Nucleation: measurements, theory, and atmospheric applications.

Annu. Rev. Phys. Chern. 1995. 46: 489-524 Copyright © 1995 by Annual Reviews Inc. All rights reserved

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NUCLEATION: Measurements,

Annu. Rev. Phys. Chem. 1995.46:489-524. Downloaded from www.annualreviews.org by Stanford University - Main Campus - Lane Medical Library on 05/13/13. For personal use only.

Theory, and Atmospheric Applications Ari Laaksonen, Vicente Talanquer, and David W. Oxtoby

James Franck Institute, University of Chicago, 5640 S. Ellis Avenue, Chicago, Illinois 60637 KEY WORDS:

condensation, aerosols, crystallization, phase transitions, water,

polar stratospheric clouds

ABSTRACT

New experiments have succeeded in measuring actual rates of nucleation and are revealing the shortcomings of classical nucleation theory, which assumes that the molecular-scale regions of the new phase may be treated using bulk thermodynamics and planar surface free energies. In response to these developments, new theories have been developed that incorporate information about molecular interactions in a more realistic fashion. This article reviews recent experimental and theoretical advances in the study of nucleation of liquids from the vapor and of crystals from the melt, with particular emphasis on phenomena that relate to particle formation in the atmosphere.

INTRODUCTION The study of nucleation has recently received a burst of attention, arising both from the development of new experimental and theoretical techniques for determining nucleation rates and from the recognition that nucleation events play a central role in many atmospheric processes such as particle formation in polar stratospheric clouds. This article reviews these recent developments, placing them in the context of earlier approaches to nucleation. Any first-order transition requires the surmounting of a barrier via a fluctuation. The driving force for the transition increases as the metastable region is penetrated, giving rise to an activation free energy that depends 489 0066-426X/95/1101-0489$05.00


490

LAAKSONEN, TALANQUER & OXTOBY

strongly on experimental conditions of temperature and pressure, Because this free energy is in turn exponentiated to obtain the actual nucleation rate, that rate is extraordinarily sensitive to many factors, including thermodynamic conditions, intermolecular potentials, and small con centrations of impurities, This sensitivity provides both the challenge and the impetus for the study of nucleation, The focus of this review differs from that of two other recent reviews (1, 2), both of which emphasized theoretical developments, especially the density functional approach. Here an attempt is made at balanced coverage of theory and experiment, with the selection of topics weighted toward those aspects that connect to atmospheric applications of nucleation science and toward those areas with the most potential for future development.

EXPERIMENTS Traditional experiments of nucleation from the vapor have concentrated on measuring the critical supersaturation Scr needed to produce a certain nucleation rate Jcr' For example, in cloud chamber experiments, researchers have typically observed the onset of nucleation with lcr >:::: 1 cm-3 S-I. During the last decade it has become possible to measure actual nucleation rates and the molecular content of nucleating clusters. We focus on these studies below; Reference 3 gives references to the earlier measurements and Reference 3a provides a comprehensive list of experi ments between 1968 and 1992. Experimental Techniques

The most common traditional nucleation instru ment is the upward thermal diffusion chamber, in which a liquid pool rests on a warm bottom plate, and vapor diffuses upward through inert carrier gas. The upper surface of the cylindrical chamber is cooler than the bottom plate. Both temperature and partial pressure of the vapor decrease nearly linearly from the liquid surface to the top plate. The saturation vapor pressure of the fluid depends exponentially on temperature, however, so the saturation ratio goes through a maximum at about 3/4 of the chamber height. The temperature gradient is adjusted so that homogeneous nucleation takes place at the maximum. The nucleated droplets grow and fall through the chamber toward the liquid pool. A laser beam near the bottom of the chamber detects falling droplets, allowing their rate of formation to be calculated. Nucleation rates that can be measured using the diffusion chamber range from 10-4 to 103 cm-3 S-I. Measurements have been performed for ethanol, n-propanol and i-propanol (4), n-nonane (5), and cesium (6). DIFFUSION CHAMBERS


NUCLEATION

491

The supersaturation and temperature profiles in the thermal diffusion chamber depend on thermodynamic and transport properties of the gas. This is the case also with the laminar flow diffusion chamber, whose operation principle is similar to that of commercially available con densation nucleus counters (CNCs). A carrier gas flow is passed through saturated vapor and led to a cooling zone where the vapor becomes supersaturated, and nucleates. (In a CNC the supersaturation is kept lower so that the vapor can only condense on aerosol particles brought in with the carrier gas flow.) The nucleated droplets then grow until they can be detected optically. Nucleation rates between 102 and 108 cm- 3 S-I have been measured for dibutylphthalate (7, 8). EXPANSION CHAMBERS A different method used in nucleation rate measurements is based on vapor cooling caused by rapid expansion. Short nucleation pulses are produced in this way, in contrast to the continuous operation of the diffusion chambers. The fast-expansion cloud chamber designed at the University of Missouri at Rolla (9, 10) contains a mixture of vapor and carrier gas. The vapor originates from a liquid pool resting on a piston at the bottom of the chamber. This vapor becomes super saturated, and nucleation starts when the piston is moved to produce an adiabatic expansion. Nucleation is then terminated after 10 ms by means of a small recompression, but the vapor remains supersaturated, and the droplets grow to a visible size. The droplet cloud is then photographed, and the nucleation rate is determined from the number of particles counted in the photograph. Nucleation rates from 102 to 105 cm-3 S-I have been measured for water (1 1), ethanol (12), toluene (13), and n-nonane (14). The method can be applied to nucleation in binary vapors as well, such as water-ethanol mixtures (15). Another version of the expansion chamber is the Gottingen chamber. In the first experiments (16) adiabatic expansion was created by moving one piston, and recompression by moving the other. The duration of the nucleation pulse is 1 ms, and nucleation rates between 105 and 1010 cm-3

S-I

can be measured by constant-angle light scattering. Because of the short duration of the nucleation pulse, the growing droplets are almost monodisperse, and therefore the scattered intensity is proportional to their number concentration. Measurements were performed for water (16), n nonane (17), and the homologous series of n-alcohols from methanol through hexanol (18). The Gottingen expansion chamber was recently reconstructed with the nucleating vapor now premixed with carrier gas in a separate plenum volume before a desired amount of the mixture is transferred to the expansion volume. The chamber is now operated by means of valves

492

LAAKSONEN, TALANQUER

& OXTOBY


instead of pistons. The amount of vapor in the chamber can be varied at constant initial temperature, unlike the earlier design in which a liquid pool inside the chamber equilibrated the vapor. The supersaturation can be varied even though the expansion ratio and, more importantly, the nucleation temperature are kept unchanged. The motivation for measuring curves of nucleation rate vs supersaturation at constant temperature comes from the Nucleation Theorem

( ) O W*

--

OJ.lgi

=

-l1n"*

1.

T,Il,j

where W* = I1G * is the work of formation of a critical nucleus in a multicomponent vapor, J.lgi is the gas-phase chemical potential of com ponent i, and I1n(is the excess number of molecules of component i in the critical nucleus over that present in the same volume in the vapor. Kash chiev (19) showed that the Nucleation Theorem holds for one-component systems whose surface energy depends only weakly on supersaturation. Strey and coworkers offered statistical mechanical arguments supporting the theorem (20) and extended these arguments as well as Kashchiev's original deviation to binary systems (21, 22). Oxtoby & Kashchiev (23) showed that the Nucleation Theorem is a general thermodynamic result not dependent on any specific model assumptions and that it holds down to the smallest nucleus sizes. The Nucleation Theorem offers a way to measure the molecular content of critical nuclei. The nucleation rate is J Jo exp ( - W* /kT), and the preexponential factor Jo varies slowly with J.lgi' It can be shown that o(kTlnJ)/OJ.lgi I1nt+m, with k the Boltzmann constant, and 0 < m < 1. The numbers of molecules in a critical cluster can therefore be determined from the slopes of nucleation rate vs gas-phase chemical potentials under isothermal conditions. The reconstructed G6ttingen expansion pulse chamber has been used to measure nucleation rates and molecular contents of critical nuclei for water (21) and butanol (24, 25) and for the binary systems ethanol-hexanol (22), water-ethanol (23), and water-n-propanol (26). The numbers of molecules found in the critical nuclei have ranged from 20 to 80. A variant of the expansion method is the shock tube. This device consists of a high-pressure section containing a vapor-carrier gas mixture, sep arated from a low-pressure section by a thin diaphragm. An adiabatic expansion is induced by breaking the diaphragm, causing the vapor to become supersaturated and nucleation to begin. At the same time, a shock wave starts to travel through the low-pressure gas. The shock wave reflects partially back from a constriction, causing a recompression and ter=

=


NUCLEAnON

493

minating nucleation after 0. 1 ms. Nucleation rates can be determined from the light scattering intensity. Peters and coworkers (see e.g. 27) have used a shock tube in measurements of critical supersaturations. Recently a group at the Eindhoven Technological University have applied this method to measure nucleation rates of n-nonane in a high-pressure non inert carrier gas (methane) that is above its critical point but is soluble and therefore affects nucleation (28). Rates between 108 and 1011 cm-3 S-1 were observed, and molecular contents of critical nuclei were determinated at total pressures from 10 to 40 bar. TURBULENT MIXING CHAMBERS A third method of nucleation measure ment, differing from both diffusion and expansion chambers, makes use of turbulent mixing of two vapors and is thus applicable to binary nucleation. The turbulent mixing method is useful for systems that are difficult to study accurately using other techniques, such as water-strong acid solutions that are important in the atmosphere. Briefly described, two carrier gas-vapor streams are led to a unit where rapid turbulent mixing takes place. The two-component vapor mixture is supersaturated and starts to nucleate immediately. The stream is passed on to a nucleation and growth tube, where its residence time is on the order of seconds. When the nucleated particle concentration is low enough, particle growth does not deplete the vapor appreciably, and the nucleation rate is assumed to be essentially constant from mixing to the end of the tube. Particles can be counted with a CNC. The turbulent mixing method has been used to measure nucleation rates of methanesulfonic acid-water and sulfuric acid water systems (29, 30).

Each nucleation measurement technique described has its own shortcomings. The modeling needed to calculate supersaturations is quite involved for both thermal and flow diffusion chambers. The thermal diffusion chamber cannot be used for accurate experimentation on binary systems, because saturation ratio maxima of the two vapors do not neces sarily occur at the same height. Furthermore, the liquid- and vapor-phase mole fractions usually differ, which causes problems: The vapor condenses continuously on the chamber walls and flows down to the liquid pool, changing the liquid composition at the bottom. The flow diffusion chamber is best suited for fluids of high molecular weight, because the nucleation zone becomes well defined only when the ratio of molecular to thermal diffusivity of the vapor is low. Expansion chambers, on the other hand, are not applicable to systems with long time lag, in which non-steady state nucleation takes place during most of the duration of the nucleation pulse (as in sulfuric acid-water). If the supersaturations necessary to produce nucleation are very high, large expansions are needed, which increases LIMITATIONS


494


uncertainties in the measurement. If the nucleating vapor is associating, the released heat of association may disturb an expansion chamber measurement. Finally, with the turbulent mixing chamber, care should be taken to ensure that the nucleation rate really does remain constant during the long nucleation period. Despite these limitations, the current arsenal of experimental techniques can be used to study a wide range of substances and nucleation rates. Different methods generally agree well for most substances. There are some exceptions, such as n-nonane and methanol, for which further studies would be desirable (note that the former requires high supersaturations to nucleate and the latter is self-associating). The most severe discrepancy between techniques concerns the pressure effect: Thermal diffusion cham ber studies have found that raising the carrier gas pressure lowers the nucleation rate substantially (31, 32). This effect is surprising from the theoretical point of view (33, 34) and has not been detected in expansion chambers. Nucleation Behavior of Molecular Fluids

To date, the nucleation of several dozen substances has been investigated experimentally. Unfortunately, the simplest molecules (such as the noble gases) are difficult to deal with because of their low condensation tem peratures, so nucleation rate data is practically nonexistent for them. Nevertheless, critical supersaturations in various molecular systems have been studied (3, 35), and a brief overview is given below. For all substances, the nucleation rate is a steep function of super saturation. This is illustrated in Figure 1, showing J vs S for water. The strong dependence of Jon S explains why large differences are often seen between theoretical and experimental nucleation rates. Secondly, in all nucleating (fluid) systems, SeT decreases as a function of temperature, and Jincreases as a function of T at constant S. Finally, critical nuclei become smaller at higher supersaturations and at lower temperatures. McGraw (3) has studied differences in nucleation behavior of various molecular types. Some vapors, such as argon and certain halogenated hydrocarbons, obey a corresponding states correlation between measured SeT and reduced temperature Tn derived for simple molecules. In contrast, two different groups of nonpolar molecules (n-alkanes and a group of aromatics) exhibited higher measures SeT than expected from the cor responding states correlation, whereas polar molecules (water, n-alcohols) were found to nucleate at lower SeT than predicted. The latter behavior was attributed to a tendency toward association in the liquid phase (hydrogen bonding). Increasing chain length of the molecules within a group was found to increase the stability of the supersaturated vapor. Finally, a 1993

NUCLEATION

495

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Figure 1

The dependence on supersaturation of the nucleation rate for the condensation of

water, as measured at a series of temperatures (in kelvins) by using a two-piston expansion chamber. (From Reference 21.)

study (35) found that highly polar aromatic molecules (benzonitrile and nitrobenzene) show even higher measured critical supersaturations than nonpolar aromatics. Fewer binary systems have been studied experimentally than one-com ponent systems, so possibilities for comparisons are somewhat limited. When two molecular species participate in nucleation, the nucleus com position (usually different from the vapor composition) becomes an additional degree of freedom, and binary systems can be classified accord ing to the nonideality they exhibit as a function of vapor composition. This nonideality can be studied using activity plots, which show lines of constant nucleation rate at varying gas-phase activities (i.e. super saturations) of the two species at fixed temperature. For ideal solutions, the lines are straight, whereas in more nonideal systems, they bend toward the origin or away from it. An example of a nearly ideal system is ethanol-



496

hexanol (22). In water-lower alcohol systems the lines of constant nucleation rate bend toward the origin, as these vapors are relatively more unstable. In these systems the nonideality is related to the strong surface enrichment of alcohols, but tendency to nucleate may also result from large mixing enthalpies, as is the case in water-acid mixtures. An example of a system where the two vapors do not help each other to nucleate is n nonane-water; the activity plot of this mixture shows an almost perfect right angle (36).

LIQUID-VAPOR NUCLEATION THEORY A complete theory of nucleation should describe the evolution of the population of clusters of the new stable phase. If clusters grow or decay by gaining or losing single molecules, the net rate at which clusters of size i become clusters of size i + I at time t is (37) J{i+ 1/2, t)

=

p{i)n(i, t)-y{i+ l)n{i+ 1, t),

2.

where n(i, t) is the number density of clusters of size i at time t; P(i) is the forward rate at which an i-cluster gains particles, and y(i) is the backward rate at which it loses particles. In steady state, the populations of different sizes of clusters no longer depend on time, and all fluxes equal a single constant flux, J(i + 1/2, t) J, for all i and t. This flux J is the nucleation rate, and by recurrence from Equation 2 it is given by =

J = n(l)

[00

1

]-1

i�l P(i)f(i)

,

3.

with f(i)

=

p . 1) ti y(j+

j= I

4.

The ratio fez) is related to the equilibrium constant for the formation of an i-cluster from isolated molecules, all at the supersaturation S of interest. In generating quantitative predictions for nucleation rate, different approximations have been made for the dependence of P(i) andf(i) or y(i) on the number of particles i in the cluster (2). Three main kinds of approaches can be identified. In phenomenological theories the effort has concentrated on obtaining the free energy of formation of small clusters from macroscopic quantities such as the surface tension and the bulk liquid density; classical nucleation theory is the basic reference for this kind of model. In kinetic approaches, nucleation rates are obtained from theories that calculate both forward and backward rate constants without

NUCLEATION

497

any need to estimate cluster formation energies. Finally, in the microscopic approaches, first-principles models are developed for the cluster structure and free energy of cluster formation; computer simulations and density functional theory are among the most successful attempts in this group. Phenomenological Approaches

Our understanding of the nucleation of liquids from the vapor has been dominated for nearly 70 years by the classical nucleation theory (CNT), as developed by Volmer & Weber (38), Becker & Doring (39), Frenkel (40), and Ze1dovich (referred to in 41). In this theory, the free energy of an i-cluster is calculated by treating the cluster as a macroscopic spherical droplet with bulk and surface free energies relative to the background vapor. The equilibrium constant for the formation of the cluster is taken as f(i) = e-I1C;/kT, the free energy is (1)


CLASSICAL NUCLEATION THEORY

IlGJkT = aA(i)

-

iln S = (}i2/3

-

iln S,

5.

where a is the surface tension of the gas-liquid interface, A(i) is the surface area of an i-cluster, which is estimated from the molecular density PI in the bulk liquid, and e (36n)I/3ap,-2/3/kT. As in many later approaches to this problem, CNT assumes that mono mers collide with the cluster surface at the gas kinetic rate for an ideal gas, resulting in a forward rate (37), =

P(i) = (kT/2nm)I/2n(I)A(i),

6.

where m is the mass of monomer molecules. The sticking probability of vapor molecules to the nucleus is assumed unity. If contributions from near the critical nucleus size dominate the sum of Equation 3, the evaluation of the rate J can be simplified and mathematical manipulations lead to the classical expression (1) 1.

CL

-

J0e-I1C*/kT..

7.

with the preexponential factor Jo

=

-l

(2a/nm)I/2n( I )2pl

8.

and with 9.

as the barrier height to nucleation.


498


The dominant role played by CNT is linked to the fact that the input to the theory is widely available for many substances, and the predictions of critical supersaturations are successful for most gases. Nevertheless, the theory is frequently in error when predicting actual nucleation rates, giving too Iow a value at low temperatures, and too high a value at high tem peratures (5, 14, 1 7). CNT also predicts significantly lower critical super saturations than experimentally observed in associated vapors (42-44) and highly polar fluids (35, 45, 46), and in general it fails to include molecular level effects relevant for small clusters. MODIFICATIONS OF THE CLASSICAL APPROACH The limitations of classical nucleation theory have stimulated the search for improved theoretical models, but most phenomenological approaches retain the basic capillarity approximation Equation 5 and only introduce correction factors to it. One extensively discussed correction to CNT (47) was proposed by Lothe & Pound (48). They consider that a cluster has translational and rotational degrees of freedom, each contributing to the free energy. These additional

terms increase the rate by factors on the order of

1017•

Reiss and coworkers

(49, 50) have argued that using the experimental surface tension in the capillarity approximation takes into account most of the translational and rotational effects. The Reiss-Katz-Cohen theory (49) proposes a new way to consistently incorporate these effects into the classical approach, leading to a much smaller change in the expected rates. Pound and coworkers (51 ) have reopened the discussion by introducing a nonlinear version of the Lothe-Pound theory, whose results are close to those of their original model. In 1 9 61 , Courtney (52) introduced a different kind of correction. He suggested that CNT neglected to specify the partial pressure of the cluster considered as a molecule in the theory, and proposed a correction factor of l / S in the preexponentia1 term in Equation 7. Blander & Katz (53) reached a similar conclusion following an independent thermodynamic approach. Girshick and coworkers (54, 55) have also stressed the role of the extra factor of l / S in the nucleation rate. Although kinetic in origin (54), their model can be thought as an ad hoc adjustment to the classical expression for the cluster formation energy to assure IlG; 0 for mono mers (self-consistency). From Equation 5 for i 1 and Equation 7, this restriction introduces a multiplicative correction factor eO/ S in the expression for JCL' Dillmann & Meier (56, 57) have proposed another extension to CNT based on Fisher's cluster theory of condensation and metastability (58). In their theory, the work of formation of an i-cluster at pressure P is taken to be =

=

NUCLEATION

AGdkT = K/H2/3 + dn i- In (qo V) - iln S,

499 1 0.

where Ki is a function of size and temperature that describes deviations of the surface free energy from that of a macroscopic liquid droplet. Dillman & Meier assumed a particular functional form for these quantities based on the work of Tolman (59): Annu. Rev. Phys. Chem. 1995.46:489-524. Downloaded from www.annualreviews.org by Stanford University - Main Campus - Lane Medical Library on 05/13/13. For personal use only.

II. with (Xl and (X2 selected to fit the saturated vapor pressure and the second virial coefficient of the fluid. The second and third terms in Equation 10 arise from differences between the molecular structure of a free i-cluster and a group of the same number of particles in the bulk liquid. In this theory rand qo are chosen to fit the critical density and pressure of each substance. The predictions of a theory with parameters determined from critical point and low-density gas properties agree surprisingly well with experimental data for nonane, water, and n-alcohols (57). Ford et al ( 60) recently pointed out an inconsistency in the Dillmann Meier theory that was corrected by considering the imperfect behavior of a real vapor. The corrections alter the predictions of the model significantly, however. Delale & Meier (61) tried to overcome the problem without spoiling the agreement with experiment but had to introduce new empirical parameters to adjust the results. The revised Dillmann-Meier theory ( 60) is independent of the value of qo and in that sense can be thought of as a self-consistent modification to CNT with three main corrections (62): the curvature dependence of the surface tension (Ki), the contribution of additional degrees of freedom (r) , and a demand for thermodynamic consistency ( lIS). Laaksonen et al (63) analyzed the model when r takes the values assigned by different theories and found that the classical value t = 0 yields good agreement between predictions and data for several substances. In the self-consistent model of Kalikmanov & van Dongen (64, 65) the functional form of K; in Equation 1 1 is rejected and substituted by the effect of the correction factor e8• This theory agrees better with experimental results for nonpolar substances than CNT. Hale (66) introduced a scaling model in 1986 based on CNT that sug gested a universal dependence of nucleation rates on ( TeIT -1) far below the critical temperature Te. Assuming that the surface tension varies lin early with temperature, the critical supersaturation is given in terms of a nearly universal constant measuring the excess surface entropy per molecule. The model works well in several cases where CNT fails. Gninasy (67, 68) recently proposed an approach to nucleation based on a parametrization of the enthalpy and entropy density profiles of droplets. By assuming these profiles to be shifted by a constant amount, regardless

500


of droplet size, the theory gives rise to curvature corrections without requiring the fitting of additional parameters in the free energy beyond those used in CNT. The model predicts nucleation rates fairly well for nonane and toluene, but it fails in describing the behavior of more polar substances.


Kinetic Approaches

In contrast with phenomenological approaches, kinetic theories of nucleation evaluate forward and backward rate constants without direct reference to formation free energies of clusters. Nucleation rates can then be calculated, in principle, from the interaction potential between particles. In practice, the uncertainties in these potentials for real systems force the theories to fit their parameters using experimental data for the surface tension, the liquid density, or even the nucleation rates themselves. This limits the testing of any theoretical result. Nowakowski & Ruckenstein (69) calculated the rate of evaporation of molecules from a cluster by using a diffusion equation in energy space. The backward rate was obtained from the mean passage time of molecules that escape from the potential well at the cluster surface. They have improved the model (70) by using a Fokker-Planck equation to describe the motion of particles on the surface. The intermolecular potential depends on two parameters that are fit to give the correct surface tension and density of the liquid. The theory predicts higher nucleation rates than CNT, with a more pronounced effect at high supersaturations. In the kinetic theory of Wilcox & Bauer (7 1 ), an excited cluster is first formed and then relaxed by collisions with monomer or background gas. Given the magnitude of all the rate constants (doubled in number by the assumptions of the theory), the time-dependent concentrations of clusters of all sizes can be calculated. The nucleation rates depend on choices of the relevant parameters that are fit to experimental data. Microscopic Approaches COMPUTER SIMULATIONS Simulation techniques avoid many of the limi tations in phenomenological approaches to nucleation. The simulation of the condensation of vapor in the thermodynamic limit is difficult, however, because nucleation occurs so rarely. Typical volumes and times in com putation work restrict simulation to supersaturations much larger than those accessible to experiment (I). Thus, research has focused on the properties of isolated clusters, where the problem is to provide a physically consistent definition of the cluster to be simulated. In a first approach, Stillinger (72) considered that a cluster in a vapor should consist of those particles that lie within a distance rc of at least one


NUCLEATION

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other molecule in the group. Rao et al (73) used this definition to study nucleation with molecular dynamics and Monte Carlo techniques. The latter method proved to be more efficient because of the long time needed to stabilize evaporation and condensation fluctuations in molecular dynamics simulations. By monitoring the cluster population at equilibrium, they identified a stable cluster corresponding to a minimum in the formation free energy. More recently, Reiss et al (74) pointed out certain incon sistencies in computer simulations based on the Stillinger cluster. Reiss et al (49) defined a cluster to be a group of molecules contained within a sphere of radius Rc whose center is fixed at the center of mass of the group. The constrained cluster is stable, but there is no natural criterion for the radius Rc of the constraining shell. Lee et al (75) followed this prescription in their Monte Carlo calculation in 1 973. They showed that at sufficiently low temperature the Helmholtz free energy of small clusters of Lennard-Jones particles is almost independent of the value of Rc, as long as the shell was not too small or too large. McGinty (76), taking a slightly different cluster definition, presented molecular dynamics cal culations for argon clusters. More recent studies have investigated the effect of curvature and temperature on the surface free energy of droplets of simple (77) and polar (78) fluids. Garcia & Soler (79) expanded the work of Lee et al in a first attempt to address the problem of nucleation. However, the restriction of an arbitrary choice for the constraining volume still remained. Recent work by Reiss and coworkers (80, 8 1 ) seems to overcome this limitation and makes an important contribution toward a true molecular theory of nucleation. These authors defined a consistent cluster as a group of i molecules con tained in a volume v defined by a spherical shell centered on the center of the mass of the group. The shell has a volume dv that always contains a molecule of the supersaturated vapor surrounding the cluster. The pres ence of the shell molecule gives physical meaning to v as an independent variable and ensures that the work of formation of the cluster includes the energy needed to create a hole of volume v in the gas. By taking the molecular content (i) and the volume (v) as independent parameters, the theory allows density fluctuations in clusters with a fixed number of particles. The nucleation rate is then determined by the growth flux on a surface representing the work of formation in the space spanned by i and v. Relevant cluster sizes and volumes are selected by the kinetics of the process. Using Monte Carlo simulations, Weakliem & Reiss (82) found that the energy surface of Lennard-Jones clusters exhibits a ridge over which developing clusters must flow in order to grow into droplets. The central feature, however, is a valley on the far side of the ridge that funnels clusters into liquid droplets. The same authors (83) developed a rate theory


502


& OXTOBY

using a modified liquid droplet model to estimate nucleation rates. Their predictions agree qualitatively with experimental trends, showing that CNT predicts too high a rate at high temperatures and too Iow a rate at low temperatures. Hoare & Pal (84) explored the potential energy surface for Lennard Jones clusters using numerical optimization techniques. The minimum energy isomers of argon microclusters (85) were then used to evaluate the translational, vibrational, and rotational contributions to the formation free energy in a harmonic-oscillator-rigid-rotor approximation. Their results exhibit a well-defined barrier to nucleation at a critical size, but the predicted nucleation data are remarkably insensitive to the details of internal cluster structure. Peters & Eggebrecht (86) recently questioned the classical idea that the formation of a critical nucleus is the rate-limiting step in nucleation. Their molecular dynamics quenching experiments suggest instead that the evolution of the phase transition is controlled by the diffusion and coalescence of clusters. Their calculations correspond to metastable states close to the spinodal, however, where the energy barrier to nucleation might be too small to detect. The basic assumption of the density func tional approach to liquid-vapor nucleation is that any nucleating entity can be considered as an inhomogeneous fluid. The theory then obtains the properties of the critical nucleus from the free energy of the nonuniform system, n[p(r)], which is a unique functional of the average density per) and whose minima determine the thermodynamically stable states at a given temperature (87). Cahn & Hilliard (88) presented the first density functional calculation of nucleation. For the grand potential of the system, they propose DENSITY FUNCTIONAL THEORY

n[p(r)]

=

f

dr{Jh[p(r)]-IlP (r)+k[Vp(r)f},

12.

where fh is the Helmholtz free energy per unit volume of a homogeneous system with density p; Il is the chemical potential, and the square-gradient term k[Vp(r)]2 accounts for nonlocal contributions to the free energy. At a given supersaturation, this functional has a saddle point in the function space where it is defined. This solution gives the density profile of the critical nucleus and the free energy cost, iln*, to form a critical liquid droplet from the unstable vapor. In contrast with CNT, this work of formation includes curvature effects in a natural way and properly vanishes at the spinodal where the radius of the droplet diverges. The nature of nucleation in this limit has been further examined by Unger & Klein


NUCLEATION

503

(89). For long-range potentials the structure of the critical nucleus at the spinodal becomes highly ramified (fractal). The square-gradient approximation is useful when the average density varies slowly over atomic distance scales. This is not necessarily the case away from the critical point or the spinodal. Thermodynamic perturbation theories have been developed to study planar (90, 91) and curved (92) interfaces under such conditions. These methods generalize the van der Waals theory of interfaces, in which the free energy of the actual non uniform fluid is expressed in terms of the free energy of a suitable reference system. Oxtoby and coworkers (93, 94) followed this approach in their work on vapor-liquid nucleation. They used hard-sphere perturbation theory to write the free energy as the sum of a hard-sphere repulsive contribution (local) and a long-range attractive contribution (nonlocal)

Q[p(r)]

=

f

dr[fh[p(r)]-JlP(r)]+

ff

drdr'cf>att(\r-r'l)p(r )p(r').

13.

Here cf>att is the long-range attractive part of the potential. From the saddle point solutions of this functional, the free energy barrier to nucleation ilQ* was evaluated. In a first approximation, these authors estimated the rate of nucleation as J = Joe-An*/kT with the same preexponential Jo as in CNT (Equation 8). Oxtoby & Evans (93) investigated a cf>att given by the Yukawa potential: rx),?e-J.r/4nr. Their results showed that the success ofCNT is extraordinarily sensitive to the range of the interactions I/X Zeng & Oxtoby (94) extended this work to a more realistic Lennard-Jones potential with a temperature dependent hard-sphere diameter. The dependence of the theoretical nucleation rates on supersaturation and temperature agreed qualitatively with experimental results for several systems. Density functional theory includes molecular-level effects crucial to describe the behavior of small nuclei. Oxtoby and coworkers have extended the density functional theory of nucleation in various directions. They have studied the bubble nucleation in simple fluids (93, 94). Deviations from CNT were more dramatic than those observed in nucleation from the vapor. The tensile strength of liquids seems to be much smaller than predicted classically. Talanquer & Oxtoby (95) applied the density functional formalism to study Stockmayer fluids: point dipoles embedded in Lennard-Jones particles. Although they found different alignment near the surface of small clusters, the effect of dipole moment on the free energy was quite small; the artificial up-down sym metry of the Stockmayer model is not realistic in describing real polar liquids. A new model based on the interaction site formalism (V Talanquer

504


& OXTOBY


& OW Oxtoby, unpublished results) looks promising for understanding

nucleation in fluids with more anisotropic molecules, as well as hetero geneous nucleation of polar fluids about ions. The work described up to this point has centered on the evaluation of the barrier to nucleation A!l*, using the classical preexponential Jo to estimate rates. Few efforts exist to develop a more complete theory to calculate Jo as well. Langer & Turski (96) proposed an interesting approach for use near the critical point. In their hydrodynamic model, the prob ability flow across the saddle point gives the preexponential as a product of a statistical factor (a measure of the phase-space volume near the saddle point) and a dynamical term (related to the growth rate of a droplet that has exceeded the critical size). Their numerical predictions, however, are nearly the same as the classical results. Recently, Talanquer & Oxtoby (97) took a step toward a full dynamical density functional theory of gas liquid nucleation. Following some ideas of Reiss and coworkers (80, 8 1 ), they calculated the free energies of noncritical and critical nuclei from equilibrium density profiles of stable clusters in closed volumes (98). In contrast with Reiss's approach (8 1 , 82), Talanquer & Oxtoby showed that the volume of the constraining sphere did not enter the final results, thus reducing the two-dimensional (i, v) dynamics back to one dimension. Nucleation rates were estimated using a consistent kinetic model from which a preexponential factor emerged naturally. This theory gave pre dictions very close to those of earlier density functional calculations (94). Density functional theory has been used to explore the range of validity of some phenomenological approaches (62). This work suggests that the success of several macroscopic theories is an accident of the range of conditions accessible to experiment, because outside this range there is no correspondence to results of the full calculations. SEMI-EMPIRICAL METHODS The sensitivity of nucleation rates to the molec ular interaction potential limits the applicability of microscopic theories for quantitative predictions on real systems. Some approaches have tried to overcome this limitation by fitting some of their basic parameters to available experimental data. One contribution in this direction was pro posed by Kobraei & Anderson (99, 1 00). They developed a model of interacting monomers and clusters, in which the partition function is approximated to estimate the cluster free energies. Their results depend on an effective range of interactions that was fit to the experimental nucleation rates for the system of interest. Unfortunately, this procedure limits the ability of experiments to test the assumptions of the theory. Nyquist et al ( 1 0 1 ) used density functional models to pursue an alter native approach. The three parameters that define the effective interaction

NUCLEATION

505

potential were fit to experimental data for bulk liquid density, equilibrium vapor pressure, and surface tension. The theory was then applied to predict nucleation rates. For nonpolar fluids (nonane, toluene) this method gave results that differed significantly from CNT and were in better accord with experiment. For water and more polar substances the results were quite close to classical calculations.


Binary Nucleation

To calculate binary nucleation rates from classical theory, the composition of the critical nucleus has to be determined. This can readily be done for model fluids because the nucleus is in unstable equilibrium, and the chemi cal potentials of both molecular species in the nucleus must equal the chemical potentials in the vapor. At fixed vapor densities this requirement is fulfilled for only one pair of liquid densities, so the nucleus composition is unique. Provided that the surface tension is known, the radius can then be calculated from the Laplace equation: /).P = 2a/r, where I1P denotes the difference in pressure inside and outside the nucleus. The work of nucleus formation is obtained from W* VI1P+ Aa, where V and A are the volume and surface area of the nucleus, respectively (102). However, with real molecular fluids the chemical potentials are usually known only at equilibrium, and the above method is not practical. Assuming that the nucleus is incompressible, the binary Gibbs-Thomson (Kelvin) equations for the critical nucleus can be derived: I1Jli 2av)r. Here l1/1i is the chemical potential difference between liquid and vapor phases, both taken at the pressure outside the nucleus, and Vi is the partial molecular volume of species i in the liquid. Both the composition and the radius of the nucleus can be obtained from these equations (103). Finally, the free energy of nucleus formation is given by an extended form of Equation 5. The history of this theory is surprisingly complex. The binary Gibbs Thomson equations were employed already in the 1 930s by Flood (104), Volmer (105), and Neumann & Doring (106); however, these studies remained relatively unknown outside Germany for a long time. In 1 950, Reiss (107) used kinetic and thermodynamic arguments to show that the binary nucleation rate was determined by passage over a saddle point in a space of droplet compositions. Doyle (108) used Reiss's theory to study nucleation in the sulfuric acid-water system and included the dependence of the surface tension on composition. Renninger et al (109) subsequently found thermodynamic inconsistencies in Doyle's treatment that were con firmed by Wilemski (103, 1 10) in his "revised classical theory" for binary systems. Wilemski showed that taking into account surface adsorption exactly cancels the surface tension derivative with respect to composition =

-

=

-


S06


present in Doyle's treatment, resulting in the binary Gibbs-Thomson equa tions. Mirabel & Reiss (I l l) and Nishioka & Kusaka ( 1 02) presented fundamental thermodynamic reasons for the derivative not to appear. Laaksonen et al ( 1 1 2) showed that the total numbers of molecules in the critical nucleus can be obtained using the Gibbs adsorption equation. Classical binary nucleation theory gives reasonable predictions for fairly ideal mixtures (22). However, for mixtures with strong surface segregation (such as water-alcohol systems) it can give rise to unphysical behavior: a decreasing nucleation rate with increasing vapor densities ( 1 S, 23, 26). For a dilute vapor, such behavior violates the Nucleation Theorem (23). Various approaches have been suggested to remedy the classical pre dictions. Rasmussen (1 1 3) introduced a dynamic surface tension and found agreement with experiments at nucleation onset conditions. However, Wilemski (1 1 4) pointed out that Rasmussen's theory is thermodynamically inconsistent in the same sense as that of Doyle ( 1 08). Flageollet-Daniel et al ( lIS) treated water-alcohol clusters in terms of a crystal lattice model, with enrichment of alcohol at the surface of the cluster and depletion in the interior. The critical cluster was found by locating the saddle point in the droplet composition space. Laaksonen and coworkers (1 1 6, 1 1 7) proposed an alternative geometric model: a spherical water-alcohol cluster composed of a bulk core and a unimolecular surface layer, both having macroscopic densities, with the surface enriched in alcohol. Although these models rely on the classical capillarity approximation (surface tension independent of cluster size), they predict reasonable critical super saturations. The latter model also yields almost quantitatively correct overall nucleus compositions in the water-ethanol system (23). The extension to binary mixtures of some of the modifications to CNT described previously is in progress. Kulmala et al ( 1 1 8) extended Gir schick's (SS) self-consistent theory, finding nucleation rates somewhat higher than those given by the classical theory. Kalikmanov & van Dongen ( 1 1 9) introduced an effective quasi-one-component model for binary sys tems based on a generalized Fisher droplet model. The first attempt at a truly microscopic description of binary nucleation was presented by Kulmala ( 120), who extended the treatment of Hoare (8S) to binary systems. However, numerical calculations were carried out only for the very nonideal H 2S04-H20 and HNOrH20 systems, for which uncertainties in molecular parameters caused tens of orders of magnitude of uncertainties in nucleation rates. Zeng & Oxtoby ( 1 2 1 ) extended the density functional theory to binary nucleation. Using this theory, Laak sonen & Oxtoby ( 1 22) showed that measurable deviations from classical theory can be seen in an almost ideal Lennard-Jones binary mixture. They also examined a more nonideal model fluid showing strong surface


NUCLEATION

507

segregation, for which the classical nucleation predictions are similarly unphysical, as in water-alcohol systems, while the density functional pre dictions are thermodynamically consistent. Density functional calculations on bubble nucleation in binary fluids (123) have shown that the interplay between the gas-liquid transition and liquid-liquid phase separation can explain certain unexpected behavior, such as the decrease in nucleation rate of some fluids with increasing temperature. Several authors (e.g. 1 24, 1 25) have examined the preexponential factor in binary nucleation. In some binary theories (notably in classical theory) only the free energy of the critical nucleus is known, not the shape of the free energy surface in composition space, so quite approximate expressions must be employed. Usually the error can be expected to be rather small, but caution should be exercised when dealing with very nonideal systems (126). Quite recently, the full binary steady-state equations have been solved numerically (127, 1 28), allowing a more detailed study of the kin etics of the phenomenon.

CRYSTAL NUCLEATION The homogeneous nucleation of crystals from the melt is much less well understood than the gas-liquid nucleation discussed up to this point. It is difficult to prepare liquids sufficiently pure to avoid heterogeneous nucleation (see next section); it is complicated to use pressure on condensed phases as a second variable (in addition to temperature), and in only a few special cases are there independent measurements of the solid-liquid surface free energy. This is in contrast to the relatively standard measure ments of surface tensions of liquids, which allow direct tests of classical nucleation theory for condensation; such tests are in general not possible for crystallization. Crystal nucleation is of interest to materials scientists, who have devoted considerable attention to the casting of metals and their alloys, and the formation of small crystallites in glassy ionic materials. Some experiments in these fields have been described in our earlier review (129). Here we focus on recent studies of atomic and molecular liquids, with a view toward the atmospheric applications to be considered in the next section. Experimental Studies

A key problem in studying homogeneous nucleation of crystals from the melt is avoiding heterogeneous nucleation on impurities. One way to do this is to emulsify a liquid sample, dividing it into micron-sized droplets. Those that contain impurities nucleate heterogeneously, while the others persist until nucleating homogeneously at a lower temperature, unless the


508


droplet surface itself catalyzes nucleation. This approach was followed in the classic study of ice nucleation by Wood & Walton (130). These authors studied the change in the rate of nucleation over a 1°C temperature range. They fit their data to classical nucleation theory by taking the surface free energy to depend either on temperature or on curvature; distinguishing between these two possibilities is a recurring problem with crystal nucleation. Broto & Clausse ( 1 3 1 ) achieved slightly larger undercoolings with smaller droplets and faster cooling rates. A second way to ensure homogeneous nucleation is to employ con ditions under which crystal growth is very slow so that the number density of crystal nuclei is very high and far exceeds any heterogeneous particle concentration. This can be achieved by studying nucleation near the glass transition as in ice crystallization from aqueous LiCI solutions ( 1 32). A different microcapillary technique was introduced (133) to study solutes, such as alcohols that are soluble in oils, for which emulsion techniques do not apply. A third way to avoid heterogeneous nucleation is to study isolated dro plets in the gas phase. Recently two groups have employed supersonic nozzle expansions to create droplets (typically containing several thousand molecules) and rapidly cool them until they crystallize. Beck et al (134) used Raman techniques to detect crystallization in nitrogen clusters, while Bartell and coworkers (135-139) employed electron diffraction for a variety of molecular liquids. As Bartell has emphasized, the small particle size and rapid cooling rates for these droplets give nucleation rates on the order 3 3 of 10 0 m- s-I, some 20 orders of magnitUde faster than in emulsion experiments. Bartell's technique also allows one to distinguish different solid phases and to see solid-solid phase transitions. The use of classical nucleation theory gives crystal-melt surface free energies for CCl4 (1 35), CH3CCl3 (136), NH3 (137), and water (138). The last is of particular interest, as under these conditions water crystallizes in the cubic ice Ie phase rather than the thermodynamically stable hexagonal ice Ih phase. The kinetic barrier to nucleation appears too high for hexagonal ice to form homogeneously, although it may do so via heterogeneous nucleation on a substrate. Computer Simulations

In view of the experimental difficulties in studying crystal nucleation, computer simulation via molecular dynamics provides an attractive alter native. Many early simulations observed such crystallization from under cooled liquids, but doubt was cast on their quantitative validity by a key study ( 1 40) that showed that nucleation rates are strongly affected by periodic boundary conditions (as revealed by the fact that nucleation


NUCLEA nON

509

became slower for larger systems). A subsequent study on systems of up to one million particles ( 1 4 1 ) showed that these small-system-size artefacts could be eliminated with about 1 5,000 particles, although a single simu lation of even this size was not enough to demonstrate bulk behavior. In this work, Swope & Andersen (141 ) carried out steepest-descent quenches of the system states to locate potential minima, followed by a Voronoi polyhedron analysis to identify solid-like regions. Under the conditions of their calculation, the critical nuclei contained only 10 to 20 Lennard-Jones particles. The small size of critical nuclei in molecular dynamics simulations reflects the large under-coolings needed to see nucleation over a reasonable computer time scale. Such undercoolings are much greater than those in typical bulk or emulsion experiments, although they approach the conditions of supersonic jet experiments (139). For a crystal to form so quickly, the barrier to nucleation must be only a few times the thermal fluctuations in the liquid. An alternative to waiting for a rare fluctuation is to force the liquid to nucleate by assigning a greater statistical weight to states between liquid and crystal. Van Duijneve1dt & Frenkel (142) used such "biased" (umbrella) Monte Carlo sampling to study the pathway from a soft-sphere liquid to the metastable bcc phase and the stable fcc phase. They found that the bcc state forms more easily because the barrier is lower, in spite of the bulk stability of the fcc phase. Clancy and coworkers (143, 1 44) used a different approach to avoid the high-barrier problem in nucleation simulations. They inserted crystalline clusters of varying sizes into liquids and determined whether the clusters grew or shrank. The borderline between the two determines the size of the critical nucleus (about 1 70 particles). However, they found that the rate of growth was not monotonic, suggesting "magic numbers" of particular stability even for defective crystals surrounded by liquid. They could fit their results only to a modified classical nucleation theory. All the simulations described so far in this review employed simple isotropic potentials. One recent molecular dynamics simulation ( 1 45) found water to crystallize into cubic ice Ie in a homogeneous static electric field. A larger system size than the one used (256 molecules) is necessary to rule out effects from periodic boundary conditions. Theory

The reference for comparison for both simulation and experiment is the classical theory of Turnbull & Fisher (146). The standing of classical nucleation theory is much less clear for crystallization than for conden sation, because most experiments use the surface free energy as a fitting parameter. There are also questions about the dynamical prefactor. The


510


dynamics of condensation is straightforward: single-particle collisions and attachment to a growing cluster. The dynamics of crystal nucleation from the melt is less easy to describe. It is difficult to identify which particles are part of the crystal and which of the surrounding liquid, and collective fluctuations can occur in which entire regions become solidlike, instead of the single-particle jumps assumed in classical theory. Within the last ten years two alternatives to classical nucleation have been proposed, although neither has been thoroughly tested. Meyer ( 1 47) suggested that crystal nucleation might occur adiabatically, rather than isothermally; this approach finds a definite stability limit for the liquid near the point at which nucleation is observed experimentally. Ruckenstein and coworkers ( 1 48, 1 49) proposed a kinetic theory in which surface particles undergo Brownian motion in a potential well; a mean-first-pass age-time analysis is then used to calculate the nucleation rate. Density functional theory provides a final theoretical approach to crystal nucleation. Applying such a theory to crystallization is complicated because an infinite set of order parameters for a crystal (the Fourier components of the density) replaces the single order parameter for a fluid (the average density). In crystal nucleation, these order parameters also vary through space. Only one early calculation of nucleation using this approach was carried out: Harrowell & Oxtoby (150) used liquid-state perturbation theory to expand the free energy of a crystal to second order about that of the uniform liquid; moreover, they assumed that the first peak in the liquid structure factor dominated and that S(k) could be set to 1 beyond this point. Within this approximate model their calculation was self consistent in comparing the density functional result with the corresponding classical prediction, and large deviations (due to curvature of the critical nucleus) were found. Future work will need to use the more accurate free-energy functionals that have been applied to planar interfaces ( 1 5 1, 152); still further off are calculations for molecular systems such as water, for which only bulk freezing behavior has been explored using density functionals (153) .

ATMOSPHERIC NUCLEATION PHENOMENA This section reviews studies conducted in the last 1 5 years that relate to nucleation in the atmosphere (for references to older studies, see 1 54, 1 55). We begin with general considerations of heterogeneous and ion-induced nucleation and homogeneous nucleation in the sulfuric acid-water system. We then address the more specialized issues of sulfate particle formation in the atmosphere and the formation of polar stratospheric cloud particles. The more traditional fields of ice nucleation and freezing in cirrus clouds

NUCLEATION

51 l

fall outside the scope of the present article (for references to recent work, see 156).


Heterogeneous Nucleation

Heterogeneous nucleation on insoluble particles may take place at much lower supersaturations than homogeneous nucleation and is therefore important in the atmosphere. The classical free energy of critical nucleus formation on a flat solid surface is AG* AG"tt,of(m), where AG"tt,o is the free energy for homogeneous nucleus formation, and f(m) (2 + m) (1 - m)2/4. The parameter m is the cosine of the contact angle between the nucleus and the substrate and is given by Young's equation: m = cos e (0'31 - 0'32)/0'21> where the O" s are interfacial energies between different phases- l denoting the mother phase, 2 the nucleus, and 3 the substrate. This description holds both for nucleation from the vapor and from the liquid. For a curved substrate, J(m) is somewhat more complicated (157). The values of f(m) are between 0 and I ; in the former case the contact angle is 0° and nucleation takes place as soon as the vapor becomes supersaturated, whereas in the latter case the contact angle is 1 80°, and the nucleation is homogeneous. The description of the interaction between substrate and nucleus using a single macroscopic parameter (m) is obviously very crude, but so far there has not been much development in more microscopic directions. Furthermore, several important aspects of heterogeneous nucleation have been omitted in the classical theory. The classical expression for the hetero geneous nucleation rate is derived assuming that vapor molecules impinge on the nucleus, whereas it was shown already in 1 954 ( 1 58) that surface diffusion of adsorbed molecules is a more effective delivery mechanism. Classical theory does not capture the role of substrate imperfections: Chemical and structural defects in the surface structure may enhance nucleation probability. The effect of line tension is also missing from the classical free-energy expression. Allowing for these features may change the classical predictions considerably (159-1 61). It is not easy to verify theoretical predictions of heterogeneous nucleation because experimental data is scarce. Measurements have been carried out only for water on partially wettable flat surfaces (see 1 62 for a review) and on nanometer sized silver particles ( 1 63). The lack of data can be attributed to the difficulty of producing smooth surfaces with well-defined characteristics. In the future, more information of heterogeneous nucleation may be gained through Monte Carlo and molecular dynamics simulations than from laboratory experiments. So far, Monte Carlo methods have been used to study ice formation on ice-like surfaces ( 1 64-1 68). =

=

=

512



Ion-Induced Nucleation

Nucleation on ions is important in the atmosphere but not well understood at the basic level. The classical theory predicts that the ion-induced nucleation rate is proportional to the ion density in the vapor. However, in the presence of both positive and negative ions, recombination may affect the nucleation rate ( 1 69). The classical expression ( l OS) for the free energy to form a critical nucleus of radius r around an ion of radius ro is AG*

==

AG�o +

e

2

-

8m, 0

(

1

1

ro

r

---

)( ) 1

1-- , s

14.

where e is the unit charge, e is the relative dielectric constant of the nucleus, and 6 0 is the vacuum permittivity. The classical theory does not predict differences in nucleation rates due to ion polarity. However, already in 1899 Wilson ( 1 70) observed that water vapor nucleates much more readily around negative ions than positive ones. Fletcher ( 1 7 1 ) attributed this sign preference to surface orientation of polar molecules. Later Rusanov & Kuni ( 1 72) formulated a thermo dynamic theory incorporating surface orientation and did find that nega tive ions should be better nucleating agents for water. Density functional calculations ( 1 73) show that asymmetry of dipolar molecules induce sign preference and influence the rates by a factor of 10-100. Rabeony & Mirabel (1 74) list early experiments that found conflicting evidence con cerning the sign preference. They noted that the electric fields applied in these studies to separate positive and negative ions may have affected the results, and used a diffusion chamber in which ions were produced with an ex-source. They studied substances with dipole moments ranging from o to 1 .8 debye and found that all of them (including water) exhibited sign preference in the presence of an electric field, but not when the field was absent. The nucleation rate depended on electric field strength, but other studies ( 1 75, 1 76) have found somewhat different dependencies. Adachi et al ( 1 77) used a different experimental setup: a unipolar ion stream mixed with dibutyl phthalate (DBP) vapor in a nucleation flow chamber with no electric fields present. They found a clear positive sign preference for the nucleation of DBP and reported a strong dependence on the chemical makeup of the ions, attributed to ion size. It is difficult to control supersaturations, ion polarities, ion concen trations, and chemical compositions of the ions simultaneously in an accurate way. These difficulties seem to have been overcome in a new experimental technique (1 78) in which ions are produced in a thermal diffusion chamber below the nucleation zone by UV-light pulses from a


NUCLEATION

513

tunable dye laser. The pulses cause resonant absorption i n dopant mol ecules that are subsequently ionized. Ions of desired polarity are directed to the nucleation zone by an electric field. Using this technique with both polar ( I -hexanol) and nonpolar (n-nonane) molecules, Katz et al (1 78) showed that the ion-induced nucleation rate is proportional to ion con centration and that it depends neither on electric field strength nor on the chemical makeup of the ions. It has been predicted that in the H20-H2S04 system, which is of con siderable atmospheric interest, ion nucleation dominates homogeneous nucleation at low relative acidities ( 1 69). One idea is to apply energetic irradiation to H20-SOz gas mixtures and to observe the resulting particle production. A severe difficulty in these studies is that the irradiation not only creates ions, but it also produces radicals and thus drives complicated chemistry, leading among other things to formation of sulfuric acid. It is difficult to separate the effect of ions on particle formation by modeling the system. In several studies (1 79-1 8 1 ) the role of ion nucleation remained unclear despite extensive model calculations. However, the dependence of ion mobility spectra on irradiation dose and on residence time indicated that ion nucleation had indeed taken place in the experiments of Makela ( 1 8 1). Unfortunately, the irradiation was too energetic and the sulfur dioxide concentrations too high to relate the results directly to atmospheric conditions. Homogeneous Nucleation of SulfuriC Acid

The study of nucleation in the sulfuric acid-water system began with Doyle (108), who pointed out that the phase change can take place at very low relative humidities, and possibly at ambient H2S04 concentrations. Several papers on the subject were published in the following 20 years; however, measurements of Roedel in 1979 (1 82) and Ayers et al ( 1 83) revealed that the sulfuric acid equilibrium vapor pressures were actually an order of magnitude lower than had been thought, invalidating many of the previous numerical conclusions. Already Doyle (108) noted that sulfuric acid forms hydrates (clusters containing one or more water molecules) in the vapor. Hydration stabilizes H2S04 vapor because the free energy of hydrate formation is negative, and it is therefore energetically more difficult to form a critical cluster from hydrates than from monomers. Reiss and coworkers ( 1 84, 185) studied the effect of hydration on nucleation, and laecker-Voirol and coworkers ( 1 86, 187) formulated a classical theory to calculate distributions of hydrates containing k acid and h water molecules. They also derived a correction to the classical free energy of cluster formation that depends on the ratio of monomer acids to the total number of acid molecules. It can


514


be calculated from estimated equilibrium constants of hydrate formation (1 86), or solved for numerically (1 88). Jaecker-Voirol & Mirabel derived an analytical expression for the classi cal nucleation rate as a function of temperature, relative humidity (RH), and relative acidity (RA) ( 1 89). However, the available vapor pressure data ranges over a factor of four, resulting in several orders of magnitude differences in calculated nucleation rates (190). Comparable differences are seen (depending on T, RH, and RA) when the extended Girshick theory is used instead of the classical theory ( 1 1 8) or when the kinetic formulation of Stauffer ( 1 24) [which yields at most one order of magnitude error in the sulfuric acid-water nucleation rate (1 27)] is replaced by more approximate expressions. Also, the theory of Doyle ( 1 08) predicts up to ten orders of magnitude higher H2S04 nucleation rates at tropospheric conditions than Wilemski's (103) revised theory (M. Kulmala, personal communication). Experiments on sulfuric acid-water nucleation have been carried out using an expansion chamber ( 1 92), a thermal diffusion chamber ( 193), and a turbulent mixing chamber (30). There is qualitative agreement between the diffusion chamber and the mixing chamber results, even though the former were obtained at RH > 1 and the latter at RH < 0.4 The mixing chamber experiments (30) revealed that classical theory predicts incorrect dependencies of nucleation rate on all three key variables. The experiments showed two to four orders of magnitude change in J per 5 K, a much stronger temperature dependence than anticipated from theory. The ratio between observed and theoretical (hydrated) nucleation rates (Jexp/Jth ) 2 varied between 1 0 - 1 and 101 3 over the whole measurement range. Jexp/Jth increased almost linearly with increasing predicted number of acid mol ecules in the critical nucleus. A separate issue, but nevertheless relevant to nucleation, is the value of H2S04 sticking coefficient. A study of H2S04-H20 aerosol growth involv ing photochemical sulfuric acid vapor production in a batch reactor sug gested that this quantity could be on the order of 0.04 ( 1 94). However, more direct experimental evidence is lacking. Rudolf & Wagner ( 195) studied sticking probabilities in the HN03/H20 system, measuring con densational growth of droplets with sizes between 0.5 and 1 2 /lm. They concluded that sticking probabilities of unity produced the best agreement with measured droplet growth rates. Nucleation of Sulfate Particles in the Atmosphere

Small sulfate particles are found in both the stratosphere and the tropo sphere. The stratospheric aerosol layer, discovered by Junge et al in 1 96 1 ( 1 9 6), i s mostly composed of aqueous sulfuric acid particles with a mean


NUCLEATION

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radius near 0. 1 11m ( 1 97). The background number concentration of the aerosol is around 1 cm- 3 , but large volcanic eruptions can cause that concentration to increase substantially until it reaches the background level again after 1-3 years. Background aerosols often incorporate small insoluble particles which may, for example, be of meteoric origin (1 98). However, Sheridan et al (199) analyzed stratospheric sulfuric acid aerosol particles after the June 1 991 eruption of Mt. Pinatubo and found that most showed no evidence of a solid or dissolved nucleus, suggesting that the volcanic H2S04 aerosol formed through homogeneous nucleation. Hamill et al ( 1 89) performed model calculations of sulfate aerosol for mation in conditions corresponding to the midlatitude stratosphere, allow ing for competition of homogeneous, heterogeneous, and ion-induced nucleation. Classical theories predicted no particle formation in the actual stratosphere, whereas in the upper troposphere the predicted homo geneous nucleation rates were high enough for new particle production. Yue & Oeepak (200) and Hamill et al (201 ) found substantial homo geneous nucleation rates at colder temperatures corresponding to polar stratospheres. Laaksonen & Kulmala (202) predicted that the stratospheric H2S04 vapor is largely hydrated, decreasing nucleation rates by several orders of magnitude; under polar conditions the rates were still high enough for considerable particle formation. At the coldest temperatures the critical nuclei became extremely small, often containing only one sulfuric acid molecule, which suggests that stratospheric particle formation may be molecular collision controlled. The formation of new sulfate particles in the troposphere has been studied actively in recent years. Much of the interest has been generated by the hypothesis which connects oceanic emissions of dimethyl sulfide (OMS) to Earth's climate (203). It is assumed that oxidation products of OMS and S02, mainly sulfuric acid and methanesulfonic acid (MSA), nucleate homogeneously and grow to act as cloud condensation nuclei (CCN). This changes the concentrations and size distributions of cloud droplets, thereby affecting cloud albedos and the global radiation budget. High concentrations of fresh particles have been measured far from obvious aerosol sources in Arctic regions (204), near clouds (205-207), in clear air in marine boundary layer (208, 209), in the upper troposphere (21 0), and at Mauna Loa, Hawaii (21 1). These observations are generally related to high RH and low concentrations of preexisting particles, and Covert et al have argued that background particle concentration is due to sporadic high production rates rather than continuous low source rates (208). Several model studies of tropospheric sulfate particle formation have incorporated oxidation chemistry, nucleation, and particle growth. The nucleation mechanism is often thought to be binary homogeneous


516


nucleation of H2S04 and H20. Kreidenweis & Seinfeld (21 2) showed that this system nucleates much more readily than the binary MSA-H20 system (although MSA vapor was predicted to enhance particle growth rates). van Dingenen & Raes (2 1 3) studied ternary nucleation of the above mentioned species using classical theory (ignoring hydration of H 2S04) and predicted that it is of minor importance in the marine boundary layer but may affect particle production in the smog chamber experiments involving DMS oxidation, such as those of Kreidenweis et al (2 14). They measured aerosol formation during the photooxidation of DMS and dimethyldisulfide in an outdoor chamber and compared the results to model calculations. The best agreement was seen with a sulfuric acid-water nucleation rate slightly less than that predicted by classical nucleation theory and a sticking coefficient of H2S04 somewhat below unity. (The sticking coefficient affects the results mostly through particle growth). However, the conclusions were limited because of uncertainties in vapor source rates. Raes et al (21 5) modeled photochemical H2S04-H20 aerosol formation in a batch reactor containing S02, NO, N02, propane, and propene. Uncertainties in sulfuric acid saturation vapor pressures and liquid activity coefficients

caused a l O-orders-of-magnitude uncertainty in the binary H2S04-H20 nucleation rate; the rates had to be at the upper limit of the range of theoretical values, substantially above those measured by Wyslouzil et al (30), to produce agreement with experiment. They speculated that the rates may be enhanced by catalysis from intermediate sulfate complexes that are later hydrolyzed to H2S04 or by ternary nucleation with unde tectable trace species such as HN03• The overall features, but not the details of sulfate aerosol concentrations in marine atmospheres can be explained by model calculations (21 6). Raes & van Dingenen (21 7) performed model calculations of aerosol formation from biogenic S02 in the marine boundary layer. Their predictions were severely limited by uncertainties in nucleation rate and sticking coefficients. However, they concluded that homogeneous nucleation can produce observed concentrations of small particles in the homogeneously mixed marine boundary layer and that clouds can enhance homogeneous nucleation. Ion nucleation was also predicted to increase concentrations of small particles. Kerminen & Wexler (21 8) suggested that the conditions most favorable to new particle production in the marine boundary layer correspond to rapid transitions from moderate to high RH during the daytime, typical, for example, to precloud conditions. Hoppel et al (209) compared model calculations with measured aerosol distributions and concluded that theoretical binary homogeneous nucleation rates together with a sulfuric acid sticking coefficient of 0.04 are not inconsistent with observations, but the experimental rates (30) are too sl ow to produce

NUCLEATION

517

enough particles. Nucleation on ions or preexisting embryos was offered as an explanation. Russell et al (21 9) modeled DMS-originating aerosol production in the marine boundary layer. They noted that uncertainty in nucleation rates and sticking coefficients is the major obstacle to veri fication of the link between DMS and CCN production.


Formation of Polar Stratospheric Clouds

The destruction of ozone in the polar lower stratosphere (220) is believed to be catalyzed by the surfaces of polar stratospheric cloud (PSC) particles. The PSCs can be divided into two classes: Type II clouds consist of water ice and form below the ice frost point (1ice, typically about 1 89 K depending on altitude), and type I clouds consist of nitric acid and water and form below about 1 95 K (22 1 ). Some type I particles (Ia) are large and aniso metric, while others (Ib) are smaller and more spherical (222). The stable form of nitric acid particles under stratospheric conditions is nitric acid trihydrate (NAT) (223). The Antarctic observations are consistent with a PSC formation mechanism involving NAT deposition onto frozen sulfuric acid particles once the temperature drops below the equilibrium tem perature TNAT (224). However, in the Arctic (where temperatures are generally not as low as in the Antarctic) substantial supercoolings relative to TNAT and supersaturations (SNAT) of nitric acid relative to the NAT equilibrium vapor pressure have been observed (225). This suggests that a nucleation barrier exists, either for freezing of binary H2S04-H20 or ter nary H2S04-HNOrH20 droplets or for deposition of NAT on frozen sulfuric acid aerosols. Sulfuric acid solutions of strato spheric composition can freeze to sulfuric acid tetrahydrate (SAT) at 4070 weight percent H2S04 or to sulfuric acid hemihexahydrate (SAH) from 35 to 45% . Water ice can freeze out below 40% . The equilibrium freezing temperatures of sulfuric acid solutions drop from 273 . 1 5 (pure water) to near 200 K at the ice-SAT eutectic and rise again to 245 K for the stoichiometric SAT solution. However, in the stratosphere, possible SAT particles will melt near 2 1 5 K as the vapor pressure of water shifts away from the SAT stability range (226). Hallett & Lewis (227) predicted that the maximum supercooling for the ice-forming solutions would be the same as the maximum supercooling of pure water, about 40 K. Steele et al (228), on the other hand, suggested that frozen hydrates might nucleate in stratospheric aerosols at around 200 K. Jensen et al (229) used classical nucleation theory to calculate freezing rates of ice-forming solutions and predicted that the freezing temperature for stratospheric sulfuric acid aerosols is always within 1 K of the ice frost point. Luo et al (230) used FREEZING OF SULFURIC ACID DROPLETS


518


classical nucleation theory to study the freezing of SAT. A difficulty in this approach is the determination of the surface energy between SAT and the mother solution and of the diffusion-activation energy across the phase boundary ( 1 55). They predicted that homogeneous nucleation of SAT will not take place in the stratosphere. MacKenzie et al (23 1 ) reached a similar conclusion using Turnbull's (232) empirical relation between nucleation rate and freezing enthalpy. The freezing of ice from dilute solutions was found to be in accord with the predictions of Jensen et al (229). Laboratory studies (226, 233-235) have confirmed that it is quite difficult to freeze SAT even out of bulk liquid solutions and thin liquid films. On the other hand, ice-forming bulk samples of H2S04 freeze readily at stratospheric temperatures. However, the role of heterogeneous nucleation remains unclear. The observations of Toon et al (236) of a sulfuric acid aerosol cloud after the eruption of Mt. Pinatubo suggest that volcanic aerosols containing relatively small amounts of HN03 might freeze near 1 90 K at an altitude near 20 km. On the other hand, Drdla et al (237) observed ternary liquid aerosols that did not freeze slightly below lice and argued that the absence of solid particles below the frost point may be due to a small bias in the temperature measurement or to a small error in the theory of Jensen et al (229). Laboratory studies offreezing of ternary H2S04-HN03-H20 bulk samples (234, 235, 238) indicate that no mixed hydrates of sulfuric acid and nitric acid form. The seeding experiments of Beyer et al (238) suggest that SAT will not induce the crystallization of NAT or vice versa. However, ice may induce the crystallization of ternary samples. Ternary samples seem to freeze more easily than binary sulfuric acid solutions at some compositions; however, atmospheric aerosol com positions may fall between the composition ranges that freeze most easily (235). MacKenzie et al (23 1 ) predicted that the freezing of NAT from a stoichiometric binary solution would not take place at stratospheric temperatures. They suggested that NAT forms more easily than SAT in laboratory because the NAT lattice is more compatible with impurity surfaces. Carslaw et al (239) applied the freezing rates of Beyer et al (238) to stratospheric droplet volumes and found that freezing would not take place at 1 95 K. They concluded that stratospheric droplets in the Arctic will remain liquid most of the time, whereas Antarctic winter temperatures are probably low and persistent enough for most of the particles to freeze. FREEZING OF TERNARY DROPLETS

Toon et al (240) studied heterogeneous gas-phase nucleation of ice on frozen aerosols in type II PSC formation. For contact parameter values below one, rapid cooling rates led to nucleation of all aerosols, but at slow cooling rates the sizeHETEROGENEOUS NUCLEATION OF ICE AND NAT


NUCLEATION

519

dependentf(m) factor gave preferential nucleation o f the largest aerosols, leading to low concentrations of relatively large particles in accord with atmospheric observations. They concluded that the value of m is close to 0.95. Wofsy et al (241 ) studied the heterogeneous nucleation of NAT on frozen aerosols using a one-component nucleation theory with SNAT as a variable and a NAT surface energy estimated by Poole & McCormick (242). They found that cooling rates below 1 K per day are required to limit the number of nucleated particles. Reasonable supercoolings and supersaturations were seen when the contact parameter was between 0.86 and 0.98. Peter et al (243) used estimates for various surface energies to predict that the contact parameter between ice and NAT should be below 0.94 and that between NAT and SAT below 0.88. Arnold (225) presented observational evidence of supercoolings up to 5 K relative to TNAT, cor responding to saturation ratios of HN03 up to 1 6. He pointed out that the one-component heterogeneous nucleation theory used by earlier inves tigators for NAT was too simplified and derived a correction that required a contact parameter between 0.88 and 0.95. MacKenzie et al (23 1) derived new values for NAT and SAT surface energies, making use of liquid-solid surface energies from Turnbull's empirical relation (232). Using these energies, they showed that the contact parameter between NAT and SAT is well below the value needed to produce substantial classical nucleation rates at 1 92 K. They also predicted that heterogeneous nucleation of nitric acid liquid solution [suggested earlier by Laaksonen et al (244) and Tabazadeh et al (245)] would not take place. Laboratory studies on the subject are few so far. Iraci et al (246) exposed thin SAT films to water and nitric acid vapors and detected NAT growth. However, the saturation ratio of HN03 was on the order of hundreds. In contrast, Marti & Mauersberger (247) used stratospheric H20 and HN03 pressures to study nucleation on clean glass surfaces. They detected an intermediate phase before NAT formation, consisting of one part nitric acid and five to six parts water. It is unknown whether this phase consisted of hydrates or solid or liquid solution. Finally, Middlebrook et al (248) detected probable surface roughness in NAT films grown by vapor deposition and suggested that the morphologies of NAT particles formed by freezing might differ from those formed by condensation. ACKNOWLEDGMENTS

This work was supported by the National Science Foundation (grant CHE 9422999) and the Petroleum Research Fund of the American Chemical Society (grant 26950-AC9). Support to AL from the Academy of Finland and to VT from the Facultad de Quimica and DGAPA at UNAM is also gratefully acknowledged.

520


& OXTOBY

Any Annual Review chapter, as well as any article cited in an Annual Review chapter, may be purchased from the Annual Reviews Preprints and Reprints service. 1-800-347-8007; 415-259-5017; email: [email protected]


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