I Homogeneous Nucleation of Carbon Dioxide (CO2) in Supersonic

Home , Nucleation

Homogeneous Nucleation of Carbon Dioxide (CO2) in Supersonic Nozzles

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of

Philosophy in the Graduate School of The Ohio State University

Kayane Kohar Dingilian, M.S.

Graduate Program in Chemical Engineering

The Ohio State University

2020

Dissertation Committee:

Barbara E. Wyslouzil, Advisor

Nicholas Brunelli

Isamu Kusaka

Kayane Kohar Dingilian

2020

ii ABSTRACT

Carbon dioxide (CO2) is an important greenhouse gas that contributes to global warming. To combat the rising emissions of CO2 into the atmosphere, scientists and researchers have devised several methods of carbon capture and storage (CCS), including the use of membranes to trap CO2 molecules, valves to condense CO2 from a mixture of gases, etc.

Supersonic separation is a novel method of gas-gas separation that has been used to separate natural gas from other gases. It has been suggested for and is being currently studied as a method for the removal of CO2 from flue gas before it enters the atmosphere. Supersonic separation relies on the condensation of CO2 clusters into particles large enough to be inertially separated, requiring a size of approximately 1 micrometer in diameter. In order to effectively design hardware to capture CO2 using this mechanism, we need to study and collect fundamental nucleation properties of CO2 and quantify the process under supersonic flow conditions.

To this end, we studied the condensation of CO2 in two supersonic nozzles of differing expansion rates, T1 and T3, and sought to quantify the onset of nucleation, particle size distributions, and aerosol number densities. First, we confirmed that homogeneous nucleation of CO2 could not take place in nozzle T1 when expansions started from a stagnation temperature of 20°C, no heat release was observed in pressure trace measurements at 7.0 to 11 mol% CO2. Lowering the stagnation temperature to 10°C showed some evidence of heat release, but it was uncertain to what extent that was due to condensation of particles or whether the nozzle

i overexpansion had caused a shock to increase the temperature and pressure of the flow. Analysis of the saturation curve suggests the nucleation event was incomplete, and lower temperatures were required for a full characterization.

Second, we performed a series of pressure trace measurements (PTM) in nozzle

T3 for concentrations of CO2 in Ar ranging from 0.5 to 39 mol% with a stagnation pressure of 458 Torr (61 kPa) and a stagnation temperature of 20°C. We successfully observed the complete nucleation event over that range and characterized the flow conditions at the onset of nucleation at pressures ranging from 7.45 to 793 Pa and temperatures between 66.5 and 92.3 K. We also observed a wide variety of saturations at onset, ranging from 2290 to 1.49 x 106. An arbitrary cutoff was made, denoting systems of 3.0 mol% CO2 and higher to be of “mid-high” flow rates, and for systems at

2.0 mol% CO2 and lower to be of “low” flow rates. We observed a consistent shift in onset conditions and nucleation rates at approximately 12 mol% CO2 as increases in the amount of CO2 in the system contributed to an increasingly softer and warmer expansion.

We further characterized the mid-high (3.0 to 39 mol%) flow region using small angle X-ray scattering (SAXS) and a limited number of Fourier transform infrared (FTIR) spectroscopy experiments. Experimental number densities were found to be on the order of 1012 cm-3 and nucleation rates on the order of 1017 cm-3s-1. Extrapolating from these experiments into the low CO2 flow regime, we obtained similar values for estimates of the number density and nucleation rates. FTIR spectra of the system of 12 mol% CO2 showed a transition from the gas phase to the solid crystalline phase, and the spectra suggest these particles are cubic in shape.

ii Third, we compared our experimental results for the onset conditions and nucleation rates with theory, choosing combinations of self-consistent nucleation theory

(SCNT) or simulation-based theory (ST) and surface tension relations by Quinn or based on fitting the data to the Lielmezs-Herrick (L-H) functional form. We also applied the nonisothermal correction factor of Feder et al. to account for incomplete thermalization between the condensing clusters and the carrier gas. The values of the nonisothermal correction factor ranged from 0.015 to 0.391, resulting in nucleation rates that ranged from approximately 1018 cm-3s-1 to 1019 cm-3s-1. In general, using the Quinn surface tension expression with either nucleation theory overestimates the pressures required to initiate nucleation, and using L-H with either nucleation theory underestimates them. We found, however, that ST theory captures the temperature dependence of the experimental data slightly better than SCNT does.

We also developed a protocol for ensuring that the nucleation we observed in our experiments was purely homogeneous or in the near-homogenous limit by flowing carrier gas through the nozzle for 1 hour at 90 SLM to remove any trace amounts of condensable vapors. An independent analysis of the dry traces from our experiments showed that we could reproduce the background flow conditions to a maximum deviation of pressure ratio of 0.00046 and temperature of 0.307 K.

Overall, we observed excellent agreement between our experiments and those of higher pressure and temperature conditions, as well as promising overlap with those at lower ones. The free energy barrier to nucleation in our experiments appears to be similar to those of other experiments in supersonic nozzles and can serve to bridge the gap between different nucleation regimes.

iii

To my family, who instilled and nurtured my passion for learning,

And to the faculty and students, who fostered my passion for teaching others.

iv ACKNOWLEDGEMENTS

First and foremost, I would like to thank my advisor, Dr. Barbara Wyslouzil, for her guidance and mentorship of my graduate career. She helped me grow and develop as a researcher and also encouraged me to explore myself as an academic professional through participation in teaching opportunities and conferences. I truly believe I could not have found a better advisor for my graduate experience.

Second, I extend my gratitude to the faculty with whom I developed my skills as a teaching assistant and lecturer at The Ohio State University – namely Dr. David

Tomasko, Dr. David Wood, Dr. Jeffrey Chalmers, Dr. Andrew Tong, and Dr. Kurt

Koelling. They have all provided me with opportunities to perform duties beyond those of grading and holding office hours, including providing full or abridged lectures and becoming involved in the courses at a deeper instructional level. I have gained valuable teaching experience and learned lessons that will help me as I progress in my career.

Third, I would like to thank those who have served on my committees, at all levels of exams, especially Dr. Nicholas Brunelli, Dr. Isamu Kusaka, and Dr. Heather

Allen, for encouraging me to think critically about my research and reflect on the greater significance of my findings.

Fourth, I would like to thank my previous advisor and mentors from the California

Institute of Technology, Dr. Richard Flagan and Dr. Michael Vicic, for their support beginning from my undergraduate years and extending through the years afterwards.

Through them, I’ve been allowed to establish a greater network of scientists and researchers as well as pursue my academic dreams.

v Fifth, I would like to thank Soenke Seifert and Randall Winans at Argonne

National Laboratory for their assistance in setting up and making measurements using the beam at the Advanced Photon Source. Thank you to Martina Lippe for visiting and helping conduct experiments as well.

Sixth, my appreciation goes out to my labmates, both former and present, including: Dr. Andrew Amaya, Dr. Yensil Park, Dr. Kehinde Ogunronbi, Tong Sun, Jiaqi

Luo, and visiting scholar Jiang Bian. Much of my training in the experiments was performed by Andrew and Kehinde. I also had the pleasure of mentoring and training two wonderful undergraduate researchers myself – Lahari Pallerla and Gabrielle

Adams.

Finally, I would like to thank the Dingilian family – my father, mother, sister

Armine, and brother Hovanness, for their continued presence and support. None of my efforts would have been possible without the strong backbone of my family.

vi VITA

October 2012 – June 2016 Bachelor of Science,

Chemical Engineering

California Institute of Technology

Pasadena, California, USA

August 2016 – December 2019 Master of Science,

Chemical Engineering

The Ohio State University

August 2017 – May 2019 Graduate Teaching Assistant

January 2020 – May 2020 The Ohio State University

August 2016 – present Graduate Research Associate

The Ohio State University

vii PUBLICATIONS

1. K. K. Dingilian, R. Halonen, V. Tikkanen, B. Reischl, H. Vehkamäki, B. E.

Wyslouzil. Homogeneous nucleation of carbon dioxide in supersonic nozzles I:

experiments and classical theories. Physical Chemistry and Chemical Physics.

2020.

2. R. Halonen, V. Tikkanen, B. Reischl, K. K. Dingilian, B. E. Wyslouzil, H.

Vehkamäki. Homogeneous nucleation of carbon dioxide in supersonic nozzles II:

molecular dynamics simulations and properties of nucleating clusters. Physical

Chemistry and Chemical Physics, submitted.

FIELDS OF STUDY

Major Field: Chemical Engineering

Aerosol Physics

Physical/Analytical Chemistry

viii TABLE OF CONTENTS

Abstract…………………………………………………………………………………………… i

Acknowledgements…………………………………………………………………………….. v

Vita………………………………………………………………………………………………. vii

List of Tables……………………………………………………………………………………. xi

List of Figures…………………………………………………………………………………. xiii

Chapter 1: Introduction………………………………………………………………………… 1

Chapter 2: Nucleation Theory…………………………………………………………………. 7

Section 2.1: Thermodynamics of Nucleating Clusters……………………………… 8

Section 2.2: The Classical Nucleation Rate………………………………………... 12

Section 2.3: Self-Consistent Nucleation Theory…………………………………… 17

Section 2.4: The Non-Isothermal Correction Factor………………………………. 18

Chapter 3: Materials and Methods………………………………………………………….. 24

Section 3.1: Physical Properties of Ar and CO2…………………………………….24

Section 3.2: Nozzles T1 and T3……………………………………………………... 26

Section 3.3: Experimental Setup…………………………………………………….. 27

Section 3.4: Pressure Trace Measurements……………………………………….. 29

Section 3.5: Small Angle X-Ray Scattering………………………………………… 34

Section 3.6: Fourier Transform Infrared Spectroscopy…………………………… 35

Section 3.7: Nucleation Rate Analysis……………………………………………… 37

Chapter 4: Results and Discussion…………………………………………………………. 39

Section 4.1: Nozzle T1……………………………………………………………….. 39

Section 4.2: Nozzle T3 – Medium and High Flow…………………………………. 42

ix Section 4.2.A: Pressure Trace Measurements…………………………….. 42

Section 4.2.B: Addressing Contamination………………………………….. 47

Section 4.2.C: Repeatability…………………………………………………. 49

Section 4.2.D: Small Angle X-Ray Scattering……………………………… 50

Section 4.2.D.i: Position-Resolved Measurements………………... 50

Section 4.2.D.ii: Fixed-Position Measurements……………………. 55

Section 4.2.E: Experimental Nucleation Rates…………………………….. 59

Section 4.2.F: Discrepancy between SAXS and PTM…………………….. 61

Section 4.2.G: Fourier Transform Infrared Spectroscopy………………… 64

Section 4.3: Nozzle T3 – Low Flow…………………………………………………. 71

Section 4.3.A: Pressure Trace Measurements…………………………….. 71

Section 4.3.B: Estimation of Nucleation Rates…………………………….. 76

Section 4.3.C: Comparison to Krohn (2020)……………………………….. 80

Section 4.4: Comparisons with Classical Nucleation and Scaling Theories……. 81

Chapter 5: Conclusions and Future Work………………………………………………….. 94

References…………………………………………………………………………………….. 98

Appendix A: Code for Calculating Onset Conditions…………………………………….. 106

Appendix B: Pressure Trace Measurements……………………………………………... 110

Appendix C: Small Angle X-Ray Scattering Spectra…………………………………….. 115

Appendix D: Position-Resolved Radius and Number Density………………………….. 119

x LIST OF TABLES

Table 3.1. Physical properties of argon and carbon dioxide. ……………………………. 24

Table 3.2. Table of positions for pressure trace measurements. Static pressure measurements began approximately 1.2 cm upstream of the throat and were taken in increments of 1.0 mm. As we approach the area near the throat, we increase the resolution of the measurements to 0.4 mm, and subsequently decrease the resolution as we measure further and further downstream. Measurements continue until the end of the nozzle………………………………………………………………………………………. 30

Table 4.1. Flow properties at the maximum nucleation rate for concentrations of 7.0 and

11 mol% CO2 in Ar in nozzle T1 at a stagnation temperature of 10°C…………………. 42

Table 4.2. Summary of initial conditions and those corresponding to the maximum nucleation rate for concentrations of 3.0 to 39 mol% CO2. All experiments started at a stagnation temperature of 20°C and a stagnation pressure of 458 Torr. Here, 푝푣0 is the initial partial pressure of CO2, and 푝퐽푚푎푥, 푇퐽푚푎푥, and 푆퐽푚푎푥 are the CO2 partial pressure, flow temperature, and saturation ratio that maximize the nucleation rate. The last column is ∆푡퐽푚푎푥, the characteristic time corresponding to the maximum nucleation rate, and was calculated using 퐽푆퐶푁푇 and the Quinn equation for surface tension…………… 46

Table 4.3. Summary of number densities and nucleation rates as calculated from position-resolved small angle X-ray scattering data………………………………………. 55

Table 4.4. Summary of number densities and nucleation rates as calculated from fixed- position small angle X-ray scattering data. Values for 푝퐽푚푎푥, 푇퐽푚푎푥, and ∆푡퐽푚푎푥 were estimated for concentrations that had no corresponding pressure trace measurement

xi data. (*) denotes: the flow properties at onset were not actually estimated for 9 and 15 mol%, but position-resolved SAXS data were not obtained, so we only have the fixed- position spectra for those concentrations…………………………………………………... 59

Table 4.5. Table of Fourier transform infrared spectroscopy peaks for a system of 12

-1 -1 mol% CO2 measured at resolutions of both 4 cm and 1 cm . Peaks of interest included

12 13 the v3 asymmetric stretch for CO2, the v3 asymmetric stretch for the CO2 isotope, and the combination bands 2v2 + v3 and v1 + v3. Gas phase and condensed phase peak locations differ by up to 5 cm-1 between the two data sets……………………………….. 69

Table 4.6. Table of conditions at the maximum nucleation rate for concentrations of 0.5 to 2.0 mol% CO2. Values were measured across a range of pressures from ~7 to ~42

Pa and temperatures from ~66 to ~79 K…………………………………………………… 74

Table 4.7. Table of estimated experimental nucleation rates based on estimated values for the number density at onset of nucleation for concentrations of 0.5 to 2.0 mol% CO2.

17 -3 -1 The resulting values for 퐽표푛푠푒푡 averaged 5 x 10 cm s …………………………………. 78

Table 4.8. Table of values used in deriving the Lielmezs-Herrick equation for the liquid surface tension of CO2………………………………………………………………………... 83

Table 4.9. Table of physical properties and parameters used in calculating the value of the nonisothermal correction factor 푓푛표푛𝑖푠표 for the range of concentrations from 0.5 to

39.3 mol% CO2………………………………………………………………………………… 87

xii LIST OF FIGURES

Figure 2.1. Change in Gibbs free energy of formation of a growing cluster plotted against the radius of the cluster. For a saturation < 1, the first term of Eqn. (2.14) is positive, and the Gibbs free energy is increasingly positive. For a saturation > 1, the first

∗ term is negative, and the Gibbs free energy reaches a maximum at 푅푝, the critical cluster radius…………………………………………………………………………………... 11

Figure 2.2. A graphical representation of the range of conditions studied in our experiments. At the lowest concentrations (0.5 mol% CO2 in Ar not depicted), the CO2 monomers have the potential to be well-thermalized by Ar molecules should they form clusters. At increasingly higher concentrations, however, there are fewer Ar molecules per CO2 molecule and thus poorer thermalization by the carrier gas to hot CO2 clusters.

…………………………………………………………………………………………………... 20

Figure 3.1. Orthographic (left) and side (right) view cutaways of nozzle T3. Nozzle T3 was constructed from the same nozzle blocks as nozzle T1, but the blocks were brought closer together vertically to decrease the area at the throat. This compression, in turn, increases the expansion rate of the nozzle. These figures were produced using

SolidWorks…………………………………………………………………………………….. 26

Figure 3.2. Schematic diagram of the continuous flow supersonic nozzle apparatus.

The nozzle cutaway views show the position of the probe or beam in each of the three characterization methods – pressure trace measurements (PTM), small angle X-ray scattering (SAXS), and Fourier transform infrared (FTIR) spectroscopy. Nozzle T3 (with

xiii CaF2 windows) was used for PTM and FTIR experiments, while T3_mica was used for

SAXS experiments……………………………………………………………………………. 28

Figure 3.3. Schematic diagram of the Fourier transform infrared (FTIR) spectroscopy setup. The infrared (IR) beam leaves the spectrometer through the side and is reflected through a series of mirrors through the desired position in the nozzle and once again into an MCT detector. The nozzle is mounted on a portable stage, allowing us to adjust the location where we measure spectra……………………………………………………. 36

Figure 3.4. The three fundamental vibrational modes of CO2 – (a) symmetric stretch, (b) bend, (c) asymmetric stretch………………………………………………………………… 37

Figure 4.1. Position-resolved pressure ratio and temperature profiles for flows of 7.0,

9.0, and 11 mol% CO2 in Ar in nozzle T1. Throughout the entire expansion, we observed no heat release, concluding that nozzle T1 does not expand quickly enough and create a large enough temperature gradient to promote the homogeneous nucleation of CO2……………………………………………………………………………… 39

Figure 4.2. Position-resolved pressure ratio and temperature profiles for flows of 7.0 and 11 mol% CO2 in Ar in nozzle T1 at a stagnation temperature of 10°C. Near the nozzle exit, we can observe some heat release, which may be due to condensation of

CO2……………………………………………………………………………………………… 40

Figure 4.3. Position-resolved saturation profiles for 7.0 and 11 mol% CO2 in Ar in nozzle T1. If this heat release corresponds to a nucleation event, it is not yet complete by the nozzle exit. We were able to extract the flow conditions at the maximum nucleation rate, listed in Table 4.1………………………………………………………….. 41

xiv Figure 4.4. Position-resolved pressure and temperature profiles for concentrations of 12

(left) and 24 (right) mol% CO2 in Ar in nozzle T3. Stagnation conditions are 458 Torr and 20°C. For all concentrations (see Appendix B), we observed significant heat release due to condensation well upstream of the nozzle exit. We also noticed that the isentrope and the wet trace align poorly at the highest CO2 concentrations……………………….. 43

Figure 4.5. Position of maximum nucleation rate 푧퐽푚푎푥 and heat capacity ratio γ of the gas mixture plotted with respect to the concentration of CO2 in Ar. As the heat capacity ratio decreases and the expansion softens, nucleation is initially allowed to move further upstream, but is forced to occur later downstream for concentrations greater than 12 mol%...... 44

Figure 4.6. (Left) Volmer plot, pressure vs. 1/temperature, of all pre-existing CO2 condensation in supersonic nozzles or free jets and the mid-high flow data from our current work. The plot suggests that free energy barriers are similar for all of the supersonic nozzle measurements except those of the high saturation extremes with

Krohn and Lippe. (Right) The measured pressures and temperatures corresponding to the onset of homogeneous and heterogeneous nucleation of CO2 in supersonic nozzles on the phase diagram of CO2. The heterogeneous data (Tanimura, Park) lie near the extrapolated liquid-vapor curve while the homogeneous data lie far to the left. The dashed line is the empirical fit in both plots………………………………………………… 45

xv than those of their clean counterparts. (Right) Selected detailed comparisons showing higher pressures and temperatures for contaminated data at the same concentration of

CO2 as their clean data counterparts………………………………………………………. 48

Figure 4.8. The average position-resolved pressure ratio (left) and temperature (right) profiles of five dry (Ar-only) traces with deviation shaded. The inset plot for each profile is centered around the maximum deviation, which is 0.00046 (0.115% of maximum) for

푝⁄푝0 and 0.307 K (0.14% of maximum) for 푇. Thus, we conclude that the conditions in the nozzle are highly repeatable, and that we have a consistent backdrop for studying homogeneous CO2 nucleation………………………………………………………………..50

Figure 4.9. Small angle X-ray scattering spectra – background, sample, background- subtracted sample, and particle-fit – for 12 (left) and 24 (right) mol% as part of position- resolved measurements. The CO2 aerosol-only curves were easily fit to a Schultz distribution of polydisperse spheres, and we obtained parameters of average particle radius, the width of the particle size distribution, and the absolute scattering intensity to calculate the number density and therefore the corresponding nucleation rate. The complete set of spectra are shown in Appendix C………………………………………… 52

Figure 4.10. Position-resolved average particle radius and average number density for

12 (left) and 24 (right) mol% CO2. We observe that, for most concentrations, particle radius steadily increases. Number density increases as particle formation occurs but begins to drop, as expected in a supersonic expansion. We probed the extent of number density decrease further in Figure 4.11 (right)…………………………………... 53

Figure 4.11. Position-resolved average particle radius for all measured concentrations of CO2. As the concentration of CO2 increases, we can observe particle formation

xvi shifting upstream and greater particle growth. For concentrations greater than 12 mol%, we see the growth curves shifting downstream and don’t observe particles larger than nearly 7.0 nm in radius……………………………………………………………………….. 54

Figure 4.12. Position-resolved specific number density for all concentrations of CO2 measured in SAXS experiments. Normalizing by density accounts for the expansion of the nozzle, so the number density should level off. We observe that it does not, and the particle radius increases, indicating that coagulation is an important mode of particle growth…………………………………………………………………………………………... 54

Figure 4.13. Small angle X-ray scattering spectra for 15 (left) and 33 (right) mol% CO2 as part of fixed-position measurements measured at 7.0 cm downstream of the physical throat. We chose the position of 7.0 cm because, according to results from pressure trace measurements, nucleation would have been quenched by that point for every system we investigated using scattering experiments……………………………………. 56

Figure 4.14. Average particle radius and aerosol number density as a function of CO2 concentration measured 7.0 cm downstream of the physical throat. As the amount of

CO2 increases and the flow warms up, particles form later in the nozzle, and we see a decrease of particle size as they take longer to develop and grow……………………… 57

Figure 4.15. (Left) Experimental nucleation rates calculated using the values of

푁(표푛푠푒푡) and 푁(7.0 푐푚) agree within the stated factor of 3 uncertainty. The latter are lower due to coagulation of the aerosol between the nucleation and measurement zones. (Right) Applying the nonisothermal correction factor to the nucleation rates reverses the trend we observe in the uncorrected plot…………………………………… 60

xvii Figure 4.16. Comparison of SAXS and PTM data for 3 (left) and 24 (right) mol% CO2.

The solid curves correspond to the normalized nucleation rates and the gray dashed lines indicate the corresponding normalized specific aerosol number densities. The circles are the measured data. In both cases, particles appeared earlier in SAXS experiments than expected based on PTM data…………………………………………... 62

Figure 4.17. A combined phase diagram for CO2 and H2O. For CO2, both the homogeneous and heterogeneous nucleation data are included. The red dotted lines indicate the isentropic expansions expected for CO2 concentrations of 39 and 3 mol%.

The red circles correspond to the conditions for 퐽푚푎푥 determined by pressure trace measurements, and the pink points are those retrieved from SAXS measurements. Both are clearly separated from the heterogeneous onset data. The blue dotted lines indicate the isentropic expansions of water for the two CO2 conditions considered above assuming the maximum allowable level of water at 10 ppm. If contamination from water vapor were an issue in these experiments, water particles should have formed when the water isentropes crossed the blue dashed lines, and we would have observed CO2 condensing close to the heterogeneous nucleation line………………………………….. 63

Figure 4.18. Position-resolved Fourier transform infrared spectroscopy data for the

-1 asymmetric stretch v3 in a system of 12 mol% CO2 measured at a resolution of 4 cm .

Measurements began at the throat and were performed in increments of 1.0 cm until a position of 7.0 cm. Spectra were normalized by the gas density to account for the nozzle expansion as we make measurements. We observe a transition from the gas phase to a distinctly solid phase starting at 5.0 cm downstream of the throat………………………. 65

xviii Figure 4.19. Raw Fourier transform infrared spectroscopy data for the asymmetric

13 stretch v3 for the CO2 isotope (left) and the combination bands 2v2 + v3 and v1 + v3

(right). Both sets of data show the development of a distinct solid phase peak as we move downstream…………………………………………………………………………….. 66

Figure 4.20. Raw (top) and density-normalized (bottom) position-resolved Fourier transform infrared spectroscopy data for the v3 asymmetric stretch in a system of 12 mol% CO2. Once again, we observe the transition from the gas phase to the condensed phase, which appears to be solid in nature. At a higher resolution, we can detect the emergence of a shoulder at ~2370 cm-1 and an additional one near ~2360 cm-1. Shown in the plot with the density-normalized spectra are the approximate average particle radii corresponding to those positions. Those values were obtained from small angle X- ray scattering measurements of the 12 mol% CO2 system………………………………. 67

13 Figure 4.21. Raw Fourier transform infrared spectroscopy data for the CO2 isotope

-1 (top) and the combination bands 2v2 + v3 and v1 + v3 (bottom). At a resolution of 1 cm and without apodization, we are far more sensitive to small fluctuations in absorbance, but we can still observe the formation of the condensed phase peaks from the gas phase…………………………………………………………………………………………… 68

Figure 4.22. Gas-phase subtracted Fourier transform infrared spectra for a system of

-1 12 mol% CO2 measured at a resolution of 1 cm . After a position of 3.6 cm downstream of the physical throat, we begin to observe the formation of the condensed phase peak.

The final shape is that of a narrow peak at 2359 cm-1 and two shoulders at 2360 cm-1 and 2368 cm-1…………………………………………………………………………………. 70

xix Figure 4.23. Infrared spectra predictions by Isenor et al. for (left) different shapes at a constant size of 10 nm in diameter and (right) different cluster sizes of a cube with rounded corners. We believe our clusters exhibit spectra most similar to those of cubes with or without rounded corners……………………………………………………………... 71

Figure 4.24. Position-resolved pressure and temperature profiles for 0.5 (left) and 2.0

(right) mol% CO2. We observed heat release due to condensation in this range of concentrations, but we still observe a significant heat release due to the nozzle overexpansion…………………………………………………………………………………. 72

Figure 4.25. Homogeneous Ar (blue) and CO2 (red) nucleation data plotted onto the phase diagrams of both gases. The dashed lines represent the Ar and CO2 isentropes at 39 mol% CO2, and the solid lines are those at 0.5 mol% CO2. The Ar line at 0.5 mol%

CO2 crosses the phase diagram but does not reach the homogeneous data, so we do not expect Ar to condense or become involved with CO2 nucleation in a heterogeneous manner…………………………………………………………………………………………. 73

Figure 4.26. Position of onset as a function of CO2 concentration. There is excellent agreement in the low flow data with the data previously collected from medium and high flow rates, as the onset of nucleation occurs further downstream in nozzle T3 as we decrease the concentration of CO2…………………………………………………………..75

Figure 4.27. Plotting the pressure vs. inverse temperature and pressure vs. temperature points for the low flow rate range of 0.5 to 2.0 mol% CO2 on the Volmer plot (left) and phase diagram (right), respectively, shows excellent agreement with both our medium and high flow rate data as well as suggests overlap with Krohn’s data….. 76

xx Figure 4.28. Average number density vs. partial pressure of CO2 for measurements at onset of nucleation and at a position 7.0 cm downstream of the physical throat for experiments conducted at the Advanced Photon Source in the summer of 2018. Data below ~30 Pa for onset points and those below ~20 Pa for points at 7.0 cm appear to follow a linear trend on the log-log axes……………………………………………………. 77

Figure 4.29. Log-log fit for number density as a function of partial pressure of CO2 for data collected at the position of onset………………………………………………………. 77

Figure 4.30. (Left) Experimental nucleation rates for both medium and high concentrations as well as estimated nucleation rates for low concentrations of CO2.

Nucleation rates, on average, are ~5 x 1017 cm-3s-1. (Right) Applying the nonisothermal correction factor results in a more distinct decrease in nucleation rate with increasing saturation………………………………………………………………………………………. 79

Figure 4.31. (a, b) We do not observe any visible trend in the experimental nucleation rate data between our measurements and those of Krohn’s, even when we allow the factor of three error on our data and the order of magnitude error on Krohn’s data. (c, d)

Applying the nonisothermal correction factor, however, we find a somewhat linear trend in the nucleation rate with respect to saturation…………………………………………… 80

Figure 4.32. The Quinn and Lielmezs-Herrick formulations for the surface tension plotted over the experimentally obtained surface tension of liquid CO2 points of Muratov and Skripov. At high temperatures, where the data was collected, both fits perform extremely well, but we are much more interested in its performance from the range of

65-90 K, where the two fits differ greatly…………………………………………………… 84

xxi Figure 4.33. 퐽푡ℎ푒표푟푦(푡)⁄퐽푡ℎ푒표푟푦,푚푎푥 plotted for four combinations of nucleation theory and surface tension for one of the experiments testing a concentration of 18 mol% CO2.

Most of the combinations yield a value for the characteristic time of nucleation close to the value we use of 8.50 μs, and the 퐽푆푇-Quinn combination yields an outlier that is only

~2.5 times higher than the smallest value obtained……………………………………….. 86

Figure 4.34. Curves corresponding to a constant non-isothermal nucleation rate of 5 x

17 -3 -1 10 cm s for all combinations of nucleation theory (퐽푆퐶푁푇 or 퐽푆푇) and surface tension correlation (Quinn or Lielmezs-Herrick) are compared to the experimental onset of nucleation data of Lettieri, Duff, and the current work…………………………………….. 89

Figure 4.35. Comparisons between experimental nucleation rates and those calculated using self-consistent classical nucleation theory (SCNT) and simulation based theory

(ST) for the surface tension correlations of Quinn or Lielmezs-Herrick (L-H) equation.

Non-isothermal corrections are included in both theories………………………………… 90

Figure 4.36. (Left) In the Hale plot, the experimental supersonic nozzle data are bounded by the lines corresponding to the nucleation rates 퐽 = 1019 cm-3s-1 on top and

17 -3 -1 loosely by 퐽 = 10 cm s on the bottom when the effective omega parameter is Ω푒푓푓 =

17 -3 -1 1.70. (Right) For a nucleation rate of 퐽 = 10 cm s , the omega parameter of Ω푒푓푓 =

1.70 best fits our data, while the value of Ω푡ℎ푒표푟푦 calculated for the liquid is a better approximation than that of the solid…………………………………………………………. 91

Krohn’s high-saturation data, shown in yellow diamonds. We are unable to capture the

xxii effects of using the nonisothermal correction factor in plotting the Volmer data of pressure and inverse temperature…………………………………………………………... 93

Figure B.1. Position-resolved pressure ratio and temperature profiles for all experiments in Nozzle T3…………………………………………………………………… 114

Figure C.1. Small angle X-ray scattering spectra for all measurements taken at onset of nucleation and at a fixed position 7.0 cm downstream of the physical throat.

…………………………………………………………………………………………………. 118

Figure D.1. Position-resolved average particle radius and average number density for

3.0, 6.0, 12, 18, 24, 30, and 39 mol% CO2……………………………………………. 120

xxiii CHAPTER 1: INTRODUCTION

Carbon capture and storage (CCS) is a developing set of technologies that aims to reduce carbon dioxide (CO2) emissions into the atmosphere. Current leading CCS methods include physical and chemical absorption, cryogenic separation (Joule-

Thomson valve or distillation), and membrane separation1-5. Although these methods are efficient in preventing CO2 release, they require complex machinery and/or chemicals that negatively impact the environment. For example, one of the leading methods of carbon capture from flue gas, absorption in monoethanolamine (MEA),

6 requires significant amounts energy to recover the CO2 and regenerate the absorbent .

Furthermore, emitted MEA can react with radicals present in the atmosphere to form hazardous compounds such as nitramine and nitrosamine7, 8. The amines, including

MEA, together with sulfuric acid, have also been shown to contribute to new particle formation, thereby increasing the concentration of fine particles in the atmosphere9.

Supersonic separation is an alternative cryogenic separation approach that has been studied in the laboratory10-15 for over 70 years and in field plants16-19 for over 20 years. In these devices, the gas mixture cools as it undergoes an isentropic expansion, promoting a transition of the condensable vapor into the condensed phase. The resulting condensate is then removed from the remaining gas via an inertial separation process. Advantages of supersonic separation over other cryogenic approaches include lower power usage, a higher temperature drop, and greater system pressure recovery19.

Several designs for supersonic separator systems have been implemented to varying degrees in full-scale plants20, 21. Of these, the separator developed by Twister

1 BV and the 3-S separator designed by TransLang Technologies Ltd. are the most mature16-19. Both of these technologies induce a swirl upstream of the supersonic nozzle in order to promote centrifugal separation of large (greater than 1 µm in diameter) condensed particles in the supersonic regime. Twister and TransLang devices have been operating in field plants, successfully separating hydrocarbons from natural gas, for over two decades. There is great interest in expanding the scope of swirling supersonic separators to the separation of other compounds, in particular CO2.

Nucleation is the process in which local density fluctuations trigger the irreversible growth of clusters of a new phase22. The phase change must occur from a supersaturated parent phase, after the free energy barrier to nucleation has been crossed22. In homogeneous nucleation, the new phase forms without making contact with other surfaces or particles22, 23. The presence of foreign objects such as pre- existing ions, droplets, debris, or a macroscopic surface can decrease the surface free energy barrier of nucleation, resulting in heterogeneous nucleation. In the presence of such entities, condensation can proceed at lower supersaturations than in their absence23. Heterogeneous nucleation dominates in natural settings, and homogeneous nucleation can require meticulous protocol to ensure a clean system.

We have chosen to study CO2 condensation in supersonic nozzles for several reasons. First, this device lets us study nucleation and particle growth on a microsecond timescale and follow particle formation and evolution as a function of position within the nozzle. Second, there are existing data on the nucleation of the simpler monatomic argon and diatomic nitrogen24, and tri-atomic carbon dioxide is a slightly more complex

25 molecule with a quadrupole . Thus, studying the behavior of CO2 not only benefits

2 carbon capture research but also provides nucleation modelers with valuable reference data.

Many research groups have attempted to model the condensation of carbon dioxide in supersonic nozzles for use in CCS12-14, 23, 26-28. Even though these groups claim they were successful in modeling a design with significant CO2 separation, many made assumptions that detracted from the accuracy of their simulations, such as assuming a saturation > 1 would be enough for promoting condensation. Experimental values for nucleation rates and particle sizes are essential for accurately predicting particle growth rates and final particle sizes, and these are the key parameters required for designing efficient machinery for supersonic separation. Ignoring important phenomena such as super-cooling, i.e. the degree of supersaturation required to initiate particle formation, will lead to faulty engineering and design.

The earliest experimental supersonic nozzle condensation studies were performed on nitrogen, humid air, and steam29-31. Duff (1966) pioneered the study15 of

CO2 in supersonic nozzles and characterized conditions at the onset of condensation for temperatures ranging from ~160-180 K and pressures between ~128-328 kPa, i.e. in the region of the phase diagram where solid is the stable phase but the supercooled liquid could also exist. Recently, Lettieri et al. (2018) performed similar experiments11 but starting from supercritical CO2 and condensing to conditions where the liquid is the stable phase. Lippe et al. (2019) performed mass spectroscopy and infrared spectrometry experiments32 on CO2 clusters formed in Laval expansions at temperatures between ~29 and 43 K, i.e. under conditions of extreme saturation with respect to the vapor or liquid. Most recently, Krohn et al. published additional onset

3 data33 for CO2 as well as experimental homogeneous nucleation rates in the range of

~31 to 63 K. In our lab, Tanimura (2015) characterized the onset of heterogeneous nucleation of carbon dioxide on water ice particles in supersonic nozzles10, and Park

34 (2019) examined heterogeneous nucleation of CO2 on n-alkane particles . Both found that in the presence of particles, heterogeneous condensation starts close to the extrapolated vapor-liquid equilibrium line. To date, Krohn’s data33 and our published data35 are the only quantitative experimental nucleation rate data available.

Molecular dynamics (MD) simulations have been established as a complementary tool to study homogeneous nucleation and particle growth from the vapor phase. Nucleation rates can be calculated using mean-first-passage-time, or threshold methods36, 37. Effects of non-isothermal nucleation38 as well as the pressure effect caused by a dense carrier gas39 have previously been studied. However, most of these simulations have been done on model Lennard-Jones systems, which may not be applicable to gases of polyatomic molecules. Horsch et al. have performed the only

40 molecular dynamics study on the nucleation of methane, ethane and CO2, and here the CO2 molecule was described by a two-center Lennard-Jones model with an embedded point quadrupole. To the best of our knowledge, no other atomistic simulations of the homogeneous nucleation of CO2 exist.

The first goal of this work is to provide quantitative rate measurements under conditions where nucleation is controlled by a free energy barrier. The second goal is to compare these rates to those calculated via classical theories and molecular dynamics simulations, where we can also gain insight as to the phase and structure of the critical clusters. A third goal is to demonstrate a smooth transition between our nucleation

4 measurements – that are dominated by a free energy barrier – with those of Lippe et al.32 and Krohn et al.33, whose measurements approach the kinetic limit.

This work was carried out in collaboration with several international groups. The specific tasks carried out by each contributor are detailed below.

At The Ohio State University, experimental studies on CO2 were conducted using two nozzles, T1 and T3, with different expansion rates. These studies included pressure trace measurements (PTM) for CO2 in T1 ranging from 7 to 11 mol%, PTM for CO2 in

T3 ranging from 0.5 to 39 mol%, small angle x-ray scattering experiments in T3 ranging from 3 to 39 mol%, and Fourier transform infrared spectroscopy experiments on 12 mol% CO2 in Ar in T3. Values that could not be measured directly, due to restricted access to the Advanced Photon Source stemming from COVID-19, were extrapolated based on the previous experiments.

Researchers at the University of Helsinki included Roope Halonen, Valtteri

Tikkanen, Bernhard Reischl, and Hanna Vehkamäki. They carried out the fully atomistic molecular dynamics simulations of CO2 nucleation in an Ar atmosphere at mixing ratios and temperatures that overlapped with those of the experiments performed at OSU.

Their analysis of the structure of critical and slightly post-critical clusters rationalized the choice of physical properties used in predicting nucleation rates within the framework of classical nucleation theory. They also provided an alternate expression for the surface tension of liquid CO2 based on the approach of Lielmezs and Herrick and modified classical nucleation theory by introducing a curvature term to the expression for cluster formation free energy, where the latter arises naturally from the simulation results.

Finally, simulations also suggested that the critical clusters were not thermalized

5 properly and thus a nonisothermal correction factor should be incorporated into the nucleation rate expressions.

Researchers at ETH Zurich included Martina Lippe, Jan Krohn, Chenxi Li, and

Ruth Signorell. They carried out post-nozzle flow experiments in the temperature range

30-65 K to provide a region of overlap with the low flow experiments conducted at OSU.

They also performed hard sphere model calculations for nucleation rates approaching the barrier-less limit.

6 CHAPTER 2: NUCLEATION THEORY

What is nucleation, and how do we describe it in a quantitative manner?

Identifying the process and putting together an appropriate theory is key to effective data analysis.

Nucleation is the formation of particles from a supersaturated parent phase.

Nucleation can occur via heterogeneous or homogeneous routes, depending on whether particles form on a pre-existing surface or substance, or if it only involves molecules of the same species, respectively41. The composition of the condensed droplets also defines the mode of nucleation; the process can be homomolecular – involving a single gaseous species, or heteromolecular – where the droplets are composed of two or more species41. In the case of the nucleation of carbon dioxide in the carrier gas argon, we are studying homogeneous, homomolecular nucleation.

This section covers a series of important topics related to nucleation theory, including Classical Nucleation Theory, Self-Consistent Nucleation Theory, and the correction factor for non-isothermal nucleation. The contents of each subsection are adapted from the following:

• Thermodynamics and Classical Nucleation Theory – Atmospheric Chemistry and

Physics of Air Pollution, John H. Seinfeld41

• Self-Consistent Nucleation Theory – Kinetic nucleation theory: A new expression

for the rate of homogeneous nucleation from an ideal supersaturated vapor,

Steven L. Girshick and Chia-Pin Chiu42

7 • Non-isothermal Nucleation – Homogeneous nucleation and growth of droplets in

vapours, J. Feder, K. C. Russell, J. Lothe, and G. M. Pound43

Section 2.1: Thermodynamics of Nucleating Clusters

The challenge in working with small clusters of molecules is that it is difficult to accurately measure their physical properties. To this end, classical nucleation theory

(CNT) assumes that clusters are spherical drops whose physical properties are those of the bulk phase. In the derivation for the fundamental thermodynamic terms of classical theory, we relate terms that are normally associated with flat liquid surfaces to those of spherical droplets.

In the bulk phase, when a liquid’s surface area is increased (for example, by apple juice spilling onto a table and spreading out), molecules that were formerly in the bulk interior of the liquid are now forced to the surface. Work is done by these molecules against the cohesive forces of the liquid to expand the surface, thus the surface tension is defined as the work per unit area done in extending its surface.

Although the physical properties of clusters are most likely not the same as those of a bulk liquid, the surface tension of a cluster is assumed to be the same as that of the bulk liquid.

We wish to characterize the thermodynamics of nucleation, and we do so by starting with the fundamental free energy for the formation of an embryo – the Gibbs free energy. This is the free energy associated with a process that can be used to do

8 work for a system at constant temperature and pressure. If the process is a change from pure vapor to a condensed phase embryo, we can write the Gibbs free energy as:

∆퐺 = 퐺푒푚푏푟푦표 푠푦푠푡푒푚 − 퐺푝푢푟푒 푣푎푝표푟 (2.1)

We assumed that the total number of molecules in the gaseous system is 푁푇, and that after the cluster embryo forms, the number of monomers remaining is

푁1 = 푁푇 − 푛 (2.2) where n is the number of molecules in the embryo.

If we set 푔푣 and 푔푙 as the Gibbs free energies per molecule in the vapor and liquid phases, respectively, we can write the Gibbs free energy change for the formation of a droplet as:

2 ∆퐺 = −푛푔푣 + 푛푔푙 + 4휋푅푝휎 (2.3) which can be simplified into

2 ∆퐺 = 푛(푔푙 − 푔푣) + 4휋푅푝휎 (2.4)

2 Here, the term 4휋푅푝휎 is the free energy associated with a liquid droplet interface with radius 푅푝 and bulk surface tension 휎. We can relate the number of molecules in this droplet to its radius using the expression

4 3 (2.5) 휋푅푝 푛 = 3 푣푙

4 The numerator 휋푅3 represents the volume of a sphere, and the denominator 푣 is the 3 푝 푙 molecular volume in the liquid phase. We can substitute the expression for 푛 into that of the Gibbs free energy to obtain

푅3 (2.6) 4휋 푝 2 ∆퐺 = ( ) (푔푙 − 푔푣) + 4휋푅푝휎 3 푣푙

9 To calculate 푔푙 − 푔푣, we recall the differential form of the Gibbs free energy at constant temperature

푑푔 = 푣푑푝 (2.7) and take the difference

푑(푔푙 − 푔푣) = (푣푙 − 푣푣)푑푝 (2.8)

Because the molecular volume of a vapor is much larger than the molecular volume of a liquid, we can neglect 푣푙, simplifying the expression to

푑(푔푙 − 푔푣) = −푣푣푑푝. (2.9)

In classical theory, the vapor phase is assumed to be ideal, so we rearrange the equation of state 푝푣푣 = 푘푇 as

푘푇 푣 = . (2.10) 푣 푝 and substitute it into the differential form of Gibbs free energy

푘푇 푑(푔 − 푔 ) = − 푑푝. (2.11) 푙 푣 푝

0 0 Integrating Eq. (2.10) from 푝퐴 to 푝퐴, where 푝퐴 is the vapor pressure over a flat surface – where condensation is expected to occur, and 푝퐴 is the equilibrium partial pressure over the liquid, we obtain

푝퐴 푝퐴 (2.12) ∫ 0 푑(푔푙 − 푔푣) = 푔푙 − 푔푣 = −푘푇 ln ( 0 ). 푝퐴 푝퐴

0 We define the supersaturation 푆 as 푝퐴⁄푝퐴, so Eq. (2.12) becomes

푔푙 − 푔푣 = −푘푇 ln 푆. (2.13)

Thus, we can write the expression for the Gibbs free energy of the formation of a droplet in terms of the radius of the cluster as

10 4 3 푘푇 2 (2.14) ∆퐺 = − 휋푅푝 ln 푆 + 4휋푅푝휎. 3 푣푙

For later use, we can also write this expression in terms of 푛 as

∆퐺 = 푏푛2/3 − 푛푘푇 ln 푆 (2.15) where

4휋 1/3 (2.16) 푏 = 3휎푣푙 ( ) . 3푣푙

What does this equation for ∆퐺 mean qualitatively? We plot Eq. (2.14).

As shown in Figure 2.1, the change in Gibbs free energy of the cluster system increases as it grows in size, making it an increasingly unfavorable process if 푆 < 1. If, however, the pressure of the vapor in the surrounding gas is such that 푆 > 1, 훥퐺 reaches a maximum and decreases with increasing size, making the transition from the vapor phase to the condensed phase favorable. The size at which the change in Gibbs

11 ∗ free energy is maximized is called the critical cluster size 푅푝, and we calculate it by setting the first derivative of the change in Gibbs free energy to 0 as such:

휕∆퐺 푘푇 ln 푆푅 ( ) = 4휋푅 (− 푝 + 2휎) = 0. (2.17) 휕푅 푝 푣 푝 푇,푝 푙

There are two solutions to this equation. The first is the trivial solution

4휋푅푝 = 0 → 푅푝 = 0, (2.18) and the second is the proper solution

2휎푣 푅∗ = 푙 . (2.19) 푝 푘푇 ln 푆

The corresponding number of molecules in the cluster at the critical size is

32휋휎3푣2 푛∗ = 푙 , (2.20) 3(푘푇 ln 푆)3 and the corresponding value of ∆퐺 is

16휋휎3푣2 ∆퐺∗ = 푙 . (2.21) 3(푘푇 ln 푆)2

∗ We can construct the Kelvin equation by writing ln 푆 in terms of 푅푝

∗ 2휎푣푙 2휎푣푙 (2.22) 푅푝 = → ln 푆 = ∗ . 푘푇 ln 푆 푘푇푅푝

Qualitatively, the Kelvin equation tells us that the vapor pressure over a curved interface is always greater than that over a flat one.

Section 2.2: The Classical Nucleation Rate

It is of interest to us to calculate theoretical nucleation rates for several reasons.

First, in our experiments, we use the nucleation rate profiles to characterize both the conditions corresponding to the peak nucleation rate and to obtain the characteristic

12 time over which nucleation is important, as will be elaborated upon further in Section

3.7. Second, we can assess the validity of the assumptions made in constructing the theory by comparing absolute values of rates to those determined experimentally.

Finally, rates are a more sensitive measure when comparing results between experiments than, for example, the conditions that initiate nucleation.

To construct the nucleation equation for classical theory, we begin by listing the primary assumptions. We assume that almost all molecules of the vapor exist as monomers or small clusters, and cluster growth occurs via the addition of monomers.

Likewise, we assume cluster size diminishes with the evaporation of monomers. There is the possibility of growth via addition (or loss via evaporation) of more than one molecule at a time, but this is deemed to be so rare that it is not considered important.

We also assume that the cluster concentration decreases with increasing size, such that clusters significantly greater than the critical size are rare and their numbers negligible.

For a vapor, we assume that the concentration of clusters obeys a Boltzmann

푒 distribution, where 푁푛 , 푛 = 2, 3, etc., the equilibrium concentration of clusters containing n molecules, can be written as

∆퐺 푁푒 = 푁 exp [− ]. (2.23) 푛 1 푘푇 where 푁1 is the total number of monomers of the vapor in the system. At the critical cluster size,

16휋휎3푣2 푁푒 = 푁 exp [− 푙 ]. (2.24) 푛 1 3(푘푇)3(ln 푆)2

Checking this equation for the monomer (푛 = 1), however, does not return 푁1. We will discuss this discrepancy in greater detail as well as a solution for it in Section 2.3.

13 The general expression for the net rate at which clusters containing 푛 – 1 molecules form clusters of 푛 molecules is written as

′ 퐽푛 = 훽푎푛−1푁푛−1 − 푒 푎푛푁푛, (2.25) where 훽 is the flux of monomer molecules to a unit area of surface, and an-1 and an are the surface areas of clusters containing 푛 – 1 and 푛 molecules, respectively. The term

푒′ is the evaporation rate of molecules from a cluster of size 푛. To obtain an expression for 훽, we recall the expression for the number of molecules striking a unit area per unit time from the kinetic theory of gases:

푁푐̅ 8푘푇 1/2 (2.26) , where 푐 = ( ) 4 휋푚 and substitute the value of 푁 from the equation of state for a monomer of an ideal gas

푝1 = 푁1푘푇 to obtain

1/2 1 8푘푇 푝1 푝1 (2.27) 훽 = ( ) = 1/2. 4 휋푚1 푘푇 (2휋푚1푘푇)

The first term of Eq. (2.27) represents monomers colliding onto clusters of 푛 – 1 size to form 푛-mers, and the second term represents monomers evaporating from clusters of 푛 size to form (푛 – 1)-mers. At equilibrium, the rates of these two processes are equal, so we can write the following

푒 ′ 푒 훽푎푛−1푁푛−1 = 푒 푎푛푁푛 . (2.28)

′ We rearrange Eq. (2.28) for 푒 and plug it into 퐽푛 to obtain

푒 푁푛−1 푁푛 (2.29) 퐽푛 = 훽푎푛−1푁푛−1 [ 푒 − 푒]. 푁푛−1 푁푛

Note that we performed this step to avoid coming up with an explicit expression for the evaporation term as we did for 훽. Here, we invoke one of the key assumptions of classical nucleation theory – that we quickly reach a pseudo-steady state so that the

14 rates of formations of clusters are equivalent regardless of size. This leaves us with the size independent form of Eqn. (2.29):

푒 푁푛−1 푁푛 (2.30) 퐽 = 훽푎푛−1푁푛−1 [ 푒 − 푒]. 푁푛−1 푁푛

Next, we wish to evaluate the nucleation rate and derive a more tangible expression for 퐽. If we sum 퐽 from 푛 = 1 to a sufficiently large cluster size 푀, successive terms cancel, and we obtain

푀−1 퐽 푁1 푁푀 (2.31) ∑푛=1 푒 = 푒 − 푒 . 훽푎푛푁푛 푁1 푁푀

We can further simplify this expression by making the approximation that the

푒 equilibrium and nonequilibrium concentrations of the monomer are equivalent, so 푁1⁄푁1

= 1. As the cluster size increases, we expect their number concentration to decrease

푒 while the equilibrium number concentration increases, so we assume 푁푀⁄푁푀 = 0. This approximation simplifies Eq. (2.31) into

푀−1 퐽 (2.32) ∑푛=1 푒 = 1. 훽푎푛푁푛

We factor 퐽 out because it is a constant, and we rearrange Eqn. (2.32) to get

1 (2.33) 퐽 = 푀−1 1 . ∑푛=1 푒 훽푎푛푁푛

We then perform two mathematical approximations to evaluate 퐽. The first is the conversion of the summation to an integral from 1 to 푀 – 1

1 (2.34) 퐽 = 푀−1 푑푛 . ∫ 푛=1 훽푎(푛)푁푒(푛)

The second is to only consider the integral in the region around 푛 = 푛∗ in a Taylor expansion of 푁푒(푛) about 푛 = 푛∗

15 ∗ 2 푒 ∆퐺 1 푑 ∆퐺 ∗ 2 (2.35) 푁 (푛) = 푁1 exp [− − ( 2 ) (푛 − 푛 ) ]. 푘푇 2푘푇 푑푛 푛∗

Here we use the expression for 훥퐺 in terms of 푛 to write out the second derivative

1 푑2∆퐺 2푏푛−4/3 ∆퐺 = 푏푛2/3 − 푛푘푇 ln 푆 → = − , (2.36) 푘푇 푑푛2 9푘푇 where 푏 was defined in Eq. (2.16). Plugging the second derivative in, Eq. (2.35) becomes

∆퐺∗ 1 2푏 푁푒(푛) = 푁 exp [− + ( ) 푛∗−4/3(푛 − 푛∗)2]. (2.37) 1 푘푇 2푘푇 9푘푇

We substitute Eqn. (2.37) back into the integral in Eqn. (2.34) to get

푀−1 푑푛 1 ∞ 휉 ∫ = ∫ exp [− (푛 − 푛∗)2] 푑푛, (2.38) 푛=1 훽푎(푛)푁푒(푛) 훽푎(푛∗)푁푒(푛∗) 0 2 where

2푏푛∗−4/3 휉 = . (2.39) 9푘푇

We can change the bounds of the integral to start from 0 since we assume 푛∗ >> 1.

Note that the integral in Eq. (2.38) is in the form of the error function, so we can write the solution for 퐽 as

휉 1/2 (2.40) 퐽 = 훽푎(푛∗)푁푒(푛∗) ( ) . 2휋

To write an expanded equation for 퐽, we substitute in the expressions for 훽, 휉, and the following two equations

∗ ∗2 푎(푛 ) = 4휋푅푝 (2.41)

∆퐺∗ 16휋휎3푣2 (2.42) 푁푒(푛∗) = 푁 exp [− ] = 푁 exp [− 푙 ] 1 푘푇 1 3(푘푇)3(ln 푆)2 to obtain

1/2 3 2 푝1 2휎 푣푙 16휋휎 푣푙 (2.43) 퐽퐶푁푇 = ( 1/2) ( 1/2) 푁1 exp [− 3 2]. (2휋푚1푘푇) (푘푇) 3(푘푇) (ln 푆)

16 Recognizing that 푝1 = 푁1푘푇 and therefore 푁1 = 푝1⁄푘푇, combining the 푘푇 terms, and by moving all the terms with an exponent of ½ into a root symbol, we obtain the more condensed and familiar expression

2 3 2 2휎 푝1 16휋휎 푣푙 (2.44) 퐽퐶푁푇 = √ 푣푙 ( ) exp [− 3 2]. 휋푚1 푘퐵푇 3(푘푇) (ln 푆)

This is the expression for the nucleation rate according to classical nucleation theory

(CNT), popularized by Becker and Döring44 in the 1930s.

Section 2.3: Self-Consistent Nucleation Theory (SCNT)

As previously mentioned, the expression for the equilibrium cluster population is assumed to be of the form of a Boltzmann distribution

∆퐺(푛) 푁푒 = 푁 exp [− ], (2.45) 푛 푓 푘푇 where 푁푓 is a normalization factor. In classical theory, 푁푓 = 푁1, the concentration of the monomer. If we insert the expression for ∆퐺(푛) into the distribution, we obtain

(푏푛2/3−푛푘푇 ln 푆) 푏푛2/3 (2.46) 푁푒 = 푁 exp [− ] = 푁 푆푛 exp [− ], 푛 푓 푘푇 푓 푘푇

For 푁푓 = 푁1, this expression does not hold true for the monomer.

Kinetic Nucleation Theory (KNT), as termed by Girshick and Chiu42, based on the work of Katz and Donohue45, changes the equilibrium reference to that of the saturated vapor. This, in turn, changes the normalization factor to 푁푆, where 푁1⁄푁푆 = 푆. Carrying this change in the distribution through the rest of the derivation for the nucleation rate leads to a prefactor for 퐽퐶푁푇,

17 푒Θ 퐽 = 퐽 , (2.47) 퐾푁푇 푆 퐶푁푇 where Θ is a dimensionless surface energy defined as

휎푎 Θ ≡ 1, (2.48) 푘푇 and 푎1 is the surface area of the monomer.

We change the notation here to SCNT, standing for self-consistent nucleation theory, to both remind ourselves how it quantitatively differs from CNT, and also to distinguish it from another popular theory, mean-field kinetic nucleation theory, or

MKNT. Unless stated otherwise, SCNT is the base theory used to analyze the data from these supersonic nozzle experiments.

Section 2.4: The Non-Isothermal Correction Factor

Even after adjusting classical theory for self-consistency in the equilibrium distribution, we notice that there are still discrepancies between theoretically calculated and experimental nucleation rates. Historically, these have been attributed to the droplet model being used, or to additional contributions to the free energy of cluster formation.

The liquid droplet model is an approximation, of course, but to date we do not have another way to easily predict nucleation rates without it. Instead, we turn to more accurately describing the energy transfer in the condensing droplet system through the use of a correction factor adjusting the nucleation rates obtained from theoretical calculations that assume all clusters have the same temperature.

We now shift our perspective to that of the condensing cluster. As a cluster or droplet grows, latent heat is released by the condensing monomers. For water, it has

18 been estimated to be approximately 20 푘푇 per condensing molecule, enough to heat up the cluster by tens of degrees. If collisions with the carrier gas are unable to efficiently transport the latent heat away from the growing clusters, they warm up, the probability of monomer loss increases, and the nucleation rate decreases. In our system of carbon dioxide in the carrier gas argon, we cover a wide range of concentrations of condensable gas, leading to vastly different frequencies of collision between CO2 and

Ar (see Figure 2.2 for a graphical representation). For low concentrations of CO2, we expect more frequent collisions between the growing clusters and the inert carrier gas

Ar, providing better heat transfer between the cluster and the surroundings. This thereby ensures that the cluster temperature is closer to that of the carrier gas. For high concentrations of CO2, at an approximately 1:2 ratio of CO2 molecules to Ar molecules, we except the carrier gas to remove latent heat more poorly, resulting in hotter critical clusters.

Whereas Eq. (2.44) and (2.47) normally assume isothermal nucleation, i.e. the carrier gas temperature is equal to the cluster temperature, it is clear that especially for systems with more CO2, imperfect heat transfer will lead to non-isothermal conditions.

How do we systematically quantify and account for this phenomenon? The answer lies in the derivation of a non-isothermal correction factor for the nucleation rate as first described by Feder et al.43

Figure 2.2. A graphical representation of the range of conditions studied in our experiments. At the

lowest concentrations (0.5 mol% CO2 in Ar not depicted), the CO2 monomers have the potential to be well-thermalized by Ar molecules should they form clusters. At increasingly higher concentrations,

however, there are fewer Ar molecules per CO2 molecule and thus poorer thermalization by the carrier

gas of hot CO2 clusters.

First, we construct a two-dimensional phase space of size (푛) and energy (휖), where movement in the +푛 direction corresponds to the addition of a monomer, and movement in the +휖 direction corresponds to the release of latent heat. As in classical theory, we assume growth and decomposition of clusters takes place one molecule at a time. We also assume that the clusters are stationary compared to the monomers such that the flux of monomers to the surface 훽 is the same as that to a plane. This is also the same approximation made in classical theory.

We use the symbol 푞 to represent the energy released upon the addition of a monomer to a cluster and ℎ to represent the latent heat of condensation. Normally, 푞 =

ℎ, but there are additional heat sinks we must account for. To keep the cluster at constant pressure, it has to do work of the magnitude 푘푇. Therefore, on average, after

20 a molecule strikes the surface with 푘푇/2 more energy than the average, the liquid droplet needs to return 푘푇/2 to the vapor. When measuring heat release, this means that 푘푇/2 of the energy released through condensation is not detected via instrumentation. The heat release is therefore modified to

푘푇 푞 = ℎ − . (2.49) 2

Another portion of the energy released will go to the creation of new surface area as the cluster grows. The energy required to create new surface is

휕퐴(푛) (2.50) 휎 푒 휕푛 where 휎푒 is the energy component in the surface free energy and 푑퐴(푛)⁄푑푛 is the change in surface area with the addition of one monomer. For the critical cluster, Eq.

(2.50) can be simplified to 푘푇 ln 푆, and the expression for the net energy release for a critical cluster becomes

푘푇 푞 = ℎ − − 푘푇 ln 푆. (2.51) 2

Simplifying the differential term allows for nonisothermal corrections to be calculated that are independent of the surface tension model/equation used to analyze the data.

The effects of such a decision are shown in Section 4.4.

Next, we derive a term to account for the extent of collisions between the carrier gas (Ar) and the condensable gas (CO2). We first consider the system without any carrier gas. The term 푏2 is the mean square of the energy fluctuation of the impinging molecules, defined as

1 (2.52) 푏2 = (푐 + 푘)푘푇2 푣 2 where 푐푣 is the molecular specific heat of the condensable gas vapor.

21 When accounting for the presence of carrier gas molecules, we assume that they are truly inert. This means that (1) when carrier gas monomers strike clusters, they do not condense and therefore do not release any heat of condensation, (2) the carrier gas molecules do not affect the surface tension of the cluster, and (3) the carrier gas does not change the rate at which monomers vaporize. The frequency of carrier gas monomers impinging upon the condensable gas clusters can be written as

푝푐 (2.53) 훽푐 = √2휋푚푐푘푇 and the mean square energy of the colliding carrier gas molecules as

1 (2.54) 푏2 = (푐 + 푘) 푘푇2 푐 푣,푐 2 where 푐푣,푐 is the molecular specific heat of the carrier gas. Thus, the mean square energy fluctuation, taking into account the presence of the carrier gas, can be written as

1 훽 1 푏2 = (푐 + 푘) 푘푇2 + 푐 (푐 + 푘) 푘푇2. (2.55) 푣 2 훽 푣,푐 2

Eq. (2.55) can be further simplified as

푁 푚 (2.56) 푏2 = 2(푘푇)2 (1 + c √ ). 푁1 푚c

Following the anisotropic diffusion tensor analysis performed by Feder et al.43, we find that taking the additional heat sinks and carrier gas collisions into account results in a nonisothermal nucleation rate that matches the isothermal one albeit with an additional prefactor 푓:

푏2 푓 = . (2.57) 푏2+푞2

Therefore, the expression for the nonisothermal nucleation rate can be written as

22 푏2 퐽 = 퐽 . (2.58) 푛표푛𝑖푠표 푏2+푞2 𝑖푠표

Though this correction factor was devised to bring theoretical values closer to experiment, we can also apply 푓 in reverse to our experimental data to estimate what nucleation rates should have been measured at the same conditions if clusters were fully thermalized. Our analysis with the nonisothermal correction factor can be found in

Section 4.4.

23 CHAPTER 3: MATERIALS AND METHODS

Section 3.1 Physical Properties of Ar and CO2

There are two gases used throughout all experiments; the carrier gas argon and the condensable gas carbon dioxide. A table of physical properties used for calculations is presented below.

Table 3.1. Physical properties of argon and carbon dioxide.

Argon (Ar)

Molecular weight (g/mol) 39.948

Heat capacity ratio 1.667

Carbon Dioxide (CO2)

Molecular weight (g/mol) 44.01

Critical temperature 푻풄 (K) 304.1282

Critical pressure 푷풄 (MPa) 7.3773

Critical density 흆풄 (kg/m3) 467.6

Triple point 216.55 K, 0.5168 MPa

Heat capacity at constant P 25.9162 + (2.93005 × 10−2)푇 + (2.3825 × 10−5)푇2

(J/mol∙K)10

1.24 Liquid surface tension, Quinn 0.0653(31.35 − (푇푐 − 273.15))

(mN/m)46

24 Liquid surface tension, fit to 푇 (푇 − 푇)푇 1.21 1.004휎 [ 푐 푚] 푚 푇 (푇 − 푇 )푇 Lielmezs-Herrick eqn 푚 푐 푚 휎 = 16.9; 푇 = 216.6 (mN/m)47 푚 푚

4 Liquid density, Span and 휌 푇 푡𝑖 ln ( ) = ∑ 푎𝑖 (1 − ) 휌푐 푇푐 Wagner 𝑖=1

푎 = 1.9245108; 푎 = -0.62385555; 푎 = - (흆 [=] kg/m3)48 1 2 3

0.32731127; 푎4 = 0.39245142; 푡1 = 0.34; 푡2 = 0.5;

푡3 = 10/6; 푡4 = 11/6

1353 Equilibrium vapor pressure, − −8.143 log 푇+0.006259푇+24.619 101325 ∗ 10 푇 liquid (Pa)49

Heat of vaporization (kJ/mol) 28.4363 − 0.021174푇 + (1.19722 × 10−4)푇2

− (4.64865 × 10−7)푇3

25 Section 3.2: Nozzles T1 and T3

Two converging-diverging supersonic nozzle designs were used in the experiments presented here, T1 and T3. Both of these nozzles were constructed with the same aluminum nozzle blocks, with T3 having a smaller distance between them, leading to a shorter throat height and a faster expansion. Boundary layers form along the walls of the nozzle as gas flows through it, leading to an effective expansion rate that is smaller than that of the physical design. T1 has been used in experiments10, 34 by

Tanimura and Park, and T3 was first constructed by to study the condensation of CO2

10 on H2O ice particles in the carrier gas nitrogen . To study the homogeneous nucleation of carbon dioxide, which requires lower temperatures, we changed the carrier gas to argon. Our nozzles are also wider than those of Tanimura; we use a width of 12 mm

26 while his nozzles were 6 mm in width. Figure 3.1 shows an orthographic and cutaway side view of nozzle T3, the primary supersonic nozzle used in our experiments.

The windows in nozzles T1 and T3 are both made of 3 mm thick calcium fluoride

-1 (CaF2). This allows for infrared spectroscopy to be carried out in the 1000 cm to 4000 cm-1 range. We considered using potassium bromide (KBr) windows to push the range

-1 -1 down to approximately 600 cm to measure the v2 bend at 667 cm , but it was deemed too costly an effort to pursue further. For small angle x-ray scattering experiments, we constructed nozzle T3_mica, which has the same effective expansion ratio as nozzle T3 but replaces the CaF2 windows with 0.25 mm thick mica windows covering a 1 mm high slit. Thin windows are required in X-ray scattering to reduce the background, but the windows must still be strong enough to hold the ~1 atm pressure difference that exists between the pressures inside the nozzle and the surroundings.

Section 3.3: Experimental Setup

A schematic of the continuous flow supersonic nozzle setup is shown in Fig. X.

Liquid argon (Ar) at 99.998% purity was purchased from Praxair in 180 L and 230 L

Dewars. Bone dry carbon dioxide (CO2) at 99.9% purity (< 10 ppm water) was purchased from Praxair in compressed gas 3.0-K cylinders. For some earlier experiments, the CO2 used was purchased from Praxair as refrigerated liquid in 180 L

Dewars, but we saw no inherent difference in the data between the two sources of CO2.

Both the Ar and CO2 are brought to room temperature by inline heaters, and their pressure is controlled by inline regulators. Multiple Dewars for Ar and multiple tanks for

CO2 were used as necessary to keep the flows stable.

Figure 3.2. Schematic diagram of the continuous flow supersonic nozzle apparatus. The nozzle cutaway views show the position of the probe or beam in each of the three characterization methods – pressure trace measurements (PTM), small angle X-ray scattering (SAXS), and Fourier transform

infrared (FTIR) spectroscopy. Nozzle T3 (with CaF2 windows) was used for PTM and FTIR experiments, while T3_mica was used for SAXS experiments.

Four unique mass flow controllers were used across all experiments. For Ar flow, the MKS 1559A at 300 SLM was used. For CO2, the following controllers were used:

• At high flow rates – MKS 1559A 400 SLM

• At medium flow rates – MKS GE50A 38 SLM (calibrated for CO2)

• At low flow rates – MKS GE50A 3.8 SLM (calibrated for CO2)

The 400 SLM mass flow controller was calibrated for nitrogen, and we adjusted the flow reading through the N2:CO2 gas correction factor.

The three gas streams (two Ar and one CO2) are combined, pass through a static mixer, and the temperature of the resulting flow is adjusted and controlled via a

28 circulating water bath. The desired initial conditions are reached in the plenum, where the stagnation temperature is measured using a platinum resistance temperature detector (RTD). Static pressures at the location of the probe and at the entrance, throat, and exit of the nozzle are all measured using absolute pressure manometers MKS 120A

Baratron (0.08% accuracy of reading).

The stagnation pressure 푝0 is determined by correcting a static pressure measurement 푝 made via a sidewall tap upstream of the converging section of the nozzle using the relation

휌푢2 푝 = 푝 + , (3.1) 0 2 where 휌 is the density of the gas mixture in kg·m-3 and 푢 is the velocity. At this point in the flow, 푢 is ~15 m·s-1 and the Mach number is ~ 0.06. The gas subsequently flows through the nozzle, either T1 or T3/T3_mica, and is exhausted to the atmosphere by two rotary vane vacuum pumps. Prior to each set of experiments, the system was purged with Ar at a flow rate of 90 SLM for one hour. This procedure was performed to minimize background contamination and the likelihood of heterogeneous nucleation occurring on particles formed from trace amounts of other condensable vapors such as water or alkanes.

Section 3.4: Pressure Trace Measurements

Position-resolved static pressure measurements were made from approximately

1.2 cm upstream of the throat to 10.9 cm downstream of the throat using a probe of length 76 cm and outer diameter 0.92 mm with holes ~31 cm from the tip. A detailed breakdown of the frequency of the measurements is shown in Table 3.2.

29 Table 3.2. Table of positions for pressure trace measurements. Static pressure measurements began approximately 1.2 cm upstream of the throat and were taken in increments of 1.0 mm. As we approach the area near the throat, we increase the resolution of the measurements to 0.4 mm, and subsequently decrease the resolution as we measure further and further downstream. Measurements continue until the end of the nozzle.

Position (mm) Step Count Step Size (mm)

0 – 10 mm 315 1.0 mm

10 – 14 mm 126 0.4 mm

14 – 40 mm 315 1.0 mm

40 – 120 mm 630 2.0 mm

Both dry (carrier gas only) and wet (carrier and condensable gas mixture) traces were performed on various concentrations of CO2 in Ar in the following experiments:

• Investigating the lack of condensation in nozzle T1 – stagnation temperatures of

20°C and 10°C – concentrations of 7 to 11 mol%

• Medium and high flow rates in nozzle T3 – stagnation temperature of 20°C –

concentrations of 3 to 39 mol %

• Low flow rates in nozzle T3 – stagnation temperature of 20°C and 12°C –

concentrations of 0.5 to 2 mol %

All of the above experiments were conducted at a stagnation pressure of 458 Torr or

61.15 ± 0.04 kPa.

The PTM data were analyzed following the methods presented by Kim et al.50.

We first measure the pressure profile with only the carrier gas argon flowing through the nozzle. This is called a dry trace, and we derive the value of 퐴∗, the area at the throat, at

30 constant stagnation temperature 푇0 and stagnation pressure 푝0 using the compressible flow relation

1⁄2 (3.2) 푚̇ ⁄휇 푅푇 퐴∗ = 𝑖 𝑖 ( 0 ) , 훾+1 −훾⁄(훾−1) 훾+1 푝 ( ) 훾∙ 0 2 2 where 푚̇ 𝑖 is the mass flow rate of carrier gas (Ar), 휇𝑖 is the molecular weight of Ar, 푅 is the universal gas constant 8.314 J/mol*K, and 훾 is the ideal gas heat capacity ratio

퐶 훾 = 푝, (3.3) 퐶푣 where 퐶푝 is the heat capacity at constant pressure and 퐶푣 is the heat capacity at constant volume. In order to convert position-resolved measurements into time-resolved data, we need to calculate the velocity of the flow. The Mach number 푀 is defined as the ratio of the local velocity 푢 to the speed of sound 푎

푢 푢 푀 = = . (3.4) 푎 √훾푅푇⁄휇𝑖

In an isentropic expansion, 푀 can be calculated from the measured pressure ratio 푝⁄푝0 by

2 푝 (1−훾)⁄훾 (3.5) 푀2 = [( ) − 1]. 훾−1 푝0

We can also derive the area ratio 퐴⁄퐴∗ from the Mach number using the relation

(훾+1)⁄(2훾−2) 퐴 1 2+(훾−1)푀2 (3.6) = [ ] . 퐴∗ 푀 훾+1

Different nozzles, such as T1 and T3, are characterized by their expansion rates, or the slope of the area ratio with respect to position

푑(퐴⁄퐴∗) (3.7)

푑푥 and is 0.147 cm-1 for T1 and 0.315 cm-1 for T3.

31 Due to its rapid expansion, nozzle T3 shocks at approximately 7.5 cm downstream of the effective throat when flowing only argon because the gas density and therefore pressure is too low for the pumps to maintain the required mass flowrate.

Historically, we have addressed nozzle shocks by using a straight line fit for

푑(퐴⁄퐴∗)⁄푑푥, but in this work, we focus on extrapolating only the end behavior of 퐴⁄퐴∗ before it shocks. This is necessary to extend the dry trace curve for proper data analysis. With the addition of CO2 in the wet traces, the system heats up and expands to a less rapid degree, resulting in the absence of shocks.

For the dry trace, we can simply use the compressible flow relations for argon.

For the wet traces, however, we need to account for the heat release due to the condensation of carbon dioxide. To do so, we integrate the diabatic flow equations for mass (continuity), momentum, and energy with inputs of the measured area ratio, the stagnation conditions 푇0 and 푝0, the wet pressure trace measurements, and the equation of state.

The continuity equation51 is

휌푢퐴 = 휌∗푢∗퐴∗ = 푐표푛푠푡푎푛푡 (3.8) where 푢 is the local flow velocity, 퐴 is the local cross-sectional area of the nozzle, and 휌 is the total density of the gas,

′ 휌 = 휌𝑖 + 휌푣 + 휌푐 (3.9)

Here, 𝑖 stands for inert, 푣 for condensable vapor, and 푐 for the condensate, the ′ symbol indicating these are the particles measured in the same volume as the gases.

The momentum equation is

휌푢푑푢 = −푑푝 (3.10)

32 where 푝 is the gas pressure.

The continuity and momentum equations are not affected by condensation, but the energy equation and equation of state must be modified to account for the heat release and depletion of condensable gas vapor due to condensation. The energy equation is

푐푝푑푇 + 푢푑푢 = 퐿(푇)푑푔 (3.11) where 퐿(푇) is the latent heat of condensation at temperature 푇, 푔 is the condensate mass fraction, and 푐푝 is the specific heat of the flow 푚 푚 푚 푐 = 𝑖 푐 + 푣 푐 + 푐 푐 (3.12) 푝 푚 푝𝑖 푚 푝푣 푚 푝푐 where 푚 is the total mass

푚 = 푚𝑖 + 푚푣 + 푚푐 (3.13)

We assume the mixture of gases and condensate particles is an ideal gas, and write the equation of state as

휌 휌 휌′ (3.14) 푝 = ( 𝑖 + 푣 + 푐) 푅푇 휇𝑖 휇푣 휇푐 where 푅 is the molar gas constant and 휇푎 refers to the molecular weight of species 푎.

The effect of condensation on the equation of state is incorporated through the pressure of the condensable 푝푣, which changes as condensable gas is depleted. We calculate 푝푣 as a function of position in the nozzle from

푝(푧) 푔(푧) (3.15) 푝푣(푧) = 푦푝0 ( ) (1 − ). 푝0 푔∞

At 퐽푚푎푥, 푔(푧)⁄푔∞is approximately 0 so the third term in Eq. (3.15) simplifies to 1. Solving

Eq. (3.8) – Eq. (3.15) results in position-resolved data for the temperature 푇, density 휌, velocity 푢, and condensate mass fraction 푔.

33 The position-resolved saturation profile is calculated via

푝 (푧) 푆(푧) = 푣 , (3.16) 푝푒푞(푇(푧)) where 푝푒푞(푇(푧)) is the equilibrium vapor pressure of CO2. These position-resolved data were converted to time-resolved data and used to calculate the conditions at the peak nucleation rate as described in Section 3.7.

Section 3.5: Small Angle X-Ray Scattering

Small angle x-ray scattering (SAXS) experiments were performed at the 12ID-C beamline at the Advanced Photon Source (APS) at Argonne National Laboratory in

Lemont, IL, USA52. Unlike in the pressure measurements, where the nozzle is stationary and the probe is movable, the nozzle in SAXS experiments is mounted on a remotely controlled translation stage. The X-ray beam is aligned with the physical throat using burn paper. The beam energy was 12 keV with a beam width of approximately 1 mm and a sample-to-detector distance of 2 m. Typically, 10 to 20 single shots with exposure times between 0.7 and 1 s were made for both the sample and the background at each position.

Two types of scattering experiments were performed. Fixed-position measurements were made 7.0 cm downstream of the physical throat over the range of

CO2 in Ar from 3 to 39 mol%. Position-resolved measurements were made starting near the end of the nozzle (8.5 cm) and moved in increments of 2-5 mm upstream until the sample scattering signal was too weak to be distinguished from the background. These measurements were made for concentrations of 3, 6, 12, 18, 24, 30, and 39 mol%. We

34 also collected data for 2 mol% but have excluded it from our analysis after PTM results indicated the flow was unstable with the mass flow controller used.

We used the APS data inversion program to convert 2D scattering patterns to 1D spectra. The absolute scattering intensity was determined using the calibration procedure described by Manka et al.53 The resulting spectra were fit assuming the aerosols followed a Schultz distribution of polydisperse spheres54. From the fit parameters, we calculated the aerosol number density 푁,

2 5 3 (푍+1) 퐼0 (3.17) 푁 = ( ) ( ) ( 6 2), 4휋 (푍+2)(푍+3)(푍+4)(푍+5)(푍+6) ⟨푟⟩ (Δ휌푆퐿퐷) where ⟨푟⟩ is the average particle radius, 퐼0 is the scattering intensity as the scattering angle approaches 0, Δ휌푆퐿퐷 is the difference in scattering length density between the

CO2 particles and the Ar-CO2 gas mixture, 푍 is given by

〈푟〉 2 (3.18) 푍 = ( ) − 1, 훿 and 훿 is the width of the particle size distribution.

Due to restrictions caused by COVID-19, despite having received beam time, we were unable to travel to APS in the summer of 2020 to take small angle X-ray scattering measurements of systems of CO2 in Ar ranging from 0.5 to 2 mol%. Instead, we extrapolated the data from the medium flow rates to estimate the values we would have measured for the low flow rates. The fits used for the data are presented in Section

4.3.B.

Section 3.6: Fourier Transform Infrared Spectroscopy

Fourier transform infrared (FTIR) spectroscopy measurements were performed using the setup described in Park and Wyslouzil34. The setup is pictured in Figure 3.3.

The IR beam (range 900 to 4000 cm-1) exits the side of the PerkinElmer Spectrum 100 and is reflected off a focusing mirror and plane mirror before it travels through the nozzle and is reflected off a second focusing mirror onto the MCT detector. The

Spectrum 10 software is used to control the spectrometer and produce the spectra. The nozzle is mounted on a moveable base so that measurements can be made as a function of axial position without changing the optical alignment. The CaF2 windows restrict the wavenumber range to ~1000 cm-1 to ~4000 cm-1.

There are three fundamental vibrational modes of CO2, shown in Figure 3.4.

36 (a) (b) (c)

Figure 3.4. The three fundamental vibrational modes of CO2 – (a) symmetric stretch, (b) bend, (c) asymmetric stretch.

-1 The first, shown in (a), is the v1 symmetric stretch at 1366 cm . Because the symmetric vibration results in no net dipole moment, however, this vibration is inactive in FTIR

-1 measurements. The second, shown in (b), is the v2 bend at 667 cm . This is outside the

CaF2 transmission range, and we considered making a nozzle with KBr windows to allow for v2 characterization. However, it was deemed too costly, so we did not investigate this vibrational band. The primary focus of the FTIR experiments, shown in

-1 (c), is the v3 asymmetric stretch at 2349 cm . This is the strongest of CO2’s vibrational modes and the one with the most extensive literature.

Preliminary spectra for a CO2 concentration of 12 mol% were measured at a resolution of 4 cm-1. A more thorough experiment was conducted at a resolution of 1 cm-1 and with apodization turned off to better observe the vapor phase absorption lines.

Section 3.7: Nucleation Rate Analysis

We obtain the characteristic time of nucleation by calculating the area underneath the curve of the normalized nucleation rate 퐽푡ℎ푒표푟푦(푡)⁄퐽푚푎푥,푡ℎ푒표푟푦 as a function of time55,

퐽푒푥푝(푡) 퐽푡ℎ푒표푟푦(푡) (3.19) ∆푡퐽푚푎푥 = ∫ 푑푡 = ∫ . 퐽푚푎푥,푒푥푝 퐽푚푎푥,푡ℎ푒표푟푦

37 According to analysis by Hämeri et al.56, we are able to use any nucleation theory that demonstrates “more or less correct behavior” with respect to temperature and saturation. We show the effects of using different theories in Section 4.4. Conceptually, the characteristic time of nucleation is the time required to nucleate the observed number of particles at the maximum nucleation rate in the absence of coagulation.

55 We define the peak nucleation rate as a function of 푆퐽푚푎푥 and 푇퐽푚푎푥 as

푁 휌NZ (3.20) 퐽푚푎푥(푆퐽푚푎푥, 푇퐽푚푎푥) = × . ∆푡퐽푚푎푥 휌VV

The second term on the right in Eq. (3.20) is a correction factor allowing for the change in gas density between the nucleation zone (NZ) and viewing volume (VV) as the flow in the nozzle expands. The value for 푁 in this formula is obtained from small angle X-ray scattering experiments, and the correction factor simplifies to unity if the measurement for 푁 is taken at the nucleation zone.

The Matlab code written and used to perform calculations using self-consistent nucleation theory to obtain the flow conditions at Jmax and the characteristic time of nucleation is attached in Appendix A.

38 CHAPTER 4: RESULTS AND DISCUSSION

Section 4.1: Nozzle T1

Our earliest experiments on the homogeneous nucleation of CO2 used nozzle

T1, which was previously used for studying the heterogeneous condensation of CO2 onto alkanes34. During SAXS measurements by Park, particle formation was observed in “background” measurements of Ar + CO2. To confirm this observation, we ran flows of CO2 in Ar at concentrations around 12 mol% in nozzle T1 to check if this was possible. To match the previous experimental conditions in nozzle T1, we used a stagnation pressure of 458 Torr and stagnation temperature of 20°C. The pressure trace measurements for 7, 9, and 11 mol% CO2 are shown in Figure 4.1.

Figure 4.1. Position-resolved pressure ratio and temperature profiles for flows of 7.0, 9.0, and 11

mol% CO2 in Ar in nozzle T1. Throughout the entire expansion, we observed no heat release, concluding that nozzle T1 does not expand quickly enough and create a large enough temperature

gradient to promote the homogeneous nucleation of CO2.

We did not observe heat release due to condensation taking place at 20°C, thus suggesting that the background signal observed by Park was not due to homogeneous nucleation of CO2 alone. In an attempt to probe nucleation further, however, we decided to test the same systems at a stagnation temperature of 10°C. The pressure and temperature traces for 7 and 11 mol% are shown in Figure 4.2.

Figure 4.2. Position-resolved pressure ratio and temperature profiles for flows of 7.0 and 11 mol%

CO2 in Ar in nozzle T1 at a stagnation temperature of 10°C. Near the nozzle exit, we can observe

some heat release, which may be due to condensation of CO2.

40 In both of these concentrations, we observe a small amount of heat release near the end of the nozzle. Because we do not expect nozzle T1 to overexpand and shock at these pressures and temperatures, we believe we are seeing heat release due to condensation of CO2. The saturation profiles are shown in Figure 4.3.

As the saturation curve does not reach zero on the right hand side of the peak, we can infer that the nucleation event is not yet complete before the nozzle exit. We can, however, generate two points of onset data from nozzle T1 at 10°C assuming that the saturation curve does, in fact, monotonically drop to zero. The pressure and temperature values are tabulated in Table 4.1.

41 Table 4.1. Flow properties at the maximum nucleation rate for concentrations of 7.0 and 11 mol% CO2 in Ar in nozzle T1 at a stagnation temperature of 10°C.

Run zJmax (cm) pJmax (Pa) TJmax (K)

T1W07B 9.08 199 86.8

T1W11B 9.12 319 89.3

Further experiments in nozzle T1 were not conducted because we could not fully characterize the nucleation pulse. A possible way of obtaining complete nucleation data in nozzle T1 would be to lower the stagnation temperature to 5°C or even 0°C, but especially at the lower temperatures, there is the risk of water condensation the surface of the nozzle windows that can cause damage. Instead, we focused our efforts on investigating homogeneous nucleation of CO2 in nozzle T3, which has a higher expansion rate and therefore can reach lower pressures and temperatures than nozzle

T1.

Section 4.2: Nozzle T3 – Medium and High Flow (3.0-39%)

Section 4.2.A: Pressure Trace Measurements

We successfully performed pressure trace measurements on systems of CO2 in

Ar from 3.0 to 39 mol%, with the results for 12 and 24 mol% displayed in Figure 4.4.

The lower bound was determined by the ability to create steady flow in the nozzle using the MKS 1559 400 SLM mass flow controller, and the CO2 concentration was increased until we could no longer characterize the complete nucleation event by the nozzle exit.

Figure 4.4. Position-resolved pressure and temperature profiles for concentrations of 12 (left) and 24

(right) mol% CO2 in Ar in nozzle T3. Stagnation conditions are 458 Torr and 20°C. For all concentrations (see Appendix B), we observed significant heat release due to condensation well upstream of the nozzle exit. We also noticed that the isentrope and the wet trace align poorly at the

highest CO2 concentrations.

Additional measurements were performed at 2.0 and 1.0 mol% using a smaller capacity mass flow controller, but the characterization of those data will be elaborated upon in

Section 4.3. The complete collection of pressure and temperature profiles from pressure trace measurements can be found in Appendix B.

The broad range of conditions measured here lead to some interesting phenomena. Notably, as the heat capacity ratio 훾 of the mixture decreases from 1.66 for

2.0 mol% to 1.54 for 39 mol% (see Figure 4.5), the expansion softens considerably, and the lowest temperature that can be reached within the nozzle at the maximum concentration increases by ~14 K relative to the lowest. Consequently, as the concentration of CO2 initially increases, the position of Jmax does not move upstream as quickly as expected if γ were constant (i.e., if the two species had more similar heat capacity ratios). In fact, as illustrated in Figure 4.5, once the concentration reaches 12 mol%, additional CO2 causes the position of Jmax to shift downstream again. The

43 continued increase of CO2 would result in the expansion no longer being able to reach low enough temperatures within the nozzle to initiate a phase transition.

Figure 4.5. Position of maximum nucleation rate 푧퐽푚푎푥 and heat capacity ratio γ of the gas mixture

plotted with respect to the concentration of CO2 in Ar. As the heat capacity ratio decreases and the expansion softens, nucleation initially moves further upstream, but is then forced to occur further downstream for concentrations greater than 12 mol%.

The CO2 partial pressures and temperatures corresponding to Jmax are illustrated in the Volmer plot, Figure 4.6 (left), the phase diagram of CO2, Figure 4.6

(right), and the complete results are tabulated in Table 4.2.

corresponding to the onset of homogeneous and heterogeneous nucleation of CO2 in supersonic

nozzles on the phase diagram of CO2. The heterogeneous data (Tanimura, Park) lie near the extrapolated liquid-vapor curve while the homogeneous data lie far to the left. The dashed line is the empirical fit in both plots.

Overall, our mid-high flow rate experiments cover a range in partial pressures from 61 to 793 Pa and corresponding temperatures of 77.8 to 92.3 K. The data exhibit a linear relationship when log 푝퐽푚푎푥 is plotted against 1⁄푇퐽푚푎푥. This is consistent with the findings of Volmer57, who established that data with a constant nucleation barrier should fall on a gently curving line when plotted on log 푝 vs. 1/푇. Over a limited temperature range, this should look like a straight line.

45 Table 4.2. Summary of initial conditions and those corresponding to the maximum nucleation rate for

concentrations of 3.0 to 39 mol% CO2. All experiments started at a stagnation temperature of 20°C and a stagnation pressure of 458 Torr. Here, 푝푣0 is the initial partial pressure of CO2, and 푝퐽푚푎푥, 푇퐽푚푎푥,

and 푆퐽푚푎푥 are the CO2 partial pressure, flow temperature, and saturation ratio that maximize the

nucleation rate. The last column is ∆푡퐽푚푎푥, the characteristic time corresponding to the maximum

nucleation rate, and was calculated using 퐽푆퐶푁푇 and the Quinn equation for surface tension.

conc. CO2 푝푣0 푝퐽푚푎푥 푇퐽푚푎푥 푆퐽푚푎푥 ∆푡퐽푚푎푥

mol% (Pa) (Pa) (K) (µs)

3.0 1800 61 77.8 29000 11.2

3.0 1800 63 78.2 25000 7.3

4.0 2400 87 80.2 16000 6.8

5.0 3100 110 81.2 13000 6.6

6.0 3700 130 82.0 12000 7.2

8.00 4890 176 82.1 15000 7.7

9.00 5500 198 82.5 14500 6.1

10.0 6100 227 83.6 11100 8.1

12.0 7330 281 86.0 5920 7.2

12.0 7330 278 85.7 6490 7.2

15.0 9160 345 86.2 6790 6.8

18.0 10991 411 87.2 5780 6.1

18.0 10991 408 87.6 5030 8.5

24.0 14700 528 88.4 5020 8.8

31.2 19100 701 91.9 2270 9.8

38.9 23800 770 91.7 2650 12.6

46 39.3 24000 793 92.3 2290 11.9

As shown in the Volmer plot (Figure 4.6 (left)) and the phase diagram (Figure 4.6

(right)), our data agree with the two research groups that have studied the homogeneous condensation of CO2 in supersonic nozzles above our range of experimental conditions. They are also consistent with the free jet data from Ramos et al, that correspond to a higher cooling rate and lower energy barrier. This suggests that the free energy barriers associated with nucleation for these three sets of supersonic nozzle experiments are likely comparable. The deviation of Krohn’s and Lippe’s onset data from the empirical fit to the Volmer plot is of great interest and is part of an ongoing collaboration discussed in Section 4.3.C.

On the phase diagram (Figure 4.6 (right)), with the exception of the data from

11 Lettieri et al. , the homogeneous CO2 data all lie to the left of the vapor-solid and extrapolated vapor-liquid equilibrium lines and are also well-separated from the

10 34 heterogeneous CO2 nucleation data of Tanimura et al. and Park and Wyslouzil .

Although the phase of the final condensate in the current experiments differs from that of Lettieri et al., where we assume they have liquid droplets instead of solid particles, it is uncertain whether the phase differs from that of Duff15. The consistency in the trend in onset data between the experiments, however, suggests that the initial critical clusters that control particle formation may be similar across all nucleation conditions, i.e. liquid- like in nature. Spectra measured by Ramos et al.58, 59 for clusters containing ~150 molecules supports this idea, whereas larger clusters exhibit spectra more consistent with a crystalline solid.

47 Section 4.2.B: Addressing Contamination

One of the biggest challenges associated with working at such low temperatures

(75-92 K for this measurement range) is the potential for contamination. According to

Heiler60, solid particle number densities of at least 108 cm-3 are required to interfere with homogeneous nucleation in a supersonic flow. Introducing such a dense aerosol from outside the nozzle is difficult in our system. Particle formation from within the nozzle, stemming from trace amounts of condensable vapors with lower vapor pressure, such as water, alcohols, alkanes, etc., is a more likely potential source of seeds for heterogeneous nucleation of CO2.

Figure 4.7. A comparison of “clean” data, shown in red circles, with “contaminated” data, shown in pink circles. (Left) A Volmer plot of all of the mid-high flow rate data. Not only can we distinguish the contaminated data from the clean data by a steeper slope, but we observe saturations for those points that are up to two orders of magnitude lower than those of their clean counterparts. (Right) Selected detailed comparisons showing higher pressures and temperatures for contaminated data at the same concentration of CO2 as their clean data counterparts.

Contamination was identified in several experiments from 2017, where repeat experiments for the same starting conditions found onset temperatures and pressures separated by up to 3.7 K and 23 Pa. In addition, a comparison of the saturations at

48 Jmax showed nucleation occurring at saturations of up to two magnitudes lower in the contaminated experiments than their clean counterparts. The entire set of contaminated data, having a different slope on the Volmer plot than the clean data, is shown in Figure

4.7 (left), and a selected point-by-point comparison is shown in Figure 4.7 (right).

To minimize the effects of unwanted trace condensable vapors in our system and to ensure consistent measurements, we established a purging protocol where we flow

90 SLM of Ar through the system for 1 hour before introducing CO2 into the system. In addition, we developed the practice of starting each set of experiments at the highest possible flow rate and working our way down, since the effect of contamination is less visible at higher concentrations of CO2. We used this protocol when taking both small angle X-ray scattering spectra at Argonne as well as Fourier transform infrared spectra.

Section 4.2.C: Repeatability

We have historically expected excellent repeatability and consistency from supersonic nozzle experiments, but we wanted to quantify this precision. The average pressure ratio and temperature profiles of the five mid-high flow dry (Ar only) traces with shaded error bars are shown in Figure 4.8.

maximum deviation, which is 0.00046 (0.115% of maximum) for 푝⁄푝0 and 0.307 K (0.14% of maximum) for 푇. Thus, we conclude that the conditions in the nozzle are highly repeatable, and that

we have a consistent backdrop for studying homogeneous CO2 nucleation.

The inset of both plots is a blown-up plot of the region near the maximum deviation. For the pressure ratio 푝⁄푝0this is equal to 0.00046, and for the temperature this is equal to

0.307 K. These numbers are a minute fraction of the values we measure, so we conclude that supersonic nozzles are extremely reliable tools for generating consistent nucleation-conducing environments. It is more difficult, however, to obtain wet (Ar +

CO2) trace data with this kind of repetition. This is to be expected as nucleation is highly sensitive to minor fluctuations in pressure and temperature.

Section 4.2.D: Small Angle X-Ray Scattering

Small angle X-ray scattering (SAXS) measurements were made both as a function of position and at a fixed position, 7.0 cm downstream of the physical throat. At this position, nucleation was completely quenched for all of the experimental conditions

50 investigated. A typical CO2-only spectrum involves obtaining a raw background spectrum and subtracting it from a background + sample spectrum. The SAXS spectrum is fit assuming that scattering is from a polydisperse collection of spherical CO2 particles that follow a Schultz distribution54. The actual size distribution functional form does not greatly affect the parameters obtained except at the smallest particle sizes50.

The number densities obtained from SAXS measurements were combined with the characteristic times from PTM experiments to calculate experimental nucleation rates from both the position-resolved and fixed-position data sets.

Section 4.2.D.i: Position-Resolved Measurements

Position-resolved SAXS measurements were made for seven systems containing

3.0-39 mol% CO2 in Ar – 3, 6, 12, 18, 24, 30, and 39 mol%. Shown in Figure 4.9 are selected spectra for the positions corresponding to the maximum experimental nucleation rate for each system; the complete set can be found in Appendix C. We took measurements of 2.0 mol% CO2 as well but decided to discard those results since, according to pressure trace measurements, the mass flow controller we used does not maintain a steady flow at that concentration.

The CO2 aerosol-only curves were easily fit assuming a Schultz distribution of polydisperse spheres, and we obtained parameters of average particle radius, the width of the particle size distribution, and the absolute scattering intensity to calculate the number density and therefore the corresponding nucleation rate. The complete set of spectra are shown in Appendix C.

The position-resolved average particle radius and average number density graphs for all seven systems measured can be found in Appendix D. Selected plots for

12 and 24 mol% are shown in Figure 4.10.

Figure 4.10. Position-resolved average particle radius and average number density for 12 (left) and 24

(right) mol% CO2. We observe that, for most concentrations, particle radius steadily increases (see Figure 4.11). Number density increases as particle formation occurs but begins to drop, as expected in a supersonic expansion. We probed the extent of number density decrease further in Figure 4.12.

We plot the average particle radii in Figure 4.11 and the specific, density- normalized number densities in Figure 4.12. Initially, specific aerosol number density and average particle size both increase as nucleation and condensational particle growth occur simultaneously. Once heat release quenches the nucleation event, no new particles are formed, and the specific aerosol number density drops but with a continued but slower increase in particle size. At this point, coagulation becomes an important mode of particle growth. For most CO2 concentrations measured, particles are still growing at the exit of the nozzle. For 3 mol% CO2 data, we observe a similar drop in specific aerosol number density, but the particle radius does not increase to the same extent.

Figure 4.11. Position-resolved average particle radius for all measured concentrations of CO2. As the

concentration of CO2 increases, we can observe particle formation shifting upstream and greater particle growth. For concentrations greater than 12 mol%, we see the growth curves shifting downstream and don’t observe particles larger than nearly 7.0 nm in radius.

54 Using the parameters obtained from position-resolved spectra, we calculated experimental nucleation rates for both the onset of nucleation as well as 7.0 cm downstream of the physical throat for comparison with results from fixed-position measurements as described in Section 4.2.D.ii. The data are tabulated in Table 4.3.

Table 4.3. Summary of number densities and nucleation rates as calculated from position-resolved small angle X-ray scattering data.

Run mol % N (onset) Jmax (onset) N (7 cm) Jmax (7 cm)

T3W03A 0.03 4.42E+12 3.94E+17 2.42E+12 2.87E+17

T3W03B 0.03 4.42E+12 6.01E+17 2.42E+12 4.39E+17

T3W06 0.06 4.07E+12 5.62E+17 2.08E+12 3.83E+17

T3W12A 0.12 3.73E+12 5.21E+17 1.93E+12 3.74E+17

T3W12B 0.12 3.73E+12 5.14E+17 1.93E+12 3.68E+17

T3W18A 0.18 3.38E+12 5.55E+17 1.82E+12 4.00E+17

T3W18B 0.18 3.38E+12 3.98E+17 1.82E+12 2.85E+17

T3W24 0.24 2.73E+12 3.09E+17 1.70E+12 2.31E+17

T3W30 0.312 2.27E+17 2.31E+17 1.57E+12 1.86E+17

T3W39A 0.389 2.57E+12 2.04E+17 1.68E+12 1.60E+17

T3W39B 0.393 2.57E+12 2.16E+17 1.68E+12 1.70E+17

Section 4.2.D.ii: Fixed-Position Measurements

Fixed-position measurements were made 7.0 cm downstream of the physical throat at concentrations not already measured in position-resolved experiments. These included 4.5, 9, 15, 21, 27, 33, 36, and 39 mol%. Selected spectra from fixed-position

55 measurements at concentrations not previously investigated in position-resolved experiments are shown in Figure 4.13. The rest of the fixed-position spectra can be found in Appendix C.

Figure 4.14 shows the measured number average particle radius and the aerosol number densities as a function of the concentration of CO2 entering the system.

At 7.0 cm downstream of the physical throat, particles ranged from 4.0 to 6.7 nm in radius. As the concentration of CO2 increases, the average particle radius increases rapidly both because fewer particles are made (as shown by the decrease in number density), and because there is more material available to condense. At a fixed position, the flow warms as the heat capacity ratio softens the expansion. Thus, above a certain concentration, particles form further and further downstream, and eventually the particle size at 7.0 cm decreases because condensational growth is not yet complete.

Figure 4.14. Average particle radius and aerosol number density as a function of CO2 concentration

measured 7.0 cm downstream of the physical throat. As the amount of CO2 increases and the flow warms up, particles form later in the nozzle, and we see a decrease of particle size as they take longer to develop and grow.

The decrease in number density with increasing CO2 concentration can be explained by the coupling between nucleation, particle growth, and changes in the expansion rate. Initially, as the partial pressure of CO2 increases, particles that nucleate can grow more rapidly, releasing heat to the flow and shutting off nucleation more rapidly in turn. This pattern mirrors the decrease of characteristic times of nucleation up to a concentration of 12 mol%. Further increases in the CO2 concentration are now associated with expansions characterized by weaker temperature gradients. Less severe changes in temperature make it possible for the phase transition to be initiated at lower nucleation rates. Although the characteristic times increase from this point,

57 nucleation rates decrease slightly more quickly, leading to a net decrease in particle concentration.

The fixed-position data were used to calculate maximum nucleation rates as measured 7.0 cm downstream of the physical throat, and these data, tabulated in Table

4.4, were added to the onset and 7.0 cm data obtained from position-resolved experiments that were tabulated in Table 4.3. For some concentrations, namely 9 and

15 mol%, we had matching PTM data from which to calculate the characteristic time of nucleation. For the rest of the concentrations measured, however, we interpolated the temperature, pressure, and characteristic times from pre-existing PTM data to obtain the experimental nucleation rates.

58 Table 4.4. Summary of number densities and nucleation rates as calculated from fixed-position small

angle X-ray scattering data. Values for 푝퐽푚푎푥, 푇퐽푚푎푥, and ∆푡퐽푚푎푥 were estimated for concentrations that

had no corresponding pressure trace measurement data. (*) denotes: the flow properties at onset

were not actually estimated for 9 and 15 mol%, but position-resolved SAXS data were not obtained, so

we only have the fixed-position spectra for those concentrations.

conc. CO2 푝퐽푚푎푥 (est) 푇퐽푚푎푥 (est) ∆푡퐽푚푎푥 (est) 푁 (7 cm) 퐽푚푎푥 (7 cm)

mol% (Pa) (K) (μs) (cm-3) (cm-3s-1)

4.5 110 80.5 6.8 2.19E+12 4.29E+17

9 198* 82.5* 6.1* 1.92E+12 4.25E+17

15 345* 86.2* 6.8* 1.90E+12 3.76E+17

21 448 88.2 8.0 1.74E+12 2.82E+17

27 572 89.6 9.3 1.64E+12 2.20E+17

33 695 90.8 10.7 1.59E+12 1.80E+17

36 757 91.4 11.4 1.51E+12 1.59E+17

39 782 91.9 12.3 1.48E+12 1.43E+17

Section 4.2.E: Experimental Nucleation Rates

Experimental nucleation rates were calculated as described in Section 3.7 and tabulated in Tables 4.3 and 4.4, with their respective sources of measurement. The experimental nucleation rates initially increase as nucleation moves further upstream and the characteristic time decreases. As the concentration of CO2 increases afterwards, however, the expansion softens and the nucleation rates level off as the characteristic time starts to increase again. There is relatively good agreement between

59 the nucleation rates measured at onset and those measured at a position of 7.0 cm, with the expected consistent lower rate at 7.0 cm due to slight coagulation of clusters resulting in lower number densities. The experimental nucleation rate data are shown as a function of saturation in Figure 4.15 for both position-resolved and fixed-position experiments.

Figure 4.15. (Left) Experimental nucleation rates calculated using the values of 푁(표푛푠푒푡) and 푁(7.0 푐푚) agree within the stated factor of 3 uncertainty. The latter are lower due to coagulation of the aerosol between the nucleation and measurement zones. (Right) Applying the nonisothermal correction factor to the nucleation rates reverses the trend we observe in the uncorrected plot.

Experimental nucleation rates range from ~1.5 x 1017 cm-3s-1 to ~6 x 1017 cm-3s-1, consistent with other measurements in supersonic nozzles23, 61-63 that range from 1016 cm-3s-1 to 1018 cm-3s-1. If we apply the nonisothermal correction factor to account for inadequate thermalization, the absolute value of the rates is increased by up to two orders of magnitude, and the trend is reversed, as shown on Figure 4.15. The error bars on both plots represent a factor of three. A more detailed analysis with the nonisothermal correction factor is presented in Section 4.4.

60 Section 4.2.F: Discrepancy between SAXS and PTM

Although all SAXS experiments used the same purging procedure before starting the measurements, directly comparing the SAXS results to the PTM data revealed a slight inconsistency between the two. On average, we observed scattering signals from

SAXS measurements about 0.5 to 0.7 cm upstream of Jmax calculated from the PTM data. This is illustrated in Figure 4.16, where the solid red curves are the theoretical normalized nucleation rate profiles, 퐽푡ℎ푒표푟푦(푥)⁄퐽푡ℎ푒표푟푦,푚푎푥, that show the expected progress of the nucleation event from start to finish. The dashed gray curves are the

′ ′ expected normalized specific aerosol number densities 푁 ⁄푁푚푎푥 derived from the nucleation rate calculations in the absence of coagulation. Particles are expected to appear slightly before the position of the maximum nucleation rate and in a manner such that the measured specific aerosol number density data (filled circles) overlap with the predicted specific aerosol number densities (dashed line). Instead, particles are first observed significantly earlier.

Figure 4.16. Comparison of SAXS and PTM data for 3 (left) and 24 (right) mol% CO2. The solid curves correspond to the normalized nucleation rates and the gray dashed lines indicate the corresponding normalized specific aerosol number densities. The circles are the measured data. In both cases, particles appeared earlier in SAXS experiments than expected based on PTM data.

We do not believe this inconsistency implies heterogeneous nucleation for the following reasons. First, for the SAXS measurements, the system was purged using the same protocol used for the PTM measurements. Second, the scattering spectra of the particles were well fit assuming a distribution of polydisperse spheres. This contrasts with SAXS spectra from Tanimura et al.’s heterogeneous nucleation experiments10 of

CO2 on H2O, where the data could not be easily fit to scattering from spherical particles.

Third, based on the homogeneous water nucleation rate data of Amaya and

64 Wyslouzil , shown with other CO2 data on the phase diagrams of CO2 and H2O in

Figure 4.17, we can estimate the temperatures at which water particles should form if

CO2 contains the maximum allowed water contamination. The red dotted lines indicate the isentropic expansions expected for CO2 concentrations of 39 mol% and 3 mol%, the upper and lower bounds of our experiments. The red circles correspond to the conditions for Jmax determined by PTM experiments, and the pink points are those

62 consistent with the SAXS measurements. The blue dotted lines indicate the isentropic expansions of water for the two CO2 conditions considered above assuming the maximum allowed water contamination in bone-dry CO2 tanks at 10 ppm is present.

퐽푚푎푥 determined by pressure trace measurements, and the pink points are those retrieved from SAXS measurements. Both are clearly separated from the heterogeneous onset data. The blue dotted lines indicate the isentropic expansions of water for the two CO2 conditions considered above assuming the maximum allowable level of water at 10 ppm. If contamination from water vapor were an issue in these experiments, water particles should have formed when the water isentropes crossed the blue dashed lines, and we would have observed CO2 condensing close to the heterogeneous nucleation line.

If water particles formed, they should have been available for heterogeneous nucleation well upstream of the expected heterogeneous nucleation boundary established by

Tanimura et al.10 as well as where it was detected in our SAXS experiments. Similar

63 arguments apply to alkanes65 and other potential contaminant molecules that condense much earlier than CO2 under the same expansion conditions. Finally, we estimated the temperatures and pressures corresponding to Jmax based on the positions where particles first appeared in SAXS. As illustrated in Figure 4.17, these points lie very close to the Jmax data calculated from PTMs and are still well-separated from the heterogeneous CO2 nucleation curve. This suggests our data do correspond to homogeneous nucleation or the near-homogeneous limit.

Another possibility is that of ion-induced nucleation. During SAXS measurements, the high-energy X-ray beam can ionize the gas phase molecules within the nozzle. If this occurs in the region just upstream of the expected location of Jmax, the ions are exposed to extremely high CO2 saturation ratios that could promote heterogeneous nucleation of CO2 onto the ions. Ionization should not, however, interfere with homogeneous nucleation below a threshold saturation ratio. Thus, aerosol number densities measured further downstream, at much lower saturation ratios, should be due to homogeneous nucleation alone.

Section 4.2.G: Fourier Transform Infrared (FTIR) Spectroscopy

We have two position-resolved sets of Fourier transform infrared spectra on the system of CO2 in Ar at a concentration of 12 mol%. The first data set, 05112018, was taken at a resolution of 4 cm-1 in increments of 1.0 cm from the throat to 7.0 cm downstream of the throat.

Figure 4.18 is the plot of the v3 asymmetric stretch normalized by the ratio of the gas density at the point of measurement to the gas density at the throat. There is a clear

64 transition from the broader, two-peak gas phase to the sharp, one-peak solid phase starting at a position around 5.0 cm.

Figure 4.18. Position-resolved Fourier transform infrared spectroscopy data for the asymmetric stretch

-1 v3 in a system of 12 mol% CO2 measured at a resolution of 4 cm . Measurements began at the throat and were performed in increments of 1.0 cm until a position of 7.0 cm. Spectra were normalized by the gas density to account for the nozzle expansion as we make measurements. We observe a transition from the gas phase to a distinctly solid phase starting at 5.0 cm downstream of the throat.

13 The plots for the v3 asymmetric peak for the CO2 isotope (left) and the

13 combination bands 2v2 + v3 and v1 + v3 (right) are shown in Figure 4.19. The CO2 isotope spectra takes the same shape as the gas phase, but the peak signifying the transition to the solid phase forms near the middle. The combination bands exhibit a single solid peak forming on the left peak of the gas phase spectrum.

Figure 4.19. Raw Fourier transform infrared spectroscopy data for the asymmetric stretch v3 for the

13 CO2 isotope (left) and the combination bands 2v2 + v3 and v1 + v3 (right). Both sets of data show the development of a distinct solid phase peak as we move downstream.

Satisfied that we were able to observe the phase transition of CO2 from the gas phase to the condensed phase using FTIR spectroscopy, we repeated the experiment

-1 on the same system of 12 mol% CO2 with a resolution of 1 cm . We also turned apodization off to better observe vapor phase absorption lines. Measurements for these spectra started 2.0 cm downstream of the physical throat and were taken in increments of 2-4 mm until we observed the development of the condensed phase peak.

Plots of the v3 asymmetric stretch spectra, the raw, baseline-corrected spectra

(top) and the spectra corrected for the change in gas density as the flow expands in the nozzle (bottom), are shown in Figure 4.20. Particle sizes corresponding to the measurements from SAXS experiments in those locations have also been noted.

scattering measurements of the 12 mol% CO2 system.

67 The spectra from the 13CO2 asymmetric stretch and the combination bands we observed in the 1 cm-1 resolution experiments are shown in Figure 4.21.

13 Figure 4.21. Raw Fourier transform infrared spectroscopy data for the CO2 isotope (top) and the

-1 combination bands 2v2 + v3 and v1 + v3 (bottom). At a resolution of 1 cm and without apodization, we are far more sensitive to small fluctuations in absorbance, but we can still observe the formation of the condensed phase peaks from the gas phase.

68 We documented the gas and condensed phase peaks observed in both sets of experiments (at resolutions of 4 cm-1 and 1 cm-1) as well as any notes on the observed peaks in Table 4.5.

Table 4.5. Table of Fourier transform infrared spectroscopy peaks for a system of 12 mol% CO2

-1 -1 measured at resolutions of both 4 cm and 1 cm . Peaks of interest included the v3 asymmetric

12 13 stretch for CO2, the v3 asymmetric stretch for the CO2 isotope, and the combination bands 2v2 + v3

-1 and v1 + v3. Gas phase and condensed phase peak locations differ by up to 5 cm between the two

data sets.

Res. Band Gas Peak Condensed Notes

Peak

-1 -1 -1 -1 4 cm v3 2347 cm 2357 cm Broad shoulder at 2366 cm

-1 13 -1 -1 4 cm CO2 v3 2282 cm 2281 cm

-1 -1 -1 4 cm 2v2 + v3 3610 cm 3598 cm

-1 -1 -1 4 cm v1 + v3 3712 cm 3706 cm

-1 -1 -1 -1 -1 1 cm v3 2350 cm 2359 cm Shoulders at 2360 cm , 2368 cm

-1 13 -1 -1 1 cm CO2 v3 2284 cm 2283 cm

-1 -1 -1 1 cm 2v2 + v3 3615 cm 3601 cm

-1 -1 -1 1 cm v1 + v3 3716 cm 3709 cm

12 For positions up to 3.2 cm downstream of the physical throat, only the gas phase CO2

-1 13 peak centered at 2350 cm is observed, as well as the CO2 counterpart. As particles form and grow, a condensed phase peak centered at 2359 cm-1 appears with a shoulder near 2370 cm-1 that grows in intensity with particle size. We subtracted the gas

69 phase from the spectra (work by Lahari Pallerla) to better observe this trend and isolate the shape of the condensed phase spectra (see Figure 4.22).

Figure 4.22. Gas phase-subtracted Fourier transform infrared spectra for a system of 12 mol% CO2 measured at a resolution of 1 cm-1. After a position of 3.6 cm downstream of the physical throat, we begin to observe the formation of the condensed phase peak. The final shape is that of a narrow peak at 2359 cm-1 and two shoulders at 2360 cm-1 and 2368 cm-1.

What do these spectra suggest about the condensed phase of CO2 clusters? If we compare our measured spectra with those calculated for different shapes and sizes by

Isenor et al.66 (reproduced in Figure 4.23), we conclude that the clusters formed are of cubic shape with or without rounded corners. There is also the possibility that they are octahedral in shape.

Figure 4.23. Infrared spectra predictions by Isenor et al. for (left) different shapes at a constant size of 10 nm in diameter and (right) different cluster sizes of a cube with rounded corners. We believe our clusters exhibit spectra most similar to those of cubes with or without rounded corners.

Section 4.3: Nozzle T3 – Low Flow (0.5-2.0%)

Section 4.3.A: Pressure Trace Measurements

Pressure trace measurements were performed on systems of CO2 in Ar ranging from 0.5 to 2.0 mol%. We started at 2.0 mol%, the previous lower bound at which we had observed condensation, and we decreased the concentration of CO2 until we could no longer detect a significant heat release. The pressure ratio and temperature profiles for 0.5 and 2.0 mol% are shown in Figure 4.24, with the complete set of pressure and temperature profiles available in Appendix B.

Figure 4.24. Position-resolved pressure and temperature profiles for 0.5 (left) and 2.0 (right) mol%

CO2. We observed heat release due to condensation in this range of concentrations, but we still observe a significant heat release due to the nozzle overexpansion.

As when the nozzle shocked in the dry traces for medium and high flow rate experiments, we observed a sharp increase in pressure and temperature at around 8 cm downstream of the throat. For pure Ar, we expect the temperature of the flow to reach approximately 50 K at the exit, and the nozzle shocks around 7.5 cm. We wanted to assume that adding a small amount of CO2 would warm up the flow enough to delay the shock, but we had to check. The isentropic condensation curves for both CO2 and

Ar onto the phase diagram of both gases as well as our previous homogeneous nucleation data for CO2 and Ar are plotted in Figure 4.25.

Figure 4.25. Homogeneous Ar (blue) and CO2 (red) nucleation data plotted onto the phase diagrams

of both gases. The dashed lines represent the Ar and CO2 isentropes at 39 mol% CO2, and the solid

lines are those at 0.5 mol% CO2. The Ar line at 0.5 mol% CO2 crosses the phase diagram but does

not reach the homogeneous data, so we do not expect Ar to condense or become involved with CO2 nucleation in a heterogeneous manner.

The dashed curves represent the highest concentration of CO2 and the solid curves represent the lowest. At 0.5 mol% CO2, the Ar isentrope crosses the solid-vapor and liquid-vapor equilibrium lines but the temperature and pressure projections of the “wet” curve remains well distanced from the homogeneous Ar nucleation data. For this reason, we do not expect to see Ar condense in T3 neither do we expect it to play a significant role in nucleation once CO2 condenses. Having established the fact that we are observing the homogeneous nucleation of CO2, we moved on with the characterization of onset data.

The results for onset data for concentrations of 0.5 to 2.0 mol% CO2 are shown in Table 4.6.

Table 4.6. Table of conditions at the maximum nucleation rate for concentrations of 0.5 to 2.0 mol%

CO2. Values were measured across a range of pressures from ~7 to ~42 Pa and temperatures from

~66 to ~79 K.

Run mol % 푧퐽푚푎푥 푝퐽푚푎푥 푇퐽푚푎푥 ∆푡퐽푚푎푥 푆퐽푚푎푥

(cm) (Pa) (K) (μs)

T3W005A 0.5 6.32 7.45 66.5 9.1 819000

T3W005B 0.5 6.32 7.14 65.81 9.3 1490000

T3W007 0.7 6 10.81 68.79 8.9 368000

T3W0085 0.85 6 13.29 69.58 11.4 288000

T3W01A 1 6.4 14.65 67.63 13.7 882000

T3W01B 1 6.2 15.57 70.66 12.0 174000

T3W012 1.2 5.28 21.47 73.36 11.8 71900

T3W015 1.5 4.88 28.94 75.79 7.7 31600

T3W017 1.7 4.72 33.94 76.46 8.1 28200

T3W02A 2 4.8 38.59 75.11 13.2 55700

T3W02B 2 4.68 41.52 79.22 10.8 10200

T3W02C 2 4.6 41.52 78.11 8.5 16800

Following the trend established by the medium and high flow rate experiments below 12 mol%, onset occurs further downstream as we decrease the concentration of CO2 (see

Figure 4.26).

nucleation occurs further downstream in nozzle T3 as we decrease the concentration of CO2.

Where do the onset points lie in comparison with the other CO2 data? The new data show remarkable overlap with not only our old data, but also Krohn’s post-nozzle data33.

Any noise from experimental uncertainties can get magnified at lower pressures especially on a logarithmic scale, but Figure 4.27 suggests that our data are beginning to deviate from the straight-line fit to the constant nucleation barrier.

Section 4.3.B: Estimation of Nucleation Rates

In order to obtain number densities and therefore experimental nucleation rates, we normally travel to Argonne National Laboratory. However, due to COVID-19 restrictions, we were unable to visit the site and perform small angle X-ray scattering experiments for the low flow rates of CO2. As such, we looked for a trend in the mid-high flow rate data we could use to extrapolate into the low flow region.

Plotted in Figure 4.28 are the number density vs. partial pressure of CO2 data for both measurements taken at the onset of nucleation and at a fixed position 7.0 cm downstream of the physical throat. We found that number density followed a linear trend on a log-log plot as we approached the lower pressure limit. As we were primarily interested in the region of approximately 7 to 40 Pa, we chose to extend the line formed by the onset data and obtained the fit shown in Figure 4.29.

Figure 4.28. Average number density vs. partial pressure of CO2 for measurements at onset of nucleation and at a position 7.0 cm downstream of the physical throat for experiments conducted at the Advanced Photon Source in the summer of 2018. Data below ~30 Pa for onset points and those below ~20 Pa for points at 7.0 cm appear to follow a linear trend on the log-log axes.

Figure 4.29. Log-log fit for number density as a function of partial pressure of CO2 for data collected at the position of onset.

The fit obtained in Figure 4.29 can be written as the following equation:

−0.111471 12.8454 푁푚푎푥 = 푝푣 × 10 . (4.1)

77 Using Eq. (4.1), we estimated values for 푁 for the low flow rate data for CO2 and projected number densities ranging from 4.6 to 5.6 x 1012 cm-3. The complete data table is shown below.

Table 4.7. Table of estimated experimental nucleation rates based on estimated values for the

number density at onset of nucleation for concentrations of 0.5 to 2.0 mol% CO2. The resulting values

17 -3 -1 for 퐽표푛푠푒푡 averaged 5 x 10 cm s .

Run mol % 푝퐽푚푎푥 푁표푛푠푒푡 (EST) ∆푡 퐽표푛푠푒푡 (EST)

(Pa) (cm-3) (s) (cm-3s-1)

T3W005A 0.5 7.45 5.60E+12 9.10E-06 6.15E+17

T3W005B 0.5 7.14 5.63E+12 9.28E-06 6.06E+17

T3W007 0.7 10.81 5.37E+12 8.94E-06 6.01E+17

T3W0085 0.85 13.29 5.25E+12 1.14E-05 4.62E+17

T3W01A 1 14.65 5.19E+12 1.37E-05 3.79E+17

T3W01B 1 15.57 5.16E+12 1.20E-05 4.31E+17

T3W012 1.2 21.47 4.98E+12 1.18E-05 4.21E+17

T3W015 1.5 28.94 4.81E+12 7.68E-06 6.27E+17

T3W017 1.7 33.94 4.73E+12 8.13E-06 5.82E+17

T3W02A 2 38.59 4.66E+12 1.32E-05 3.52E+17

T3W02B 2 41.52 4.62E+12 1.08E-05 4.28E+17

T3W02C 2 41.52 4.62E+12 8.49E-06 5.45E+17

How do these estimated nucleation rates compare with our measured ones? The data overlap well with the higher flow rate measurements (see Figure 4.30 (left)), with

78 the nucleation rates centered around 5 x 1017 cm-3s-1. The relationship between the measured Jmax and saturation appears to be constant past saturations of 104. When comparing our results to those of theory, however, we have to account for improper thermalization between condensing clusters and the carrier gas. At the lowest concentrations, there are approximately 200 argon molecules per CO2 monomer, so we expect clusters to be well-thermalized. At the highest concentrations, however, the same ratio is about 2:1, so we expect a much more severe correction factor will be applied to obtain the nucleation rate should the clusters be properly thermalized.

Applying the nonisothermal correction factor, we obtain Figure 4.30 (right), where intrinsic rates start at approximately 2 x 1019 cm-3s-1 and decrease by an order of magnitude as we reach the measurements at our highest saturation values.

Figure 4.30. (Left) Experimental nucleation rates for both medium and high concentrations as well as

17 estimated nucleation rates for low concentrations of CO2. Nucleation rates, on average, are ~5 x 10 cm-3s-1. (Right) Applying the nonisothermal correction factor results in a more distinct decrease in nucleation rate with increasing saturation.

79 Section 4.3.C: Comparison to Krohn (2020)

At first glance, we do not see a coherent trend between our nucleation rate data and Krohn’s nucleation rate data33 in the region of overlap (see Figure 4.31 (a), (b)), with rates differing by about three orders of magnitude.

Figure 4.31. (a, b) We do not observe any visible trend in the experimental nucleation rate data between our measurements and those of Krohn’s, even when we allow for the factor of three error on our data and the order of magnitude error on Krohn’s data. (c, d) Applying the nonisothermal correction factor, however, we find a monotonic decrease in the nucleation rate with respect to saturation and much better agreement between the two data sets.

80 However, when we apply the nonisothermal correction factor to adjust for different degrees of thermalization across the two experiments (see Figure 4.31 (c),

(d)), we see a seamless transition between our nucleation rate data and that of Krohn’s.

The error bars on our data represent a factor of three, and those on Krohn’s data represent an order of magnitude.

Section 4.4: Comparisons with Classical Nucleation and Scaling Theories

Comparing experimental results with the predictions of any nucleation theory is often hampered by the lack of appropriate physical property data in the experimentally relevant temperature range. For the highly supercooled conditions investigated here, where temperatures are 125-142 K below the triple point, even the nature of the critical cluster, i.e. whether it is crystalline, liquid, or amorphous, is uncertain. In our FTIR experiments, we were not able to observe a change in the spectra that indicated a transition in the condensate from a liquid to a solid primarily due to strong interference from the gas phase spectra. Raman spectra58 measured by Ramos et al. and MD simulations by our collaborators at the University of Helsinki35 both support the presence of liquid-like critical clusters. This evidence, combined with the already-not- dependable incorporation of cluster properties into classical nucleation theory via the bulk values of the material, led us to test several alternate hypotheses for CO2 cluster formation.

The liquid surface tension equation used for data analysis throughout the majority of our work was formulated by Quinn46, and the equation is reproduced below:

81 휎 = 0.0653(31.35 − (푇 − 273.15))1.24 (4.2) where 휎 is in units of mN/m and 푇 is the temperature in units of K.

Our collaborators at the University of Helsinki suggested trying another equation for the surface tension based on a phenomenological scaling law proposed by Lielmezs and Herrick47. This equation for the liquid surface tension of a substance uses the critical point and boiling point as reference points. The reduced surface tension is expressed as

휎∗ = 푎푇∗푏, (4.3) where * denotes a reduced quantity, i.e. reduced surface tension and reduced temperature (in K), and 푎 and 푏 are constant fitting parameters. The reduced coordinates are

푇 ⁄푇−1 푇∗ = 푐 , (4.4) 푇푐⁄푇푏−1 where 푇푐 is the critical temperature and 푇푏 is the temperature at the boiling point of CO2, and

휎⁄푇 휎∗ = , (4.5) 휎푏⁄푇푏 where 휎푏 is the surface tension in mN/m at the boiling point. Shown in Table 4.8 are the values we used in applying the Lielmezs-Herrick formulation to CO2:

82 Table 4.8. Table of values used in deriving the Lielmezs-Herrick equation for the liquid surface tension

of CO2.

Symbol Meaning Value

푇푐 critical temperature 304.2 K

푇푏 temperature at boiling point 216.6 K

휎푏 surface tension at boiling point 16.9 mN/m

Substituting these values and parametrizing the liquid CO2 surface tension data by

Muratov and Skripov67 yields

휎∗ = 1.004 × 푇∗1.21. (4.6)

It has been shown that the relation by Lielmezs and Herrick performs relatively well for simple liquids, yielding smaller root-mean-square (rms) errors than other commonly used methods47, and the deviation between the data and Eq. (4.6) is about

5%. The relation by Quinn46 has an rms error of 8%, but we chose this relation for our analysis because we believe it better captures the behavior of the surface tension as we approach the low-temperature limit. We have plotted the two expressions for surface tension along with the Muratov and Skripov data in Figure 4.32.

Based on their MD work, our Helsinki collaborators also developed a simulation- based correction to classical theory. Classical theory uses the capillarity approximation, which is the assumption that the surface tension of a cluster equals that of an infinite planar bulk interface, i.e. 휎 = 휎∞. The surface properties of a cluster containing a relatively small number of monomers can, however, differ significantly from that of a planar surface. The curvature dependency of the surface tension has typically been implemented using the Tolman equation for a cluster68

휎 휎(푟) = ∞ , (4.7) 1+2훿푇⁄푟 where 훿푇 is the Tolman length. Truncating the Taylor expansion after the second term yields

2훿 휎(푟) = 휎 (1 − 푇). (4.8) ∞ 푟

84 Following the work of Tanaka et al.69, our Helsinki collaborators constructed a nucleation rate equation that incorporates an empirically-derived simulation-based correction to the formation free energy expression using an effective Tolman length 훿푇,푒.

This value is independent of cluster size and equal to 푟1⁄2, where 푟1 is the radius of the monomer. Unlike the usual value for the Tolman length, however, this value is positive, so they argue that this correction accounts for a collection of microscopic phenomena that have the same dependence on radius as the curvature term.

We denote nucleation rate equation implementing this correction as 퐽푆푇, and the critical cluster sizes corresponding to 퐽푆푇 are significantly smaller than those corresponding to 퐽푆퐶푁푇.

To probe the effects of implementing an alternative surface tension relation and/or nucleation theory, we decided to investigate four pairs of expressions for σ and J

– JSCNT-Quinn, JSCNT-LH, JST-Quinn, and JST-LH – in greater detail.

To investigate the effect of changing the surface tension and nucleation theory on the characteristic time of nucleation, we plotted the normalized theoretical nucleation rates as a function of time, i.e. 퐽푡ℎ푒표푟푦(푡)⁄퐽푡ℎ푒표푟푦,푚푎푥, for all four combinations, shown in

Figure 4.33.

Figure 4.33. 퐽푡ℎ푒표푟푦(푡)⁄퐽푡ℎ푒표푟푦,푚푎푥 plotted for four combinations of nucleation theory and surface

tension for one of the experiments testing a concentration of 18 mol% CO2. Most of the combinations

yield a value for the characteristic time of nucleation close to the value we use of 8.50 μs, and the 퐽푆푇- Quinn combination yields an outlier that is only ~2.5 times higher than the smallest value obtained.

If we calculate the characteristic time from each of the curves, we find that three of the four curves yield ΔtJmax values (7.3, 8.50, 11.1 μs) that differ by ~25% from the average of these three. Although the combination of JST and the Quinn surface tension appears to be an outlier, the associated ΔtJmax (18.6 μs) is still only ~2.5 times higher than the smallest value. Assuming constant experimental number densities obtained from SAXS measurements, this combination would also yield the smallest experimental nucleation rates, but still within our established error of a factor of three.

For direct comparison between the predictions of nucleation theory and experimental nucleation rates, both the physical properties of the clusters and the effects of thermalization are important. Thus, all of the comparisons with theory assume liquid-like critical clusters that are not fully thermalized by the carrier gas.

86 For the CO2-Ar mixtures used in these experiments, at the conditions corresponding to the maximum nucleation rates, the values of the non-isothermal correction factor vary between 0.015 and 0.391, with the strongest corrections corresponding to the highest CO2 concentrations. The complete set of parameters and nonisothermal correction factors is tabulated in Table 4.9.

Table 4.9. Table of physical properties and parameters used in calculating the value of the

nonisothermal correction factor 푓푛표푛𝑖푠표 for the range of concentrations from 0.5 to 39.3 mol% CO2.

2 mol % 푝퐽푚푎푥 푇퐽푚푎푥 푁푐⁄푁 푏 푞 푓푛표푛𝑖푠표

(Pa) (K)

0.5 7.45 66.5 199 3.54E-40 2.35E-20 0.391

0.5 7.14 65.81 199 3.47E-40 2.31E-20 0.393

0.7 10.81 68.79 142 2.70E-40 2.36E-20 0.327

0.85 13.29 69.58 117 2.28E-40 2.36E-20 0.290

1 14.65 67.63 99 1.83E-40 2.31E-20 0.256

1 15.57 70.66 99 2.00E-40 2.38E-20 0.261

1.2 21.47 73.36 82 1.79E-40 2.40E-20 0.237

1.5 28.94 75.79 66 1.53E-40 2.43E-20 0.206

1.7 33.94 76.46 58 1.37E-40 2.42E-20 0.190

2 38.59 75.11 49 1.13E-40 2.38E-20 0.165

2 41.52 79.22 49 1.25E-40 2.47E-20 0.170

2 41.52 78.11 49 1.22E-40 2.44E-20 0.170

3 61.4 77.8 32 8.06E-41 2.40E-20 0.123

87 3 62.5 78.2 32 8.15E-41 2.41E-20 0.123

4 87 80.2 24 6.42E-41 2.41E-20 0.099

4.5 109 80.5 21 5.75E-41 2.39E-20 0.091

5 111 81.2 19 5.26E-41 2.41E-20 0.083

6 134 82 16 4.47E-41 2.40E-20 0.072

8 176 82.1 12 3.36E-41 2.37E-20 0.056

9 198 82.5 10 3.01E-41 2.37E-20 0.051

10 227 83.6 9 2.78E-41 2.37E-20 0.047

12 281 86 7 2.45E-41 2.39E-20 0.041

12 278 85.7 7 2.44E-41 2.39E-20 0.041

15 345 86.2 6 1.97E-41 2.37E-20 0.034

18 411 87.2 5 1.68E-41 2.37E-20 0.029

18 408 87.6 5 1.69E-41 2.38E-20 0.029

21 448 88.2 4 1.47E-41 2.38E-20 0.025

24 528 88.4 3 1.29E-41 2.36E-20 0.023

27 572 89.6 3 1.17E-41 2.38E-20 0.020

31.2 701 91.9 2 1.07E-41 2.40E-20 0.018

33 695 90.8 2 9.84E-42 2.38E-20 0.017

36 757 91.4 2 9.13E-42 2.38E-20 0.016

38.9 770 91.7 2 8.49E-42 2.38E-20 0.015

39 782 91.9 2 8.51E-42 2.38E-20 0.015

39.3 793 92.3 2 8.49E-42 2.39E-20 0.015

88 In the context of the experiments, we are first interested in determining the vapor pressure corresponding to an experimental nucleation rate of 5 x 1017 cm-3s-1 as a function of temperature. To do so, we numerically solve for the pressure corresponding to this rate, assuming an average constant nonisothermal factor of 푓푛표푛𝑖푠표 = 0.05.

Figure 4.34 compares the available supersonic nozzle data with the computed pressure values as a function of temperature for all four combinations of nucleation theory and surface tension equation.

Figure 4.34. Curves corresponding to a constant non-isothermal nucleation rate of 5 x 1017 cm-3s-1 for

all combinations of nucleation theory (퐽푆퐶푁푇 or 퐽푆푇) and surface tension correlation (Quinn or Lielmezs- Herrick) are compared to the experimental onset of nucleation data of Lettieri, Duff, and the current work.

On the scale of Figure 4.34, all of the theories do reasonably well in predicting the

15 pressure and temperature pairs for the CO2 condensation experiments of Duff and

Lettieri et al11. For the mid-high flow rate data, the predictions of self-consistent nucleation theory 퐽푆퐶푁푇 yields onset pressure above the data for either surface tension expression, whereas simulation-based theory 퐽푆푇 yields onset pressures that are slightly

89 below the experimental data for either surface tension expression. The low flow rate data touches or overlaps reasonably well with the simulation-based theory and

Lielmezs-Herrick surface tension combination. None of the combinations reproduce the experimentally obtained temperature dependency over the entire range of measurements.

We performed a more stringent comparison of the experimentally derived mid- high flow nucleation rate data with the two theories as illustrated in Figure 4.35, where we directly examined the ratio of 퐽푒푥푝⁄퐽푡ℎ푒표푟푦 using all four combinations of nucleation theory and surface tension expression.

Figure 4.35. Comparisons between experimental nucleation rates and those calculated using self- consistent classical nucleation theory (SCNT) and simulation based theory (ST) for the surface tension correlations of Quinn or Lielmezs-Herrick (L-H) equation. Non-isothermal corrections are included in both theories.

All theoretical values were corrected for nonisothermal effects based on the experimental conditions at nucleation. Although none of the theory and surface tension combinations is quantitatively correct, the temperature dependence of the ST theory matches that of the experiments slightly better than SCNT does.

90 Finally, an alternate way to test the consistency of all the available data, and to predict nucleation rates empirically, is to consider the scaling laws first proposed by

Hale et al.70 As illustrated in Figure 4.36, all of the data measured in supersonic nozzles at pressures and temperatures at or above our conditions are bounded by the 퐽푒푥푝 =

19 -3 -1 17 -3 -1 10 cm s at the upper limit, and loosely bounded by 퐽푒푥푝 = 10 cm s at the lower limit. We do not expect Krohn’s data to follow this trend.

Figure 4.36. (Left) In the Hale plot, the experimental supersonic nozzle data are bounded by the lines corresponding to the nucleation rates 퐽 = 1019 cm-3s-1 on top and loosely by 퐽 = 1017 cm-3s-1 on the

17 bottom when the effective omega parameter is Ω푒푓푓 = 1.70. (Right) For a nucleation rate of 퐽 = 10

-3 -1 cm s , the omega parameter of Ω푒푓푓 = 1.70 best fits our data, while the value of Ω푡ℎ푒표푟푦 calculated for the liquid is a better approximation than that of the solid.

The straight lines were calculated for an effective omega parameter of Ω푒푓푓 = 1.70. The value of Ω푒푓푓 is about 11% lower than the value of Ω푡ℎ푒표푟푦 = 1.91 for the liquid critical cluster calculated using

2⁄3 2⁄3 (4.9) 푘Ω ≈ 퐾퐸⁄푁퐴 = 휎푣 ⁄(푇푐 − 푇),

91 where 푁퐴 is Avogadro’s number, 퐾퐸 is the Eötvös constant, and the extrapolated surface tension is that of Quinn. The deviation observed here is comparable to that

24, 71 observed for other small molecules we have studied ; for Ar, we found Ω푒푓푓: Ω푡ℎ푒표푟푦

= 1.5:2 and for N2 = 1.54:2.07. Assuming a solid critical cluster (see Fig 4.36 (right)) yields Ω푡ℎ푒표푟푦 = 3.28, a value that implies a much greater deviation between the effective and theoretical values of 훺. Although this is not proof that the critical clusters are liquid-like, the difference between Ω푒푓푓 and Ω푡ℎ푒표푟푦 is much larger than we have ever observed.

We also compared our low flow rate data and that of Krohn’s to two theories –

SCNT, as with our medium and high flow rate data, and the hard sphere (HS) theory described in Krohn et al.33 The HS model assumes that every collision between molecules of the condensable gas leads to sticking and cluster growth, so it is a representation of the maximum nucleation rate possible33. As a result, the pressure at which nucleation occurs with respect to temperature in the HS model is fairly constant.

We find that SCNT (see Figure 4.37 (red)) does a better job than HS (see Figure 4.37

(black)) in capturing the behavior of the CO2 nucleation onset data through the high saturation region, though there is agreeable overlap with HS for some of the data from

Krohn, shown in yellow diamonds. At extremely high saturations, the HS model is adequate for describing nucleation, but as the temperature increases, we see a deviation in SCNT and HS that corresponds to the appearance of a free energy barrier.

We cannot capture the effect of our experimental nucleation rate data undergoing the nonisothermal factor correction on this kind of plot, however.

Figure 4.37. A comparison of experimental onset data to SCNT (left) and HS (right) theories. SCNT better captures the behavior of the data at higher pressures and temperatures, while ST, a collision- based model, provides reasonable agreement for Krohn’s high-saturation data, shown in yellow diamonds. We are unable to capture the effects of using the nonisothermal correction factor in plotting the Volmer data of pressure and inverse temperature.

In summary, a comparison of experimental data with nucleation theory reveals the following. While the formulation for the surface tension has a slight effect on the characteristic time used for calculating experimental nucleation rates, self-consistent nucleation theory (SCNT) is adequate, though not exact, for describing CO2 nucleation data across the range of experiments presented here. Simulation-based theory (ST) appears to provide a better temperature dependence, but we have concerns about the use of the Lielmezs-Herrick (L-H) surface tension extrapolation to low temperatures. As the saturation increases, the data approach the curves predicted by hard sphere (HS) model. Applying the nonisothermal correction factor from Feder et al.43 results in intrinsic nucleation rates increased by up to two orders of magnitude and unifies data across different experiments and nucleation regimes.

93 CHAPTER 5: CONCLUSIONS AND FUTURE WORK

In our experiments, we studied the homogeneous nucleation of carbon dioxide in nozzles T1 and T3. We found that nozzle T1 does not reach temperatures cold enough to promote the condensation of CO2 at a stagnation temperature of 20°C, but we can observe heat release near the nozzle exit when the stagnation pressure is lowered to

10°C.

In nozzle T3, we performed pressure trace measurements, small angle X-ray scattering experiments, and obtained Fourier transform infrared spectra to characterize the nucleation of CO2. We obtained pressure trace measurements for concentrations of

CO2 in Ar ranging from 0.5 to 39 mol% and characterized the onset of nucleation. We found that our data were in excellent agreement with both measurements taken at higher pressures and temperatures11, 15, 58, 59 as well as data measured at the lower extremes32, 33. We found that, due to the large difference in the heat capacity ratio between that of CO2 and that of Ar, nucleation first shifts upstream and changes direction after reaching a concentration of ~12 mol% as additional CO2 causes the expansion to soften significantly and leads to warmer temperatures at the same location in the nozzle. In addition, we found that the flow conditions in our nozzle are highly repeatable and developed a protocol for minimizing contamination from trace amounts of condensable gases other than CO2.

We performed two kinds of small angle X-ray scattering experiments – position- resolved and fixed-position measurements at a location 7.0 cm downstream of the physical throat. We found that our scattering spectra fit well to a Schultz distribution of

94 polydisperse spheres54 and that coagulation is a significant mode of particle growth. We also observed the trend in particle size mirrors that of the position of onset; as the concentration of CO2 increases, particles initially grow to a larger size by the time the flow reaches 7.0 cm, but the particle size we observe decreases as the flow warms up and nucleation starts further downstream. Due to a discrepancy in the position of onset of nucleation derived from PTM and SAXS experiments, we suspected heterogeneous nucleation may be occurring but ruled it out via evidence from previous data from both heterogeneous CO2 nucleation experiments as well as pure water data that suggested that even the SAXS-derived data lie far to the left of the heterogeneous nucleation curve on the phase diagram.

We calculated experimental nucleation rates for 3.0 to 39 mol% CO2 and estimated those for 0.5 to 2.0 mol% and found absolute values to be on the order of 5 x

1017 cm-3s-1. To compare our experimental results with predictions by theory, we applied the nonisothermal correction factor to the experimental nucleation rates to account for improper thermalization between the condensing CO2 clusters and the decreasing ratio of carrier gas molecules to CO2 monomers. We found that the correction factor resulted in values for intrinsic nucleation rates of up to two orders of magnitude greater than what we measured. Testing two theories – self-consistent classical theory42 and simulation-corrected theory69 – with two equations for the surface tension of the cluster

– that of Quinn46 and that of Lielmezs and Herrick47 – indicated that our experimental data are bounded by the combinations of SCNT with Quinn and ST with L-H. We conclude that the critical clusters are of liquid-like nature for several reasons. Analysis with scaling theory70 suggests that the empirical line for a nucleation rate of 1017 cm-3s-1

95 is closest to that of the Eötvös constant corresponding to the liquid critical cluster.

Additionally, both Raman spectra58 from Ramos et al. and MD simulations from our collaborators in Helsinki suggest that the critical clusters are liquid-like in form.

Fourier transform infrared spectroscopy experiments showed a transition from the gas phase to the crystalline solid condensed phase, and we believe these particles are of cubic shape, with or without rounded corners.

Our work is part of an ongoing collaboration with researchers at ETH Zurich and the University of Helsinki. At ETH Zurich, researchers from the Signorell group32, 33 are working on homogeneous nucleation of CO2 studies in uniform post-nozzle flow with a two-fold aim. The first is to collect onset data to fill in the gaps between our low flow data and Lippe’s32 measurements at extremely high supersaturations. The second is to calculate the nucleation rates in their experimental setup and compare them with our calculated rates, both with and without the application of the nonisothermal correction factor. At the University of Helsinki, researchers from the Vehkamäki group38 are working on MD simulations to develop a greater understanding of the critical clusters of

CO2 and improve modeling for CO2 condensation in supersonic flows.

We propose additional methods of advancing the research on the homogeneous nucleation of carbon dioxide in our own laboratory. First, collecting experimental SAXS data from the Advanced Photon Source at Argonne National Laboratory to replace the low flow rate estimates would provide us with more accurate results as to number densities and nucleation rates. It would also be of interest to collect particle sizes for the low flow rate region, especially at the lower limit of our measurement range. Second, obtaining more sensitive measurement equipment, for pressure for example, would

96 allow us to more reliably collect and analyze data for extremely low pressures of CO2 and increase our probing into the subsequent nucleation regime. Purchasing another pump to further decrease the pressure we can reach at the exit of the nozzle before the expansion shocks would allow us to test concentrations of CO2 higher than 39 mol% or pressure of 793 Pa. Third, constructing an additional nozzle with a distinct expansion rate somewhere between that of T1 and T3 would allow us to calculate experimentally derived critical cluster sizes for CO2. Nozzle T1 does not expand quickly enough for experiments to be performed at the same ones as those in nozzle T3, but a slightly faster nozzle might be sufficient. Finally, we would be interested in performing FTIR spectroscopy experiments on the system of heterogeneous CO2 nucleation on water.

Both homogeneous CO2 and homogeneous water spectra have been obtained in our laboratory, as well as clusters formed from heterogeneous nucleation of other molecules on water, but none of CO2 on water.

97 REFERENCES

1. Gibbins, J.; Chalmers, H., Carbon capture and storage. Energy Policy 2008, 36, 4317-

4322.

2. Hart, A.; Gnanendran, N., Cryogenic CO2 Capture in Natural Gas. Energy Procedia

2009, 1, 697-706.

3. Haszeldine, R. S., Carbon Capture and Storage: How Green Can Black Be? Science

2009, 325 (5948), 1647-1652.

4. Pires, J. C. M.; Martins, F. G.; Alvim-Ferraz, M. C. M.; Simoes, M., Recent developments on carbon capture and storage: An overview. Chemical Engineering Research and Design 2011, 89, 1446-1460.

5. Markewitz, P.; Kuckshinrichs, W.; Leitner, W.; Linssen, J.; Zapp, P.; Bongartz, R.;

Schreiber, A.; Muller, T. E., Worldwide innovations in the development of carbon capture technologies and the utilization of CO2. Energy & Environmental Science 2012, 5, 7281-

7305.

6. Luis, P., Use of monoethanolamine (MEA) for CO2 capture in a global scenario:

Consequences and alternatives. Desalination 2016, 380, 93-99.

7. Xie, H.-B.; Li, C.; He, N.; Wang, C.; Zhang, S.; Chen, J., Atmospheric Chemical

Reactions of Monoethanolamine Initiated by OH Radical: Mechanistic and Kinetic Study.

Environmental Science & Technology 2014, 48, 1700-1706.

8. Borduas, N.; Abbatt, J. P. D.; Murphy, J. G., Gas Phase Oxidation of

Monoethanolamine (MEA) with OH Radical and Ozone: Kinetics, Products, and Particles.

Environmental Science & Technology 2013, 47, 6377-6383.

98 9. Xie, H.-B.; Elm, J.; Halonen, R.; Myllys, N.; Kurten, T.; Kulmala, M.; Vehkamaki, H.,

Atmospheric Fate of Monoethanolamine: Enhancing New Particle Formation of Sulfuric Acid as an Important Removal Process. Environmental Science & Technology 2017, 51, 8422-

8431.

10. Tanimura, S.; Park, Y.; Amaya, A.; Modak, V.; Wyslouzil, B. E., Following heterogeneous nucleation of CO2 on H2O ice nanoparticles with microsecond resolution.

RSC Advances 2015, 5, 105537-105550.

11. Lettieri, C.; Paxson, D.; Spakovszky, Z.; Bryanston-Cross, P., Characterization of

Nonequilibrium Condensation of Supercritical Carbon Dioxide in a de Laval Nozzle. Journal of Engineering for Gas Turbines and Power 2018, 140, 041701.

12. Hernandez Nogales, A. Analysis of low temperature carbon dioxide capture in a supersonic nozzle. Escola Tècnica Superior d'Enginyeria Industrial de Barcelona, 2011.

13. Ahmad Samawe, R.; Rostani, K.; Mohd Jalil, A.; Esa, M.; Othman, N., Concept Proofing of Supersonic Nozzle Separator for CO2 Separation from Natural Gas using a Flow Loop. In

Offshore Technology Conference Kuala Lumpur, Malaysia, 2014.

14. Wen, C.; Yang, Y., Novel CO2 Separation Using Supersonic Separator for Carbon

Capture and Storage (CCS). In 79th EAGE Conference and Exhibition, 2017; pp 1-5.

15. Duff, K. M. Non-equilibrium condensation of carbon dioxide in supersonic nozzles.

Massachusetts Institute of Technology, 1966.

16. Okimoto, F. T.; Brouwer, J. M., Twister Supersonic Gas Conditioning - Studies,

Applications, and Results. In GPA, San Antonio, 2003.

17. Brouwer, J. M.; Epsom, H. D., Twister Supersonic Gas Conditioning for Unmanned

Platforms and Subsea Gas Processing. In Offshore Europe, Society of Petroleum Engineers:

Aberdeen, UK, 2003; p 7.

99 18. Borissov, A.; Mirzoev, G.; Shtern, V. Supersonic Swirling Separator 2. 2014.

19. Alfyorov, V.; Bagirov, L.; Dmitriev, L.; Feygin, V.; Imayev, S.; Lacey, J. R., Supersonic nozzle efficiently separates natural gas components. Oil & Gas Journal 2005, 53-58.

20. Altam, R. A.; Lemma, T. A.; Jufar, S. R., Trends in Supersonic Separator design development. In MATEC Web of Conferences, 2017; Vol. 131.

21. Haghighi, M.; Hawboldt, K. A.; Abdi, M. A., Supersonic gas separators: Review of latest developments. Journal of Natural Gas Science and Engineering 2015, 27, 109-121.

22. Kashchiev, D., Nucleation: Basic Theory with Applications. 2000.

23. Wyslouzil, B. E.; Wölk, J., Overview: Homogeneous nucleation from the vapor phase -

The experimental science. The Journal of Chemical Physics 2016, 145, 211702.

24. Bhabhe, A.; Wyslouzil, B., Nitrogen nucleation in a cryogenic supersonic nozzle. The

Journal of Chemical Physics 2011, 135 (24), 244311.

25. Leyssale, J.-M.; Delhommelle, J.; Millot, C., Molecular simulation of the homogeneous crystal nucleation of carbon dioxide. The Journal of Chemical Physics 2005, 122, 10.

26. Diemand, J.; Angelil, R.; Tanaka, K. K.; Tanaka, H., Large scale molecular dynamics simulation of homogeneous nucleation. The Journal of Chemical Physics 2013, 139,

074309.

27. Toxvaerd, S., Molecular-dynamics simulation of homogeneous nucleation in the vapor phase. The Journal of Chemical Physics 2001, 115 (19), 8913-8920.

28. Hammer, M.; Wahl, P. E.; Anantharaman, R.; Berstad, D.; Lervag, K. Y., CO2 capture from off-shore gas turbines using supersonic gas separation. Energy Procedia 2014, 63,

243-252.

29. Binnie, A. M.; Green, J. R., An electrical detector of condensation in high-velocity steam. Proceedings of the Royal Society A 1941, 181 (985), 134-154.

100 30. Faro, I.; Small, T. R.; Hill, F. K., The Supersaturation of Nitrogen in a Hypersonic Wind

Tunnel. Journal of Applied Physics 1952, 23 (40), 40-43.

31. Wegener, P. P., Condensation Phenomena in Nozzles. Progress in Astronautics and

Rocketry 1964, 15, 701-724.

32. Lippe, M.; Szczepaniak, U.; Hou, G.-L.; Chakrabarty, S.; Ferreiro, J. J.; Chasovskikh, E.;

Signorell, R., Infrared Spectroscopy and Mass Spectrometry of CO2 Clusters during

Nucleation and Growth. The Journal of Physical Chemistry A 2019, 123, 2426-2437.

33. Krohn, J.; Lippe, M.; Li, C.; Signorell, R., Carbon dioxide and propane nucleation: the emergence of a nucleation barrier. Physical Chemistry and Chemical Physics 2020, 22,

15986-15998.

34. Park, Y.; Wyslouzil, B. E., CO2 condensation onto alkanes: unconventional cases of heterogeneous nucleation. Physical Chemistry and Chemical Physics 2019, 21, 8295-8313.

35. Dingilian, K. K.; Halonen, R.; Tikkanen, V.; Reischl, B.; Vehkamäki, H.; Wyslouzil, B. E.,

Homogeneous nucleation of carbon dioxide in supersonic nozzles I: experiments and classical theories. Physical Chemistry and Chemical Physics 2020, 22, 19282-19298.

36. Wedekind, J.; Strey, R., New method to analyze simulations of activated processes.

Journal of Chemical Physics 2007, 126 (13), 134103.

37. Yasuoka, K.; Matsumoto, M., Molecular dynamics of homogeneous nucleation in the vapor phase. I. Lennard-Jones fluid. Journal of Chemical Physics 1998, 109 (19), 8451-

8462.

38. Halonen, R.; Zapadinsky, E.; Vehkamäki, H., Deviation from equilibrium conditions in molecular dynamic simulations of homogeneous nucleation. Journal of Chemical Physics

2018, 148 (16), 164508.

101 39. Wedekind, J.; Hyvärinen, A.-P.; Brus, D.; Reguera, D., Unraveling the "Pressure Effect" in Nucleation. Physical Review Letters 2008, 101 (12).

40. Horsch, M.; Vrabec, J.; Bernreuther, M.; Grottel, S.; Reina, G.; Wix, A.; Schaber, K.;

Hasse, H., Homogeneous nucleation in supersaturated vapors of methane, ethane, and carbon dioxide predicted by brute force molecular dynamics. Journal of Chemical Physics

2008, 128, 164510.

41. Seinfeld, J. H., Atmospheric Chemistry and Physics of Air Pollution. Wiley

Interscience: 1986.

42. Girshick, S. L.; Chiu, C.-P., Kinetic nucleation theory: A new expression for the rate of homogeneous nucleation from an ideal supersaturated vapor. Journal of Chemical Physics

1990, 93 (2), 1273-1277.

43. Feder, J.; Russell, K. C.; Lothe, J.; Pound, G. M., Homogeneous nucleation and growth of droplets in vapours. Advances in Physics 1966, 15 (57), 111-178.

44. Becker, R.; Doring, W., Kinetische Behandlung der Keimbildung in übersättigten

Dämpfen. Annalen der Physik 1935, 416 (8), 34.

45. Katz, J. L.; Donohue, M. D., A Kinetic Approach to Homogeneous Nucleation Theory.

Advances in Chemical Physics 1979, 40, 137-155.

46. Quinn, E. L., The Surface Tension of Liquid Carbon Dioxide. Journal of the American

Chemical Society 1927, 49 (11), 2704-2711.

47. Lielmezs, J.; A., H. T., Correlation of the thermal conductivity of saturated vapours.

Thermochimica Acta 1989, 141, 113-124.

102 48. Span, R.; Wagner, W., A New Equation of State for Carbon Dioxide Covering the Fluid

Region from the Triple‐Point Temperature to 1100 K at Pressures up to 800 MPa. Journal of Physical and Chemical Reference Data 1996, 25, 1509-1596.

49. Michels, A.; Wassenaar, T.; Zwietering, T.; Smits, P., The vapor pressure of liquid carbon dioxide. Physica 1950, 16, 501-504.

50. Kim, Y. J.; Wyslouzil, B. E.; Wilemski, G.; W ö lk, J.; Strey, R., Isothermal Nucleation

Rates in Supersonic Nozzles and the Properties of Small Water Clusters. The Journal of

Physical Chemistry A 2004, 108 (20), 4365-4377.

51. Wyslouzil, B. E.; Wilemski, G.; Beals, M. G.; Frish, M. B., Effect of carrier gas pressure on condensation in a supersonic nozzle. Physics of Fluids 1994, 6, 2845-2854.

52. Seifert, S.; Winans, R. E.; Tiede, D. M.; Thiyagarajan, P., Design and performance of a

ASAXS instrument at the Advanced Photon Source. Journal of Applied Crystallography 2000,

33, 782-784.

53. Manka, A.; Pathak, H.; Tanimura, S.; W ö lk, J.; Strey, R.; Wyslouzil, B. E., Freezing water in no-man's land. Physical Chemistry and Chemical Physics 2012, 14, 4505-4516.

54. Kotlarchyk, M.; Chen, S.-H., Analysis of small angle neutron scattering spectra from polydisperse interacting colloids. The Journal of Chemical Physics 1983, 79, 2461-2469.

55. Ogunronbi, K. E. Investigating the Phase Transitions of lower n-alkanes - pentane, hexane, and heptane - in a supersonic nozzle. Ohio State University, Columbus, OH, 2019.

56. Hämeri, K.; Kulmala, M., Homogeneous nucleation in a laminar flow diffusion chamber: The effect of temperature and carrier gas on dibutyl phthalate vapor nucleation rate at high supersaturations. The Journal of Chemical Physics 1996, 105, 7696-7704.

57. Volmer, M., Kinetik der Phasenbildung. Dresden u. Leipzig: 1939.

103 58. Ramos, A.; Fernández, J. M.; Tejeda, G.; Montero, S., Quantitative study of cluster growth in free-jet expansions of CO2 by Rayleigh and Raman Scattering. Physical Review A

2005, 72 (5), 053204.

59. Ramos, A.; Tejeda, G.; Fernández, J. M.; Montero, S., Cluster growth in supersonic jets of CO2 through a slit nozzle. AIP Conference Proceedings 2012, 1501 (1), 1383-1389.

60. Heiler, M. Instationare Phanomene in Homogen/Heterogen Kondensierenden Dusen- und Turbinenstromungen. Universitat Karlsruhe, 1999.

61. Mullick, K.; Bhabhe, A.; Manka, A.; W ö lk, J.; Strey, R.; Wyslouzil, B. E., Isothermal

Nucleation Rates of n-Propanol, n-Butanol, and n-Pentanol in Supersonic Nozzles: Critical

Cluster Sizes and the Role of Coagulation. The Journal of Physical Chemistry B 2015, 119

(29), 9009-9019.

62. Ghosh, D.; Bergmann, D.; Schwering, R.; Wölk, J.; Strey, R.; Tanimura, S.; Wyslouzil, B.

E., Homogeneous nucleation of a homologous series of n-alkanes (CiH2i+2, i = 7-10) in a supersonic nozzle. Journal of Chemical Physics 2010, 132 (2), 024307.

63. Ogunronbi, K. E.; Sepehri, A.; Chen, B.; Wyslouzil, B. E., Vapor phase nucleation of the short-chain n-alkanes (n-pentane, n-hexane and n-heptane): Experiments and Monte Carlo simulations. Journal of Chemical Physics 2018, 148 (14), 144312.

64. Amaya, A. J.; Wyslouzil, B. E., Ice nucleation rates near ~225 K. The Journal of

Chemical Physics 2018, 148, 084501.

65. Ogunronbi, K.; Wyslouzil, B. E., Vapor-phase nucleation of n-pentane, n-hexane, and n-heptane: Critical cluster properties. Journal of Chemical Physics 2019, 151 (15),

154307.

66. Isenor, M.; Escribano, R.; Preston, T. C.; Signorell, R., Predicting the infrared band profiles for CO2 cloud particles on Mars. Icarus 2013, 223 (1), 591-601.

104 67. Muratov, G. N.; Skripov, V. P., Поверхностное натяжение двуокиси углерода.

Teplofizika Vysokikh Temperatur 1982, 20 (3), 596-598.

68. Tolman, R. C., The Effect of Droplet Size on Surface Tension. Journal of Chemical

Physics 1949, 17 (3), 333-337.

69. Tanaka, K. K.; Diemand, J.; Angelil, R.; Tanaka, H., Free energy of cluster formation and a new scaling relation for the nucleation rate. Journal of Chemical Physics 2014, 140

(19), 194310.

70. Hale, B. N., Application of a scaled homogeneous nucleation-rate formalism to experimental data at T << Tc. Physical Review A 1986, 33 (6), 4156-4163.

71. Sinha, S.; Bhabhe, A.; Laksmono, H.; W ö lk, J.; Strey, R.; Wyslouzil, B., Argon nucleation in a cryogenic supersonic nozzle. The Journal of Chemical Physics 2010, 132 (6),

064304.

105 APPENDIX A: CODE FOR CALCULATING ONSET CONDITIONS

106 107 108

109 APPENDIX B: PRESSURE TRACE MEASUREMENTS

Figure B.1. Position-resolved pressure ratio and temperature profiles for all experiments in Nozzle T3.

110

111

112

113

114 APPENDIX C: SMALL ANGLE X-RAY SCATTERING SPECTRA

Figure C.1. Small angle X-ray scattering spectra for all measurements taken at onset of nucleation and at a fixed position 7.0 cm downstream of the physical throat.

115

116

117

118 APPENDIX D: POSITION-RESOLVED RADIUS AND NUMBER DENSITY

Figure D.1. Position-resolved average particle radius and average number density for

3.0, 6.0, 12, 18, 24, 30, and 39 mol% CO2.

119

120