The Problem of Measuring Homogeneous Nucleation Rates and the

7748

J . Phys. Chem. 1994, 98, 7748-7758

FEATURE ARTICLE The Problem of Measuring Homogeneous Nucleation Rates and the Molecular Contents of Nuclei: Progress in the Form of Nucleation Pulse Measurements R. Strey,' P. E. Wagner,' and Y. Viisaned Max-Planck- Institut fur Biophysikalische Chemie, Postfach 2841, 0-37018 Gottingen, Germany Received: March 23, 1994; In Final Form: May 27, 1994"

This articles focuses on the very difficult but increasingly important problem of measuring nucleation rates as opposed to only the onset conditions or limits of metastability. Past efforts are reviewed, and recent progress is described including a dramatic new advance consisting of a technique developed by the authors in recent years. This is a nucleation pulse method for experimental study of homogeneous nucleation in vapors. We highlight the specific features of the experimental setup and procedure allowing time-resolved measurements of nucleation rates J i n the range lo5 < J / c ~s-l< - ~lo9 as functions of supersaturation S and temperature T . Homogeneous nucleation is induced in an expansion chamber by a single pressure pulse. Pressure, temperature, and vapor supersaturation are maintained uniform and practically constant during the nucleation period of about 1 ms defining the nucleation pulse. The number concentration of nucleated droplets as well as the condensational growth following the nucleation pulse is monitored by light scattering, permitting one to discriminate heterogeneously from homogeneously nucleated droplets. We illustrate the proper function of the technique with new measurements of homogeneous nucleation rates for 1-butanol from the vapor phase in the temperature range 225 < T / K < 265. The growth of the 1-butanol droplets formed was observed to be comparatively slow, ensuring well-defined constant conditions during the nucleation pulse. Furthermore, at the temperatures considered the equilibrium vapor pressure of 1-butanol is sufficiently high to permit its precise experimental determination. Since from past measurements wealso know the surface tension of 1-butanol with sufficient accuracy, a quantitative comparison of the experimentally obtained nucleation rates with the predictions of theoretical models can be performed. The new feature for expansion chambers is that the individual J-S curves are obtained at selectable constant temperatures. From the slopes of these isothermal J-S curves, the molecular content of the critical clusters can be determined without reference to any specific nucleation theory. This allows a direct experimental test of the Gibbs-Thomson (or Kelvin) equation for very small droplets.

Where you can measure what you are speaking about and express it in numbers, you know something about it, and when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind." Lord Kelvin' 1. Introduction

The kinetics of first-order phase transitions, such as crystallization, melting, vapor phase condensation, boiling, solid-state precipitation, glass and polymer crystallization, and crystal formation from solutions and binary phase separations, proceeds through nucleation, growth, and aging stages. The theoretical description of the entire process is a formidable task. Therefore, it is advisable to decouple the processes and to look for "simple systems". Studying these quantitatively, one can hope to learn how to describe the more complicated systems. Fortunately, in all cases the new phase forms within a mother phase, so that the theoretical descriptions are similar. The structure of the new phase (Le., particle number density and size distribution) is determined by many parameters, most importantly, however, by the first step, the nucleation event. Due to the technical and ~

* To whom correspondence should be addressed.

~~

Permanent address: Institut for Experimentalphysik, Universitst Wien, Boltzmanngasse 5, A-1090 Wien, Austria. t Permanentaddress: Finnish Meteorological Institute, Sahaajankatu 22E, SF-00810 Helsinki, Finland. 0 Abstract published in Advance ACS Absrracfs, July 1, 1994.

environmentalrelevance of disperse phases, there is broad interest in understanding the details of nucleation processes. Generally, the vapor phase offers the simplest experimental conditions. In his pioneering investigation on water vapor with expansion chambers, C. R. T. Wilson almost 100 years ago2 qualitatively observed the two essential features of nucleation experiments to be discussed in this article: He saw the onset of the condensation process and the associated light scattering phenomena. The advance we can report here is that nowadays one can measure with modern technique pressures and scattered light fluxes. Today, a quantitative interpretation of the latter is possible according to the Lorenz-Mie theory3v4which was not available at his time. Further, it deserves to be mentioned that Wilson was the first researcher to deny the necessity of condensation nuclei for condensation to occur, as F. Becker stated in 1912.5 The fundamental observation was that in sufficiently supersaturated vapors spontaneous density fluctuations can act as condensation nuclei. One refers to this kind of nucleation as homogeneous nucleation [for reviews see refs 6-1 11, otherwise as heterogeneous nucleation. Understanding the nature of the fluctuations in homogeneous nucleation has remained a theoretical and experimental challenge. Experimentally,by confining nucleation to a short time interval and determining the number density of nucleated particles, one can obtain the nucleation rate. Nucleation pulses so defined are a special feature of the expansion chamber described below. Inadvertently, again Wilson was the first to apply a nucleation pulse in the vapor phase, because of the specific construction of

0022-3654/94/2098-7748$04.50/0 0 1994 American Chemical Society

Feature Article

The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7749

his chamber.2 The deliberate application of nucleation pulses is a well-known procedure in metal physics or for studies of phase transitions in melts. Contemporary with Wilson, G. Tammann experimentally studied the nucleation p " s in melts of organic substances.69'2 Organic liquids, such as borneol or camphor, were cooled to temperatures far below the respective melting points. After a certain time interval the samples were heated again to temperatures closeto, but still below, the melting temperature. By cooling the system nucleation was initiated; the nuclei, however, forming at a steady rate were unable to grow substantially because of the low mobility of the molecules. Bringing the samples close to the melting temperature, still somewhat undercooled, caused quenching of the nucleation process, but the nuclei still could grow to crystals of macroscopic size and were counted. Thus, Tammann already applied nucleation pulses to the systems. Generally, the most transparent experimental results are obtained from studies of homogeneous nucleation in supersaturated vapors. The necessary supersaturations are generated by nonisothermal diffusion currents in ~taticl3-'~ or steady-stateflow 18 diffusion cloud chambers, by turbulent mixing,19 or by adiabatic expansionchamber~,?.2~3~ shock t ~ b e s , 3 or ~ Jsupersonic ~ nozzles.40.41 In order to facilitate a unique interpretation of the experimental results and a direct test of nucleation theories, it is important to observe steady-state nucleation at uniform and constant thermodynamic conditions. For steady-state nucleation to occur it is required to decouple nucleation from subsequent growth of the condensingparticles and thus prevent self-quenching of the nucleation process. A description of the complex phase transition process involving simultaneousnucleation, growth, and aging processes, if decoupling is not performed, has been given, for instance, by Kahlweit42.43 and by Langer and S c h w a r t ~ . ~ ~ Decoupling can essentially be achieved either by removing the growing particles from the sensingvolume rapidly after nucleation or by terminating thenucleation process deliberately after a short, well-defined time interval before substantial droplet growth had occurred. In diffusion chambers the growing droplets are removed by gravitational sedimentation or by convectiveflow, and practically constant steady-state conditions are achieved at low nucleation rates. However, nonlinear temperature and vapor pressure profiles are present in diffusion cloud chambers leading to substantial nonuniformitiesof temperature and supersaturationsin the sensing volume. Accordingly, it is difficult to assign single values of supersaturation and temperature to the nucleation rateobserved. Nevertheless,many useful studies using diffusioncloud chambers have been performed (refs 13-17 and references therein). The steady-state nature of the nucleation process can convincingly be seen in a diffusion cloud chamber. In expansion chambers, on the other hand, nucleation occurs at uniform conditions in the sensing volume, but generally selfquenching will occur. In order to prevent self-quenching and to ensurepractically constant conditionsduring the nucleation period in expansion chambers, the nucleation can be terminated by a slight recompression after the adiabatic expansion process. Steady-state nucleation is thus occurring during a short pulse of selectable duration of the order 1 ms, the nucleation pulse. The time dependence of the supersaturation during a single nucleation experiment is schematically represented in Figure 1. One starts with a saturated or a somewhat undersaturated system (S I 1). At first the supersaturation is increased by adiabatic expansion. Subsequently, it is kept constant for a desired time interval corresponding to the nucleation pulse. The nucleation period is terminated by a small adiabatic recompression inducing a slight reduction of supersaturation. At excessive nucleation rates and/or droplet growth rates a substantial vapor depletion could already occur during the nucleation pulse; however, this can experimentally be avoided. Well after the recompression the

nucleation pulse

1-

.-s

s

1

1

t

4 v,

17

1Srl

01 0

c

time

Figure 1. Schematic diagram showing the time dependence of the supersaturation in the expansion chamber during a single nucleation experiment. The experimentis generallystarted at well-defined somewhat undersaturated conditions. Adiabatic expansion leads to vapor supersaturation, and nucleationoccurs during the time interwalof the nucleation pulse. After the nucleation process is quenched by recompression, condensational drop growth causes further reduction of the supersaturation.

supersaturation decreases further due to droplet growth. According to the described procedure, nucleation occurs at welldefined supersaturation and temperature. By virtue of the shortness of the nucleation pulsegenerally rather high nucleation rates can be considered. It is important to note that the calculations of vapor supersaturations (vapor phase activities) and temperatures in expansion chambers are quite straightforward as long as the vapor partial pressures are small compared to the total gas pressure in the chamber. Hence, experimental errors due to insufficient knowledge of the thermodynamical properties of the condensable vapors are minimized. The nucleation pulse method, based on essentially the same principle as employed in Tammann's early studies, was employed in a two-piston expansion chamber developed by Wagner and Strey28in 1979. With this chamber nucleation rates up to 109 cm-3 s-l could be studied as a function of supersaturation. However, due to the operation characteristics only nonisothermal J S c u r v e s were obtained. A detailed description of the chamber was given in 1984 in connectionwith measurements for n-nonane.N In 1986 experimental data on the homologous series of alcohols followed.3' The number density of droplets was measured by constant angle Mie scattering (CAMS)4Sutilizing characteristic light scattering patterns for growing droplets related to the color effects seen already by Wilson. It is appropriate to mention that Kassner and co-workers22 developed in 1965 a single-piston expansion chamber at the University of Missouri in Rolla, exploiting for the first time the nucleation pulse idea for the vapor phase. Using this chamber, Miller23947 measured water nucleation in 1976. Starting with a well-defined saturated vapor, increasing supersaturations were achieved by increasing the expansion ratios, similarly as later by Wagner and Strey.28J'J However, as already mentioned, this procedure only yields J S curves that are nonisothermal. The chamber was further developed by S ~ h m i t t .Schmitt ~~ and coworkers measured nucleation rates for ethanol,25 toluene,Z6 and n-11onane.2~ The nucleated droplets were photographed and then counted. The pressure pulse generated was not flat but parabolically shaped. It was numerically integrated, and an effective time interval during which nucleation occurred was determined. Using this interval, the drop counts were converted to nucleation rates. We generally found good agreement with nucleation rate data obtained in Rolla.Z8J6 The effort to improve the measurement of nucleation rates continues, and as an important part of this feature article we describe below a dramatic improvement in technique that we have been able to developduring the past few years. The advance includes a further development of the two-piston chamber into a chamber featuring two rapidly operating valves. The basic

Strey et al.

7750 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994

incentive to reconstruct the two-piston chamber came from theoretical work of Kashchiev,46 who drew attention to the fact that isothermal J - S curves yield the number of molecules n* in the critical cluster. That is, a direct measurement of the molecular content of the nuclei is obtained from the experimental nucleation rate data without recourse to any specific nucleation theory permitting a direct test of the Gibbs-Thomson equation for very small droplets. We will present such a test toward the end of this paper. The advantageous features of isothermal J S curves have been utilized already by Viisanen et al. in 1993 reinvestigating water nucleation.36 Furthermore, we applied the technique to a series of binary systems. First, in order to facilitate the complex interpretation and to test the evaluation procedure, a simple nearly ideal binary system ethanol-1 -hexanol was a n a l y ~ e d and 3 ~ then the nonideal system ~ a t e r - e t h a n o l .The ~ ~ nucleation rates for the water-propanol system were recently presented in a review article in comparison to previous literature results.33 In addition, the water-n-nonane system was ~ t u d i e dexhibiting 3~ a pronounced miscibility gap in the liquid state. In this paper we confine ourselves to a one-component system and demonstrate with new measurements of homogeneous nucleation rates for 1-butanol the proper function and features of the nucleation pulse chamber. The choice of 1-butanol is motivated by the observation during past measurements3’ that 1-butanol is a well-suited substance for condensation experiments. On the one hand, it has a sufficiently high vapor and on the other hand, the droplet growth is sufficiently slow. Yet, it is a substance that differs by a special feature from substances like water or n-nonane that have been studied numerous times before: The 1-butanol molecule contains a polar and nonpolar part, likely to display surface orientation effects (ref 8, Chapter 8.5). Such effects might play a role in the dynamic process of nucleus formation.

energy AG is calculated assuming macroscopic surface tensions and densities. Clarification of the surfate free energy and the properties of the clusters has remained an unsolved problem in nucleation theory. The nucleation rate J can be expressed in terms of the formation free energy AG* of critical clusters, Le., clusters in unstable equilibrium with the surrounding vapor, by means of a Boltzmann-type expression

J = K exp(-AG*/kT)

(2)

where the kinetic prefactor Kdepends mainly on the impingement rate of vapor molecules and the effective surface area A of the critical cluster. In the classical theory by Becker Ddring66from 1935

K=

(3)

and AG* = -n*Ap

+ uA

(4)

where

n* =

3 2 w m2u3 3Ap3

and

P”

AN= kTln-=kTlnS

(6)

PVC

a, pv, pvo vm, and m are surface tension, actual vapor pressure, equilibrium vapor pressure, molecular volume, and mass, respectively. n* denotes the number of molecules in the critical cluster. The classical theory has the important advantage that 2. Theory it allows to calculate sizes of critical clusters and nucleation rates A comprehensive theoretical description of nucleation processes for arbitrary systems by only using handbook values of the could provide insight into the detailed process leading to the corresponding substance properties. formation of nuclei of the new phases and allow quantitative The Gibbs-Thomson equation (eq 5) accounts for the depenpredictions of the nucleation rates for various practically important denceof the equilibriumvapor pressure on curvature. Obviously, >plications. In a satisfactory theoretical approach any ad hoc the macroscopic classical concept breaks down for clusters of assumptions about the actual structure of the clusters formed only a few molecules, and a sufficiently accurate description for should be avoided, and a consistent description of the clusters clusters is not achieved. This has led several authors to reconsider should be provided for all cluster sizes down to transient dimers the macroscopic classical model. and even single molecules. Such a description is still lacking Modifications of the Classical Theory. Using a modified despite the attention the theory of homogeneous nucleation has treatment of nucleation kinetics, Girshick and ChiuS7s8 have received during the past decades. Although various theoretical obtained an additional factor 1 / S in the classical expression for developments have been reported in recent years,49-63~70~7k77~80 it the nucleation rate, where S is the vapor supersaturation. A is notable that the classical theory formulated by Volmer and similar result had earlier been obtained by Courtney68and Blander Weber,64 Farkas,65 and Becker and Ddring,66as it is described and K a t ~ . 6In~ addition, Girshick and Chiu attempted to achieve in considerable detail by Volmer6 in 1939, is still among the most self-consistency of the classical theory.57 To this end they successful models for quantitative prediction of nucleation suggested an ad hoc adjustment of the expression for the free phenomena. Several authors have presented advanced treatments energy AG of cluster formation yielding AG = 0 for the case of in order to avoid some of the idealizing assumptions of the classical monomers. This results in a further factor ee in the expression theory. A useful review was recently provided by Oxtoby.’’ for the homogeneous nucleation rate, so that the nucleation rate Quite generally, the formation free energy of clusters in according to Girshick and Chius’ is given as supersaturated vapors is expressed in terms of bulk and surface contributions ee (7) JGC = ~ J B D AG = -AGb AGs (1) where where the bulk term AGb corresponds to the free energy gain of the condensing molecules, which drives the transition. The surface 8 = (36?r)’/’~,~/~u/kT (8) term AGs corresponds to the free energy loss by forming the surface of the cluster. is a dimensionless surface tension and JBDis the classical nucleation Classical Theory. In the classical work by Volmer,6~6~ Beckerrate. In this connection it should be pointed out that the factor Ddring,66 and Zeldovich,6’ the clusters are viewed as stationary e e l s , shown in eq 7,appears already in an expression derived in spherical liquid drops with macroscopic properties a t isothermal 1939 by Volmer [ref 6, p 125 (eqs 44 and 45)]. Volmer has conditions. In this classical theory the cluster formation free provided even a numerical estimate for this factor in agreement

+

Feature Article

The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7751

with the more recent ca1c~lations.s~ Furthermore, Volmer6 has discussed the influence of the condensation coefficient for the smallest clusters on the steady-state flux of nuclei. He has argued that incomplete energy dissipation during collisions of single molecules and dimers will lead to reductions of the nucleation rate approximately compensating the factor ee/S. This effect, however, he expected to depend on the structureof the condensing molecules.6 In this connection influence of the inert carrier gas on the nucleation process might deserve further consideration. Here, also recent work by Bauer and Wilcox considering inert carrier gas effects should be m e n t i ~ n e d .The ~ ~ usefulness of these approaches has to be tested by comparison to experimental data. A scaling model of homogeneous nucleation based on the classical theory has been derived by Hale.70 An important assumption in this model is a general linear dependenceof surface tension on temperature. The scaling model has the attractive feature that it provides a universal description of nucleation for arbitrary substances and allows simultaneous comparison of different experimental results. On the basis of Fisher’s drop Dillmann and Meier74-76 have introduced a new semiphenomenological approach, where the free energy of cluster formation is expressed in the form

+

AG = ~ , 8 n ~ ’ TkT ~ In n - kTln q,, - nkT In S

(9)

They try to account for the curvature dependence of the surface tension with prefactors

In this approach the cluster distribution and consequently the density of the vapor considered are expressed in terms of the free energy ACof cluster formation. Detailed quantitative comparison with the equation of state of the vapor, requiring the knowledge of the temperature-dependent secondvirial coefficient, then yields the unknown parameters 7 , 40, al, and CYZ in the expression for AG, and thus an expressionfor the nucleation rate can be obtained. Similarly as for the classical theory, this semiphenomenological approach allows one to calculate nucleation rates by only using handbook values for the corresponding substance properties. Although some of the parameters in the expression for AC are determined from critical point properties, generally the DillmannMeier approach appears to provide a good approximation for the free energy of formation of clusters far away from the critical point. Good agreement with experimentaldata has been observed in several cases.36J6 Recently, Ford et pointed out an inconsistency within the Dillmann-Meier approach, which is connected to the description of the imperfect behavior of a real vapor. In order to provide a proposed consistent treatment of real vapor behavior, Ford et a modification of the Dillmann-Meier theory. This revised approach, however, appears to be less successful in explaining experimental data. Granasy62 has recently presented an analysis of the interfacial enthalpy and entropy profile for small clusters. Comparing published nucleation rate data to the predictions of his model, he found improved agreement of the modified classical nucleation theory for nonpolar substances. For polar alcohols a further modification accounting for dipole orientation was concluded to be necessary. Kalikmanov and van D ~ n g e nrejected ~~ eq 10, but rather combined Fisher’s droplet model with the ee/S factor. They found improved agreement of experimental rates for some nonpolar substances with the classical theory so modified. As pointed out earlier, the classical nucleation theory applies to isothermal systems. Accordingly, the clusters are assumed to remain at constant temperature while the energy liberated during cluster formation is assumed to be instantaneously removed by

collisions with carrier gas molecules. Nonisothermal effects in the theory of homogeneous nucleation have been considered by K a n t r o ~ i t zand ~ ~ by Feder et ~ 1 . 7Recently, ~ Barrett et a1.80 have studied the influence of fluctuations of the energy of clusters in a nucleating system in thermal equilibrium with an inert carrier gas. It has been predicted that these energy fluctuations will cause some dependence of the nucleation rates on pressure and nature of the carrier gas. As will be shown below, however, significant dependence of nucleation rates on the nature of the carrier gas has not been observed in the present study, nor could a significant pressure dependence be 0bserved3~181in expanding vapors. In diffusion cloud however, the nucleation rates appear to depend on the carrier gas pressure. Microscopic Approaches. The limitations inherent in the classical theory and its modifications could be avoided by using fully microscopic methods to describe the detailed process leading to nucleation. The results of numerical simulations depend on the actual definition of molecular clusters. In early studies83184 clusters were defined consideringdistancesbetween two molecules or between one molecule and the center of mass of a group of molecules. More recently, Reiss and co-workersS6pointed out that a proper definition of “physical clusters” is essential for the applicationof a molecular model. Their cluster definitioninvolves the cluster volume in addition to the number of molecules. Unfortunately, even with presently available fast computers it is impossible to perform detailed simulations of nucleation events under realistic conditions. Furthermore, it turns out that the predictions by microscopic simulations are sensitively dependent on the molecular interaction potentials, which are generally not known with sufficient accuracy for the systems usually studied experimentally. Thus, quantitative calculationsof nucleation rates for practically important applications cannot be provided so far by fully microscopic methods. Cahn and Hilliarda5introduced a density functional approach for describing nucleation. Recently, Oxtoby and co-workers53@J extended the density functional approach to perform comparisons with experimentalobservations. Using a Lennard-Jonespotential, ratios of classical to nonclassical nucleation rates were obtained, and a similar behavior was found as observed experimentally. For quantitative comparisons with experimental data more realistic interaction potentials would have to be considered. The problemswith applicationof nonclassicaland microscopicmodels, as mentioned above, to quantitatively predict the actual nucleation behavior of realisticsystemshave made it difficultso far to perfbrm the corresponding experimental tests. Measurement of Molecular Content of Critical Clusters. As pointed out above, isothermal nucleation rate curves allow the determination of the number of molecules in the critical cluster n*

While this is obvious within the framework of classical theory, as noted already by Nielsen,86 more general arguments have been given.6 An analysis based on the statistical mechanics of flu~tuations36.~~ led to the conclusion that eq 11 actually yields the excess number An* of molecules in the critical cluster over that present in the same volume before the fluctuation occurred, rather than the total number n* of molecules in the cluster. This result has recently been confirmed using a different approach.*8 For the case of formation of liquid (or solid) clusters in a dilute vapor, however, An* is to a very good approximation equal ton*. Accordingly, in this case eq 11 allows one to determine the total number of molecules n* in the critical cluster without reference to any specific nucleation theory, if only the nucleation rate is known as function of the vapor pressure at constant temperature. Due to the dependence of the kinetic prefactor K on the vapor pressure, the value of n*, as obtained by eq 11, will actually be

Strey et al.

7752 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 pressure

chomber volume socket for pressure transducer

support for tetlon membrane

volume

expansion membrone tef'on valve exponsion volume

for exponsion volume

Figure 2. Cross section of the expansion chamber showing the main design features to scale. Expansion and recompression valve are released by a rotating-trigger unit in a well-defined time sequence.

overestimating n* by about 1 or 2 m o l e c ~ l e s . ~In~ conclusion, ~~6 it should be emphasized that according to eq 11 measurements of homogeneous nucleation rates at constant temperature as a function of supersaturation allow obtaining directly the molecular content of the nuclei and thus allow a direct quantitative test of the Gibbs-Thomson equation for droplets as small as 30 molecules or fewer.

3. Experiment Experiments on homogeneous nucleation from vapors are generally performed in two basic steps: (A) preparation of particle-free supersaturated vapor and (B) detection of the nucleation process. For preparing well-defined vapors or vapor mixtures the following requirementsareessential: (Al) Supersaturatedvapors are generated in a simple and reproducible process allowing straightforward calculations of partial vapor pressures and temperature without requiring data on equilibrium vapor pressures, etc. (A2) Contaminations are excluded from the system to the extent that significant heterogeneous nucleation is avoided. (A3) Vapor pressures and temperatures can be selected independently. Hence, particularly nucleation rates can be measured us supersaturations at fixed temperature facilitating direct comparisons to corresponding theoretical predictions as well as toother experiments and allowing to determine isothermal slopes. (A4) Vapor mixtures can be obtained regardless of the miscibility of the condensing liquids. (A5) Vapor pressures and temperature are uniform over the considered volume. (A6) Vapor pressures and temperature are constant during the observed nucleation process. The detection of the nucleation process can generally not be performed directly, but only droplets growing subsequent to the formation of critical clusters can be observed. In order to facilitate a unique interpretation of nucleation experiments, the following requirements for the detection of nucleation should be fulfilled: (Bl) The observation process has negligible influence on temperature and supersaturation in the nucleating system. (B2) The interaction of nucleation and subsequent condensational growth is negligible. (B3) Steady-state conditions prevail during nucleation. (B4) The number of nucleated drops is measured soon after the nucleation process in order to avoid loss of droplets, e.g., by coagulation. (B5) The number of nucleated droplets is measured by an absolute method allowing absolute determination of the nucleation rates. (B6) The detection method allows to discriminate heterogeneous from homogeneous nucleation. In recent years we have developed an advanced version of an expansion chamber for the study of homogeneous nucleation by means of the nucleation pulse method. As described in the

following, this measurement system allows to meet all of the above-listed requirements to a good approximation. The Nucleation Pulse Chamber. The nucleation pulse measurements of the rate of homogeneous nucleation were performed using an expansion chamber with a cross section shown in Figure 2 featuring the essential parts on a quantitative scale. The actual measuring volume of the expansion chamber amounts to about 25 cm3. The chamber described in this paper differs from the previous two-piston chamber described earlier30 by two essential features. First, the pistons are replaced by valves connecting additional volumes to the chamber. This has the advantage of producing smoother pressure pulses with higher reproducibility. Second, vapors and carrier gas are now premixed in a separate plenum volume before the desired amount of the vapor/carrier gas mixture is transferred to the expansion chamber. The vapor supply unit is shown in Figure 3. The advantage with respect to the previous procedure is that the vapor fraction w in the vapor-carrier gas mixture is experimentally fixed by corresponding preparation of this mixture in the plenumvolume. Therefore, the actual partial vapor pressure pv in the measuring chamber before expansion can be directly determined from measurement of the initial total pressure po in the chamber by

(see requirement Al). This allows to vary the actual initial vapor pressure pv and thus the achieved supersaturation ratio by corresponding variation of the initial total pressure po in the chamber while keeping the expansion ratio unchanged. Previously, we had to rely on the equilibration of the vapor over a liquid pool inside the chamber. Therefore, it is now possible to measure the nucleation rate at constant temperature as a function of supersaturation (see requirement A3). The second feature, furthermore, allows vapor mixtures to be studied where the corresponding liquids exhibit a miscibility gap, e.g., water and alkanes or long-chain alcohols (see requirement A4). Some details of the expansion chamber, experimental system, and procedure will be described in the following. The Vapor Supply Unit. The schematic diagram of the measuring system, as shown in Figure 3, illustrates the relative arrangement of the components. A thermostated acryl-glass box contains the receptacle R with a volume of about 5000 cm3 and the vaporizers V1 and V2 connected by stainless steel tubings and a manifold of valves V, and needle (metering) valves N,. The pressure tranducers are Baratrons (MKS Instruments, Inc.) with full scale ranges 725 kPa (PI), 1.45 kPa (Pz), and 145 kPa (P3). The flow from the receptacle to the chamber can be regulated by an electronically controlled regulation valve R, keeping the

The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7753

Feature Article

homoneneous nucleation

in f: .-M

: I

, 0

Vback

10

2 v, 20

RP

Figure 3. Schematic diagram of the measuring system: G1, G2, compressedinert gases;V1,V2,vaporizers; R, plenum volume containing the gas mixture considered; CH, expansion chamber; DP, RP, vacuum pumps; P1, P2, P3, precision pressure transducers; Rv, electronically controlled regulationvalve ensuringconstant pressure (P3) during flushing of the expansion chamber; V, N, valves (see text). pressure in the tubing and the expansion chamber during the flushing and filling procedure at a constant value monitored by the pressure transducer P3. The vaporizers are separately thermostated to temperatures above the chamber temperature to supply sufficiently high vapor pressures in order to approach the saturation limit in the receptacle R as closely as required. Before the start of an experiment the whole system is evacuated, the receptacle reaching about 2 X 10-4 Pa (see requirement A2). Then the vaporizers V Iand V2 are partly filled with the liquids to be studied,dissolved gases being removed by thevacuum pumps. Next the vapor (or, vapors, if binary nucleation is studied) from the vaporizer(s) is (are sequentially) admitted to the receptacle R, and then filtered carrier gas (usually a noble gas, e.g., argon) is added in excess (see requirement A2). The pressures of the gases admitted are monitored by the pressure transducers PI and P1. This procedure allows a precise setting of the vapor fraction(s) of the vapor(s) considered. The vapor(s) and the carrier gas are allowed to mix in the receptacle. Then the expansion chamber is filled with the vapor/carrier gas mixture. Part of the vapor@) tends to adsorbon the wallsofthesystem. Therefore,thechamber, the tubings, and the receptacle are pretreated by introducing vapor with the approximate vapor pressure to be considered and letting it come into contact with the walls for at least 15 min. Furthermore, the chamber is flushed at a well-defined selectable pressure which is kept constant by means of the regulation valve R, in connection with a regulator electronics type 250 (MKS Instruments, Inc.). The flushing time is typically 5 min, sufficient to flush with at least three chamber volumes. While flushing the recompression valve V, is closed and the recompression volume (see Figure 2) is evacuated via V,, so that the Teflon membrane rests against the support. Furthermore, the back-pressure required for recompression is set in the pressure reservoir (see Figure 2) and finely adjusted by the pumpvolume Pb. The carrier gas-vapor mixture flows through the open inlet valve Vi,,, the chamber, the open expansion valve Vcxp,and exit valve V,. The flushing speed is determined by the needle valve Ne. Flushing is terminated by closing Ne and R,. Then valve Vi, is closed, and the pressure is finely adjusted by the pump volume P, and read from P3. This pressure is the initial total pressurepo determining the vapor saturation ratio in the chamber (see below). After closing the expansion valve Vcxprthe pressure in the expansion volume (see Figure 2) is set and finely adjusted by the pump volume P,. Finally V, and V, are closed. The Pressure Pulse. With the pressures set in the respective volumes, determining the amount of expansion and recompression in the chamber, subsequent expansions can be performed. For that purpose the expansion valve and the recompression valve are opened in a well-defined time sequence by means of a specially

30

40

50

60

t [msl

Figure 4. Result of a typical single homogeneousnucleation experiment showing the total pressure and the light flux scattered at an angle of 0 = 15' normalized with respect to the transmitted light flux as functions of time. Significant light scattering is observed only ufter the nucleation pulse has occurred.

51

45

-

10

11

12

13

14

15

time [ms]

Figure 5. Detailed presentation of a typical single nucleation pulse illustrating the method of pulse evaluation applied. By averaging over the central portion (open squares)of the nucleationpulse, the experimental

pressure drop Apw is obtained,linear interpolationsyield the experimental duration A?exptof the nucleation pulse.

designed rotating trigger unit, which limits variations of the nucleation pulse duration to less than 50 ps. The pressure in the expansion chamber is measured as a function of time during the pressure pulse by means of a piezoelectric pressure transducer (Model 7031, Kistler Instruments). The expansionis performed within a few milliseconds. A typical pressure pulse is shown in Figure 4 along with the scattered light signal (see below). The expanded, supersaturated state, during which nucleation occurs, is maintained for about 1 ms. Then a small recompression is performed to quench the nucleation process. The rate of the pressure drop depends on the viscosity of the gases and the heat conduction to the chamber walls. Since we are able to control the pressure in the expansion volume and the time sequence of opening the valves, pulses of desired depth and length can be produced. The reproducibility, a very important feature, has been demonstrated for different carrier gases recently.36 In that paper it was also indicated how the pressure pulse is evaluated by a trapezoidal construction. In the following we describe the pulse evaluation in some detail. The time dependence of the total pressure drop Ap during a typical expansion pulse is shown in Figure 5 over a time interval just encompassing the nucleation period. As can be seen from Figure 5 , the pressure drop increases at first roughtly linearly with time. Subsequently Ap exhibits a plateau, during which it remains approximatelyconspnt. Finally, a fairly linear decrease of Ap is observed. According to the wellknown steep decrease of the nucleation rate with decreasing supersaturation, it can be expected that noticeable nucleation rates are only observed for Ap's within a few percent of the plateau value. Accordingly,most of theobserved nucleation processoccurs

1154 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994

during the pulse plateau at well-defined and practically constant temperature and supersaturation (see requirement A6). It is useful to characterize each experimental nucleation pulse by a plateau value Apcxptof the pressure drop, by the plateau duration Atcxptrand by the total pulse length Atc, during which Ap remains above Apc,which is by a selectable percentage (usually 2%)lower than Apcxpt.The plateau value Apexpt is calculated by averaging Ap over the central portion, typically about 20 data points (corresponding to the open squares in Figure 5 ) , of the pulse plateau. Thenoise level in the plateau region can becharacterized by the corresponding relative standard deviation of Ap, which usually remains below 0.2%. The plateau duration time interval Atexptis calculated by fitting straight lines through the increasing and the decreasing portions of the expansion pulse within the time interval Atc. Atc thus serves as an upper bound for Atexpt as can be seen in Figure 5 and is typically about 30% larger than Atexpt.The lines defining the trapezoid are shown on a computer screen together with the corresponding data points. Interactively, the trapezoid in Figure 5 can be properly adjusted. This procedure has the advantage that any spurious spikes or irregular pulse shapes which might occur in a few cases are immediately recognized, and the corresponding run can be discarded and repeated. Since the initial total pressure po in the chamber has been measured before the expansion as described above, the average expansion ratio

during the nucleation pulse is obtained. This allows to calculate the nucleation temperature Texpt occurring during the nucleation pulse from Poisson's law

where TOis the chamber temperature. The ratio K of the specific heats is obtained using the Richarz formula89accounting for the influence of the vapor on the value of K . During one series of measurements the vapor fraction w and thus K remain constant. Furthermore, usually TOand fi are kept constant, and thus a constant nucleation temperature TcXnis obtained (see requirement A3). In the actual experiments a reproducibility of Apexptwithin a few hundredth of a percent and thus a reproducibility of Texpt within a few hundredth of a degree are achieved. The vapor fraction w is known from the preparation of the vapor/carrier gas mixture in the plenum volume R and will remain unchanged during expansion. Therefore, one can easily calculate the actual vapor pressure in the expanded state. Then, from the known equilibriumvapor pressurep,, a t the nucleation temperature Texpt the supersaturation Scxpt occurring during the nucleation pulse is obtained as

(see requirement Al). As w, p, and Tcxpt are usually unchanged is simply selected by proper during one measuring series, Scxpt setting of po, typical values ranging f20% around ambient pressure. It is important to note that pressures and temperature are uniform inside the consideredsensingvolume (see requirement A5). The recompression (usually 4% of Apexpt)is chosen to reduce the nucleation rate by at least 2 orders of magnitude but at the same time to ensure that the nuclei formed during the nucleation pulse can grsw and develop into droplets of micron size, which can be observed by light scattering. The Light Scattering Detection. In order to determine the drop number concentration C,,, and condensational growth rates, the drops are illuminated by a He-Ne laser beam and the transmitted light flux as well as the light flux scattered at fixed

Strey et ai. angle of 8 = 15O is monitored simultaneously as functions of time by two sensors (constant angle Mie scattering, CAMS). The condensing droplets are practically nonabsorbing for the light wavelength considered, and thus the observation process has negligible influence on the nucleating system (requirement Bl). The scattered light flux is normalized relative to the transmitted light flux and exhibits a series of maxima and minima (see Figure 4) in good agreement with Lorenz-Mie theory (see Figure 7), allowing to simultaneously determine drop radius and number concentration at various times during the growth process. A more detailed description of the CAMS method has been given by Wagner.45 An important feature of the CAMS method is that after relative calibration of the respective light sensors the number density of the particles can be quantitatively obtained without reference to external standards from the height of the first maximum in the normalized scattered light curve by comparison to the Lorenz-Mie theory (requirement B5). Accordingly, the number density is measured only a few milliseconds after nucleation has occurred, and thus particle losses by coagulation, etc., are excluded (see requirement B4). The accuracy of the CAMS method has recently been confirmed by comparison to a different technique for absolute number concentration measurements.m Assuming the nucleation process to be stationary, the nucleation rate is then given by

In a previous study Wagner and st re^'^ have verified experimentally that eq 16 does indeed hold by varying the pulse length between 1 and 8 ms. A strict proportionality between the pulse length and the number density of droplets was observed (see requirement B3). Nucleation rates ranging from about lo5 to lo9 cm-3 s-l can be measured with the present experimental system. The lower limit of the measuring range is related to the minimum number of droplets required inside the sensing volume to prevent excessive noise in the light scattering. On the other hand, beyond the upper limit vapor depletion during droplet growth restricts the size of the droplets to the extent that no maximum of the light scattering curve can be observed, and thus the droplet concentration cannot be determined anymore. It should be noted that light extinction occurring in the chamber at comparatively high drop concentrations is accounted for by normalizing the scattered relative to the transmitted light flux, both being nearly equally affected by light extinction for the present scattering geometry. For studies of homogeneous nucleation it is essential to consider particle-free systems in order to exclude heterogeneous effects. While static diffusion chambers are self-cleaning, care has to be taken in expansion chamber experiments to avoid a significant influence due to heterogeneous nucleation. The expansion pulse chamber described in the present paper is evacuated prior to the measuring series, and only filtered noble gases are used as carrier gases in order to minimize contamination. A special feature of the light scattering technique is that the CAMS method applied in this study allows time-resolved droplet growth measurements, and thus any influences due to heterogeneousnucleation can safely be detected (see requirement B6). In order to demonstrate the effect of impurities, aerosol particles have deliberately been introduced into the expansion chamber during a series of test measurements. As can be seen from the corresponding experimental result shown in Figure 6, in this case light scattering and thus droplet growth are observed already soon after the start of the expansion process well before the maximum supersaturation occurs in the expansion chamber. This is due to the fact that the aerosol particles are activated to growth by heterogeneous nucleation at already comparatively low supersaturations. The results obtained for a particle-free system, shown in Figure 4, exhibit a qualitatively different behavior. In the absence of impurities significant light scattering and correspondinglydroplet

Feature Article

x

1

The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7755 n-butanol

heterogeneous nucleation I

1

0

10

20

30

50

40

60

.lo5 6

8

10

12

14

16

16

20

t [msl

Figure 6. As in Figure 4, but for heterogeneousnucleation on deliberately introduced aerosol particles. Significant light scattering is observed already before the nucleation pulse has occurred. Lorenz-Mie theory

I

4 ,

e- I

I

\

I

Sexpt

Figure 8. Nucleation rates Japt is supersaturated 1-butanol vapor measured as functions of vapor supersaturation S,, for various constant nucleation temperatures ranging from 225 to 265 K. Different symbols

refer to different carrier gases.

insignificant level so that only particle growth at constant number density can occur (see requirement B2). The time independence of the number density (over the monitored time frame) is also evident from the light scattering curve, because the ratios of the individual peak heights remain in agreement with the predictions by the Lorenz-Mie theory calculated for constant drop concentration.

4. Results and Discussion 01 0.0

”

0.5

1.0

1.5

U

2.0

bml F l p e 7. Light flux scattered by 1-butanoldrops at an angle of 8 = 15’ normalized relative to the transmitted light flux, calculated by means of Lorenz-Mie theory”‘ as a function of drop radius for a refractive index n = 1.414. The actual scatteringgeometry is taken into account. A drop number concentration of 1 ~ m is- considered. ~

growth are found only after the nucleation pulse has occurred, and thus the maximum supersaturation has been reached. During homogeneous nucleation measurements usually heterogeneous effects were not observed. In the rare cases, where light scattering was measured already before the nucleation pulse has occurred, the corresponding data were discarded. For comparison, the corresponding theoretical scattered light flux us particle size curves, as calculated according to the Lorenz-Mie t h e ~ r y ,are ~,~ presented in Figure 7. The scattered flux shown in Figure 7 is normalized relative to the transmitted light flux and refers to an assumed particle number concentration of 1 cm-3. Calculations were performed accounting for the actual scattering geometry and assuming absence of multiple scattering. Comparison with Figures 4 and 6 shows that the sequence of maxima and minima observed with increasing radius is clearly evident in the experimental light scattering curves, and thus growth of nearly monodispersed droplet aerosols is taking place both for homogeneous and for heterogeneous nucleation. The feasibility of using light scattering in the Mie theory range for the study of droplet growth has been demonstrated independently by Vietti and Schustergl and by Wagner?* The method for absolute droplet concentration measurement, by means of the CAMS technique, as used in the present study, has been introduced by Wagner.45 For the present experiments it is only necessary to make sure that the first Mie maximum occurs; Le., the droplets have to be able to grow to about 0.7 pm in radius (see Figure 7). It is important to realize that for the homogeneous nucleation experiment shown in Figure 4 nucleation and growth are decoupled: While nucleation occurs, only a negligible vapor fraction is consumed, leaving the supersaturation practically unchanged. By the recompression nucleation is suppressed to an

The expansion chamber system described above can be used for studies of homogeneous nucleation in one-component as well as multicomponent vapors. The chamber as described above allows measurements of homogeneousnucleation rates as opposed to onset conditions of nucleation. Furthermore, the nucleation rates can be measured over an extended rate range at constant selectable temperature. In the present paper we report a detailed study of homogeneous nucleation in supersaturated 1-butanol vapor. As mentioned earlier, this substance was selected because the thermodynamic properties of 1-butanol are well-known over a comparatively wide temperature range, allowing accurate calculations of the homogeneous nucleation rates according to various theories and a detailed comparison to experimental data. Furthermore, as seen from the light scattering curves shown in Figure 4, the growth rates for the experimental conditions considered are comparatively low, ensuring practically complete decoupling of nucleation and subsequent condensational growth. Measurement of isothermalnucleation rate curvespermits a direct determination of the molecular content of nuclei and a test of the Gibbs-Thomson equation for very small droplets. Homogeneous Nucleation Rates. Homogeneous nucleation rates J ranging from about los to lo9 ~ m s-l - were ~ measured quantitatively as functions ofvapor supersaturationsat constant nucleation temperature T. Temperatures were chosen between 225 and 265 K in steps of 5 K. Generally argon was used as the carrier gas. In order to check a possible influence of the nature of the carrier gas on the nucleation process, measurements at one temperature (240 K) were repeated with helium and with xenon as carrier gases. The experimental results are shown in Figure 8. It can be seen that the data obtained for different carrier gases do not show a significant dependence of nucleation on carrier gas properties. Thus, it appears that the carrier gas molecules are not taking part in thecluster formation process during theobserved homogeneous nucleation. It is worth noting that considerable differences of the drop growth rates in the measuring chamber are observed for different carrier gases at otherwise unchanged conditions. These findings arein agreement with recentlyreported results for water vapor.36 The apparent nondependence of the observed nucleation rates from the nature of the carrier gas thus

7756 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994

Strey et al. n-butanol

B-D theory

5

to the predictions of the classical nucleation theory.6-66 '

1

'

I

'

1

'

1

'.

I

\

'.

'

1

'

'

'

I

20 .

'wit

10

0

910

20

Figure 11. Experimentalnucleation rates for 1-butanol vapor in argon, as shown in Figure 8. Linear regressions yield the slopes of the In J u s

In S curves.

critical supersaturation '\',,I

7

Sexpt

S

Figure 9. Experimentalnucleation rates as shown in Figure 8 compared

301

6

-

9 8. 7. 6. 5 .

4 ' " " " " ' ' ' " " ~ 200 210 220 230 240 250 260 270 280

T [KI

Figure 10. Critical supersaturations Sdt corresponding to a nucleation rate of lo7 c ~ n -s-l~ for 1-butanol vapor at the various nucleation

temperatures T considered. The predictions according to the classical theory (B-D),66 the Girshick-Chiu theory (G-C),s7and the DillmannMeier theory (D-M)76 are shown for comparison.

also provides clear evidence for the fact that the nucleation process occurring during the supercritical nucleation pulse is not significantly affected by condensational drop growth. The symbols shown in Figure 8 refer to single measurements, and no averages were taken. From repeated measurements with unchanged experimental conditions, we observed a statistical scatter of the supersaturations Scxpt below 0.5%. The scatter of the measured nucleation rates JexPt was typically no more than 15%, a maximum scatter of 30% being observed in a few cases. Sources of possible systematic errors were discussed in ref 36 in detail and are smaller than the symbol width in Figure 8. Tables of the experimental data and a more detailed error analysis will be presented elsewhereeg3 Comparison with Some Theories. The observed experimental data have been compared to different theoretical models. A comparison of the measured nucleation rates to the predictions by the classical nucleation theory6,66is presented in Figure 9. It can be seen that the classical nucleation theory is consistently underestimating the nucleation rates by a maximum of about 2 orders of magnitude, exhibiting a quite similar temperature trend over the entire range of temperatures considered. The smallness of this discrepancy is quite surprising in view of the various simplifying assumptions inherent in the classical theory. It should be noted that the macroscopic surface tension and vapor pressure entering in eqs 2-6 have precisely been determined pre~iously.~8+94 From the measured homogeneous nucleation rates we have determined the critical supersaturations referring to a nucleation rate of lo7 ~ m s-l. - ~Figure 10 shows the dependence of the critical supersaturation on the nucleation temperature. Experimental data as well as the corresponding predictions by the

classical nucleation the self-consistent kinetic approach by Girshick and Chiu,S7 and the semiphenomenological version of Fisher's drop model of Dillmann and Meier7'j are indicated. The theoretical curves are calculated using the thermophysical parameters compiled by Dillmann and Meier.''j While the classical theory predicts consistently higher critical supersaturations (as can be seen in Figure 9 ) , both other theories result in too low values for most of the temperature considered. It appears that for the present system the classical theory is providing the best overall approximation. In this connection it should be mentioned that previous measurements of nucleation rates for 1-butanol by means of a different experimental system30.31seemed to indicate close agreement to the Dillmann-Meier the0ry.~6 Detailed inspection of the previous data shows that they are consistent with the present study except for the lowest temperature considered in the previous study?' where errors due the low chamber temperature cannot be excluded. In the recent paper on water nucleati0n3~an almost perfect agreement with the Dillmann-Meier approach had been observed while for water the classical theory showed much poorer predictions. Since in both cases, the previous water nucleation and the butanol nucleation studied here, the identical experimental setup and procedure were used and the same kind of comparison with the theories was performed, it can be concluded that there exist inconsistenciesin the macroscopic theories. Accordingly, further experimental studies of nucleation rates are needed. Such work is in progress. Molecular Content of Nuclei. For determination of the sizes of the critical clusters according to eq 11, the slopes of the In J us In S curves are required. In Figure 11 the experimental values of log J are plotted us log S. It can be seen that the data are not showing any significant deviations from the indicated empirical linear fits. Therefore, it can be concluded that inside the range of observed nucleation rates the size of the critical clusters is practically constant for each of the nucleation temperatures considered. The critical cluster size is dependent on supersaturation S and temperature T. We have determined the critical cluster sizes n* from the slopes of the experimental In J us In S curves for those values of S and T where a nucleation rate of lo7 cm-3 s-l was experimentally observed as indicated in Figure 10. The experimental critical cluster sizes are shown in Figure 12 us both the supersaturation S and the nucleation temperature T . A quite smooth monotonic decrease of cluster size with increasing supersaturation can be observed. Particularly, for the higher supersaturations considered the clusters are quite small. The corresponding predictions by the classical theory and the Dillmann-Meier theory, referring to the same values of S and T which were considered experimentally, are shown as well. As noted by Girshick and Chiu,S7 the critical cluster sizes obtained

The Journal of Physical Chemistry, Vol. 98, No. 32, I994 7757

Feature Article

T [Kl

T [KI 280

60

260

240

220

225

30

245

265

Gibbs-Thomson 50 40

n

I

\

30

10

-

9 .

20

6 7 . 6 .

r critical nucleus size 4

5

5 .

20

6 76910

I 30

41 9

S

Figure 12. Number of molecules n* in the critical 1-butanolclusters as function of supersaturation S and nucleation temperature T as obtained from the slopcs of the corresponding In J us In S curves shown in Figure 1 1, The corresponding predictions according to the classical theory (E D)66 and the Dillmann-Meier theory (D-M)76 are shown for comparison. Of course, the prediction by the classicaltheory correspondsto the GibbsT h o m s ~ nequation ~ ~ , ~ ~(cq 5 ) .

in their model and in the classical nucleation theory are the same. At low supersaturations both theories agree quite well with experiment. Increasing deviationsbetween theory and experiment can be observed, however, for increasing supersaturations and correspondinglydecreasing critical cluster sizes. This is consistent with the fact that macroscopicdescriptionsof clusters are expected to become unreliable for smaller clusters, and thus increasing deviations of the nucleation rates predicted by theories based on the macroscopicsurface tension and density can be expected with increasing S. A quite similar behavior is found for both theories considered, the classical theory deviating somewhat less than the Dillmann-Meier theory. While the classical theory seems to provide a quite consistent prediction of the nucleation rates, obviously significant deviations regarding the cluster sizes (and correspondingly the slopes of the J us S curves) are observed indicating inaccuracies in the cluster description. Experimental Test of the Gibbs-Thomson Equation. The present experiments provide a direct test of the Gibbs-Thomson95.96 equation for small cluster sizes. As can be seen from Figure 12,at measured cluster sizes below 40 molecules the GibbsThomson equation (eq 5 ) , which is used in the classical theory, for given S a n d T and thereby fixed macroscopic surface tension and density, predicts too small numbers of molecules and thus too strong curvature of the cluster surface. Accordingly, if the equilibriumsupersaturation for the critical cluster of experimental size n* at experimental temperature Tis calculated by means of the Gibbs-Thomson equation, a value below the experimental supersaturation S is obtained. This can be seen more clearly, if we rewrite eq 5 in the form

Pv Pvc

kTln-=-

2QY*

r

which expresses ther dependenceofthe equilibriumvaporpressure pV(r) of the curved droplet surface. We obtain from our measurements the supersaturation ratio S = pv(r*)/pwof critical clusters which are,per definitionem, in unstable equilibrium with the surrounding supersaturated vapor phase. The critical cluster radius r* is calculated from the measured n* as r* = (3n*u,/ 4 ~ ) * /valid 3 for the spherical droplets considered in the macroscopic picture. As shown in Figure 13, it is found that the experimental equilibrium vapor pressure at the surface of small 1-butanolclusters e x d s the correspondingprediction according to the Gibbs-Thomson equation by a maximum of 19%. At this point we can close the circle to Lord Kelvin's (Sir W. Thomson's) epigraph at the outset and conclude that "the equilibrium vapor

I 10

11

12

13

r* [I1

Figure 13. Equilibrium supersaturations=pv(r*)/pv.for critical clusters of critical radius r* at temperature T. The experimental data as shown in Figure 12 are presented together with the correspondingprediction by the Gibbs-Thomson equation in the form of eq 17.

pressure at curved surfaces of liquid" (the title of ref 96) he was speaking about in 1871 has been measured for extremely small droplets and expressed in numbers. It is still surprising that the macroscopicsurface tension and density sufficeto describe clusters of as few as 40 molecules or radii of curvature as small as 10 A. We are presently continuing with experimental studies of other systems. Results of experiments on nucleation for different substances including detailed tables providing quantitative experimental data will be presented in later papers. Furthermore, studies on various binary systems and a ternary system are in progress.

5. Conclusions The two most important observationsmade by Wilson,? namely, the homogeneous nucleation of droplets and the characteristic colors of scattered light, have been turned into a quantitative determination method for homogeneous nucleation rates. An expansion pulse leading to well-defined nucleation conditions for a short time interval is generated in a compact expansion chamber equipped with two valves and pressure supply units. From simultaneous measurement of pressure and scattered as well as transmitted light fluxes, the supersaturation in the expansion chamber and the number density of nucleated droplets are determined. This feature also allows one to discriminate heterogeneously from homogeneously nucleated droplets. Knowledge of the nucleation rate as a function of the thermodynamic driving force permits direct assessment of the number of molecules in the critical clusters, Le., objects of typically 10-100 molecules transiently existing in the sub-microsecond range, before they either growth further or decay again. Quantitative comparisons of experimental data on nucleation in n-butanol vapor to the classical and to more recent approaches, based on macroscopic drop models for the clusters formed, reveal that the 60 year old classical nucleation theory69" appears to be most successful in explaining the observed critical supersaturations and nucleation rates. Furthermore, direct measurements of the critical cluster size reveals remarkably close agreement with the Gibbs-Thomson equation for droplets with curvature radii as small as 10 A. However, the descriptionby the Gibbs-Thomson equation clearly deteriorates for clusters containing fewer than 40 molecules, calling for truly molecular based theories.56 Acknowledgment. The experiments were carried out in the laboratory of Prof. M. Kahlweit. We thank him for initiating and supporting the present research. References and Notes (1) The applicability of this phrase to nucleation research was noted by

P.Andre in: Nucleation;Zettlemoyer, A. C.,Ed.; Dekker: New York, 1969; p 69.

7758 The Journal of Physical Chemistry, Vol. 98, No. 32, I994 (2) Wilson, C. R. T. Philos. Trans. R. SOC.London, A 1897,189,265. (3) Mie, G. Ann. Phys. 1908, 25, 377. (4) Kerker, M. The Scattering of Light; Academic Press: New York, 1969. (5) Becker, F. Z . Phys. Chem. (Munich) 1912, 78, 39. (6) Volmer, M. Kinetik der Phasenbildung, Steinkopff: Dresden, 1939. (7) Nucleation Phenomena; Zettlemoyer, A. C., Ed.; Elsevier: Amsterdam, 1977. (8) Abraham, F. F. In Homogeneous Nuclearion Theory; Academic Press: New York, 1974. (9) Nucleation; Zettlemoyer, A. C., Ed.; Dekker: New York, 1969. (10) Wegener, P. P.; Mirabel, P. Natunvissenschaften 1987, 74, 111. (11) Oxtoby, D. W. J . Phys.: Condens. Matter 1992, 4 , 7627. (12) Tammann, G. Kristallisieren and Schmelzen; Steinkopff; Leipzig, 1903. (13) Katz, J. L.; Ostermier, M. J . Chem. Phys. 1967, 47, 478. (14) Katz, J. L. J . Chem. Phys. 1970, 52, 4733. (15) Heist, R.; Reiss, H. J . Chem. Phys. 1973, 59, 665. (16) Kacker, A.; Heist, R. H. J . Chem. Phys. 1985,82, 2734. (17) Hung, C.; Kransnopoler, M. J.; Katz, J. L. Chem. Phys. 1989, 90, 1856. (18) Anisimov, M. P.; Cherevko, A. G. J. Aerosol Sci. 1985, 16, 97. (19) Wyslouzil, B. E.; Seinfeld, J. H.; Flagan, R. C. J. Chem. Phys. 1991, 94, 6842. (20) Volmer, M.; Flood, H. 2.Phys. Chem. A 1934, 190, 273. (21) Sander, A.; Damkbhler, G. Natunvissenschaften 1943, 31, 460. (22) Allard, E. F.; Kassner, J. L., Jr. J . Chem. Phys. 1965, 42, 1401. (23) Miller, R.; Anderson, R. J.; Kassner, J. L.; Hagen, D. E. J . Chem. Phys. 1983, 78, 3204. (24) Schmitt, J. L. Rev. Sci. Instrum. 1981, 52, 1749. (25) Schmitt, J. L.; Adams,G. W.; Zalabsky, R. A. J. Chem. Phys. 1982, 77, 2089. (26) Schmitt, J. L.; Zalabsky, R. A.; Adams. G. W. J . Chem. Phys. 1983, 79, 4496. (27) Adams, G. W.; Schmitt, J. L.; Zalabsky, R. A. J. Chem. Phys. 1984, 81, 5074. (28) Wagner, P. E.; Strey, R. J. Phys. Chem. 1981,85,2694. (29) Strey, R.; Wagner, P. E. J . Phys. Chem. 1982, 86, 1013. (30) Wagner, P. E.; Strey, R. J. Chem. Phys. 1984, 80, 5266. (31) Strey, R.; Wagner, P. E.; Schmeling, T. J . Chem. Phys. 1986, 84, 2325. Strey, R.;Schmeling,T.; Wagner, P. E.J. Chem. Phys. 1986,85,6192. (32) Wagner, P. E.; Strey, R. In Aerosols, Science, Industry, Health and Environment; Masuda, S., Takahashi, K., Eds.; Pergamon: New York, 1990; VOl. 1, p 201. (33) Strey, R.; Wagner,P. E.;Viisanen, Y. In NucleationandAtmarpheric Aerosols; Fukuta, N., Wagner, P. E., Eds.; Deepak Publishing: Hampton, VA, 1992; p 111. (34) Wagner, P. E.;Strey, R.;Viisanen, Y. In NucleationandAtmarpheric Aerosols; Fukuta, N., Wagner, P. E., Eds.; Deepak Publishing: Hampton, 27. VA: -, -1992: D (35) Strey, R.; Viisanen, Y. J . Chem. Phys. 1993, 99, 4693. (36) Viisanen, Y.; Strey, R.; Reiss, H. J . Chem. Phys. 1993, 99, 4680. (37) Viisanen, Y.; Strey. R.; Laaksonen, A.; Kulmala, M. J. Chem. Phys. 1994,100, 6062. (38) Peters, F.; Paikert, B. J . Chem. Phys. 1989, 91, 5672. (39) Peters, F.; Paikert, B. Exp. Fluids 1989, 7, 521. (40) Wegener, P. P.; Wu, B. J. C. In NucleationPhenomena;Zettlemoyer, A. C., Ed.; Elsevier: Amsterdam, 1977; p 325. (41) Frish, M. B.; Wilemski, G. In Atmospheric AerosolsandNucleation; Wagner, P. E., Vali, G., Eds.; Springer: Berlin, 1988; p 527. (42) Kahlweit, M. In Physical Chemistry, An Advanced Treatise; Jost, W., Ed.; 1970; Vol. 10, p 719. (43) Kahlweit, M. In Reactiuity of Solids; Mitchell, J. W., et al., Eds.; Wiley: New York, 1969; p 93. (44) Langer, J. S.; Schwartz, A. J. Phys. Rev.A 1980, 21, 948. (45) Wagner, P. E. J. Colloid Interface Sei. 1985, 105, 456. (46) Kashchiev, D. J . Chem. Phys. 1982, 76, 5098. (47) Miller, R. Thesis, University of Rolla, Missouri, 1976.

-.

Strey et al. (48) Schmeling, T.; Strey, R. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 871. (49) Bauer, S. H.; Frurip, D. J. J. Phys. Chem. 1977, 81, 1015. (50) Mou, C. Y.; Lovett, R. J . Chem. Phys. 1979, 70, 3488. (51) Lovett, R. J. Chem. Phys. 1984,81, 6191. (52) Yang, C. H.; Qiu, H. J. Chem. Phys. 1986,84, 416. (53) Oxtoby, D. W.; Evans, R. J. Chem. Phys. 1988,89, 7521. (54) Wilcox, C. F.; Bauer, S . H. J . Chem. Phys. 1991,94,8302. Bauer, S. H.; Wilcox, C. F. J . Phys. Chem. 1993, 97, 271. (55) Nowakowski, B.; Ruckenstein, E. J . Chem. Phys. 1991, 94, 8487. (56) Ellerby, H. M.; Weakliem, C. L.; Reiss, H. J . Chem. Phys. 1991,95, 9209. (57) Girshick, S. L.; Chiu, C.-P. J . Chem. Phys. 1990, 93, 1273. (58) Girshick, S. L. J. Chem. Phys. 1991, 94, 826. (59) Kobraei, H. R.; Anderson, B. R. J . Chem. Phys. 1991, 95, 8398. (60) Zeng, X.C.; Oxtoby, D. W. J . Chem. Phys. 1991, 94,4472. (61) Rasmussen, D. H.; Liang, Esen, E.; Appelby, M. R. Lungmuir 1992, 8, 1868. (62) Granasy, L. Europhys. Lett. 1993, 24, 121. (63) Kalikmanov, V. I.; van Dongen, M. E. H. Europhys. Lett. 1993,21, 645. (64) Volmer, M.; Weber, A. 2.Phys. Chem. (Leipzig) 1926, 119, 227. (65) Farkas, L. Z . Phys. Chem. (Leipzig) 1927, 125, 236. (66) Becker, R.; Dbring, W. Ann. Phys. 1935, 24, 719. (67) Zeldovich, J. B. Sou. Phys. JETP 1942, 12, 525. (68) Courtney, W. G. J . Chem. Phys. 1961, 35, 2249. (69) Blander, M.; Katz, J. L. J. Stat. Phys. 1972, 4, 55. (70) Hale, B. Phys. Rev.A 1986, 33, 4156. (71) Fisher, M. E. Physics 1967, 3, 255. (72) Stauffer, D.; Kiang, C. S.; Walker, G. H.;Puri, 0. P.; Wise, Jr., J. D.; Patterson, E. M. Phys. Lett. A 1971, 35, 172. (73) Stauffer, D.; Kiang, C. S. In Nucleation Phenomena; Zettlemoyer, A. C., Ed.; Elsevier: New York, 1977; p 103. (74) Dillmann, A. PhD Thesis, Gijttingen, 1989. (75) Dillmann, A,; Meier, G. E. A. Chem. Phys. Lett. 1989, 160, 71, (76) Dillmann, A.; Meier, G. E. A. J . Chem. Phys. 1991, 94, 3872. (77) Ford, I. J.; Laaksonen, A,; Kulmala, M. J . Chem. Phys. 1993, 99, 764. (78) Kantrowitz, A. J. Chem. Phys. 1951, 19, 1097. (79) Feder, J.; Russell, K. C.; Lothe, J.; Pound, G. M. Ado. Phys. 1966, 15, 111. (80) Barrett, J. C.; Clement, C. F.; Ford, I. J. Presented at the 7th International Conferenceon Surface and ColloidScience, Compiegne, France, 1991. (81) Wilemski, G.; Wyslouzil, Gauthier, M.; Frish, M. In Nucleation and Atmospheric Aerosols; Fukuta, N., Wagner, P. E.,Eds.; Deepak Publishing: Hampton, VA, 1992; p 23. (82) Katz, J. L.; Fisk, J. A.; Charakov, V. In Nucleation andAtmospheric Aerosols; Fukuta, N., Wagner, P. E., Eds.; Deepak Publishing: Hampton, VA. 1992 D 11. (83) Siilinger, F. H. J. Chem. Phys. 1963, 38, 1486. (84) Reiss, H.; Katz, J. L.; Cohen, E. R. J . Chem. Phys. 1968,48,5553. (85) Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1958, 28. 258. (86) Nielsen, A. E. Kinetics of Preclpitation; Pergamon: Oxford, 1964. (87) Hill, T. L. StatisticalMechanics; McGraw-Hill Bookcompany; New York, 1956; Longmans Green: 1906; Vol. 1. (88) Oxtoby, D. W.; Kashchiev, D. J. Chem. Phys., submitted. (89) Richarz, F. Ann. Phys. 1906, 19, 639. (90) Szymanski, W. W.; Wagner, P. E . J. Aerosol Sci. 1990, 21, 441. (91) Vietti, S.; Schuster, B. G. J . Chem. Phys. 1973, 58, 434. (92) Wagner, P. E. J. Colloid Interface Sei. 1973, 44, 181. (93) Viisanen, Y.; Strey, R. J . Chem. Phys., submitted. (94) Strey, R.; Schmeling, T. R. Ber. Bunsen-Ges. Phys. Chem. 1983,87, 324. (95) Gibbs, J. W. Trans. Connect. Acad. 1876, 3, 108; 1878, 3, 343; Scientific Papers. (96) Thomson, W. Proc. R. Soc. Edinburgh, 1870, 7, 63; Philos. Mag. 1871, 42, 448. ~I

~