Aerosol size and relative humidity

JOURNAL

OF COLLOID

SCIENCE

13, 472-482 (1958)

AEROSOL SIZE AND RELATIVE HUMIDITY 1 Clyde Orr, Jr., F. Kenneth Hurd, and William J. Corbett Georgia Institute of Technology, Atlanta, Georgia Received April 18, 1958 ABSTRACT A theoretical and experimental investigation was made of the gain and loss of water with humidity change for particles of NaC1, (NH~)2SO4, CaC12.6H20, AgI, PbI:, and KCI having radii between 0.01 and 0.1 micron. Predictions were made using a combination of adsorption theory, the Kelvin equation, Ostwald's theory, and other thermodynamic considerations. Where actual data were nonexistent, empirical relations were employed to arrive at reasonable approximations. The indicated behavior of hygroscopic particles exposed to increasing humidity is to adsorb moisture amounting to a few molecular layers at low relative humidities; to dissolve as humidity increases, becoming saturated droplets and at the same time undergoing an abrupt size increase; and thereafter, as humidity increases still further, to grow larger and more dilute. The indicated behavior for decreasing humidity is for the droplet size to decrease as humidity decreases and, at a humidity considerably lower than that at which the particle initially dissolved, to recrystallize, undergoing an abrupt size decrease. For nonhygroscopic compounds, no abrupt solution or recrystallization irregularities are predicted. The predictions are in satisfactory agreement with experimental findings. INTRODUCTION

The behavior of aerosol particles and droplets under varying humidities is of practical concern in studies of air pollution, visibility, microwave energy propagation, aircraft icing, fog formation, artificial weather modification, and cloud physics in general. For certain conditions the characteristics to be expected have been established by previous investigators. La Mer and associates (1, 2) showed that liquid droplets follow the Kelvin equation at least to sizes as small as 0.03 g radius. Insoluble solid particles pick up small amounts of moisture by physical adsorption in accordance with one of several equations (3). Soluble aerosol particles, as shown by Junge (4) for the case of very small atmospheric ions, are solids at low 1 The work reported herein was conducted at the Georgia Institute of Technology, Engineering Experiment Station, through sponsorship of the Geophysics Research Directorate, Air Force Cambridge Research Center, Air Research and Developnmnt Command, under Contract No. AF 19(604)-1086. Reproduction of this article in whole or in part is permitted for any purpose of the United States Government. 472

AEROSOL SIZE AND HUMIDITY

473

humidities and dissolved droplets at high humidities, undergoing a transition at some intermediate condition. The purpose of this article is to describe the behavior of solid (at low humidities) hygroscopic aerosol particles over a wide range of humidities and to show that results in reasonable agreement with experimental ones m a y be estimated, although extension of data into regions of supersaturation are required. THEORETICAL

A. Solid Particles at Low Relative Humidities All solids in contact with the atmosphere attract a part of the atmosphere's gases to their surface because of unsatisfied molecular forces in their surface layers. The most condensable gases are adsorbed in the greatest quantities, so, even at low relative humidities, some water vapor is present on aerosol particles. For adsorption occurring on a free surface, i.e., one not exhibiting capillary cracks or pores, physical adsorption is expressed by the relation (5)

~_ v~

c

~

[11

(l_~0)[l+(C_l)~01'

where v = volume of gas adsorbed at pressure P, v,~ = volume of adsorbed gas to give a unimolecular film over the entire adsorbing surface, P = adsorbing gas pressure, P0 = vapor pressure of adsorbing gas at experimental temperature, and C = e (~I-E L)/Rr ; where E1 = heat of adsorption of gas in first layer, EL = heat of liquefaction of gas, R = gas constant, and T = absolute temperature, °K. The quantity E1 -- EL is considered to be the energy released by the adsorbing gas or vapor above that normally released by condensation. Ordinarily it is calculated from adsorption measurements. Lacking experimental data for water vapor adsorbed on the materials of interest, values for the quantity were obtained by assuming that, if the solid dissolves in the condensed vapor, the adsorbed vapor satisfies the surface forces of the solid, or that E~ - EL is approximately equal to the surface energy of the solid. Surface energies have been measured and also calculated from theoretical considerations for some of the more common materials. Granted the validity of the assumption regarding the quantity E~ - E~, the ratio v/vm of Eq. [1] m a y be calculated for any relative humidity and temperature when surface energy data are available. For example, sodium chloride, having a surface energy (6) of 276 ergs/cm. 2, exposed to water vapor, each molecule of which, according to Livingston (7), occupies

474

ORR, tIURD AND CORBETT

10.8 A 2, would at 25 ° C. and 30 % relative humidity be covered with a film of water averaging 1.42 molecules thick. Therefore a single NaC1 particle with a diameter of 0.05 t~ when dry is predicted to increase in size by approximately 0.001 u or about 2 % owing to adsorbed water. The contribution of physically adsorbed water is thus not great as far as size is involved. Figure 1 presents curves (segment A) calculated in this fashion for the materials studied. Surface energy data were taken from the literature or estimated from Fig. 2, which shows surface energies as a function of cold water solubility.

B. Transition from Solid Particles to Droplets If a particle is soluble in water, its adsorption of water vapor will cause it to gain a water envelope which may then dissolve the particle at some relative humidity less than 100 %. To predict the humidity at which a solid surrounded b y its liquid film will shift into a completely dissolved droplet requires that assumptions be made regarding the state of the liquid film. Very probably a concentration gradient actually exists across it during the transition, the inner layer being nearly a concentrated solution while the outer layer is more nearly pure water. For present purposes, it will be 100

90

7o

100

~°~,

>"? / '

__i j

20

I IIII ii

i

_~: ,o c~ 00.01 ~-

I 0.02

0.04

60

i

0,02

0.04

,=,2.~o,,. ,/ /

li,

i

60

0.04

0.06

70

0.1

,o, ol 0.01

0.02

0.04

~

//

" iI

i

i

8o I 70 J

601

50

i ]

,o

/

20

iI

'

III '°ii •

0.02

0.I

0.06

0.I

0.06

0.I

I00.

90

,o

0.05

100

I

i> i

,ol

tll

0,01

Bo

20

//

~o, 2oi

If ll

,0

_~

901

~o,

oo! ~o i

I,tl I

50

00.0l

/~

I.~-~/ Ill/ill/|///

/ !

/-FIWT1 70

0.I _

-

FT[III

~0<~"'~s°']'",i ~°

100,

~

oil'

0.05

100

0o

~

-

~

/

,

il

~°

.L-lIT]

.

.01 '

r~--t,D ~

!

i'

,0, 0.04

0.06

0.I

00i.01

V

II

20 I

,i 0.02 '

/

ii II !1

I r

,

--

"

i

i

....

II

~o, , i' 40 1

l

'

lIc

'I 0.02

0.04

PARTICLE RADIUS, #

FIG. 1. C a l c u l a t e d aerosol size as a f u n c t i o n of r e l a t i v e h u m i d i t y .

475

AEROSOL SIZE AND HUMIDITY

1000

100

10

d --" E

0

o o

E ~

0.1

....I

0.01 _1

o

0.001

(:}.0001 1

10

100

1,000

10,000

100,000

SURFACE ENERGY., ergs/cm 2 FIG. 2. S u r f a c e e n e r g y as ~ f u n c t i o n of cold w a t e r s o l u b i l i t y .

assumed that there is no concentration gradient across the liquid phase. The undissolved particle with its film of liquid surrounded by air containing water vapor will be considered as a three-phase system in equilibrium. The undissolved particle will be considered as in equilibrium with a saturated solution, the concentration of which is discussed below. The liquid film surrounding the particle will be considered to be in equilibrium with the atmosphere. The film of liquid surrounding a particle will exhibit a vapor pressure like any other liquid, and this pressure will determine whether more vapor condenses from the atmosphere or whether some of the film evaporates. The vapor pressure will, however, be altered by two independent effects. The dissolved component will reduce the vapor pressure as shown by Raoult's law and the film's very great curvature will increase its vapor pressure. This increase in vapor pressure is given by the Kelvin equation. The concentrations of solutions in equilibrium with small particles are dependent on particle size and surface energy, as well as temperature and the other usual considerations. The smaller the size, the greater is the concentration of the solution with which the particle is in equilibrium. The relationship between particle size, surface energy, and equilibrium concen-

478

ORRt ttURD AND CORBETT

tration can be obtained for ionic solutions from considerations of the first law of thermodynamics and the Gibbs free energy. This fundamental expression is

1)

where al = activity of solute in a solution produced with particles of radius rl (rl is usually chosen sufficiently large for the term to be negligible), a2 = activity of solute in a solution produced with particles of radius r2, = surface free energy of solute, M = molecular weight of solute, p = density of solute, R = gas constant, 8.32 X 107 dynes cm./gram-mole °K, and T = absolute temperature, °K. This relation is quite general and holds regardless of the extent of dissociation of the solute into ions. When the necessary data are available, the activity of the solute in a film of liquid adhering to a particle can be calculated. From this solute activity the vapor pressure of the solution can be determined as outlined in a following section. In the absence of activity data, an approximate equation of Dundon and Mack (8)

1/rl

iln

E

= ~

~--~

1)

'

[31

where S~ = solubility of a small particle of radius r2, $1 = solubility of a large particle (usually taken as one of infinite size) of radius r~, and i = van't Hoff factor, can be used to approximate the concentration of a solution that would exist in equilibrium with a small solute particle. Because of insufficient data it was necessary to use Eq. [3] instead of Eq. [2] in the case of (NH4)2SO4, for example. Curvature of surface is related to vapor pressure by the Kelvin equation ln(P)

-

RTo~r'2]f~

[4]

where p = vapor pressure of a solution droplet of radius r ( = partial pressure of water vapor in air = water vapor pressure X per cent relative humidity), P0 --- vapor pressure of a solution of infinite extent, M~ = molecular weight of solvent, ~/ = surface tension of solution, R = gas constant, 8.32 X 107 dynes cm./gram-mole °K, T = absolute temperature, p~ = solution density, and r = droplet radius. These relations predict, as may easily be seen, that a soluble particle will shift rather suddenly into a liquid droplet. As relative humidity increases, the particle which already was enveloped in a film of absorbed water, picks up a little more water from the surroundings. The extra solvent dilutes the liquid of the film so that it is no longer saturated with respect to the solid inside. As a result, more material is dissolved from the surface

AEROSOL

SIZE AND HUMIDITY

477

of the particle, thereby reducing its size. But as particle size decreases, the concentration of solution with which the particle can be in equilibrium increases. Increased solution concentration reduces vapor pressure, and causes more moisture to be condensed from the atmosphere; moisture condensing causes the particle to grow in size, reducing surface curvature and reducing vapor pressure still further. At a certain relative humidity the situation clearly is unstable, and equilibrium is re-established only after the complete dissolution of the particle. The relative humidity just below that at which a particle dissolves may be calculated, assuming, of course, that the conditions postulated and the relations expressing them mathematically, as given above, are reasonably correct. For example, a cubical particle of NaC1 0.04 ~ on an edge has a surface area equal to, and for purposes of this calculation equivalent to, a spherical particle having a radius of 0.0277 ~. Using a surface energy of 276 ergs/cm. ~, a solute density of 2.163 g./cm2, an activity of 0.986 for a saturated solution of bulk material at 25° C., solution of Eq. [2] gives the change in activity due to particle size, yielding a value of 1.22 for a2. From a plot of solute activity versus solution vapor pressure, the vapor pressure of this solution, assuming it to be of infinite extent, is 15.70 mm. Hg. With the vapor pressure established, other quantities were then evaluated for substitution into the Kelvin equation (Eq. [4]) to arrive at the corresponding relative humidity. This equation requires information on concentration, surface tension, and solution density among other things. Such data are not easily obtainable for supersaturated solutions. They usually must be obtained by extrapolation and estimation; the details of these methods are also discussed in a following section. Once obtained, these values, when substituted into Eq. [4], yielded a vapor pressure of 16.25 ram. Hg for the droplet. Compared with a water vapor pressure at 25 ° C. of 23.75 ram. Hg, this represents a relative humidity of 68.5 %. Thus it is predicted that a cubical particle of NaC1 0.04 ~ on an edge will dissolve in the film condensed about it when the relative humidity reaches 68.5 %. The solubility, the surface curvature, and all the other factors determining the condition of solubility change with particle size. As a result the shift from particle to droplet occurs at lower and lower relative humidities as particle size decreases. As size increases the change-point becomes asymptotic to a line drawn, in the case of NaC1, at about 75 % relative humidity, indicating that large particles would all go into solution at this point. This value compares favorably with the 75.5 % given by Twomey (9) for NaC1 particles observable in the microscope. Segments B of Fig. 1 present the calculated transition points. Predicted curves are not given for CaC12.6H20, AgI, and PbI2 because the necessary data could not be found. However, rough estimates show that the CaCl:. 6H20 transition would fall below about 15 %, whereas the AgI and PbI2

478

ORR, HURD AND CORBETT

transition would lie well above the highest relative humidity employed experimentally, about 95 %.

C. Droplets and Increasing Relative Humidity Once a particle is dissolved, the droplet's further growth with increasing relative humidity is described by the Kelvin equation using the properties of the solution and paying particular attention to the vapor pressure lowering due to dissolved components. Segments C of Fig. 1 were calculated in this manner. The necessary data in the cases of AgI and PbI2 could not be found, and, if they were available, would undoubtedly have applied only to extremely high relative humidities.

D. Droplets and Decreasing Relative Humidity As relative humidity decreases, droplet size decreases. The Kelvin equation is equally applicable to decreasing relative humidity and size conditions as it is to increasing ones. For concentrations greater than the normal limits of solubility, this equation was employed to obtain segments D in Fig. 1 using extrapolated vapor pressure, surface tension, and density data. A droplet subjected to decreasing relative humidity conditions is predicted to follow first down segment C, then down segment D for an undetermined distance, and finally shift suddenly back to A. Below the humidity condition established by the intersection of lines B, C, and D, i.e., line D, conditions must be regarded as metastable because the droplet solution is supersaturated. At present there are no good methods for predicting the point at which a supersaturated solution will crystallize. When erystallization does occur, it will occur spontaneously and the particle size will shift over to line A. This explains the often-noted lingering of smogs (2) below the humidities at which they first appeared.

E. Methods or Arriving at Supersaturated Solution Data Vapor pressures of aqueous NaC1 solutions at 25° C., the concentrations of Which were greater than the normal limit of solubility (26.4 % by weight), were predicted by extrapolation of the activity coefficients of NaC1 in unsaturated solutions. The values of these coefficients up to concentrations of 6.0 molal were taken from Robinson and Stokes (10). They were plotted versus concentration and the resulting curve was extrapolated to 15.0 molal using as a guide for the extrapolation similar curves for LiCI and CaC12. The activity coefficient curves for these latter materials are similar to that of NaC1 below 6.0 molal, and, owing to the high solubility of these compounds, data are available for them over a greater range. The validity of this extrapolation for NaC1 is supported by the fact that experimental data are available for some materials, in particular CaC12, in the super-

AEROSOL SIZE AND HUMIDITY

479

saturated region, and for such materials, the curves extend smoothly from the unsaturated to the supersaturated region . . . . From the activity coefficients of NaC1 in solution obtained by the extrapolation, NaC1 activities were calculated using the relationship a~/2 = v s m ,

[5]

where as = the activity of NaC1 in solution, % = the activity coefficient of NaC1 in solution, and M = the molality of the solution. The solvent, or water, activity was calculated from t h e activities of NaC1 by the relationship In \ ~ - /

a.

~w d in as,

[6]

where a'~ = the activity of the solvent at the mole fraction x'~ of the solvent, ~ = the activity of the solvent at the mole fraction Xw of the solvent, a'~ = the activity of NaC1 at the mole fraction x'~ of NaC1, and a~ = the activity of NaC1 at the mole fraction xs of NaC1. Since the solvent activity, a~, is defined as the ratio of the partial pressure of the solvent p~ at concentration, x~, to the vapor pressure of the pure solvent, p0, the relationship aw -

P~

[7]

In (p'l)

[81

can be written. Hence,

in

(a'w

\G/

Equation [6], requiring graphical integration for evaluation, was written in the form in (p,l~ = _ ~ a , ~ ~-1]

as

x~ a~ (1 -- x~) das,

[9]

where p'l = the partial pressure of the solvent at the concentration where the NaC1 activity -- a'~, and P1 = the partial pressure of the solvent at the concentration where the NaC1 activity is a~. B y means of Eq. [9] and the data of Robinson and Stokes (10) for the activity of aqueous solutions of NaCl, curves of vapor pressure versus solute activity and vapor pressure versus solution concentration were predicted for solutions up to 50 % b y weight of NaC1. T h e curve of vapor pressure versus solution concentration checked very closely with data from the I n t e r n a t i o n a l Critical Tables for the unsaturated region (11), lending confidence in the accuracy of the supersaturated portion of the curves.

480

ORR, H U R D

AND CORBETT

The vapor pressures of (NH4)2SO4 solutions were predicted from the equation of Edgar and Swann (12), Q

1

where pl = the vapor pressure of the solution of mole fraction Xl at temperature T1, p2 = the vapor pressure of the solution of mole fraction x2 at temperature T2, Q = the heat of vaporization of H20, R = gas constant, 1.987 cal./gram-mole °C., and i = the van't Hoff factor, using the data of these same investigators for saturated solutions. The vapor pressures of aqueous (NH4)2SO4 solutions, at 25° C., were predicted in this manner from zero to 75 % concentration by weight. The van't Hoff factor, calculated as shown below, was assumed to be constant for these calculations. The vapor pressure curve calculated in this manner checked very well in the unsaturated region with data from the International Critical Tables (13), indicating that vapor pressures for the supersaturated region predicted in this manner are reasonably correct. Vapor pressures of aqueous CaCl~ solutions at 25°C. were obtained by means of Eq. [8], these activities having been determined well into the supersaturated region (14). Activity and vapor pressure data for KC1 solutions at 25° C. were obtained from Robinson et al. (15). The van't Hoff factor for (NH4)2SO4 was calculated from data on the vapor pressure lowering (16), boiling point elevation (17), and freezing point lowering (18). Because of the lack of data in the supersaturated region, the van't Hoff factor was evaluated from data at saturation. Other properties of solutions show no discontinuity in going from the saturated to the supersaturated region; therefore it was assumed that the van't Hoff factors behave in a similar manner. Also, these factors do not change value appreciably with concentration in the unsaturated region, thus indicating that use of the values obtained at saturation for slightly supersaturated solutions does not greatly affect the accuracy of the calculations. Data for the surface tension (19, 20) and solution density (21, 22) for NaC1 and (NH4)~SO4 in the unsaturated region were obtained from the International Critical Tables. The data of Harkins and Gilbert (23) for surface tension and solution density of unsaturated CaC12 solutions were used. The surface tension data for unsaturated KC1 solutions were obtained from the work of Jones and Ray (24). The solution densities of unsaturated KC1 solutions were obtained from the International Critical Tables (25). Surface tension and solution density curves, for the compounds considered, were extended into the supersaturation region by simple extrapolation. The normal limit of solubility for all compounds considered was obtained from the data of Seidell (26).

AEROSOL lo0

AND

-!

80 70 60

/ l

--

100

oo?H4,2~°4 I 2,~-ll-ITl ,0

i'

7ol

!l

oot

50

501 I

00

3OI

20

201

10

10~ 0.04

0.06

0.1

•-"--

90

I

6o

40

0,02

481

HUMIDITY

IOO

I I#lll I/[ i L_#,,"I I

NoCI 90

50

SIZE

:°° 2ol

%;Ol

90

0.04

0.06

0.1

i I~"~trIll

CQCI2'6H20

80 1

00

0.02

'Ool 90

b Ill II11[

0.02

0.01

0.04

KC{

60

60 50

0,1

. "

'

00

50

0.~

f

"40

40

30 20

-

10 °0.01

0~

0 04

0 06

0 1

~

40

0o

iI

ao

20

!

20

10

!

10 0

°o o]

0 02

0 04

0 06

0.1

0.01

I

0.02

0.04

0.06

0.1

PARTICLE RADIUS,# --EXPERIMENTAL

FIG.

3.

......

THEORETICAL

Comparison of experimental and calculated results. EXPERIMENTAL

Since the experimental apparatus and technique are described elsewhere (27, 28), they will not be dwelt on here. Both were patterned after the descriptions of Israel (29) and Junge (4). Briefly it may be stated that particle (and droplet) size and size change were determined from mobility measurements, i.e., the movement of the particles under imposed electric fields. Refinements in the method of interpretation of the mobility data allowed specific size information to be obtained in contrast to earlier work which yielded only average values for the entire size distribution. Crystal or lowhumidity particle size was confirmed with electron photomicrographs. The results in every case checked the predicted curves. Typical correspondence of calculated and experimental data is shown by curves in Fig. 3. CONCLUSIONS

Making assumptions which were supported by experimental data from similar systems or comparable conditions, the behavior of aerosol particles was calculated under changing humidity conditions. With better data for

482

ORR~ HURD AND CORBETT.

the properties of surfaces and supersaturated solutions, more exact predictions could undoubtedly have been obtained. The methods employed are believed valid for particles of compositions other than those studied. REFERENCES 1. LAMER, V. K., AND GRUEN, R., Trans. Faraday Soc. 48, 410 (1952). 2. I~A MER, V. K., AND COTSON, S., Science 118, 516 (1953). 3. BRUNAUER, S., "The Adsorption Of Gases and Vapors." Princeton University Press, Princeton, New Jei~sey, 1945. 4. JUNGE, C., Bet. deut. Wetterdienstes, U. S. Zone 38, 264 (1952). 5. BRUNAUER, S., EMMETT, P. H., AND TELLER, E.. J. Am. Chem. Soc. 60, 309 (1938). 6. VAN ZEGGEREN, F., AND BENSON, G. C., Can. J. Chem. 35, 1150 (1957). 7. LIVINGSTON,H. I(., J. Colloid Sci. 4, 447 (1949). 8. DUNDON,M. L., AND MACK, E., JR., J. Am. Chem. Soc. 45, 2479 (]923). 9. TWOMEY, S., J. Appl. Phys. 24, 1099 (1953). 1O. ROBINSON,R. A., AND STOKES, R. H., "Electrolyte Solutions." Academic Press, New York, 1955. 11. "International Critical Tables," Vol. 3, p. 297. McGraw-Hill, New York, 1933. 12. EDGAR, GRAHAM,AND SWANN, W. 0., J. A m . Chem. Soc. 44, 570 (1928). 13. ';International Critical Tables," Vol. 3, p. 363. McGraw-Hill, New York, 1933. 14. STOKES, R. It., Trans. Faraday Soc. 41, 637 (1945). 15. ROBINSON, R. A., AND STOKES, R. H., "Electrolyte Solutions." Academic Press, New York, 1955. 16. ROBINSON,R. A., AND STOKES,R. H., "Electrolyte Solutions," p. 363. Academic Press, New York, 1955. 17. ROBINSON,R. A., AND STOKES,R. H., "Electrolyte Solutions," p. 325. Academic Press, New York, 1955. 18. ROBINSON,R. A., AND STOKES,R. H., "Electrolyte Solutions," p. 255. Academic Press, New York, 1955. 19. "International Critical Tables," Vol. 4, p. 465. McGraw-Hill, New York, 1933. 20. "International Critical Tables," Vol. 4, p. 464. McGraw-Hill, New York, 1933. 21. "International Critical Tables," Vol. 3, p. 80. McGraw-Hill, New York, 1933. 22. "International Critical Tables," Vol. 3, p. 60. McGraw-Hill, New York, 1933. 23. HARKINS, W. D., AND GILBERT, E. C., J. Am. Chem. Soc. 43, 604 (1926). 24. JONES, GRINNEL, AND RAY, W. A., J. Am. Chem. Soc. 59, 187 (1937). 25. "International Critical Tables," Vol. 3, p. 87. McGraw-Hill, New York, 1933. 26. SEIDELL, A., "Solubilities of Inorganic and Metal Organic Compounds," Vol. i, 3rd ed. Van Nostrand, New York, 1940. 27. ORR, CLYDE, JR., HURD, F. KENNETH, HENDRIX, WARREN P., AND CORBETT, WlLL~AM J., "An Investigation into the Growth of Small Aerosol Particles with Humidity Change." Final Report, Project No. A-162, Engineering Experiment Station of the Georgia Institute of Technology, Atlanta, Georgia, December 31, 1956. 28. OaR, CLYDE, JR., HURD, F. KENNETH, HENDRIX, W. P., AND JUNGE, CHRISTIAN, "The Behavior of Condensation Nuclei under Changing Humidities." To be published by the J. Meteorol. 15, No. 2 (1958). 29. ISRAEL, H., Gerlands Beitr. Geophys. 31, 173 (1931).