a laminar flow diffusion chamber for homogeneous nucleation studies †

A Laminar Flow Diffusion Chamber for Homogeneous Nucleation Studies†

Heikki Lihavainen and Yrjo1 Viisanen*Finnish Meteorological Institute, Sahaajankatu 20 E, FIN-00810 Helsinki, Finland

ReceiVed: March 29, 2001; In Final Form: September 5, 2001

A new version of laminar flow diffusion chamber was developed for unary homogeneous nucleation fromvapor to liquid studies. The developed laminar flow diffusion chamber can quantitatively measure nucleationrates as a function of saturation ratio and temperature. A numerical model was developed for calculations oftemperature and saturation ratio profiles inside the laminar flow diffusion chamber. Several experiments weremade to ensure the proper and reliable operation of the device and the numerical model. Nucleation rates ofn-hexanol were measured using helium as a carrier gas from 265.0 to 295.0 K. The measured nucleation raterange was from 103 to 107 cm-3 s-1. Temperature and saturation ratio dependencies were compared to theclassical nucleation theory. The saturation ratio dependency was well predicted by the theory. Results werecompared to the measurements made by others. They were in reasonable agreement at the lower end of thetemperature range of this study. The carrier gas effect was also investigated using argon as another carriergas. The nucleation was stronger when argon was used as a carrier gas.

Introduction

The formation of a liquid droplet from vapor is referred toas nucleation. This first-order phase transition has many forms.Homogeneous nucleationis the formation of liquid phase frompure vapor. It can be further divided into unary, binary, ternary,etc. homogeneous nucleation, depending on how many sub-stances take part in the nucleation process. Heterogeneousnucleation is referred to when nucleation occurs on foreignsubstances or surfaces. Nucleation always needs supersaturatedconditions in order to occur.

Even though nucleation has been studied over 100 years, thephenomenon is still poorly understood. To gain more under-standing on nucleation phenomena, more experimental data withdifferent kinds of devices are needed. There exist over twentydifferent kinds of devices to study nucleation.1 Most of thesedevices are variations of only a few methods. To date,experimental homogeneous nucleation rate studies have beendone usually with an expansion cloud chamber and with a staticdiffusion chamber. The nucleation rate range of these devicesdoes not overlap. The static diffusion chamber is applicable tomeasure low nucleation rates, and the expansion cloud chamberis used to measure quite high nucleation rates. The nucleationrate range of the laminar flow diffusion chamber fills this gapbetween these two devices.

The laminar flow diffusion chamber (from now on referredto as LFDC) has been applied to homogeneous nucleationstudies with variable success. The laminar flow diffusionchamber for generating fine particles is based on a methodutilized at the end 1970s by Anisimov et al.2 Anisimov andCherevko3 studied the nucleation of dibutyl phthalate in a gasflow diffusion chamber very similar to LFDC. Nguyen et al.4

named their device a laminar flow aerosol generator. Theoperational principle of it is the same as in the LFDC. Theystudied both homogeneous and heterogeneous nucleation ofdibutyl phthalate in nitrogen and air. They investigated carefullythe operational characteristics and the effects of different

experimental parameters. They proposed that the laminar flowaerosol generator is better suited for heterogeneous nucleationstudies than for homogeneous nucleation studies because of largeloss of the vapor onto the walls. This is not the case in a mixingtype of instrument they developed.5 Anisimov et al.6 studiedhomogeneous nucleation rates of dibutyl phthalate andn-hexanolin nitrogen. They concluded that the LFDC works well withless volatile substances and when the nucleation temperature isabove the room temperature. They measured nucleation ratesbetween 102 and 108 cm-3 s-1. Vohra and Heist7 presented aversion of a LFDC they built and studied its operationalcharacteristics. They measured critical saturation ratios for1-propanol in four different carrier gases: helium, argon,nitrogen, and hydrogen. They concluded that their device iscapable of quantitative investigations of vapor nucleation,although nucleation rate measurements would be desirable.Hameri et al.8 studied the operational characteristics of theLFDC and the ability of the model they used to describe theactual events inside the condenser. They used dibutyl phthalateas the nucleating vapor and both nitrogen and helium as carriergas. They found out that the equations they presented were inreasonable agreement with the experiments. Ha¨meri and Kul-mala9 measured homogeneous nucleation rates from 10 to 4.5× 107 cm-3 s-1 over a temperature range from 246 to 317 K.They used nitrogen as the carrier gas and dibutyl phthalate asthe nucleating vapor. They also studied the effects of differentcarrier gases on the nucleation. They concluded that the LFDCis well suited for measuring nucleation rates of substances oflow volatility.

The main aim of this study was to develop a version of alaminar flow diffusion chamber that can reliably measurehomogeneous nucleation rate isotherms. Nucleation rates ofn-hexanol in helium and argon were measured over a temper-ature range from 265 to 295 K and over a nucleation rate rangefrom 103 to 107 cm-3 s-1 and compared with the results withthe classical nucleation theory and also to measurements madeby others.† Part of the special issue “Howard Reiss Festschrift”.

11619J. Phys. Chem. B2001,105,11619-11629

10.1021/jp011189j CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 11/06/2001

Theory

The dominating theory of unary homogeneous nucleation hasbeen for over 70 years the classical nucleation theory. It hasbeen developed over a period of time by Volmer and Weber,10

Volmer,11,12 Becker and Do¨ring,13 Zeldovich,14 and Frenkel.15

The classical nucleation theory predicts the nucleation rate as afunction of both temperature and saturation ratio. In this theorythe properties of a liquid droplet are assumed to be the same asthe bulk liquid properties. This enables the use of the bulk liquiddensity and macroscopic flat film surface tension in thecalculations. This is generally called capillary approximation.The classical nucleation theory is still widely used even thoughit predicts nucleation rates that differ from experimental resultsby orders of magnitudes. It is still useful for qualitativeunderstanding of nucleation. A number of different theoreticalalternatives have been proposed. None of these has been capableof explaining experimental results any better than the classicalnucleation theory. Nucleation rate predicted by the classicalnucleation theory can be written in a form

whereC* is the rate of arrival of monomer molecules at thecritical cluster,

whererd/ is the size of the critical radius of the cluster,p is the

vapor pressure,m is the mass of the molecule,kB is theBoltzmann constant, andT is the temperature.Z is the so-calledZeldovich nonequilibrium factor,

whereσ is the surface tension andVliq is the volume occupiedby a liquid molecule. It is the correction factor for the differencebetween the equilibrium cluster concentrationNn*

e and thepseudo-steady-state cluster concentration

whereN1 is the number of monomers andS is the saturationratio. The size of the critical cluster,rd

/, is determined by so-called Gibbs-Thompson or Kelvin equation

Experimental Section

Principle of Operation. The used experimental setup ispresented in Figure 1. It consists of three main parts; thesaturator, the preheater, and the condenser. A steady and filteredflow of carrier gas is introduced into the saturator. In thesaturator the carrier gas is fully saturated with the nucleatingsubstance by evaporation from the liquid phase. The saturatoris a horizontally placed round tube. The nucleating substanceis as a liquid pool at the bottom of the saturator. The amountof the vapor in the carrier gas is controlled with the temperatureinside the saturator. After the saturator, mixture is led into the

preheater. The preheater, as well as condenser, is also a round,temperature controlled tube. The preheater and condenser havethe same inner diameter. They are placed vertically downward.The preheater is always at a higher temperature than thesaturator. In the preaheter a laminar flow profile is achieved.From the preheater the gas stream enters into the condenser,which is at a lower temperature than the preheater and thesaturator. The stream is rapidly cooled by heat exchange withthe wall of the condenser. Supersaturation is achieved becauseequilibrium vapor pressure is a rather strong, rising exponentialfunction of temperature. If this supersaturation is high enough,nucleation occurs. Formed droplets are counted after thecondenser by light scattering.

Because the boundary conditions in the device are well-known, a mathematical model can be used to calculate thevelocity, temperature, vapor pressure, and saturation ratioprofiles inside the condenser.

The temperatures of the preheater and the condenser are keptconstant during a measurement of one nucleation rate isotherm.This means that the boundary conditions are constant and thenucleation temperature is almost constant over one nucleationrate isotherm measurement. The flow is kept laminar in thepreheater and condenser to ensure that the mixing characteristicsare well characterized. Typical Reynols numbers in the mea-surements were around 100.

Experimental Setup. n-Hexanol (Acros, purity 98%) wasused without further purification as the nucleating substance.Helium (AGA, purity 99.9999%) and argon (AGA, purity99.9999%) were used as the carrier gases. Lines from the carriergas container to the saturator are made from Teflon and stainlesssteel to avoid any desorption problems. The carrier gas is runtrough a HEPA-filter before it enters into the saturator. The flowrate was measured with a mini-BUCK M-5 soap bubble flowcalibrator, from two places; before the gas enters into thesaturator and after it comes out from the condenser.

The saturator is a 1.0 m long horizontal tube with an innerdiameter of 0.035 m. The inner surface of the saturator tube ismade from Teflon. Constant temperature over the saturator isachieved by circulating temperature-controlled circulating liquidthrough a jacket surrounding the inner tube. The temperatureof the circulating liquid is controlled with a LAUDA RK 20circulation liquid bath. The temperature of the nucleating vaporand carrier gas mixture is measured with a (HERAEUS QUAT100/200) temperature sensor. It is situated on the surface of theliquid pool near the end of the saturator next to the preheater.The place of the temperature sensor was chosen so that thetemperature change of the liquid surface caused by the heat ofevaporation is negligible. The residence time of the carrier gasin the saturator was roughly 20 s.

J ) C*ZNn*e (1)

C* )4πrd

/ p

(2πmkBT)1/2(2)

Z )σ1/2Vliq

2πrd/2 (kBT)1/2

(3)

Nn*e ) N1 exp[-

16πσ3Vl2

3(kBT)3(ln S)2] (4)

rd/ )

2σVliq

kBT ln S(5)

Figure 1. Experimental setup used in then-hexanol studies.

11620 J. Phys. Chem. B, Vol. 105, No. 47, 2001 Lihavainen and Viisanen

Both the preheater and the condenser are about 0.3 m longtubes. Inner diameters of the preheater and the condenser are4.0 mm. The inner tubes, where the gas flows, are made fromstainless steel and the heating and the cooling jackets are madefrom brass. The temperatures of the tubes were controlled bycirculating temperature-controlled (preheater: LAUDA RC6 CS,condenser: LAUDA RK20 KS) circulating liquid through thejackets surrounding the tubes. The temperature drop at the wallof the tubes between the preheater and the condenser has to beas steep as possible. This is achieved with the following features;see Figure 2. In the connection between the preheater and thecondenser, the inner tubes are about 1 mm longer than thecooling jackets. The preheater and the condenser are separatedthermally with a Teflon insulator. The Teflon insulator is about1 mm thick between the inner tubes and about 3 mm thick inother parts. Because the heat conductivity of stainless steel isabout 100 times higher than the heat conductivity of Teflon, itis assumed that the shoulder from the preheater is at the sametemperature as the preheater and the shoulder from the condenseris at the temperature of condenser. Special construction at theends of the preheater’s heating and the condenser’s coolingjackets allows the circulating liquid to flow uniformly and closeto the connection; see Figure 2. Special care was taken that theinner surface at the connection between the preheater and thecondenser is as smooth as possible, so that the flow profile wasnot affected. The temperatures of the preheater and thecondenser are measured from the circulating liquid. Temperaturesensors (HERAEUS QUAT 100/200) are located at the inletsof the circulating liquid into the heating or the cooling jacket,which in turn were located about 4 cm from the end of thepreheater or from the beginning the condenser; see Figure 2.Both the preheater and the condenser are thermally insulatedfrom the environment. The preheater and the condenser arevertically placed so that the settling of the particles due togravitation does not take place. The direction of the flow isvertically downward so that the condensed liquid from thecondenser does not fall back into the preheater and interferewith the boundary conditions.

After the condenser, the liquid condensed onto the wall ofthe condenser is separated from the gas stream. This is done in

a so-called T-crossing section. The stream of gas mixtureincluding the formed particles is forced to make a slightly over90° turn into the optical head where the particles are counted.The liquid cannot make this turn and so it falls into the liquiddrainage. The T-crossing section is made from aluminum. TheT-crossing section was separately cooled with a cooling bath(LAUDA RC6 CS). It was kept at the same or coldertemperature than the condenser to avoid evaporation of theformed droplets.

The formed droplets are counted with a TSI 3010 Condensa-tion Particle Counter’s (CPC) optical head. The optical headwas separated from rest of the CPC and connected into thedeveloped LFDC. Inside the CPC’s optical head the flow istaken through a laser beam. The droplets scatter laser light asthey pass through the beam. These scattered laser light pulsesare counted. The counts were continuously recorded with a PC.

Pressure inside the LFDC was measured from the T-crossingsection with the MKS barometer connected into the MKS Type270 D signal conditioner. The temperature of the laboratory wasmeasured with a HERAEUS QUAT 100/200 temperature sensor.

Mathematical Model. Measuring temperature, vapor pres-sure, and flow profiles reliably from a laminar flow is animpracticable task. They have to be calculated. This meanssolving five coupled partial differential equations: equationsfor heat and mass transfer, equations of motion (both axial andradial direction), and an equation of continuity. Several as-sumptions were made to simplify these equations. The equationscan be solved in a steady state. In all the equations, the effectof radial velocity was assumed to be negligible and it was setto zero. In calculations of the velocity profile, the flow wasassumed to be incompressible. These assumptions reduced theequations of motion to a quite simple problem that has beenanalytically solved,16 resulting in parabolic velocity profile. Theassumptions mentioned above reduce the amount of coupledpartial differential equations to only two. Because of the axialsymmetry, two-dimensional equations for heat and mass transferare solved in cylindrical coordinates, giving

whereF is the density of the carrier gas and vapor mixture,uis the velocity in axial direction,cp is the specific heat of themixture,k is the thermal conductivity of the mixture,ω is massfraction of nucleating vapor,D is binary diffusion coefficient,R is thermal diffusion factor,M is the molecular weight of thevapor carrier gas mixture,Ptot is the total pressure, andR is thegeneral gas constant. In the temperature profile calculations,convective heat transfer in the axial direction and heat transferby conduction in the axial and radial direction were taken intoaccount. In the mass fraction profile calculation convective masstransfer in the axial direction, mass transfer by diffusion, andthermal diffusion in the axial and radial directions were takeninto account.

Figure 2. Schematic cross section of the connection between preheaterand the condenser.

∂

∂z(uFcpT) ) 1

r∂

∂r[kr∂T∂r ] + ∂

∂zk∂T∂z

(6)

∂

∂z(uFω) ) 1

r∂

∂r[rFD∂ω∂r ] + ∂

∂z[FD∂ω∂z] + 1

rd∂r

×

[rRDFω∂ ln T∂r + ∂

∂z[RDFω∂ ln T∂z ] (7)

F )MPtot

RT(8)

Laminar Flow Diffusion Chamber for Homogeneous Nucleation J. Phys. Chem. B, Vol. 105, No. 47, 200111621

Equations 6-8 are coupled with temperature and concentra-tion. Equations are solved with the following boundary condi-tions,

where TS2 is the temperature of the preheater,TC is thetemperature of the condenser,TS1 is the temperature of thesaturator,ωeq(Ti) is the mass fraction defined by equilibriumvapor pressure at the temperatureTi, z ) 0 is the boundarybetween the preheater and the condenser,z0 is a chosen startingpoint of calculations in the preheater (z0 is negative or zero),and R0 is the radius of the tube. The starting point of thecalculations was set in the preheater, because calculationsrevealed that at slower flow rates the temperature started to varyalready inside the preheater. The boundary conditions for massfraction were calculated usingω ) Fvap/(Fcar + Fvap) and theideal gas law, where subscript “vap” refers to “vapor” andsubscript “car” refers to “carrier gas”. Equations 6-10 weresolved using finite element method, e.g. 17.

In Figure 3 calculated profiles of temperature, equilibriumvapor pressure, vapor pressure, saturation ratio, and nucleationrate (according to the classical nucleation theory) at the centerof the tube with boundary conditions typical in measurementsof this study are presented. It can be seen that the temperatureand therefore also the equilibrium vapor pressure drop muchfaster than the actual vapor pressure toward the boundary valuesresulting supersaturation. The behavior of the profiles in theradial direction is similar than in axial direction resulting inthe nucleation zone being at the center of the tube. In the axialdirection the nucleation rate maximum is located slightly beforethe saturation ratio maximum because nucleation is a functionof both temperature and saturation ratio. The size of thenucleation peak is quite small. In the volume where about 90%of the particles are formed, the temperature drops about 1.5 Kand the saturation ratio changes about 3%. After the nucleationzone, the vapor is still clearly supersaturated. This enables thenucleated particles to grow to a size in which they are opticallydetectable.

Data Interpretation of Experimental Nucleation Rate.TheLFDC experiments yield the total number of particles nucleatedper unit time in the chamber. However, rates of nucleation areusually expressed in units of cm-3 s-1. The results yielded fromthe LFDC were converted to conventional nucleation rate unitsusing the similar method that is used to convert static diffusioncloud chamber data.18 Wagner and Anisimov19 suggested thatthis method be used in LFDC. The maximum experimentalnucleation rateJexp

max can be calculated from the relationship

where the integrals are over the volume of the condenser tubeandJtheor

max is the theoretical nucleation rate maximum.Jexp dV isthe measured particle concentration multiplied by the flow rate.The relationship (11) results from the fact that the experimentaland the theoretical nucleation rates have usually the same slopeeven though there is a difference in the absolute values.18 Ithas been found that the use of different theories changes thecalculated experimental results no more than 10-50%.8 In thisstudy the classical nucleation theory was used. The experimentalnucleation temperature and saturation ratio are those at thetheoretical nucleation rate maximum.

Thermodynamic Data. To be able to calculate nucleationrates in the LFDC, molecular weights, thermal conductivities,binary diffusion coefficients, equilibrium vapor pressures,densities, viscosities, specific heats, thermal diffusion factors,and surface tensions of the substances as functions of temper-ature are needed. They are presented in Table 1. The densitiesof the substances in a gas phase were calculated using the idealgas law.

The available experimental values for the binary diffusioncoefficient20 were determined at much higher temperatures thanthe measurements in this study were performed, therefore simplecurve fitting was out of the question. It was decided to use themethod of Fuller et al. described in Reid et al.21 It describesquite well the experimental data. The viscosity and thermalconductivity of the gaseousn-hexanol were calculated usingformulas based on theory.21

Verification of the Operation of the Laminar FlowDiffusion Chamber. To make reliable homogeneous nucleationmeasurements, the following experimental features should befulfilled:30,31 (1) Nucleation agents should be eliminated. Theycan be, e.g., ions or preexisting particles. Nucleation on thesesurfaces needs lower saturation ratios for its occurrence than inhomogeneous nucleation. (2) The supersaturation of the vaporshould be achieved by a well-defined and reproducible process.(3) Droplet growth should have a negligible influence on thesaturation ratio. (4) The nucleation rate should be measured asa function of the saturation ratio over a sufficiently large rangeso that the slope of the nucleation rate as a function of saturationratio can be determined. (5) To have sensitive evaluation ofnucleation theories both the temperature and the saturation ratiodependencies of the nucleation rate have to be determined.Several experiments were made to ensure the proper operationof LFDC. These experiments made it also possible to compareresults from LFDC to results from model calculations.

Figure 4 shows a typical measured particle concentrationisotherm as a function of the temperature of the saturator. Atthe low end of the range it can clearly be seen that nucleationagents exist. The vapor condenses on them at lower saturationratios than needed for homogeneous nucleation to occur. Theconcentration of nucleation agents was less than 0.1 cm-3. At

Figure 3. Temperature,T, equilibrium vapor pressure,Peq, vaporpressure,P, saturation ratio,S, and nucleation rate,J, profiles as afunction of the length of the condenser,z. The flow rate is 17 cm3/s.

T(r, z ) z0) ) TS2

T(R0, z0 < z e 0) ) TS2

T(R0, z > 0) ) TC (9)

ω(r, z ) z0) ) ωeq(TS1)

ω(R0, z0 < z e 0) ) ωeq(TS1)

ω(R0, z > 0) ) ωeq(TC) (10)

Jexpmax

Jexp dV)

Jtheormax

JtheordV(11)


the high end of the range the effects of the depletion of vaporand the heat released by condensation on the nucleated particlescan be seen. These effects were negligible at concentrationsbelow 103 cm-3. The particle concentration range was alsolimited by the optical head, the optical head was not able tocount all the particles in particle concentrations higher than 104

cm-3 because of the coincidence effect.The experimental temperature range withn-hexanol and

helium as a carrier gas spans from 265 to 295 K. The uppertemperature limit is set by the clear effect of depletion of vaporand heat released by condensation. The lower temperature limitis set by the counting efficiency of the optical head. At lowtemperatures the particles reaching the optical head are probablysmaller and are not entirely counted by the optical head.

The stability of the developed LFDC is good. Figure 5 showsthe output particle concentration ofn-hexanol in argon as afunction of time. All the temperatures and the flow rate werekept constant during the measurement. The nucleation rate is

quite constant over a time range of 21 h. The standard deviationfrom the average value is l3%. Temperatures of the saturator,preheater, and the condenser were constant within the accuracyof the temperature sensors ((0.05 K) during this time. Thetemperature of the laboratory varied about(2 K.

Reproducibility was tested by measuring the rate of particleformation at same temperatures and flow rate in 10 differentdays. The average deviation of particle formation was about(10%. Most of this deviation is due to the fact that thetemperature of the saturator was not actually the same in the10 different measurements. The temperature of the saturatorvaried about 0.1 K.

To ensure the proper operation of the saturator, two differentkinds of experiments were made. First the temperatures of thepreheater and the condenser and the flow rate were kept constantwhile the temperature of the saturator was increased by smallsteps. When the temperature of the saturator was raised abovethe temperature of the preheater, the particle formation rate didnot increase anymore. Even though the temperature of thesaturator was raised, the amount of the vapor entering thecondenser was constant. It corresponded to the equilibrium vapor

TABLE 1: Thermodynamic Properties for Helium, Argon, and n-Hexanola

HeliumM ) 0.004003 kg/molk ) 418.68× (-5.85430× 10-5 + 2.686056× 10-6T - 0.700113× 10-9T2 + 1.07396× 10-11T3 -6.01768× 10-15T4) W/mK bη ) 1.4083× 10-6T1.5/(T + 70.22) Ns/m2 bcp ) 5200 J/(kg K)

ArgonM ) 0.039948 kg/molk ) exp(-8.7978+ 0.83517 lnT) W/mK cη ) exp(-15.712+ 0.88153 lnT) Ns/m2 ccp ) 520 J/(kg K)

N-HexanolM ) 0.102177 kg/molpeq ) 133.322× exp[135.4577- (12092.97/T) - 16.7 lnT] Pa dσ ) 10-3 × (27.84- 0.08381× (T - 273.15)) N/m eηvap ) 0.0269[(MT)1/2/ωLΩ], whereσL is the hard-sphere diameter andΩ is the collision integral and it is calculated from empirical equationΩ ) 1.16145T* -0.14874+ 0.52487 exp(-0.77320T*) + 2.16178 exp(-2.43787T*), whereT* ) kBT/ε andε is the characteristic energy fkvap ) 2.63× 10-23[(T/M)1/2/σ2Ω] W/mK fFliq ) 1000× (0.8269- 5.1191× 10-4((T - 273.15)) kg/m3 gD ) 1.12189× 10-4(T1.75/P) m2/s, in helium hD ) 3.22780× 10-5(T1.75/P) m2/s, in argon hcp ) (4.811+ 5.981× 10-1T - 3.010× 10-4T2 + 5.4261× 10-8T3)/0.102177 J/(kg K) f(1/R) ) [-0.62264+ [T/(-23.4305+ 0.30537)]](y - 0.1548)+ 0.0964, in helium wherey is mole fraction i(1/R) ) [25.4071- [T/(-297.6230+ 1.33707)]](y - 2.7573)+ 70.0550, in argon i

Mixed Propertiesk ) ∑i)1

n (yiki/yjAij), whereAij ) ε[1 + (ktri/ktrj)1/2(Mi/Mj)1/4]/[8(1 + Mi/Mj)]1/2 andktri/ktrj ) (ηi/ηj)(Mj/Mi) fcp ) ∑i)1

n yicp,i f

a The parameters listed are:M molar mass,k thermal conductivity,D binary diffusion coefficient,peq equilibrium vapor pressure,F density,ηviscosity,cp the specific heat of the substance in a vapor phase,R thermal diffusion factor,σ surface tension. Subscript “liq” refers to “liquid phase”and “vap” to “vapor phase”. The letters at the end of each property refer to the list of references at the end of the table.b Reference 18.c Reference22. d Reference 23.e Reference 24.f Reference 21.g Fitted to data from refd 25, 26, and 27.h Method by Fuller et al. described in ref 21.i Estimatedusing the method described by Mason28 and Vasakova and Smolı´k.29

Figure 4. Measured particle concentration,N, as a function oftemperature of the saturator,TS1. Nucleation temperature is 280 K.

Figure 5. Particle concentration,N, as a function of time. Nucleationtemperature is 270 K.


pressure at the temperature of the preheater. The temperatureof the saturator, at which the particle formation rate stoppedincreasing, was the same as the temperature of the preheater.

The second method was to shorten the residence time of thecarrier gas in the saturator by increasing the flow rate. Becausethe nucleation rate is a strong function of the saturation ratio,the nucleation rate will drop drastically if the residence time inthe saturator is too short for full saturation. This did not happen,as can be seen from Figure 6. As a comparison, the theoreticalnucleation rates predicted by the classical nucleation theorymultiplied by a factor of 2× 105 are presented.

Figure 6 is also a test between the operation of theexperimental device and the used mathematical model. Theexperimental nucleation rates as a function of flow rate doesnot change much at flow rates higher than 10 cm3/s. This isalso the case with the theoretical nucleation rate. The nucleationrate measurements were made at about 16.7 cm3/s when heliumwas used as a carrier gas. The experimental nucleation rate dropstoward zero at a higher flow rate than predicted by the scaledtheoretical nucleation rate. This is probably due to two reasons.When the flow rate is slower, the particles grow bigger and arepartly lost in the T-crossing section because of gravitation. Atlower flow rates the effect of the vapor depletion is also higherbecause of the longer residence time in the nucleation zone. Aswas mentioned above, the mathematical model does not takeinto account the aerosol dynamics. The saturation ratio increases,according to the model, as a function of the flow rate becausethe effect of the axial heat conduction becomes negligible asthe flow rate increases. At low flow rates the effect of axialheat conduction becomes more significant and therefore thetemperature of the stream reaches the temperatures of thecondenser slower than in the case where the axial conductionis not taken into account. Thus the axial heat conduction raisesthe equilibrium vapor pressure and lowers the saturation ratioin the nucleation zone. The axial diffusion of molecules is muchslower than the axial heat conduction. Therefore the effect ofaxial diffusion is negligible. At flow rates higher than 10 cm3/swith helium as a carrier gas, the temperature and the saturationratio at the theoretical nucleation rate maximum are constant.

It can be concluded that the experimental device operates asexpected. The mathematical model describes well the actualevents inside the LFDC. The conditions for making reliablenucleation measurements described in the beginning of thischapter are fulfilled.

Experimental Procedure.For one nucleation rate isotherm,the temperatures of the preheater and the condenser were keptconstant. Only the temperature of the saturator was changed.The temperatures of the saturator, preheater, and the condenserwere first set to their initial temperatures. These temperatures

were determined with the mathematical model. When thetemperatures were stabilized, a flow of carrier gas wasintroduced into the system. The volume flow rate was set toabout 16.7 cm3/s with helium and 5 cm3/s with argon as a carriergas. After a steady nucleation rate was achieved, the PC wasset to record particle concentration. The volume flow rate,temperatures, and the pressure were recorded at the same time.The particle concentration for every different saturation ratiowas recorded for at least 15 min after the nucleation rate wasstabilized. In Figure 7 a typical measurement event is presented.When the saturation ratio (the temperature of the saturator) waschanged, the volume flow rate, temperatures, and the pressurewere recorded. Isotherms were measured using 5 K steps innucleation temperature. The temperature difference betweenpreheater and condenser was the same with each isotherm. Afterthe nucleation rate isotherm measurement, the liquid drainagewas emptied and it was checked that the temperature sensor inthe saturator was on the surface of the pool. If not, liquid wasadded into the saturator. Usually one nucleation rate isothermwas measured during different days, sometimes raising thetemperature of the saturator and sometimes lowering it.

Results and Discussion

The experimental data, the calculated experimental nucleationrate maxima, the saturation ratios, and temperatures at thetheoretical nucleation rate maxima are listed in Table 2. Theexperimental temperature range was from 265.0 to 295.0 K withhelium as a carrier gas and from 268.9 to 289.7 K with argonas a carrier gas. The measurements were made at ambientatmospheric pressure. The experimental particle concentrationrange, where the effects of nucleation agents, vapor depletionand heat released by condensation were negligible, was roughlyfrom 10-1 to 103 cm-3 with helium as a carrier gas and from100 to 102 cm-3 with argon as a carrier gas. There are twoexceptions to these limits. With helium as a carrier gas at 265.0K the lower concentrations are not valid probably because partthe particles are too small to be counted, and at 295.0 K theupper limit is reduced by the vapor depletion and heat releasedby this. Data outside these ranges are left out from all the figuresand tables in this section.

Comparison with the Classical Nucleation Theory.Theclassical nucleation theory was chosen because it predicts thenucleation rate as a function of saturation ratio and temperature,it is widely used, and it uses macroscopic measurable quantities.In Figure 8 experimental results ofn-hexanol in helium areplotted as a function of theoretical predictions by the classicalnucleation theory. Experimental particle concentrations are about5 orders of magnitude higher at 265.0 K than predicted by the

Figure 6. Nucleation rate,J, as a function of particle concentration,N, as a function volume flow rate,Q. The circles are measuredconcentrations and the solid line is theoretical concentration multipliedby a factor of 2× 105. Figure 7. Typical measurement event. Particle concentrationN as a

function of measuring time.


TABLE 2: Homogeneous Nucleation Rates ofn-Hexanol Measured in the LFDCa

n-Hexanol-Helium, 15-02-2000

TS2 ) 296.52 K,TC ) 263.69 K,P ) 99900 Pa,Q ) 16.76 cm3/s,Tlab ) 294.15 K

TS1 N Tnuc Snuc Jexp

293.48 17.36 265.04 12.81 1.59E5293.92 93.92 265.04 13.28 6.22E5294.36 471.01 265.06 13.76 2.50E6294.72 1502.18 265.07 14.16 7.55E6294.50 715.38 265.06 13.92 4.08E6294.19 291.60 265.05 13.57 1.73E6293.70 46.32 265.04 13.05 2.91E5293.25 9.97 265.02 12.58 6.64E4




297.11 0.29 270.07 10.11 2.42E3297.54 1.05 270.08 10.45 8.34E3297.84 3.74 270.08 10.73 2.84E4298.28 16.34 270.10 11.06 1.18E5




298.27 15.42 270.09 11.06 1.13E5298.68 83.66 270.11 11.42 5.83E5299.14 377.38 270.13 11.83 2.48E6299.51 1371.47 270.13 12.17 8.63E6299.29 600.63 270.13 11.96 3.88E6298.90 153.92 270.11 11.61 1.04E6298.51 37.64 270.11 11.27 2.68E5298.09 7.98 270.09 10.91 5.99E4297.70 1.64 270.09 10.58 1.29E4297.30 0.38 270.07 10.26 3.14E3




297.30 0.39 270.08 10.26 3.26E3




301.68 0.14 274.86 8.92 1.28E3302.11 0.74 274.87 9.21 6.32E3302.50 2.98 274.89 9.48 2.42E4302.83 16.82 274.90 9.71 1.31E5




304.4 922.86 274.93 10.62 6.32E6




303.89 512.54 274.93 10.48 3.62E6303.48 131.14 274.93 10.17 9.74E5303.08 29.28 274.92 9.88 2.29E5302.72 6.29 274.91 9.62 5.13E4303.11 32.92 274.91 9.90 2.56E5302.37 1.77 274.90 9.37 1.51E4301.89 0.30 274.88 9.05 2.91E3




306.75 0.23 279.99 7.99 2.31E3307.09 0.77 279.99 8.18 7.41E3307.48 3.12 280.01 8.41 2.86E4307.91 17.88 280.02 8.67 1.55E5308.28 63.01 280.04 8.89 5.21E5308.71 284.00 280.04 9.16 2.23E6


TABLE 2 (Continued)




308.73 318.68 280.05 9.17 2.52E6309.12 1193.76 280.06 9.42 9.00E6308.95 696.81 280.06 9.31 5.36E6




308.93 632.50 280.06 9.30 4.88E6308.52 139.50 280.04 9.04 1.13E6308.11 39.43 280.03 8.79 3.36E5307.70 7.16 280.01 8.54 6.43E4307.31 1.60 279.97 8.32 1.51E4306.96 0.49 279.96 8.11 4.83E3306.53 0.11 279.96 7.87 1.12E3




311.54 0.15 285.03 7.07 1.72E3311.86 0.54 285.04 7.22 5.80E3312.27 2.35 285.05 7.42 2.39E4312.65 10.24 285.07 7.60 9.93E4313.10 45.28 285.08 7.83 4.15E5313.46 195.30 285.11 8.02 1.71E6313.85 758.23 285.12 8.22 6.35E6




313.65 438.39 285.12 8.12 3.78E6313.29 118.99 285.09 7.93 1.07E6312.85 24.50 285.08 7.70 2.34E5312.46 5.70 285.06 7.51 5.68E4312.12 1.15 285.05 7.34 1.19E4311.67 0.23 285.03 7.13 2.55E3




312.46 6.30 285.06 7.51 6.21E4313.23 91.86 285.09 7.90 8.24E5314.00 1271.93 285.13 8.31 1.04E7




316.49 0.25 289.93 6.44 3.03E3316.87 1.02 289.94 6.60 1.16E4317.32 5.81 289.97 6.78 6.23E4317.61 21.09 289.98 6.91 2.18E5318.08 111.48 290.00 7.11 1.09E6318.44 406.85 290.01 7.27 3.81E6




318.44 361.65 289.98 7.28 3.41E6318.67 908.97 289.99 7.38 8.33E6318.27 247.86 289.97 7.20 2.38E6317.91 64.42 289.96 7.04 6.47E5




317.91 53.43 289.99 7.04 5.35E5317.46 11.28 289.97 6.85 1.19E5317.06 2.33 289.96 6.68 2.59E4316.68 0.63 289.93 6.52 7.32E3316.27 0.15 289.92 6.36 1.81E3316.08 0.09 289.91 6.28 1.14E3


theory. The difference is slightly rising as a function oftemperature, being about 6 orders of magnitude higher thantheoretical predictions at 295.0 K. Isotherms plotted this wayform a straight line. Straight curve, logNexp ) intercept+ slope× log Ntheor, was fitted to each isotherm. If the theory predictscorrectly the experimental results, the intercepts should be 0and slopes should be 1.18 Slopes and intercepts as a function oftemperature at the nucleation maximum are presented in Figure9. The average of the slopes is 0.98 with the statistical averageerror of 0.02. This implies that the saturation ratio dependencyis well predicted by classical nucleation theory. It is also seen

that slopes have a clear tendency to get slightly smaller as afunction of temperature. Values of the intercepts are almostconstant, all values are between 5 and 6. Intercepts have a veryslight tendency to get bigger as the temperature rises, whichwould imply a slightly stronger temperature dependency thantheory predicts.

Relationship (11) was applied to convert the experimentalresults from this study into conventionally used nucleation rateunits. The results are shown in Figure 10. It is seen that thenucleation rates as a function of saturation ratio form almost astraight line. This can be used in determining the number of

TABLE 2 (Continued)




321.23 0.19 294.92 5.77 2.57E3321.70 1.09 294.94 5.93 1.38E4322.07 5.02 294.96 6.06 6.08E4322.50 26.00 294.98 6.21 2.99E5322.893 89.81 295.00 6.33 9.93E5




322.26 10.66 294.97 6.14 1.20E5321.84 2.23 294.95 5.99 2.64E4321.44 0.51 294.94 5.85 6.34E3321.10 0.14 294.92 5.73 1.82E3

n-Hexanol-Argon, 15-06-2000



300.32 403.09 268.91 10.68 5.49E5300.10 199.72 268.88 10.51 2.80E5299.93 103.51 268.88 10.38 1.48E5299.73 52.94 268.85 10.23 7.77E4299.53 26.31 268.85 10.07 3.96E4299.36 15.01 268.78 9.96 2.31E4299.14 8.72 268.78 9.79 1.38E4




310.49 10.31 280.25 7.65 1.92E4310.92 38.29 280.28 7.88 6.75E4311.35 172.97 280.31 8.11 2.89E5311.67 502.25 280.34 8.28 8.06E5




311.48 327.14 280.31 8.18 5.40E5311.16 103.86 280.28 8.01 1.79E5310.71 21.96 280.26 7.77 4.00E4310.32 6.64 280.23 7.57 1.27E4




320.78 47.29 290.01 6.53 9.55E4321.21 211.12 290.02 6.70 4.04E5321.65 821.92 290.05 6.88 1.49E6321.65 831.11 290.05 6.88 1.51E6321.43 430.06 290.05 6.79 8.01E5321.00 115.86 290.01 6.62 2.28E5320.60 27.57 289.98 6.46 5.70E4320.22 8.42 290.94 6.31 1.82E4

a The temperature of the preheaterTS2 (K), the temperature of the condenserTC (K), total pressureP (Pa), the average flow rateQ (cm3/s), thetemperature of the laboratoryTlab (K), the temperature of the saturatorTS1 (K), particle concentrationN (cm-3), the calculated temperature at thenucleation maximumTnuc (K), saturation ratio at the nucleation maximumSnuc, experimental nucleation rateJexp (cm-3 s-1). Temperatures of thepreheater, condenser, and laboratory as well as pressure and flow rate are averages over one measurement session of an isotherm.


molecules in the critical cluster. The molecular content of thecritical cluster was calculated by using a so-called nucleationtheorem. It was first derived by Kashchiev32 and later in moregeneral ways by Viisanen et al.33 and Oxtoby and Kashchiev.34

The nucleation theorem states

Equation 12 means that the size of the critical cluster at aconstant temperature can be determined from the experimentalresults by calculating the slopes of the nucleation rates as afunction of the saturation ratio.

In Figure 11 the number of molecules in the critical clusteris presented as a function of saturation ratio. The saturation ratiothat is used is the experimental critical saturation ratio, at anucleation rate of 105 cm-3 s-1. The theoretical predictions werecalculated by using the Gibbs-Thompson equation (5) at theexperimental critical saturation ratios. The number of moleculesin the critical cluster determined from experimental resultsdecreases as the saturation ratio rises, varying from 41 at 265K to 64 at 295 K. The experimentally determined size of thecritical cluster is well predicted by theory. This is quitesurprising because in Gibbs-Thompson equation the bulk liquidproperties are used.

Comparison with Other Measurements.The results of thenucleation rates ofn-hexanol from this study were comparedto data published by Strey et al.35 They used a device wherethe supersaturation is achieved by adiabatic expansion. Streyet al.35 measured nucleation rates of series of a alcoholsincludingn-hexanol in argon at temperatures around 258, 267,277, 286, and 295 K in the two piston expansion chamber. Thenucleation rate range is from about 106 to 109 cm-3 s-1.

The temperature dependency is clearly different: Strey et al.35

report a weaker temperature dependency than predicted by theclassical nucleation theory. In this study the temperaturedependency is slightly stronger than the theory predicts. At thelower temperature range of this study the results are similarwith Strey et al.,35 but the difference increases with temperature.Results from this study are about 5 orders of magnitude higherthan results by Strey et al.35 at 295 K.

Carrier Gas Effect. In Figure 12 the nucleation rates as afunction of saturation ratio using both helium and argon as acarrier gas are presented. It is seen that nucleation rates in argon

Figure 8. Experimental particle concentration,Nexp, of n-hexanol as afunction of theoretical predictions,Ntheor. Mean temperatures at thetheoretical nucleation rate maxima are presented in the figure. The solidlines are fitted curves.

Figure 9. Intercepts and slopes from curve fitting to isotherms in Figure8 as a function of the temperature,T, at the theoretical nucleationmaximum. The white squares are the intercepts, and the black circlesare the slopes.

Figure 10. Homogeneous nucleation rate,J, of n-hexanol as a functionof saturation ratio,S. Experimental results are presented with blacksymbols. The mean nucleation temperatures corresponding to the seriesA-G are 265.0 (A), 270.1 (B), 274.9 (C), 280.0 (D), 285.1 (E), 290.0(F), and 295.0 (G). Predictions of the classical nucleation theory areshown as dotted lines and symbols with primes corresponding toexperimental temperature. The solid black lines are regression lines.

∂ ln J∂ ln S|T ≈n* (12)

Figure 11. Molecular content of the critical cluster,n*, of n-hexanolas a function of the critical saturation ratio,Scri, at 105 cm-3 s-1. Theblack circles are the experimental results calculated from eq 12. Thesolid line is a theoretical prediction from the Gibbs-Thompsonequation.

Figure 12. Experimental nucleation rates,J, as a function of saturationratio, S. Temperatures when helium is as a carrier gas (the blacksymbols) from right to left: 270.1, 280.0, and 290.0 K. Temperatureswhen argon is as a carrier gas (the white symbols) from right to left:268.9, 280.3, 290.0 K.


are higher. The difference is reduced though at higher temper-atures. A similar kind of behavior has been reported also byothers. Vohra and Heist7 studied the critical saturation ratio ofn-propanol in various carrier gases including helium and argon.The critical saturation ratios in helium were about 10% higherthan in argon. Ha¨meri and Kulmala9 studied nucleation ratesof dibutyl phthalate in helium, argon, and nitrogen. They alsoreported that nucleation in argon was stronger than in helium.The carrier gas effect has not been found in expansion typedevices.33 This effect needs to be further studied.

Accuracy of Experimental Results.The experimental resultsmay be inaccurate for many different reasons. Most critical,considering the accuracy of the results, are the temperatures ofthe saturator and the condenser. Other possible sources of errorin experiments are the flow rate, the total pressure, and thetemperatures of the preheater and the laboratory. The temper-ature measurements were accurate within(0.05 K (accordingto the manufacturer). Calibration of the temperature sensors wasdone by the manufacturer. Temperature sensors were testedagainst each other in a heating bath. The results were withinthe accuracy given by the manufacturer. The temperaturevariation of the saturator, preheater, and condenser were withinthe accuracy of the temperature sensors during the measure-ments. Temperature variation in the laboratory during themeasurements was typically about(2 K, and pressure variationwas about(1%. Variation of the flow rate was less than(5%.The accuracy of the flow calibrator is(0.5% (according to themanufacturer). All these combined caused a change of about(70% to the calculated theoretical total particle output in theworst case. Converted into nucleation rate units this means thatthe experimental nucleation rate maximum changed about(1%,the saturation ratio at the nucleation rate maximum changedabout(1%, and the temperature at the nucleation rate maximumchanged less than(0.1%. The maximum deviation in particleconcentration from the average value was less than(50%,which was determined from tests of repeatability and steady-state measurements. This deviation can be explained by theinaccuracy of the experimental parameters. As can be seenabove, the inaccuracy of the parameters caused(70% changein the theoretical particle concentration. The variation inmeasured particle concentration caused the same amount ofvariation to the calculated experimental nucleation rate.

Error in the calculated saturation ratio depends mainly onthe accuracy of the binary diffusion coefficient. The deviationbetween the calculated and the measured binary diffusioncoefficient is less than 3%. This causes less than a 2% errorinto the calculated saturation ratio. The total inaccuracy in thecalculated saturation ratio was less than(3%. The accuracy ofthe critical saturation ratio consists of statistical error from linefitting, experimental error, and error from model calculations.The inaccuracy of the critical saturation ratio is less than 10%.The inaccuracy of the molecular contents of critical cluster wasapproximated to be less than 4%. This is the statistical error ina calculation of the slope of the log(S) - log(J) curve.

Summary and Conclusions

A laminar flow diffusion cloud chamber was developed forunary homogeneous nucleation rate studies. The main goal ofthis study was to build a device for producing reliablehomogeneous nucleation rate data. Several experiments weremade to verify and understand the operation of the device. Theseexperiments showed that the different parts of the device workas was expected. It was shown that the developed mathematicalmodel describes well the events inside the laminar flow diffusionchamber.

Homogeneous nucleation rates ofn-hexanol were measured.The results were compared to the classical nucleation theory.The saturation ratio dependency of the experiments is in goodagreement with the theory. Experimental nucleation rates havea slightly stronger temperature dependency than the classicalnucleation theory predicts. The molecular content of the criticalcluster was well predicted by the Gibbs-Thompson equation.

Homogeneous nucleation rates were compared to results byStrey et al.35 At lower temperatures the data agree quite well.However, the temperature dependency is different: This studyshows a slightly stronger temperature dependency than theclassical nucleation theory predicts. In contrast, the temperaturedependency measured by Strey et al. is weaker than the theorypredicts.

In this study the carrier gas effect was found. The nucleationrates in argon were orders of magnitude higher than in helium.A similar behavior has been found also by others in a laminarflow diffusion chamber.7,9

References and Notes

(1) Heist, R. H.; He, H.J. Phys. Chem. Ref. Data1994, 23, 781.(2) Anisimov, M. P.; Kostrovskii, V. G.; Shtein, M. S.Colloid J. USSR

1978, 40, 90.(3) Anisomov, M. P.; Cherevko, A. G.J. Aerosol Sci.1985, 16, 97.(4) Nguyen, H. V.; Okuyama, K.; Mimura, T.; Kousaka, Y.; Flagan,

R. C.; Seinfeld, J. H.J. Colloid Interface Sci.1987, 119, 491.(5) Okuyama, K.; Kousaka, Y.; Warren, D. R.; Flagan, R. C.; Seinfeld,

J. Aerosol Sci. Technol. 1987, 6, 15.(6) Anisimov, M. P.; Ha¨meri, K.; Kulmala, M.J. Aerosol Sci.1994,

1, 23.(7) Vohra, V.; Heist, R. H.J. Chem. Phys. 1996, 104, 382.(8) Hameri, K.; Kulmala, M.; Krissinel, E.; Kodenyov, G.J. Chem.

Phys.1996, 105, 7683.(9) Hameri, K.; Kulmala, M.J. Chem. Phys.1996, 105, 7696.

(10) Volmer, M.; Weber, A.Z. Phys. Chem. 1926, 62, 277.(11) Volmer, M.Z. Elektrochem. 1929, 35, 555.(12) Volmer, M.; Flood, H.Z. Phys. Chem., A1934, 170, 273.(13) Becker, R.; Do¨ring, W. Ann. Phys. (Leipzig), 1935, 24, 719.(14) Zeldovich, B.Zh. Exp. Teor. Ph. (USSR), 1942, 12, 525.(15) Frenkel, J.Kinetic Theory of Liquids; Dover: New York, 1955.(16) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N.Transport Phenomena;

John Wiley & Sons: New York, 1960.(17) Henwood, D.; Bonet, J.Finite Elements, A Gentle Introduction;

MacMillan Press: London, 1996.(18) Hung, C.-H.; Krasnopoler, M. J.; Katz, J. L.J. Chem. Phys. 1989,

90, 1856.(19) Wagner, P. E.; Anisimov, M. P.J. Aerosol Sci.1993, 24, s103.(20) Seager, S. L.; Geertson, L. R.; Giddings, J. C.J. Chem. Eng. Data

1963, 8, 168.(21) Reid, R. C.; Prausnitz, J. M.; Boling, B. E.The Properties of Gases

and Liquids, 4th ed.; McGraw-Hill: New York, 1987.(22) Hameri, K. Homogeneous Nucleation in a Laminar Flow Diffusion

Chamber. Ph.D. Thesis, Department of Physics, University of Helsinki,Helsinki, 1995.

(23) Schmelling, T.; Strey, R.Ber. Bunsen-Ges. Phys. Chem. 1983, 87,871.

(24) Strey, R.; Schmeling, T.Ber. Bunsen-Ges. Phys. Chem.1983, 87,324.

(25) Handbook of Chemistry and Physics, 79th ed.; CRC Press: NewYork, 1998.

(26) Landolt-Bornstein, Zahlenwerte und Funktionen aus Physik-Chemie- Astronimie- Geophysik- Technik; Springer: Berlin, 1960.

(27) Timmermans, J.Physico Chemical Constant of Pure OrganicCompounds; Elsevier: New Yord, 1950; Vol. I.

(28) Mason, E. A.J. Chem. Phys.1957, 27, 75.(29) Vasakova, J.; Smolik, J.Rep. Ser. Aerosol Sci.1994, 25, 1.(30) Wagner, P. E.; Strey, R.J. Chem. Phys.1984, 80, 5266.(31) Strey, R.; Wagner, P. E.; Viisanen, Y.J. Phys. Chem.1994, 98,

7748.(32) Kashchiev, D.J. Chem. Phys. 1982, 76, 5098.(33) Viisanen, Y.; Strey, R.; Reiss, H.J. Chem. Phys. 1993, 99, 4680.(34) Oxtoby, D. W.; Kashchiev, D.J. Chem. Phys.1994, 100, 7665.(35) Strey, R.; Wagner, P. E.; Schmeling, T.J. Chem. Phys. 1986, 84,

2325.


a laminar flow diffusion chamber for homogeneous nucleation studies †

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