^{1,a)}, Shinobu Tanimura

^{1,b)}and Barbara E. Wyslouzil

^{1,2,c)}

### Abstract

We determined the partial pressures*p* _{ J } _{max}, temperatures*T* _{ J } _{max}, monomer supersaturations *S* _{ J } _{max}, and characteristic times Δ*t* _{ Jmax } corresponding to the maximum nucleation rates of methanol in a supersonic nozzle. We found that *T* _{ J } _{max} increased from 202.2 K to 223.7 K as *p* _{ J } _{max} increased from 67.1 to 413.2 Pa, while the maximum nucleation rate *J* _{max} changed by less than a factor of 4 over the measurement range. Our nucleation rates appear reasonably consistent with measurements in other devices and are within one order of magnitude of the nucleation rates predicted by classical nucleation theory.

This work was supported by the National Science Foundation (NSF) under Grant No. CHE-0911144 and by the Donors of the Petroleum Research Fund administered by the American Chemical Society. Use of the Advanced Photon Source was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, under Contract No. DE AC02 06CH11357. Preliminary pressure measurements were conducted by Uta M. Dieregsweiler. We thank D. Ghosh, A. Manka, D. Bergmann, J. Wölk, K. Mullick, S. Seifert, and R. Winans for help conducting the SAXS experiments. We thank G. Wilemski for useful discussions.

I. INTRODUCTION

II. EXPERIMENTS

A. Flow apparatus and supersonic nozzles

B. Pressure trace, TDLAS, and SAXS measurements

C. Chemicals

III. DATA ANALYSIS

A. Characterizing supersonic flow with vapor association and condensation

B. Nucleation conditions and characteristic times

C. SAXS

D. Nucleation rates

IV. RESULTS AND DISCUSSION

A. The effect of boundary layer compression on the nucleation parameters

B. SAXS experiments

C. Nucleation rates

D. Comparison to results from other measurement devices

E. Comparison to classical nucleation theory

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Nucleation
- 93.0
- Polymers
- 29.0
- Fluid drops
- 20.0
- Carbon nanotubes
- 19.0
- Temperature measurement
- 14.0

## Figures

The measured pressure ratios and temperatures for (a) the condensing flow of a dilute MeOH mixture and (b) the condensing flow of a D_{2}O mixture under comparable experimental conditions (Ref. 26). The solid gray lines are the measured pressure ratios *p*/*p* _{0}, while the dashed gray lines are those expected for an isentropic expansion of the gas mixture in the absence of vapor phase association. The open circles, solid black lines, and dashed black lines, represent *T* _{TDLAS}, the centerline temperature *T*, and the temperature expected for an isentropic expansion of the gas in the absence of clustering, respectively. The shaded regions correspond to the nucleation zone.

The measured pressure ratios and temperatures for (a) the condensing flow of a dilute MeOH mixture and (b) the condensing flow of a D_{2}O mixture under comparable experimental conditions (Ref. 26). The solid gray lines are the measured pressure ratios *p*/*p* _{0}, while the dashed gray lines are those expected for an isentropic expansion of the gas mixture in the absence of vapor phase association. The open circles, solid black lines, and dashed black lines, represent *T* _{TDLAS}, the centerline temperature *T*, and the temperature expected for an isentropic expansion of the gas in the absence of clustering, respectively. The shaded regions correspond to the nucleation zone.

The nucleation conditions for the *n*-alcohols are summarized in a Volmer plot. For each alcohol the data lie along a straight line. Data from shorter chain length *n*-alcohols lie above and to the right of those of longer chain length alcohols. For methanol, the legend indicates the nozzle used (H or H2), and shows if the analysis was made with (w/) or without (w/o) considering BLC. Data from Ghosh *et al.* (Ref. 30) are analyzed without BLC.

The nucleation conditions for the *n*-alcohols are summarized in a Volmer plot. For each alcohol the data lie along a straight line. Data from shorter chain length *n*-alcohols lie above and to the right of those of longer chain length alcohols. For methanol, the legend indicates the nozzle used (H or H2), and shows if the analysis was made with (w/) or without (w/o) considering BLC. Data from Ghosh *et al.* (Ref. 30) are analyzed without BLC.

The one-dimensional SAXS spectra measured in 2005 as a function of *p* _{ v } _{0} where *p* _{0} and *T* _{0} were maintained at 59.67 kPa and 35 °C, respectively. The spectrum for 1982.7 Pa MeOH is at the true absolute intensity. The spectra at *p* _{ v } _{0} = 1677.2, 1223.6, 816.3, and 510.9 Pa are offset by factors 10^{−1}, 10^{−2}, 10^{−3}, and 10^{−4}, respectively. The solid line is the fit to the spectrum at *p* _{ v } _{0} = 1982.7 Pa assuming the droplets are a collection of polydisperse spheres that follow a Schulz distribution.

The one-dimensional SAXS spectra measured in 2005 as a function of *p* _{ v } _{0} where *p* _{0} and *T* _{0} were maintained at 59.67 kPa and 35 °C, respectively. The spectrum for 1982.7 Pa MeOH is at the true absolute intensity. The spectra at *p* _{ v } _{0} = 1677.2, 1223.6, 816.3, and 510.9 Pa are offset by factors 10^{−1}, 10^{−2}, 10^{−3}, and 10^{−4}, respectively. The solid line is the fit to the spectrum at *p* _{ v } _{0} = 1982.7 Pa assuming the droplets are a collection of polydisperse spheres that follow a Schulz distribution.

The mean radius 〈*r*〉, the width of the distribution function *σ*, and the average droplet number density *N* _{avg} are shown as a function of the initial methanol partial pressure *p* _{ v } _{0}. Here, 〈*r*〉 and *σ* are shown as a function of *p* _{ v } _{0} corresponding to the SAXS measurements, while *N* _{avg} values are shown as a function of *p* _{ v } _{0} from the corresponding PTM/TDLAS experiments. The legend indicates the nozzle used (H or H2) and shows whether the analysis included (w/) or ignored (w/o) BLC. The open circles lie almost directly on top of the circles with crosses.

The mean radius 〈*r*〉, the width of the distribution function *σ*, and the average droplet number density *N* _{avg} are shown as a function of the initial methanol partial pressure *p* _{ v } _{0}. Here, 〈*r*〉 and *σ* are shown as a function of *p* _{ v } _{0} corresponding to the SAXS measurements, while *N* _{avg} values are shown as a function of *p* _{ v } _{0} from the corresponding PTM/TDLAS experiments. The legend indicates the nozzle used (H or H2) and shows whether the analysis included (w/) or ignored (w/o) BLC. The open circles lie almost directly on top of the circles with crosses.

The predicted specific particle concentration, *Ñ* _{theory}, is calculated by integrating the nucleation pulse and scaling the results to match the experimental value at the exit of the nozzle. Measured values of *Ñ* from independent SAXS experiments are compared to each other and to *Ñ* _{theory}. Downstream of the nucleation pulse a constant value of *Ñ* implies that coagulation and Ostwald ripening are not important physical processes on the timescale of these experiments.

The predicted specific particle concentration, *Ñ* _{theory}, is calculated by integrating the nucleation pulse and scaling the results to match the experimental value at the exit of the nozzle. Measured values of *Ñ* from independent SAXS experiments are compared to each other and to *Ñ* _{theory}. Downstream of the nucleation pulse a constant value of *Ñ* implies that coagulation and Ostwald ripening are not important physical processes on the timescale of these experiments.

The experimental nucleation rates *J* _{max} of methanol as a function of the monomer supersaturation *S* _{ J } _{max}. The nucleation temperature range is also indicated here. The legend for methanol data indicate the nozzle used (H or H2) and show if the analysis is with (w/) or without (w/o) BLC.

The experimental nucleation rates *J* _{max} of methanol as a function of the monomer supersaturation *S* _{ J } _{max}. The nucleation temperature range is also indicated here. The legend for methanol data indicate the nozzle used (H or H2) and show if the analysis is with (w/) or without (w/o) BLC.

The experimental nucleation conditions in the SN and NPC (Refs. 22 and 23) are compared to those predicted by CNT (Eq. (2)). For methanol, the legend indicates the nozzle used (H or H2), and whether the analysis is with (w/) or without (w/o) BLC.

The experimental nucleation conditions in the SN and NPC (Refs. 22 and 23) are compared to those predicted by CNT (Eq. (2)). For methanol, the legend indicates the nozzle used (H or H2), and whether the analysis is with (w/) or without (w/o) BLC.

The scaled nucleation rates are plotted as a function of the scaled free energy of formation in a Hale plot. Data from the SN is compared to the NPC data of Strey *et al.* (Refs. 22 and 23). For SN data, H, or H2 indicates the nozzle used, w/BLC or w/o BLC indicates whether the analysis included BLC or not.

The scaled nucleation rates are plotted as a function of the scaled free energy of formation in a Hale plot. Data from the SN is compared to the NPC data of Strey *et al.* (Refs. 22 and 23). For SN data, H, or H2 indicates the nozzle used, w/BLC or w/o BLC indicates whether the analysis included BLC or not.

The experimental and theoretical values Ω increase with the chain length of the *n*-alcohol. The experimental values are consistently higher than the theoretical values by 0.23–0.3.

The experimental and theoretical values Ω increase with the chain length of the *n*-alcohol. The experimental values are consistently higher than the theoretical values by 0.23–0.3.

The ratio log(*J* _{exp}/*J* _{BD}) is plotted as a function of 1/*T* to compare the experimental nucleation rates to those calculated from CNT using Eq. (2). The dashed line represent 1 order of magnitude agreement between nucleation rates determined in experiments to those predicted by CNT.

The ratio log(*J* _{exp}/*J* _{BD}) is plotted as a function of 1/*T* to compare the experimental nucleation rates to those calculated from CNT using Eq. (2). The dashed line represent 1 order of magnitude agreement between nucleation rates determined in experiments to those predicted by CNT.

## Tables

The methanol nucleation parameters and corresponding nucleation rates were determined by either including BLC (w/BLC) or ignoring BLC (w/o BLC) in the analysis. The stagnation pressure *p* _{0} for all experiments is 59.67 kPa, *p* _{ v } _{0} is the partial pressure of the condensible species at the nozzle inlet, *T* _{0} is the stagnation temperature, *p* _{ J } _{max}, *T* _{ J } _{max}, *S* _{ J } _{max}, and Δ*t* _{ Jmax } are the partial pressure of monomer, temperature, monomer supersaturation, and characteristic time corresponding to maximum nucleation rate *J* _{max}, respectively, ρ_{NZ}/ρ_{VV} is the density ratio of the gas in the nucleation zone to that in viewing volume, *T* _{vv} is the temperature at the viewing volume, and *N* _{avg} is the average particle number density.

The methanol nucleation parameters and corresponding nucleation rates were determined by either including BLC (w/BLC) or ignoring BLC (w/o BLC) in the analysis. The stagnation pressure *p* _{0} for all experiments is 59.67 kPa, *p* _{ v } _{0} is the partial pressure of the condensible species at the nozzle inlet, *T* _{0} is the stagnation temperature, *p* _{ J } _{max}, *T* _{ J } _{max}, *S* _{ J } _{max}, and Δ*t* _{ Jmax } are the partial pressure of monomer, temperature, monomer supersaturation, and characteristic time corresponding to maximum nucleation rate *J* _{max}, respectively, ρ_{NZ}/ρ_{VV} is the density ratio of the gas in the nucleation zone to that in viewing volume, *T* _{vv} is the temperature at the viewing volume, and *N* _{avg} is the average particle number density.

The size distribution parameters derived by fitting scattering from a Schulz distribution of polydisperse spheres to the SAXS spectra. The stagnation pressure *p* _{0} for all experiments is maintained at 59.67 kPa, *p* _{ v } _{0} is the partial pressure of the condensible species at nozzle inlet, *T* _{0} is the stagnation temperature, 〈*r*〉 is the mean radius, *σ* is the width of the size distribution function, *I* _{0} is the scattering intensity as *q* _{SAXS} → 0, and the ratio *σ*/〈*r*〉 is the polydispersity index.

The size distribution parameters derived by fitting scattering from a Schulz distribution of polydisperse spheres to the SAXS spectra. The stagnation pressure *p* _{0} for all experiments is maintained at 59.67 kPa, *p* _{ v } _{0} is the partial pressure of the condensible species at nozzle inlet, *T* _{0} is the stagnation temperature, 〈*r*〉 is the mean radius, *σ* is the width of the size distribution function, *I* _{0} is the scattering intensity as *q* _{SAXS} → 0, and the ratio *σ*/〈*r*〉 is the polydispersity index.

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