Hydrocarbon Nucleation and Aerosol Formation in Neptune's [PDF]

ICARUS

99,

318-346 (1992)

Hydrocarbon JULIANNE Division

OfGeological

Nucleation and Aerosol Formation Neptune’s Atmosphere I. MOSES,’ MARK ALLEN,~

and Planetary

Sciences,

California

Institute

in

AND YUK L. YUNG of Technology,

Pasadena,

California

91125

Received January 7, 1992; revised June 8, 1992

1. INTRODUCTION Photodissociation of methane at high altitude levels in Neptune’s atmosphere leads to the production of complex hydrocarbon species such as acetylene (C2H2), ethane (C,&), methylacetylene (CH,C,H), propane (C,Hs), diacetylene (C,H,), and butane (C,H,). These gases diffuse to the lower stratosphere where temperatures are low enough to initiate condensation. Particle formation may not occur readily, however, as the vapor species become supersaturated. We present a theoretical analysis of particle formation mechanisms at conditions relevant to Neptune’s troposphere and stratosphere and show that hydrocarbon nucleation is very inefficient under Neptunian conditions: saturation ratios much greater than unity are required for aerosol formation by either homogeneous, heterogeneous, or ion-induced nucleation. Homogeneous nucleation will not be important for any of the hydrocarbon

species considered; however, both heterogeneous and ion-induced nucleation should be possible on Neptune for most of the above species. The relative effectiveness of heterogeneous and ioninduced nucleation depends on the physical and thermodynamic properties of the particular species, the abundance of the condensable species, the temperature at which the vapor becomes supersaturated, and the number and type of condensation nuclei or ions available. Typical saturation ratios required for observable particle formation rates on Neptune range from -3 for heterogeneous nucleation of methane in the upper troposphere to greater than 1000for heterogeneous nucleation of methylacetylene, diacetylene, and butane in the lower stratosphere. Thus, methane clouds may form slightly above, and stratospheric hazes far below, their saturation levels. When used in conjunction with the results of detailed models of atmospheric photochemistry, our nucleation models place realistic constraints on the altitude levels at which we expect hydrocarbon hazes or clouds to form on Neptune. Q 1%~Academic Press, Inc.

t Now an NRC Associate at NASA Ames Research Center, Space Sciences Division, Moffett Field, California 94035. * Also at Earth and Space Sciences Division, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109. Presented at Neptune/T&on Conference in Tucson, Arizona, during January 6-10, 1992.

Hydrocarbon hazes and clouds in the upper troposphere and lower stratosphere of Neptune constitute a major stage in the evolution of carbon-bearing molecules in the atmosphere. Methane, which is relatively abundant in Neptune’s deep atmosphere, provides the main source of carbon found in the photochemical hazes. Ultraviolet photolysis of methane in Neptune’s upper atmosphere initiates the production of more complex hydrocarbon molecules. These molecules diffuse to the cold lower stratosphere where the less volatile species can condense; consequently, haze layers can form in the lower stratosphere. Carbon is ultimately lost from the stratosphere through precipitation of these haze particles. Evidence confirming the presence of particulate layers in Neptune’s atmosphere originates from groundbased and Earth-orbiting ultraviolet, visible, and near-infrared observations of Neptune (see reviews by Trafton 1981, Caldwell et al. 1984, Bergstralh and Baines 1984, and Orton and Appleby 1984). Results from the Voyager Photopolarimeter Subsystem (PPS) provide further evidence for an ultraviolet-absorbing haze on Neptune (Lane et al. 1989, Pryor and Hord 1991). Spatially resolved images of Neptune at 0.26 pm obtained with the PPS instrument indicate that high-altitude hazes extend almost uniformly across the planet; haze absorption seems especially prevalent at equatorial latitudes (30” S to -5” N) (Lane et al. 1989, Pryor and Hord 1991). At longer wavelengths, the Voyager imaging subsystem results at visible and near infrared wavelengths (Smith et al. 1989) and the PPS images at 0.75 pm (Lane et al. 1989) reveal many spatial inhomogeneities and other features suggestive of clouds and lowaltitude hazes. Both the Voyager and Earth-based results are consistent with the suggestion that a thin but widespread UV-absorbing hydrocarbon haze exists in the Neptune stratosphere while an underlying region con-

318 0019-1035/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

HYDROCARBON

AEROSOL

taining spatially distinct methane clouds and hazes is present in the upper troposphere. The Earth-orbiting satellite and groundbased observations have been used to analyze the vertical structure and optical properties of the clouds and hazes on Neptune (e.g., Pollack et al. 1986, Hammel et al. 1989, and Baines and Smith 1990). However, the numerous unknown free parameters involved in such radiative transfer algorithms make detailed quantitative modeling very difficult. By simultaneously analyzing a multiwavelength dataset, Baines and Smith (1990) are able to constrain some of the physical and optical properties of the separate cloud and haze layers in Neptune’s atmosphere. For example, the column density, imaginary refractive index, and average radius of the stratospheric haze particles; the single-scattering albedo, optical depth, and pressure levels of the tropospheric clouds; and the methane mixing ratio of the deep atmosphere are derived in the models of Baines and Smith. Although such analyses are useful and ambitious, the results are quite sensitive to assumptions concerning the background atmosphere (e.g., the stratospheric methane abundance, the atmospheric temperature profile, and the stratospheric haze condensation levels). Some of the free parameters needed for aerosol modeling can be estimated from models of hydrocarbon photochemistry. For instance, Romani and Atreya (1988, 1989) develop theoretical models of Neptune atmospheric photochemistry to help place constraints on the stratospheric methane abundance and the location and composition of the photochemical hazes. In particular, Romani and Atreya (1988) predict that diacetylene (C,H,), acetylene (C,H,), and ethane (C,H,) will condense in the lower stratosphere at their saturation levels, at pressures of a few mbar. Romani and Atreya (1989) update their earlier model and deal with the condensation process in a more realistic manner. They assume that embryos of the condensed phase already exist in the condensation regions and calculate the condensation rate of the different vapor species onto these preexisting embryos. We believe that photochemical models such as these may not be reliable indicators of the actual levels at which hazes will form on Neptune. We suggest that hydrocarbon haze and cloudparticle formation on Neptune is less efficient than previously believed and that haze condensation levels cannot easily be predicted without an examination of the details of nucleation and particle formation under relevant Neptune atmospheric conditions. The ultimate explanation for the suggested inefficiency of haze-particle formation on Neptune arises from the planet’s cold atmosphere. Temperatures in Neptune’s lower stratosphere and upper troposphere are sufficiently below the triple points of the condensable hydrocarbon species that we would not expect supercooled liquids to be present in the haze layers; instead, aerosol formation

FORMATION

ON NEPTUNE

319

would proceed by direct nucleation of the ice phase. Formation of ice nuclei occurs at greater supersaturations than is typical with liquids because particle formation can proceed only by heterogeneous nucleation onto ions or insoluble particles. Nucleation onto soluble particles, usually the most efficient particle-formation mechanism available, does not occur with nucleation of ices because the soluble particles tend to break apart and do not have the structural integrity required for efficient ice-particle formation (Pruppacher and Klett 1978). Furthermore, ice nucleation on insoluble particles is only effective if the insoluble nuclei have a favorable crystal structure and/or morphology. Methane, ethane, and propane are the only condensable species in Neptune’s atmosphere that have triple points just 10 K or so warmer than the temperatures at their expected condensation levels; thus, these species may nucleate initially as supercooled liquids, perhaps followed by later crystallization and freezing. The requirement of insoluble particles for heterogeneous nucleation is then relaxed; however, nucleation of liquids will still be inefficient in Neptune’s atmosphere because the low temperatures and low abundances of the condensable vapors greatly inhibit the kinetics of molecular-cluster formation. When both temperatures and vapor abundances are low, individual molecules rarely encounter each other, and nucleation rates are small. A good example of the inefficiency of nucleation at cold temperatures is the situation in the Earth’s polar mesosphere where noctilucent clouds form only at saturation ratios of - 100 (Arnold 1980, Gadsden 1981, Keesee 1989). Because nucleation at stratospheric conditions on Neptune may be inefficient, the hazes may form at altitude levels significantly below those predicted by the assumption that the hydrocarbon species condense as soon as the vapor becomes saturated. The purpose of this paper is to explore this hypothesis further. This work was originally motivated by a study of nucleation and particle formation in the Earth’s mesosphere by Keesee (1989), and the structural organization of our paper parallels that of Keesee. First (Section 2), we consider nucleation theory in some detail and develop the equations we will need to predict particle formation rates on Neptune. The sensitivity of the nucleation rate equations to various thermodynamic and particle properties is briefly reviewed at this time. Then (Section 3), we examine the gas-phase composition of Neptune’s atmosphere and identify the hydrocarbon species that may condense in the stratosphere and upper troposphere. Next (Section 4), we examine the sources and concentrations of ions and condensation nuclei in Neptune’s atmosphere. Then (Section 5), we apply nucleation theory to tropospheric and stratospheric conditions on Neptune. We consider the relative efficiencies of the various nucleation mechanisms, determine the critical

320

MOSES. ALLEN. AND YUNG

saturation ratios required to obtain observable particle formation rates, and estimate the levels at which we expect the methane clouds and stratospheric hazes to form on Neptune. Finally (Section 6), we summarize our principal conclusions and suggest directions for future research. 2. NUCLEATION

THEORY

Classical nucleation theory can be used to predict particle formation rates on Neptune. The thermodynamic and kinetic aspects of nucleation theory are reviewed in this section; note that we use the term “nucleation” to refer specifically to the formation of condensed particles from a vapor phase. Our discussion relies heavily on the nucleation rate derivations of McDonald (1962, 1963), Twomey (1977), Pruppacher and Klett (1978), and Seinfeld (1986). In this section, we focus on two broad types of nucleation phenomena-homogeneous nucleation, or the formation of new particles (“embryos” of a condensed phase) from the gas phase, and heterogeneous nucleation, or nucleation of vapor onto foreign material or surfaces such as ions, preexisting aerosols, or container walls. We distinguish between heterogeneous nucleation about ions, which we call ion-induced nucleation, and heterogeneous nucleation about spherical insoluble particles, for which we use the general term heterogeneous nucleation. Although most of the following discussion can be found in the above references, the key concepts and equations are introduced in this section so that the reader can better understand the peculiarities of the nucleation behavior of hydrocarbons on Neptune. 2.1. Homogeneous

Nucleation

As discussed by Twomey (1977) homogeneous nucleation occurs when vapor molecules develop associations or clusters of more than one molecule; for example, dimers, trimers, etc. These associations evolve during the frequent random encounters between individual molecules in a gas, and the clusters formed in this manner can exist for a finite period of time before being broken up by further collisions. At any one time, the probability that a grouping of g molecules can be found in a local region is finite, but generally very small, and depends on the change in energy of the system that is experienced upon the formation of the particle. If the molecular clusters grow to some arbitrary size (containing tens or hundreds of molecules), then they can be considered as embryos of a completely new phase rather than as vapor molecules. The classical theory of homogeneous nucleation (often called the spherical or liquid drop model) is based on work by W. Thomson (later Lord Kelvin, 1870) who used the fact that the vapor pressure of a spherical drop of finite radius in equilibrium with vapor is greater than that of a flat-surfaced liquid at the same temperature. In this the-

ory, the formation of a spherical drop (either liquid or solid) of radius r from a system of pure vapor involves a change in free energy of the system of an amount AG(r) = %rr3 AG,,, + 4vr2cr,

(1)

where AG,,, = - (plm,)kT In S is the bulk free energy change per unit volume of the transformation to the condensed phase, p is the mass density of the droplet, m, is the mass of one molecule of the condensable species, k is the Boltzmann constant, T is the temperature of the system, S is the saturation ratio of the vapor (the actual partial pressure of the condensable vapor divided by its saturation vapor pressure), and o is the surface free energy of the condensed droplet (e.g., surface tension in the case of liquids). The first term on the right-hand side of Eq. (1) is proportional to the droplet volume and represents a decrease in the free energy of a system of supersaturated vapor resulting from the decrease in the volume energy occurring during the phase transformation. The second term on the right-hand side of Eq. (1) is proportional to the surface area of the droplet and represents the free energy increase caused by the creation of an interface between the condensed and vapor phases. For saturation ratios greater than 1, the change in free energy of the system can be either positive or negative. During the initial formation of a condensed embryo, the growing cluster is small, and the surface energy term will dominate the behavior of AG(r) such that the free energy change is positive. However, as more and more monomers are added to the cluster, the embryo radius is increased, and the volume energy term becomes progressively more important. If the vapor is supersaturated, AG can become negative for large embryo radii (see Fig. l), and AG will be a maximum at some critical radius r = r* (with some critical number of molecules in the embryo g = gJ for S > 1. Particle growth beyond the critical size is thermodynamically favored; that is, the addition of another monomer results in a decrease in the total free energy of the system. The critical embryo radius r, can be easily found by taking the derivative of AG with respect to the radius and setting the result equal to zero: *

-2a

r =a=

2mp

pkTln S’

For a fixed temperature and droplet parameters, the value of r* depends only on the saturation ratio. The more supersaturated the vapor is, the smaller the critical radius r, and the more likely the formation of stable critical-sized embryos. The free energy of a critical-sized embryo is

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321

AEROSOL FORMATION ON NEPTUNE

inverse cube of the temperature. However, the argument of the exponent also varies inversely with the square of the log of the saturation ratio. If the total vapor density is held constant rather than the saturation ratio S, large nucleation rates are more likely at cold temperatures because S generally increases drastically with decreasing temperature. Descriptions of the behavior of the nucleation rate as a function of temperature and saturation ratio will be presented when we directly apply these equations to the conditions in Neptune’s atmosphere (Section 5). 0

2.2. Zon-Znduced Nucleation 0

10

20

30

RADIUS OF CLUSTER (A)

FIG. 1. The free energy change as a function of particle radius for homogeneous nucleation of a system of vapor with saturation ratio S. This particular case describes nucleation of methane at 66 K. The critical embryo radius r,, or the radius at which the free energy change is a maximum, is indicated. Embryos with radii larger than r, are stable.

AG* acts as a barrier to the formation of stable clusters. For fixed temperature and particle properties, AG* is sensitive only to the saturation ratio. For a dilute mixture of condensable vapor in an atmosphere, the homogeneous nucleation rate (particles cm-3 see-‘) can be written (Seinfeld 1986) l/2

(2) where n, is the number density of monomers of condensable vapor in the system, p is the flux of monomers onto a unit surface area (equal to n,(kT/2~m,)“*), a(g*) is the surface area of a critical-sized cluster (equal to 4~ r z if we assume the growing cluster to be spherical), n, exp( - AG*/kT) is the equilibrium concentration of embryos of critical radius r*, ($27~)“~ is the Zeldovich factor Z which accounts for the nonequilibrium nature of the cluster distribution, and 1 y=

-kT

[ ag* 1g=g,’ a*AG

Although for a fixed saturation ratio, p and n, are sensitive functions of temperature through the expression for the saturation vapor density, the exponential term in Eq. (2) is even more sensitive to temperature and generally dominates the nucleation rate. For a given saturation ratio, lower temperatures lead to slower nucleation rates because the argument of the exponent varies with the

Ion-induced nucleation is a type of heterogeneous nucleation in which molecules of the condensable vapor species cluster about a gaseous ion. Because of an ion’s electrostatic potential, molecules can cluster more efficiently about ions than about other vapor molecules. Wilson’s cloud chamber experiments first demonstrated this effect (Wilson 1897). Clustering about ions is so efficient that small clusters of vapor molecules about ions are stable in an atmosphere and are thermodynamically preferred over the case of unclustered ions (see Russell 1969, Castleman and Tang 1972, Castleman and Keesee 1986). The following discussion is based on material presented in Russell (1969) and Castleman (1979). J. J. Thomson (1888) was the first to theorize that ions could promote the nucleation process. He expanded classical homogeneous nucleation theory to include nucleation about an electrically charged spherical droplet. In this theory (later called the classical theory of ion-induced nucleation), the growth of a spherical droplet of radius r consisting of g molecules of the condensable vapor surrounding a metastable ion cluster of radius r, and charge 4 is regulated by the free energy change AG between the condensed phase and the pure vapor state (Volmer and Flood 1934, Tohmfor and Volmer 1938, Frenkel 1946, Russell 1969): AGi,,(r) = - zkTlnS(r3

- ri) + 4r~(r*

- rf)

1 _ql

-~)(i-/),

(3)

where E is the dielectric constant of the condensed droplet, and the other parameters are the same as in homogeneous nucleation. The first two terms on the right-hand side of Eq. (3) are equivalent to the homogeneous nucleation case, and the third term is the classical electrostatic energy term describing the interaction between the centrally located ion cluster and the surrounding molecules. When a dielectric material condenses about an ion cluster, the system free energy is reduced by an amount equal to the electric potential energy of the droplet.

MOSES, ALLEN, AND YUNG

where n, is the number density of stable ion clusters of radius r,. In a dilute system, many of the ions are found in small clusters, so n, is often approximated by nion, the total ion number density (Castleman and Tang 1972). The resulting ion-induced nucleation rate is

Jim = zP48i*)nion ew 0

10

20

30

RADIUS OF CLUSTER (9 FIG. 2. The shape of the free energy curve for ion-induced nucleation as a function of particle radius (for methane at 66 K). Note that we define AGi,n(r) relative to the free energy of a small metastable ion cluster of radius r,. The free energy change is a sensitive function of the saturation ratio S. Increasing S causes both a decrease in the critical cluster size and a decrease in the magnitude of the free energy barrier. For S = 13, the radius r, at the free energy minimum and the critical radius ri* at the free energy maximum are marked in the figure.

Figure 2 illustrates the behavior of AGion as a function of particle radius. Note that AGion exhibits both a maximum and a minimum for a certain range of saturation ratios. The thermodynamically preferred stable cluster size r, (i.e., at the minimum of AG,,“) and the critical embryo size ri* (i.e., at the maximum of AGi,,) can be determined by evaluating the roots of the derivative of the free energy equation (Eq. (3)) with respect to r. Because of the reduction in the system free energy caused by the electrostatic energy term, the free energy maximum for ion-induced nucleation is located at a radius r = ri* that is smaller than the critical radius (r = r,) for homogeneous nucleation. The difference in free energy between a cluster of size ra (containing g, molecules) and a cluster or embryo of size ri* (containing gi* molecules) acts as an energy barrier and restricts the rate of formation of large clusters-only particles with radii greater than ri* can grow spontaneously; smaller-radii clusters are unstable relative to evaporation. Figure 2 also shows the effect of variations in the saturation ratio on the free energy curve. The larger the saturation ratio, the smaller the effective barrier. Note that both the critical embryo size and the critical free energy barrier AG:,,, decrease with increasing saturation ratio S. The derivation of the ion-induced nucleation rate Ji,” parallels that of homogeneous nucleation. The population of ion clusters is described by a Boltzmann distribution defined in terms of the number of small stable ion clusters; that is,

where AG;O, is the free energy at g = gi*, a(gi*) is the surface area of the critical cluster (47rr$ fi is the same as in homogeneous nucleation, and Z is the Zeldovich factor (~/27r)“~, where

For both homogeneous and ion-induced nucleation, the Zeldovich factor ranges from -10e4 to 10-l for a wide variety of conditions and is usually 10m25 Z s 10-i. Since the exponential terms can vary by hundreds of orders of magnitude with changes in atmospheric conditions, variations in Z are not important, and the nucleation rates are controlled by the size of the energy barriers AG* and AGjr,, . For any given saturation ratio and particle radius, nucleation about charged particles is more efficient than homogeneous nucleation. In fact, as S becomes large, the energy barrier to ion-induced nucleation can disappear completely. At this point, the formation of stable clusters is not thermodynamically inhibited; however, cluster formation is still limited by kinetics considerations, i.e., by the probability of encounters between vapor molecules and the growing clusters. 2.3. Heterogeneous

Nucleation

The presence of dust and other foreign particles in an atmosphere allows supersaturated vapor to nucleate relatively efficiently. This process, called heterogeneous nucleation, can occur for ices only if the foreign particles are insoluble in the condensed species. The surface energy per given volume of ice exhibited with condensation about a preexisting particle is smaller than the surface energy that the same volume would have as a homogeneous sphere; thus, heterogeneous nucleation on insoluble particles is more efficient than homogeneous nucleation. We now examine the theory of heterogeneous nucleation for the simple case of an insoluble, partially wettable spherical substrate in some detail; this discussion is based on

HYDROCARBON

AEROSOL FORMATION

material presented by Pruppacher and Klett (1978) and Sigsbee (1969), and the reader is referred to these reviews for more thorough discussions of heterogeneous nucleation. If an insoluble substrate (foreign particle) is immersed in a supersaturated vapor, individual molecules of the condensable vapor can impinge on, adsorb to, and desorb from the substrate surface. Eventually, a steady state of adsorbing and desorbing monomers will be obtained such that the temperature and chemical potential of a molecule adsorbed on the substrate are the same as a molecule in the vapor phase. Once adsorbed monomers are present on the substrate, stable embryos of the condensed phase can form on the substrate surface by impingement of one monomer at a time, either directly from the vapor or by surface diffusion of adsorbed monomers across the substrate. Because of the lack of relevant laboratory data for condensed hydrocarbons, we will consider only the simpler direct-deposition case at this time. We assume that a nucleating embryo acquires a spherical cap shape on the substrate. The embryo free energy can be determined from the geometry of this situation (see Pruppacher and Klett 1978). If r, is the radius of the foreign substrate, r is the radius of the nucleating embryo, and 8 is the contact angle that the embryo makes with the substrate, then the free energy of formation of the embryo on this insoluble substrate can be written

g

r

323

ON NEPTUNE

1.. ’ ’ . . . . ’ . . . . ‘7-i’ I

’ ‘I. ’ . ’

homogeneous /

9’.

w 52

a 4

i*“-Y+, -

JR &

VP/

/

/

/

,

-.z.

0

-

\

/

/

/

.p

\

‘-;iy‘., s

/’

..r.:..... ,’

‘\

‘.‘\‘.

QB.’

,/ *?.’

.:

\

&$-g...~.;;b.. ....:‘.,

,I

. ‘...\

,’

I

,.’

/

/

&

&??Y 6’

\ \

.,‘\ \

,:

‘..,.\\

.’

:\ \ ‘.,‘\ \ ‘.,‘\ \ I,\ \

I....,...,,....,....,..,

0

5

10

15

20

25

30

RADIUS OF CLUSTER (ii)

FIG. 3. The shape of the free energy curve for heterogeneous nucleation on a spherical insoluble substrate as a function of contact angle and embryo radius (for methane at 66 K, S = 5, and rN = 1 nm). Also shown for comparison are the free energy curves for homogeneous and ion-induced nucleation under the same atmospheric conditions.

find that AC,,, is a maximum at r = rr = - 2u/AG,,, , the same result as for homogeneous nucleation (see Fig. 3). However, the free energy barrier for heterogeneous nucleation at r* is smaller, in general, than that for homogeneous nucleation. The critical free energy barrier to stable embryo formation is

3(1 - mx) _ (1 - r&x)3

where

where x = TN/r, m = cos 8, cf~ = (1 - 2mx + x*)l’*, AG,,, = - plm,kT In S as in the case for homogeneous nucleation, and cCcvis the surface tension of the condensate with respect to the vapor. The contact angle 8 is related to the surface tensions of the substrate/condensate/vapor system by Young’s relation,

cos 8 =

U”N

-

cr,N

ucv

.

where the subscripts v, c, and N refer respectively to the vapor, condensate, and substrate. The expression for AGhetis similar to the free energy change found in homogeneous nucleation in the sense that both a volume and surface energy term are present and will compete when s> 1. If we take the derivative of AGhet with respect to r, we

2&,X)

= 1+

( 1 mx 3

1

-

4

+x3[2-3(y)

+tyy] +)mx’&%)

with x, m, r#~as in the previous equation (Eq. (5)) for AGhet(r). The shape of AGhet versus embryo radius is illustrated in Fig. 3 for various values of 8. In this figure, we also compare the free energy barriers for ion-induced nucleation and homogeneous nucleation with those of heterogeneous nucleation. Note that heterogeneous nucleation has a much smaller barrier than that of homogeneous nucleation, but the relative magnitude of the ion-induced nucleation barrier relative to that of heterogeneous nucleation depends on the value of the contact angle. Although rN

324

MOSES, ALLEN,

was held fixed in these cases, the magnitude of the energy barrier is also quite sensitive to the insoluble particle radius r, . While the critical cluster radius for ion-induced nucleation (ri*) is not much different from that of homogeneous or heterogeneous nucleation (r,) in this case, the number of molecules required to form a critical-sized cluster is proportional to r? or r,‘* so that differences between heterogeneous and homogeneous nucleation show up more readily in plots of AG versus the number of molecules in a critical-sized cluster. The heterogeneous nucleation rate, or the rate at which stable critical-sized embryos form on a substrate surface per unit time per unit surface area multiplied by the surface area of the substrate, can be written (Pruppacher and Klett 1978) Jhet = Z4n2rf&k,

exp

AGtt [ 1, - k~

where ci is the surface concentration of adsorbed monomers (molecules cm-‘), and the Zeldovich factor is now

The Zeldovich factor can be approximated by [AG&(3~kTg~)l”2 (cf. Sigsbee 1969, Moses 1991). The adsorbed monomer concentration c1 is generally not known but can be estimated by equating the flux of monomers to the surface (p) with the desorption flux from the surface; thus,

C, =

:($J’exp

[+I

where AGdes is the desorption energy of a monomer from the substrate and Y, is the vibration or jump frequency of an adsorbed molecule normal to the substrate surface (-kT/h). The determination of the energy term AGdes is experimentally difficult. Seki and Hasegawa (1983) report that the desorption energy of water from silicate surfaces is approximately 0.18 eV. Since we have no information concerning AGdesfor the hydrocarbon species we are considering, we adopt this value. As with the other nucleation mechanisms, the heterogeneous nucleation rate is dominated by the exponential term. Under the same background conditions, the critical nucleation barrier is always smaller for heterogeneous nucleation than for homogeneous nucleation. The relative effectiveness of heterogeneous versus ion-induced nucleation depends on the size of the insoluble particle and on the “wettability” of its surface. For instance, large rN’s

AND YUNG

and small 13’spromote nucleation while small rN’s and large 8’s inhibit it. Although 8 has little meaning when we are dealing with solid crystals forming on a solid substrate, m = cos 8 still is useful in defining a compatibility parameter when discussing the effectiveness of heterogeneous ice nucleation (Pruppacher and Klett 1978). More detailed descriptions of the variation of the nucleation rate with saturation ratio, substrate size, and contact angle will be presented in the next section when we consider nucleation under conditions directly relevant to Neptune’s atmosphere. At that time, we will also discuss in more detail the relative efficiencies of homogeneous, ion-induced, and heterogeneous nucleation and will give specific examples of hydrocarbon nucleation rates on Neptune. 2.4. Limitations of Classical Theory A major assumption in the derivation of the classical theory of nucleation (heterogeneous and ion-induced as well as homogeneous) is that a macroscopic description can be used in formulating the free energy change. This assumption is most certainly invalid for small clusters of say less than tens of molecules. For the case of small clusters, macroscopic parameters such as surface tension and droplet density have very little meaning. Intricacies of the structure of a small droplet are ignored as are the rotational and translational energies of the droplet. For ion-induced nucleation, classical theory also ignores any perturbation effects that the ion may produce in the droplet itself. Classical heterogeneous nucleation theory is fraught with even more difficulties. For instance, the concept of a contact angle is a macroscopic one and oversimplifies the situation for small embryos. In addition, the surface of a potential condensation nucleus is not homogeneous; molecules adsorb preferentially at certain active sites on the substrate, and embryos are more likely to form at these locations (Pruppacher and Klett 1978). Modifications to the classical theory have been presented by various researchers (see review by Pruppacher and Klett 1978). Most of these modifications involve corrections based on statistical mechanics, and these theories predict nucleation rates that are several orders of magnitude different from classical theory. For instance, Lothe and Pound (1969) predict that classical theory is off by a factor of -10 i7. Surprising 1y , however, classical theory agrees better with laboratory results than most of these modified theories (Pruppacher and Klett 1978). The Lothe-Pound theory, in particular, is difficult to reconcile with experiments dealing with the nucleation of water and many other compounds. Because of uncertainties inherent in nucleation theory, laboratory studies are helpful in establishing nucleation

HYDROCARBON

rates. Unfortunately, we have been unable to find any published results concerning nucleation of any of the hydrocarbon species we are considering; the closest studies we could find concerned nucleation of some high-molecular-weight alkanes such as hexane, heptane, octane, and nonane (Katz and Virkler 1976). Katz and Virkler find that their laboratory results for these alkanes are in good agreement with classical theory and in poor agreement with Lothe-Pound theory. Because of the lack of data on hydrocarbon nucleation and because the classical theories have had reasonable success in matching H,O and alkane experimental data, we have decided to use the classical models in this paper. Large errors in the calculated nucleation rates will lead to only slight errors in the estimated saturation ratios at which homogeneous, heterogeneous, or ion-induced nucleation will begin to become significant because the nucleation rates vary exponentially with S. Thus, any errors in our calculated nucleation rates resulting from our use of the classical theories will slightly raise or lower our estimates of the altitude levels at which we expect the hydrocarbon species to nucleate but will not affect the qualitative conclusions of this paper. However, we would like to emphasize that the lack of experimental data on hydrocarbons at low temperatures combined with our use of classical theory makes our quantitative results uncertain. 3. ABUNDANCES

OF CONDENSABLE

325

AEROSOL FORMATION ON NEPTUNE

HYDROCARBONS

Our first step in establishing particle-formation rates on Neptune involves the determination of the gas-phase abundances of the condensable hydrocarbon species in Neptune’s troposphere and stratosphere. The photochemical model used to estimate these abundances is described in detail in Moses (1991). The background atmosphere is constructed assuming hydrostatic equilibrium in an atmosphere composed primarily of H, with volume mixing ratios of 19% He and 2% CH, in the troposphere below the base of the methane condensation region, and 0.02% CH, above the methane condensation region. The hydrocarbon chemistry is taken from Gladstone et al. (1991) and describes the sequence through which methane in the upper atmosphere is converted to 35 other different hydrocarbon species. The thermal structure employed in this model (see Fig. 4) is slightly different from that of Moses (1991). From 5 bar to 2 mbar, we adopt a temperature profile consistent with the Voyager 2 ingress radio occultation (RSS) experiment (Linda1 et al. 1990); however, we shift the profile to slightly higher temperatures to remain consistent with our assumption of a higher helium abundance (19% according to Conrath et al. (1991)). At pressure levels below 10F2 mbar (high altitudes), we use the profile deduced from

0 : 8 0 0

100

200

300

400

500

TEMPERATURE (K)

FIG. 4. The temperature cleation models.

profile used in our photochemical

and nu-

the ingress solar occultation experiment of the Voyager ultraviolet spectrometer (UVS) (Broadfoot et al. 1989). Since no information currently exists concerning the thermal structure in the 1 to lo-’ mbar region on Neptune, we adopt a smoothly varying profile that provides a hydrostatic model consistent with the H, densities reported by Bishop et al. (1991). The eddy diffusion coefficient in our model is derived in a manner similar to that of Moses (1991). We use mixinglength theory to determine the diffusion coefficient (K) in the troposphere and let the stratospheric diffusion coefficient vary with the total atmospheric number density (n) by

where p = 0.66, n, = 2.45 x 10” cme3, and K, = 1.1 x lo8 cm2 sect’. The diffusion coefficient is 7.8 x lo2 cm2 set-’ at the tropopause (110 mbar) and is lo8 cm2 set-’ at the CH, homopause (10-j mbar). Moses (1991) and Bezard et al. (1991) demonstrate that the photochemical model results are very sensitive to the stratospheric eddy diffusion coefficient. The above choice of a diffusion profile provides a photochemical model that fits constraints imposed by Voyager and Earth-based observations of C,H, and CH, in Neptune’s upper atmosphere (see Moses 1991). We have attempted to reproduce the conditions relevant to the Voyager 2 encounter with Neptune as closely as possible. In particular, we use solar flux values representative of those observed near sunspot maximum and examine the chemistry at latitudes corresponding to the

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i

FIG. 5. The volume mixing ratios of the major condensable species in our photochemical model (at 61” N latitude). The small horizontal lines indicate the pressure levels at which the species become saturated.

ultraviolet spectrometer and radio science occultations (e.g., 61” N latitude). Details of the solar spectrum used are presented in Moses (1991). The contribution from solar radiation scattered from H atoms in the local interstellar medium is included and is found to be very important to the photochemical model results. Steady-state solutions to the diurnally averaged photochemical model discussed above are presented in Fig. 5 for the most abundant condensable hydrocarbon species. A comparison of our photochemical model with the vapor pressures of the hydrocarbon species listed in Appendix A suggests that at least seven hydrocarbon species become supersaturated in Neptune’s atmosphere and have the capability to condense. The saturation levels of each of the species are indicated by a small horizontal bar on the mixing ratio profiles of Fig. 5. Methane has the ability to condense in the upper troposphere while acethylene (C,H,), ethane (C,H,), methylacetylene (CH,C,H), propane (C,H,), diacetylene (C,H,), and butane (C,H,,J have the ability to condense in the lower stratosphere. Although etylene (C,H,) is abundant, it never exceeds its saturation vapor abundance and is not included in the figure. Propylene (C,H,), allene (CH,CCH2), and ethylacetylene (I-C,H,) become supersaturated but will probably not form hazes in appreciable numbers because of their low abundances and unfavorable physical properties; consequently, these species are also not included in the figure. The photochemical model does not allow for nucleation and condensation; therefore, the mixing ratio profiles in Fig. 5 do not reflect the partitioning between gas and condensed phases and overestimate the gas phase abundances below the -10 mbar pressure level. Although the eddy diffusion profile and stratospheric

methane abundance in the photochemical model are chosen so that the resulting methane and acetylene concentrations compare well with ultraviolet and infrared observations (Broadfoot et al. 1989, Bishop et al. 1991, Bezard et al. 1991), the resulting ethane model abundance seems smaller by a factor of 4 to 15 than groundbased and spacecraft observations (Kostiuk et al. 1990, BCzard et al. 1991, Orton et al. 1992). Other researchers have noted similar difficulties in fitting both the ethane and acetylene observations (e.g., Bezard et al. 1991). Models based on our current understanding of hydrocarbon photochemistry seem to be unable to reproduce the large C,H, to C2H, ratio observed on Neptune and the outer planets. Thus, the ethane abundance in our model may be as much as an order of magnitude lower than the actual ethane abundance on Neptune. If this is true, then ethane will actually become saturated at -10 mbar, about 6 km above the saturation level in our current model (Fig. 5). Since the nucleation rates are sensitive to the vapor abundance, we will later examine the effect of increasing the ethane abundance by a factor of 10 from that illustrated in Fig. 5. A comparison of the downward fluxes of the different hydrocarbon compounds in the stratosphere can help determine the relative importance of the different species in resupplying carbon atoms to the troposphere as well as help determine the relative contribution of the different condensable species to the stratospheric haze layers. At the saturation levels for each species, the downward flux of C,H, is 3.7 x 10’ cmm2 see-’ (or perhaps even higher if our photochemical model underestimates the ethane abundance), of C,H, is 5.4 x 106, of C,H, is 5.2 x lo’, of C4H, is 1.5 x lo’, of CH,C,H is 1.1 x 105, and of C,H,, is 1.5 x lo3 cm-* sec- ‘. For all the species except CH,C,H, the fluxes are approximately constant in the lower stratosphere, implying that photochemistry is not active in the condensation region. Ethane clearly dominates the condensation of carbon-bearing species in the stratosphere. 4. CONCENTRATIONS OTHER

OF FOREIGN

NUCLEATION

PARTICLES

AND

CENTERS

The predicted low concentrations of the hydrocarbons in the stratosphere and the cold temperatures found in Neptune’s lower stratosphere and upper troposphere severely limit the effectiveness of homogeneous nucleation of any of the hydrocarbon species considered (see Section 5 below). Consequently, ion-induced or heterogeneous nucleation (or both) must be responsible for particle formation in these regions of Neptune’s atmosphere. Our second step in deriving particle formation rates on Neptune thus involves a determination of whether a sufficient source of ions or foreign nuclei with the appropriate physi-

HYDROCARBON

cal properties nucleation.

exists in Neptune’s atmosphere

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AEROSOL FORMATION ON NEPTUNE

to facilitate

4.1. Source of Ionization High energy galactic cosmic rays (GCR’s) provide a source of ionization in Neptune’s lower stratosphere. On Earth, galactic and solar cosmic rays are responsible for ion pair production at low levels in the terrestrial ionosphere (e.g., the D layer). The possible importance of cosmic ray ionization in inducing nucleation and forming aerosols in the terrestrial atmosphere has been discussed in the literature (e.g., Dickinson 1975, Arnold 1980, 1982, Turco et al. 1982, Hofmann and Rosen 1983). To determine the importance of ion-induced nucleation as a source of particle formation in Neptune’s atmosphere, we need to determine whether GCR’s can produce enough ions to make ion-induced nucleation a viable option. Moses et al. (1989) first considered this option for Neptune, and our deviation of GCR-induced ionization is similar to theirs. Complete, detailed models of cosmic ray ionization in the atmospheres of Saturn, Uranus, and Neptune presented by Capone et al. (1977) show that ionization by GCR’s in the stratospheres of the outer planets is considerable. Unfortunately, the models of Capone and his colleagues were based on pre-Voyager views of the composition and temperature profiles of the outer planets and are no longer sufficiently accurate for use in examining the importance of ion-induced nucleation in the hydrocarbon condensation regions on Neptune. Moses et al. (1989) also based their model on pre-Voyager information. We have therefore constructed our own model ionosphere of Neptune based on some of the same physical principles as the ionosphere models of Capone et al. (1976, 1977, 1979) but have included in our model updated estimates of the temperature structure and composition of Neptune. A complete description of our GCR ionization model for Neptune is presented in Appendix B. Figure 6 illustrates the results of that model computed using the onedimensional kinetics and diffusion algorithm described in Allen et al. (1981). In Fig. 6, the GCR-produced total ion density profiles are shown as a function of altitude in Neptune’s atmosphere. Our ion density profile reproduces the essential features of Capone et al. (1977); however, our updated model atmosphere and GCR flux results in a peak electron density that is a factor of 6 less than that reported by Capone et al. The results shown in Fig. 6 also illustrate the differences between the ion profile at sunspot maximum versus sunspot minimum. The production rate at 60” magnetic latitude at any altitude is -40% higher at solar minimum than at solar maximum. A substantial source of ions from galactic cosmic rays exists in the lower stratosphere of Neptune; total ion number densities are on the order of a couple thousand per

TOTAL ION NUMBER

DENSIM

(VII-~)

FIG. 6. The total number density of ions produced from ionization by galactic cosmic rays at stratospheric levels on Neptune. The two different curves are for sunspot minimum and sunspot maximum.

cubic centimeter in the lower stratosphere. Ion-induced nucleation might therefore be possible in the hydrocarbon condensation regions if other conditions are met. As an interesting sidelight, since more ions are produced at solar minimum than at solar maximum and more at high magnetic latitudes than at low magnetic latitudes, haze production might vary with the 11-year solar cycle and with location (magnetic latitude) on the planet if ion-induced nucleation is an important mechanism for particle formation on Neptune (see Moses et al. 1989). 4.2. Source of Condensation

Nuclei

For heterogeneous nucleation to operate on Neptune, a source of condensation nuclei, or particles that act as sites for growth of the condensed phase, must be present. On Earth, such particles are abundant-soil and clay particles, organic debris, sea salt, soot, resin, volcanic aerosols, and pollutants all act as condensation nuclei (CN). Most terrestrial CN are derived from sources at the Earth’s surface as is evident from the decrease in CN with altitude above the surface (Pruppacher and Klett 1978). On Neptune, no “surface” as such exists, and a large source of CN from the lower atmosphere is unlikely to be important. Some mixing of tropospheric cloud particles or nonequilibrium species of low volatility from deeper tropospheric regions to the hydrocarbon condensation regions may occur; however, the major source of CN on Neptune will probably be from high altitudes, either from chemical products raining down or from extraplanetary sources. A large percentage of aerosol material in the stratosphere and thermosphere of Neptune could originate from the passage of meteors through the planet’s atmosphere.

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Most interplanetary particles will ablate during their passage through the atmosphere; metals and refractory vapor lost from the meteoroids can then recondense in the trail of the meteor to form small smoke-like dust particles (see Hunten et al. (1980) for a terrestrial analogy). In addition, some micrometeoroid particles are small enough to survive the ablation process either relatively untouched or as residual meteoroids that do not fully evaporate. Thus, an extraplanetary source of condensation nuclei might be present in the hydrocarbon condensation regions in the lower stratosphere of Neptune. Meteoroid ablation and recondensation have been studied extensively in the context of the Earth’s atmosphere (e.g., Rosinski and Snow 1961, Hunten et al. 1980), but direct comparisons of terrestrial meteoroid studies with the situation on Neptune are problematical. For instance, the characteristics of the meteoroid population in the outer Solar System are unknown. Based on certain assumptions concerning the mass and velocity distribution of meteoroids at 30 AU, Moses (1992) presents meteoric dust ablation and recondensation calculations for Neptune. Although the dust concentration profiles presented by Moses (1992) are uncertain by an order of magnitude or so, the particle number density estimates are sufficient for our nucleation calculations. Uncertainties in the dust concentration translate directly to uncertainties in the nucleation rate; however, the exponential growth of the nucleation rate with small changes in the saturation ratio enable us to predict the levels at which particle formation is likely to occur with good accuracy despite uncertainties in particle concentration. Moses (1992) assumes that the refractory vapor molecules that have ablated from silicate-rich meteoroids in Neptune’s atmosphere recondense to form very small silicate particles. Since the details of this recondensation process have not been worked out, three different initial particle sizes are examined. To simplify the calculations, Moses assumes that the particle radius remains constant throughout the lifetime of these particles. The particles then evolve by sedimentation, coagulation, and eddy diffusion. The steady-state solution to the particle concentration from the Moses (1992) model is reproduced in Fig. 7. The particle concentrations are sharply peaked near the maximum in the ablation profile; however, diffusion and sedimentation act to further distribute the particles with altitude. Since gravitational settling becomes increasingly important for larger particles, the maximum particle concentration of the larger particles is located at lower altitudes. Figure 7 illustrates that the dust concentrations for particles of all three assumed sizes are similar in the lower stratosphere. A comparison of Figs. 6 and 7 shows that, at 10 mbar, the concentration of meteoric dust nuclei is approximately two orders of magnitude smaller than the concentration of ions produced from galactic cosmic rays.

?

H

$t

10

100

1000

IO

PARTICLE NUMBER DENSllY (cm-‘)

FIG. 7. The concentration of dust particles produced form the recondensation of ablated meteoric vapor. Three curves are shown for different assumed particle radii of 1, 3, and 10 nm. The dust production rate profile used to generate these curves is taken from an intermediate case between the Oort-cloud and Neptune-family dust population ablation curves presented in Moses (1992).

However, the properties of the dust particles may still conspire to make heterogeneous nucleation more efficient than ion-induced nucleation. The recondensed refractory material (e.g., silicates or metals) from the ablated meteoroids should have good structural integrity, should be insoluble in the condensed hydrocarbon species, and should have surface sites capable of adsorbing hydrocarbons; hence, meteoric dust should have at least some of the properties required for good ice-forming nuclei (see Pruppacher and Klett 1978). However, the predicted small sizes of the recondensed material (a few nanometers, according to Hunten et al. 1980) might hamper heterogeneous nucleation. The ability of the dust particles to act as good condensation nuclei also depends on the compatibility of the substrate with the condensate. For example, the dust particles should have high surface energies with respect to the hydrocarbon vapor and low surface energies with respect to the hydrocarbon condensate in order to have a low contact angle and a high degree of “wettability.” Contact angles for liquid water on most metals, silicates, and natural terrestrial condensation nuclei are usually greater than 45”. We could not find data for the contact angle of condensed hydrocarbons on different substrates, so we examine a wide range of contact angles in our nucleation calculations. Another possible source of condensation nuclei on Neptune is from photochemical products that condense in the upper stratosphere and fall to the lower altitude haze and cloud condensation regions. For instance, C6H2, &HZ,

HYDROCARBON

AEROSOL

other polyacetylenes, or complex hydrocarbons with low vapor pressures might be abundant enough to condense and form small amounts of condensation nuclei at high altitudes. These particles will settle through the atmosphere and eventually encounter regions that are supersaturated with respect to some of the vapor species we are considering (e.g., diacetylene and butane). Each of the less volatile hydrocarbon species that condenses at a relatively high altitude can act as a source of CN for the more volatile species below it. For instance, diacetylene can act as a source of CN for acetylene, which can be a source of CN for ethane. All these species can be sources of CN for methane in the upper troposphere provided the particles do not evaporate before they reach the methane condensation region. Products from interactions of magnetospheric and other high-energy charged particles with atmospheric constituents may be a third source of CN on Neptune. This source is similar to the photochemical source in the sense that the material is probably hydrocarbon in nature and is formed at higher altitudes in the atmosphere. Laboratory experiments of plasma discharges in simulated Uranian and Neptunian atmospheres reveal that some condensed material is produced from the irradiation of mixtures of methane, helium, and hydrogen vapor (e.g., Khare et al. 1987). Tholin, soot, or other hydrocarbon condensates produced by charged-particle impact might be present in quantities significant enough to provide a source of CN on Neptune. In addition, charged-particle impact might be a source of Cs and higher hydrocarbon vapor that might condense in Neptune’s middle stratosphere. In summary, several possible sources of condensation nuclei exist for heterogeneous nucleation in Neptune’s atmosphere. Although we have focused primarily on the production of dust particles from the ablation and recondensation of meteoroids, other sources may be present. In particular, the actual process of nucleation and condensation of the relatively involatile species at high altitudes on Neptune might provide a source of CN in the lower regions of the stratosphere. Our use of the meteoric dust profile in our nucleation rate calculations may cause us to underestimate the concentration and perhaps even the range of sizes of the possible insoluble nuclei available for heterogeneous nucleation. 5.

AEROSOL

FORMATION

IN NEPTUNE’S

ATMOSPHERE

The nucleation equations developed in Section 2 are now applied to a study of hydrocarbon aerosol formation in Neptune’s atmosphere. The surface tensions, dielectric constants, bulk densities, and vapor pressures of the important hydrocarbons at low temperatures are discussed in Appendix A. No data are available for the physical properties of diacetylene (except the vapor pressure). We

FORMATION

ON NEPTUNE

329

assume that C4Hz has the same density, surface tension, and dielectric constant as C,H, and discuss the sensitivity of the diacetylene nucleation rates to the adopted physical properties. Since many of the hydrocarbon physical properties are poorly known at low temperatures, our quoted nucleation rates are uncertain. Surface energy is the single most sensitive, but poorly known, quantity. 5.1. Ejjiciency of Hydrocarbon Low Temperatures

Nucleation

at

Although none of the nucleation mechanisms discussed in Section 2 are particularly efficient at forming hydrocarbon aerosol particles at low temperatures, each mechanism has the potential to operate somewhere in Neptune’s atmosphere. In this section, we examine the efficiencies of the different nucleation processes. Although we are primarily interested in the situation in Neptune’s atmosphere, most of the results presented here are general. They depend only on the temperature, saturation ratio, and physical properties of the hydrocarbons; thus, they can be applied to similar conditions found in laboratory experiments or in other planetary atmospheres. The ioninduced and heterogeneous nucleation rates are calculated first in terms of a frequency (e.g., a rate per ion or per condensation nucleus), and then compared directly to Neptune by multiplying the given frequencies by the concentration of ions or condensation nuclei found in Sections 4.1 and 4.2. When we apply the nucleHomogeneous nucleation. ation rate equations from Section 2 to the nucleation of hydrocarbons at low temperatures (using the physical data from Appendix A), we find that homogeneous nucleation is very inefficient. Enormous saturation ratios are required to obtain observable homogeneous nucleation rates at the low temperatures we are examining. Since ethane is the most abundant condensable molecule in Neptune’s lower stratosphere and since condensed ethane will probably be the dominant stratospheric aerosol mass component, we examine the details of homogeneous nucleation of ethane at low temperatures (Fig. 8). Note the enormous vertical scale of Fig. 8. In this figure, the homogeneous nucleation rate versus the saturation ratio of ethane is plotted at several temperatures below the ethane saturation level on Neptune (13 mbar, 65 K with our current photochemical model). The exponential term in the expression for the homogeneous nucleation rate (Eq. (2)) dominates the overall shape of the curves in Fig. 8. Although the preexponential term is proportional to S2 and contributes somewhat to the behavior of J, the nucleation barrier AG* and the exponential term exp( - AG*IkT) will dominate the expression for the nucleation rate provided that the saturation ratio is not so large that the nucleation barrier is

330

MOSES, ALLEN, AND YUNG

recombine with a species of opposite charge (e.g., an electron) is independent of the size of the cluster and can be estimated as (Bauer 1973) r r =-

1 anion'

8 ‘I

where (Yis the recombination rate of the hydrocarbon ion clusters described in Appendix B (CZ- 4 x lO-‘j cm3 set-I) and nion is the electron number density (assumed to be equal to the ion number density). The time required for a critical number gi* of condensable hydrocarbon molecules to impinge on the cluster is approximately 0

2000

4000

6000

ETHANE SATURATION RATIO

FIG. 8. The homogeneous nucleation rate of ethane (new particles formed per cubic centimeter per second) as a function of saturation ratio and temperature. Note that the ordinate is scaled logarithmically. Homogeneous nucleation is minimal unless large supersaturations develop.

negligible (i.e., so large that AG* + 0). For any particular saturation ratio, the nucleation rate decreases rapidly with decreasing temperature due to the inverse cube dependence of AG*IkT with temperature. In our photochemical model discussed in Section 3, ethane becomes saturated at 13 mbar (-65 K). However, homogeneous nucleation of ethane will be minimal unless very large saturation ratios build up. At 21 mbar (60 K), the ethane saturation ratio can reach -30 if no condensation has occurred; however, the homogeneous nucleation rate is still negligible (1O-25ocmm3 set-‘). Even at 77 mbar (53 K) where the ethane saturation ratio can reach 5500, J is only 10ess cmP3 set-‘. In other words, not a single condensed ethane particle can form by homogeneous nucleation in the entire Neptune stratosphere over the age of the Solar System. Homogeneous nucleation is extremely inefficient for all the hydrocarbon species except methane. Ion-induced nucleation. The previous discussion of ion-induced nucleation in Section 2.2 does not completely describe the situation that would be encountered on Neptune. The classical description of ion-induced nucleation inherently assumes that the ions have infinite lifetimes. In real atmospheres, the ions may recombine with free electrons and be neutralized before the ion cluster has had a chance to reach a thermodynamically stable size. To account for this possibility, we add a correction factor to the nucleation rate; this correction factor describes the probability that a critical-sized cluster can be formed before recombination occurs. We assume that the time constant for an ion cluster to

Timp

=-

gi*

4n-rf*p’

where 0 is again the flux of condensable molecules encountering a surface area 4~rl. We can estimate the probability that an ion survives long enough to acquire a critical number of hydrocarbon molecules in a manner similar to that of Hamill et al. (1982),

where S is the saturation ratio, n, is the saturation vapor density, p is the bulk density of the condensed phase, and ml is the mass of a single molecule of the condensed phase. We then get a rough determination of the ioninduced nucleation rate by multiplying the classical nucleation rate (Eq. (4)) by this probability factor. The total ion number density in Neptune’s lower stratosphere never exceeds a few thousand per cubic centimeter (see Fig. 6). Thus, electron recombination times are fairly long, typically > 100 sec. For most supersaturated conditions in the stratosphere and upper troposphere, 7imp4 T,, and the correction factor is near unity. Nevertheless, conditions do exist where the probability for critical cluster formation is negligibly small. These conditions develop when nucleation is limited kinetically; that is, when the encounter probability between condensable monomers and a growing cluster is quite small. Low temperatures and low vapor abundances trigger these kinetically limited situations. At the temperatures that we are considering, methane cluster formation never falls in the category of nucleation that is kinetically limited. Under tropospheric conditions on Neptune, the clusters grow rapidly compared to the lifetime of an ion. However, nucleation of a relatively heavy, involatile, and nonabundant species such as diacetylene can be more affected by the correction factor. At the region in which C4H2just becomes supersaturated,

HYDROCARBON

AEROSOL FORMATION ON NEPTUNE

I

0

1000

2000 mANE

3000

4000

5000

J

6000

SATURATION RATIO

FIG. 9. The ion-induced nucleation rate of ethane (particles formed per ion per second) as a function of saturation ratio and temperature. The scale is the same as in Fig. 7.

the diacetylene abundance is low, causing the correction factor to fall below 10-i’ and impeding ion-induced nucleation. However, the increasing concentration of C,H, with decreasing altitude allows the correction factor to build up to ~10-~ in the lower stratosphere, thus making ion-induced nucleation more feasible. Because of the importance of ethane as a potentially major aerosol source on Neptune, we examine the efficiency of ion-induced nucleation of ethane in some detail (Fig. 9). Note that Fig. 9 has the same scale as Fig. 8. In Fig. 9, we plot the nucleation rate per ion, and one must multiply the plotted values by the ion number density (-2000 cmw3 in the ethane condensation region) to truly compare these nucleation rates with those in Fig. 8. However, even without the factor of -2000, ion-induced nucleation is much more effective than homogeneous nucleation for the same conditions. Figures 8 and 9 emphasize that relatively small increases in the saturation ratio can lead to huge increases in the nucleation rate. This behavior of nucleation rates allows us to define a parameter called the critical saturation ratio (e.g., Twomey 1977, Pruppacher and Klett 1978). We use this term to describe the saturation ratio at which significant or observable nucleation rates are obtained. The concept of a critical saturation ratio is a useful one for comparing nucleation theory with laboratory experiments or with real atmospheres. Small clusters of molecules will form throughout an atmosphere at finite, but usually negligible, rates. The critical saturation ratio (S,,,) provides a good indication of where particle formation will actually become important. If the abundance of a condensable vapor species exceeds Scrit, then particle formation is likely to occur at a significant rate. A large

331

Scrit implies an inefficient nucleation process while a small Scrit implies an efficient one. The choice of an “observable nucleation rate” is somewhat arbitrary. According to Keesee (1989), nucleation rates of at least 1O-2 to 10e3 particles cmm3 set-’ are required to produce visible clouds in the terrestrial atmosphere; thus, we define the critical saturation ratio (S,,,) to be where the nucleation rate reaches 10m2cme3 set-i. As long as the nucleation rate remains steep in plots of J versus S, our choice of a particular nucleation rate for the determination of Scrit is not crucial. Changes of several orders of magnitude in our choice of J will usually still correspond to saturation ratios in the neighborhood of &it * If we assume that lZion= 2000, we see from Fig. 9 that Scrit is less than 1000 for 60 K and less than lo4 for 50 K. When we correctly include the ion-density results of Section 4.1 and also consider the actual saturation ratios of ethane in Neptune’s atmosphere, we find that observable ion-induced nucleation rates for ethane are achieved at altitudes below 42 mbar (55 K) in Neptune’s stratosphere. Ethane and several of the other hydrocarbon species have the ability to nucleate efficiently by ion-induced nucleation in Neptune’s atmosphere. We will discuss this point in further detail in Section 5.2. Heterogeneous nucleation. The efficiency of heterogeneous nucleation depends strongly on the properties of the insoluble particle upon which the vapor condenses. In particular, the heterogeneous nucleation rate depends on the size of the particle and its “compatibility” with the condensed phase (which is described by the contact angle between the condensed phase and the substrate). As already discussed, the recondensation of ablated meteor vapor might provide particles that range from 1 to 10 nm in radius (with perhaps some smaller and some larger particles). Contact angles for condensed hydrocarbons on appropriate substances are not known, so we examine the sensitivity of the nucleation rate to contact angle. Calculations of the heterogeneous nucleation frequency (i.e., the nucleation rate per particle) versus saturation ratio for three different hydrocarbon species are shown in Figs. IOa-10c. The calculations are performed for different temperatures-a convention necessitated by the different vapor pressures and abundances of the different species. Each temperature is chosen to be near the temperature at which we expect particle formation to become significant on Neptune (i.e., where the actual vapor saturation ratio exceeds Scrit). The insoluble particle radius rN is fixed at 10 nm in all three cases, and the contact angle 0 is allowed to vary. The ion-induced nucleation frequency (nucleation rate per ion) is shown in each figure for comparison. Figures lOa-10c help illustrate differences between nu-

332

MOSES, ALLEN, AND YUNG

-

0

2

4 METHANE

6

6

SATURATION

10

RATIO

0

ETH4NE

200 DIACETYLENE

400 SATURATION

SATURATION

RATIO

600

mno

FIG. 10. Heterogeneous nucleation of vapor on IO-nm particles as a function of saturation ratio and contact angle: (a) shows nucleation of methane at 75 K, (b) shows nucleation of ethane at 60 K, and (c) shows nucleation of diacetylene at 104 K. The nucleation rate is plotted in terms of the number of stable embryos condensing on the insoluble particle per particle per second. The ion-induced nucleation frequency (number of stable embryos formed per ion per second) is shown for comparison.

cleation of three different classes of hydrocarbons. Figure 10a is illustrative of a lightweight, volatile species in an atmosphere; methane is the only hydrocarbon species that fits into this category on Neptune. Figure 10a shows that methane particle formation proceeds relatively efficiently at low saturations (i.e., Scrit < 10) at 75 K for both ioninduced and heterogeneous nucleation. Although the nucleation rate is very sensitive to the assumed contact angle, all reasonable assumed values for 13give relatively efficient nucleation rates at saturation ratios less than -10. At -1.1 bar in Neptune’s troposphere, T = 75 K, and the methane saturation ratio is 2.7. Thus, if the contact angle for condensed methane on an insoluble particle is less than 45”, heterogeneous nucleation will take place readily on any particles present at this location. The extreme sensitivity of J to S allows us to pinpoint with reasonable accuracy (within a couple of kilometers) the

methane particle formation region on Neptune even though we have little knowledge of 8 and r,. Methane nucleation becomes significant for all choices of r, and 8 while methane is in a small range of saturation ratios. Figure lob is illustrative of nucleation of a second class of hydrocarbons (e.g., ethane and acetylene) that nucleate with a more moderate efficiency. The ethane nucleation rate at 60 K is shown in Fig. lob. Depending on the properties of the insoluble particle, large saturation ratios might be required to initiate observable particle formation at this temperature. The calculated nucleation rates are very sensitive to the assumed contact angle: small contact angles (