 1972 The atmosphere may be regarded as a carrier gas (the permanent gases) containing water vapour C02and trace substances. The latter comprise gaseous compounds of nitrogen sulphur chlorine carbon and oxygen together with an aerosol of fine particles. The sources of these particles are diverse ; winds lift many from the earth ; smoke from forest fires contributes inorganic ash and carbon compounds; the bursting of bubbles in the sea surface provides salt particles ; some come from volcanic eruptions and meteorites and yet others are of biological origin. Nor must we forget radioactive fallout.Each year domestic and industrial activities introduce hundreds of millions of tons of pollutants gaseous and particulate into the atmosphere. Much of this discharge is invisible but fogs and smokes may often be seen when readily condensible vapours ranging from water vapour and partly burnt fuels to metallic oxides are emitted from cooling towers chimneys and exhausts to undergo cooling by turbulent diffusive mixing with the atmosphere. Fumes from smelting contain volatile metallic salts which condense as fine particles. Sulphuric mist and alkali mist are other undesir- able aerosols. When fired heavy guns often produce smoke composed of water droplets and metallic particles. On the other hand signal smokes serve a useful purpose and the manufacture of carbon black of titanium dioxide and silica involves the production of very fine particles.The knowledge gained by atmospheric scientists is very relevant to the technology of such processes. SIZE DISTRIBUTION OF THE ATMOSPHERIC AEROSOL When the relative humidity is below saturation the particles in the atmosphere may be grouped according to size as follows ; " small ions " (radius rw pm) which consist generally of a singly charged molecule with a cluster of a few neutral molecules ; Aitken particles (5 x pm<r< 10-1pm) ; " large particles " (10-l pm <r< 1 pm) and " giant particles " (1 pm <r <20 pm). For the size range 0.1 pm<r< 10pm Junge found that the size distribution measured at Frankfurt and on the Zugspitze could be expressed by n(r) = Alr4 (1) where n(r)dr is the number of particles with sizes between r and r +dr ; A = 0.054 (ref.(3)) with 4 the volume fraction of the disperse phase. This remarkable law has been confirmed for " country " and " industrial " atmo~pheres.~-~ The size distri- bution for salt particles in marine air deviates from Junge's law.'. Eqn (1) has been found to describe the size distributions in certain artificial aerosol^.^ 7 AEROSOL FORMATION BASIC PROCESSES The following basic processes must be considered prenucleation kinetics nucleation growth ripening and coagulation sedimentation impaction and dispersal. PRENUCLEATION KINETICS In the atmosphere chemical and photo-chemical reactions may take place in the gas phase in water droplets or on the surface of particles and as a result new particles may be nucleated or changes in preformed particles result.'O Hydrogen sulphide and SO are oxidized to form sulphuric acid and sulphates in particular (NH4),S04.The photolysis of NOz gives atomic oxygen which reactswith unsaturated and aromatic hydrocarbons to form aldehydes ketones peroxyacyl nitrates and ozone. Salt particles react with nitrogen oxides and sulphuric acid droplets to give hydrogen chloride nitrates and sulphates. l In industrial processes the mechanism by which supersaturation is generated may be relatively simple for example the admixture of a cool gas or it may involve a complex sequence of elementary reactions as in the production of Ti02. NUCLEATION Condensation may take place on surfaces,12 on insoluble l3 and soluble par- ticle~,~~ on positive and negative ions l5 and even on other molecules in the gas phase.16 In the atmosphere condensation occurs at supersaturations ranging from a few tenths of one per cent to a few tens per cent." Heterogeneous nucleation on "foreign " particles is also common in industrial processes.Attrition may produce particles of the product which then act as centres for further growth. In the absence of foreign nuclei condensation of a vapour make take place by homogeneous nucleation but this requires much larger supersaturations. There are technical processes in which homogeneous nucleation is predominant and the theories of homogeneous nucleation ripening and coagulation are in any case branches of a single comprehensive theory.Chance collisions of single vapour molecules A form dimers A which in turn form trimers A3 and so on. The sequence of reactions may be represented by +A +A . ..A[- +Ai +A,, .. . (i 2) -A -A implying the assumption that only single molecules and not clusters are gained or lost. The rate of formation of the cluster A is given by ht(t)ldt = Pi-14-10) -alw -PrntO)+%+*ni+10) (3) where ni(t) is the concentration of Ai and /Ii and oli are respectively the frequencies of capture and escape from an i-mer. Clusters which are not too small are considered to be very similar to small droplets containing the same number of molecules. There is a critical size of cluster-droplet containing i = K molecules the vapour pressure of which is just equal to the partial pressure pr of the supersaturated vapour and PI = Pa3 exp (2avlllkTrK) (4) where pImis the vapour pressure over a plane liquid surface (r-+a),4 is the surface tension oI1the liquid molecular volume and rK the radius of the critical nucleus W.J. DUNNING Droplets smaller than this tends to evaporate those larger to grow. The steady state rate at which critical nuclei are formed and become free growing is given by J = ZfiKflK (5) where 2 is the Zeldovich l8 factor and classical theory l9 gives Thus J is a very strong function of the supersaturation ratio pr/prm; there is a critical supersaturation ratio below which J is negligibly small. How closely clusters resemble droplets endowed with macroscopic properties is questionable.A "revision " of the theory by Lothe and Pound 2o predicts that the nucleation rate should be higher by a factor of 1017 than that resulting from classical theory. Dunning approached the problem in a different manner and predicted a factor of about lo4 instead of 1017. Further studies by Reiss 22 support the view that the classical expression is effectively valid. REVIEW OF EXPERIMENTAL METHODS AND RESULTS The experimental techniques available for testing the theory are piston cloud chambers diffusion cloud chambers supersonic nozzles and shock tubes. CLOUD CHAMBER EXPERIMENTS Successive expansions of the piston cloud chamber to increasing volumes leads to the appearance of condensation. The results of Wilson 23 and of Powell 24 show a straight line dependence 25 between log (pr/prco)crit and T-s as predicted by classical theory.Lothe and Pound 26 consider that the data of Wilson,23 of Powell 24 and of Volmer and Flood 27 are in agreement with classical theory if 0 = om. In the diffusion cloud chamber,28 vapour and an inert gas lie between the surface of the liquid and a cooler horizontal plate. At a certain height within the gas the upwardly diffusing vapour condenses and drops descend into the pool. The super- saturation at the condensation level may be calculated. Katz and Ostermeier 22 have found remarkably good agreement with classical theory and state that the Lothe- Pound revision does not fit their results at all. NOZZLE EXPERIMENTS When a vapour flowing through a convergent-divergent nozzle reaches super- sonic speeds it expands adiabatically and its temperature falls.At a critical super- saturation condensation occurs and the heat released causes the pressure to increase above the value it would have had in the absence of condensation. Oswatitsch 30 showed how gas dynamics and the kinetics of nucleation and growth may be combined to furnish a detailed description of the pressure changes during the whole course of condensation. Experiments by Wegener and Pouring,31 Stein,32 Barschdorff 33 and Jaeger et al.34show that the experimental nucleation rates for water vapour in air agree roughly with classical theory. Similar agreement has been found for C02 in air,35 for benzene in air 36 and for C2H50Hin air.37 On the other hand measurements on NH3,34CHC13 38 and CC13F 34p 38 agree with the Lothe-Pound revision.For steam 39 (i.e. pure water vapour without a carrier gas) condensation occurs at a rate slower (by a factor of about than classical theory predicts and about 10-1times more slowly than that predicted by Lothe and Pound. Wegener et al.39point out that AEROSOL FORMATION in the steam case all effects which may be due to heterogeneous nuclei can be categoric- ally ruled out. Barschdorff et aL4' have shown that data 41 for the condensation of pure steam is in accord with classical theory modified to include the effect of non-isothermal nucleation. Pure nitrogen shows the same effect.42 SHOCK TUBE EXPERIMENTS In a shock tube (fig.l) a thin diaphragm divides a long tube into two sections. In the " driver " section the gas is at a higher pressure than in the other section. On bursting the diaphragm a shock wave propagates into the low pressure section while an expansion wave passes into the high pressure section. The use of the shock wave P (i) Low a High I FIG.1 .-(i) Shock tube before bursting diaphragm a. Pressure in tube shown above. (ii) Shock tube a short time after bursting diaphragm ; b shock wave c contact surface d limit of expansion wave. Pressure and temperature in tube shown below. for observing relaxation effects and reaction rates is familiar to chemists. Less well known is the technique of Wegener and Lundquist 43 in which the expansion is used to study condensation phenomena.This technique allows a wide range of cooling rates to be investigated in the same experiment 44 and the closed system has advantages. Some preliminary results of such experiments carried out at Yale have been DIRECT INVESTIGATIONS ON CLUSTERS Stein and Wegener 45 have measured the relative intensity of Rayleigh scattered light from a free jet placed in the cavity of an argon laser and found for example in one experiment that there were 10l2 particles ~m-~ and that their average size was about 45A. These figures are in agreement with classical theory. When a supersonic jet of vapour emerges from a nozzle and is collimated by a skimmer and a slit the resulting molecular beam may be examined. Bentley 46 and also Henkes 47 passed the beam into the source of a mass spectrometer and measured the ion currents for the various cluster masses.48* 49 High energy electron diffraction patterns have been obtained from clusters by Anderson and Stein.5o W.J. DUNNING NUCLEATION TIME-LAG When a system suddenly becomes supersaturated clusters must be built up to critical size.51- 52 During this time lag the rate of nucleation J(t) is given approxi- mately by 53 J(t) = J exp (-z/t) (7) where J is the steady-state rate (eqn (5)) and z = K2/pKwhere PK is the frequency of monomer capture by the critical nucleus of size K. In cloud chamber and supersonic nozzle experiments z for water vapour is appreciably less than the " time of observa-tion " during which the supersaturation persists and the steady state approximation is valid.54 When the growth process of the embryos is not simple but say the result of chemical reactions at the surface of the embryos the possibility that the nucleation rate is non-steady must be considered.GROWTH OSTWALD RIPENING AND SMOLUCHOWSKI COAGULATION Fig. 2 illustrates schematically the change in the size distribution n(r) with time. The initial cluster distribution ab relaxes to the steady state distribution cd during the build up period (w 107). The strip ec corresponds to n(r,)dr and to the number of critical nuclei. In the next interval of time these nuclei are born and become free- growing. In the second-next interval the first-born grow and another lot of nuclei are born.In the third-next interval the first-born continue growing the second-born grow and a third batch of nuclei appear and so on until the distribution is that of A. Size r r FIG.2.-Schematic representation of the development with time of the size distribution. As a result of growth the supersaturation decreases with time hence the rate of nucleation is greatest at t-1Oz and decreases to become negligible at the metastable limit ; beyond this limit the continued decrease in the supersaturation is solely the AEROSOL FORMATION result of growth. Clearly the first born are the most numerous class and the largest in size. The foot of the leading edge of the distribution tracing the curve r,, in the (r t) plane shows the growth of the first-born.The number dnuclei being born at any time is related to the height of the ordinate at the trailing edge of the distribution the foot of which initially follows the curve r,. When the supersaturation reaches the metastable limit the height of the trailing edge is negligible (curve E). The supersaturation continues to decrease and in consequence rK continues to increase since r and supersaturation are related by the Gibbs- Thomson relation (eqn (4)); r overtakes the size of the smallest drops and then Ostwald ripening occurs (curves F G). Droplets Iarger than rK continue to grow but those which have become smaller than r begin to evaporate and there is a flux of droplets in the direction r = 0. The distribution now begins to spread out on both sides of r which is itself changing with time.At all times rK must remain smaller than r,,, otherwise the whole precipitate would evaporate contrary to thermodynamic principles. OSTWALD RIPENING The sequence of reactions in eqn (2) and the corresponding rate eqn (3) apply not only to subcritical clusters but also to the drops of condensate so long as the assump- tion remains valid that only single molecules are gained or lost and that reactions between droplets (i.e. coagulation) may be neglected. When we change from discrete distributions niwith integral i to continuous distributions n(r)dr eqn (3) becomes the equation of continuity an(r,t) a[h( r,t)] =O (8) at + ar where i = dr/dt is the rate of growth. Gyarmathy 55 has derived an expression for i,the rate of growth of a droplet from a supersaturated vapour in a carrier gas Here L is the latent heat of condensation per kg A the gas constant per kg of vapour, Pthe total pressure andp the partial pressure of the carrier gas far from the droplet ; D is the diffusion coefficient and I the mean free path the thermal conductivity OJ the medium p the density of the liquid and rK is given by eqn (4).Size distributions during nucleation and growth (e.g. A B C D,E in fig. 2) may be computed from the expressions for i and J and the equation for continuity. When nucleation ceases this source of particles is replaced by a sink for particles near r = 0. Some time after this still assuming that coagulation is absent net growth from the vapour becomes unimportant and the supersaturation changes only very slowly with time; this is the stage of secular ripening.Lifshitz and Slezov 56 obtained an “asymptotic ” solution of the ripening eqns (4) (8) and (9). Wagner 57 extended their results and Dunning 58 further simplified the procedure. If we use Gyarmathy’s eqn (9) for growth (r>rK)and evaporation (Y <rK),the size distribution during secular ripening is of the form Nr t) = dt) h(P1 ’P (10) in which g(t) depends only on the time and h(p) only on the relative size p (p = r/rK) Further s(t>= dto) [I+ (f -fo)l%1-2 (11) W. J. DUNNING 13 where the ripening time constant rR is given by ZR = [I(L2/;CR,T)+(R,TpJPDpI,)I 1.5lR Tp (12) and g(to)and rKO are the values of g(t) and rKat a" start "time to within the period of secular ripening and t>to.The expression for h@) p is o)P = ~[2/(2-p)15 ~XP PP/(~-P)I for (13) = 0 for p22. The total number of particles N(0 t) present during secular ripening varies with time as N(0,t) = N(0 to)[I +(t-tO)/TR]4 (14) where N(0 to)is the number present at the start time. The quasi-stationary size distribution is independent of the original size distri- bution. Its form depends upon the growth-evaporation law. SMOLUCHOWSKI COAGULATION When in addition to the gain and loss of single molecules reactions between all size classes are taken into consideration e.g. the problem becomes more complex. It may be simplified by assuming that Smoluch- owski coagulation occurs for which only the forward reactions in 15 take place.We then have for this process where p (ui u,) is the coefficient of coagulation for particles of volumes ui and L:~and n is the concentration (time dependent) of particles in the volume size class ui. On passing from a discrete to a continuous distribution eqn (16) becomes 59* 6o The left hand side of this equation with 6 = dv/dt corresponds to the terms in eqn (8) and the right hand side to Smoluchowski coagulation. Friedlander and his collabor- ators 60-62 have sought a solution to this equation of the form 44 t)=dt) $(r) (18) in which g(t)is a function only of the time and $ afunction only of q. The dimension- less number y is equal to u/u* where the mean particle volume v* = V(0,t)/N(O,t)and V(0,t) N(0 t) are respectively the total volume and total number of all the particles.Friedlander and Wang 6o have shown that in the case when B is assumed constant an " asymptotic " solution is obtained with g(0 = "(0 t>l"~(O t) (19) and for $(q) analytical expressions 62have been obtained for the lower and upper ends of the distribution. Numerical solutions over the whole distribution have been obtained by Hidy 63 and by Pich Friedlander and Lai.62 These results suggest AEROSOL FORMATION strongly that a quasi-stationary distribution of size is obtained after a prolonged time but a general proof is not available. A size distribution for the hypothetical steady state resulting from coagulation is shown schematically as curve H in fig. 2. CONCLUSION Although with the possible exception of prenucleation kinetics the basic processes are aspects of a single conceptual scheme it is still necessary to treat them as separate stages in the development of an aerosol.In the simplest production systems these processes develop and follow each other in sequence. For example when gases flow into a tube diffusive mixing or chemical reactions generate an increasing supersaturation as the gases move downstream. Further downstream nucleation or growth on foreign nuclei takes place and these are followed by the other processes in overlapping sequence. Should the mixture emerge and mix with the atmosphere a simple first approach would be to suppose that the concentration fields of the components depend only on location and not on time.A steady state is conceived in which a cloud is centred on the source of partly reacted gases and partly condensed products and within it all concentrations and process rates depend only on position. When realistic factors such as turbulence wind convection topography and climate are introduced the complexity of the problem becomes great. Another conceptually simple system would be an analogy with the continuous stirred tank reactor into which reactants enter at a steady rate and the contents are removed at a rate to balance the input. 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