Evaporation of Fine Atmospheric Particles BY C. N. DAVIES Dept of Chemistry University of Essex Wivenhoe Park Colchester Essex Received 6th December 1972 The decrease in size of aerosol particles due to evaporation is calculated for two situations. In the first the air is saturated with vapour so that evaporation is due to the Kelvin effect and proceeds at the minimal rate ;in the second the air is free from vapour and the rate of evaporation is maximal. The calculations are based upon an interpolation formula covering the transition with decreasing particle size from diffusion control to free molecular flow. 1. INTRODUCTION The rate of evaporation of a pure substance in the form of a spherical particle of radius a when subject to diffusion control in a gas at rest is equal to 4o = -drn/dt = 4naV(n,-n,) (1.1) where n is the concentration of vapour in equilibrium with the surface of the particle n the vapour concentration at a distance from the surface and V the coefficient of diffusion of vapour molecules through the surrounding gas.This equation is valid when the Knudsen number Kn = 2/a (14 1 being the mean free path of the gas molecules is near to zero. Increase of the rate of evaporation due to the particles falling under gravity can be seen from the equation of Frossling to be negligible for radii below 10 pm. If the gas pressure is low or the particle is very small so that Kn is very large the rate of evaporation is 4K = 4na2a(n,-n,)ta (1.3) where Z is the mean velocity of the evaporating molecules and the fraction a is the evaporation coefficient.For intermediate values of Kn the rate of evaporation has been calculated by Fuchs on the assumption that free molecular flow of molecules of vapour as in (1.3) proceeds from the surface of the particle outwards for a distance A and diffusive flow continues from radius a+A to infinity as in eqn (1.1). A is a length near to the mean free path of the molecules of the surrounding gas. Suppose that the concentration of vapour at distance A from the surface is n and that n = 0. The rates of transport across the zones inside and outside radius a+A must be the same hence 4 = 4n(a+A)Vn1 = na2(n,-n,)Za (1-4) which gives after eliminating n C. N. DAVIES In the opinion of Wright,3 his experiments indicate that the thickness of the free molecule region is A = 2V/C (1 4 and from the kinetic theory of gases the coeffiicent of diffusion is given by v = a/3.(1.7) The coefficient of diffusion of the evaporating molecules can therefore be eliminated from eqn (1.5) which becomes 4 = &/(4Kn/3a +1/( 1+2Kn/3)). (1.8) For small values of Kn this reduces to 4 = 40/(1+2Kn(2/a -1)/3) (1.9) which is considered by Fuchs and Sutugin to be accurate for Kn< 1. From eqn (1.1) and (1.2) 4o = &4V/aZa = &4Kn/3u (1.10) so that (1 .8) can be written for large values of Kn as 4 = &/( 1+9a/8Kn2). (1.11) However this equation involves Kn in a manner which Fuchs and Sutugin show to be incompatible with current theory of free molecular flow.The A concept is thus invalid at high values of Kn. They have pointed out that a theoretical solution exists of a mathematically analogous problem which brings the value of 6 to the correct limits as Kn tends to zero and to infinity. They have given an interpolation formula which fits the exact solution closely. This formula results in eqn (1.8) being replaced by the expression 4 = $o/(l +Kn(1.333Kn+0.71)/(Kn+ 1)) (1.12) which reduces to 4 = &o/(l +0.71Kn) for Kn< 1 (1.13) and to 4 = (4K/a)/(1+0.283Kn-I) for Kn9 1. (1.14) Eqn (1.13) is the same as (1.9) when a = 0.97. For other values of 01 the difference between them decreases as Kn decreases and increases as adecreases. At Kn = 0.25 they agree better than 1 % for a = 1 and eqn (1.13) is 18 % low at a = 0.6.At Kn = 0.1 eqn (1.13) predicts a value of 4 low by 8.4 %. Eqn (1.14) is in reasonable agreement with free molecule theory. As far as is known eqn (1.12) is satisfactory for calculating the rate of evaporation of aerosol particles particularly so because for most substances ais probably equal to unity. Hitherto this calculation has been tiresome because of the need for making exploratory determinations of the values of the terms in the denominator of eqn (1.5) or (1.8) as evaporation proceeds and their relative importance changes. This necessity can be avoided when eqn (1.12) is used as explained below. 2. GENERAL FORMULAE FOR THE RATE OF EVAPORATION OF AEROSOL PARTICLES From eqn (1.1) and (1.12) dm/dt = -4naV(1z,-nm)(Kn+ 1)/(1+1.71Kn + 1.333Kn2).(2.1) EVAPORATION OF PARTICLES and the saturation ratio of vapour in equilibrium with the surface of the particle be S = n,/no (2.3) where no is the saturation vapour pressure of the evaporating substance in bulk. Then dY -VndS -n,lno) Y+l dt PA2 y2+ 1.71y+ 1.333’ The saturation ratio S is determined by Kelvin’s equation which for low values of S-1 approximates to S-1 2 2yMIRTap (2.5) where y is the surface tension of the evaporating substance M is its molecular weight R is the gas constant per gram molecule and p is the density of the particle; Tis the absolute temperature assumed to be the same throughout the particle and the surrounding gas. A water droplet evaporating in air cools to the wet bulb tempera- ture which considerably slows down the rate of evaporation; this is on account of the high latent heat of evaporation and high saturation vapour concentration.Substances of lower volatility such as the others shown in table 2 below cool only to a negligible extent which can be calculated by the methods described by Fuch~.~ Two limiting cases arise which correspond to maximal and minimal rates of evaporation. The maximal rate occurs during evaporation into gas which is free from vapour so that n = 0. The minimal rate is for evaporation into gas which is saturated with vapour so that n = no. For evaporation into saturated vapour eqn (2.4) and (2.5) give dY -Vno 2YM Y+l __ -~~ dt p12 RMpA y(y2+ 1.71y+ 1.333) = -DK Y+l y(y2+ 1.71~ +1.333) where the diffusion factor D = Vno/pA2s-’ (2.7) and the Kelvin factor K = 2yM/RTpE.,dimensionless I.Suppose that the particle size initially corresponds to y = yo and reduces to y = y at t = t, then y(y2+1.71y+1.333) dy [It] = -Y+l therefore 0.333(y~-~~)+0.355(y~-y~)+0.623(yo-y,)-0.623 In DK Y+l When the particle is evaporating into vapour-free gas n = 0 and in place of (2.6) eqn (2.4) and (2.5) give Y+l --Y+l D(l +K’Y)y2+1.71~ +1.333’ (2.10) C. N. DAVIES whence [t]? = 1 y(yz+ 1.71y+ 1.333)dy -lYo 9 D yI (Y+ 1)(Y+K) so that yoy2dy 0.377-K D[r]; = s ___ + yI ~+1 1-K 1-K and 1.333-1.71K+Kz 1-K Y +K Owing to the approximation of the Kelvin vapour pressure formula which was used for eqn (2.5) expressions (2.9)and (2.1 I) are only strictly valid for particles which are initially of a sufficiently large size so that In S2 S-I.(2.22) The times tl should not be so long as to allow the particle to diminish to a size beyond which this is no longer true. Table 1 compares values of In S and S-1 for droplets of dibutylphthalate a substance of moderate vapour pressure in air at 20°C; Kelvin's equation gives the exact value of Sas In S = 2yM/RTap. (2.13) It will be seen that the approximation used for formulae (2.9) and (2.1 1) predicts too low a value for S so that the actual size of the droplet after evaporating for a certain time will be smaller than the size calculated by the formulae. The theoretical TABLE1 radius of droplet In S s-1 /m 1.o 0.00787 0.00787 0.5 0.01 59 0.0157 0.1 0.0819 0.0787 0.05 0.1705 0.1574 0.01 1.197 0.787 0.005 3.826 1.574 lifetimes are thus too long.However this is not often of practical importance even for quite small drops because the rate of evaporation increases rapidly as the size diminishes; the error in the calculated lifetime of a drop which was initially greater than about 0.1pm radius is not too great especially wherl it is evaporating into a saturated atm9sphere. 3. LIFETIMES OF AEROSOL PARTICLES The lifetime of a particle is calculated by making y1 = 0 in eqn (2.9)and (2.1 l) yo being the initial value ao/L EVAPORATION OF PARTICLES For particles in a saturated atmosphere this gives the lifetime -L( DK 0.333~:+0.355yg +0.623y0 -0.623 In (yo+ 1)) (3.1) and for particles in a vapour-free atmosphere [filvf = $0.5Yg-( 0.623 -)(Yo-ln YO+^])+ 1-K 1.333-1.71K +K2 1-K (yo-K (3.2) K In E)] where D and K as defined by (2.7) depend on the substance of which the particle is composed and on the gas temperature and pressure.In eqn (2.9) and (3.1) it will be seen that if the time is measured in units of (DK)-' there is a relation between the dimensionless quantity tDK and the dimensionless quantity y which is the radius measured in units of the gas mean free path. For evaporation into vapour free space the situation is more complicated because tD in eqn (2.11) and (3.2) is a function of both the dimensionless particle size y and the dimensionless Kelvin factor K,the latter depending on the nature of the substance of the particle and also on the temperature and pressure of the gas.The progression of y with tD is therefore specific when evaporation takes place into a vapour-free atmosphere. Values of D and K have been calculated for five pure substances and are shown with the basic properties required in table 2; the data are for pure air at one atmos-phere and 20"C,the mean free path being taken as 6 x cm. TABLE 2 substanu Y/dy cm- M p/g cm-3 V/crnZ s-1 sat. v.p./Torr no/g cm-3 vapour mole-cules/cm3 Dls-1 sodium chloride 124 58.5 2.165 0.1 3x 10-25 9.6~10-31 1.21x 10-21 0.0459 diethylhexyl-sebacate 30 426 0.92 0.024 1.49XlO-9 35x10-14 5x10' 234x10-5 0.19 dibutyl-phthalate sulphuricacid 36 55 278 98 1.048 1.84 0.031 0.09 3x 10-5 7.1 X lO.-S 4.6~10-1O 3.8~10-10 1012 2.3~1012 0.38 0.52 0.131 0.04 water 72 18 1.00 0.26 17.54 1.72X 10-5 5.7X 1012 1.24~105 0.0178 Fig.1 shows the decrease in size of aerosol particles with time when evaporating into a saturated atmosphere according to eqn (2.9) and (3.1). The curves are common to all substances and all gases. By dividing the values on the scale of abscissa by DK times in seconds for specific conditions are obtained. In fig. 2 similar curves of particle radius against the dimensionless time Dt in this case are shown for evaporation into vapour-free gas. It turns out that the curve for a particular initial radius is not very sensitive to the value of K and is there-fore only slightly dependent on the specific circumstances.Curves for a = 0.24 pm are shown for K = 0.04 0.13 and 0.19 corresponding to sulphuric acid dibutyl-phthalate and diethylhexylsebacate respectively evaporating into dry air at 20°C and 1 atm. pressure. There is not a lot of difference between these curves. The big differencein absolute time comes about when the dimensionless times are divided by the values of D appropriate to the systems. For example dibutylphthalate evaporates 15 0o0 times more rapidly than does diethylhexylsebacate and this sub-stance in turn goes 2.1 x 1OI6 times faster than sodium chloride. The figures assume a 2 1 in all cases. C. N. DAVIES dimensionless time DKt FIG.1.-Evaporation of particles into vapour-saturated air at 20°C and 1 atm pressure.- eqn (2.9) and (3.1); -- eqn (3.3) evaporation without allowance for free molecular flow. These curves are common to all substances. dimensionless time Dt FIG.2.-Evaporation of particles into vapour-free air at 20°C and 1 atm. pressure. - eqn (2.11) and (3.2) K = 0.1 ;--,eqn (2.11) and (3.2) K = 0.04,0.13and 0.19 ;--,eqn (3.4) evaporation with diffusion control only. EVAPORATION OF PARTICLES From eqn (1.1) and (2.5) the lifetime of a particle in saturated vapour which is subject to diffusion control and the Kelvin effect can be calculated from which is the first term of eqn (2.9); the remaining terms come from the tendency towards control by molecular flow with decreasing radius. Similarly for a particle in vapour-free gas subject only to diffusion control eqn (1.1) gives (3.4) which is the first term of eqn (2.11).In this case the Kelvin and free flow effects come in at about the same radius so that the remaining terms of eqn (2.1I) involve both. The curve of eqn (3.3) for a = 0.3 pm is plotted on fig. 1 showing of course a more rapid evaporation than the curve of eqn (2.9). The exact curve using the correct Kelvin expression instead of expanding the logarithm will lie between the two but very much closer to that of eqn (2.9). A similar curve for diffusion control alone with a = 0.3 pm is plotted on fig. 2 according to eqn (3.4). 4. APPLICATION TO PARTICLES IN THE ATMOSPHERE Saturation vapour concentration with substances such as diethylhexylsebacate or sodium chloride is equivalent to only a low concentration of small particles 1000/cm3 of radius 0.03 pm for the former and 10-'/cm3 of radius 0.0005 pm with the latter.Evaporation of aerosols of such substances therefore invariably takes place into a saturated atmosphere. This might not always be true for substances of moderate vapour pressure such as dibutylphthalate for which a concentration of 1000 particles/cm3 of radius 0.5 pm is equivalent to saturation vapour concentration. Such aerosols generated in the laboratory in containers will be in equilibrium with saturated vapour. In this case evaporation results in the isothermal distillation of the substance from small particles to larger ones and to the walls of the container. The walls of the container usually present a very much greater area than the particles.In the open atmopshere large particles grow at the expense of small ones which therefore require to be con- tinuously generated to maintain a concentration the rate of generation being high for particles of short lifetime. The lifetime of small particles which would otherwise evaporate in minutes is extended if they carry an electric charge. Normally too few charges are available to account for the stabilisation of concentrations of the order of 106/cm3 in this way though the stability of small ions in clean air up to 1000/cm3 is due to the electrical effect. Fine particulate substances in the atmosphere which are present in high anumbe concentration therefore need to be substances of low vapour pressure such s ionicr crystals oxides and a few organic compounds of high molecular weight.The degree of supersaturation necessary for fine atmospheric particles to grow into fog or mist droplets is not attained in the atmosphere. For water vapour in the atmoshpere the absolute maximum value of S is 1.01 and 1,003 is rarely exceeded; particles which are insoluble in water need to be greater than 0.2pm in radius to condense growing water droplets and soluble parfjcles must be greater than 0.06pm radius. Electric charge is of no significance. The life-times of these particles in dry air C. N. DAVIES can be calculated by the equations given above. The possible constituents of con-densation nuclei are limited by the evaporating tendency.N. Frossling Gerlands Beitr. z. Geophys. 1938 52 170. N. Fuchs Phys Z. Sowjetunion 1934 6 (3) 224. P. G. Wright Disc. Faraday SOC.,1960 30 100. N. A. Fuchs and A. G. Sutugin Topics in Current AerosoZ Research Ed. G. M. Hidy and J. R. Brock. (Pergamon Press Oxford 1971). See H&h Dispersed Aerosols chap. 3.2 p. 31. N. A. Fuchs Evaporation and Droplet Growth in Gaseous Media (Pergamon Press Oxford 1959) chap. 1.6 p. 11.