Isothermal Nucleation Rates of n-Propanol, n-Butanol, and n-Pentanol

Article pubs.acs.org/JPCB

Isothermal Nucleation Rates of n‑Propanol, n‑Butanol, and n‑Pentanol in Supersonic Nozzles: Critical Cluster Sizes and the Role of Coagulation K. Mullick,† A. Bhabhe,† A. Manka,‡ J. Wölk,‡ R. Strey,‡ and B. E. Wyslouzil*,†,§ †

William G. Lowrie Department of Chemical and Biomolecular Engineering and §Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States ‡ Institut für Physikalische Chemie, Universität zu Köln, Luxemburger Str. 116, D-50939 Köln, Germany S Supporting Information *

ABSTRACT: We follow the nucleation of n-alcohols, n-propanol through n-pentanol, in two sets of supersonic nozzles having differing linear expansion rates. Combining the data from static pressure trace measurements with small-angle X-ray scattering we report the experimental nucleation rates and critical cluster sizes. For n-propanol, position resolved measurements clearly confirm that coagulation of the 2−10 nm size (radius) droplets occurs on the time scale of the experiment but that the effect of coagulation on the results is minimal. Under the conditions of the current experiments, our results suggest that alcohols have critical clusters that range from the dimer (n-pentanol) to the hexamer (n-propanol). We then compare the experimental results with classical nucleation theory (CNT), the Girshick-Chiu variant of CNT (GC), and the mean field kinetic nucleation theory (MKNT). Both CNT and MKNT underestimate the nucleation rates by up to 5 and 7 orders of magnitude, respectively, while GC theory predicts rates within 2 orders of magnitude. Correspondingly, the critical cluster size for all alcohols is overpredicted by factors of 2−9 with agreement improving with increasing chain length. An interesting byproduct of our experiments is that we find that the coagulation rate is enhanced by a factor of 3 over the value one would calculate for the free molecule regime. Blander and Katz,3 Girshick and Chiu,4 and Wilemski.5 These theories remain popular because they provide an intuitive picture of the nucleation process and require only macroscopic physical property data to calculate the nucleation rates and the critical cluster properties. Newer theories that incorporate molecular level information have also been developed and include the density functional theory of Oxtoby and Evans,6 the extended modified liquid droplet model of Reguera and Reiss,7 and the mean-field kinetic nucleation theory (MKNT) of Kalikmanov.8 These theories are generally more computationally intensive than CNT and its variants, and require information on intermolecular potentials to evaluate the nucleation rate.

I. INTRODUCTION First order phase transitions like melting, condensation, evaporation, and crystallization play important roles in nature, as well as in many technical applications. Nucleation, the formation of the first fragments of a new phase from the supersaturated mother phase, initiates these phase transitions via a homogeneous or heterogeneous pathway. In heterogeneous nucleation the phase transition occurs on a surface or a foreign particle, while in homogeneous nucleation the new phase emerges from density fluctuations within the supersaturated mother phase. In order to create the new phase, the nucleation process has to overcome a free energy barrier, where the maximum in the free energy barrier occurs at the critical cluster size n*. In the last century many theoretical descriptions have been developed to predict the rate at which critical clusters form. The theories most often used are the classical nucleation theory (CNT),1 or its modifications, including those of Courtney,2 © XXXX American Chemical Society

Special Issue: Branka M. Ladanyi Festschrift Received: August 18, 2014 Revised: October 30, 2014

A

dx.doi.org/10.1021/jp508335p | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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compare our results to the predictions of CNT, the GirshickChiu variant of CNT, and MKNT. The current paper is organized as follows. In section II, we briefly describe the experiments and the data analysis methods, the theory is summarized in section III, and we present the experimental results and our comparisons with the available literature data and theory in section IV. Section V summarizes our results and conclusions.

One key assumption in CNT is that the surface energy of a cluster can be described in terms of the macroscopic surface tension. Although this assumption is valid for large clusters, it is most likely invalid for very small clusters, especially clusters where all of the molecules reside on the surface. To address this problem, Kalikmanov8 developed MKNT, a nonperturbative semiphenomenological nucleation model that incorporates a statistical mechanics approach to describe the free energy of very small clusters and a classical approach to describe the free energy of large clusters. For simple substances, such as argon8−10 and nitrogen,11 the nucleation rates predicted by MKNT agree much better with experimental data than those predicted by CNT even though the critical cluster sizes estimated experimentally are significantly smaller than the predictions of either theory.10,11 How well MKNT predicts the nucleation rates for more complex molecules is still an open question and is one of the goals of the current work. The n-alcohols are an interesting class of compounds with a rich database of nucleation rate measurements made using experimental devices that span ∼20 orders of magnitude. Under most conditions, nucleation takes place at temperatures higher than the triple point where the physically properties of the n-alcohols have been measured and are readily available. Recently, Manka et al.12 and Iland et al.13 measured nucleation rates J in the nucleation pulse chamber (106 < J/cm−3 s−1 < 109; 235 < T/K < 266) for the homologous n-alcohol series ethanol through n-pentanol and derived estimates for the critical cluster sizes using the nucleation theorem. They compared their rate results to those predicted by CNT and the extended modified liquid drop-dynamical nucleation theory (EMLD-DNT) theory7 and found for n-propanol, CNT predicted rates ∼2 orders of magnitude too low, while EMLD-DNT matched the experimental results. For n-butanol and n-pentanol, CNT underestimated the nucleation rate by 3−5 orders of magnitude, while the EMLD-DNT predictions are ∼2 orders of magnitude lower at all measured nucleation temperatures. The experimental critical cluster sizes were also in better agreement with the predictions of EMLD-DNT than with those of CNT. Very recently, Görke et al.14 investigated the homogeneous nucleation of n-propanol in the laminar flow diffusion chamber (100 < J/cm−3 s−1 < 106; 270 < T/K < 300) at different total pressures. In addition to analyzing the effect of carrier gas pressure on nucleation rates, they compared their nucleation rate measurements to data from the literature, as well as to predictions of classical nucleation theory and EMLD-DNT. They found very good agreement between their experimental results and CNT at 280 K, but CNT underpredicted the measured rates at higher temperatures and overpredicted them at lower temperatures. In contrast, EMLD-DNT predicted their experimental results at 300 K and overestimated the experimental results at lower nucleation temperatures. Here we extend our earlier work measuring the nucleation rates of the n-alcohols in a supersonic nozzle (nozzle H)15 by measuring nucleation rates of n-propanol through n-pentanol in two additional nozzles (nozzles A and C). The nozzles in the current work have expansion rates higher and lower than the nozzle used by Ghosh et al.,15 and enable us to make isothermal rate measurements in the range ∼1017 < J/cm−3 s−1 < ∼1018 for temperatures between 210 and 240 K. Using these data and the first nucleation theorem,16 we estimate the number of molecules in the critical cluster, n*, under conditions where n* is less than 10. For n-propanol, we also estimate the effect that particle coagulation has on both the nucleation rates and n*. Finally, we

II. EXPERIMENTS AND DATA ANALYSIS The experimental apparatus, procedures, and data analysis methods used in this work have been discussed extensively in our earlier publications,15,17 and only the key concepts will be reviewed here. We use the apparatus described in our previous work15,17 to prepare the gas mixtures and maintain continuous stable supersonic flow in our nozzles. Summarizing briefly, the condensable of interest is vaporized, diluted with the carrier gas, and flows through the supersonic nozzle before it is exhausted to the atmosphere by vacuum pumps. As the condensable vaporcarrier gas mixture expands across the nozzle, droplets form by homogeneous nucleation and grow, depleting the vapor and adding heat to the flow. Heat addition is detected by making position resolved static pressure trace measurements (PTM) along the centerline of the nozzle. The aerosol is characterized by making small-angle X-ray scattering (SAXS) measurements near the nozzle exit. Figure S1 in the Supporting Information summarizes a typical experimental data set and shows how p, T, g, S, ⟨r⟩, and N change as a function of position in the flow. Experiments are conducted in matched sets of nozzles where each set of nozzles has the same linear expansion rate d(A/A*)/dx but different windows in the flat sidewalls. In particular, the SAXS experiments require 25 μm thick mica windows to ensure acceptable signal-to-noise ratio. Given the fragility of the mica windows, we use nozzles with CaF2 windows for the pressure trace measurements. The characteristics of the nozzles used in this work are summarized in Table 1. Nozzles A Table 1. Linear Expansion Rates of Nozzles A, C, and Ha nozzle

window

d(A/A*)/dx (cm−1)

A (A2) C3 (C2) H (H2)

CaF2 (Mica) CaF2 (Mica) CaF2 (Mica)

0.045 0.079 0.059

a

Nozzles A and C were used in the nucleation rate measurements presented here. Nozzles H and C were used in position-resolved measurements of coagulation.

and C were used to measure the nucleation rates from which we determine the critical cluster sizes, and nozzle C was also used in the position resolved measurements. Nozzle H was used in the experiments of Ghosh et al.,15 as well as in the position resolved measurements presented here. To establish the effective area ratio of the nozzle, we first measure the pressure profile of the pure carrier gas. Next we measure the pressure profile of the condensable−carrier gas mixture. The other state variables of the condensing flow, that is, the condensate mass fraction g, density ρ, temperature T, and velocity u, are determined by solving the diabatic flow equations18 using the effective area ratio and the condensing flow pressure ratio as input. The SAXS measurements are performed at the 12 ID_C beamline, Advanced Photos Source, Argonne National Laboratories, Argonne, IL. We use a ∼0.04 mm2 beam of 12 keV B



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X-rays with a wavelength spread of Δλ/λ of 10−4 and a sample to detector distance of 0.851 m. The calibration procedures used to determine the wave vector range and the absolute intensity are the same as those described in Ghosh et al.15 The one-dimensional, background subtracted, SAXS scattering spectra are analyzed assuming the aerosol follows a Schulz distribution of polydisperse spheres.19 Fits to the data determine the average particle size ⟨r⟩, the width of the distribution δ, and the intensity as the momentum transfer vector q → 0, I0. The number density N, is determined from the fit parameters using

JCNT =

−H(n) = − n ln S + θ∞(n2/3 − 1)

JGC =

2 ⎛ 16πvm2σ 3 ⎞ 2σ ⎛⎜ pv ⎞⎟ ⎛ e θ∞ ⎞ ⎟ vm ⎜ ⎟ exp⎜ − 3 2 πm ⎝ kT ⎠ ⎝ S ⎠ ⎝ 3(kT ) (ln S) ⎠

(6)

where eqs 4 and 6 differ only by the factor (eθ∞/S). In both CNT and the GC approach, the number of molecules in the critical cluster, n*, is the same and is given by the Gibbs−Thomson equation,

* = nGT (2)

32πvm2σ 3 3(kT ln S)3

(7)

The rate expression for the mean-field kinetic nucleation theory of Kalikmanov8,9 is given by

where ρNZ/ρVV corrects for the change in gas density between the nucleation zone (ρNZ), that is, the region where the particles are formed, and the viewing volume (ρVV), that is, the region where the particle number density is measured. Estimates for the uncertainties in S and J(S,T) are also based on the analysis of Kim et al.20 The condensables include n-propanol (≥99.9% purity, SigmaAldrich CHROMASOLV), n-butanol (≥99.9% purity, SigmaAldrich CHROMASOLV), and n-pentanol (≥99% purity, Sigma-Aldrich, ACS Reagent). The carrier gas is nitrogen (>99.99 purity, Airgas and Praxair). The physical properties including vapor pressure, liquid density, heat capacity, surface tension, and the critical properties, that are used to calculate theoretical predictions, critical cluster sizes, and to determine the state variables of the flow, are those used by Ghosh et al.15 Additional parameters used in the calculation of nucleation rates from MKNT are summarized in Appendix A.

JMKNT

⎡∞ ⎤−1 − H ( n ) ⎥ = Akin ⎢∑ e ⎢⎣ n = 1 ⎥⎦

(8)

where Akin is the kinetic prefactor proposed by Katz and Wiedersich.21 The expression for H(n) is −H(n) = −n ln S + θmicro[n ̅ s(n) − 1]

(9)

In eq 9, θmicro and n ̅ (n) are the reduced microscopic surface tension and number of surface molecules in a cluster containing n molecules. There is no surface contribution to the free energy of cluster formation for the monomer in this theory because n ̅ s (1) = 1. The details of MKNT theory, including the procedure to calculate θmicro and n ̅ s (n), are available in the papers by Kalikmanov.8,9 The approach we have adopted to evaluate this theory when the critical cluster sizes are small, is detailed in the Supporting Information of Sinha et al.10 For MKNT, the critical cluster size n* corresponds to the minimum of H(n), as described in Sinha et al.10 The usefulness of comparing the predictions of JGC to JMKNT is that the kinetic prefactors differ at most by (n*GC/n*MKNT)2/3 a factor that is close to 1, and both theories return the monomer concentration for n = 1. Thus, differences in the nucleation rates primarily reflect differences in n* and the formation energy of the clusters of size n, H(n). s

III. THEORY The experimental results in this paper will be compared to the predictions of three nucleation theories: the Classical Nucleation Theory (JCNT) of Becker and Döring,1 the modification of CNT proposed by Girshick and Chiu4 (JGC), and the Mean Field Kinetic Nucleation Theory (JMKNT) of Kalikmanov.8 CNT was chosen because many papers compare experimental data to this expression, and thus, it is easier to determine whether the current measurements are consistent with other existing data sets. In CNT, the formation energy of the clusters of size n, H(n) can be written in terms of the reduced macroscopic surface tension θ∞ = σs1/(kT) as −H(n) = − n ln S + θ∞n2/3

(5)

Incorporating these change into the rate expressions yields

(1)

where Z = (⟨r⟩/ δ)2 − 1 and Δρ is the contrast factor related to the difference in the scattering length density between the condensed alcohol droplets and the gas mixture. We follow the procedure outlined in Kim et al.20 to compute the characteristic times Δt and conditions (T, S) corresponding to the maximum nucleation rate, and evaluate the experimental nucleation rate using

N ρNZ J (S , T ) = Δt ρVV

(4)

where m is the molecular mass. Equation 4 has been criticized for leading to a cluster distribution function that does not satisfy the law of mass action and is not self-consistent in the sense that the cluster distribution function does not return the monomer concentration for n = 1. To correct these inconsistencies, Girshick and Chiu4 included the factor S−1 in the prefactor and suggested the following modified expression for H(n)

⎛ 4π ⎞2 ⎛ (Z + 6)(Z + 5)(Z + 4)(Z + 3)(Z + 2) ⎞ I0 = N ⎜ ⎟ ⎜ ⎟ ⎝ 3 ⎠⎝ (Z + 1)5 ⎠ × ⟨r ⟩6 (Δρ)2

2 ⎛ 16πvm2σ 3 ⎞ 2σ ⎛⎜ pv ⎞⎟ ⎟ exp⎜ − vm 3 2 πm ⎝ kT ⎠ ⎝ 3(kT ) (ln S) ⎠

IV. RESULTS AND DISCUSSION The experiments in nozzles A and C used to determine n*, started from a stagnation temperature T0 = 40.0 ± 1.0 °C and a stagnation pressure p0 = 30.00 ± 0.02 kPa. In the figures that follow, data measured in nozzle A are shown as filled symbols, while data measured in nozzle C are shown as open symbols. The position resolved experiment started from p0 = 30.00 ± 0.02 kPa with T0 = 45.0 ± 1.0 °C in nozzle H and T0 = 50.0 ± 1.0 °C in nozzle C.

(3)

(36πν2m)1/3

where s1 = is the surface area of a monomer, k is the Boltzmann constant, and σ and νm are the surface tension of the bulk vapor−liquid interface and the molecular volume, respectively. The expression for JCNT is given by C



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Table 2. Conditions Corresponding to the Maximum Nucleation Rate Jmax in Nozzle Aa alcohol n-propanol n-propanol n-propanol n-propanol n-butanol n-butanol n-butanol n-pentanol n-pentanol n-pentanol n-pentanol

pvo, kPa 0.09 0.17 0.33 0.61 0.07 0.13 0.23 0.02 0.04 0.06 0.11

T, K 209.87 220.13 229.78 240.47 220.63 229.54 239.69 209.72 219.96 229.94 240.45

pJmax, kPa −02

2.10 × 10 4.82 × 10−02 1.07 × 10−01 2.38 × 10−01 2.02 × 10−02 4.18 × 10−02 8.91 × 10−02 4.28 × 10−03 9.14 × 10−03 2.03 × 10−02 4.35 × 10−02

SJmax 50 27 17 11 58 35 20 436 170 85 44

Δt, s −05

1.57 × 10 1.33 × 10−05 9.95 × 10−06 9.87 × 10−06 1.18 × 10−05 9.29 × 10−06 8.40 × 10−06 2.40 × 10−05 2.15 × 10−05 1.26 × 10−05 9.11 × 10−06

ρNZ/ρVV

N, cm−3

Jmax, cm−3·s−1

1.02 1.14 1.25 1.35 1.17 1.28 1.40 1.04 1.16 1.30 1.44

1.35 × 10 2.88 × 1012 1.45 × 1012 8.59 × 1011 3.36 × 1012 2.05 × 1012 1.22 × 1012 4.00 × 1012 2.88 × 1012 2.48 × 1012 1.72 × 1012 12

8.81 × 1016 2.49 × 1017 1.82 × 1017 1.18 × 1017 3.32 × 1017 2.82 × 1017 2.03 × 1017 1.73 × 1017 1.56 × 1017 2.55 × 1017 2.72 × 1017

a All expansions started from T0 = 40 ± 1.0 °C and p0 = 30 ± 0.02 kPa. The reported variables include the initial partial pressure of the condensible pvo, the temperature T, pressure p, and supersaturation S corresponding to Jmax, the characteristic time for nucleation Δt, the ratio of the gas mixture densities in the nucleation zone ρNZ and the viewing volume ρVV, the number density N in the viewing volume, and the measured nucleation rate Jmax. The average particle radius and distribution widths are given in Figure S2 of the Supporting Information.

Table 3. Conditions Corresponding to the Maximum Nucleation Rate in Nozzle Ca alcohol n-propanol n-propanol n-propanol n-propanol n-butanol n-butanol n-butanol n-butanol n-pentanol n-pentanol n-pentanol

pvo, kPa 0.11 0.21 0.39 0.72 0.05 0.09 0.16 0.28 0.04 0.08 0.14

T, K 209.80 219.75 230.11 240.01 210.24 220.04 229.72 239.89 220.57 229.42 239.73

pJmax, kPa −02

2.50 × 10 5.68 × 10−02 1.31 × 10−01 2.70 × 10−01 1.15 × 10−02 2.42 × 10−02 5.32 × 10−02 1.16 × 10−01 1.27 × 10−02 2.54 × 10−02 5.31 × 10−02

Δt, s

SJmax 60 33 20 13 162 75 43 26 215 114 59

−05

1.10 × 10 9.00 × 10−06 6.49 × 10−06 6.02 × 10−06 1.17 × 10−05 8.82 × 10−06 5.26 × 10−06 5.44 × 10−06 1.22 × 10−05 7.22 × 10−06 5.50 × 10−06

ρNZ/ρVV

N, cm−3

Jmax, cm−3·s−1

1.33 1.47 1.62 1.72 1.35 1.50 1.66 1.84 1.52 1.67 1.85

2.41 × 10 2.29 × 1012 1.52 × 1012 1.11 × 1012 6.48 × 1012 2.88 × 1012 1.91 × 1012 1.43 × 1012 3.29 × 1012 2.08 × 1012 1.58 × 1012 12

2.92 × 1017 3.75 × 1017 3.79 × 1017 3.19 × 1017 7.45 × 1017 4.90 × 1017 6.01 × 1017 4.82 × 1017 4.09 × 1017 4.79 × 1017 5.31 × 1017

a All expansions started from T0 = 40 ± 1.0 °C and p0 = 30 ± 0.02 kPa. The variables measured are the stagnation pressure pvo, the temperature T, pressure p, and supersaturation S according to the maximum nucleation rate Jmax, the characteristic time for nucleation Δt, the ratio of the gas mixture densities in the nucleation zone ρNZ and the viewing volume ρVV, the number density N in the viewing volume, and the measured nucleation rate Jmax. The average particle radius and distribution widths are given in Figure S2 of the Supporting Information.

A. Pressure Trace Measurements. Tables 2 and 3 summarize the key PTM and SAXS results obtained in nozzles A and C for all three alcohols, as well as the associated nucleation rates. Figure 1a illustrates the pressure and temperature corresponding to the maximum nucleation rate (pJmax,TJmax) data in a Volmer plot, a plot of pressure (p) versus inverse temperature (1/T), where the lines in Figure 1a are meant to guide the eye. Given the higher expansion rate of the gas mixture in nozzle C, experiments conducted in this nozzle can probe the metastable region more deeply than experiments in nozzle A. Thus, in nozzle C nucleation should occur at higher condensable partial pressures at a given temperature or at lower temperatures at a given pressure, and the maximum nucleation rates should be higher than in nozzle A. Although the (pJmax,TJmax) data for the two nozzles are close, the expected trend with respect to temperature and pressure is clearly observed. Furthermore, since the vapor pressures of the condensing species decrease from n-propanol through n-pentanol, the values of pJmax also decrease from n-propanol through n-pentanol. Figure 1b demonstrates that when the pressures and temperatures are normalized by their critical values, the data corresponding to the different alcohols are extremely consistent. In fact data for different alcohols measured in a single nozzle are generally closer together than data for a single alcohol in different nozzles.

The experiments were conducted so that the values of TJmax were close to the desired values 220, 230, and 240 K, and all points were within 0.5 K of the targets. To determine the values of pJmax that correspond exactly to T = 220, 230, and 240 K, we interpolated the data using log(1/T) coordinates in Figure 1a. B. Nucleation Rates. Figure 2 illustrates the variation in the measured nucleation rates (symbols) with the experimental supersaturations (data from Tables 2 and 3). Error bars were estimated based on the analysis of Kim et al.,20 and errors in S only reflect the uncertainty in T, not any underlying uncertainty in the equilibrium vapor pressure correlations. For a given nozzle and alcohol, the measured nucleation rates vary by less than a factor of 10 over the range of temperatures and supersaturations investigated. The solid lines in Figure 2 are power law fits to the isothermal (Jmax,S) values after we corrected S for the slight differences in T between the experimental and target values as described in section IV.D. These corrections are the reason why the lines do not always pass through the center of the symbols. The gray symbols in Figure 2 correspond to the nucleation rate measurements of Ghosh et al.,15 where we obtained the rates at 220, 230, and 240 K by interpolating the published values. Ghosh et al.’s experiments were conducted in nozzle H, a nozzle with an expansion rate that lies between those of nozzle A and nozzle C, and we therefore expect the nucleation rates to lie between the current measurements. In most cases the older data agree with D



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Figure 1. (a) pJmax is plotted as a function of the inverse of T measured for three alcohols (CnH2n+1OH, n = 3−5) in two nozzles with different expansion rates. The closed symbols are data measured in nozzle A and the open symbols are data measured in nozzle C. (b) The data scale well when normalized by the critical properties. The differences between data from nozzles A and C, corresponding to different nucleation rates, are generally larger than the differences between the alcohols in the same nozzle.

Figure 2. Experimental nucleation rates Jmax as a function of the maximum supersaturation SJmax. The horizontal error bars represent a ±10% change in supersaturation. The vertical bars represent a factor of 2 in the experimental nucleation rate. The lines are fits to the isothermal nucleation rate data (see Table 5).

measured near the nozzle exit (x = 6.5 cm in these experiments) may underestimate the number of particles produced during the nucleation pulse if coagulation is severe. Furthermore, corrections depend both on the values of N0 and the time available for coagulation between the end of the nucleation pulse and the measurement location (viewing volume). Since the experimental critical cluster sizes are calculated from the slope of the nucleation rate curves, it is important to consider the effect that coagulation has on nucleation rates and, thereby, on experimentally determined n* values. In general, the change in particle number density can be written as

the new data given the factor of 2 uncertainty quoted for all of the rates. In Figure 2 we have omitted the error bars associated with the Ghosh data to maintain clarity. Given the challenges associated with these experiments, the overall agreement, although not perfect, is still quite reasonable. C. Coagulation. One potential issue in supersonic nozzle nucleation experiments is the effect that coagulation may have on the estimated nucleation rates when aerosol number densities are above ∼1012 cm−3. As illustrated in Figure 3a, the position resolved measurements made with n-propanol in nozzles H and C, show that after reaching a maximum the number densities decrease by as much as a factor of 3 as the aerosol flows downstream. For these experiments, the stagnation conditions in nozzle H were p0 = 30 kPa, T0 = 45 °C, and pv0 = 0.39 kPa, and in nozzle C were p0 = 30 kPa, T0 = 50 °C, and pv0 = 0.43 kPa. Part of the decrease in N simply reflects the continued expansion of the gas mixture, an effect that we account for when calculating the nucleation rate. If this were the only effect then N/N0 should follow ρ/ρ0 exactly. Figure 3b, where the triangles correspond to N/N0 and the squares correspond to the density ratio, illustrates that this is not the case for the two position resolved measurements made here. Since the number density directly enters the nucleation rate calculation, number densities

N (t0) dρ K dN = − coa N 2 dt ρ(t0) dt 2

(10)

In eq 10, the first term describes the effect of dilution, the second term accounts for the effect of coagulation, and Kcoa is the coagulation coefficient.22,23 This equation can be integrated numerically with N(t0) and Kcoa adjusted to fit the experimental data. As illustrated in Figure 3b, good fits over the entire range are possible. For these two cases, the values of Kcoa are very close to each other; 6.0 × 10−9 cm3 s−1 for nozzle H and 6.5 × 10−9 cm3 s−1 E



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Figure 3. (a) Number densities and particle sizes as a function of the distance from the position corresponding to Jmax denoted as xJmax. After an initial period of rapid growth, particles grow more slowly driven in part by coagulation. (b) Normalized particle number densities N/N(t0) from SAXS and the normalized gas densities ρ/ρ(t0) from the corresponding PTM as a function of time-of-flight t − t0 through the nozzle. The reference time, t0 corresponds to the time when N is maximized. The solid lines correspond to a model that incorporates coagulation and dilution, and is used to determine reasonable values for N0 and the coagulation coefficient Kcoa, as described in the text.

Table 4. Comparison of the Propanol Nucleation Rates Calculated Correcting for Density Alone (Jmax) and for Density and Coagulation (Jmax,corrected) nozzle A

nozzle C

T, K

Jmax, cm−3 s−1

Jmax,corrected, cm−3 s−1

209.87 220.13 229.78 240.47

8.81 × 10 2.49 × 1017 1.82 × 1017 1.18 × 1017

8.81 × 10 3.08 × 1017 2.06 × 1017 1.18 × 1017

16

16

for nozzle C. We note our approach assumes Kcoa is constant and that more sophisticated approaches are possible.24,25 Under the conditions found in the nozzles, coagulation occurs in the free molecular regime. Furthermore, the width of the particle size distribution δ around the mean radius is small (δ/⟨r⟩ ≈ 25%), and thus, the aerosol can be considered monodisperse. The coagulation coefficient due to Brownian motion in the free molecular regime can therefore be calculated theoretically using26,27 ⎛ 6kTddrop ⎞ ⎟⎟ Kcoa = 4β ⎜⎜ ⎝ ρl ⎠

Jmax, cm−3 s−1

Jmax,corrected, cm−3 s−1

209.80 219.75 230.11 240.01

2.92 × 10 3.75 × 1017 3.79 × 1017 3.19 × 1017

3.74 × 1017 5.46 × 1017 4.26 × 1017 3.19 × 1017

17

To examine the effect of coagulation on propanol nucleation rates and critical cluster sizes we integrated eq 10 using Kcoa = 6.0 × 10−9 cm3 s−1. We started the integration at the point where droplet growth is essentially complete and the flow begins to expand again. This criterion corresponds to ∼1 (0.6) cm downstream of the position where the nucleation rate is maximized in nozzle H (C). We then adjusted N0 until the value predicted at x = 6.5 cm matches the value measured by SAXS. These values of N0, and the appropriate values of the density ratio, were then used to estimate the new values of J. As demonstrated in Table 4 for propanol, the nucleation rates increase at most by ∼50% and are, therefore, well within our estimated factor of 2 error bars. Furthermore, in estimating the critical cluster size, the effect of increases in the nucleation rate of nozzle C are offset in part by the corresponding increases in the nucleation rate of nozzle A. Thus, the increase in the experimental critical cluster sizes due to coagulation is less than ∼1. We note that when nucleation occurs close to the viewing volume (T ∼ 210 K, nozzle A), the rates do not change because there is no time for coagulation. Surprisingly, at the highest nucleation temperatures, rates are also quite constant. This is because the characteristic time is short and, thus, the aerosol number density is low enough that coagulation is minimal. D. Critical Cluster Sizes. To calculate the critical cluster sizes we followed the procedure outlined by Kim et al.20 and first corrected the supersaturation for the slight differences between the actual temperatures and the desired values. The corrected values of S are reported in Table 5 and were used to determine the black lines in Figure 2. We did not correct the nucleation rates, because for a given nozzle, these very minor changes in

(11)

where ρl is the density of the liquid droplet and β is the coalescence efficiency. For the droplets in the nozzle it is assumed that every collision results in coagulation of the two colliding particles, since van der Waals forces should guarantee the sticking, and thus, β is taken to equal one.28 The fact that Kcoa does not vary significantly between the two experiments is consistent with the fact that in these experiments the droplet sizes are close (Figure 3a), as is the temperature range over which coagulation occurs (240−220 K). The available experimental literature data on coagulation for particles in this size range28−32 generally find an enhancement in the coagulation rate over that predicted by eq 11, that is, the experimental coagulation rates are higher than the predicted rates, with enhancement factors EF given by EF = Kcoa(exp)/Kcoa(theory)

T, K

(12)

in the range of 2−9. In our experiments we find EF ∼ 3. F



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form hydrogen-bonded aggregates in the gas phase, like the n-alcohols, that are not compact spherical objects. Recent work by Nellas et al.33 illustrates clearly that the change in free energy between cluster sizes, δΔG(n), for n-pentanol is not a linear function of n2/3 − (n − 1)2/3, as predicted by CNT once n < ∼15, and that the deviations from the expected behavior become more severe as the temperature decreases to the range of those used in the current experiments. Furthermore, Girshick34 showed that the free energies of the smallest clusters can affect both the height of the free energy barrier and its location. Both observations are consistent with the mismatch we observe between the experimental values of n* and those predicted by CNT. In contrast to CNT, MKNT was designed to calculate free energies down to a single monomer.8 Practically speaking, however, in our current scheme for implementing MKNT,10 the critical cluster size is always larger than the coordination number N1, typically 6 or 7 molecules. This is because when n < N1 all of the molecules in the cluster are considered to be surface molecules n ̅ s (n) = n, and H(n) is a linear function of n. Thus, for the smallest clusters, the maximum in −H(n) always corresponds to N1. In Figure 4, the experimental critical cluster size for n-propanol is plotted against the value determined by the Gibbs−Thomson

Table 5. Experimental Critical Cluster Sizes (n*) at 220, 230, and 240 K Based on the Measurements Made in Nozzles A and C are Compared to Those Calculated Using the Gibbs− Thomson Equation (n*GT) and MKNT (n*MKNT)a alcohol

T, K

SJmax (A)

SJmax (C)

n*

n*GT

n*MKNT

n-propanol n-propanol n-propanol n-butanol n-butanol n-butanol n-pentanol n-pentanol n-pentanol

220 230 240 220 230 240 220 230 240

28.81 17.74 10.92 58.73 34.12 19.83 168.43 87.23 45.17

34.13 20.81 12.68 74.57 43.68 25.58 220.48 112.48 57.39

2.4 4.6 6.7 1.6 3.1 3.4 3.6 2.5 2.8

10.6 13.3 17.7 9.1 11.4 14.9 7.0 8.8 7.0

16 21 26 14 18 22 14 16 21

a The temperature (T) and supersaturation (S) correspond to the maximum nucleation rate (Jmax).

temperature do not affect the rate significantly. We then determined the critical cluster sizes via the first nucleation theorem,16 and our results are summarized in Table 5 for the three n-alcohols. As expected, the critical clusters are very small, with n* always less than 10 molecules and, in most cases, less than 5 molecules. As detailed in Appendix B, we estimate that the uncertainties in the values of n* are at most a factor of 2. The number of molecules in the critical clusters increases slightly with increasing temperature and decreases as the alcohol chain length increases. In contrast, the critical cluster radii, calculated using ⎛ 3n* v ⎞1/3 exp m * ⎟⎟ r = ⎜⎜ ⎝ 4π ⎠

(13)

is more constant with all values lying in the range 0.43 < r*/nm

Isothermal Nucleation Rates of n-Propanol, n-Butanol, and n-Pentanol

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